Some fixed point and stability results in $ b $-metric-like spaces with an application to integral equations on time scales

Zeynep Kalkan; Aynur Şahin; Ahmad Aloqaily; Nabil Mlaiki; Zeynep Kalkan; Aynur Şahin; Ahmad Aloqaily; Nabil Mlaiki

doi:10.3934/math.2024556

AIMS Mathematics

2024, Volume 9, Issue 5: 11335-11351. doi: 10.3934/math.2024556

Previous Article Next Article

Research article Special Issues

Some fixed point and stability results in $b$ -metric-like spaces with an application to integral equations on time scales

1.
Department of Mathematics, Faculty of Sciences, Sakarya University, Sakarya, 54050, Turkey
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia
3.
School of Computer, Data and Mathematical Sciences, Western Sydney University, Sydney, 2150, Australia

Received: 26 January 2024 Revised: 06 March 2024 Accepted: 18 March 2024 Published: 22 March 2024
MSC : 47H10, 45B05

This paper presents the stability theorem for the $T$ -Picard iteration scheme and establishes the existence and uniqueness theorem for fixed points concerning $T$ -mean nonexpansive mappings within $b$ -metric-like spaces. The outcome of our fixed point theorem substantiated the existence and uniqueness of solutions to the Fredholm-Hammerstein integral equations defined on time scales. Additionally, we provided two numerical examples from distinct time scales to support our findings empirically.

Keywords:

Citation: Zeynep Kalkan, Aynur Şahin, Ahmad Aloqaily, Nabil Mlaiki. Some fixed point and stability results in $b$ -metric-like spaces with an application to integral equations on time scales[J]. AIMS Mathematics, 2024, 9(5): 11335-11351. doi: 10.3934/math.2024556

Related Papers:

[1]	Xiaogang Liu, Muhammad Ahsan, Zohaib Zahid, Shuili Ren . Fault-tolerant edge metric dimension of certain families of graphs. AIMS Mathematics, 2021, 6(2): 1140-1152. doi: 10.3934/math.2021069
[2]	Maryam Salem Alatawi, Ali Ahmad, Ali N. A. Koam, Sadia Husain, Muhammad Azeem . Computing vertex resolvability of benzenoid tripod structure. AIMS Mathematics, 2022, 7(4): 6971-6983. doi: 10.3934/math.2022387
[3]	Pradeep Singh, Sahil Sharma, Sunny Kumar Sharma, Vijay Kumar Bhat . Metric dimension and edge metric dimension of windmill graphs. AIMS Mathematics, 2021, 6(9): 9138-9153. doi: 10.3934/math.2021531
[4]	Chenggang Huo, Humera Bashir, Zohaib Zahid, Yu Ming Chu . On the 2-metric resolvability of graphs. AIMS Mathematics, 2020, 5(6): 6609-6619. doi: 10.3934/math.2020425
[5]	Meiqin Wei, Jun Yue, Xiaoyu zhu . On the edge metric dimension of graphs. AIMS Mathematics, 2020, 5(5): 4459-4465. doi: 10.3934/math.2020286
[6]	Lyimo Sygbert Mhagama, Muhammad Faisal Nadeem, Mohamad Nazri Husin . On the edge metric dimension of some classes of cacti. AIMS Mathematics, 2024, 9(6): 16422-16435. doi: 10.3934/math.2024795
[7]	Mohra Zayed, Ali Ahmad, Muhammad Faisal Nadeem, Muhammad Azeem . The comparative study of resolving parameters for a family of ladder networks. AIMS Mathematics, 2022, 7(9): 16569-16589. doi: 10.3934/math.2022908
[8]	Dalal Awadh Alrowaili, Uzma Ahmad, Saira Hameeed, Muhammad Javaid . Graphs with mixed metric dimension three and related algorithms. AIMS Mathematics, 2023, 8(7): 16708-16723. doi: 10.3934/math.2023854
[9]	Syed Ahtsham Ul Haq Bokhary, Zill-e-Shams, Abdul Ghaffar, Kottakkaran Sooppy Nisar . On the metric basis in wheels with consecutive missing spokes. AIMS Mathematics, 2020, 5(6): 6221-6232. doi: 10.3934/math.2020400
[10]	Ahmed Alamer, Hassan Zafar, Muhammad Javaid . Study of modified prism networks via fractional metric dimension. AIMS Mathematics, 2023, 8(5): 10864-10886. doi: 10.3934/math.2023551

Abstract

1. Introduction

The fractional derivatives with constant or variable order ^[3,9] are excellent mathematical tools for the description of memory and the hereditary properties of various processes and materials^[12,19]. In fractional calculus, these derivatives are defined through fractional integrals. There are several approaches to fractional derivatives including Riemann-Liouville ^[10,14,15], Caputo, Hadamard derivatives, ^[4,6,13,17].

Efforts have been dedicated to generalizations concerning mappings of bounded variation, absolute continuity, various classes of convex functions, and their extension to fractional calculus, involving Riemann-Liouville integrals and their generalizattions as referenced in ^[1,2,12,15].

In ^[8], the author proved some integral inequalities for functions whose $k$ th $(k\in\mathbb{N})$ derivatives are convex involving Caputo derivatives and obtain the following results for $a, \Delta\in I, a < \Delta, \, \alpha, \beta\in\mathbb{R},$ $\alpha, \beta\geq 1,$ and $\psi:I\to \mathbb{R}$ :

● If $\psi^{(k)}\, (k\in\mathbb{N})$ exists and is positive and convex, then

$\begin{align} &\Gamma(k-\alpha+1)^CD^{\alpha-1}_{a_{+}}\psi(\xi)+(-1)^k\Gamma(k-\beta+1)^CD^{\beta-1}_{\Delta-}\psi(\xi) \\ &\leq (\xi-a)^{k-\alpha+1}\dfrac{\psi^{(k)}(a)+\psi^{(k)}(\xi)}{2} +(\Delta-\xi)^{k-\beta+1}\dfrac{\psi^{(k)}(\Delta)+\psi^{(k)}(\xi)}{2}. \end{align}$

(1.1)

● If $\psi^{(k)}$ exists and is positive, convex and symmetric about $\frac{a +\Delta}{2},$ then

$\begin{align} & \frac{1}{2}\left(\dfrac{1}{k-\alpha+1}+\dfrac{1}{k-\beta+1}\right)\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right) \\ &\le \dfrac{\Gamma(k-\beta+1)^CD_{\Delta_-}^{\beta-1}\psi(\alpha)}{2(\Delta-a)^{k-\beta+1}}+(-1)^k\dfrac{\Gamma(k-\alpha+1) ^CD_{a_{+}}^{\alpha-1}\psi(\Delta)}{2(\Delta-a)^{k-\alpha+1}} \\ & \leq\dfrac{\psi^{(k)}(\Delta)+\psi^{(k)}(a)}{2}. \end{align}$

(1.2)

In ^[11], the authors gave a version of Hadamard's inequality using the Caputo derivative. In ^[7], the authors proved Hadamard inequalities for strongly $\alpha, m$ -convex functions via Caputo fractional derivatives. In this paper, we consider the Caputo derivatives of a real valued function $\psi$ whose derivatives $\psi^{(k)}\, (k \in \mathbb{N})$ are genaralized modified $h$ -convex. Some Caputo fractional versions of Hermite-Hadamard inequalities are obtained. From which particular cases are revealed, we have also established a new integral inequality between Caputo derivatives $^{C}D^{\alpha}_{.}\psi$ and the Riemann-Liouville integrals $R^{k-\alpha}_{.}(\psi^{(k)})^{2}.$ By deriving new differential inequalities in this context, we aim to extend the applicability of fractional calculus to problems involving generalized convex functions. These results have significance in various fields, including mathematics, physics, and engineering, where fractional calculus plays a crucial role in modeling complex phenomena with memory and long-range dependence.sts Our results generalize those cited in ^[8] and unify several classes of functions, like convex and $s$ -convex functions.

2. Preliminaries

This section deals with some definitions of convexity ^[2,5,8], generalized $h$ -convexity ^[20], fractional integrals and derivatives ^[6,18].

Let $I\subset\mathbb{R}$ be an interval and $h:[0, 1]\to (0, \infty), \, \psi:I \to (0, \infty)$ be two real valued functions, then

● $\psi$ is said to be $h$ -convex, if

$\begin{equation} \psi (\rho c + (1-\rho)d)\leq h(\rho) \psi (c)+h(1-\rho)\psi (d) \end{equation}$

(2.1)

holds for all $c, d \in I$ and $\rho\in (0, 1].$ If (2.1) is reversed, then $\psi$ is said to be $h$ -concave.

