Ulam–Hyers–Rassias Mittag-Leffler stability of ϖ–fractional partial differential equations

Abstract

This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results.

1 Introduction

Fractional calculus extends the classical calculus from integer order to any order. Although the fractional derivation has been given by several mathematicians like Riemann–Liouville, Grünwald–Letnikov, Caputo, this concept was defined in the seventeenth century when Leibniz defined the symbol of the positive integer order derivation, L’Hôpital questioned him on the possibility of having a derivative of order $\frac{1}{2}$. This question attracted the attention of mathematicians, including Euler and Lagrange, in the eighteenth century, followed by Liouville, Riemann, and Grünwald. For more historical details, see [1–3].

Fractional differential equations (FDEs) describe the observed evolution phenomena in many scientific fields, such as engineering, physics, electrochemistry, medicine, control theory, and other domains. The importance of these equations appears in the modeling of various real-world applications and motivated many mathematicians to study their qualitative and quantitative theories (see [2–6]).

In 1940, Ulam introduced a new notion for the stability of functional equations (see [7]). The first answer to the question proposed by Ulam was given by Hyers in 1941 [8]. Subsequently, this concept is called stability in the sense of Ulam–Hyers. In 1978, Rassias [9] suggested an extension called the Ulam–Hyers stability. Considerable attention has been paid to the study of stability in the sense of Ulam–Hyers and in the sense of Ulam–Hyers–Rassias of various types of differential equations; for more details, see [10] and [11, 12].

Motivated by works [11, 13–15], we extend paper [11] devoted to the solutions to the Darboux problem of ϖ-fractional partial differential equations in the linear and nonlinear cases. The main highlights of the paper are the following:

• 1. A representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions is given.

• 2. Study the Ulam–Hyers–Rassias Mittag–Leffler stability result using the generalized Gronwall lemma.

In this work, the Ulam–Hyers–Rassias Mittag–Leffler stability for the following partial fractional differential equation is introduced and investigated:

$$ ^{C}D_{a}^{\chi , \left (\varpi _{1}, \varpi _{2}\right )}\mathcal{U} \left (\varsigma _{1},\varsigma _{2}\right ) + \lambda \mathcal{U} \left (\varsigma _{1},\varsigma _{2}\right )= F\left (\varsigma _{1}, \varsigma _{2}, \mathcal{U}(\varsigma _{1},\varsigma _{2}) \right ), \qquad \left (\varsigma _{1},\varsigma _{2}\right ) \in J, $$

(1.1)

where $a=(e_{1}, e_{2})\in \mathbb{R}^{2}$, $J=[e_{1}, b_{1}]\times [e_{2}, b_{2}]$, $\lambda \in \mathbb{R}$, is the Caputo $(\varpi _{1}, \varpi _{2})$-fractional derivative of order $\chi =(\chi _{1}, \chi _{2}) \in (0,1] \times (0,1]$, and the function $F:J \times \mathbb{R}\longrightarrow \mathbb{R}$, satisfying some conditions, will be specified later.

In Sect. 2, we give some preliminary results. In Sect. 3, we give the representation of the solutions for the Darboux problem of partial fractional differential equations in the linear case in the space of continuous functions. Section 4 is devoted to the investigation of the results of the Ulam–Hyers–Rassias Mittag–Leffler stability. Finally, we provide some illustrative examples in Sect. 5 and present conclusions in Sect. 6.

2 Preliminarily

In this section, we review some fundamental concepts in fractional calculus, employing the notations of the ϖ-fractional integral and derivative.

Definition 2.1

[10, 16] Let $a=(e_{1}, e_{2})$ and $\chi = (\chi _{1}, \chi _{2})$. Assume that $\varpi _{1}\in C^{1}([e_{1}, b_{1}],\mathbb{R})$ and $\varpi _{2}\in C^{1}([e_{2}, b_{2}],\mathbb{R})$ are positive, strictly increasing functions, satisfying $\varpi '_{1}(\varsigma _{1}), \varpi '_{2}(\varsigma _{2})\neq 0$ for all $\varsigma _{1}\in [e_{1},b_{1}]$ and $\varsigma _{2}\in [e_{2}, b_{2}]$. The partial Riemann–Liouville $\left (\varpi _{1}, \varpi _{2}\right )$-fractional integral of order χ for $f\in L^{1}(J,\mathbb{R})$ with respect to $\varpi _{1}$ and $\varpi _{2}$ is defined as

$$\begin{aligned}& \left ( I_{a}^{\chi , \left ( \varpi _{1}, \varpi _{2}\right ) }f \right ) \left (\varsigma _{1},\varsigma _{2}\right ) \\& \quad = \dfrac{1}{\Gamma ( \chi _{1}) \Gamma ( \chi _{2})} \int _{e_{1}}^{ \varsigma _{1}}\int _{e_{2}}^{\varsigma _{2}} \varpi _{1}'(s)\varpi _{2}'(t) \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(s)\right ) ^{\chi _{1}-1} \\& \qquad{}\times \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t)\right ) ^{\chi _{2}-1} f(s,t)dtds. \end{aligned}$$

Definition 2.2

[10, 16] Let $f\in L^{1}(J,\mathbb{R})$ and $\chi \in (0,1] \times (0,1]$. The partial Riemann–Liouville $(\varpi _{1}, \varpi _{2})$-fractional derivative of order χ for f is defined when $I_{a}^{1-\chi , \left ( \varpi _{1},\varpi _{2}\right )} f \in AC(J, \mathbb{R})$ by

where

$$ \left ( D_{\varsigma _{1}\varsigma _{2}}^{\left ( \varpi _{1},\varpi _{2} \right )} f \right )(\varsigma _{1},\varsigma _{2}) = \left ( \dfrac{1}{\varpi '_{1}(\varsigma _{1})\varpi '_{2}(\varsigma _{2})} \dfrac{\partial ^{2}}{\partial \varsigma _{1} \partial \varsigma _{2}} f\right ) (\varsigma _{1},\varsigma _{2}). $$

Definition 2.3

[10, 16] Let $\chi \in (0,1] \times (0,1]$. The partial Caputo $(\varpi _{1}, \varpi _{2})$-fractional derivative of order χ for a function f is defined as

$$ \left ( ^{C}D_{a}^{\chi , \left ( \varpi _{1},\varpi _{2}\right ) }f \right ) \left (\varsigma _{1},\varsigma _{2}\right )=\left ( ^{RL}D_{a}^{ \chi , \left ( \varpi _{1},\varpi _{2}\right ) }g\right )\left ( \varsigma _{1},\varsigma _{2}\right ), $$

where $g\left (\varsigma _{1},\varsigma _{2}\right )=f(\varsigma _{1}, \varsigma _{2})-f(e_{1}, \varsigma _{2})-f(\varsigma _{1}, e_{2})+f(e_{1},e_{2})$.

Lemma 2.1

[10, 16] If $\delta _{1}, \delta _{2} \in \left ( -1, \infty \right )$ and $\chi =\left ( \chi _{1},\chi _{2}\right ) \in \left ( 0, \infty \right ) \times \left ( 0, \infty \right )$, then for all $\left (\varsigma _{1},\varsigma _{2}\right )\in J$, we have

$$\begin{aligned}& I_{a}^{\chi , \left ( \varpi _{1}, \varpi _{2} \right ) } \left ( \left (\varpi _{1}(\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{ \delta _{1}} \left (\varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\delta _{2}} \right ) \\& = \dfrac{\Gamma (\delta _{1}+1)\Gamma (\delta _{2}+1)}{\Gamma (\chi _{1}+\delta _{1}+1)\Gamma (\chi _{2}+ \delta _{1}+1)} \left (\varpi _{1}(\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}+ \delta _{1}} \left (\varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}+\delta _{2}}. \end{aligned}$$

Lemma 2.2

[16] If $\chi =\left ( \chi _{1},\chi _{2}\right ) \in \left ( 0, \infty \right ) \times \left ( 0, \infty \right )$, $\vartheta =\left ( \vartheta _{1},\vartheta _{2}\right ) \in \left ( 0, \infty \right ) \times \left ( 0, \infty \right )$ and $\left (\varsigma _{1},\varsigma _{2}\right )\in J$, then the following properties are satisfied

(i):

For $f\in L^{1}(J,\mathbb{R})$, we have $\left ( I_{a}^{\chi , \left ( \varpi _{1}, \varpi _{2}\right ) } I_{a}^{ \vartheta , \left ( \varpi _{1}, \varpi _{2}\right ) } f\right ) ( \varsigma _{1},\varsigma _{2})= \left ( I_{a}^{\chi +\vartheta , \left ( \varpi _{1}, \varpi _{2}\right ) }f\right ) (\varsigma _{1}, \varsigma _{2})$.

