On Lennard-Jones systems with finite range interactions and their asymptotic analysis

Mathias Schäffner; Anja Schlömerkemper; Mathias Schäffner; Anja Schlömerkemper

doi:10.3934/nhm.2018005

Networks and Heterogeneous Media

2018, Volume 13, Issue 1: 95-118. doi: 10.3934/nhm.2018005

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On Lennard-Jones systems with finite range interactions and their asymptotic analysis

Mathias Schäffner ^{1
,},
Anja Schlömerkemper ^{2
,}

1.
TU Dresden, Department of Mathematics, Zellescher Weg 12-14, 01069 Dresden, Germany
2.
University of Würzburg, Institute of Mathematics, Emil-Fischer-Straẞe 40, 97074 Würzburg, Germany

Received: 01 June 2017 Revised: 01 October 2017
Primary: 49J45, 74R10, 74G10, 74G65

The aim of this work is to provide further insight into the qualitative behavior of mechanical systems that are well described by Lennard-Jones type interactions on an atomistic scale. By means of $Γ$-convergence techniques, we study the continuum limit of one-dimensional chains of atoms with finite range interactions of Lennard-Jones type, including the classical Lennard-Jones potentials. So far, explicit formula for the continuum limit were only available for the case of nearest and next-to-nearest neighbour interactions. In this work, we provide an explicit expression for the continuum limit in the case of finite range interactions. The obtained homogenization formula is given by the convexification of a Cauchy-Born energy density.

Furthermore, we study rescaled energies in which bulk and surface contributions scale in the same way. The related discrete-to-continuum limit yields a rigorous derivation of a one-dimensional version of Griffith' fracture energy and thus generalizes earlier derivations for nearest and next-to-nearest neighbors to the case of finite range interactions.

A crucial ingredient to our proofs is a novel decomposition of the energy that allows for refined estimates.

Keywords:

Citation: Mathias Schäffner, Anja Schlömerkemper. 2018: On Lennard-Jones systems with finite range interactions and their asymptotic analysis, Networks and Heterogeneous Media, 13(1): 95-118. doi: 10.3934/nhm.2018005

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Abstract

A crucial ingredient to our proofs is a novel decomposition of the energy that allows for refined estimates.

1. Introduction

Lately, fractional calculus has played a very significant role in various scientific fields; see for instance ^[1,2] and the references cited therein. As a result of this, fractional differential equations have caught the attention of many investigators working in different desciplines ^{[3,4,5,6,7,8,9,10,11]}. However, most of researchers works have been conducted by using fractional derivatives that mainly rely on Riemann-Liouville, Hadamard, Katugampola, Grunwald Letnikov and Caputo approaches.

Fractional derivatives of a function with respect to another function have been considered in the classical monographs ^[1,12] as a generalization of Riemann-Liouville derivative. This fractional derivative is different from the other classical fractional derivative as the kernel appears in terms of another function $ \psi $. Thus, this type of derivative is referred to as $ \psi $-fractional derivative. Recently, this derivative has been reconsidered by Almeida in ^[14] where the Caputo-type regularization of the existing definition and some interesting properties are provided. Several properties of this operator could be found in ^{[1,12,13,15,16]}. For some particular cases of $ \psi $, one can realize that $ \psi $-fractional derivative can be reduced to the Caputo fractional derivative ^[1], the Caputo-Hadamard fractional derivative ^[17] and the Caputo-Erdélyi-Kober fractional derivative ^[18].

On the other hand, the investigation of qualitative properties of solutions for different fractional differential (and integral) equations is the key theme of applied mathematics research. Numerous interesting results concerning the existence, uniqueness, multiplicity, and stability of solutions or positive solutions by applying some fixed point techniques are obtained. However, most of the proposed problems have been handled concerning the classical fractional derivatives of the Riemann-Liouville and Caputo ^{[19,20,21,22,23,24,25,26,27,28,29]}.

In parallel with the intensive investigation of fractional derivative, a normal generalization of the Langevin differential equation appears to be replacing the classical derivative by a fractional derivative to produce fractional Langevin equation (FLE). FLE was first introduced in ^[30] and then different types of FLE were the object of many scholars ^{[31,32,33,34,35,36,37,38,39,40]}. In particular, the authors studied a nonlinear FLE involving two fractional orders on different intervals with three-point boundary conditions in ^[40], whereas FLE involving a Hadamard derivative type was considered in ^[33,34,35].

Alternatively, the stability problem of differential equations was discussed by Ulam in ^[41]. Thereafter, Hyers in ^[42] developed the concept of Ulam stability in the case of Banach spaces. Rassias provided a fabulous generalization of the Ulam-Hyers stability of mappings by taking into account variables. His approach was refered to as Ulam-Hyers-Rassias stability ^[43]. Recently, the Ulam stability problem of implicit differential equations was extended into fractional implicit differential equations by some authors ^{[44,45,46,47]}. A series of papers was devoted to the investigation of existence, uniqueness and U-H stability of solutions of the FLE within different kinds of fractional derivatives.

Motivated by the recent developments on $ \psi $-fractional calculus, in the present work, we investigate the existence, uniqueness and stability in the sense Ulam–Hyers–Rassias of solutions for the following FLE within $ \psi $-Caputo fractional derivatives of different orders involving nonlocal boundary conditions

$\begin{equation} \left \{ \begin{array}{l} \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left( ^{c}\mathfrak{D}_{a+,t}^{\beta ,\psi }+\lambda \right) \left[ u\right] = \mathfrak{F}(t,u(t),^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi } \left[ u\right] ),\; t\in \left( a,T\right) , \\ u(a) = 0,\; u(\eta ) = 0,\; u(T) = \mu \left( J_{a+,\xi }^{\gamma ,\psi }\right) \left[ u\right] ,\; \mu > 0, \end{array} \right. \end{equation}$ $

(1.1)

where $ \left(J_{a+, \xi }^{\gamma, \psi }\right) $ and $ \left(^{c}\mathfrak{D}_{a+, t}^{\theta, \psi }\right) $ are $ \psi $-fractional integral of order $ \gamma $, $ \ \psi $-Caputo fractional derivative of order $ \theta \in \{ \alpha, \beta, \gamma \} $ respectively, $ 0\leq a < \eta < \xi < T < \infty, \ 1 < \alpha \leq 2, \ 0 < \gamma < \beta \leq 1, \; \lambda $ is a real number and $ \mathfrak{F}:[a, T]\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{\ R}^{+} $ is a continuous function. We observe that problem (1.1) is designated within a general platform in the sense that general fractional derivative is considered with respect to different fractional orders, the forcing function depends on the general fractional derivative and boundary conditions involve integral fractional operators. Furthermore, the stability analysis in the sense of Ulam is investigated by the help of new versions of $ \psi $-fractional Gronwall inequality and $ \psi $-fractional integration by parts. It is worth mentioning here that the proposed results in this paper which rely on $ \psi $-fractional integrals and derivatives can generalize the existing results in the literature ^[31,40] and obtain them as particular cases.

The major contributions of the work are as follows: Some lemmas and definitions on $ \psi $-fractional calculus theory are recalled in Section $ 2 $. In Section $ 3 $, we prove the existence and uniqueness of solutions for problem (1.1) via applying fixed point theorems. Section $ 4 $ devotes to discuss different types of stability results for the problem (1.1) by the help of generalized $ \psi $-Gronwall's inequality ^[49] and $ \psi $-fractional integration by parts. The proposed results are examined via Maple using several numerical examples for different values of function $ \psi $, presented in several tables in Section $ 5 $, to check the applicability of the theoretical findings. We end the paper by a conclusion in Section $ 6 $.

2. Preliminaries and essential lemmas

The standard Riemann-Liouville fractional integral of order $ \alpha $, $ \Re(\alpha) > 0 $, has the form

$\begin{equation*} \left( J_{a+,t}^{\alpha }\right) \left[ u\right] = \frac{1}{\Gamma \left( \alpha \right) }\int_{a}^{t}\left( t-\tau \right) ^{\alpha -1}u\left( \tau \right) \mathrm{d}\tau ,\ t > a. \label{EQ2} \end{equation*}$ $

The left-sided factional integrals and fractional derivatives of a function $ u $ with respect to another function $ \psi $ in the sense of Riemann-Liouville are defined as follows ^[13,14]

$\begin{equation*} \left( J_{a+,t}^{\alpha ,\psi }\right) \left[ u\right] = \frac{1}{\Gamma \left( \alpha \right) }\int_{a}^{t}\psi ^{\prime }\left( \tau \right) \left( \psi \left( t\right) -\psi \left( \tau \right) \right) ^{\alpha -1}u\left( \tau \right) \mathrm{d}\tau \label{EQ3}, \end{equation*}$ $

and

$\begin{equation*} \left( \mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left[ u\right] = \left( \frac{1}{\psi ^{\prime }\left( t\right) }\frac{\mathrm{d}}{\mathrm{d}t}\right) ^{n}\left( J_{a+,t}^{n-\alpha ,\psi }\right) \left[ u\right] , \label{EQ4} \end{equation*}$ $

respectively, where $ n = [\alpha ]+1. $

Analogous formulas can be offered for the right fractional (integral and derivative) as follows:

$\begin{equation*} \left( J_{t,b-}^{\alpha ,\psi }\right) \left[ u\right] = \frac{1}{\Gamma \left( \alpha \right) }\int_{t}^{b}\psi ^{\prime }\left( \tau \right) \left( \psi \left( \tau \right) -\psi \left( t\right) \right) ^{\alpha -1}u\left( \tau \right) \mathrm{d}\tau , \label{EQ5} \end{equation*}$ $

and

$\begin{equation*} \left( \mathfrak{D}_{t,b-}^{\alpha ,\psi }\right) \left[ u\right] = \left( -\frac{1}{ \psi ^{\prime }\left( t\right) }\frac{\mathrm{d}}{\mathrm{d}t}\right) ^{n}\left( J_{t,b-}^{n-\alpha ,\psi }\right) \left[ u\right] . \label{EQ6} \end{equation*}$ $

The left (right) $ \psi $-Caputo fractional derivatives of $ u $ of order $ \alpha $ are given by

$\begin{equation*} \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left[ u\right] = \left( J_{a+,t}^{n-\alpha ,\psi }\right) \left( \frac{1}{\psi ^{\prime }\left( t\right) }\frac{\mathrm{d}}{\mathrm{d}t}\right) ^{n}\left[ u\right] \label{EQ7} \end{equation*}$ $

and

$\begin{equation*} \left( ^{c}\mathfrak{D}_{t,b-}^{\alpha ,\psi }\right) \left[ u\right] = \left( J_{t,b-}^{n-\alpha ,\psi }\right) \left( \frac{-1}{\psi ^{\prime }\left( t\right) }\frac{\mathrm{d}}{\mathrm{d}t}\right) ^{n}\left[ u\right] , \label{EQ8} \end{equation*}$ $

respectively. In particular, when $ \alpha \in (0, 1) $, we have

$\begin{equation*} \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left[ u\right] = \left( J_{a+,t}^{1-\alpha ,\psi }\right) \left( \frac{1}{\psi ^{\prime }\left( t\right) }\frac{\mathrm{d}}{\mathrm{d}t}\right) \left[ u\right] \label{EQ9} \end{equation*}$ $

and

$\begin{equation*} \left( ^{c}\mathfrak{D}_{t,b-}^{\alpha ,\psi }\right) \left[ u\right] = -\left( J_{t,b-}^{1-\alpha ,\psi }\right) \left( \frac{1}{\psi ^{\prime }\left( t\right) }\frac{\mathrm{d}}{\mathrm{d}t}\right) \left[ u\right] , \label{EQ10} \end{equation*}$ $

where $ u, \psi \in \mathcal{C}^{n}[a, b] $ two functions such that $ \psi $ is increasing and $ \psi ^{\prime }(t)\neq 0 $, for all $ t\in \lbrack a, b] $.

Remark 2.1. We propose the remarkable paper ^[16] in which some generalizations using $ \psi $-fractional integrals and derivatives are described. In particular, we have

$\begin{equation*} \left \{ \begin{array}{c} if\ \ \psi (t)\longrightarrow t,{\rm{ \ \ then \ }}J_{a+,t}^{\alpha ,\psi }\longrightarrow J_{a+,t}^{\alpha },\qquad \qquad \qquad \qquad \ \ \\ if\ \ \psi (t)\longrightarrow \ln t,{\rm{ \ \ then \ }}J_{a+,t}^{\alpha ,\psi }\longrightarrow {\rm{ }}^{H}J_{a+,t}^{\alpha },\qquad \qquad \qquad \ \ \\ if\ \ \psi (t)\longrightarrow t^{\rho },{\rm{ \ \ then }},{\rm{ \ }} J_{a+,t}^{\alpha ,\psi }\longrightarrow {\rm{ }}^{\rho }J_{a+,t}^{\alpha },\ \rho > 0,\qquad \qquad \ \end{array} \right. \end{equation*}$ $

where $ J_{a+, t}^{\alpha }, ^{H}J_{a+, t}^{\alpha }, ^{\rho }J_{a+, t}^{\alpha } $ are classical Riemann-Liouville, Hadamard, and Katugampola fractional operators.

Lemma 2.2. ^[14] Given $ u\in \mathcal{C}([a, b]) $ and $ v\in \mathcal{C} ^{n}([a, b]) $, we have that for all $ \alpha > 0 $

$\begin{eqnarray*} \int_{a}^{b}v\left( \tau \right) \left( ^{c}\mathfrak{D}_{a+,\tau }^{\alpha ,\psi }\right) \left[ u\right] \mathrm{d}\tau & = &\int_{a}^{b}u\left( \tau \right) \left( ^{c}\mathfrak{D}_{\tau ,b-}^{\alpha ,\psi }\right) \left[ \frac{v}{\psi ^{\prime }}\right] \frac{\mathrm{d}}{\mathrm{d}\tau }\psi \left( \tau \right) \mathrm{ \ d}\tau \\ &&+\left. \sum\limits_{k = 0}^{n-1}\left( -\frac{1}{\psi ^{\prime }\left( t\right) } \frac{\mathrm{d}}{\mathrm{d}t}\right) ^{k}\left( J_{\tau ,b-}^{n-\alpha ,\psi }\right) \left[ \frac{v}{\psi ^{\prime }}\right] u_{\psi }^{\left[ n-k-1\right] }\left( \tau \right) \right \vert _{\tau = a}^{\tau = b}, \end{eqnarray*}$ $

where

$\begin{equation*} u_{\psi }^{\left[ k\right] }\left( t\right) = \left( \frac{1}{\psi ^{\prime }\left( t\right) }\frac{\mathrm{d}}{\mathrm{d}t}\right) ^{k}u\left( t\right) . \end{equation*}$ $

Lemma 2.3. ^[1] Let $ \alpha > 0 $ and $ u, \ \psi \in \mathcal{C}([a, b]) $. Then

$\begin{equation*} \left \Vert J_{a+,t}^{\alpha ,\psi }\left[ u\right] \right \Vert _{C}\leq K_{\psi }\left \Vert u\right \Vert _{C},\ K_{\psi } = \frac{1}{\Gamma \left( 1+\alpha \right) }\left( \psi \left( b\right) -\psi \left( a\right) \right) ^{\alpha }. \end{equation*}$ $

For all $ n-1 < \alpha < n $,

$\begin{equation*} \left \Vert ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\left[ u\right] \right \Vert _{C}\leq K_{\psi }\left \Vert u\right \Vert _{C^{\left[ n\right] }\psi },\ K_{\psi } = \frac{1}{\Gamma \left( n+1-\alpha \right) }\left( \psi \left( b\right) -\psi \left( a\right) \right) ^{n-\alpha } \end{equation*}$ $

where $ ||\cdot||_C $ is the Chebyshev norm defined on $ C([a, b]) $.

The following results are well known and one can see ^[1,14] for further details.