● The function $\psi$ is said to be modified $h$ -convex if

$\begin{equation} \psi (\rho c + (1-\rho)d)\leq h(\rho) \psi (c)+ (1- h(\rho))\psi (d). \end{equation}$

(2.2)

● The function $\psi$ is said to be generalized modified $h$ -convex if

$\begin{equation} \psi (\rho c + (1-\rho)d)\leq \psi (d)+ h(\rho) \theta(\psi (c),\psi (d)). \end{equation}$

(2.3)

Definition 2.1 (Additivity). ^[20] A continuous bifunction $\theta$ is said to be additive, if

$\theta(a_{1}, b_{1}) + \theta(a_{2}, b_{2}) = \theta(a_{1} + a_{2}, b_{1} + b_{2}),\quad \forall a_{1}, a_{2}, b_{1},b_{2 }\in\mathbb{R}.$

Definition 2.2 (Nonnegative homogeneity). ^[20] A continuous bifunction $\theta$ is said to be nonnegatively homogeneous if, for all $\lambda > 0,$

$\theta(\lambda a_{1},\lambda a_{2}) = \lambda \theta(a_{1}, a_{2}),\quad \forall a_1, a_{2} \in\mathbb{R}.$

Remark 2.1. For different functions $h, \theta$ one can obtain various classes of generalized modified convex functions:

● By taking in (2.1) $h(z) = z^{s} (0 < s\le 1),$ we have the definition of modified generalized $s$ -convex functions.

● If, we take $\theta(r, z) = r-z,$ then we obtain the definition of a modified $h$ -convex function.

Let $[a, \Delta]\, (-\infty < a < \Delta < +\infty)$ be a finite interval on the real axis $\mathbb{R}.$ For any function $\psi\in L_{1}([a, \Delta]),$ the Riemann-Liouville fractional integrals $R^{\alpha}_{a_{+}}$ and $R^{\alpha}_{\Delta_-}$ of order $\alpha\in\mathbb{R}$ $(\alpha > 0)$ of $\psi$ are defined by

$\begin{equation} R^{\alpha}_{a_{+}}\psi(s) = \frac{1}{\Gamma(\alpha)} \displaystyle {\int}_{a}^{s}(s-t)^{\alpha-1}\psi(t)dt ,\quad s > a \quad (left) \end{equation}$

(2.4)

and

$\begin{equation} R^{\alpha}_{\Delta_-}\psi(s) = \frac{1}{\Gamma(\alpha)}\displaystyle {\int}_{s}^{\Delta}(t-s)^{\alpha-1} \psi(t)dt ,\,\,\, s < \Delta \quad (right), \end{equation}$

(2.5)

respectively. Here $\Gamma(\alpha) = {\int}_{0}^{\infty}\, t^{\alpha-1}\, e^{-t}\, dt, \; \alpha > 0$ is the gamma function. We set $R^{0}_{a+}\psi = R^{0}_{\Delta-}\psi = \psi.$

Let $[a, \Delta]$ be a finite interval of the real line $\mathbb{R}$ . Let $\alpha > 0, k\in\mathbb{N},$ $k = [\alpha]+1$ and $\psi\in AC^{k}([a, \Delta])$ ( $AC^{k}([a, \Delta])$ means the space of complex-valued functions $\psi(x)$ which have continuous derivatives up to order $k-1$ on $[a, b]$ such that $\psi^{(k-1)}(x) \in AC([a, \Delta]):$ i.e., absolutely continuous) see Lemma 2.4 ^[18]. The left and right Caputo fractional derivatives of order $\alpha\, (\alpha\ge 0)$ of $\psi$ are given by the following formulas (see ^[1,4,10,13])

$^CD^{\alpha}_{a_{+}}\psi(\xi) = \dfrac{1}{\Gamma(k-\alpha)}\displaystyle {\int}_{a}^{\xi}\psi^{(k)}(t)(\xi-t)^{k-\alpha-1}dt,\qquad \xi > a$

and

$^CD^{\alpha}_{\Delta_-}\psi(\xi) = \dfrac{(-1)^k}{\Gamma(k-\alpha)}\displaystyle {\int}_{\xi}^{\Delta}\psi^{(k)}(t)(t-\xi)^{k-\alpha-1}dt,\qquad \xi < \Delta,$

respectively.

If $\alpha = k\in\mathbb{N},$ then

$^CD^{\alpha}_{a+}\psi(\xi) = \psi^{(k)}(\xi) \quad \mbox{and}\quad ^CD^{\alpha}_{\Delta-}\psi(\xi) = (-1)^k\, \psi^{(k)}(\xi).$

In particular, if $k = 1,$ $\alpha = 0,$ then

$^CD^{0}_{a_{+}}\psi(\xi) = ^CD^{0}_{\Delta_-}\psi(\xi) = \psi(\xi).$

Lemma 2.1. ^[16] The following formulas for Caputo fractional derivatives of order $\alpha > 0, k-1 < \alpha < k\, (k\in\mathbb{N})$ of a power function at $t = a$ and $t = b$ hold

$\begin{equation} ^CD^{\alpha}_{a_{+}}(t-a)^{p} = \dfrac{\Gamma(p+1)}{\Gamma(p-\alpha+1)}(t-a)^{p-\alpha},\,\qquad t > a \end{equation}$

(2.6)

and

$\begin{equation} ^CD^{\alpha}_{b_-}(b-t)^{p} = \dfrac{\Gamma(p+1)}{\Gamma(p-\alpha+1)}(b-t)^{p-\alpha}, \,\qquad t < b. \end{equation}$

(2.7)

Our objective in this work, is to prove some fractional integral inequalities for functions whose $k$ th $(k \in\mathbb{N})$ derivatives are generalized modified $h$ -convex functions involving the Caputo derivative operator.

3. Results

Theorem 3.1. Let $I$ be an interval of $\mathbb{R},$ $a, \Delta\in I, a < \Delta$ and $\alpha, \beta > 0,$ such that $k-1 < \alpha, \beta < k, k\in\mathbb{N}.$ Let $\psi:I\to\mathbb{R}$ be differentiable function. If, $\psi^{(k)}(k\in\mathbb{N})$ exists and is a positive generalized modified $h$ -convex function and $\theta$ is a continuous bifunction, then the following integral inequality

$\begin{eqnarray} \Gamma(k-\alpha+1)\,\left( ^CD^{\alpha-1}_{a_{+}}\psi\right) (\xi)+ (-1)^k\Gamma(k-\beta+1)\left( ^C D^{\beta-1}_{\Delta_-}\psi\right) (\xi) & &\\ \leq ( \Delta-\xi)^{k-\beta+1}\left[\psi^{(k)}(\xi)+\theta\,( \psi^{(k)}(\Delta),\psi^{(k)}(\xi))\displaystyle {\int}_{0}^{1}h(z)dz\right]& &\\ +\,(\xi-a)^{k-\alpha+1}\left[\psi^{(k)}(\xi)+ \theta\,( \psi^{(k)}(a),\psi^{(k)}(\xi))\displaystyle {\int}_{0}^{1}h(z)dz\right] \end{eqnarray}$

(3.1)

holds.

Proof. For all $\xi\in[a, \Delta]$ and for all $t\in[a, \xi],$ we have

$\begin{equation} (\xi-t)^{k-\alpha}\leq(\xi-a)^{k-\alpha}, \end{equation}$

(3.2)

and

$t = \frac{\xi-t}{\xi-a}a +\frac{t-a}{\xi-a}\xi.$

Since $\psi^{(k)}$ is generalized modified $h$ -convex, (2.3) implies that

$\begin{equation} \psi^{(k)}(t)\leq \psi^{(k)}(\xi)+h\left( \frac{\xi-t}{\xi-a}\right)\theta\,(\psi^{(k)}(a),\psi^{(k)}(\xi)). \end{equation}$

(3.3)

Multiplying inequalities (3.2) and (3.3) on both side and integrating, we obtain

$\begin{align} & \displaystyle {\int}_{a}^{\xi}(\xi-t)^{k-\alpha}\, \psi^{(k)}(t)dt\\ &\leq \displaystyle {\int}_{a}^{\xi}(\xi-a)^{k-\alpha} \times \left[ \psi^{(k)}(\xi)+h\left( \dfrac{\xi-t}{\xi-a}\right)\theta\,(\psi^{(k)}(a),\psi^{(k)}(\xi)) \right] dt. \end{align}$

(3.4)