(ii):

For $f\in AC(J,\mathbb{R})$, we have $\left ( I_{a}^{1, \left ( \varpi _{1}, \varpi _{2}\right ) } D_{ \varsigma _{1}\varsigma _{2}}^{ \left ( \varpi _{1},\varpi _{2} \right ) }f\right ) (\varsigma _{1},\varsigma _{2})= f(\varsigma _{1}, \varsigma _{2})-f(e_{1}, \varsigma _{2})-f(\varsigma _{1}, e_{2})+f(e_{1},e_{2})$.

Definition 2.4

[1] Let $m \in \mathbb{N}^{*}$, and $\chi _{j}, \vartheta _{j}, z, \rho \in \mathbb{C}$, satisfying $Re(\chi _{j}), Re(\vartheta _{j}) > 0$ for $j=1,2,\dots ,m$. The generalized Mittag–Leffler function is defined by

$$ \mathbb{E}_{\rho}\left (\left (\chi _{j}, \vartheta _{j} \right )_{j=1,m}; \left (z \right ) \right )= \sum \limits _{k=0}^{\infty} \dfrac{(\rho )_{k}}{\prod _{j=1}^{m}\Gamma \left ( k\chi _{j}+ \vartheta _{j}\right ) } \dfrac{z^{k}}{k!}, $$

where

$$ (\rho )_{k}= \rho (\rho +1)\dots (\rho +k-1)= \dfrac{\Gamma (\rho +k)}{\Gamma (\rho )}. $$

In particular, if $m=2$ and $\rho =1$, we have

$$ \mathbb{E}_{\rho}\left (\left (\chi _{j}, \vartheta _{j} \right )_{j=1,2}; \left (z \right ) \right )= \mathbb{E}\left (\left (\chi _{j}, \vartheta _{j} \right )_{j=1,2}; \left (z \right ) \right )= \sum \limits _{k=0}^{\infty} \dfrac{z^{k}}{\Gamma \left ( k\chi _{1}+ \vartheta _{1}\right ) \Gamma \left ( k\chi _{2}+ \vartheta _{2}\right ) }. $$

Theorem 2.3

Let $e_{1}, e_{2} \in \mathbb{R}$, and $\chi , \vartheta > 0$. Assume that $\varpi _{1}\in C^{1}([e_{1}, b_{1}],\mathbb{R})$ and $\varpi _{2}\in C^{1}([e_{2}, b_{2}],\mathbb{R})$ are positive, strictly increasing functions, with $\varpi '_{1}(\varsigma _{1}), \varpi '_{2}(\varsigma _{2})\neq 0$ for all $\varsigma _{1}\in [e_{1}, b_{1}]$ and $\varsigma _{2}\in [e_{2}, b_{2}]$. Additionally, let $\mathcal{A}(\varsigma _{1},\varsigma _{2})$ and $\mathcal{W}(\varsigma _{1},\varsigma _{2})$ be nonnegative functions that are locally integrable on $D=\{(\varsigma _{1},\varsigma _{2})\in \mathbb{R}^{2}, e_{1}\leq \varsigma _{1}< X, e_{2}\leq \varsigma _{2} < Y \}$. Define $\mathcal{B}(\varsigma _{1},\varsigma _{2})$ as a nonnegative, nondecreasing, and continuous function on D such that $\mathcal{B}(\varsigma _{1},\varsigma _{2})\leq M$ for some positive number M. If the following inequality holds

$$\begin{aligned}& \begin{gathered}[b] \mathcal{W}(\varsigma _{1},\varsigma _{2}) \\ \quad \leq \mathcal{A}(\varsigma _{1},\varsigma _{2})+ \mathcal{B}( \varsigma _{1},\varsigma _{2})\int _{e_{1}}^{\varsigma _{1}}\int _{e_{2}}^{ \varsigma _{2}} \varpi _{1}'(s)\varpi _{2}'(t)\left ( \varpi _{1}( \varsigma _{1})-\varpi _{1}(s)\right ) ^{\chi -1} \\ \qquad{}\times \left ( \varpi _{2}( \varsigma _{2})-\varpi _{2}(t)\right ) ^{\vartheta -1} \mathcal{W}(s,t)dtds, \end{gathered} \end{aligned}$$

(2.1)

for all $(\varsigma _{1},\varsigma _{2})\in D$, then

$$\begin{aligned} \mathcal{W}(\varsigma _{1},\varsigma _{2}) \leq & \mathcal{A}( \varsigma _{1},\varsigma _{2})+ \sum _{n=1}^{\infty} \dfrac{\left ( \Gamma (\chi )\Gamma (\vartheta )\mathcal{B}(\varsigma _{1},\varsigma _{2})\right )^{n}}{\Gamma (n\chi )\Gamma (n\vartheta )} \int _{e_{1}}^{\varsigma _{1}}\int _{e_{2}}^{\varsigma _{2}} \varpi _{1}'(s) \varpi _{2}'(t) \\ & \times \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(s)\right ) ^{n \chi -1}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t)\right ) ^{n \vartheta -1} \mathcal{A}(s,t)dtds. \end{aligned}$$

(2.2)

Proof

Let us define the operator $\mathcal{K}$ for all $(\varsigma _{1},\varsigma _{2})\in D$ as follows

$$ \begin{aligned} \mathcal{K}\mathcal{S}(\varsigma _{1},\varsigma _{2}) & = \mathcal{B}( \varsigma _{1},\varsigma _{2})\int _{e_{1}}^{\varsigma _{1}}\int _{e_{2}}^{ \varsigma _{2}} \varpi _{1}'(s)\varpi _{2}'(t)\left ( \varpi _{1}( \varsigma _{1})-\varpi _{1}(s)\right ) ^{\chi -1} \\ &\quad{}\times \left ( \varpi _{2}( \varsigma _{2})-\varpi _{2}(t)\right ) ^{\vartheta -1} \mathcal{S}(s,t)dtds. \end{aligned} $$

This implies that

$$ \mathcal{W}(\varsigma _{1},\varsigma _{2})\leq \mathcal{A}(\varsigma _{1}, \varsigma _{2})+\mathcal{K}\mathcal{W}(\varsigma _{1},\varsigma _{2}), \,\, \forall (\varsigma _{1},\varsigma _{2})\in D. $$

(2.3)

Through multiple iterations of (2.3), we obtain

$$ \mathcal{W}(\varsigma _{1},\varsigma _{2})\leq \sum _{k=0}^{n-1} \mathcal{K}^{k}\mathcal{A}(\varsigma _{1},\varsigma _{2})+\mathcal{K}^{n} \mathcal{W}(\varsigma _{1},\varsigma _{2}), \,\, \forall (\varsigma _{1}, \varsigma _{2})\in D. $$

(2.4)