Lemma 2.4. ^[1] Let $ \alpha, \beta > 0 $, consider the functions

$\begin{equation*} \left( J_{a+,t}^{\alpha ,\psi }\right) \left[ \left( \psi \left( \tau \right) -\psi \left( a\right) \right) ^{\beta -1}\right] = \frac{\Gamma \left( \beta \right) }{\Gamma \left( \alpha +\beta \right) }\left( \psi \left( t\right) -\psi \left( a\right) \right) ^{\alpha +\beta -1}, \end{equation*}$ $

$\begin{equation*} \left( J_{a+,t}^{\alpha ,\psi }\right) \left[ 1\right] = \frac{1}{\Gamma \left( 1+\alpha \right) }\left( \psi \left( t\right) -\psi \left( a\right) \right) ^{\alpha } \end{equation*}$ $

and

$\begin{equation*} \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left[ \left( \psi \left( \tau \right) -\psi \left( a\right) \right) ^{\beta -1}\right] = \frac{\Gamma \left( \beta \right) }{\Gamma \left( \beta -\alpha \right) }\left( \psi \left( t\right) -\psi \left( a\right) \right) ^{\beta -\alpha -1}, \end{equation*}$ $

$\begin{equation*} \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left[ 1\right] = \frac{1}{\Gamma \left( 1-\alpha \right) }\left( \psi \left( t\right) -\psi \left( a\right) \right) ^{-\alpha },\mathit{{\rm{}}}\alpha > 0. \end{equation*}$ $

Note that

$\begin{equation*} \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left[ \left( \psi \left( \tau \right) -\psi \left( a\right) \right) ^{k}\right] = 0,\ k = 0,..,n-1. \end{equation*}$ $

The subsequent properties are valid: If $ \alpha, \beta > 0 $, then

$\begin{equation*} \left( J_{a+,t}^{\alpha ,\psi }\right) \left( J_{a+,t}^{\beta ,\psi }\right) \left[ u\right] = \left( J_{a+,t}^{\alpha +\beta ,\psi }\right) \left[ u \right] {\rm{ and\ }}\left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left( ^{c}\mathfrak{D}_{a+,t}^{\beta ,\psi }\right) \left[ u\right] = \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha +\beta ,\psi }\right) \left[ u\right] , \label{EQ11} \end{equation*}$ $

$\begin{equation} \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left( J_{a+,t}^{\beta ,\psi }\right) \left[ u\right] = \left( J_{a+,t}^{\beta -\alpha ,\psi }\right) \left[ u\right] {\rm{.}} \end{equation}$ $

(2.1)

Lemma 2.5. ^[1] Given a function $ u\in \mathcal{C}^{n}[a, b] $ and $ \alpha > 0 $, we have

$\begin{equation*} J_{a+,t}^{\alpha ,\psi }\left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left[ u \right] = u\left( t\right) -\sum\limits_{j = 0}^{n-1}\left[ \frac{1}{j!}\left( \frac{1 }{\psi ^{\prime }\left( t\right) }\frac{\mathrm{d}}{\mathrm{d}t}\right) ^{j}u\left( a\right) \right] \left( \psi \left( t\right) -\psi \left( a\right) \right) ^{j}. \end{equation*}$ $

In particular, given $ \alpha \in (0, 1) $, we have

$\begin{equation*} J_{a+,t}^{\alpha ,\psi }\left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left[ u \right] = u\left( t\right) -u\left( a\right) . \end{equation*}$ $

Lemma 2.6. Given a function $ u\in \mathcal{C}^{n}[a, b] $ and $ 1 > \alpha > 0 $, we have

$\begin{equation*} \left \Vert \left( J_{a+,t_{2}}^{\alpha ,\psi }\right) \left[ u\right] -\left( J_{a+,t_{1}}^{\alpha ,\psi }\right) \left[ u\right] \right \Vert \leq \frac{2\left \Vert u\right \Vert _{\infty }}{\Gamma \left( \alpha +1\right) }\left( \psi \left( t_{2}\right) -\psi \left( t_{1}\right) \right) ^{\alpha }. \end{equation*}$ $

Proof. Using Lemmas 2.3 and 2.4, we have

$\begin{eqnarray*} \left \vert \left( J_{a+,t_{2}}^{\alpha ,\psi }\right) \left[ u\right] -\left( J_{a+,t_{1}}^{\alpha ,\psi }\right) \left[ u\right] \right \vert & = & \frac{1}{\Gamma \left( \alpha \right) }\left \vert \int_{a}^{t_{2}}\psi ^{\prime }\left( \tau \right) \left[ \left( \psi \left( t_{2}\right) -\psi \left( \tau \right) \right) ^{\alpha -1}-\left( \psi \left( t_{1}\right) -\psi \left( \tau \right) \right) ^{\alpha -1}\right] u\left( \tau \right) \mathrm{d}\tau \right \vert \\ &&+\frac{1}{\Gamma \left( \alpha \right) }\left \vert \int_{t_{1}}^{t_{2}}\psi ^{\prime }\left( \tau \right) \left( \psi \left( t_{2}\right) -\psi \left( \tau \right) \right) ^{\alpha -1}u\left( \tau \right) \mathrm{d}\tau \right \vert \\ &\leq &\frac{\left \Vert u\right \Vert _{\infty }}{\Gamma \left( \alpha +1\right) }\left[ \left( \psi \left( t_{2}\right) -\psi \left( t_{1}\right) \right) ^{\alpha }+\left( \psi \left( t_{1}\right) -\psi \left( a\right) \right) ^{\alpha }-\left( \psi \left( t_{2}\right) -\psi \left( a\right) \right) ^{\alpha }\right] \\ &&+\frac{\left \Vert u\right \Vert _{\infty }}{\Gamma \left( \alpha +1\right) } \left( \psi \left( t_{2}\right) -\psi \left( t_{1}\right) \right) ^{\alpha } \\ &\leq &\frac{2\left \Vert u\right \Vert _{\infty }}{\Gamma \left( \alpha +1\right) }\left( \psi \left( t_{2}\right) -\psi \left( t_{1}\right) \right) ^{\alpha }. \end{eqnarray*}$ $

Now we state here two important fixed point theorems, namely Banach and Krasnoselskii's fixed point theorems. These will help us to develop sufficient conditions for the existence and uniqueness of solutions.

Theorem 2.7. ^[48] Let $ \mathcal{B}_{r} $ be the closed ball of radius $ r > 0 $, centred at zero, in a Banach space $ X $ with $ \Upsilon :\mathcal{B} _{r}\rightarrow X $ a contraction and $ \Upsilon (\partial \mathcal{\ B} _{r})\subseteq \mathcal{B}_{r} $. Then, $ \Upsilon $ has a unique fixed point in $ \mathcal{B}_{r} $.

Theorem 2.8. ^[48] Let $ \mathcal{M} $ be a closed, convex, non-empty subset of a Banach space $ X\times X $. Suppose that $ \mathbb{E} $ and $ \mathbb{ \ F} $ map $ \mathcal{M} $ into $ X $ and that

(i) $ \mathbb{E}u+\mathbb{F}v\in \mathcal{M} $ for all $ u, v\in \mathcal{ M} $;

(ii) $ \mathbb{E} $ is compact and continuous;

(iii) $ \mathbb{F} $ is a contraction mapping.

Then the operator equation $ \mathbb{E}w+\mathbb{F}w = w $ has at least one solution on $ \mathcal{M} $.

Definition 2.9. The problem (1.1) is U-H stable if there exists a real number $ c_{f} > 0 $ such that for each $ \epsilon > 0 $ and for each solution $ \tilde{u}\in \mathcal{C}\left([ a, T]\right) $ of the inequality

$\begin{equation} \left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left( ^{c}\mathfrak{D}_{a+,t}^{\beta ,\psi }+\lambda \right) \left[ \tilde{u}\right] -\mathfrak{F}(t,u(t),^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\left[ \tilde{u}\right] )\right \vert \leq \epsilon ,\ t\in \left[ a,T\right] , \end{equation}$ $

(2.2)

there exists a solution $ u\in \mathcal{C}\left[ a, T\right] $ of the problem (1.1) with

$\begin{equation*} \left \vert \tilde{u}(t)-u(t)\right \vert \leq \epsilon c_{f}. \end{equation*}$ $

Definition 2.10. The problem (1.1) is generalized U-H stable if there exists $ \Phi \left(t\right) \in C\left(\mathbb{R} ^{+}, \mathbb{R} ^{+}\right) $, $ \Phi \left(0\right) = 0 $ such that for each $ \epsilon > 0 $ and for each solution $ \tilde{u}\in \mathcal{C}\left[ a, T\right] $ of inequality (2.2), there exists a solution $ u\in \mathcal{C}\left[ a, T\right] $ of problem (1.1) with

$\begin{equation*} \left \vert \tilde{u}(t)-u(t)\right \vert \leq \Phi \left( \epsilon \right) ,\ t\in \left[ a,T\right] , \end{equation*}$ $

where $ \Phi \left(\epsilon \right) $ is only dependent on $ \epsilon $.

Definition 2.11. The problem (1.1) is U-H-R stable if there exists a real number $ c_{f} > 0 $ such that for each $ \epsilon > 0 $ and for each solution $ \tilde{u}\in \mathcal{C}\left[ a, T\right] $ of the inequality

$\begin{equation*} \left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left( ^{c}\mathfrak{D}_{a+,t}^{\beta ,\psi }+\lambda \right) \left[ \tilde{u}\right] -\mathfrak{F}(t,u(t),^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\left[ \tilde{u}\right] )\right \vert \leq \epsilon \Phi \left( t\right) ,\ t\in \left[ a,T\right] , \label{EQ14} \end{equation*}$ $

there exists a solution $ u\in \mathcal{C}\left[ a, T\right] $ of the problem (1.1) with

$\begin{equation*} \left \vert \tilde{u}(t)-u(t)\right \vert \leq \epsilon c_{f}\Phi \left( t\right) . \end{equation*}$ $

Definition 2.12. The problem (1.1) is generalized U-H-R stable with respect to $ \Phi $ if there exists $ c_{f} > 0 $ such that for each solution $ \tilde{u} \in \mathcal{C}\left[ a, T\right] $ of the inequality

$\begin{equation*} \left \vert ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\left( ^{c}\mathfrak{D}_{a+,t}^{\beta ,\psi }+\lambda \right) \left[ u\right] -\mathfrak{F}(t,u(t),^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi } \left[ u\right] )\right \vert \leq \Phi \left( t\right) ,\ t\in \left[ a,T \right] , \label{EQ15} \end{equation*}$ $

there exists a solution $ u\in \mathcal{C}\left[ a, T\right] $ of the problem (1.1) with

$\begin{equation*} \left \vert \tilde{u}(t)-u(t)\right \vert \leq c_{f}\Phi \left( t\right) . \end{equation*}$ $

We adopt the following conventions:

$\begin{equation*} \mathfrak{F}_{u}(t) = \mathfrak{F}(t,u(t),^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\left[ u\right] )\ \;\; {\rm{and }} \;\; \mathcal{K}\left( t;a\right) = \psi \left( t\right) -\psi \left( a\right) . \label{EQ17} \end{equation*}$ $

We remark that, the following generalized $ \psi $-Gronwall Lemma is an important tool in proving the main results of this paper.

Lemma 2.13. ^[49] Let $ u, v $ be two integrable functions on $ [a, b] $. Let $ \psi \in \mathcal{C}^{1}[a, b] $ be an increasing function such that $ \psi ^{\prime }\left(t\right) \neq 0 $, $ \forall\; t\in \lbrack a, b] $. Assume that

(i) $ u $ and $ v $ are nonnegative;

(ii) The functions $ \left(g_{i}\right) _{i = 1\cdots n}\ $are bounded and monotonic increasing functions on $ [a, b]; $

(iii) The constants $ \alpha _{i} > 0\ \left(i = 1, 2, \ldots, n\right).\ $If

$\begin{equation*} u\left( t\right) \leq v\left( t\right) +\sum\limits_{i = 1}^{n}g_{i}\left( t\right) \int_{a}^{t}\psi ^{\prime }\left( \tau \right) \left( \mathcal{K}\left( t;\tau \right) \right) ^{\alpha _{i}-1}u\left( \tau \right) \mathrm{d}\tau , \end{equation*}$ $

then

$\begin{equation*} u\left( t\right) \leq v\left( t\right) +\sum\limits_{k = 1}^{\infty }\left( \sum\limits_{1^{\prime },2^{\prime },3^{\prime },\dots,k^{\prime } = 1}^{n}\tfrac{\prod \limits_{i = 1}^{k}\left( g_{i^{\prime }}\left( t\right) \Gamma \left( \alpha _{i^{\prime }}\right) \right) }{\Gamma \left( \sum\limits_{i = 1}^{k}\alpha _{i^{\prime }}\right) }\int_{a}^{t}\left[ \psi ^{\prime }\left( \tau \right) \left( \mathcal{K}\left( t;\tau \right) \right) ^{\sum_{i = 1}^{k}\alpha _{i^{\prime }}-1}\right] v\left( \tau \right) \mathrm{\ d}\tau \right) . \end{equation*}$ $

Remark 2.14. ^[49] For $ n = 2 $ in the hypotheses of Lemma 2.13. Let $ v(t) $ be a nondecreasing function for $ a\leq t < T $. Then we have

$\begin{equation*} u\left( t\right) \leq v\left( t\right) \left[ E_{\alpha _{1}}\left( g_{1}\left( t\right) \Gamma \left( \alpha _{1}\right) \left( \mathcal{K} \left( t;a\right) \right) ^{\alpha _{1}}\right) +E_{\alpha _{2}}\left( g_{2}\left( t\right) \Gamma \left( \alpha _{2}\right) \left( \mathcal{K} \left( t;a\right) \right) ^{\alpha _{2}}\right) \right] , \end{equation*}$ $

where $ E_{\alpha _{i}}(i = 1, 2) $ is the Mittag-Leffler function defined below.

Definition 2.15. ^[50] The Mittag-Leffler function is given by the series

$\begin{equation} E_{\alpha }\left( z\right) = \sum\limits_{k = 0}^{\infty }\frac{z^{k}}{\Gamma \left( \alpha k+1\right) }, \end{equation}$ $

(2.3)

where $ \Re\left(\alpha \right) > 0 $ and $ \Gamma \left(z\right) $ is a Gamma function. In particular, if $ \alpha = 1/2 $ in (2.3) we have

$\begin{equation*} E_{1/2}\left( z\right) = \exp \left( z^{2}\right) \left[ 1+ erf\left( z\right) \right] , \end{equation*}$ $

where $ erf \left(z\right) $ is the error function.

In the remaining portion of the paper, we make use of the next suppositions:

(A₁) For each $ t\in \left[ a, T\right] $, there exist a constant $ L_{i} > 0\ \left(i = 1, 2\right) $ such that

$\begin{equation*} \left \vert \mathfrak{F}(t,u_{1},v_{1})-\mathfrak{F}(t,u_{2},v_{2})\right \vert \leq L_{1}\left \vert u_{1}-v_{2}\right \vert +L_{2}\left \vert u_{1}-v_{2}\right \vert , {\rm{ for \;all }}\;u_{i},v_{i}\in \mathbb{R}. \end{equation*}$ $

(A₂) There exists an increasing function $ \chi \left(t\right) \in \left(\mathcal{C}\left[ a, T\right], \mathbb{R} ^{+}\right) $, for any $ t\in \left[ a, T\right], $

$\begin{equation*} \left \vert \mathfrak{F}(t,u,v)\right \vert \leq \chi \left( t\right) ,{\rm{ }}u,v\in \mathbb{R}. \end{equation*}$ $

(A₃) There exist a constant $ L > 0 $ such that

$\begin{equation*} \left \vert \mathfrak{F}(t,u,v)\right \vert \leq L,{\rm{ for\; any }}\;t\in \left[ a,T \right] ,{\rm{ }}u,v\in \mathbb{R}. \end{equation*}$ $

(A₄) There exists an function $ \Phi \left(t\right) \in \left(\mathcal{C}\left[ a, T\right], \mathbb{R} ^{+}\right) $ and there exists $ l_{\alpha, \psi } > 0 $ such that for any $ t\in \left[ a, T\right], $

$\begin{equation*} \left( J_{a+,t}^{\alpha ,\psi }\right) \left[ \Phi \right] \leq l_{\alpha ,\psi }\Phi \left( t\right) ,\ \alpha > 0. \end{equation*}$ $

Denoting

$\begin{equation*} \sigma _{11} = \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ 1\right] {\rm{ and}}\ \sigma _{12} = \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ \mathcal{K}\left( \tau ;a\right) \right] , \label{EQ18} \end{equation*}$ $

and

$\begin{equation*} \sigma _{21} = \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\beta +\gamma ,\psi }\right) \left[ 1\right] {\rm{ and}}\ \sigma _{22} = \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ \mathcal{K}\left( \tau ;a\right) \right] -\mu \left( J_{a_{+},\xi }^{\beta +\gamma ,\psi }\right) \left[ \mathcal{K}\left( \tau ;a\right) \right] \right) . \label{EQ19} \end{equation*}$ $

Further, we assume

$\begin{equation*} \left \vert \sigma _{11}\sigma _{22}-\sigma _{21}\sigma _{12}\right \vert \neq 0, \label{EQ20} \end{equation*}$ $

where $ \sigma _{ij} $ are constants.

3. Existence and uniqueness of solution

In order to study the nonlinear FLE (1.1), we first consider the linear associated FLE and conclude the form of the solution.

3.1. Linear boundary problem

The following lemma regards a linear variant of problem

$\begin{equation} \left \{ \begin{array}{l} \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left( ^{c}\mathfrak{D}_{a+,t}^{\beta ,\psi }+\lambda \right) \left[ u\right] = F(t),\; t\in \left( a,T\right) , \\ u(a) = 0,\; u(\eta ) = 0,\; u(T) = \mu \left( J_{a+,\xi }^{\gamma ,\psi }\right) \left[ u\right] ,\; a < \eta < \xi < T, \end{array}\right. \end{equation}$ $

(3.1)

where $ F\in \mathcal{C}([a, T], \mathbb{R}). $

Lemma 3.1. The unique solution of the $ \psi $-Caputo linear problem (3.1) is given by the integral equation

$\begin{aligned} u(t) &=-\lambda\left(J_{a+, t}^{\beta, \psi}\right)[u]+\left(J_{a+, t}^{\alpha+\beta, \psi}\right)[F] \\ &+\frac{(\mathcal{K}(t ; a))^{\beta}(\mathcal{K}(t ; \eta)}{\Gamma(\beta+2) \Delta}\left\{\left(J_{a+, T}^{\alpha+\beta, \psi}\right)[F]-\lambda\left(J_{a+, T}^{\beta, \psi}\right)[u]-\mu\left(J_{a+, \xi}^{\alpha+\beta+\gamma, \psi}\right)[F]+\mu \lambda\left(J_{a+, \xi}^{\beta+\gamma, \psi}\right)[u]\right\} \\ &-\frac{(\mathcal{K}(t ; a))^{\beta}}{\Delta(\mathcal{K}(\eta ; a))^{\beta}}\left(\frac{(\mathcal{K}(T ; a))^{\beta}(\mathcal{K}(T ; t))}{\Gamma(\beta+2)}-\frac{\mu(\mathcal{K}(\xi ; a))^{\beta+\gamma}[(\beta+1)(\mathcal{K}(\xi ; t))-\gamma(\mathcal{K}(t ; a))]}{\Gamma(\beta+\gamma+2)(\beta+1)}\right) \\ & \times\left\{\left(J_{a+, \eta}^{\alpha+\beta, \psi}\right)[F]-\lambda\left(J_{a+, \eta}^{\beta, \psi}\right)[u]\right\}, \end{aligned}$ $

(3.2)

where

$ \Delta=\left[\frac{(\mathcal{K}(T ; a))^{\beta} \mathcal{K}(T ; \eta)}{\Gamma(\beta+2)}-\frac{\mu(\mathcal{K}(\xi ; a))^{\beta+\gamma}\;[(\beta+1) \mathcal{K}(\xi ; \eta)-\gamma \mathcal{K}(\eta ; a)]}{\Gamma(\beta+\gamma+2)(\beta+1)}\right] \neq 0. $