That is

$\begin{align} \Gamma(k-\alpha+1)\left(^CD^{\alpha-1}_{a_{+}}\psi\right)(\xi)\leq(\xi-a)^{k-\alpha+1} \times \left[ \psi^{(k)}(\xi) +\theta\,( \psi^{(k)}(a),\psi^{(k)}(\xi))\displaystyle {\int}_{0}^{1}h(z)dz \right]. \end{align}$

(3.5)

Let $\xi\in [a, \Delta], t\in[\xi, \Delta],$ thus

$\begin{equation} (t-\xi)^{k-\beta}\leq(\Delta-\xi)^{k-\beta}. \end{equation}$

(3.6)

We have

$t = \frac{t-\xi}{\Delta-\xi}\Delta +\frac{\Delta-t}{\Delta-\xi}\xi.$

Since $\psi^{(k)}$ is generalized modified $h$ -convex on $[\alpha, \Delta],$ then

$\begin{equation} \psi^{(k)}(t)\leq \psi^{(k)}(\xi)+ h\left( \dfrac{t-\xi}{\Delta-\xi}\right) \theta\,(\psi^{(k)}(\Delta),\psi^{(k)}(\xi)). \end{equation}$

(3.7)

Similarly, we obtain

$\begin{align} (-1)^k\Gamma(k-\beta+1)\left( ^C D^{\beta-1}_{\Delta_-}\psi\right) (\xi)\leq(\Delta-\xi)^{k-\beta+1} \times \left[ \psi^{(k)}(\xi) + \theta\,( \psi^{(k)}(\Delta),\psi^{(k)}(\xi))\displaystyle {\int}_{0}^{1}h(z)dz \right]. \end{align}$

(3.8)

Adding (3.5) and (3.8), the claim follows. □

Corollary 3.1. If, we set $\alpha = \beta$ in (3.1), then we obtain

$\begin{align*} &\Gamma(k-\alpha+1)\left[\left(^CD^{\alpha-1}_{a_{+}}\psi\right)(\xi)+(-1)^k\left(^CD^{\alpha-1}_{\Delta_-}\psi\right)(\xi)\right]\\ & \leq (\Delta-\xi)^{k-\alpha+1} \left[\psi^{(k)}(\xi)+\theta\left(\psi^{(k)}(\Delta),\psi^{(k)}(\xi)\right)\displaystyle {\int}_{0}^{1}h(z)dz\right]\\ & + (\xi-a)^{k-\alpha+1} \left[\psi^{(k)}(\xi)+ \theta\left( \psi^{(k)}(a),\psi^{(k)}(\xi)\right)\displaystyle {\int}_{0}^{1}h(z)dz \right]. \end{align*}$

Corollary 3.2. By setting $\theta\, (r, z) = r-z, h(t) = t^{s}, s\in[0, 1]$ in (3.1), we obtain

$\begin{align} &\Gamma(k-\alpha+1)\left(^CD^{\alpha-1}_{a_{+}}\psi\right)(\xi)+(-1)^k\Gamma(k-\beta+1)\left(^CD^{\beta-1}_{\Delta_-}\psi\right)(\xi)\\ &\leq \dfrac{(\Delta-\xi)^{k-\beta+1}\,\psi^{(k)}(\Delta)+(\xi-a)^{k-\alpha+1}\,\psi^{(k)}(a)}{s+1}\\ &+ \dfrac{(\xi-a)^{k-\alpha+1}+(\Delta-\xi)^{k-\beta+1}}{s+1}\, s\,\psi^{(k)}(\xi). \end{align}$

(3.9)

In particular, if $h(z) = z,$ then we have

$\begin{align} &\Gamma(k-\alpha+1)\left(^CD^{\alpha-1}_{\alpha_{+}}\psi\right)(\xi)+(-1)^k\Gamma(k-\beta+1)\left(^CD^{\beta-1}_{\Delta_-}\psi\right)(\xi)\\ &\leq \dfrac{(\Delta-\xi)^{k-\beta+1}\,\psi^{(k)}(\Delta)+(\xi-a)^{k-\alpha+1}\,\psi^{(k)}(a)}{2}\\ &+ \dfrac{(\xi-a)^{k-\alpha+1}+(\Delta-\xi)^{k-\beta+1}}{2}\,\psi^{(k)}(\xi). \end{align}$

(3.10)

Taking $\alpha = \beta$ in (3.10), we obtain

$\begin{align} &\Gamma(k-\alpha+1)\left[\left( ^CD^{\alpha-1}_{a_{+}}\psi\right) (\xi)+(-1)^k\left(^CD^{\alpha-1}_{\Delta_-}\psi\right)(\xi)\right]\\ &\leq \dfrac{(\Delta-\xi)^{k-\alpha+1}\,\psi^{(k)}(\Delta)+(\xi-a)^{k-\alpha+1}\psi^{(k)}(a)}{2}\\ &+ \dfrac{(\xi-a)^{k-\alpha+1}+(\Delta-\xi)^{k-\alpha+1}}{2}\,\psi^{(k)}(\xi). \end{align}$

(3.11)

Example 3.1. Let $\psi:[a, \Delta]\to [0, \infty),$ $\psi(\xi) = \frac{2}{(k+2)!}(\xi-a) ^{k+2},$ $a < \xi\le \Delta.$ Let $h:[0, 1]\to (0, \infty),$ $h(t)\ge t,$ $\theta(x, y) = 2x+y.$ We verify easly that $\psi^{(k)}(\xi) = (\xi-a)^{2}$ is generalized modified $h$ -convex on $[a, \Delta].$ From Corollary 3.1 and Lemma 2.1, we obtain

$\begin{equation} lhs : = \Gamma(k-\alpha+1)\left( ^CD^{\alpha-1}_{a_{+}}\psi\right) (\xi) = \frac{2(\xi-a)^{k-\alpha+3}}{(k-\alpha+1)(k-\alpha+2)(k-\alpha+3)}, \end{equation}$

(3.12)

and

$\begin{align} rhs: & = (\xi-a)^{k-\alpha+1} \left[(\xi-a)^{2} +( 0+(\xi-a)^{2})\displaystyle {\int}_{0}^{1}h(z)dz\right]\\ & = (\xi-a)^{k-\alpha+3}\left( 1+\displaystyle {\int}_{0}^{1}h(z)dz\right). \end{align}$

(3.13)

For the right derivative $\left(^CD^{\alpha-1}_{\Delta_-}\psi\right) (\xi),$ we consider the function $\psi(\xi) = \frac{2(\Delta-\xi) ^{k+2}}{(k+2)!},$ $a\le \xi < \Delta.$

$\begin{equation} (-1)^{k}\Gamma(k-\alpha+1)\left( ^CD^{\alpha-1}_{\Delta_-}\psi\right) (\xi) = \frac{2(\Delta-\xi)^{k-\alpha+3}}{(k-\alpha+1)(k-\alpha+2)(k-\alpha+3)} \end{equation}$

(3.14)

and

$\begin{equation} rhs: = (\Delta-\xi)^{k-\alpha+3}\left( 1+\displaystyle {\int}_{0}^{1}h(z)dz\right). \end{equation}$

(3.15)

Now let $I$ be an interval of $\mathbb{R},$ $a, \Delta\in I, (a < \Delta)$ and $\alpha, \beta > 0,$ such that $k-1 < \alpha, \beta < k, (k\in\mathbb{N}).$ Let $\psi:I\to\mathbb{R}.$ Assume that $|\psi^{(k+1)}|$ is generalized modified $h$ - convex on $[a, \Delta].$

It is clear that for all $\xi\in [a, \Delta], t\in [a, \xi],$ we have

$\begin{equation} (\xi-t)^{k-\alpha}\leq (\xi-a)^{k-\alpha},\qquad t\in[a,\xi]. \end{equation}$

(3.16)

Since $|\psi^{(k+1)}|$ is generalized modified $h$ -convex, we have for $t\in[a, \xi],$

$\begin{eqnarray} \qquad \qquad \textbf{Lhs = }-\left[|\psi^{(k+1)}(\xi)|+ \theta\left(|\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right) h\left( \dfrac{t-a}{\xi-a} \right) \right]\\ \leq |\psi^{(k+1)}(t)|\leq |\psi^{(k+1)}(\xi)|+ \theta\left(|\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right) h\left( \dfrac{t-a}{\xi-a} \right) = \textbf{Rhs}. \end{eqnarray}$

(3.17)

Multiplying (3.16) by the Rhs of inequality (3.17) and integrating the resulting inequality over $[a, \xi],$ we obtain