To complete the proof of the remaining part of Theorem 2.3, it is necessary to establish the following inequality

$$\begin{aligned} \mathcal{K}^{n}(\varsigma _{1},\varsigma _{2})\leq & \dfrac{\left ( \Gamma (\chi )\Gamma (\vartheta )\mathcal{B}(\varsigma _{1},\varsigma _{2})\right )^{n}}{\Gamma (n\chi )\Gamma (n\vartheta )} \int _{e_{1}}^{\varsigma _{1}}\int _{e_{2}}^{\varsigma _{2}} \varpi _{1}'(s) \varpi _{2}'(t) \\ & \times \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(s)\right ) ^{n \chi -1}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t)\right ) ^{n \vartheta -1} \mathcal{A}(s,t)dtds. \end{aligned}$$

(2.5)

Clearly, (2.5) holds for $n=1$. Now, we suppose that (2.5) holds for $n=k$. Thus, for $n=k+1$, we obtain

$$\begin{aligned} \mathcal{K}^{k+1}(\varsigma _{1},\varsigma _{2}) =& \mathcal{K} \left ( \mathcal{K}^{k}(\varsigma _{1},\varsigma _{2})\right ) \\ \leq & \dfrac{\left (\Gamma (\chi )\Gamma (\vartheta )\right )^{k}}{\Gamma (k\chi )\Gamma (k\vartheta )} \mathcal{B}(\varsigma _{1},\varsigma _{2}) \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi _{1}'(s)\varpi _{2}'(t) \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(s)\right ) ^{\chi -1} \\ &{}\times \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t)\right ) ^{\vartheta -1} \\ &\times \int _{e_{1}}^{s}\int _{e_{2}}^{t} \mathcal{B}^{k}(s,t) \varpi '_{1}(\tau )\varpi '_{2}(\zeta ) \left ( \varpi _{1}(s)- \varpi _{1}(\tau )\right ) ^{k\chi -1} \left ( \varpi _{2}(t)-\varpi _{2}( \zeta )\right ) ^{k\vartheta -1} \\ &{}\times \mathcal{A}(s,t)d\zeta d\tau dtds \\ \leq & \dfrac{\left (\Gamma (\chi )\Gamma (\vartheta )\right )^{k}}{\Gamma (k\chi )\Gamma (k\vartheta )} \mathcal{B}^{k+1}(\varsigma _{1},\varsigma _{2})\int _{e_{1}}^{ \varsigma _{1}}\int _{e_{2}}^{\varsigma _{2}} \int _{e_{1}}^{s}\int _{e_{2}}^{t} \varpi '_{1}(\tau )\varpi '_{2}(\zeta ) \varpi _{1}'(s)\varpi _{2}'(t) \\ &{}\times \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(s)\right ) ^{\chi -1} \\ &\times \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t)\right ) ^{ \vartheta -1} \left ( \varpi _{1}(s)-\varpi _{1}(\tau )\right ) ^{k \chi -1}\left ( \varpi _{2}(t)-\varpi _{2}(\zeta )\right ) ^{k \vartheta -1} \\ &{}\times \mathcal{A}(s,t)d\zeta d\tau dtds \\ \leq & \dfrac{\left (\Gamma (\chi )\Gamma (\vartheta )\right )^{k}}{\Gamma (k\chi )\Gamma (k\vartheta )} \mathcal{B}^{k+1}(\varsigma _{1},\varsigma _{2}) \int _{e_{1}}^{ \varsigma _{1}}\int _{e_{2}}^{\varsigma _{2}} \int _{\tau}^{ \varsigma _{1}}\int _{\zeta}^{\varsigma _{2}} \varpi '_{1}(\tau ) \varpi '_{2}(\zeta ) \varpi _{1}'(s)\varpi _{2}'(t) \\ &{}\times \left ( \varpi _{1}( \varsigma _{1})-\varpi _{1}(s)\right ) ^{\chi -1} \\ & \times \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t)\right ) ^{ \vartheta -1} \left ( \varpi _{1}(s)-\varpi _{1}(\tau )\right ) ^{k \chi -1}\left ( \varpi _{2}(t)-\varpi _{2}(\zeta )\right ) ^{k \vartheta -1} \\ &{}\times \mathcal{A}(s,t)dtds d\zeta d\tau \\ \leq & \dfrac{\left (\Gamma (\chi )\Gamma (\vartheta )\right )^{k}}{\Gamma (k\chi )\Gamma (k\vartheta )} \mathcal{B}^{k+1}(\varsigma _{1},\varsigma _{2}) \int _{e_{1}}^{ \varsigma _{1}}\int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(\tau ) \varpi '_{2}(\zeta ) \\ &\times \left (\int _{\tau}^{\varsigma _{1}}\varpi _{1}'(s) \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(s)\right ) ^{\chi -1}\left ( \varpi _{1}(s)-\varpi _{1}(\tau )\right ) ^{k\chi -1}ds\right ) \\ &\times \left ( \int _{\zeta}^{\varsigma _{2}} \varpi _{2}'(t) \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t)\right ) ^{ \vartheta -1} \left ( \varpi _{2}(t)-\varpi _{2}(\zeta )\right ) ^{k \vartheta -1} dt\right ) \\ &{}\times \mathcal{A}(s,t) d\zeta d\tau . \end{aligned}$$

(2.6)

Moreover, using the fact

$$\begin{aligned}& \int _{\tau}^{\varsigma _{1}}\varpi _{1}'(s) \left ( \varpi _{1}( \varsigma _{1})-\varpi _{1}(s)\right ) ^{\chi -1}\left ( \varpi _{1}(s)- \varpi _{1}(\tau )\right ) ^{k\chi -1}ds \\& \quad = \dfrac{\left (\Gamma (\chi )\Gamma (k\chi ) \right )^{k}}{\Gamma ((k+1)\chi )} \left (\varpi _{1}(\varsigma _{1})-\varpi _{1}(\tau ) \right )^{(k+1) \chi - 1}, \end{aligned}$$

and from inequality (2.6), (2.5) holds for $n=k+1$, and this proves (2.5) for all $n\geq 1$. Moreover, we have

$$\begin{aligned} \mathcal{K}^{n}(\varsigma _{1},\varsigma _{2})& \leq M^{n} \dfrac{\left ( \Gamma (\chi )\Gamma (\vartheta )\right )^{n}}{\Gamma (n\chi )\Gamma (n\vartheta )} \int _{e_{1}}^{\varsigma _{1}}\int _{e_{2}}^{\varsigma _{2}} \varpi _{1}'(s) \varpi _{2}'(t) \\ &\quad{} \times \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(s)\right ) ^{n \chi -1}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t)\right ) ^{n \vartheta -1} \mathcal{A}(s,t)dtds. \end{aligned}$$

Thus, we see that $\mathcal{K}^{n}(\varsigma _{1},\varsigma _{2}) \rightarrow 0$ as $n\rightarrow \infty$. Consequently, the inequality (2.2) is shown. □

Corollary 2.4

Under the hypothesis of Theorem 2.3, if $\mathcal{A}(\varsigma _{1},\varsigma _{2})$ is a nondecreasing function on D, then

$$\begin{aligned} \mathcal{W}(\varsigma _{1},\varsigma _{2}) & \leq \mathcal{A}(\varsigma _{1}, \varsigma _{2})\mathbb{E}\big( (\chi , 1), (\vartheta , 1); \Gamma ( \chi )\Gamma (\vartheta )\mathcal{B}(\varsigma _{1},\varsigma _{2}) \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1})\right )^{\chi} \\ &\quad{}\times \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2})\right )^{ \vartheta} \big). \end{aligned}$$

Remark 2.5

Theorem 2.3 and Corollary 2.4 are the generalizations of the Gronwall inequalities in [11].

3 Darboux problem for linear partial fractional differential equations

Our goal in this section is to provide a representation of solutions for the linear Darboux problem within the space of continuous functions.

