Langevin equation with nonlocal boundary conditions involving a $\psi$-Caputo fractional operators of different orders

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

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

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]

and

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

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

and

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

and

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

and

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

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 $\alpha > 0$

where

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

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

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

and

Note that

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

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

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

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

Proof. Using Lemmas 2.3 and 2.4, we have

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

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

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

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

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

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

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

We adopt the following conventions:

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

then

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

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

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

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

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

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

(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 $t\in \left[ a, T\right],$

Denoting

and

Further, we assume

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

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

where

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

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

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

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

where

and

We note

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

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

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

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

and

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

where

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.

with

and

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

equipped with the norm

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

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

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

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

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

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

Then

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

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

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

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

Using (3.21) and (3.24), we obtain

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

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

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

where

and

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

Hence, we get

Consequently,

A similar argument shows that

where

Combining (3.27) and (3.28), we obtain

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

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

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

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

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

and

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

and

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

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

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

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

we obtain

Then, we have

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

which implies

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

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

Then

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

where

From (3.39), we can write

On the other hand

which yields

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

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 $t\in \left[ a, T\right]$

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

It follows that

By (3.42), we have

Consequently, by (3.43), we have

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

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

On the other hand

Using (3.44) and (3.45), we get

where

and

Consequently, we have

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

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

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

where

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

where

with

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

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

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

with

where

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

The solution of the problem (4.4) is given by

which implies that

where

with

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

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

and

where

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

By setting

and

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

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

where

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

and

where

Set

where

with

This means that

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

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

By Remark 2.14, one can obtain

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

with

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

with the conditions (4.14), we have

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

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

and

where

Using Lemma 4.2, we have

where

with

From the above, it follows

By Remark 2.14, one can obtain

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

where

We conclude that

By Remark 2.14, one can obtain

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

where

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

Using Lemma 2.3 and (4.15), we obtain

where

We conclude that

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

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

By taking

we have

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

A simple computation shows that

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

and

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

is

and

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

is satisfied and

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,

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

and

With (5.2), we obtain

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

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

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 $\psi$.

and

Example 5.2. Let

and

Then, we have

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

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

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

Thus

Then by the above inequality, we obtain

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

(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

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

(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

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

then

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

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

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

where

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:

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.