$\begin{align} \displaystyle {\int}_{a}^{\xi}(\xi-t)^{k-\alpha}\,\psi^{(k+1)}(t)dt \leq (\xi-a)^{k-\alpha} \left( |\psi^{(k+1)}(\xi)|+\theta\left( |\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right)\displaystyle {\int}_{0}^{1}h(z)dz \right), \end{align}$

(3.18)

by integration by parts, we have

$\begin{align*} \displaystyle {\int}_{a}^{\xi}(\xi-t)^{k-\alpha}\,\psi^{(k+1)}(t)dt & = \psi^{(k)}(t)(\xi-t)^{k-\alpha}\;|_a^\xi +(k-\alpha)\displaystyle {\int}_{a}^{\xi}(\xi-t)^{k-\alpha-1}\,\psi^{(k)}(t)dt\\ & = \Gamma(k-\alpha+1)\left(^CD^{\alpha}_{a+}\psi\right)(\xi) -\psi^{(k)}(a)(\xi-a)^{k-\alpha}. \end{align*}$

Hence

$\begin{align} & \Gamma(k-\alpha+1)\left( ^CD^{\alpha}_{a_{+}}\psi\right) (\xi)-\psi^{(k)}(a)(\xi-a)^{k-\alpha} \\ &\leq\left(\psi^{(k+1)}(\xi)+\theta\left( |\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right) \displaystyle {\int}_{0}^{1}h(z)dz \right)(\xi-a)^{k-\alpha}. \end{align}$

(3.19)

In a similar way, if we proceed with the Lhs of (3.17) as we did for the Rhs, it follows that

$\begin{align} \psi^{(k)}(a)(\xi-a)^{k-\alpha}-\Gamma(k-\alpha+1)&\left( ^CD^{\alpha}_{a_{+}}\psi\right) (\xi)\\ &\leq \left(|\psi^{(k+1)}(\xi)|+\theta\left(|\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)| \right)\displaystyle {\int}_{0}^{1}h(z)dz \right)(\xi-a)^{k-\alpha}. \end{align}$

(3.20)

From (3.19) and (3.20), we obtain

$\begin{align} |\Gamma(k-\alpha+1)\left( ^CD^{\alpha}_{a_{+}}\psi\right) (\xi)-\psi^{(k)}&(a)(\xi-a)^{k-\alpha}|\\ &\leq \left(|\psi^{(k+1)}(\xi)|+\theta\left( |\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right) \displaystyle {\int}_{0}^{1}h(z)dz \right)(\xi-a)^{k-\alpha}. \end{align}$

(3.21)

Doing the same for $t\in [\xi, \Delta]$ and $\beta > 0, k-1 < \beta < k,$ and taking into acount that $|\psi^{(k+1)}|$ is generalized modified $h$ -convex, we have

$\begin{align} \textbf{Lhs = } -&\left[ |\psi^{(k+1)}(\xi)|+ \theta\left(|\psi^{(k+1)}(\Delta)|,|\psi^{(k+1)}(\xi)|\right) h\left( \dfrac{t-\xi}{\Delta-\xi}\right) \right] \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \leq \,\psi^{(k+1)}(t)\,\le \psi^{(k+1)}(\xi)+ \theta\left(|\psi^{(k+1)}(\Delta)|,|\psi^{(k+1)}(\xi)|\right) h\left( \dfrac{t-\xi}{\Delta-\xi}\right) = \textbf{Rhs}. \end{align}$

(3.22)

Hence

$\begin{align} &|\Gamma(k-\beta+1)\left( ^CD^{\beta}_{\Delta_-}\psi\right) (\xi)-\psi^{(k)}(\Delta)(\Delta-\xi)^{k-\beta}|\\ &\leq (\Delta-\xi)^{k-\beta} \times\left[|\psi^{(k+1)}(\xi)|+ \theta\left( |\psi^{(k+1)}(\Delta)|,|\psi^{(k+1)}(\xi)|\right) \displaystyle {\int}_{0}^{1}h(z)dz \right]. \end{align}$

(3.23)

Combine (3.21) and (3.23) via triangular inequality, and we obtain the double inequality

$\begin{eqnarray} \left|\Gamma(k-\alpha+1)\left( ^CD^{\alpha}_{a_{+}}\psi\right) (\xi)+\Gamma(k-\beta+1)\left( ^CD^{\beta}_{\Delta_-}\psi\right) (\xi)\right.\\ \left. -\left(\psi^{(k)}(a)(\xi-a)^{k-\alpha}+\psi^{(k)}(\Delta)(\Delta-\xi)^{k-\beta}\right)\right|& &\\ \le (\Delta-\xi)^{k-\beta}\left[|\psi^{(k+1)}(\xi)|+ \theta\left( |\psi^{(k+1)}(\Delta)|,|\psi^{(k+1)}(\xi)|\right)\displaystyle {\int}_{0}^{1}h(z)dz \right]& &\\ +\, (\xi-a)^{k-\alpha} \left[|\psi^{(k+1)}(\xi)|+\theta( |\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|) \displaystyle {\int}_{0}^{1}h(z)dz \right]. \end{eqnarray}$

(3.24)

Which leads to the following result:

Theorem 3.2. Let $I$ be an interval of $\mathbb{R},$ $a, \Delta\in I\, (a < \Delta)$ and $\alpha, \beta > 0,$ such that $k-1 < \alpha, \beta < k,$ $(k\in\mathbb{N})$ . Let $\psi:I\to\mathbb{R}$ be a function such that $\psi\in AC^{k+1}.$ Assume that $|\psi^{(k+1)}|$ is a generalized modified $h$ -convex function and $\theta$ a continuous bifunction, then

$\begin{eqnarray} \left|\Gamma(k-\alpha+1)\left( ^CD^{\alpha}_{a_{+}}\psi\right) (\xi)+\Gamma(k-\beta+1)\left( ^CD^{\beta}_{\Delta_-}\psi\right) (\xi)\right.\\ \left. -\left(\psi^{(k)}(a)(\xi-a)^{k-\alpha}+\psi^{(k)}(\Delta)(\Delta-\xi)^{k-\beta}\right)\right| & &\\ \le (\Delta-\xi)^{k-\beta}\left[|\psi^{(k+1)}(\xi)|+ \theta\left( |\psi^{(k+1)}(\Delta)|,|\psi^{(k+1)}(\xi)|\right)\displaystyle {\int}_{0}^{1}h(z)dz \right] & &\\ +\, (\xi-a)^{k-\alpha} \left[|\psi^{(k+1)}(\xi)|+\theta\left( |\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right) \displaystyle {\int}_{0}^{1}h(z)dz \right] \end{eqnarray}$

(3.25)

holds.

As a consequences, we have

Corollary 3.3. If in (3.25), we set $\alpha = \beta,$ then

$\begin{align} &\left|\Gamma(k-\alpha+1)(^CD^{\alpha}_{a_{+}}\psi(\xi)+^CD^{\alpha}_{\Delta_-}\psi(\xi)) - \left(\psi^{(k)}(a)(\xi-a)^{k-\alpha}+\psi^{(k)}(\Delta) (\Delta-\xi)^{k-\alpha}\right)\right| \\ &\le (\Delta-\xi)^{k-\alpha}\left[|\psi^{(k+1)}(\xi)|+ \theta\left( |\psi^{(k+1)}(\Delta)|,|\psi^{(k+1)}(\xi)|\right)\displaystyle {\int}_{0}^{1}h(z)dz \right] \\ &+\,(\xi-a)^{k-\alpha}\left[|\psi^{(k+1)}(\xi)|+\theta\left( |\psi^{(k+1)}(a)|,|\psi^{(k+1)}(\xi)|\right)\displaystyle {\int}_{0}^{1}h(z)dz\right] \end{align}$

(3.26)

holds.