Lemma 3.1

Consider vectors $a=(e_{1}, e_{2})$ and $\chi = (\chi _{1}, \chi _{2})$, and let $\varpi _{1}\in C^{1}\left ( [e_{1}, b_{1}],\mathbb{R}\right )$ and $\varpi _{2} \in C^{1}\left ( [e_{2}, b_{2}],\mathbb{R}\right )$ be two strictly increasing positive functions such that $\varpi '_{1}(\varsigma _{1}),\varpi '_{2}(\varsigma _{2})\neq 0$ for all $\left (\varsigma _{1},\varsigma _{2}\right )\in J$. Let $\varphi \in C^{1}\left ( \left [ e_{1}, b_{1}\right ], \mathbb{R} \right )$, $\psi \in C^{1}\left ( \left [ e_{2}, b_{2}\right ], \mathbb{R}\right )$ and $h \in C\left ( J , \mathbb{R}\right )$. Then, we have:

(i):

$$\begin{aligned}& \lim _{n \rightarrow \infty} \sum \limits _{k=0}^{n-1} (-\lambda )^{k} \left ( I_{a}^{(k+1)\chi , \left ( \varpi _{1}, \varpi _{2}\right ) } \varphi \right ) \left (\varsigma _{1} \right ) \\& = \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2})\right )^{ \chi _{2}} \int _{e_{1}}^{\varsigma _{1}} \varpi _{1}'(s) \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{\chi _{1}-1} \\& \quad \times \mathbb{E}\left ( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{1}(e_{2}) \right )^{\chi _{2}} \right ) \varphi (s)ds, \end{aligned}$$

(ii):

$$\begin{aligned}& \lim _{n \rightarrow \infty} \sum \limits _{k=0}^{n-1} (-\lambda )^{k} \left ( I_{a}^{(k+1)\chi , \left ( \varpi _{1}, \varpi _{2}\right ) } \psi \right ) \left (\varsigma _{2}\right ) \\& = \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(e_{1})\right )^{ \chi _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi _{2}'(t) \left ( \varpi _{2} (\varsigma _{2})-\varpi _{2}(t)\right )^{\chi _{2}-1} \\& \quad \times \mathbb{E}\left ( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \right ) \psi (t)dt, \end{aligned}$$

(iii):

$$\begin{aligned}& \lim _{n \rightarrow \infty} \sum \limits _{k=0}^{n-1} (-\lambda )^{k} \left ( I_{a}^{(k+1)\chi , \left ( \varpi _{1}, \varpi _{2}\right ) } \left ( 1\right ) \right ) \\& = \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(e_{1})\right )^{ \chi _{1}} \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \\& \quad \times \mathbb{E}\left ( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \right ), \end{aligned}$$

and

(iv):

$$\begin{aligned}& \lim _{n \rightarrow \infty} \sum \limits _{k=0}^{n-1} (-\lambda )^{k} \left ( I_{a}^{(k+1)\chi , \left ( \varpi _{1}, \varpi _{2}\right ) } h \right ) \left (\varsigma _{1},\varsigma _{2}\right ) \\& = \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right )^{\chi _{2}-1} \\& \quad \times \mathbb{E}\left ( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \right ) h(s,t)dtds. \end{aligned}$$

Proof

(i) We have

$$\begin{aligned}& \lim _{n \rightarrow \infty} \sum \limits _{k=0}^{n-1} (-\lambda )^{k} \left ( I_{a}^{(k+1)\chi , \left ( \varpi _{1}, \varpi _{2} \right ) } \varphi \right ) \left (\varsigma _{1} \right ) \\& =\lim _{n \rightarrow \infty} \sum \limits _{k=0}^{n-1} (-\lambda )^{k} \int _{e_{1}}^{\varsigma _{1}}\int _{e_{2}}^{\varsigma _{2}} \dfrac{1}{\Gamma ( (k+1)\chi _{1}) \Gamma ( (k+1)\chi _{2})} \varpi _{1}'(s) \varpi _{2}'(t) \\& \quad \times \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(s) \right ) ^{(k+1)\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right ) ^{(k+1)\chi _{2}-1} \varphi (s)dtds \\& =\lim _{n \rightarrow \infty} \sum \limits _{k=0}^{n-1} (-\lambda )^{k} \int _{e_{2}}^{\varsigma _{2}} \dfrac{1}{ \Gamma ( (k+1)\chi _{2})} \varpi _{2}'(t) \left ( \varpi _{1}(\varsigma _{2})-\varpi _{1}(t) \right ) ^{(k+1)\chi _{2}-1} dt \\& \quad \times \int _{e_{1}}^{\varsigma _{1}} \dfrac{1}{\Gamma ( (k+1)\chi _{1}) } \varpi _{1}'(s)\left ( \varpi _{1}( \varsigma _{1})-\varpi _{1}(s)\right ) ^{(k+1)\chi _{1}-1} \varphi (s)ds \\& =\lim _{n \rightarrow \infty} \sum \limits _{k=0}^{n-1} (-\lambda )^{k} \dfrac{1}{ \Gamma ( (k+1)\chi _{2}+1)} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(e_{2})\right ) ^{(k+1)\chi _{2}} \\& \quad \times \int _{e_{1}}^{\varsigma _{1}} \dfrac{1}{\Gamma ( (k+1)\chi _{1}) } \varpi _{1}'(s)\left ( \varpi _{1}( \varsigma _{1})-\varpi _{1}(s)\right ) ^{(k+1)\chi _{1}-1} \varphi (s)ds \\& =\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2})\right ) ^{ \chi _{2}} \int _{e_{1}}^{\varsigma _{1}} \varpi _{1}'(s)\left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(s)\right ) ^{\chi _{1}-1} \\& \quad {}\times \lim _{n \rightarrow \infty} \sum \limits _{k=0}^{n-1} \dfrac{1}{\Gamma ( k\chi _{1}+\chi _{1}) \Gamma ( k\chi _{2}+\chi _{2}+1)} \\& \quad \times \left ( -\lambda \left ( \varpi _{1}(\varsigma _{1})- \varpi _{1}(s)\right ) ^{\chi _{1}} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(e_{2})\right ) ^{\chi _{2}} \right ) ^{k}\varphi (s)ds \\& = \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2})\right )^{ \chi _{2}} \int _{e_{1}}^{\varsigma _{1}} \varpi _{1}'(s) \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{\chi _{1}-1} \\& \quad \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \varphi (s)ds. \end{aligned}$$

Similarly, we can establish (ii), (iii), and (iv), and this completes the proof.  □

Lemma 3.2

Let $a=(e_{1}, e_{2})$ and $\chi = (\chi _{1}, \chi _{2})$. Consider $\varpi _{1}\in C^{1}\left ( [e_{1}, b_{1}],\mathbb{R}\right )$ and $\varpi _{2} \in C^{1}\left ( [e_{2}, b_{2}],\mathbb{R}\right )$ two strictly increasing, positive functions with $\varpi '_{1}(\varsigma _{1}),\varpi '_{2}(\varsigma _{2})\neq 0$ for all $\left (\varsigma _{1},\varsigma _{2}\right )\in J$. Let $\varphi \in C^{1}\left ( [e_{1}, b_{1}], \mathbb{R}\right )$, $\psi \in C^{1}\left ( [e_{2}, b_{2}], \mathbb{R}\right )$, and $h \in C\left ( J , \mathbb{R}\right )$. Then, the solution to the system

$$ \left \lbrace \textstyle\begin{array}{l@{\quad}l} ^{C}D_{a}^{\chi , \left ( \varpi _{1},\varpi _{2}\right ) } \mathcal{U} \left (\varsigma _{1},\varsigma _{2}\right ) + \lambda \mathcal{U}\left (\varsigma _{1},\varsigma _{2}\right )= h\left ( \varsigma _{1},\varsigma _{2}\right ), & \left (\varsigma _{1}, \varsigma _{2}\right ) \in [e_{1}, b_{1}]\times [e_{2}, b_{2}], \vspace{0.3cm} \\ \mathcal{U}(\varsigma _{1},e_{2})=\varphi (\varsigma _{1}), & \varsigma _{1}\in [e_{1}, b_{1}], \vspace{0.3cm} \\ \mathcal{U}(e_{1},\varsigma _{2})=\psi (\varsigma _{2}), & \varsigma _{2} \in [e_{2}, b_{2}], \vspace{0.3cm} \\ \varphi (e_{1})=\psi (e_{2}), \end{array}\displaystyle \right . $$