(3.3)

Proof. Applying $ \left(J_{a+, t}^{\alpha, \psi }\right) $ on both sides of (3.1-a), we have

$\begin{equation} \left( ^{c}\mathfrak{D}_{a+,t}^{\beta ,\psi }+\lambda \right) \left[ u\right] = \left( J_{a+,t}^{\alpha ,\psi }\right) \left[ F\right] +c_{1}+c_{2}(\psi (t)-\psi (a)), \end{equation}$ $

(3.4)

for $ c_{1}, \; c_{2}\in \mathbb{R}. $

Now applying $ \left(J_{a+, t}^{\beta, \psi }\right) $ to both sides of (3.4), we get

$\begin{equation*} u(t) = -\lambda \left( J_{a+,t}^{\beta ,\psi }\right) \left[ u\right] +\left( J_{a+,t}^{\alpha +\beta ,\psi }\right) \left[ F\right] +c_{1}\left( J_{a+,t}^{\beta ,\psi }\right) \left[ 1\right] +c_{2}\left( J_{a+,t}^{\beta ,\psi }\right) \left[ \mathcal{K}\left( \tau ;a\right) \right] +c_{3}, \label{EQ25} \end{equation*}$ $

where $ c_{3}\in \mathbb{R}. $

Using the boundary conditions in (3.1-b), we obtain $ c_{3}: = c_{3}\left(F\right) = 0 $ and

$\begin{equation} J_{a+,t}^{\delta ,\psi }\left[ u\right] = -\lambda \left( J_{a+,t}^{\beta +\delta ,\psi }\right) \left[ u\right] +\left( J_{a+,t}^{\alpha +\beta +\delta ,\psi }\right) \left[ F\right] +c_{1}\left( J_{a+,t}^{\beta +\delta ,\psi }\right) \left[ 1\right] +c_{2}\left( J_{a+,t}^{\beta +\delta ,\psi }\right) \left[ \mathcal{K}\left( \tau ;a\right) \right] . \end{equation}$ $

(3.5)

Further, we get a system of linear equations with respect to $ c_{1} $, $ c_{2} $ as follows

$\begin{equation*} \left( \begin{array}{cc} \sigma _{11} & \sigma _{12} \\ \sigma _{21} & \sigma _{22} \end{array} \right) \left( \begin{array}{c} c_{1} \\ c_{2} \end{array} \right) = \left( \begin{array}{c} b_{1} \\ b_{2} \end{array} \right) , \label{EQ27} \end{equation*}$ $

where

$\begin{equation*} b_{1} = \lambda \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ u\right] -\left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ F\right] \label{EQ28} \end{equation*}$ $

and

$\begin{equation*} b_{2} = \lambda \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ u\right] -\mu \left( J_{a_{+},\xi }^{\beta +\delta ,\psi }\right) \left[ u\right] \right) -\left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ F \right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ F\right] \right) . \label{EQ29} \end{equation*}$ $

We note

$\begin{equation*} \Delta \equiv \det \left( \sigma \right) = \left \vert \sigma _{11}\sigma _{22}-\sigma _{21}\sigma _{12}\right \vert . \label{EQ30} \end{equation*}$ $

Because the determinant of coefficients for $ \Delta \neq 0.\ $Thus, we have

$\begin{equation*} c_{1}: = c_{1}\left( F\right) = \frac{\sigma _{22}b_{1}-\sigma _{12}b_{2}}{ \Delta }\;{\rm{ and }}\;c_{2}: = c_{2}\left( F\right) = \frac{\sigma _{11}b_{2}-\sigma _{21}b_{1}}{\Delta }. \label{EQ31} \end{equation*}$ $

Substituting these values of $ c_{1} $ and $ c_{2} $ in (3.5), we finally obtain (3.2) as

$\begin{equation} u\left( t\right) = -\lambda \left( J_{a+,t}^{\beta ,\psi }\right) \left[ u \right] +\left( J_{a+,t}^{\alpha +\beta ,\psi }\right) \left[ F\right] + \tfrac{\sigma _{22}b_{1}-\sigma _{12}b_{2}}{\Delta }\left( J_{a+,t}^{\beta ,\psi }\right) \left[ 1\right] +\tfrac{\sigma _{11}b_{2}-\sigma _{21}b_{1}}{ \Delta }\left( J_{a+,t}^{\beta ,\psi }\right) \left[ \mathcal{K}\left( \tau ;a\right) \right]. \end{equation}$ $

(3.6)

That is, the integral equation (3.6) can be written as (3.2) and

$\begin{eqnarray*} \nonumber \left( J_{a+,t}^{\delta ,\psi }\right) \left[ u\right]& = &-\lambda \left( J_{a+,t}^{\beta+\delta ,\psi }\right) \left[ u\right]\\ &+&\left( J_{a+,t}^{\alpha +\beta +\delta ,\psi }\right) \left[ F\right] +\tfrac{ \sigma _{22}b_{1}-\sigma _{12}b_{2}}{\Delta }\left( J_{a+,t}^{\beta +\delta ,\psi }\right) \left[ 1\right] +\tfrac{\sigma _{11}b_{2}-\sigma _{21}b_{1}}{ \Delta }\left( J_{a+,t}^{\beta +\delta ,\psi }\right) \left[ \mathcal{K} \left( \tau ;a\right) \right]. \label{EQ33} \end{eqnarray*}$ $

Differentiating the above relations one time we obtain (3.1-a), also it is easy to get that the condition (3.1-b) is satisfied. The proof is complete.

For convenience, we define the following functions

$\begin{equation} d_{11}\left( t\right) = -\tfrac{1}{\Delta }\left( \sigma _{22}\left( J_{a+,t}^{\beta ,\psi }\right) \left[ 1\right] -\sigma _{21}\left( J_{a+,t}^{\beta ,\psi }\right) \left[ \mathcal{K}\left( \tau ;a\right) \right] \right) {\rm{, }}\;\;d_{21}\left( t\right) = -d_{11}\left( t\right) \end{equation}$ $

(3.7)

and

$\begin{equation} d_{12}\left( t\right) = \tfrac{1}{\Delta }\left( \sigma _{12}\left( J_{a+,t}^{\beta ,\psi }\right) \left[ 1\right] -\sigma _{11}\left( J_{a+,t}^{\beta ,\psi }\right) \left[ \mathcal{K}\left( \tau ;a\right) \right] \right) {\rm{, }}\;\;d_{22}\left( t\right) = d_{12}\left( t\right) . \end{equation}$ $

(3.8)

3.2. Nonlinear problem

The following result is an immediate consequence of Lemma 3.1.

Lemma 3.2. Let $ \ \lambda \in \mathbb{R}. $ Then problem (1.1) is equivalent to the integral equation

$\begin{equation} u\left( t\right) = -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ u \right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] +\phi _{u}\left( \mathfrak{F}\right) , \end{equation}$ $

(3.9)

where

$\begin{eqnarray} \phi _{u}\left( \mathfrak{F}\right) & = &d_{11}\left( t\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] +d_{12}\left( t\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\gamma ,\psi }\right) \left[ \mathfrak{F}_{u} \right] \right) \\ &&+\lambda d_{21}\left( t\right) \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ u\right] -\lambda d_{22}\left( t\right) \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ u\right] -\mu \left( J_{a_{+},\xi }^{\beta +\gamma ,\psi }\right) \left[ u\right] \right) \end{eqnarray}$ $

(3.10)

and $ d_{ij} $ are defined in (3.7) and (3.8).

From the expression of (1.1-a) and (3.9), we can see that if all the conditions in Lemmas 3.1 and 3.2 are satisfied, the solution is a $ \mathcal{C}\left[ a, T\right] $ solution of the $ \psi $-Caputo fractional boundary value problem (1.1).

In order to lighten the statement of our result, we adopt the following notation.

$\begin{equation} \varsigma _{11} = \sup\limits_{t\in \left[ a,T\right] }\left \vert \lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ 1\right] +\rho _{11}+L_{1}\left( \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ 1\right] +\rho _{12}\right) \right \vert , \end{equation}$ $

(3.11)

$\begin{equation} \varsigma _{12} = L_{2}\varsigma _{13}\;{\rm{ where }}\;\varsigma _{13} = \sup\limits_{t\in \left[ a,T\right] }\left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ 1\right] \right \vert +\rho _{12}, \end{equation}$ $

(3.12)

$\begin{equation} \varsigma _{21} = \sup\limits_{t\in \left[ a,T\right] }\left \vert \lambda \left( J_{a+,t}^{\beta -\gamma ,\psi }\right) \left[ 1\right] +\rho _{21}+L_{1}\left( \left( J_{a+,t}^{\alpha +\beta -\gamma ,\psi }\right) \left[ 1\right] +\rho _{22}\right) \right \vert , \end{equation}$ $

(3.13)

$\begin{equation*} \varsigma _{22} = L_{2}\varsigma _{23}\;{\rm{ where }}\;\varsigma _{23} = \sup\limits_{t\in \left[ a,T\right] }\left \vert \left( J_{a+,t}^{\alpha +\beta -\gamma ,\psi }\right) \left[ 1\right] \right \vert +\rho _{22}, \label{EQ41} \end{equation*}$ $

with

$\begin{equation} \rho _{11} = \left \vert \lambda \right \vert \sup\limits_{t\in \left[ a,T\right] }\left( \left \vert d_{21}\left( t\right) \right \vert \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ 1\right] +\left \vert d_{22}\left( t\right) \right \vert \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\beta +\gamma ,\psi }\right) \left[ 1\right] \right) \right) , \end{equation}$ $

(3.14)

$\begin{equation} \rho _{12} = \sup\limits_{t\in \left[ a,T\right] }\left( \left \vert d_{11}\left( t\right) \right \vert \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ 1\right] +\left \vert d_{12}\left( t\right) \right \vert \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\gamma ,\psi }\right) \left[ 1\right] \right) \right) , \end{equation}$ $

(3.15)

$\begin{equation} \rho _{21} = \left \vert \lambda \right \vert \sup\limits_{t\in \left[ a,T\right] }\left( \left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ d_{21} \right] \right \vert \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ 1 \right] +\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ d_{22} \right] \right \vert \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\beta +\gamma ,\psi }\right) \left[ 1\right] \right) , \end{equation}$ $

(3.16)

and

$\begin{equation} \rho _{22} = \sup\limits_{t\in \left[ a,T\right] }\left( \left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ d_{11}\right] \right \vert \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ 1\right] +\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ d_{12}\right] \right \vert \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\gamma ,\psi }\right) \left[ 1\right] \right). \end{equation}$ $

(3.17)

We are now in a position to establish the existence and uniqueness results. Fixed point theorems are the main tool to prove this.

Let $ \mathcal{C} = \mathcal{C}([a, T], \mathbb{R}) $ be a Banach space of all continuous functions defined on $ [a, T] $ endowed with the usual supremum norm$. $ Define the space

$\begin{equation} E = \{u:u\in \mathcal{C}^{3}([a,T],\mathbb{R}),\ \left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ u\right] \in \mathcal{C}\}, \end{equation}$ $

(3.18)

equipped with the norm

$\begin{equation*} \left \Vert u\right \Vert _{E} = \max \left \{ \left \Vert u\right \Vert _{\infty },\left \Vert \left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ u \right] \right \Vert _{\infty }\right \} . \label{EQ47} \end{equation*}$ $

Then, we may conclude that $ \left(E, \left \Vert.\right \Vert _{E}\right) $ is a Banach space.

To introduce a fixed point problem associated with (3.9) we consider an integral operator $ \Psi :E\rightarrow E $ defined by

$\begin{equation} \left( \Psi u\right) \left( t\right) = -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ u\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] +\phi _{u}\left( \mathfrak{F}\right) . \end{equation}$ $

(3.19)

Theorem 3.3. Assume that $ \mathfrak{F}:[a, T]\times \mathbb{R} \times \mathbb{R} \longrightarrow \mathbb{R} ^{+} $ is a continuous function that satisfies (A$ _{1} $). If we suppose that

$\begin{equation} 0 < \varsigma = \max \left \{ \varsigma _{11},\varsigma _{12},\varsigma _{21},\varsigma _{22}\right \} < 1, \end{equation}$ $

(3.20)

holds. Then the problem (1.1) has a unique solution on $ E. $

Proof. The proof will be given in two steps.

Step 1. The operator $ \Psi $ maps bounded sets into bounded sets in $ E $.

For our purpose, consider a function $ u\in E. $ It is clear that $ \Psi u\in E. $ Also by (2.1), (3.10) and (3.19), we have

$\begin{equation} \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi u\right) = -\lambda \left( J_{a+,t}^{\beta -\delta ,\psi }\right) \left[ u\right] +\left( J_{a+,t}^{\alpha +\beta -\delta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] +\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ \phi _{u}\left( \mathfrak{F}\right) \right] . \end{equation}$ $

(3.21)

Indeed, it is sufficient to prove that for any $ r > 0 $, for each $ u\in B_{r} = \left \{ u\in E:\left \Vert u\right \Vert _{E}\leq r\right \}, $ we have $ \left \Vert \Psi u\right \Vert _{E}\leq r. $

Denoting

$\begin{equation*} L_{0} = \sup\limits_{t\in \left[ a,T\right] }\left \{ \left \vert \mathfrak{F}(t,0,0\right \vert :t\in \left[ a,T\right] \right \} < \infty \;{\rm{ and }}\;L_{B} = L_{1}\sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) \right \vert +L_{2}\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ u\right] \right \vert +L_{0}. \label{EQ51} \end{equation*}$ $

By (A$ _{1} $) we have for each $ t\in \left[ a, T\right] $

$\begin{equation*} \left \vert \mathfrak{F}_{u}(t)\right \vert = \left \vert \mathfrak{F}_{u}(t)-\mathfrak{F}_{0}(t)+\mathfrak{F}_{0}(t)\right \vert \leq \left \vert \mathfrak{F}_{u}(t)-\mathfrak{F}_{0}(t)\right \vert +\left \vert \mathfrak{F}_{0}(t)\right \vert \leq L_{B}. \label{EQ52} \end{equation*}$ $

Firstly, we estimate $ \left \vert \phi _{u}\left(\mathfrak{F}\right) \right \vert $ as follows

$\begin{eqnarray*} &&\left \vert \phi _{u}\left( \mathfrak{F}\right) \right \vert \notag \\ & = &\left \vert \lambda d_{21}\left( t\right) \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ u\right] \right \vert +\left \vert d_{12}\left( t\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ \left \vert \mathfrak{F}_{u}-\mathfrak{F}_{0}\right \vert +\mathfrak{F}_{0}\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ \left \vert \mathfrak{F}_{u}-\mathfrak{F}_{0}\right \vert +\mathfrak{F}_{0} \right] \right) \right \vert \notag \\ &&+\left \vert \lambda d_{22}\left( t\right) \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ u\right] -\mu \left( J_{a_{+},\xi }^{\beta +\delta ,\psi }\right) \left[ u\right] \right) \right \vert +\left \vert d_{11}\left( t\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ \left \vert \mathfrak{F}_{u}-\mathfrak{F}_{0}\right \vert +\mathfrak{F}_{0}\right] \right \vert . \label{EQ53} \end{eqnarray*}$ $

Then

$\begin{eqnarray*} \left \vert \phi _{u}\left( \mathfrak{F}\right) \right \vert &\leq &\left \vert d_{11}\left( t\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ L_{B}\right] \right \vert +\left \vert d_{12}\left( t\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ L_{B}\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ L_{B} \right] \right) \right \vert \notag \\ &&+\left \vert \lambda d_{21}\left( t\right) \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ u\right] \right \vert +\left \vert \lambda d_{22}\left( t\right) \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ u\right] -\mu \left( J_{a_{+},\xi }^{\beta +\delta ,\psi }\right) \left[ u\right] \right) \right \vert . \label{EQ54} \end{eqnarray*}$ $

Taking the maximum over $ \left[ a, T\right], $ we get

$\begin{equation} \sup\limits_{t\in \left[ a,T\right] }\left \vert \phi _{u}\left( \mathfrak{F}\right) \right \vert \leq \rho _{11}\sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) \right \vert +\rho _{12}\left( L_{1}\sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) \right \vert +L_{2}\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ u\right] \right \vert +L_{0}\right) , \end{equation}$ $

(3.22)

where $ \phi _{u} $, $ d_{ij}\left(t\right) $ and $ \rho _{ij} $ defined by (3.10), (3.7), (3.8) and (3.14–3.17) respectively. Using (3.19) and (3.22), we obtain

$\begin{equation} \left \Vert \left( \Psi u\right) \right \Vert _{\infty }\leq \varsigma _{11}\sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) \right \vert +\varsigma _{12}\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ u\right] \right \vert +\varsigma _{13}L_{0}, \end{equation}$ $

(3.23)

where $ \varsigma _{ij} $ defined by (3.11) and (3.12). On the other hand

$\begin{eqnarray*} &&\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ \phi _{u}\left( \mathfrak{F}\right) \right] \notag \\ & = &\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{11}\right] \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] +\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{12}\right] \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] \right) \notag \\ &&+\lambda \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{21}\right] \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ u\right] -\lambda \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{22}\right] \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ u\right] -\mu \left( J_{a_{+},\xi }^{\beta +\delta ,\psi }\right) \left[ u\right] \right) . \label{EQ57} \end{eqnarray*}$ $

Taking the maximum over $ \left[ a, T\right], $ we get

$\begin{eqnarray} &&\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ \phi _{u}\left( \mathfrak{F}\right) \right] \right \vert \\ &\leq &\rho _{21}\sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) \right \vert +\rho _{22}\left( L_{1}\sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) \right \vert +L_{2}\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ u\right] \right \vert +L_{0}\right) . \end{eqnarray}$ $

(3.24)

Using (3.21) and (3.24), we obtain

$\begin{equation} \left \Vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi u\right) \right \Vert _{\infty }\leq \varsigma _{21}\sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) \right \vert +\varsigma _{22}\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ u \right] \right \vert +\varsigma _{23}L_{0}. \end{equation}$ $

(3.25)

Consequently, by (3.23) and (3.25), we have

$\begin{equation*} \left \Vert \left( \Psi u\right) \right \Vert _{E}\leq \varsigma \left \Vert u\right \Vert _{E}+L_{0}\max \left \{ \varsigma _{13},\varsigma _{23}\right \} \leq \varsigma r+\left( 1-\varsigma \right) r = r, \label{EQ60} \end{equation*}$ $

where $ \varsigma $ is defined by (3.20) and choose

$\begin{equation*} r > \frac{L_{0}\max \left \{ \varsigma _{13},\varsigma _{23}\right \} }{\left( 1-\varsigma \right) },\ 0 < \varsigma < 1. \label{EQ61} \end{equation*}$ $

The continuity of the functional $ \mathfrak{F}_{u} $ would imply the continuity of $ \left(\Psi u\right) $ and $ \left(^{c}\mathfrak{D}_{a+, t}^{\delta, \psi }\right) \left(\Psi u\right) $ Hence, $ \Psi $ maps bounded sets into bounded sets in $ E $.