Corollary 3.4. By taking $\theta(z, r) = z-r, h(t) = t^{s}, s\in [0, 1]$ in (3.26), we obtain

$\begin{align} |\Gamma(k-\alpha+&1)[\left(^CD^{\alpha}_{a_{+}}\psi\right)(\xi)+\left(^CD^{\alpha}_{\Delta_-}\psi\right)(\xi)]- \left( \psi^{(k)}(a)(\xi-a)^{k-\alpha}+\psi^{(k)}(\Delta)(\Delta-\xi)^{k-\alpha}\right) |\\& \leq \dfrac{s \left((\xi-a)^{k-\alpha}+(\Delta-\xi)^{k-\alpha}\right)|\psi^{(k+1)}(\xi)|}{s+1} +\dfrac{(\xi-a)^{k-\alpha}|\psi^{(k+1)}(a)|+(\Delta-\xi)^{k-\alpha}|\psi^{(k+1)}(\Delta)|}{s+1}. \end{align}$

(3.27)

In particular for $s = 1,$ we have

$\begin{align} |\Gamma(k-\alpha+&1)[\left(^CD^{\alpha}_{a_{+}}\psi\right)(\xi)+\left(^CD^{\alpha}_{\Delta_-}\psi\right)(\xi)] - \left( \psi^{(k)}(a)(\xi-a)^{k-\alpha}+\psi^{(k)}(\Delta)(\Delta-\xi)^{k-\alpha}\right) | \\ &\leq \dfrac{\left((\xi-a)^{k-\alpha}+(\Delta-\xi)^{k-\alpha}\right)|\psi^{(k+1)}(\xi)|}{2} +\, \dfrac{(\xi-a)^{k-\alpha}|\psi^{(k+1)}(a)|+(\Delta-\xi)^{k-\alpha}|\psi^{(k+1)}(\Delta)|}{2}. \end{align}$

(3.28)

Example 3.2. Let $\psi, h, \theta$ as in the Example 3.1. We verify easily that $\psi^{(k+1)}(\xi) = 2(\xi-a)$ is generalized modified $h$ -convex on $[a, \Delta].$ From Corollary 3.3 and Lemma 2.1, we obtain

$\begin{equation} lhs : = \Gamma(k-\alpha+1)^CD^{\alpha}_{a_{+}}\psi(\xi) = \frac{2(\xi-a)^{k-\alpha+2}}{(k-\alpha+1)(k-\alpha+2)}, \end{equation}$

(3.29)

and

$\begin{align} \label{aex01} rhs: = (\xi-a)^{k-\alpha} \left[2(\xi-a) +( 0+2(\xi-a))\displaystyle {\int}_{0}^{1}h(z)dz\right] = 2\,(\xi-a)^{k-\alpha+1}\left( 1+ \displaystyle {\int}_{0}^{1}h(z)dz\right). \end{align}$

For the right derivative $^CD^{\alpha}_{\Delta_-}\psi(\xi),$ we have

$\begin{equation} lhs: = (-1)^{k}\Gamma(k-\alpha+1)^CD^{\alpha}_{\Delta_-}\psi(\xi) = \frac{2(\Delta-\xi)^{k-\alpha+2}}{(k-\alpha+1)(k-\alpha+2)} \end{equation}$

(3.30)

and

$\begin{equation} rhs: = 2\,(\Delta-\xi)^{k-\alpha+1}\left( 1+\displaystyle {\int}_{0}^{1}h(z)dz\right). \end{equation}$

(3.31)

Now suppose that $\psi : [a, \Delta]\to (0, \infty)$ is a generalized modified $h$ -convex function and symmetric about $\dfrac{a+\Delta}{2},$ then for all $\xi\in[a, \Delta]$ the inequality

$\begin{equation} \psi\left( \dfrac{a+\Delta}{2}\right) \leq \psi(\xi)\left( 1+h\left( \dfrac{1}{2}\right)\theta(1,1)\right) \quad \end{equation}$

(3.32)

is valid. Here $\theta$ is assumed to be nonnegatively homogeneous. Indeed, set

$r = a\dfrac{\xi-a}{\Delta-a}+\Delta\dfrac{\Delta-\xi}{\Delta-a},\quad z = \Delta\dfrac{\xi-a}{\Delta-a}+\alpha\dfrac{\Delta-\xi}{\Delta-a}.$

Hence

$\dfrac{a+\Delta}{2} = \dfrac{r}{2}+\dfrac{z}{2}.$

Since $\psi$ is generalized modified $h$ -convex, symmetric about $\frac{a+\Delta}{2},$ and the bifunction $\theta$ is assumed to be nonnegatively homogeneous, it results in

$\begin{eqnarray*} \psi\left(\dfrac{a+\Delta}{2} \right) & = & \psi\left( \frac{r}{2}+\frac{z}{2}\right)\\ &\leq&\psi(z)+ h\left( \dfrac{1}{2}\right)\theta(\psi(r),\psi(z))\\ & = & \psi(\xi)+h\left( \dfrac{1}{2}\right)\theta(\psi(\xi),\psi(\xi)) \\ & = &\psi(\xi)\left( 1+h\left( \dfrac{1}{2}\right)\theta(1,1)\right). \end{eqnarray*}$

Theorem 3.3. Let $I$ be an interval of $\mathbb{R},$ $a, \Delta\in I$ $(a < \Delta)$ and $\alpha, \beta\geq 1,$ $k-1 < \alpha, \beta < k, k\in\mathbb{N}$ . Let $\psi:I\to \mathbb{R}$ be a real valued function such that $\psi\in AC^{k}.$ If $\psi^{(k)}$ is a positive, generalized modified $h$ -convex and symmetric about $\frac{a+\Delta}{2}$ and furthermore the bifunction $\theta$ is nonnegatively homogeneous, then the following inequality holds

$\begin{eqnarray} N_{\theta}^{-1} \left\lbrace\dfrac{\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\beta+1}+\dfrac{\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\alpha+1}\right\rbrace \leq \dfrac{\Gamma(k-\beta+1)\left(^CD_{\Delta_-}^{\beta-1}\psi\right)(a)}{(\Delta-a)^{k-\beta+1}}+ \dfrac{\Gamma(k-\alpha+1)\left(^CD_{a_{+}}^{\alpha-1}\psi\right)(\Delta )}{(\Delta-a)^{k-\alpha+1}}\\ \leq \psi^{(k)}(\Delta)+\psi^{(k)}(a)+ \left[\theta( \psi^{(k)}(\Delta),\psi^{(k)}(a))+ \theta(\psi^{(k)}(a),\psi^{(k)}(\Delta))\right]\displaystyle {\int}_{0}^{1}h(z)dz. \end{eqnarray}$

(3.33)

If, furthermore, $\theta$ is additive, then

$\begin{align} N_{\theta}^{-1} \left\lbrace\dfrac{\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\beta+1}+\dfrac{\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\alpha+1}\right\rbrace & \leq \dfrac{\Gamma(k-\beta+1)\left(^CD_{\Delta_-}^{\beta-1}\psi\right)(\alpha)}{(\Delta-a)^{k-\beta+1}}+ \dfrac{\Gamma(k-\alpha+1)\left(^CD_{a_{+}}^{\alpha-1}\psi\right)(\Delta )} {(\Delta-a)^{k-\alpha+1}} \\ &\leq M_{\theta} (\psi^{(k)}(\Delta)+\psi^{(k)}(a)) \end{align}$

(3.34)

holds. Here

$N_{\theta} = 1+h\left( \dfrac{1}{2}\right)\theta(1,1),\; M_{\theta} = 1+\theta(1,1)\displaystyle {\int}_{0}^{1}h(z)dz.$

Proof. For all $\xi\in [a, \Delta], k-1 < \alpha < k,$ we have $\xi = \frac{\Delta-\xi}{\Delta-a}a +\frac{\xi-a}{\Delta-a}\Delta$ and

$\begin{equation} (\xi-\alpha)^{k-\alpha}\leq(\Delta-a)^{k-\alpha} \end{equation}$

(3.35)

and $\psi^{(k)}$ satisfies

$\begin{equation} \psi^{(k)}(\xi)\leq \psi^{(k)}(a)+h\left( \dfrac{\xi-a}{\Delta-a}\right) \theta\,(\psi^{(k)} (\Delta),\psi^{(k)}(a)). \end{equation}$

(3.36)

Multiplying (3.35) and (3.36) and proceeding as above, we obtain

$\begin{align} &\Gamma(k-\alpha+1)\left(^CD_{\Delta-}^{\alpha-1}\psi\right)(a) \\ &\leq \left[ \psi^{(k)}(a)+ \theta\left( \psi^{(k)}(\Delta),\psi^{(k)}(a)\right) \displaystyle {\int}_{0}^{1}h(z)dz\right] \times\,(\Delta-\alpha)^{k-\alpha+1}. \end{align}$

(3.37)

Also, we have for $\xi\in [a, \Delta], k-1 < \beta < k,$

$\begin{equation} (\Delta -\xi)^{k-\beta}\leq (\Delta-a)^{k-\beta} \end{equation}$

(3.38)

and

$\begin{equation} \psi^{(k)}(\xi)\leq \psi^{(k)}(\Delta)+h\left( \dfrac{\Delta-\xi}{\Delta-a}\right) \theta(\psi^{(k)} (a),\psi^{(k)}(\Delta)) . \end{equation}$

(3.39)