(3.1)

is given by

$$\begin{aligned} \mathcal{U}(\varsigma _{1},\varsigma _{2}) =& \varphi (\varsigma _{1})+ \psi (\varsigma _{2})-\varphi (e_{1})-\lambda \left ( \varpi _{2}( \varsigma _{2})-\varpi _{2}(e_{2})\right )^{\chi _{2}} \\ &{}\times \int _{e_{1}}^{ \varsigma _{1}} \varpi '_{1}(s) \left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \\ &{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{ \chi _{1}} \\ &{}\times\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \varphi (s)ds \\ &- \lambda \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{2}(t) \left ( \varpi _{2} (\varsigma _{2})-\varpi _{2}(t)\right )^{\chi _{2}-1} \\ &{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}} \\ &{}\times \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) \psi (t)dt \\ &{}+ \lambda \varphi (e_{1}) \left ( \varpi _{1}(\varsigma _{1})- \varpi _{1}(e_{1})\right )^{\chi _{1}} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(e_{2})\right )^{\chi _{2}} \\ &{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \\ & + \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right )^{\chi _{2}-1} \\ & \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{ \chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t)\right )^{ \chi _{2}} \big) h(s,t)dtds. \end{aligned}$$

Proof

Problem (3.1) is equivalent to the following integral equation (see [16, Lemma 10])

$$\begin{aligned} \mathcal{U}(\varsigma _{1},\varsigma _{2}) &= \varphi (\varsigma _{1})+ \psi (\varsigma _{2})-\varphi (e_{1}) - \lambda I_{a}^{\chi , \left ( \varpi _{1}, \varpi _{2}\right ) }\mathcal{U} \left (\varsigma _{1}, \varsigma _{2}\right ) \\ &\quad{} +I_{a}^{\chi , \left ( \varpi _{1}, \varpi _{2} \right ) }h \left (\varsigma _{1},\varsigma _{2}\right ),\ \left ( \varsigma _{1},\varsigma _{2}\right )\in J. \end{aligned}$$

(3.2)

Using the Picard method, we can find the solution by considering the following provided approximate formula

$$\begin{aligned}& \begin{aligned} \mathcal{U}_{n+1}(\varsigma _{1},\varsigma _{2}) &= \varphi ( \varsigma _{1})+\psi (\varsigma _{2})-\varphi (e_{1}) - \lambda I_{a}^{ \chi , \left ( \varpi _{1}, \varpi _{2}\right )}\mathcal{U}_{n} \left (\varsigma _{1},\varsigma _{2}\right ) \\ &\quad{} +I_{a}^{\chi , \left ( \varpi _{1}, \varpi _{2}\right )}h \left (\varsigma _{1},\varsigma _{2} \right ), \qquad n \geq 0, \end{aligned} \\& \mathcal{U}_{0}(\varsigma _{1},\varsigma _{2}) = \varphi (\varsigma _{1})+ \psi (\varsigma _{2})-\varphi (e_{1}). \end{aligned}$$

Then, we obtain

$$\begin{aligned}& \begin{aligned} \mathcal{U}_{1}(\varsigma _{1},\varsigma _{2}) &= \varphi (\varsigma _{1})+ \psi (\varsigma _{2})-\varphi (e_{1}) - \lambda I_{a}^{\chi , \left ( \varpi _{1}, \varpi _{2}\right )} \left ( \varphi (\varsigma _{1})+ \psi (\varsigma _{2})-\varphi (e_{1})\right ) \\ &\quad{} +I_{a}^{\chi , \left ( \varpi _{1}, \varpi _{2}\right )}h\left (\varsigma _{1},\varsigma _{2} \right ) \vspace{0.3cm} \\ &= \varphi (\varsigma _{1})+\psi (\varsigma _{2})-\varphi (e_{1}) - \lambda I_{a}^{\chi , \left ( \varpi _{1}, \varpi _{2}\right )} \varphi (\varsigma _{1}) - \lambda I_{a}^{\chi , \left ( \varpi _{1}, \varpi _{2}\right )} \psi (\varsigma _{2}) \vspace{0.3cm} \\ &\quad{} + \lambda \varphi (e_{1}) I_{a}^{\chi , \left ( \varpi _{1}, \varpi _{2} \right )} (1) +I_{a}^{\chi , \left ( \varpi _{1}, \varpi _{2}\right )}h \left (\varsigma _{1},\varsigma _{2}\right ), \end{aligned} \\& \begin{aligned} \mathcal{U}_{2}(\varsigma _{1},\varsigma _{2}) &= \displaystyle{ \varphi (\varsigma _{1})+\psi (\varsigma _{2})-\varphi (e_{1}) + \sum _{k=0}^{1} (-\lambda )^{k+1} I_{a}^{(k+1)\chi , \left ( \varpi _{1}, \varpi _{2}\right )} \varphi (\varsigma _{1})} \\ &\quad{} + \sum _{k=0}^{1} (- \lambda )^{k+1} I_{a}^{(k+1)\chi , \left ( \varpi _{1}, \varpi _{2} \right )} \psi (\varsigma _{2}) \\ &\quad{} - \displaystyle{\varphi (e_{1}) \sum _{k=0}^{1} (-\lambda )^{k+1} I_{a}^{(k+1) \chi , \left ( \varpi _{1}, \varpi _{2}\right )} (1) + \sum _{k=0}^{1} (-\lambda )^{k} I_{a}^{(k+1)\chi , \left ( \varpi _{1}, \varpi _{2} \right )} h \left (\varsigma _{1},\varsigma _{2}\right )}, \end{aligned} \end{aligned}$$

and

$$ \begin{aligned} \mathcal{U}_{n}(\varsigma _{1},\varsigma _{2}) &= \displaystyle{ \varphi (\varsigma _{1})+\psi (\varsigma _{2})-\varphi (e_{1}) + \sum _{k=0}^{n-1} (-\lambda )^{k+1} I_{a}^{(k+1)\chi , \left ( \varpi _{1}, \varpi _{2}\right )} \varphi (\varsigma _{1})} \\ &\quad{} + \sum _{k=0}^{n-1} (-\lambda )^{k+1} I_{a}^{(k+1)\chi , \left ( \varpi _{1}, \varpi _{2} \right ) } \psi (\varsigma _{2}) \\ &\quad{} - \displaystyle{\varphi (e_{1}) \sum _{k=0}^{n-1} (-\lambda )^{k+1} I_{a}^{(k+1) \chi , \left ( \varpi _{1}, \varpi _{2}\right )} (1) + \sum _{k=0}^{n-1} (-\lambda )^{k} I_{a}^{(k+1)\chi , \left ( \varpi _{1}, \varpi _{2} \right )} h \left (\varsigma _{1},\varsigma _{2}\right )}. \end{aligned} $$