Step 2.Now we show that $ \Psi $ is a contraction. By (A$ _{1} $) and (3.19), for $ u, v\in E $ and $ t\in \left[ a, T\right] $, we have

$\begin{equation*} \left \vert \left( \Psi u\right) \left( t\right) -\left( \Psi v\right) \left( t\right) \right \vert \leq \left \vert \lambda \right \vert \left \vert \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ u-v\right] \right \vert +\left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}-\mathfrak{F}_{v}\right] \right \vert +\left \vert \phi _{u}\left( \mathfrak{F}\right) -\phi _{v}\left( \mathfrak{F}\right) \right \vert , \label{EQ62} \end{equation*}$ $

where

$\begin{equation*} \left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}-\mathfrak{F}_{v}\right] \right \vert \leq \left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ 1\right] \right \vert \left( L_{1}\left \vert u-v\right \vert +L_{2}\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ u-v\right] \right \vert \right) \label{EQ63} \end{equation*}$ $

and

$\begin{eqnarray*} &&\phi _{u}\left( \mathfrak{F}\right) -\phi _{v}\left( \mathfrak{F}\right) \quad \quad \qquad \quad \notag \\ & = &d_{11}\left( t\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}-\mathfrak{F}_{v}\right] +d_{12}\left( t\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}-\mathfrak{F}_{v}\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ \mathfrak{F}_{u}-\mathfrak{F}_{v}\right] \right) \notag \\ &&+\lambda d_{21}\left( t\right) \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ u-v\right] -\lambda d_{22}\left( t\right) \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ u-v\right] -\mu \left( J_{a_{+},\xi }^{\beta +\delta ,\psi }\right) \left[ u-v\right] \right) , \label{EQ64} \end{eqnarray*}$ $

for all $ t\in \left[ a, T\right] $, which implies

$\begin{equation*} \left \vert \phi _{u}\left( \mathfrak{F}\right) -\phi _{v}\left( \mathfrak{F}\right) \right \vert \leq \left( \rho _{11}+L_{1}\rho _{12}\right) \sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) -v\left( t\right) \right \vert +\rho _{12}L_{2}\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ u-v\right] \right \vert . \label{EQ65} \end{equation*}$ $

Hence, we get

$\begin{eqnarray*} &&\left \vert \left( \Psi u\right) \left( t\right) -\left( \Psi v\right) \left( t\right) \right \vert \quad \quad \qquad \quad \notag \\ &\leq &\left \vert \lambda \right \vert \left \vert \sup\limits_{t\in \left[ a,T\right] }\left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ 1\right] \right \vert \sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) -v\left( t\right) \right \vert +\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ 1\right] \right \vert \notag \\ &&\left( L_{1}\sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) -v\left( t\right) \right \vert +L_{2}\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ u-v\right] \right \vert \right) +\sup\limits_{t\in \left[ a,T\right] }\left \vert \phi _{u}\left( \mathfrak{F}\right) -\phi _{v}\left( \mathfrak{F}\right) \right \vert .\ \ \ \ \ \ \label{EQ66} \end{eqnarray*}$ $

Consequently,

$\begin{equation} \left \Vert \left( \Psi u\right) -\left( \Psi v\right) \right \Vert _{\infty }\leq \varsigma _{11}\sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) -v\left( t\right) \right \vert +\varsigma _{12}\sup\limits_{t\in \left[ a,T \right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ u-v \right] \right \vert . \end{equation}$ $

(3.26)

A similar argument shows that

$\begin{eqnarray} &&\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ \left( \Psi u\right) -\left( \Psi v\right) \right] \right \vert \quad \quad \qquad \quad \\ & = &\left \vert \lambda \right \vert \left \vert \left( J_{a+,t}^{\beta -\delta ,\psi }\right) \left[ u-v\right] \right \vert +\left \vert \left( J_{a+,t}^{\alpha +\beta -\delta ,\psi }\right) \left[ \mathfrak{F}_{u}-\mathfrak{F}_{v}\right] \right \vert +\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ \phi _{u}\left( \mathfrak{F}\right) -\phi _{v}\left( \mathfrak{F}\right) \right] \right \vert , \end{eqnarray}$ $

(3.27)

where

$\begin{eqnarray} &&\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ \phi _{u}\left( \mathfrak{F}\right) -\phi _{v}\left( \mathfrak{F}\right) \right] \right \vert \quad \quad \qquad \quad \\ &\leq &\left( \rho _{21}+L_{1}\rho _{22}\right) \sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) -v\left( t\right) \right \vert +\rho _{22}L_{2}\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ u-v\right] \right \vert +\rho _{22}L_{0}. \end{eqnarray}$ $

(3.28)

Combining (3.27) and (3.28), we obtain

$\begin{equation} \left \Vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi u\right) -\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi v\right) \right \Vert _{\infty }\leq \varsigma _{21}\sup\limits_{t\in \left[ a,T\right] }\left \vert u\left( t\right) -v\left( t\right) \right \vert +\varsigma _{22}\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ u-v\right] \right \vert . \end{equation}$ $

(3.29)

Consequently, by (3.26) and (3.29), we have

$\begin{equation*} \left \Vert \left( \Psi u\right) -\left( \Psi v\right) \right \Vert _{E}\leq \varsigma \left \Vert u-v\right \Vert _{E} \label{EQ71} \end{equation*}$ $

and choose $ \varsigma = \max \left \{ \varsigma _{11}, \varsigma _{12}, \varsigma _{21}, \varsigma _{22}\right \} < 1.\ $Hence, the operator $ \Psi $ is a contraction, therefore $ \Psi $ maps bounded sets into bounded sets in $ E $. Thus, the conclusion of the theorem follows by the contraction mapping principle.

For simplicity of presentation, we let

$\begin{equation*} \Lambda _{11} = \left( J_{a+,T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] +\left( J_{a+,T}^{\alpha +\beta -\gamma ,\psi }\right) \left[ 1\right] +\left( d_{11}\left( T\right) +\left( ^{c}\mathfrak{D}_{a+,T}^{\gamma ,\psi }\right) \left[ d_{11}\right] \right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ 1\right] , \label{EQ72} \end{equation*}$ $

$\begin{equation*} \Lambda _{12} = \left( d_{12}\left( T\right) +\left( ^{c}\mathfrak{D}_{a+,T}^{\gamma ,\psi }\right) \left[ d_{12}\right] \right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\gamma ,\psi }\right) \left[ 1\right] \right) , \label{EQ73} \end{equation*}$ $

$\begin{equation*} \Lambda _{21} = \left( J_{a+,T}^{\beta ,\psi }\right) \left[ 1\right] +d_{21}\left( T\right) \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ 1 \right] +d_{22}\left( T\right) \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\beta +\gamma ,\psi }\right) \left[ 1\right] \right) , \label{EQ74} \end{equation*}$ $

$\begin{equation*} \Lambda _{22} = \left( J_{a+,T}^{\beta -\gamma ,\psi }\right) \left[ 1\right] +\left( ^{c}\mathfrak{D}_{a+,T}^{\gamma ,\psi }\right) \left[ d_{21}\right] \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ 1\right] +\left( ^{c}\mathfrak{D}_{a+,T}^{\gamma ,\psi }\right) \left[ d_{22}\right] \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\beta +\gamma ,\psi }\right) \left[ 1\right] \right) . \label{EQ75} \end{equation*}$ $

We consider the space defined by (3.18) equipped with the norm

$\begin{equation} \left \Vert u\right \Vert _{E} = \left \Vert u\right \Vert _{\infty }+\left \Vert \left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ u\right] \right \Vert _{\infty }. \end{equation}$ $

(3.30)

It is easy to know that $ \left(E, \left \Vert.\right \Vert _{E}\right) $ is a Banach space with norm (3.30). On this space, by virtue of Lemma 3.2, we may define the operator $ \Psi :E\longrightarrow E $ by

$\begin{equation*} \left( \Psi u\right) \left( t\right) = \left( \Psi _{1}u\right) \left( t\right) +\left( \Psi _{2}u\right) \left( t\right) = \left( -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ u\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] +\phi _{u}\left( \mathfrak{F}\right) \right) , \label{EQ77} \end{equation*}$ $

where $ \Psi _{1} $ and $ \Psi _{2} $ are the two operators defined on $ B_{r} $ by

$\begin{equation} \left( \Psi _{1}u\right) \left( t\right) = \left( J_{a+,t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] +d_{11}\left( t\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] +d_{12}\left( t\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\gamma ,\psi }\right) \left[ \mathfrak{F}_{u}\right] \right) \end{equation}$ $

(3.31)

and

$\begin{equation} \left( \Psi _{2}u\right) \left( t\right) = -\lambda \left( J_{a+,t}^{\beta ,\psi }\right) \left[ u\right] +\lambda d_{21}\left( t\right) \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ u\right] -\lambda d_{22}\left( t\right) \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ u\right] -\mu \left( J_{a_{+},\xi }^{\beta +\gamma ,\psi }\right) \left[ u\right] \right) , \end{equation}$ $

(3.32)

where $ d_{ij}\left(t\right) $ are defined by (3.7) and (3.8).

Applying $ \left(^{c}\mathfrak{D}_{a+, t}^{\gamma, \psi }\right) $ on both sides of (3.31) and (3.32), we have

$\begin{eqnarray*} \left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ \Psi _{1}u\right] & = &\left( J_{a+,t}^{\alpha +\beta -\gamma ,\psi }\right) \left[ \mathfrak{F}_{u}\right] +\left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ d_{11}\right] \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] \notag \\ &&+\left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ d_{12}\right] \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u}\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\gamma ,\psi }\right) \left[ \mathfrak{F}_{u} \right] \right) \label{EQ80} \end{eqnarray*}$ $

and

$\begin{eqnarray} \left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ \Psi _{2}u\right] & = &-\lambda \left( J_{a+,t}^{\beta -\gamma ,\psi }\right) \left[ u\right] +\lambda \left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ d_{21}\right] \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ u\right] \\ &&-\lambda \left( ^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\right) \left[ d_{22}\right] \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ u\right] -\mu \left( J_{a_{+},\xi }^{\beta +\gamma ,\psi }\right) \left[ u\right] \right). \end{eqnarray}$ $

(3.33)

Thus, $ \Psi $ is well-defined because $ \Psi _{1} $ and $ \Psi _{2} $ are well-defined. The continuity of the functional $ \mathfrak{F}_{u} $ confirms the continuity of $ \left(\Psi u\right) (t) $ and $ \left(^{c}\mathfrak{D}_{a+, t}^{\gamma, \psi }\right) \left[ \Psi u\right] (t) $, for each $ t\in \left[ a, T\right]. $ Hence the operator $ \Psi $ maps $ E $ into itself.

In what follows, we utilize fixed point techniques to demonstrate the key results of this paper. In light of Lemma 3.2, we rewrite problem (3.9) as

$\begin{equation} u = \Psi u,\ u\in E. \end{equation}$ $

(3.34)

Notice that problem (3.9) has solutions if the operator $ \Psi $ in (3.34) has fixed points. Conversely, the fixed points of $ \Psi $ are solutions of (1.1). Consider the operator $ \Psi :E\longrightarrow E.\ $ For $ u, v\in B_{r}, $ we find that

$\begin{equation*} \left \Vert \Psi u\right \Vert _{E} = \left \Vert \Psi _{1}u\right \Vert _{E}+\left \Vert \Psi _{2}u\right \Vert _{E}. \label{EQ83} \end{equation*}$ $

Theorem 3.4. Assume that $ \mathfrak{F}:[a, T]\times \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}^{+} $ is a continuous function and the assumption (A$ _{3} $) holds. If

$\begin{equation} 0 < \left \vert \lambda \right \vert \left( \Lambda _{21}+\Lambda _{22}\right) < 1, \end{equation}$ $

(3.35)

then, problem (1.1) has at least one fixed point on $ [a, T] $.

Proof. The proof will be completed in four steps:

Step 1. Firstly, we prove that, for any $ u, v\in B_{r} $, $ \Psi _{1}u+\Psi _{2}v\in B_{r} $, it follows that

$\begin{eqnarray*} &&\left \Vert \left( \Psi _{1}u\right) \right \Vert _{E} = \left \Vert \left( \Psi _{1}u\right) \right \Vert _{\infty }+\left \Vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi _{1}u\right) \right \Vert _{\infty } \quad \quad \qquad \quad \\ &\leq &\left[ \left( J_{a+,T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] +d_{11}\left( T\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ 1\right] +d_{12}\left( T\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ 1\right] \right) \right] \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty } \\ &&+\left[ \left( J_{a+,T}^{\alpha +\beta -\delta ,\psi }\right) \left[ 1 \right] +\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{11}\right] \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ 1\right] \right] \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty } \\ &&+\left[ \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{12}\right] \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ 1\right] \right) \right] \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty }, \end{eqnarray*}$ $

we obtain

$\begin{eqnarray*} &&\left \Vert \left( \Psi _{1}u\right) \right \Vert _{E}\times \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty }^{-1} \quad \quad \qquad \quad \notag \\ & = &\left( J_{a+,T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] +\left( J_{a+,T}^{\alpha +\beta -\delta ,\psi }\right) \left[ 1\right] +\left( d_{11}\left( T\right) +\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{11}\right] \right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ 1\right] \notag \\ &&+\left( d_{12}\left( T\right) +\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{12}\right] \right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ 1\right] \right) . \label{EQ86} \end{eqnarray*}$ $

Then, we have

$\begin{equation} \left \Vert \left( \Psi _{1}u\right) \right \Vert _{E}\leq \left( \Lambda _{11}+\Lambda _{12}\right) \times \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty },\ \Lambda _{11}+\Lambda _{12} < \infty , \end{equation}$ $

(3.36)

which yields that $ \Psi _{1} $ is bounded. On the opposite side

$\begin{eqnarray*} &&\frac{1}{\left \vert \lambda \right \vert }\left \Vert \left( \Psi _{2}v\right) \right \Vert _{E}\times \left \Vert v\right \Vert _{E}^{-1} \quad \quad \qquad \quad \label{EQ88} \\ &\leq &\left( J_{a+,T}^{\beta ,\psi }\right) +d_{21}\left( T\right) \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ 1\right] +d_{22}\left( T\right) \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\beta +\delta ,\psi }\right) \left[ 1\right] \right) \notag \\ &&+\left( J_{a+,T}^{\beta -\delta ,\psi }\right) \left[ 1\right] +\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{21}\right] \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ 1\right] +\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{22}\right] \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\beta +\delta ,\psi }\right) \left[ 1\right] \right) , \notag \end{eqnarray*}$ $

which implies

$\begin{equation} \left \Vert \left( \Psi _{2}v\right) \right \Vert _{E}\leq \left \vert \lambda \right \vert \left( \Lambda _{21}+\Lambda _{22}\right) \times \left \Vert v\right \Vert _{E}. \end{equation}$ $

(3.37)

Then, from (3.36) and (3.37), it follows that

$\begin{equation} \left \Vert \Psi \left( u,v\right) \right \Vert _{E}\leq \left( \Lambda _{11}+\Lambda _{12}\right) \times \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty }+\left \vert \lambda \right \vert \left( \Lambda _{21}+\Lambda _{22}\right) \times \left \Vert v\right \Vert _{E}. \end{equation}$ $

(3.38)

By (A$ _{3} $) and (3.38), we have that

$\begin{equation*} \left( \Lambda _{11}+\Lambda _{12}\right) \times \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty }+\left \vert \lambda \right \vert \left( \Lambda _{21}+\Lambda _{22}\right) \times \left \Vert v\right \Vert _{E}\leq \left( \Lambda _{11}+\Lambda _{12}\right) \times L+\left \vert \lambda \right \vert \left( \Lambda _{21}+\Lambda _{22}\right) \times r\leq r. \label{EQ91} \end{equation*}$ $

Then

$\begin{equation*} r > \frac{\left( \Lambda _{11}+\Lambda _{12}\right) \times L}{1-\left \vert \lambda \right \vert \left( \Lambda _{21}+\Lambda _{22}\right) },\ 0 < \left \vert \lambda \right \vert \left( \Lambda _{21}+\Lambda _{22}\right) < 1, \label{EQ92} \end{equation*}$ $

which concludes that $ \Psi _{1}u+\Psi _{2}v\in B_{r} $ for all $ u, v\in B_{r}. $

Step 2. Next, for $ u, v\in B_{r}, \ \Psi _{2} $ is a contraction. From (3.32) and (3.33), we have

$\begin{equation*} \left \Vert \left( \Psi _{2}u\right) -\left( \Psi _{2}v\right) \right \Vert _{E} = \left \Vert \left( \Psi _{2}u\right) -\left( \Psi _{2}v\right) \right \Vert _{\infty }+\left \Vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi _{2}u\right) -\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi _{2}v\right) \right \Vert _{\infty }, \label{EQ93} \end{equation*}$ $

where

$\begin{eqnarray} &&\left \Vert \left( \Psi _{2}u\right) -\left( \Psi _{2}v\right) \right \Vert _{\infty }\leq \\ &&\left \vert \lambda \right \vert \left( \left( J_{a+,t}^{\beta ,\psi }\right) \left[ 1\right] +d_{21}\left( T\right) \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ 1\right] +d_{22}\left( T\right) \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\beta +\delta ,\psi }\right) \left[ 1\right] \right) \right) \left \Vert u-v\right \Vert _{\infty }. \end{eqnarray}$ $

(3.39)

From (3.39), we can write

$\begin{equation} \left \Vert \left( \Psi _{2}u\right) -\left( \Psi _{2}v\right) \right \Vert _{\infty }\leq \left \vert \lambda \right \vert \Lambda _{21}\times \left \Vert u-v\right \Vert _{\infty }. \end{equation}$ $

(3.40)

On the other hand

$\begin{eqnarray*} &&\left \Vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi _{2}u\right) -\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi _{2}v\right) \right \Vert _{\infty }\times \left \Vert u-v\right \Vert _{\infty }^{-1}\leq \\ &&\left \vert \lambda \right \vert \left( \left( J_{a+,t}^{\beta -\delta ,\psi }\right) \left[ 1\right] +\left( ^{c}\mathfrak{D}_{a+,T}^{\delta ,\psi }\right) \left[ d_{21}\right] \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ 1\right] +\left( ^{c}\mathfrak{D}_{a+,T}^{\delta ,\psi }\right) \left[ d_{22}\right] \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\beta +\delta ,\psi }\right) \left[ 1\right] \right) \right) , \end{eqnarray*}$ $

which yields

$\begin{equation} \left \Vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi _{2}u\right) -\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi _{2}v\right) \right \Vert _{\infty }\leq \left \vert \lambda \right \vert \Lambda _{22}\times \left \Vert u-v\right \Vert _{\infty }. \end{equation}$ $

(3.41)

Thus, using (3.40) and (3.41), it follows that

$\begin{equation*} \left \Vert \left( \Psi _{2}u\right) +\left( \Psi _{2}v\right) \right \Vert _{E}\leq \left \vert \lambda \right \vert \left( \Lambda _{21}+\Lambda _{22}\right) \times \left \Vert u-v\right \Vert _{E}, \label{EQ98} \end{equation*}$ $

and choose $ 0 < \left \vert \lambda \right \vert \left(\Lambda _{21}+\Lambda _{22}\right) < 1.\ $Hence, the operator $ \Psi _{2} $ is a contraction.