Multiplying (3.39) and (3.38) and integrating over $[a, \Delta],$ we get

$\begin{align} \Gamma(k-\beta+1)\left(^CD_{\Delta_-}^{\beta-1}\psi\right)(a)\leq \left[\psi^{(k)}(\Delta)+\theta\left( \psi^{(k)}(a),\psi^{(k)}(\Delta)\right) \displaystyle {\int}_{0}^{1}h(z)dz\right] (\Delta-a)^{k-\beta+1}. \end{align}$

(3.40)

Adding (3.37) and (3.40), we obtain

$\begin{align} &\dfrac{\Gamma(k-\beta+1)\left(^CD_{\Delta_-}^{\beta-1}\psi\right)(\alpha)}{(\Delta-a)^{k-\beta+1}}+ \dfrac{\Gamma(k-\alpha+1)\left(^CD_{a_{+}}^{\alpha-1}\psi\right)(\Delta )}{(\Delta-a)^{k-\alpha+1}} \end{align}$

(3.41)

$\begin{align} & \leq\psi^{(k)}(\Delta)+\psi^{(k)}(a) +\,\left[\theta(\psi^{(k)}(\Delta),\psi^{(k)}(a))+\theta(\psi^{(k)}(a),\psi^{(k)}(\Delta))\right]\displaystyle {\int}_{0}^{1}h(z)dz. \end{align}$

(3.42)

Set $N_{\theta} = 1+h\left(\dfrac{1}{2}\right)\theta(1, 1),$ thus (3.32) is written as

$\begin{equation} \psi^{(k)}\left(\dfrac{a+\Delta}{2}\right) \leq N_{\theta}\,\psi^{(k)}(\xi), \quad \xi\in[a,\Delta]. \end{equation}$

(3.43)

Multiplying by $(\xi-a)^{k-\alpha}$ on both sides of (3.43) and integrating the result over $[a, \Delta],$ it results that

$\begin{equation} N_{\theta}^{-1}\,\dfrac{\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\alpha+1} \leq \dfrac{\Gamma(k-\alpha+1)\left(^CD_{\Delta_-}^{\alpha-1}\psi\right)(a)}{(\Delta-a)^{k-\alpha+1}}. \end{equation}$

(3.44)

Multiplying (3.43) by $(\Delta-\xi)^{k-\beta},$ and integrating over $[a, \Delta],$ we obtain

$\begin{equation} N_{\theta}^{-1}\dfrac{\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\beta+1}\leq \dfrac{\Gamma(k-\beta+1)\left(^CD_{a_{+}}^{\beta-1}\psi\right)(\Delta )}{(\Delta-a)^{k-\beta+1}}. \end{equation}$

(3.45)

Adding (3.44) and (3.45), we obtain the first inequality. By combining the resulting inequality with (3.41), we obtain (3.33). Using the fact that $\theta$ is additive and nonnegatively homogeneous (3.34) results. That proves the claim. □

Corollary 3.5. By taking $\alpha = \beta$ in (3.33), then

$\begin{eqnarray} N_{\theta}^{-1}\, \dfrac{2\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)}{k-\alpha+1}&\leq& \dfrac{\Gamma(k-\alpha+1)\left(^CD_{a_{+}}^{\alpha-1} \psi(\Delta) +^CD_{\Delta_-}^{\alpha-1}\psi(a)\right)}{(\Delta-a)^{k-\alpha+1}}\\ &\leq& \psi^{(k)}(\Delta)+\psi^{(k)}(a)\\ &+& \left[\theta(\psi^{(k)}(\Delta),\psi^{(k)}(a))+\theta(\psi^{(k)}(a),\psi^{(k)}(\Delta))\right]\displaystyle {\int}_{0}^{1}h(z)dz \end{eqnarray}$

(3.46)

holds.

If, $\theta$ is additive, then

$\begin{eqnarray} \dfrac{2N_{\theta}^{-1}\,\psi^{(k)}\left(\dfrac{a+\Delta}{2}\right)} {k-\alpha+1} &\leq&\dfrac{\Gamma(k-\alpha+1)\left(^CD_{a_{+}}^{\alpha-1}\psi(\Delta) + ^CD_{\Delta_-}^{\alpha-1}\psi(a)\right)}{(\Delta-a)^{k-\alpha+1}} \\ &\leq& M_{\theta}\,(\psi^{(k)}(\Delta)+\psi^{(k)}(a)). \end{eqnarray}$

(3.47)

Corollary 3.6. By setting $h(t) = t^{s}, s\in[0, 1]$ in (3.47), it results that

$\begin{align*} &\dfrac{2^{s}\psi^{(k)}\left(\frac{a+\Delta}{2}\right)}{(2^{s}+\theta(1,1))(k-\alpha+1)} \\ &\leq \dfrac{\Gamma (k-\alpha+1)\left[ \left( ^CD_{\Delta_-}^{\alpha+1}\psi\right) (a)+ \left( ^CD_{a_{+}}^{\alpha+1}\psi\right)(\Delta)\right] }{(\Delta-a)^{k-\alpha+1}} \\& \leq \dfrac{\psi^{(k)}(a)+\psi^{(k)}(\Delta)}{s+1}(s+1+\theta(1,1)). \end{align*}$

In particular, if $h(t) = t,$ then

$\begin{align*} & \dfrac{2\psi^{(k)}\left(\frac{a+\Delta}{2}\right)}{(2+\theta(1,1))(k-\alpha+1)} \\ & \leq \dfrac{\Gamma (k-\alpha+1)\left[ \left(^CD_{\Delta_-}^{\alpha+1}\psi\right) (a)+ \left( ^CD_{a_{+}}^{\alpha+1}\psi\right)(\Delta)\right] }{(\Delta-a)^{k-\alpha+1}} \\& \leq \dfrac{\psi^{(k)}(a)+\psi^{(k)}(\Delta)}{2}(2+\theta(1,1)). \end{align*}$

Theorem 3.4. Let $\psi\in AC^{k}(a, \Delta),$ $k\in N; k-1 < \alpha < k.$ Assume that $\psi^{(k)}$ is positive, generalized modified $h$ -convex on $[a, \Delta]$ and symmetric to $\dfrac{a+\Delta}{2}.$ Assume that $\theta$ is nonnegatively homogeneous. Then

$\begin{align} \frac{\psi^{(k)}\left( \dfrac{a+\Delta}{2}\right)}{ 1+h(\frac{1}{2})\theta(1,1)} \left[ \left( ^CD_{\Delta_-}^{\alpha}\psi\right)(a)+ \left( ^CD_{a_{+}}^{\alpha}\psi\right)(\Delta)\right] \le R_{\Delta-}^{k-\alpha}(\psi^{(k)})^{2}( a)+ R_{a+}^{k-\alpha}(\psi^{(k)})^{2}(\Delta) \end{align}$

(3.48)

holds. Where $R_{.}^{k-\alpha}$ is the Riemann-Liouville integral operator of order $k-\alpha.$

Proof. Since $\psi^{(k)}$ is generalized modified $h$ -convex and $\theta$ is nonnegatively homogeneous, then we have for $\mu\in [0, 1]$

$\begin{align} \psi^{(k)}\left( \dfrac{a+\Delta}{2}\right) & = \psi^{(k)}\left(\dfrac{\mu \Delta+(1-\mu )a + \mu a+(1-\mu)\Delta}{2}\right)\\ &\le \psi^{(k)}(\mu \Delta+(1-\mu )a )+ h(\frac{1}{2})\,\theta\,(\psi^{(k)}(\mu a+(1-\mu )\Delta ),\psi^{(k)}(\mu \Delta+(1-\mu )a)) \\ & = (\psi^{(k)})^{2}(\mu \Delta+(1-\mu )a )\left[ 1+h\left( \frac{1}{2}\right) \theta(1,1)\right] . \end{align}$

(3.49)

Multiplying (3.49) by $\mu^{k-\alpha-1}\psi^{(k)}(\mu \Delta+(1-\mu)a)$ and integrating over $[0, 1],$ with respect to $\mu,$ we obtain