From Lemma 3.1 and letting $n\rightarrow \infty$, we obtain

$$\begin{aligned} \mathcal{U}(\varsigma _{1},\varsigma _{2})&= \varphi (\varsigma _{1})+ \psi (\varsigma _{2})-\varphi (e_{1}) \\ &\quad{} -\lambda \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \int _{e_{1}}^{\varsigma _{1}} \varpi '_{1}(s) \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{\chi _{1}-1} \\ &\quad{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{ \chi _{1}} \\ &\quad {}\times \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \varphi (s)ds \\ &\quad{} - \lambda \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{2}(t) \left ( \varpi _{2} (\varsigma _{2})-\varpi _{2}(t)\right )^{\chi _{2}-1} \\ &\quad{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}} \\ &\quad {}\times\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) \psi (t)dt \\ &\quad{} + \lambda \varphi (e_{1}) \left ( \varpi _{1}(\varsigma _{1})- \varpi _{1}(e_{1})\right )^{\chi _{1}} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(e_{2})\right )^{\chi _{2}} \\ &\quad{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \\ &\quad{} + \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right )^{\chi _{2}-1} \\ &\quad{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{ \chi _{1}} \\ &\quad {}\times\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t)\right )^{ \chi _{2}} \big) h(s,t)dtds, \end{aligned}$$

and the proof is completed. □

4 Ulam–Hyers–Rassias Mittag-Leffler stability result

In this section, we explore the stability of equation (1.1) in the sense of the Ulam–Hyers–Rassias Mittag–Leffler theory, focusing on the interval J. To delve into this analysis, we assume that ϵ and λ are positive constants, with F belonging to the class $C\left (J \times \mathbb{R}, \mathbb{R}\right )$ and $\Phi \in C\left (J, \mathbb{R}_{+}\right )$. Our investigation encompasses both the equation (1.1) itself and the corresponding inequality:

where $\epsilon >0$. For $(\varsigma _{1},\varsigma _{2})\in J$, and $C_{F}>0$, let us consider

$$ \mathcal{H}=\mathbb{E}\left ( (\chi _{1}, 1), (\chi _{2}, 1); C_{F} \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1})\right )^{ \chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \right ). $$

Definition 4.1

Equation (1.1) is said to be Ulam–Hyers–Rassias Mittag–Leffler stable with respect to Φ (UHRML-Φ stability) if there exists $k>0$ such that for every $\epsilon >0$ and for every solution $\mathcal{V}$ of (4.1) such that , there exists a solution $\mathcal{U} \in C\left ( J, \mathbb{R}\right )$ of (1.1) with

$$ \left | \mathcal{U}(\varsigma _{1},\varsigma _{2})-\mathcal{V}( \varsigma _{1},\varsigma _{2})\right | \leq \epsilon k \Phi ( \varsigma _{1},\varsigma _{2})\mathcal{H}(\varsigma _{1},\varsigma _{2}), \qquad (\varsigma _{1},\varsigma _{2} )\in J. $$

(4.1)

Remark 4.1

A function $\mathcal{V} \in C\left ( J, \mathbb{R} \right )$ is a solution of (4.1) if and only if there exists a function $\mathcal{G} \in C\left ( J, \mathbb{R} \right )$ such that for all $(\varsigma _{1},\varsigma _{2})\in J$, we have

• (i) $\left | \mathcal{G}(\varsigma _{1},\varsigma _{2})\right | \leq \epsilon \Phi \left (\varsigma _{1},\varsigma _{2}\right )$,

• (ii) .

Lemma 4.2

Let $\mathcal{V} \in C\left (J, \mathbb{R} \right )$ be a solution of (4.1). Then, $\mathcal{V}$ is a solution of the following integral inequality

$$\begin{aligned}& \bigg| \mathcal{V}(\varsigma _{1},\varsigma _{2})-\mathcal{V}( \varsigma _{1},e_{2}) -\mathcal{V}(e_{1},\varsigma _{2})+\mathcal{V}(e_{1},e_{2}) \\& \quad{} + \lambda \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2})\right )^{ \chi _{2}} \int _{e_{1}}^{\varsigma _{1}} \varpi '_{1}(s) \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{\chi _{1}-1} \\& \quad{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \mathcal{V}(s, e_{2})ds \\& \quad{} + \lambda \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{2}(t) \left ( \varpi _{2} (\varsigma _{2})-\varpi _{2}(t)\right )^{\chi _{2}-1} \\& \quad{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) \mathcal{V}(e_{1},t)dt \\& \quad{} - \lambda \mathcal{V}(e_{1},e_{2}) \left ( \varpi _{1}( \varsigma _{1})-\varpi _{1}(e_{1})\right )^{\chi _{1}} \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2})\right )^{\chi _{2}} \\& \quad{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \\& \quad{} - \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right )^{\chi _{2}-1} \\& \quad{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) F(s,t, \mathcal{V}(s,t))dtds \bigg| \\& \leq M\epsilon \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{ \varsigma _{2}}\varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} ( \varsigma _{1})-\varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}( \varsigma _{2})-\varpi _{2}(t)\right )^{\chi _{2}-1} \Phi (s,t)dtds, \end{aligned}$$

where M is a positive constant.

Proof

Let $\mathcal{V} \in C\left (J, \mathbb{R} \right )$ be a solution of (4.1), from Remark 4.1(ii) we have

$$ ^{C}D_{a}^{\chi , \varpi}\mathcal{V} \left (\varsigma _{1},\varsigma _{2} \right ) + \lambda \mathcal{V}\left (\varsigma _{1},\varsigma _{2} \right )= F\left (\varsigma _{1},\varsigma _{2}, \mathcal{V}( \varsigma _{1},\varsigma _{2}) \right )+ \mathcal{G} \left ( \varsigma _{1},\varsigma _{2}\right ). $$

Using Lemma 3.2, we find

$$\begin{aligned} \mathcal{V}(\varsigma _{1},\varsigma _{2}) =& \mathcal{V}(\varsigma _{1},e_{2}) +\mathcal{V}(e_{1},\varsigma _{2}) -\mathcal{V}(e_{1},e_{2}) \\ &{}-\lambda \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2})\right )^{\chi _{2}} \int _{e_{1}}^{\varsigma _{1}} \varpi '_{1}(s) \left ( \varpi _{1} ( \varsigma _{1})-\varpi _{1}(s)\right )^{\chi _{1}-1} \\ &{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \\ &{}\times\mathcal{V}(s, e_{2})ds \\ &{} - \lambda \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{2}(t) \left ( \varpi _{2} (\varsigma _{2})-\varpi _{2}(t)\right )^{\chi _{2}-1} \\ &{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) \\ &{}\times\mathcal{V}(e_{1}, t)dt \\ &{} +\lambda \mathcal{V}( e_{1}, e_{2}) \left ( \varpi _{1}( \varsigma _{1})-\varpi _{1}(e_{1})\right )^{\chi _{1}} \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2})\right )^{\chi _{2}} \\ &{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \\ &{} + \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right )^{\chi _{2}-1} \\ &{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) \\ &{}\times F(s,t, \mathcal{V}(s,t))dtds \\ &{} + \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right )^{\chi _{2}-1} \\ &{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) \\ &{}\times\mathcal{G}(s,t)dtds. \end{aligned}$$

This implies that

$$\begin{aligned}& \bigg| \mathcal{V}(\varsigma _{1},\varsigma _{2})-\mathcal{V}( \varsigma _{1},e_{2}) -\mathcal{V}(e_{1},\varsigma _{2})+\mathcal{V}(e_{1},e_{2}) \\& \quad {} +\lambda \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \int _{e_{1}}^{\varsigma _{1}} \varpi '_{1}(s) \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{\chi _{1}-1} \\& \quad{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \mathcal{V}( s,e_{2} )ds \\& \quad{} + \lambda \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{2}(t) \left ( \varpi _{2} (\varsigma _{2})-\varpi _{2}(t)\right )^{\chi _{2}-1} \\& \quad{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) \mathcal{V}(e_{1},t)dt \\& \quad{} - \lambda \mathcal{V}( e_{1},e_{2} ) \left ( \varpi _{1}( \varsigma _{1})-\varpi _{1}(e_{1})\right )^{\chi _{1}} \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2})\right )^{\chi _{2}} \\& \quad{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \\& \quad{} - \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right )^{\chi _{2}-1} \\& \quad{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) F(s,t, \mathcal{V}(s,t))dtds \bigg| \\& \leq \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right )^{\chi _{2}-1} \\& \quad{} \times \bigg| \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big)\bigg| \bigg| \mathcal{G}(s,t)\bigg| dtds \\& \leq M\epsilon \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{ \varsigma _{2}}\varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} ( \varsigma _{1})-\varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}( \varsigma _{2})-\varpi _{2}(t)\right )^{\chi _{2}-1} \Phi (s,t)dtds, \end{aligned}$$

where the constant M satisfies

$$ \left | \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{ \chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t)\right )^{ \chi _{2}} \big) \right |\leq M. $$