Step 3. The continuity of $ \Psi _{1} $ follows from that of $ \mathfrak{F}_{u}.\ $Let $ \left \{ u_{n}\right \} $ be a sequence such that $ u_{n} \longrightarrow u $ in $ E $. Then for each $ t\in \left[ a, T\right] $

$\begin{eqnarray*} \left \vert \left( \Psi _{1}u_{n}\right) \left( t\right) -\left( \Psi _{1}u\right) \left( t\right) \right \vert & = &\left( J_{a+,t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u_{n}}-\mathfrak{F}_{u}\right] +d_{11}\left( t\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u_{n}}-\mathfrak{F}_{u}\right] \\ &&+d_{12}\left( t\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u_{n}}-\mathfrak{F}_{u}\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ \mathfrak{F}_{u_{n}}-\mathfrak{F}_{u}\right] \right) . \end{eqnarray*}$ $

By last equality with Eq (3.31), we can write

$\begin{eqnarray*} &&\left \vert \left( \Psi _{1}u_{n}\right) \left( t\right) -\left( \Psi _{1}u\right) \left( t\right) \right \vert \leq \\ &&\left[ \left( J_{a+,t}^{\alpha +\beta ,\psi }\right) \left[ 1\right] +d_{11}\left( t\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ 1\right] +d_{12}\left( t\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ 1\right] \right) \right] \sup\limits_{t\in \left[ a,T\right] }\left \vert \mathfrak{F}_{u_{n}}-\mathfrak{F}_{u}\right \vert . \end{eqnarray*}$ $

It follows that

$\begin{eqnarray} &&\left \Vert \left( \Psi _{1}u_{n}\right) -\left( \Psi _{1}u\right) \right \Vert _{E} = \left \Vert \left( \Psi _{1}u_{n}\right) -\left( \Psi _{1}u\right) \right \Vert _{\infty }+\left \Vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \left( \Psi _{1}u_{n}\right) -\left( \Psi _{1}u\right) \right) \right \Vert _{\infty }\leq \\ &&\left[ \left( J_{a+,T}^{\alpha +\beta ,\psi }\right) +d_{11}\left( T\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) +d_{12}\left( T\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \right) \right] \left \Vert \mathfrak{F}_{u_{n}}-\mathfrak{F}_{u}\right \Vert _{\infty } \\ &&+\left[ \left( J_{a+,T}^{\alpha +\beta -\delta ,\psi }\right) +\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{11}\right] \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \right. \\ &&+\left. \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{12}\right] \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \right) \right] \left \Vert \mathfrak{F}_{u_{n}}-\mathfrak{F}_{u}\right \Vert _{\infty }. \end{eqnarray}$ $

(3.42)

By (3.42), we have

$\begin{eqnarray} &&\left \Vert \left( \Psi _{1}u_{n}\right) -\left( \Psi _{1}u\right) \right \Vert _{E}\left \Vert \mathfrak{F}_{u_{n}}-\mathfrak{F}_{u}\right \Vert _{\infty }^{-1}\leq \\ &&\left( J_{a+,T}^{\alpha +\beta ,\psi }\left[ 1\right] \right) +\left( J_{a+,T}^{\alpha +\beta -\delta ,\psi }\left[ 1\right] \right) +\left( d_{11}\left( T\right) +\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{11}\right] \right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\left[ 1 \right] \right) \\ &&+\left( d_{12}\left( T\right) +\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ d_{12}\right] \right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi } \left[ 1\right] \right) -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\left[ 1\right] \right) \right) . \end{eqnarray}$ $

(3.43)

Consequently, by (3.43), we have

$\begin{equation*} \left \Vert \left( \Psi _{1}u_{n}\right) -\left( \Psi _{1}u\right) \right \Vert _{\infty }\leq \left( \Lambda _{11}+\Lambda _{12}\right) \times \left \Vert \mathfrak{F}_{u_{n}}-\mathfrak{F}_{u}\right \Vert _{\infty },\ \Lambda _{11}+\Lambda _{12} < \infty . \label{EQ104} \end{equation*}$ $

Since $ \mathfrak{F}_{u} $ is a continuous function, then by Lebesgue's dominated convergence theorem it follows that

$\begin{equation*} \left \Vert \left( \Psi _{1}u_{n}\right) -\left( \Psi _{1}u\right) \right \Vert _{\infty}\longrightarrow 0{\rm{ as }}n\longrightarrow \infty . \label{EQ105} \end{equation*}$ $

Furthermore, $ \Psi _{1} $ is uniformly bounded on $ B_{r} $ as $ \left \Vert \left(\Psi _{1}u\right) \right \Vert _{E}\leq \left(\Lambda _{11}+\Lambda _{12}\right) \times \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty }, $ due to (3.36).

Step 4. Finally, we establish the compactness of $ \Psi _{1}. $ Let $ u, v\in B_{r} $, for $ t_{1}, t_{2}\in \left[ a, T\right], \; t_{1} < t_{2}, $ we have

$\begin{eqnarray} &&\left \Vert \left( \Psi _{1}u\right) \left( t_{2}\right) -\left( \Psi _{1}u\right) \left( t_{1}\right) \right \Vert _{\infty } \\ &\leq &\left[ \left( J_{a+,t_{2}}^{\alpha +\beta ,\psi }\right) \left[ 1 \right] +d_{11}\left( t_{2}\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ 1\right] +d_{12}\left( t_{2}\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ 1\right] \right) \right] \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty } \\ &&-\left[ \left( J_{a+,t_{1}}^{\alpha +\beta ,\psi }\right) \left[ 1\right] +d_{11}\left( t_{1}\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ 1\right] +d_{12}\left( t_{1}\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \right) \left[ 1\right] \right] \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty } \\ &\leq &\left[ \left( \left( J_{a+,t_{2}}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\left( J_{a+,t_{1}}^{\alpha +\beta ,\psi }\right) \left[ 1\right] \right) +\Lambda _{41}\left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ 1\right] \right. \\ &&\left. +\Lambda _{42}\left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ 1\right] \right) \right] \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty }. \end{eqnarray}$ $

(3.44)

On the other hand

$\begin{eqnarray} &&\left \Vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi _{1}u\right) \left( t_{2}\right) -\left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left( \Psi _{1}u\right) \left( t_{1}\right) \right \Vert _{\infty } \\ &\leq &\left[ \left( J_{a+,t_{2}}^{\alpha +\beta -\delta ,\psi }\right) +\left( ^{c}\mathfrak{D}_{a+,t_{2}}^{\delta ,\psi }\right) \left[ d_{11}\right] \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) +\left( ^{c}\mathfrak{D}_{a+,t_{2}}^{\delta ,\psi }\right) \left[ d_{12}\right] \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \right) \right] \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty } \\ &&-\left[ \left( J_{a+,t_{1}}^{\alpha +\beta -\delta ,\psi }\right) +\left( ^{c}\mathfrak{D}_{a+,t_{1}}^{\delta ,\psi }\right) \left[ d_{11}\right] \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) +\left( ^{c}\mathfrak{D}_{a+,t_{1}}^{\delta ,\psi }\right) \left[ d_{12}\right] \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \right) \right] \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty } \\ &&\left[ \left( \left( J_{a+,t_{2}}^{\alpha +\beta -\delta ,\psi }\right) -\left( J_{a+,t_{1}}^{\alpha +\beta -\delta ,\psi }\right) \right) +\Lambda _{43}\left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) +\Lambda _{44}\left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \right) \right] \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty }. \end{eqnarray}$ $

(3.45)

Using (3.44) and (3.45), we get

$\begin{eqnarray} &&\left \Vert \left( \Psi _{1}u\right) \left( t_{2}\right) -\left( \Psi _{1}u\right) \left( t_{1}\right) \right \Vert _{E} \\ &\leq &\left[ \left( \left( J_{a+,t_{2}}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\left( J_{a+,t_{1}}^{\alpha +\beta ,\psi }\right) \left[ 1\right] \right) +\left( \left( J_{a+,t_{2}}^{\alpha +\beta -\delta ,\psi }\right) -\left( J_{a+,t_{1}}^{\alpha +\beta -\delta ,\psi }\right) \right) \right. \\ &&\left. +\left( \Lambda _{41}+\Lambda _{43}\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ 1\right] +\left( \Lambda _{42}+\Lambda _{44}\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ 1\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ 1\right] \right) \right] \left \Vert \mathfrak{F}_{u}\right \Vert _{\infty }, \end{eqnarray}$ $

where

$\begin{equation*} \Lambda _{41} = d_{11}\left( t_{2}\right) -d_{11}\left( t_{1}\right), \Lambda _{42} = d_{12}\left( t_{2}\right) -d_{12}\left( t_{1}\right) \end{equation*}$ $

and

$\begin{equation*} \Lambda _{43} = \left( ^{c}\mathfrak{D}_{a+,t_{2}}^{\delta ,\psi }\right) \left[ d_{11} \right] -\left( ^{c}\mathfrak{D}_{a+,t_{1}}^{\delta ,\psi }\right) \left[ d_{11}\right], \Lambda _{44} = \left( ^{c}\mathfrak{D}_{a+,t_{2}}^{\delta ,\psi }\right) \left[ d_{12}\right] -\left( ^{c}\mathfrak{D}_{a+,t_{1}}^{\delta ,\psi }\right) \left[ d_{11}\right] . \end{equation*}$ $

Consequently, we have

$\begin{equation*} \left \Vert \left( \Psi _{1}u\right) \left( t_{2}\right) -\left( \Psi _{1}u\right) \left( t_{1}\right) \right \Vert \times \left \Vert \mathfrak{F}_{u}\right \Vert ^{-1}\longrightarrow 0{\rm{ as }}t_{1}\rightarrow t_{2}. \label{EQ113} \end{equation*}$ $

Thus, $ \Psi _{1} $ is relatively compact on $ B_{r} $. Hence, by the Arzela-Ascoli Theorem, $ \Psi _{1} $ is completely continuous on $ B_{r} $. Therefore, according to Theorem 2.8, the Problem (1.1) has at least one solution on $ B_{r} $. This completes the proof.

4. Stability of solutions

Hereafter, we discuss the Ulam–Hyers and Ulam–Hyers–Rassias stability of solutions of the FLE (1.1). In the proofs of Theorems 4.4 and 4.9, we use integration by parts in the settings of $ \psi $-fractional operators. Denoting

$\begin{equation} \varphi _{1}\left( t\right) = \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ 1\right] \;{\rm{ and }}\;\varphi _{2}\left( t\right) = \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \mathcal{K}\left( \tau ;a\right) \right] . \end{equation}$ $

(4.1)

Remark 4.1. For every $ \epsilon > 0, $ a function $ \tilde{u}\in \mathcal{C} $ is a solution of of the inequality

$\begin{equation} \left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left( ^{c}\mathfrak{D}_{a+,t}^{\beta ,\psi }+\lambda \right) \left[ \tilde{u}\right] -\mathfrak{F}(t, \tilde{u}(t),^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\left[ \tilde{u}\right] )\right \vert \leq \epsilon \Phi \left( t\right) ,\ t\in \left[ a,T\right] , \end{equation}$ $

(4.2)

where $ \Phi \left(t\right) \geq 0 $ if and only if there exists a function $ g\in \mathcal{C}, $ (which depends on $ \tilde{u} $) such that

$ (\textbf{i}) \; \left \vert g\left(t\right) \right \vert \leq \epsilon \Phi \left(t\right), \ \forall t\in \left[ a, T\right]; $

$ (\textbf{ii}) \; \left(^{c}\mathfrak{D}_{a+, t}^{\alpha, \psi }\right) \left(^{c}\mathfrak{D}_{a+, t}^{\beta, \psi }+\lambda \right) \left[ \tilde{u}\right] = \mathfrak{F}(t, \tilde{u}(t), ^{c}\mathfrak{D}_{a+, t}^{\gamma, \psi }\left[ \tilde{u}\right])+g\left(t\right). $

Lemma 4.2. If $ \tilde{u}\in \mathcal{C} $ is a solution of the inequality (4.2) then $ \tilde{u} $ is a solution of the following integral inequality

$\begin{equation} \left \vert \tilde{u}(t)-\left( -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}\right] +\phi _{\tilde{u}}\left( \mathfrak{F}\right) \right) \right \vert \leq C_{\Phi }\left( t\right) , \end{equation}$ $

(4.3)

where

$\begin{equation*} C_{\Phi }\left( t\right) = \epsilon \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \Phi \right] +c_{1}\left( \epsilon \Phi \right) \varphi _{1}\left( t\right) +c_{2}\left( \epsilon \Phi \right) \varphi _{2}\left( t\right) +c_{3}\left( \epsilon \Phi \right) , \label{EQ117} \end{equation*}$ $

where $ c_{1}\left(\epsilon \Phi \right)-c_{3}\left(\epsilon \Phi \right) $ are real constants with $ \mathfrak{F}_{ \tilde{u}} = \Phi $ and $ C_{\Phi } $ is independent of $ \tilde{u}\left(t\right) $ and $ \mathfrak{F}_{\tilde{u}}. $

Proof. Let $ \tilde{u}\in \mathcal{C} $ be a solution of the inequality (4.2). Then by Remark 4.1-$ (ii) $, we have that

$\begin{equation} \tilde{u}\left( t\right) = -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}+g\right] +\phi _{\tilde{u}}\left( \mathfrak{F}_{\tilde{u} }+g\right) , \end{equation}$ $

(4.4)

where

$\begin{equation*} \phi _{\tilde{u}}\left( \mathfrak{F}_{\tilde{u}}+g\right) = c_{1}\left( \mathfrak{F}_{\tilde{u} }+g\right) \varphi _{1}\left( t\right) +c_{2}\left( \mathfrak{F}_{\tilde{u}}+g\right) \varphi _{2}\left( t\right) +c_{3}\left( \mathfrak{F}_{\tilde{u}}+g\right) , \label{EQ119} \end{equation*}$ $

with

$\begin{equation*} c_{j}\left( \mathfrak{F}_{\tilde{u}}+g\right) = c_{j}\left( \mathfrak{F}_{\tilde{u}}\right) +c_{j}\left( g\right) ,\ j = 1,2,3. \label{EQ120} \end{equation*}$ $

In view of (A$ _{1} $) and (4.3), we obtain

$\begin{eqnarray*} &&\left \vert \tilde{u}\left( t\right) -\left( -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}\right] +\phi _{\tilde{u}}\left( \mathfrak{F}\right) \right) \right \vert \quad \notag \\ & = &\left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ g \right] +c_{1}\left( g\right) \varphi _{1}\left( t\right) +c_{2}\left( g\right) \varphi _{2}\left( t\right) +c_{3}\left( g\right) \right \vert \notag \\ &\leq &\left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \epsilon \Phi \right] +c_{1}\left( \epsilon \Phi \right) \varphi _{1}\left( t\right) +c_{2}\left( \epsilon \Phi \right) \varphi _{2}\left( t\right) +c_{3}\left( \epsilon \Phi \right) \right \vert = C_{\Phi }\left( t\right) . \label{EQ121} \end{eqnarray*}$ $

As an outcome of Lemma 4.2, we have the following result:

Corollary 4.3. Assume that $ \mathfrak{F}_{\tilde{u}} $ is a continuous function that satisfies $ (A_{1}). $ If $ \tilde{u}\in \mathcal{C} $ is a solution of the inequality

$\begin{equation} \left \vert ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\left( ^{c}\mathfrak{D}_{a+,t}^{\beta ,\psi }+\lambda \right) u(t)-\mathfrak{F}(t,u(t),^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi }\left[ u\right] (t))\right \vert \leq \epsilon ,\ t\in \left[ a,T\right] , \end{equation}$ $

(4.5)

then $ \tilde{u} $ is a solution of the following integral inequality

(4.6)

with

$\begin{equation} \tilde{u}(a) = 0,\; \tilde{u}(\eta ) = 0,\; \tilde{u}(T) = \mu \left( J_{a+,\xi }^{\gamma ,\psi }\right) \left[ \tilde{u}\right] ,\; a < \eta < \xi < T,\ 0 < \mu , \end{equation}$ $

(4.7)

where

$\begin{equation} C_{\epsilon } = \epsilon \varsigma _{13}, \end{equation}$ $

(4.8)

where $ \varsigma _{13} $ is given by (3.12).