$\begin{eqnarray} \psi^{(k)}\left( \dfrac{a+\Delta}{2}\right)\displaystyle {\int}_{0}^{1}\mu^{k-\alpha-1}\psi^{(k)}( \mu \Delta+(1-\mu )a )d\mu = \frac{\psi^{(k)}\left( \frac{a+\Delta}{2}\right)}{(\Delta-a)^{k-\alpha}}\Gamma(k-\alpha)(^CD_{a_{+}}^{\alpha}\psi)(\Delta), \end{eqnarray}$

and

$\begin{align*} & \left[ 1+h\left( \frac{1}{2}\right)\, \theta(1,1)\right]\displaystyle {\int}_{0}^{1}\mu^{k-\alpha-1} (\psi^{(k)})^{2}(\mu \Delta+(1-\mu )a ) d\mu\\ & = \frac{ 1+h(\frac{1}{2})\theta(1,1)}{(\Delta-a)^{k-\alpha}}\displaystyle {\int}_{a}^{\Delta}(x-a)^{k-\alpha-1}(\psi^{(k)})^{2}(x )dx\\ & = \left[ 1+h(\frac{1}{2})\theta(1,1)\right]\frac{\Gamma(k-\alpha)}{(\Delta-a)^{k-\alpha}} R_{a_{+}}^{k-\alpha}(\psi^{(k)})^{2}(\Delta). \end{align*}$

Hence

$\begin{equation} \frac{\psi^{(k)}\left( \frac{a+\Delta}{2}\right)}{ 1+h(\frac{1}{2})\theta(1,1)} (^{C}D_{a_{+}}^{\alpha}f)(\Delta) \le R_{a_{+}}^{k-\alpha}(\psi^{(k)})^{2}(\Delta). \end{equation}$

(3.50)

And similarly

$\begin{equation} \psi^{(k)}\left( \dfrac{a+\Delta}{2}\right) \le \psi^{(k)}(\mu a +(1-\mu )\Delta )\left[ 1+h\left( \frac{1}{2}\right) \theta(1,1)\right] \end{equation}$

(3.51)

by multiplying (3.51) by $\mu^{k-\alpha-1}\psi^{(k)}(\mu a+(1-\mu)\Delta),$ integration yields to

$\begin{eqnarray} \psi^{(k)}\left( \frac{a+\Delta}{2}\right)\displaystyle {\int}_{0}^{1}\mu^{k-\alpha-1}\psi^{(k)}(\mu a+(1-\mu )\Delta )d\mu = \frac{\psi^{(k)}\left( \frac{a+\Delta}{2}\right)} {(\Delta-a)^{k-\alpha}}\,\Gamma(k-\alpha)(^{C}D_{\Delta_-} ^{\alpha}\psi)(a) \end{eqnarray}$

and

$\begin{align} &\left[1+h\left( \frac{1}{2}\right)\, \theta(1,1)\right]\displaystyle {\int}_{0}^{1}\mu^{k-\alpha-1}(\psi^{(k)})^{2}(\mu a +(1-\mu )\Delta ) d\mu\\ & = \left[1+h\left( \frac{1}{2}\right) \theta(1,1)\right]\frac{\Gamma(k-\alpha)}{(\Delta-a)^{k-\alpha}} R_{\Delta_-}^{k-\alpha}(\psi^{(k)})^{2}(a), \end{align}$

(3.52)

it results that

$\begin{equation} \frac{\psi^{(k)}\left( \frac{a+\Delta}{2}\right)}{ 1+h(\frac{1}{2})\,\theta(1,1)}\, (^{C}D_{\Delta-}^{\alpha}\psi)(a) \le \,R_{\Delta_-}^{k-\alpha}(\psi^{(k)})^{2}(a). \end{equation}$

(3.53)

By adding (3.50) and (3.53), we get (3.48). That proves the claim. □

Corollary 3.7. Under the same assumptions as Theorem 3.4, if $h(t) = t^{s}, s\in [0, 1],$ then

$\begin{eqnarray} \label{eca1} \frac{2^{s}\,\psi^{(k)}\left( \dfrac{a+\Delta}{2}\right)}{ 2^{s}+\theta(1,1)} \left[ \left( ^CD_{\Delta-}^{\alpha}\psi\right)(a)+ \left( ^CD_{a_{+}}^{\alpha}\psi\right)(\Delta)\right] \le R_{\Delta_-}^{k-\alpha}(\psi^{(k)})^{2}(a)+ R_{a+}^{k-\alpha}(\psi^{(k)})^{2}(\Delta). \end{eqnarray}$

If $\theta(u, v) = -\theta(v, u)$ , then

$\begin{equation} \psi^{(k)}\left( \dfrac{a+\Delta}{2}\right) \left(^CD_{\Delta_-}^{\alpha}\psi(a) +^CD_{a_{+}}^{\alpha}\psi(\Delta) \right) \le R_{\Delta_-}^{k-\alpha}(\psi^{(k)})^{2}(\alpha)+ R_{a_{+}}^{k-\alpha}(\psi^{(k)})^{2}(\Delta) \end{equation}$

(3.54)

is valid.

4. Conclusions

In this work, we have established some estimates including once the derivatives of Caputo and another time the integrals of Riemann-Liouville and the derivatives of Caputo for a function whose derivative order $k$ th $(k \in \mathbb{N})$ is generalized modified $h$ -convex and symmetrical in the middle. Estimates of consequences for special classes of convex functions and $s$ -convex functions in $[0, 1]$ were obtained. The estimates we have just made are compared to those presented in the results ^[8].

Future research could focus on extending these results to variable order or other types of convex functions or exploring inequalities for functions that do not necessarily have symmetry. Furthermore, the application of derived inequalities to concrete problems in applied mathematics, physics, or engineering could still validate the practical significance of our theoretical contributions. Taking these limitations into account could lead to a more complete understanding and wider applicability of fractional inequalities.

Author contributions

HB: conceptualization, writing original draft preparation, writing review and editing, supervision; MSS: conceptualization, writing original draft preparation, writing review and editing, supervision; HG: conceptualization, writing review and editing, supervision; UFG: funding, writing review and editing. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The work of U.F.-G. was supported by the government of the Basque Country for the ELKARTEK24/78 and ELKARTEK24/26 research programs, respectively.

Conflict of interest

The authors declare no competing interests.