 □

In the following, let us consider the following assumptions:

• (H1) The function F is continuous in each of its variables, and there exists a positive constant $L_{F}$ such that

  $$ \left | F(\varsigma _{1},\varsigma _{2}, \mathcal{U}_{1})-F( \varsigma _{1},\varsigma _{2}, \mathcal{U}_{2}) \right | \leq \dfrac{L_{F}}{\Gamma (\chi _{1})\Gamma (\chi _{2})}\left |\mathcal{U}_{1}- \mathcal{U}_{2} \right |, $$

  for all $(\varsigma _{1},\varsigma _{2})\in J$ and $\mathcal{U}_{1}, \mathcal{U}_{2}\in \mathbb{R}$.

• (H2) The function Φ is nondecreasing in each of its variables.

Remark 4.3

It follows from (H2) that there exists a positive constant C such that

$$\begin{aligned}& \int _{e_{1}}^{\varsigma _{1}}\int _{e_{2}}^{\varsigma _{2}}\varpi '_{1}(s) \varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}-1} \Phi (s,t)dtds \\& \quad \leq C\Phi (\varsigma _{1}, \varsigma _{2}), \end{aligned}$$

for all $(\varsigma _{1},\varsigma _{2}) \in J$.

Theorem 4.4

Assume that (H1) and (H2) hold. Then, (1.1) is Ulam–Hyers–Rassias Mittag–Leffler stable with respect to Φ.

Proof

In order to prove the UHRML-Φ stability of the system (1.1), one must prove that there exists a constant $k>0$ such that for any $\varepsilon >0$ and for any solution $\mathcal{V}$ of (4.1), the unique solution $\mathcal{U}\in C(J,\mathbb{R})$ to the problem

should satisfy the inequality (4.2). Thus, let $\mathcal{V}\in C(J, \mathbb{R})$ be a solution of (4.1), and let $\mathcal{U}$ be the unique solution of (4.3). Then,

$$\begin{aligned} \mathcal{U}(\varsigma _{1},\varsigma _{2}) = & \mathcal{V}( \varsigma _{1},e_{2}) +\mathcal{V}(e_{1},\varsigma _{2}) -\mathcal{V}(e_{1},e_{2}) \\ &{}\times -\lambda \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \int _{e_{1}}^{\varsigma _{1}} \varpi '_{1}(s) \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{\chi _{1}-1} \\ &{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \\ &{}\times \mathcal{V}(s, e_{2})ds \\ &{} - \lambda \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{2}(t) \left ( \varpi _{2} (\varsigma _{2})-\varpi _{2}(t)\right )^{\chi _{2}-1} \\ &{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) \\ &{}\times \mathcal{V}(e_{1}, t)dt \\ &{} +\lambda \mathcal{V}( e_{1}, e_{2}) \left ( \varpi _{1}( \varsigma _{1})-\varpi _{1}(e_{1})\right )^{\chi _{1}} \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2})\right )^{\chi _{2}} \\ &{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \\ &{} + \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right )^{\chi _{2}-1} \\ &{} \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) \\ &{}\times F(s,t,\mathcal{U}(s,t))dtds. \end{aligned}$$

Furthermore, we get

$$\begin{aligned}& \bigg|\mathcal{V}(\varsigma _{1},\varsigma _{2})-\mathcal{U}( \varsigma _{1},\varsigma _{2}) \bigg| \\& \leq \bigg|\mathcal{V}(\varsigma _{1},\varsigma _{2}) - \mathcal{V}( \varsigma _{1},e_{2}) -\mathcal{V}(e_{1},\varsigma _{2}) +\mathcal{V}(e_{1},e_{2}) \\& \quad + \lambda \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \int _{e_{1}}^{\varsigma _{1}} \varpi '_{1}(s) \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{\chi _{1}-1} \\& \quad \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \mathcal{V}(s, e_{2})ds \\& \quad +\lambda \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{2}(t) \left ( \varpi _{2} (\varsigma _{2})-\varpi _{2}(t)\right )^{\chi _{2}-1} \\& \quad \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) \mathcal{V}(e_{1}, t)dt \\& \quad -\lambda \mathcal{V}( e_{1}, e_{2}) \left ( \varpi _{1}( \varsigma _{1})-\varpi _{1}(e_{1})\right )^{\chi _{1}} \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2})\right )^{\chi _{2}} \\& \quad \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \\& \quad - \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right )^{\chi _{2}-1} \\& \quad \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) F(s,t, \mathcal{U}(s,t))dtds\bigg| \\& \quad \leq \bigg|\mathcal{V}(\varsigma _{1},\varsigma _{2}) - \mathcal{V}(\varsigma _{1},e_{2}) -\mathcal{V}(e_{1},\varsigma _{2}) + \mathcal{V}(e_{1},e_{2}) \\& \quad + \lambda \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \int _{e_{1}}^{\varsigma _{1}} \varpi '_{1}(s) \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{\chi _{1}-1} \\& \quad \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \mathcal{V}(s, e_{2})ds \\& \quad +\lambda \left ( \varpi _{1}(\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{2}(t) \left ( \varpi _{2} (\varsigma _{2})-\varpi _{2}(t)\right )^{\chi _{2}-1} \\& \quad \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) \mathcal{V}(e_{1}, t)dt \\& \quad -\lambda \mathcal{V}( e_{1}, e_{2}) \left ( \varpi _{1}( \varsigma _{1})-\varpi _{1}(e_{1})\right )^{\chi _{1}} \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2})\right )^{\chi _{2}} \\& \quad \times \mathbb{E}\big( (\chi _{1}, \chi _{1}+1), (\chi _{2}, \chi _{2}+1); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(e_{1}) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(e_{2}) \right )^{\chi _{2}} \big) \\& \quad - \int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right )^{\chi _{2}-1} \\& \quad \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) F(s,t, \mathcal{V}(s,t))dtds\bigg| \\& \quad +\int _{e_{1}}^{\varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}} \varpi '_{1}(s)\varpi '_{2}(t)\left ( \varpi _{1} (\varsigma _{1})- \varpi _{1}(s)\right )^{\chi _{1}-1} \left ( \varpi _{2}(\varsigma _{2})- \varpi _{2}(t)\right )^{\chi _{2}-1} \\& \quad \times \mathbb{E}\big( (\chi _{1}, \chi _{1}), (\chi _{2}, \chi _{2}); -\lambda \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s) \right )^{\chi _{1}}\left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}} \big) \\& \quad \times \bigg|F(s,t, \mathcal{V}(s,t))-F(s,t, \mathcal{U}(s,t)) \bigg|dtds. \end{aligned}$$

Then, from (H1), Lemma 4.2, and Remark 4.3, we get

$$\begin{aligned} \bigg|\mathcal{V}(\varsigma _{1},\varsigma _{2})-\mathcal{U}( \varsigma _{1},\varsigma _{2}) \bigg| \leq & \epsilon M C\Phi ( \varsigma _{1},\varsigma _{2}) \\ &{}+ \dfrac{ML_{F}}{\Gamma (\chi _{1})\Gamma (\chi _{2})} \int _{e_{1}}^{ \varsigma _{1}} \int _{e_{2}}^{\varsigma _{2}}\varpi '_{1}(s)\varpi '_{2}(t) \left ( \varpi _{1} (\varsigma _{1})-\varpi _{1}(s)\right )^{\chi _{1}-1} \\ &{} \times \left ( \varpi _{2}(\varsigma _{2})-\varpi _{2}(t) \right )^{\chi _{2}-1} \big| \mathcal{V}(s,t)-\mathcal{U}(s,t)\big|dtds. \end{aligned}$$