Proof. By Remark 4.1-$ (ii), $ (4.4), and by using (3.10) with the conditions (4.7), we have

$\begin{eqnarray*} \phi _{\tilde{u}}\left( \mathfrak{F}+g\right) & = &d_{11}\left( t\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}\right] +d_{12}\left( t\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}\right] \right) \notag \\ &&+\lambda d_{21}\left( t\right) \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ \tilde{u}\right] -\lambda d_{22}\left( t\right) \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ \tilde{u}\right] -\mu \left( J_{a_{+},\xi }^{\beta +\delta ,\psi }\right) \left[ \tilde{u}\right] \right) \notag \\ &&+d_{11}\left( t\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ g\right] +d_{12}\left( t\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ g\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ g\right] \right) . \label{EQ126} \end{eqnarray*}$ $

The solution of the problem (4.4) is given by

$\begin{eqnarray*} &&\left \vert \tilde{u}\left( t\right) -\left( -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}\right] +\phi _{\tilde{u}}\left( \mathfrak{F}\right) \right) \right \vert \quad \notag \\ &\leq &\left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ g \right] +d_{11}\left( t\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ g\right] +d_{12}\left( t\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ g\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ g\right] \right) \right \vert , \label{EQ127} \end{eqnarray*}$ $

which implies that

$\begin{equation*} \left \vert \tilde{u}\left( t\right) -\left( -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}\right] +\phi _{\tilde{u}}\left( \mathfrak{F}\right) \right) \right \vert \leq \Phi _{\epsilon }\left( t\right) , \label{EQ128} \end{equation*}$ $

where

$\begin{equation*} \Phi _{\epsilon }\left( t\right) = \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \epsilon \right] +c_{1}\left( \epsilon \right) \varphi _{1}\left( t\right) +c_{2}\left( \epsilon \right) \varphi _{2}\left( t\right) +c_{3}\left( \epsilon \right) \label{EQ130} \end{equation*}$ $

with

$\begin{equation*} C_{\epsilon }\equiv \sup\limits_{t\in \left[ a,T\right] }\left \vert \Phi _{\epsilon }\left( t\right) \right \vert = \epsilon \varsigma _{13}, \label{EQ131} \end{equation*}$ $

which is the desired inequality (4.6).

This corollary is obtained from Lemma 4.2 by setting $ \Phi \left(t\right) = 1 $, for all $ \ t\in \left[ a, T\right], $ with (4.7).

Theorem 4.4. Assume that $ \mathfrak{F}_{\tilde{u}} $ is a continuous function that satisfies (A$ _{1} $) and (A$ _{4} $). The Eq (1.1-a) is H-U-R stable with respect to $ \Phi \ $if there exists a real number $ l_{\alpha, \psi } > 0 $ such that for each $ \epsilon > 0 $ and for each solution $ \tilde{u} \in \mathcal{C}^{3}\left(\left[ a, T\right], \mathbb{R} \right) $ of the inequality (4.2)$, \ $there exists a solution $ u^{\ast }\in \mathcal{C}^{3}\left(\left[ a, T\right], \mathbb{R} \right) $ of (1.1-a) with

$\begin{equation} \left \vert \tilde{u}(t)-u^{\ast }(t)\right \vert \leq \epsilon l_{\alpha ,\psi }\Phi \left( t\right) . \end{equation}$ $

(4.9)

Proof. Using (4.2) and (1.1), we obtain

$\begin{eqnarray} \tilde{u}\left( t\right) & = &-\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}+g\right] +c_{1}\left( \mathfrak{F}_{\tilde{u}}+g\right) \varphi _{1}\left( t\right) +c_{2}\left( \mathfrak{F}_{\tilde{u}}+g\right) \varphi _{2}\left( t\right) +c_{3}\left( \mathfrak{F}_{\tilde{u}}+g\right) \\ & = &\theta _{\tilde{u}}\left( t,\mathfrak{F}_{\tilde{u}}+g\right) +c_{1}\left( \mathfrak{F}+g\right) \varphi _{1}\left( t\right) +c_{2}\left( \mathfrak{F}+g\right) \varphi _{2}\left( t\right) +c_{3}\left( \mathfrak{F}+g\right) \end{eqnarray}$ $

(4.10)

and

$\begin{eqnarray*} u^{\ast }\left( t\right) & = &-\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ u^{\ast }\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u^{\ast }}\right] +c_{1}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \varphi _{1}\left( t\right) +c_{2}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \varphi _{2}\left( t\right) +c_{3}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \notag \\ & = &\theta _{u^{\ast }}\left( t,\mathfrak{F}\right) +c_{1}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \varphi _{1}\left( t\right) +c_{1}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \varphi _{1}\left( t\right) +c_{2}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \varphi _{2}\left( t\right) +c_{3}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) , \label{EQ134} \end{eqnarray*}$ $

where

$\begin{equation} \theta _{\tilde{u}}\left( t,\mathfrak{F}\right) = -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}\right] . \end{equation}$ $

(4.11)

By using (4.9) and (4.10), we have the following inequalities

$\begin{eqnarray*} \left \vert \tilde{u}(t)-u^{\ast }(t)\right \vert &\leq &\left \vert \tilde{u} \left( t\right) -\left( \theta _{u^{\ast }}\left( t,\mathfrak{F}\right) +c_{1}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \varphi _{1}\left( t\right) +c_{2}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \varphi _{2}\left( t\right) +c_{3}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \right) \right \vert \notag \\ &\leq &\left \vert \tilde{u}\left( t\right) -\left( \theta _{\tilde{u}}\left( t,\mathfrak{F}\right) +c_{1}\left( \mathfrak{F}_{\tilde{u}}\right) \varphi _{1}\left( t\right) +c_{1}\left( \mathfrak{F}_{\tilde{u}}\right) \varphi _{1}\left( t\right) +c_{2}\left( \mathfrak{F}_{\tilde{u}}\right) \varphi _{2}\left( t\right) +c_{3}\left( \mathfrak{F}_{\tilde{u} }\right) \right) \right \vert \notag \\ &&+\left \vert \theta _{\tilde{u}}\left( t,\mathfrak{F}\right) -\theta _{u^{\ast }}\left( t,\mathfrak{F}\right) \right \vert +\left \vert \left( c_{1}\left( \mathfrak{F}_{ \tilde{u}}\right) -c_{1}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \right) \varphi _{1}\left( t\right) \right \vert \notag \\ &&+\left \vert \left( c_{2}\left( \mathfrak{F}_{\tilde{u}}\right) -c_{2}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \right) \right \vert \varphi _{2}\left( t\right) +\left \vert c_{3}\left( \mathfrak{F}_{\tilde{u}}\right) -c_{3}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \right \vert . \label{EQ136} \end{eqnarray*}$ $

By setting

$\begin{equation*} c_{33} = c_{3}\left( \mathfrak{F}_{\tilde{u}}\right) -c_{3}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) = \tilde{u}\left( a\right) -u^{\ast }\left( a\right) ,\ c_{11} = c_{1}\left( \mathfrak{F}_{\tilde{u}}\right) -c_{1}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) \;{\rm{ and }}\;c_{22} = c_{2}\left( \mathfrak{F}_{\tilde{u} }\right) -c_{2}^{\prime }\left( \mathfrak{F}_{u^{\ast }}\right) , \label{EQ137} \end{equation*}$ $

and

$\begin{equation*} w\left( \eta \right) = \tilde{u}\left( \eta \right) -u^{\ast }\left( \eta \right) +\theta _{\tilde{u}}\left( \eta ,\mathfrak{F}\right) -\theta _{u^{\ast }}\left( \eta ,\mathfrak{F}\right) \;{\rm{ and }}\;w\left( T\right) = \tilde{u}\left( T\right) -u^{\ast }\left( T\right) +\theta _{\tilde{u}}\left( T,\mathfrak{F}\right) -\theta _{u^{\ast }}\left( T,\mathfrak{F}\right) . \label{EQ138} \end{equation*}$ $

It follows from (4.9) and (4.10), that

$\begin{equation*} \left( \begin{array}{cc} \varphi _{1}\left( \eta \right) & \varphi _{2}\left( \eta \right) \\ \varphi _{1}\left( T\right) & \varphi _{2}\left( T\right) \end{array} \right) \left( \begin{array}{c} c_{11} \\ c_{22} \end{array} \right) = \left( \begin{array}{c} w\left( \eta \right) \\ w\left( T\right) \end{array} \right) , \label{EQ139} \end{equation*}$ $

Applying Lemma 4.2 and from estimation (4.11), it follows

$\begin{equation*} \left \vert \tilde{u}(t)-u^{\ast }(t)\right \vert \leq C_{\Phi }\left( t\right) +\left \vert \theta _{\tilde{u}}\left( t,\mathfrak{F}\right) -\theta _{u^{\ast }}\left( t,\mathfrak{F}\right) \right \vert +c_{11}\varphi _{1}\left( t\right) +c_{22}\varphi _{2}\left( t\right) +c_{33}, \label{EQ140} \end{equation*}$ $

where

$\begin{eqnarray*} \left \vert \theta _{\tilde{u}}\left( t,\mathfrak{F}\right) -\theta _{u^{\ast }}\left( t,\mathfrak{F}\right) \right \vert & = &\left \vert -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}\right] -\left( -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ u^{\ast }\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u^{\ast }}\right] \right) \right \vert \notag \\ &\leq &\left \vert -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}-\mathfrak{F}_{u^{\ast }}\right] \right \vert . \label{EQ141} \end{eqnarray*}$ $

Using Lemma 2.4 and (A$ _{1} $), we have

$\begin{equation*} \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \left \vert \left( ^{c}\mathfrak{D}_{a_{+},t}^{\delta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] \right \vert \right] = z_{0}\left( t\right) -\left( J_{a_{+},t}^{\alpha +\beta -\delta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] \label{EQ142} \end{equation*}$ $

and

$\begin{equation*} \left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{ u}}-\mathfrak{F}_{u^{\ast }}\right] \right \vert \leq \left \vert L_{1}\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] \right \vert +L_{2}\left \vert z_{0}\left( t\right) -\left( J_{a_{+},t}^{\alpha +\beta -\delta ,\psi }\right) \left[ \tilde{u}-u^{\ast } \right] \right \vert , \label{EQ143} \end{equation*}$ $

where

$\begin{equation*} z_{0}\left( t\right) = \frac{\left \vert \tilde{u}\left( a\right) -u^{\ast }\left( a\right) \right \vert }{\Gamma \left( \alpha +\beta \right) }\times \left( J_{a,t-}^{1-\delta ,\psi }\right) \left[ \left( \mathcal{K}\left( t;a\right) \right) ^{\alpha +\beta -1}\right] . \label{EQ144} \end{equation*}$ $

Set

$\begin{equation*} q\left( t\right) = \mathcal{G}\left( t\right) +L_{2}\frac{\left \vert \tilde{u} \left( a\right) -u^{\ast }\left( a\right) \right \vert }{\Gamma \left( \alpha +\beta \right) }\times \left( J_{a,t-}^{1-\delta ,\psi }\right) \left[ \left( \mathcal{K}\left( t;a\right) \right) ^{\alpha +\beta -1}\right] , \label{EQ145} \end{equation*}$ $

where

$\begin{equation*} \mathcal{G}\left( t\right) = C_{\Phi }\left( t\right) +c_{11}\varphi _{1}\left( t\right) +c_{22}\varphi _{2}\left( t\right) +c_{33}, \label{EQ146} \end{equation*}$ $

with

This means that

$\begin{equation} p\left( t\right) \leq q\left( t\right) +\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] +L_{1}\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] +L_{2}\left( J_{a_{+},t}^{\alpha +\beta -\delta ,\psi }\right) \left[ \tilde{ u}-u^{\ast }\right] . \end{equation}$ $

(4.12)

Using Lemma 2.13, the above inequality implies the estimation for $ p\left(t\right) $ such as

$\begin{equation*} p\left( t\right) \leq q\left( t\right) +\sum\limits_{k = 1}^{\infty }\left( \begin{array}{l} \tfrac{\left( \lambda \Gamma \left( \beta \right) \right) ^{k}}{\Gamma \left( k\beta \right) }\int_{a}^{t}\left[ \psi ^{\prime }\left( \tau \right) \left( \mathcal{K}\left( t;\tau \right) \right) ^{k\beta -1}\right] q\left( \tau \right) \mathrm{d}\tau \\ + \\ \tfrac{\left( L_{1}\Gamma \left( \alpha +\beta \right) \right) ^{k}}{\Gamma \left( k\left( \alpha +\beta \right) \right) }\int_{a}^{t}\left[ \psi ^{\prime }\left( \tau \right) \left( \mathcal{K}\left( t;\tau \right) \right) ^{k\left( \alpha +\beta \right) -1}\right] q\left( \tau \right) \mathrm{d}\tau \\ + \\ \tfrac{\left( L_{2}\Gamma \left( \alpha +\beta -\delta \right) \right) ^{k}}{ \Gamma \left( k\left( \alpha +\beta -\delta \right) \right) }\int_{a}^{t} \left[ \psi ^{\prime }\left( \tau \right) \left( \mathcal{K}\left( t;\tau \right) \right) ^{k\left( \alpha +\beta -\delta \right) -1}\right] q\left( \tau \right) \mathrm{d}\tau \end{array} \right) . \label{EQ149} \end{equation*}$ $

Therefore, with (A$ _{4} $), the inequality (4.12) can be rewritten as

$\begin{equation*} p\left( t\right) = \left \vert \tilde{u}(t)-u^{\ast }(t)\right \vert \leq \epsilon l_{\alpha ,\psi }\Phi \left( t\right) . \label{EQ150} \end{equation*}$ $

By Remark 2.14, one can obtain

$\begin{eqnarray*} p\left( t\right) &\leq &q\left( t\right) \left[ E_{\beta }\left( \lambda \Gamma \left( \beta \right) \left( \mathcal{K}\left( t;a\right) \right) ^{\beta }\right) +E_{\alpha +\beta }\left( \lambda \Gamma \left( \alpha +\beta \right) \left( \mathcal{K}\left( t;a\right) \right) ^{\alpha +\beta }\right) \right. \\ &&+\left. E_{\alpha +\beta +\delta }\left( \lambda \Gamma \left( \alpha +\beta +\delta \right) \left( \mathcal{K}\left( t;a\right) \right) ^{\alpha +\beta +\delta }\right) \right] . \end{eqnarray*}$ $

Thus, we complete the proof.

Theorem 4.5. Assume that the assumptions (A$ _{1} $) and (A$ _{4} $). If a continuously differentiable function $ \tilde{u}:\left[ a, T\right] \longrightarrow \mathbb{R} $ satisfies (4.2), where $ \Phi :\left[ a, T\right] \longrightarrow \mathbb{R} ^{+} $ is a continuous function with (A$ _{3} $), then there exists a unique continuous function $ u^{\ast }:\left[ a, T\right] \longrightarrow \mathbb{R} $ of problem (1.1) such that

$\begin{equation} \left \vert \tilde{u}\left( t\right) -u^{\ast }\left( t\right) \right \vert \leq \epsilon l_{\alpha ,\psi }\Phi \left( t\right) , \end{equation}$ $

(4.13)

with

$\begin{equation} \left \vert \tilde{u}\left( a\right) -u^{\ast }\left( a\right) \right \vert = \left \vert \tilde{u}\left( \eta \right) -u^{\ast }\left( \eta \right) \right \vert = \left \vert \tilde{u}\left( T\right) -u^{\ast }\left( T\right) \right \vert = 0. \end{equation}$ $

(4.14)

Proof. Assume that $ \tilde{u}\in \mathcal{C}^{3}\left(\left[ a, T\right], \mathbb{R} \right) $ is a solution of the (4.2). In view of proof of Theorem 4.4, we get

$\begin{equation*} \mathcal{G}\left( t\right) = C_{\Phi }\left( t\right) +c_{11}\varphi _{1}\left( t\right) +c_{22}\varphi _{2}\left( t\right) +c_{33} = C_{\Phi }\left( t\right), \label{EQ153} \end{equation*}$ $

with the conditions (4.14), we have

$\begin{equation*} C_{\Phi }\left( t\right) = \epsilon \left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \Phi \right] +\left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ \Phi \right] d_{11}\left( t\right) +\left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ \Phi \right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ \Phi \right] \right) d_{12}\left( t\right) \right \vert . \label{EQ154} \end{equation*}$ $

Set $ q\left(t\right) = C_{\Phi }\left(t\right).\ $Using Theorem 4.4 and (A$ _{4} $), we conclude that, the estimation for $ p\left(t\right) = \left \vert u\left(t\right) -\tilde{u}\left(t\right) \right \vert $ such as (4.12).\ So the inequality (4.12) can be rewritten as

$\begin{equation*} p\left( t\right) = \left \vert u\left( t\right) -\tilde{u}\left( t\right) \right \vert \leq \epsilon l_{\alpha ,\psi }\Phi \left( t\right) . \label{EQ156} \end{equation*}$ $

By Remark 2.14, one can obtain

This proves that the problem (1.1) is, Ulam–Hyers–Rassias stable.

Theorem 4.6. Assume that the assumptions (A$ _{2} $), (A$ _{4} $) and (4.2) hold. Then Eq (1.1-a) is H-U-R stable.