References

[1]	S. Banach, Sur les opérations dans les ensembles abstraites et leurs applications, Fund. Math., 3 (1922), 133–187. https://doi.org/10.4064/fm-3-1-133-181 doi: 10.4064/fm-3-1-133-181
[2]	R. P. Agarwal, Ü. Aksoy, E. Karapınar, İ. M. Erhan, $F$ -contraction mappings on metric-like spaces in connection with integral eqautions on time scales, RACSAM, 114 (2020). https://doi.org/10.1007/s13398-020-00877-5 doi: 10.1007/s13398-020-00877-5
[3]	D. R. Kumar, Common fixed point results under $w$ -distance with applications to nonlinear integral equations and nonlinear fractional differential equations, Math. Slovaca, 71 (2021), 1511–1528. https://doi.org/10.1515/ms-2021-0068 doi: 10.1515/ms-2021-0068
[4]	M. Younis, I. Altun, V. Chauhan, Graphical structure of extended $b$ -metric spaces: An application to the transverse oscillations of a homogeneous bar, Int. J. Nonlin. Sci. Num., 23 (2022), 1239–1252. https://doi.org/10.1515/ijnsns-2020-0126 doi: 10.1515/ijnsns-2020-0126
[5]	D. R. Kumar, Common solution to a pair of nonlinear Fredholm and Volterra integral equations and nonlinear fractional differential equations, J. Comput. Appl. Math., 404 (2022). https://doi.org/10.1016/j.cam.2021.113907 doi: 10.1016/j.cam.2021.113907
[6]	W. Shatanawi, T. A. M. Shatnawi, New fixed point results in controlled metric type spaces based on new contractive conditions, AIMS Math., 8 (2023), 9314–9330. https://doi.org/10.3934/math.2023468 doi: 10.3934/math.2023468
[7]	A. Z. Rezazgui, A. A. Tallafha, W. Shatanawi, Common fixed point results via $A\nu$ - $\alpha$ -contractions with a pair and two pairs of self-mappings in the frame of an extended quasi $b$ -metric space, AIMS Math., 8 (2023), 7225–7241. https://doi.org/10.3934/math.2023363 doi: 10.3934/math.2023363
[8]	M. Joshi, A. Tomar, T. Abdeljawad, On fixed points, their geometry and application to satellite web coupling problem in $S$ -metric spaces, AIMS Math., 8 (2023), 4407–4441. https://doi.org/10.3934/math.2023220 doi: 10.3934/math.2023220
[9]	M. Younis, D. Singh, L. Chen, M. Metwali, A study on the solutions of notable engineering models, Math. Model. Anal., 27 (2022), 492–509. https://doi.org/10.3846/mma.2022.15276
[10]	M. Younis, H. Ahmad, L. Chen, M. Han, Computation and convergence of fixed points in graphical spaces with an application to elastic beam deformations, J. Geom. Phys., 192 (2023). https://doi.org/10.1016/j.geomphys.2023.104955 doi: 10.1016/j.geomphys.2023.104955
[11]	F. E. Browder, Nonexpansive nonlinear operators in a Banach spaces, P. Natl. Acad. Sci. USA, 54 (1965), 1041–1044. https://doi.org/10.1073/pnas.54.4.1041 doi: 10.1073/pnas.54.4.1041
[12]	D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr., 30 (1965), 251–258. https://doi.org/10.1002/mana.19650300312 doi: 10.1002/mana.19650300312
[13]	S. Zhang, About fixed point theory for mean nonexpansive mapping in Banach spaces, J. Sichuan Univ., 2 (1975), 67–68.
[14]	A. A. Mebawondu, C. Izuchukwu, H. A. Abass, O. T. Mewomo, Some results on generalized mean nonexpansive mapping in complete metric spaces, Bol. Soc. Parana. Mat., 40 (2022), 1–16. https://doi.org/10.5269/bspm.44174 doi: 10.5269/bspm.44174
[15]	J. Morales, E. Rojas, Some results on $T$ Zamfirescu operators, Rev. Notas Mat., 5 (2009), 64–71.
[16]	I. A. Bakhtin, The contraction principle in quasimetric spaces, Funct. Anal., 30 (1989), 26–37.
[17]	S. Czerwik, Contraction mappings in $b$ -metric spaces, Acta Math. Inform. Univ. Ostrav., 1 (1993), 5–11.
[18]	S. G. Matthews, Partial metric topology, Ann. NY Acad. Sci., 728 (1994), 183–197. https://doi.org/10.1111/j.1749-6632.1994.tb44144.x
[19]	A. A. Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory A., 2012 (2021). https://doi.org/10.1186/1687-1812-2012-204 doi: 10.1186/1687-1812-2012-204
[20]	M. A. Alghamdi, N. Hussain, P. Salimi, Fixed point and coupled fixed point theorems on $b$ -metric-like spaces, J. Inequal. Appl., 2013 (2013). https://doi.org/10.1186/1029-242X-2013-402 doi: 10.1186/1029-242X-2013-402
[21]	N. Hussain, J. R. Roshan, V. Parvaneh, Z. Kadelburg, Fixed points of contractive mappings in $b$ -metric-like spaces, The Scientific World J., 2014 (2014). https://doi.org/10.1155/2014/981578 doi: 10.1155/2014/981578
[22]	H. H. Alsulami, S. Gülyaz, E. Karapınar, İ. M. Erhan, An Ulam stability result on quasi- $b$ -metric-like spaces, Open Math., 14, (2016), 1087–1103. https://doi.org/10.1515/math-2016-0097 doi: 10.1515/math-2016-0097
[23]	K. Jain, J. Kaur, Some fixed point results in $b$ -metric spaces and $b$ -metric-like spaces with new contractive mappings, Axioms, 10 (2021). https://doi.org/10.3390/axioms10020055 doi: 10.3390/axioms10020055
[24]	S. K. Prakasam, A. J. Gnanaprakasam, Ö. Ege, G. Mani, S. Haque, N. Mlaiki, Fixed point for an $\mathbb{O}g\mathfrak{F}$ -c in $\mathbb{O}$ -complete $b$ -metric-like spaces, AIMS Math., 8 (2023), 1022–1039. https://doi.org/10.3934/math.2023050 doi: 10.3934/math.2023050
[25]	Z. Kalkan, A. Şahin, Some new results in partial cone $b$ -metric space, Commun. Adv. Math. Sci., 3 (2020), 67–73. https://doi.org/10.33434/cams.684102 doi: 10.33434/cams.684102
[26]	D. R. Kumar, M. Prabavathy, S. Radenovic, On existence and approximation of common fixed points in $b$ -metric spaces, Asian-Eur. J. Math., 15 (2022). https://doi.org/10.1142/S1793557122500309 doi: 10.1142/S1793557122500309
[27]	I. D. Arandelovic, Z. D. Mitrovic, A. Aloqaily, N. Mlaiki, The results of common fixed points in $b$ -metric spaces, Symmetry, 15 (2023), 1344. https://doi.org/10.3390/sym15071344 doi: 10.3390/sym15071344
[28]	A. Beiranvand, S. Moradi, M. Omid, H. Pazandeh, Two fixed point theorems for special mappings, arXiv Preprint, 2009.
[29]	M. Öztürk, M. Başarır, On some common fixed point theorems for $f$ -contraction mappings in cone metric spaces, Int. J. Math. Anal., 5 (2011), 119–127. https://doi.org/10.1186/1687-1812-2011-93 doi: 10.1186/1687-1812-2011-93
[30]	C. T. Aage, P. G. Golhare, On fixed point theorems in dislocated quasi $b$ -metric spaces, Int. J. Adv. Math., 2016 (2016), 55–70. https://doi.org/10.1186/s13663-016-0565-9 doi: 10.1186/s13663-016-0565-9
[31]	K. Zoto, P. S. Kumari, Fixed point theorems for $s$ - $\alpha$ contractions in dislocated and $b$ -dislocated metric spaces, Thai J. Math., 17 (2019), 263–276.
[32]	A. M. Zaki, A. O. Ismail, A note on cone metric spaces, Curr. Sci. Int., 11 (2022), 319–328.
[33]	H. Aydi, A. Felhi, S. Sahmim, On common fixed points for $\alpha -\psi$ contractions and generalized cyclic contractions in $b$ -metric-like spaces and consequences, J. Nonlinear Sci. Appl., 9 (2016), 2492–2510. https://doi.org/10.22436/jnsa.009.05.48 doi: 10.22436/jnsa.009.05.48
[34]	K. Calderón, A. Padcharoen, J. M. Moreno, Some stability and strong convergence results for the algorithm with perturbations for a $T$ -Ciric quasicontraction in CAT(0) spaces, J. Inequal. Appl., 2023 (2023). https://doi.org/10.1186/s13660-022-02911-z doi: 10.1186/s13660-022-02911-z
[35]	A. M. Harder, T. L. Hicks, Some stability results for fixed point iteration procedures, Math. Japon., 33 (1988), 693–706.
[36]	V. Berinde, Iterative approximation of fixed points, Berlin: Springer-Verlag, 2007. https://doi.org/10.1109/SYNASC.2007.49
[37]	S. Hilger, Analysis on measure chains a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18–56. https://doi.org/10.1007/BF03323153 doi: 10.1007/BF03323153
[38]	M. Bohner, A. Peterson, Dynamic equations on time scales, Boston/Berlin: Birkhauser, 2001. https://doi.org/10.1007/978-1-4612-0201-1
[39]	R. Agarwal, M. Bohner, A. Peterson, Inequalities on time scales: A survey, Math. Inequal. Appl., 4 (2001), 535–557. https://doi.org/10.7153/mia-04-48 doi: 10.7153/mia-04-48
[40]	G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl., 285 (2003), 107–127. https://doi.org/10.1016/S0022-247X(03)00361-5 doi: 10.1016/S0022-247X(03)00361-5
[41]	S. Georgiev, Integral equations on time scales, Paris: Atlantis Press, 2016. https://doi.org/10.2991/978-94-6239-228-1
[42]	Z. Kalkan, A. Şahin, Some new stability results of Volterra integral equations on time scales, Maltepe J. Math., 4 (2022), 44–54. https://doi.org/10.47087/mjm.1145159 doi: 10.47087/mjm.1145159
[43]	X. Hu, Y. Li, Multiplicity result to a system of over-determined Fredholm fractional integro-differential equations on time scales, AIMS Math., 7 (2022), 2646–2665. https://doi.org/10.3934/math.2022149 doi: 10.3934/math.2022149
[44]	A. Pouria, The numerical solution of Fredholm-Hammerstein integral equations by combining the collocation method and radial basis functions, Filomat, 33 (2019), 667–682. https://doi.org/10.2298/FIL1903667A doi: 10.2298/FIL1903667A
[45]	U. Kohlenbach, Some logical metatheorems with applications in functional analysis, T. Am. Math. Soc., 357 (2004), 89–128. https://doi.org/10.1090/S0002-9947-04-03515-9 doi: 10.1090/S0002-9947-04-03515-9

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(1692) PDF downloads(128) Cited by(4)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

Some fixed point and stability results in $b$ -metric-like spaces with an application to integral equations on time scales

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Results

4. Conclusions

Author contributions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Mathematics

Some fixed point and stability results in b b -metric-like spaces with an application to integral equations on time scales

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Results

4. Conclusions

Author contributions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Some fixed point and stability results in $b$ -metric-like spaces with an application to integral equations on time scales