It follows from [11, Lemma 2] that

$$ \bigg|\mathcal{V}(\varsigma _{1},\varsigma _{2})-\mathcal{U}( \varsigma _{1},\varsigma _{2}) \bigg|\leq \epsilon k \Phi (\varsigma _{1}, \varsigma _{2})\mathcal{H}(\varsigma _{1},\varsigma _{2}), $$

where $C_{F}= ML_{F}$ and $k= MC$. Thus, (1.1) is Ulam–Hyers–Rassias Mittag–Leffler stable with respect to Φ. □

5 Illustrative examples

In this section, we present numerical examples that substantiate our theoretical results. Following that, we delve into examining the stability of the next system with respect to a function Φ, a concept we refer to as UHRML-Φ stability.

$$ \left \lbrace \textstyle\begin{array}{l@{\quad}l} ^{C}D_{a}^{\chi , \left ( \varpi _{1},\varpi _{2}\right ) } \mathcal{V} \left (\varsigma _{1},\varsigma _{2}\right ) + \lambda \mathcal{V}\left (\varsigma _{1},\varsigma _{2}\right ) \\ \quad = F\left ( \varsigma _{1},\varsigma _{2}, \mathcal{V}(\varsigma _{1},\varsigma _{2}) \right ) + \mathcal{G}(\varsigma _{1},\varsigma _{2}), & \left ( \varsigma _{1},\varsigma _{2}\right ) \in [e_{1}, b_{1}]\times [e_{2}, b_{2}], \vspace{0.3cm} \\ \mathcal{V}(\varsigma _{1},e_{2})=\varphi (\varsigma _{1}), & \varsigma _{1}\in [e_{1}, b_{1}], \vspace{0.3cm} \\ \mathcal{V}(e_{1},\varsigma _{2})=\psi (\varsigma _{2}), & \varsigma _{2} \in [e_{2}, b_{2}], \vspace{0.3cm} \\ \varphi (e_{1})=\psi (e_{2}). \end{array}\displaystyle \right . $$

(5.1)

5.1 Example 1

We consider the system (5.1) with the following parameters: $[e_{1},b_{1}]=[e_{2},b_{2}]=[0,1]$, $\chi _{1}=0.6$, $\chi _{2}=0.3$, $\varpi _{1}=\varpi _{2}=id$, $\lambda = 0$, $F(\cdot ,\cdot ,t) = \sin t$, $\mathcal{G}\equiv \epsilon$, with $\epsilon >0$, $\varphi (t) = t(1-t)$, $\psi (t) = \ln (1+t)$ and $\Phi \equiv 1$. According to Theorem 4.4, the solution is UHRML-Φ stable. Figure 1 shows the residue $R := \epsilon k \Phi \mathcal{H} - |\mathcal{U}-\mathcal{V}|$, where $\mathcal{U}$ is the solution of (5.1) with $\mathcal{G}\equiv 0$, $\mathcal{H}$ is the function appearing in the Definition 4.1, and $k=\frac{1}{\Gamma (1+\chi _{1})\Gamma (1+\chi _{2})}$. One can notice that this residue is nonnegative, and subsequently the solution $\mathcal{V}$ is UHRML-Φ stable for all the values of ϵ. In Table 1, we report the value $\text{argmin}_{\varsigma _{1},\varsigma _{2}}\{\epsilon k \Phi \mathcal{H} - |\mathcal{U}-\mathcal{V}|\}$ for various parameters ϵ and χ. One can remark that this misfit criterion remains nonnegative for all the tests performed, which confirms the UHRML-Φ stability of the system (5.1).

Figure 1

The residue $R(\varsigma _{1},\varsigma _{2}):=\epsilon k \Phi (\varsigma _{1}, \varsigma _{2}) \mathcal{H}(\varsigma _{1},\varsigma _{2}) - | \mathcal{U}(\varsigma _{1},\varsigma _{2})-\mathcal{V}(\varsigma _{1}, \varsigma _{2})|$ for various values of ϵ

Table 1 $\text{argmin}_{\varsigma _{1},\varsigma _{2}}\{\epsilon k \Phi \mathcal{H} - |\mathcal{U}-\mathcal{V}|\}$ for various parameters ϵ and χ

5.2 Example 2

We perform the same tests as in Example 1, but for several values of λ. In Table 2, Table 3, and Table 4, we report the value $\text{argmin}_{\varsigma _{1},\varsigma _{2}}\{\epsilon k \Phi \mathcal{H} - |\mathcal{U}-\mathcal{V}|\}$ for various parameters ϵ and χ. Once again, one can notice that the misfit criterion remains nonnegative for all the tests performed, which is in total agreement with our theoretical study.

Table 2 $\text{argmin}_{\varsigma _{1},\varsigma _{2}}\{\epsilon k \Phi \mathcal{H} - |\mathcal{U}-\mathcal{V}|\}$ for various parameters ϵ and χ. Case $\lambda = 1$

Table 3 $\text{argmin}_{\varsigma _{1},\varsigma _{2}}\{\epsilon k \Phi \mathcal{H} - |\mathcal{U}-\mathcal{V}|\}$ for various parameters ϵ and χ. Case $\lambda = 10$

Table 4 $\text{argmin}_{\varsigma _{1},\varsigma _{2}}\{\epsilon k \Phi \mathcal{H} - |\mathcal{U}-\mathcal{V}|\}$ for various parameters ϵ and χ. Case $\lambda = -1$

5.3 Example 3

In this example, we consider the system (5.1) with the following parameters: $[e_{1},b_{1}]=[e_{2},b_{2}]=[0,1]$, $\chi _{1}=0.7$, $\chi _{2}=0.4$, $\varpi _{1}(t)=e^{t}-1$, $\varpi _{2}(t)=\sqrt{t}$, $\lambda = 2$, $F(\varsigma _{1},\varsigma _{2},t) = \frac{\varsigma _{1}-\varsigma _{2}}{1+t^{2}}$, $\mathcal{G}\equiv \epsilon$, with $\epsilon >0$, $\varphi (t) = \sin (\pi t)$, $\psi (t) = 1-\cos (\pi t)$ and $\Phi \equiv 1$. In Fig. 2, we plotted the norm $|\mathcal{U}(\varsigma _{1},\varsigma _{2})-\mathcal{V}(\varsigma _{1}, \varsigma _{2})|$ for different values of ϵ. In view of the choice of the function $\mathcal{G}$ and according to Theorem 4.4 and Definition 4.1, we must find that $|\mathcal{U}(\varsigma _{1},\varsigma _{2})-\mathcal{V}(\varsigma _{1}, \varsigma _{2})|$ is bounded by a constant that depends on ϵ. One can notice that the numerical solutions we found agree with the expected results since all the trajectories remain below a constant that goes to zero when ϵ tends to zero.

Figure 2

$|\mathcal{U}(\varsigma _{1},\varsigma _{2})-\mathcal{V}(\varsigma _{1}, \varsigma _{2})|$ for various values of ϵ

6 Conclusion

In this study, we explored solution representations and demonstrated Ulam–Hyers–Rassias Mittag–Leffler stability for the Darboux problem using the generalized Gronwall inequality. A representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions is presented. Looking forward, we plan to deepen our understanding by exploring alternative mathematical approaches and validating our findings through numerical simulations. Combining these results with those obtained in the literature, we can extend this work to the stochastic concept.