Proof. By (A$ _{2} $) and (4.11), we have

$\begin{equation*} \left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{ u}}\right] -\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u} \right] \right \vert \leq \left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \tilde{\chi}\right] -\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \chi \right] \right \vert \label{EQ157} \end{equation*}$ $

and

$\begin{equation*} \left \vert \tilde{u}(t)-u^{\ast }(t)\right \vert \leq C_{\Phi }\left( t\right) +\left \vert \theta _{\tilde{u}}\left( t,\mathfrak{F}+g\right) -\theta _{u^{\ast }}\left( t,\mathfrak{F}\right) \right \vert +c_{11}\left \vert \varphi _{1}\left( t\right) \right \vert +c_{22}\left \vert \varphi _{2}\left( t\right) \right \vert +c_{33}, \label{EQ158} \end{equation*}$ $

where

$\begin{eqnarray*} &&\left \vert \theta _{\tilde{u}}\left( t,\mathfrak{F}+g\right) -\theta _{u^{\ast }}\left( t,\mathfrak{F}\right) \right \vert \notag \\ & = &\left \vert -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{ \tilde{u}}+g\right] -\left( -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ u^{\ast }\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u^{\ast }}\right] \right) \right \vert \notag \\ &\leq &\left \vert -\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] +\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}-\mathfrak{F}_{u^{\ast }}\right] \right \vert +\left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ g\right] \right \vert . \label{EQ159} \end{eqnarray*}$ $

Using Lemma 4.2, we have

$\begin{equation*} p\left( t\right) \leq q\left( t\right) +\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] , \label{EQ160} \end{equation*}$ $

where

$\begin{equation*} q\left( t\right) = \mathcal{G}\left( t\right) +\left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \tilde{\chi}\right] -\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \chi \right] \right \vert , \label{EQ161} \end{equation*}$ $

with

$\begin{equation*} \mathcal{G}\left( t\right) = C_{\Phi }\left( t\right) +c_{11}\varphi _{1}\left( t\right) +c_{22}\varphi _{2}\left( t\right) +c_{33}. \label{EQ162} \end{equation*}$ $

From the above, it follows

$\begin{equation*} p\left( t\right) \leq q\left( t\right) +\sum\limits_{k = 1}^{\infty } \begin{array}{l} \dfrac{\left( \lambda \Gamma \left( \beta \right) \right) ^{k}}{\Gamma \left( k\beta \right) }\int_{a}^{t}\left[ \psi ^{\prime }\left( \tau \right) \left( \mathcal{K}\left( t;\tau \right) \right) ^{k\beta -1}\right] q\left( \tau \right) \mathrm{d}\tau \end{array} . \label{EQ163} \end{equation*}$ $

By Remark 2.14, one can obtain

$\begin{equation*} p\left( t\right) \leq q\left( t\right) E_{\beta }\left( \lambda \Gamma \left( \beta \right) \left( \mathcal{K}\left( t;a\right) \right) ^{\beta }\right) . \end{equation*}$ $

Remark 4.7. If $ \Phi \left(t\right) $ is a constant function in the inequalities (4.2), then we say that (1.1-a) is Ulam–Hyers stable.

Corollary 4.8. Assume that the assumptions (A$ _{2} $), (A$ _{4} $) and (4.2) hold. Then Eq (1.1-a) with (4.13) is Ulam–Hyers–Rassias stable.

Proof. Using Theorem 4.6, we have

$\begin{equation*} p\left( t\right) \leq q\left( t\right) +\lambda \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] , \label{EQ164} \end{equation*}$ $

where

$\begin{equation*} p\left( t\right) = \left \vert \tilde{u}\left( t\right) -u\left( t\right) \right \vert {\rm{ and}}\ q\left( t\right) = C_{\Phi }\left( t\right) +\left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \tilde{ \chi}\right] -\left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \chi \right] \right \vert . \label{EQ165} \end{equation*}$ $

We conclude that

By Remark 2.14, one can obtain

$\begin{equation*} p\left( t\right) \leq q\left( t\right) E_{\beta }\left( \lambda \Gamma \left( \beta \right) \left( \mathcal{K}\left( t;a\right) \right) ^{\beta }\right) . \end{equation*}$ $

Theorem 4.9. Assume that the assumptions (A$ _{1} $) and (4.2) with (4.14) hold. Then problem (1.1) is Ulam–Hyers stable and consequently generalized Ulam–Hyers stable.

Proof. Let $ u^{\ast } $ be a unique solution of the fractional Langevin type problem (1.1), that is, $ u^{\ast }(t) = \left(\Psi u^{\ast }\right) (t) $. Assume that $ \tilde{u}\in \mathcal{C}\left(\left[ a, T\right], \mathbb{R} \right) $ is a solution of the (4.2). By using the estimation

$\begin{equation*} \left \vert \left( \Psi \tilde{u}\right) \left( t\right) -\left( \Psi u^{\ast }\right) \left( t\right) \right \vert \leq \lambda \left \vert \left( J_{a_{+},t}^{\beta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] \right \vert +\left \vert \left( J_{a_{+},t}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{u^{\ast }}-\left( \mathfrak{F}_{\tilde{u}}+g\right) \right] \right \vert +\left \vert \phi _{\tilde{u}}\left( \mathfrak{F}+g\right) -\phi _{u^{\ast }}\left( t,\mathfrak{F}\right) \right \vert , \label{EQ168} \end{equation*}$ $

where

$\begin{eqnarray} &&\left \vert \phi _{\tilde{u}}\left( \mathfrak{F}+g\right) -\phi _{u^{\ast }}\left( t,\mathfrak{F}\right) \right \vert \\ & = &d_{11}\left( t\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}-\mathfrak{F}_{u^{\ast }}\right] +d_{12}\left( t\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u} }-\mathfrak{F}_{u^{\ast }}\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ \mathfrak{F}_{\tilde{u}}-\mathfrak{F}_{u^{\ast }}\right] \right) \\ &&+\lambda d_{21}\left( t\right) \left( J_{a_{+},\eta }^{\beta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] -\lambda d_{22}\left( t\right) \left( \left( J_{a_{+},T}^{\beta ,\psi }\right) \left[ \tilde{u}-u^{\ast } \right] -\mu \left( J_{a_{+},\xi }^{\beta +\delta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] \right) \\ &&+d_{11}\left( t\right) \left( J_{a_{+},\eta }^{\alpha +\beta ,\psi }\right) \left[ g\right] +d_{12}\left( t\right) \left( \left( J_{a_{+},T}^{\alpha +\beta ,\psi }\right) \left[ g\right] -\mu \left( J_{a_{+},\xi }^{\alpha +\beta +\delta ,\psi }\right) \left[ g\right] \right) . \end{eqnarray}$ $

(4.15)

Taking the maximum over $ \left[ a, T\right], $ we get

$\begin{equation*} \sup\limits_{t\in \left[ a,T\right] }\left \vert \left( \Psi \tilde{u}\right) \left( t\right) -\left( \Psi u^{\ast }\right) \left( t\right) \right \vert \leq \varsigma _{11}\sup\limits_{t\in \left[ a,T\right] }\left \vert \tilde{u} \left( t\right) -u^{\ast }\left( t\right) \right \vert +\varsigma _{12}\sup\limits_{t\in \left[ a,T\right] }\left \vert \left( ^{c}\mathfrak{D}_{a+,t}^{\delta ,\psi }\right) \left[ \tilde{u}-u^{\ast }\right] \right \vert +\epsilon \varsigma _{13}. \label{EQ170} \end{equation*}$ $

Using Lemma 2.3 and (4.15), we obtain

$\begin{equation*} \sup\limits_{t\in \left[ a,T\right] }\left \vert \left( \Psi \tilde{u}\right) \left( t\right) -\left( \Psi u^{\ast }\right) \left( t\right) \right \vert \leq \left( \varsigma _{11}+\varsigma _{12}\kappa _{0}\right) \sup\limits_{t\in \left[ a,T\right] }\left \vert \tilde{u}\left( t\right) -u^{\ast }\left( t\right) \right \vert +\epsilon \varsigma _{13}, \label{EQ172} \end{equation*}$ $

where

$\begin{equation*} \kappa _{0} = \frac{1}{\Gamma \left( 2-\delta \right) }\left( \psi \left( T\right) -\psi \left( a\right) \right) ^{1-\delta }. \label{EQ173} \end{equation*}$ $

We conclude that

$\begin{equation*} \left \Vert \tilde{u}-u^{\ast }\right \Vert _{\infty }\leq \frac{\epsilon \varsigma _{13}}{\left( 1-\varsigma _{11}-\varsigma _{12}\kappa _{0}\right) } ,\ 0 < 1-\varsigma _{11}-\varsigma _{12}\kappa _{0} < 1. \label{EQ174} \end{equation*}$ $

Thus problem (1.1) is Ulam–Hyers stable. Further, using Theorem 4.5 implies that solution of (1.1) is generalized Ulam–Hyers stable. This completes the proof.

Corollary 4.10. Let the conditions of Theorem 4.9 hold. Then Problem (1.1) is generalized Ulam–Hyers–Rassias stable.

Proof. Set $ \epsilon = 1 $ in the proof of Theorem 4.9, we get

$\begin{equation} \left \Vert \tilde{u}-u^{\ast }\right \Vert _{\infty }\leq \frac{\varsigma _{13}}{\left( 1-\varsigma _{11}-\varsigma _{12}\kappa _{0}\right) },\ \ 0 < 1-\varsigma _{11}-\varsigma _{12}\kappa _{0} < 1. \end{equation}$ $

(4.16)

Remark 4.11. (ⅰ) Considering (1.1) and inequality (4.2), then under the assumptions of Theorem 4.5, one can follow the same procedure to confirm that (1.1) is Ulam–Hyers stable.

(ⅱ) Other stability results for the Eq (1.1) can be discussed in a similar manner.

5. Applications

In this section, we provide some test problems to illustrate the applicability of the established results.

Example 5.1. Without loss of generality, we only consider the following $ \psi $ -Caputo Langevin equations

$\begin{equation*} \left( ^{c}\mathfrak{D}_{a+,t}^{\alpha ,\psi }\right) \left( ^{c}\mathfrak{D}_{a+,t}^{\beta ,\psi }+\lambda \right) \left[ u\right] = \mathfrak{F}(t,u(t),^{c}\mathfrak{D}_{a+,t}^{\gamma ,\psi } \left[ u\right] ). \end{equation*}$ $

By taking

$\begin{equation} \mathfrak{F}\left( t,u,v\right) = \frac{\kappa }{20}E_{1/2}\left( t^{1/2}\right) u+\frac{ \kappa }{10}v,\ \kappa \in \left[ 0,+\infty \right) \end{equation}$ $

(5.1)

we have

$\begin{equation*} L_{1} = \frac{\kappa }{20}\sup \left \{ E_{1/2}\left( t^{1/2}\right) :t\in \left[ a,T\right] \right \} {\rm{ \ and \ }}\ L_{2} = \frac{\kappa }{10}. \end{equation*}$ $

For $ \psi \left(t\right) = t, \ $we shall show that condition (3.3) holds with

$\begin{equation} \alpha = 3/2,\ \beta = 6/7,\ \gamma = 5/7,\ \lambda = 1/6,\ \mu = 2,\ a = 0,\eta = 4/7,\ \xi = 4/5,\ T = 1. \end{equation}$ $

(5.2)

A simple computation shows that

$\begin{equation*} \sigma _{11} = 0.653,\ \sigma _{12} = 0.201,\ \sigma _{21} = 0.0483\;{\rm{ and}}\ \sigma _{22} = 0.255. \end{equation*}$ $

$\begin{equation*} \Delta \equiv \left \vert \sigma _{11}\sigma _{22}-\sigma _{21}\sigma _{12}\right \vert = 0.157. \end{equation*}$ $

(ⅰ) Thus, the hypotheses ($\mathrm{A}_{1}$) and (3.20) are satisfied with

$\begin{equation*} d_{11}\left( T\right) = 1.54,{\rm{ }}d_{12}\left( T\right) = 1.02,\ d_{21}\left( T\right) = 1.54\;{\rm{ and }}\;d_{22}\left( T\right) = 1.02, \end{equation*}$ $

$\begin{equation*} \rho _{11} = 0.217,\ \rho _{12} = 0.345,\ \rho _{21} = 0.280\;{\rm{ and }}\;\rho _{22} = 0.635 \end{equation*}$ $

and

$\begin{equation*} \varsigma _{11} = 0.393+0.174\kappa ,\ \varsigma _{12} = 0.0696\kappa ,\ \varsigma _{21} = 0.458+0.328\kappa \;{\rm{ and }}\;\varsigma _{22} = 0.131\kappa, \end{equation*}$ $

where $ \rho _{11}, \ \rho _{12}, \ \rho _{21} $, $ \rho _{22}, \ \varsigma _{11}, \ \varsigma _{12} $ and $ \varsigma _{21} $ are given by (3.14)–(3.17) and (3.11)–(3.13) respectively.

Thus condition (3.20), with

$\begin{equation*} 0 < \kappa \leq 1.648 \end{equation*}$ $

$\begin{equation*} \varsigma \equiv \max \left \{ \varsigma _{11},\varsigma _{12},\varsigma _{21},\varsigma _{22}\right \} = 0.998 < 1,\ {\rm{with}}\ \kappa = 1.648 \end{equation*}$ $

and

$\begin{equation*} L_{0} = 0,\ \varsigma _{13} = 0.696,\ \varsigma _{23} = 1.31\;{\rm{ and }}\;\max \left \{ \varsigma _{13},\varsigma _{23}\right \} = 1.31. \end{equation*}$ $

Hence, by Theorem 3.3, the Problem (1.1) with (5.1) and (5.2) has a unique solution.

(ⅱ) On the other hand, using (3.35), the condition

$\begin{equation*} \Lambda _{21} = 2.11,\ \Lambda _{22} = 2.14,\ 0 < \left \vert \lambda \right \vert \left( \Lambda _{21}+\Lambda _{22}\right) = 0.708 < 1, \end{equation*}$ $

is satisfied and

$\begin{equation*} r > \frac{\left( \Lambda _{11}+\Lambda _{12}\right) \times L}{1-\left \vert \lambda \right \vert \left( \Lambda _{21}+\Lambda _{22}\right) } = 6.88L\;{\rm{ with }}\;\Lambda _{11} = 1.31,\Lambda _{12} = 0.700. \end{equation*}$ $

So, by Theorem 3.4, the Problem (1.1) with (5.1) and (5.2) has at least one fixed point on $ [0, 1] $.

(ⅲ) It is easy to check that the condition (4.8) is satisfied. Indeed,

$\begin{equation*} C_{\epsilon } = \epsilon \varsigma _{13} = .696\epsilon . \end{equation*}$ $

Then by Corollary 4.3, we have, if $ \tilde{u}\in \mathcal{C} $ is a solution of the inequality (4.5), then $ \tilde{u} $ is a solution of the integral inequality (4.6).

(ⅳ) Let $ \Phi \left(t\right) = \psi \left(t\right) -\psi \left(a\right) $ in Remark 2.14 satisfy $ (A_{4}). $

From (4.2) and the condition ($\mathrm{A}_{4}$), we get

$\begin{equation} \left( J_{a+,t}^{\alpha ,\psi }\right) \left[ \tau \right] = \frac{\Gamma \left( 2\right) }{\Gamma \left( \alpha +2\right) }t^{\alpha +1}\leq \frac{ \Gamma \left( 2\right) T^{\alpha }}{\Gamma \left( \alpha +2\right) } t,l_{\alpha ,t} = \frac{\Gamma \left( 2\right) T^{\alpha }}{\Gamma \left( \alpha +2\right) } \end{equation}$ $

(5.3)

and

$\begin{equation*} \sup\limits_{t\in \left[ a,T\right] }\left \vert C_{\epsilon }\left( t\right) \right \vert \leq \epsilon \left \vert \tfrac{\Gamma \left( 2\right) T^{\alpha +\beta }}{\Gamma \left( \alpha +\beta +2\right) }+\tfrac{\Gamma \left( 2\right) \eta ^{\alpha +\beta }}{\Gamma \left( \alpha +\beta +2\right) } d_{11}\left( T\right) +\left( \tfrac{\Gamma \left( 2\right) T^{\alpha +\beta }}{\Gamma \left( \alpha +\beta +2\right) }-\mu \tfrac{\Gamma \left( 2\right) \xi ^{\alpha +\beta +\gamma }}{\Gamma \left( \alpha +\beta +\gamma +2\right) }\right) d_{12}\left( T\right) \right \vert . \end{equation*}$ $

With (5.2), we obtain

$\begin{equation*} \epsilon l_{\alpha ,\psi }\equiv \sup\limits_{t\in \left[ a,T\right] }C_{\epsilon }\left( t\right) = 0.216\epsilon . \end{equation*}$ $

By Lemma 2.13 and Remark 2.14, there exists $ l_{\alpha, \psi }\equiv 0.216 > 0 $ such that for each $ \varepsilon > 0 $, we have

$\begin{equation*} p\left( t\right) \leq .216\epsilon t\left[ E_{6/7}\left( 1/6\Gamma \left( 6/7\right) t^{6/7}\right) +E_{33/14}\left( 1/6\Gamma \left( 33/14\right) t^{33/14}\right) +E_{43/14}\left( 1/6\Gamma \left( 43/14\right) t^{43/14}\right) \right] . \end{equation*}$ $

Therefore, by Theorem 4.5, the Problem (1.1) with (5.1) and (5.2) is generalized Ulam–Hyers–Rassias-Mittag-Leffler stable.

(ⅴ) The condition (4.16) is satisfied with

$\begin{equation*} \kappa _{0} = 1.11,\ \varsigma _{11}+\varsigma _{12}\kappa _{0} = 0.807 < 1. \end{equation*}$ $

By Theorem 4.9, this implies that Problem (1.1) with (5.1) and (5.2) has Ulam–Hyers–Rassias stability.

Further in the below tables (as Tables 1 and 2), we list the consequences of proposed theorems for different values of functions $ \psi $.

Table 1. Numerical values for different parameters.

		Theorem 3.3		Theorem 3.4		Corollary 4.3	Theorem 4.9
$ \psi(t) $	$ \kappa $	$ \varsigma $	$ r > $	$ \vert\lambda\vert(\Lambda _{21}+\Lambda _{22}) $	$ r > $	$ C_{\epsilon } $	$ \varsigma _{11}+\varsigma _{12}\kappa _{0} $
$ t $	$ 1.647 $	$ 0.998 $	$ 0 $	$ 0.708 $	$ 6.88L $	$ 0.696\varepsilon $	$ 0.807 $
$ t^{1/3} $	$ 1.825 $	$ 0.997 $	$ 0 $	$ 0.708 $	$ 6.86L $	$ 0.693\varepsilon $	$ 0.723 $
$ \ln(t+1) $	$ 1.262 $	$ 0.999 $	$ 0 $	$ 0.594 $	$ 2.49L $	$ 0.292\varepsilon $	$ 0.511 $
$ \exp(t) $	$ 0.749 $	$ 0.998 $	$ 0 $	$ 0.950 $	$ 111L $	$ 2.52\varepsilon $	$ 0.919 $
$ \sin t, \; [0, \frac{\pi}{2}] $	$ 0.406 $	$ 0.999 $	$ 0 $	$ 0.708 $	$ 6.87L $	$ 0.696\varepsilon $	$ 0.679 $

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Table 2. Numerical values for different parameters.

				Theorem 4.5	Corollary 4.8
$ \psi(t) $	$ [a, T] $	$ \kappa $	$ (d_{11}, d_{12}) $	$ l_{\alpha, \psi } $	$ l_{\alpha, \psi } $
$ t $	$ [0, 1] $	$ 1.647 $	$ (1.54, 1.020) $	$ 0.216 $	$ 0.105 $
$ t^{1/3} $	$ [0, 1] $	$ 1.825 $	$ (1.29, 0.470) $	$ 0.212 $	$ 0.105 $
$ \ln(t+1) $	$ [0, 1] $	$ 1.262 $	$ (1.17, 0.914) $	$ 0.0901 $	$ 0.00442 $
$ \exp(t) $	$ [0, 1] $	$ 0.749 $	$ (2.39, 1.16) $	$ 0.789 $	$ 0.378 $
$ \sin t, $	$ [0, \frac{\pi}{2}] $	$ 0.406 $	$ (1.37, 0.962) $	$ 0.213 $	$ 0.105 $

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and

Example 5.2. Let

$\begin{equation} \mathfrak{F}\left( t,u,v\right) = E_{1/2}\left( t^{1/2}\right) +\frac{\kappa }{20} E_{1/2}\left( t^{1/2}\right) u+\frac{\kappa }{10}v,\ \kappa \in \left[ 0,+\infty \right) \end{equation}$ $

(5.4)

and

$\begin{equation} \alpha = 5/3,\ \beta = 3/4,\ \gamma = 1/2,\ \lambda = 1/8,\ \mu = 2,\ \eta = 5/4,\ \xi = 5/3. \end{equation}$ $

(5.5)

Then, we have

$\begin{equation*} L_{0} = \sup E\left \{ _{1/2}\left( t^{1/2}\right) :t\in \left[ a,T\right] \right \} ,\ L_{1} = \frac{\kappa }{20}\sup E\left \{ _{1/2}\left( t^{1/2}\right) :t\in \left[ a,T\right] \right \} ,\ L_{2} = \frac{\kappa }{10}. \end{equation*}$ $

In the below tables (as Tables 3 and 4), we list the consequences of proposed theorems for different values of functions $ \psi $.

Table 3. Numerical values for different parameters.

			Theorem 3.3		Theorem 3.4		Corollary 4.3	Theorem 4.9
$ \psi(t) $	$ [a, T] $	$ \kappa $	$ \varsigma $	$ r > $	$ \vert\lambda\vert(\Lambda _{21}+\Lambda _{22}) $	$ r > $	$ C_{\epsilon } $	$ \varsigma _{11}+\varsigma _{12}\kappa _{0} $
$ \ln t $	$ [1, e] $	$ 0.131 $	$ 0.999 $	$ 31600 $	$ 0.548 $	$ 3.82L $	$ 0.674\varepsilon $	$ 0.750 $
$ \sqrt{t} $	$ [1, 2] $	$ 2.439 $	$ 0.988 $	$ 213 $	$ 0.144 $	$ 0.315L $	$ 0.0781\varepsilon $	$ 0.455 $
$ t^{2} $	$ [1/2, 1] $	$ 4.911 $	$ 0.996 $	$ 746 $	$ 0.476 $	$ 5.27L $	$ 0.213\varepsilon $	$ 0.885 $
$ \sin t $	$ [0, \pi/2] $	$ 0.406 $	$ 0.999 $	$ 12100 $	$ 0.708 $	$ 6.87L $	$ 0.696\varepsilon $	$ 0.679 $
$ 2^{t} $	$ [1, 2] $	$ 0.1512 $	$ 0.998 $	$ 30200 $	$ 0.785 $	$ 37.4L $	$ 3.82\varepsilon $	$ 0.967 $
$ \exp (t^{2}) $	$ [0, 1] $	$ 0.686 $	$ 0.998 $	$ 6330 $	$ 0.723 $	$ 89.5L $	$ 3.77\varepsilon $	$ 1.22 $

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Table 4. Numerical values for different parameters.

				Theorem 4.5	Corollary 4.8
$ \psi(t) $	$ [a, T] $	$ \kappa $	$ (d_{11}, d_{12}) $	$ l_{\alpha, \psi } $	$ l_{\alpha, \psi } $
$ \ln t $	$ [1, e] $	$ 0.131 $	$ (2.00, 1.19) $	$ 0.199 $	$ 0.096 $
$ \sqrt{t} $	$ [1, 2] $	$ 2.439 $	$ (1.46, 1.34) $	$ 0.0233 $	$ 0.0111 $
$ t^{2} $	$ [1/2, 1] $	$ 4.911 $	$ (1.07, 0.116) $	$ 0.0865 $	$ 0.0479 $
$ \sin t $	$ [0, \pi/2] $	$ 0.406 $	$ (1.37, 0.962) $	$ 0.213 $	$ 0.105 $
$ 2^{t} $	$ [1, 2] $	$ 0.1525 $	$ (4.58, 1.66) $	$ 1.15 $	$ 0.514 $
$ \exp (t^{2}) $	$ [0, 1] $	$ 0.686 $	$ (0.688, 0.00777) $	$ 0.009 $	$ 0.357 $

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If $ \Phi \left(t\right) = \psi \left(t\right) -\psi \left(a\right) $ then

Example 5.3. (ⅰ) If we set $ \Phi \left(t\right) = \exp \left(\theta \left(\psi \left(t\right) -\psi \left(a\right) \right) \right) $ for every $ \theta \neq 0 $, then by the changing of variables $ \theta \left(\psi \left(t\right) -\psi \left(a\right) \right) = u $, we obtain

$\begin{equation} \left( J_{a+,t}^{\alpha ,\psi }\right) \left[ \Phi \left( t\right) \right] = \frac{\gamma \left( \alpha ,\theta \left( \psi \left( t\right) -\psi \left( a\right) \right) \right) }{\theta ^{\alpha }\Gamma \left( \alpha \right) } \exp \left( \theta \left( \psi \left( t\right) -\psi \left( a\right) \right) \right), \end{equation}$ $

(5.6)

where $ \gamma \left(\alpha, t\right) $ is the incomplete Gamma function defined by

$\begin{equation} \gamma \left( \alpha ,t\right) = \int_{0}^{t}\tau ^{\alpha -1}e^{-\tau } \mathrm{d}\tau ,\ \Re(t) > 0,\ \left \vert \arg \left( t\right) \right \vert < \pi. \end{equation}$ $

(5.7)

Thus

$\begin{equation} \frac{t^{\alpha }}{\alpha }e^{-t}\leq \gamma \left( \alpha ,t\right) \leq \frac{t^{\alpha }}{\alpha }. \end{equation}$ $

(5.8)

Then by the above inequality, we obtain

$\begin{equation*} \left( J_{a+,t}^{\alpha ,\psi }\right) \left[ \Phi \left( t\right) \right] \leq \frac{\left( \theta \left( \psi \left( t\right) -\psi \left( a\right) \right) \right) ^{\alpha }}{\theta ^{\alpha }\Gamma \left( \alpha +1\right) } \exp \left( \theta \left( \psi \left( t\right) -\psi \left( a\right) \right) \right) ,\ {\rm{for \;all }}\;t\in \left[ a,T\right] . \end{equation*}$ $

Hence function $ \Phi \left(t\right) $ satisfies the condition ($\mathrm{A}_{4}$) with

$\begin{equation*} l_{\alpha ,\psi } = \frac{\left( \theta \left( \psi \left( t\right) -\psi \left( a\right) \right) \right) ^{\alpha }}{\theta ^{\alpha }\Gamma \left( \alpha +1\right) }. \end{equation*}$ $

(ⅱ) If we set $ \Phi \left(t\right) = E_{\alpha }\left(\theta \left(\psi \left(t\right) -\psi \left(a\right) \right) ^{\alpha }\right) $ for every $ \theta \neq 0 $, then

$\begin{equation*} \left( J_{a+,t}^{\alpha ,\psi }\right) \left[ \Phi \left( t\right) \right] = \frac{1}{\theta }\left( E_{\alpha }\left( \theta \left( \psi \left( t\right) -\psi \left( a\right) \right) ^{\alpha }\right) -1\right) \leq \frac{1}{ \theta }E_{\alpha }\left( \theta \left( \psi \left( t\right) -\psi \left( a\right) \right) ^{\alpha }\right) . \end{equation*}$ $

Thus function $ \Phi \left(t\right) $ satisfies condition ($\mathrm{A}_{4}$) with

$\begin{equation*} l_{\alpha ,\psi } = \frac{1}{\theta }. \end{equation*}$ $

(ⅲ) The function $ \Phi \left(t\right) $ is positive and there exists a constant $ l_{\alpha, \psi } $ such that the condition ($\mathrm{A}_{4}$) is satisfied.

Indeed, for each $ t\leq \left[ a, T\right] $, we get

$\begin{equation*} \left( \psi \left( t\right) -\psi \left( \tau \right) \right) ^{\alpha -1}\leq \left( \psi \left( t\right) -\psi \left( a\right) \right) ^{\alpha -1}. \end{equation*}$ $

Where as $ \tau \in \left[ a, t\right], \ \alpha \geq 1 $ and $ \psi ^{\prime }\left(\tau \right) \geq 0 $

$\begin{eqnarray} \left( J_{a+,t}^{\alpha ,\psi }\right) \left[ \Phi \right] & = &\frac{1}{ \Gamma \left( \alpha \right) }\int_{a}^{t}\psi ^{\prime }\left( \tau \right) \left( \psi \left( t\right) -\psi \left( \tau \right) \right) ^{\alpha -1}\Phi \left( \tau \right) \mathrm{d}\tau \\ &\leq &\frac{\left( \psi \left( t\right) -\psi \left( a\right) \right) ^{\alpha -1}}{\Gamma \left( \alpha \right) }\int_{a}^{t}\psi ^{\prime }\left( \tau \right) \Phi \left( \tau \right) \mathrm{d}\tau, \end{eqnarray}$ $

(5.9)

then

$\begin{eqnarray*} \int_{a}^{t}\psi ^{\prime }\left( \tau \right) \Phi \left( \tau \right) \mathrm{d}\tau & = &\left( \psi \left( t\right) \Phi \left( t\right) -\psi \left( a\right) \Phi \left( a\right) \right) -\int_{a}^{t}\psi \left( \tau \right) \Phi ^{\prime }\left( \tau \right) \mathrm{d}\tau \\ &\leq &\left( \psi \left( t\right) \Phi \left( t\right) -\psi \left( a\right) \Phi \left( a\right) \right) -\psi \left( a\right) \int_{a}^{t}\Phi ^{\prime }\left( \tau \right) \mathrm{d}\tau \\ & = &\left( \psi \left( t\right) -\psi \left( a\right) \right) \Phi \left( t\right) . \end{eqnarray*}$ $

$\begin{equation*} \left( J_{a+,t}^{\alpha ,\psi }\right) \left[ \Phi \right] \leq \frac{\left( \psi \left( t\right) -\psi \left( a\right) \right) ^{\alpha }}{\Gamma \left( \alpha \right) }\Phi \left( t\right) . \end{equation*}$ $

Thus function $ \Phi \left(t\right) $ satisfies the condition ($\mathrm{A}_{4}$) with

$\begin{equation*} l_{\alpha ,\psi } = \frac{\left( \psi \left( t\right) -\psi \left( a\right) \right) ^{\alpha }}{\Gamma \left( \alpha \right) }. \end{equation*}$ $

Example 5.4. Let $ h\in \mathcal{C}^{2}\left(\mathbb{R}^{2}\right) $ be bounded and let $ g\in L_{1}\left[ 0, T\right] $. Then the functions

$\begin{equation*} \mathfrak{F}\left( t,u,v\right) = g\left( t\right) h\left( u,v\right)\;\; {\rm{ or }}\;\;\mathfrak{F}\left( t,u,v\right) = g\left( t\right) +h\left( u,v\right) , \end{equation*}$ $

satisfies (5.4). In view of condition ($\mathrm{A}_{2}$), we consider the different values of function $ \mathfrak{F} $ (as Table 5),

Table 5. Upper bounds.

$ \mathfrak{F}\left(t, u, v\right) $	$ \left \vert \mathfrak{F}\left(t, u, v\right) \right \vert \leq $	$ \mathfrak{F}\left(t, u, v\right) $	$ \left \vert \mathfrak{F}\left(t, u, v\right) \right \vert \leq $
$ g\left(t\right) \left(\frac{\left \vert u\right \vert +\left \vert v\right \vert }{\left \vert u\right \vert +\left \vert v\right \vert +1}\right) $	$ \left \vert g\left(t\right) \right \vert $	$ g\left(t\right) + \frac{\left \vert u\right \vert }{\left \vert u\right \vert +1}+\frac{\left \vert v\right \vert }{\left \vert v\right \vert +1} $	$ \left \vert g\left(t\right) \right \vert +2 $
$ g\left(t\right) \sin u+\frac{2}{\pi }\arctan v $	$ 2\left \vert g\left(t\right) \right \vert $	$ g\left(t\right) \tanh u+sgn\left(v\right) $	$ \left \vert g\left(t\right) \right \vert +1 $

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where

$ {sgn}(v) = \left\{

$\begin{array}{ccc} \frac{v}{|v|} & \text { if } & v \neq 0, \\ 0 & \text { if } & v = 0. \end{array}$ \right. $

6. Conclusions

The Langevin equation has been introduced to characterize dynamical processes in a fractal medium in which the fractal and memory features with a dissipative memory kernel are incorporated. Therefore, the consideration of Langevin equation in frame of fractional derivatives settings would be providing better interpretation for real phenomena. Consequently, scholars have considered different versions of Langevin equation and thus many interesting papers have been reported in this regard. However, one can notice that most of existing results have been carried out with respect to the classical fractional derivatives.

In this paper, we have tried to promote the current results and considered the FLE in a general platform. The boundary value problem of nonlinear FLE involving $ \psi $- fractional operators of different orders was investigated. One of the major differences in the problem considered in this work and relevant work already published in literature, is that, we are dealing with general fractional operator. Secondly, the forcing function depends on fractional derivative of unknown function. We employ the newly accommodated $ \psi $- fractional calculus to prove the following for the considered problem:

(ⅰ) The existence and uniqueness of solutions: Techniques of fixed point theorems are used to prove the results. Prior to the main theorems, the forms of solutions are derived for both linear and nonlinear problems.

(ⅱ) Stability in sense of Ulam: We adopt the required definitions of Ulam–Hyers stability with respect to $ \psi $- fractional derivative. The Ulam–Hyers–Rassias and generalized U-H-R stability of the solution are discussed. Gronwall inequality and integration by parts in frame of $ \psi $- fractional derivative are also employed to complete the proofs.

(ⅲ) Applications: Particular examples are addressed at the end of the paper to show the consistency of the theoretical results.

We claim that the results of this paper are new and generalize some earlier results. Moreover, by fixing the parameters involved in the given problem, we can obtain some new results as special cases of the ones presented in this paper. For example, letting $ \psi = t, \mu = 0, a = 0 $ and $ T = 1 $ in the results of Section $ 3 $, we get the ones derived in ^[40]. Besides, the existence results for the initial value problem of nonlinear classical Langevin equation of the form:

$\begin{equation*} \dddot{u}+\lambda \ddot{u} = \mathfrak{F}\left( t,u,\dot{u}\right) ,\ u\left( 0\right) = 0,\ u\left( \eta \right) = 0,\ u\left( 1\right) = 0,\ 0 < \eta < 1, \end{equation*}$ $

can be addressed by fixing a $ \alpha = 2 $ and $ \beta = 1 $ in the results of this paper.

For further investigation, one can propose to study the properties of the solution of the considered problem via some numerical computations and simulations. We leave this as promising future work. Results obtained in the present paper can be considered as a contribution to the developing field of fractional calculus via generalized fractional derivative operators.

Conflict of interest

The authors declare that they have no conflict of interest.

Acknowledgement

J. Alzabut would like to thank Prince Sultan University for funding and supporting this work.

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Networks and Heterogeneous Media

On Lennard-Jones systems with finite range interactions and their asymptotic analysis

Related Papers:

Abstract

1. Introduction

2. Preliminaries and essential lemmas

3. Existence and uniqueness of solution

3.1. Linear boundary problem

3.2. Nonlinear problem

4. Stability of solutions

5. Applications

6. Conclusions

Conflict of interest

Acknowledgement

References

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Networks and Heterogeneous Media

On Lennard-Jones systems with finite range interactions and their asymptotic analysis

Related Papers:

Abstract

1. Introduction

2. Preliminaries and essential lemmas

3. Existence and uniqueness of solution

3.1. Linear boundary problem

3.2. Nonlinear problem

4. Stability of solutions

5. Applications

6. Conclusions

Conflict of interest

Acknowledgement

References

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