Ulam–Hyers–Rassias Stability of ψ-Hilfer Volterra Integro-Differential Equations of Fractional Order Containing Multiple Variable Delays

Abstract

The authors consider a nonlinear $\psi$-Hilfer fractional-order Volterra integro-differential equation ($\psi$-Hilfer FrOVIDE) that incorporates N-multiple variable time delays into the equation. By utilizing the $\psi$-Hilfer fractional derivative, they investigate the Ulam–Hyers–Rassias and Ulam–Hyers stability of the equation by using fixed-point methods. Their results improve existing ones both with and without delays by extending them to nonlinear $\psi$-Hilfer FrOVIDEs that incorporate N-multiple variable time delays.

1. Introduction

In recent decades, fractional calculus has attracted increased attention in the scientific community and plays a vital role in various fields, including engineering and medicine, owing to its importance in both theoretical and physical applications. In this regard, we refer the reader to [1,2,3,4]. The numerous definitions of fractional derivatives and integrals that have been developed in recent years demonstrate the growing interest of researchers in this continuously expanding field (see [3,5,6]). The investigation of Ulam–Hyers (U-H) and Ulam–Hyers–Rassias (U-H-R) stability has become a focal point for a large number of researchers, and this area of study has evolved into one of central importance in scientific analysis; see, for example, [7,8,9].

Recently, significant attention has been devoted to the study of U-H and U-H-R stability in the context of $\psi$-Hilfer fractional-order Volterra integro-differential equations (FrOVIDEs) both with and without delays (see [10,11,12,13] as examples).

As a brief review of some work devoted to the U-H and the U-H-R stability of $\psi$-Hilfer FrOVIDEs, we wish to mention that Sousa and Oliveira [6] obtained results involving convergent sequences of functions. In 2019, motivated by [6], Luo, Shah, and Luo [11] considered the $\psi$-Hilfer FrOVIDE

$$\left\{\begin{array}{c}{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\vartheta \left(s\right)=A\vartheta \left(s\right)+B\vartheta (s-\tau \left(s\right))+f\left(s,\vartheta \left(s\right),\vartheta (s-\tau (s\left)\right)\right),\hfill \\ \vartheta \left(s\right)=\varphi \left(s\right),s\in [-h,0],\hfill \end{array}\right.$$

(1)

and, using a generalized Gronwall inequality, they obtained results on the H-U stability of (1). In a notable paper, Sousa and Oliveira [12] investigated the U-H-R and U-H stability of the $\psi$-Hilfer FrOVIDE problem

$$\left\{\begin{array}{c}{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\mu \left(t\right)=f(t,\mu \left(t\right))+\underset{0}{\overset{t}{\int }}k(t,s,\mu (t\left)\right)\mathrm{d}s,\hfill \\ I_{0^{+}}^{1-\gamma }\mu \left(0\right)=\sigma ,\hfill \end{array}\right.$$

(2)

via fixed-point methods.

In [10], Sousa and Oliveira studied the time-varying delay Hilfer FrOVIDE

$$\left\{\begin{array}{c}{}^{H}D_{a^{+}}^{\alpha ,\beta ;\psi }\nu \left(x\right)=f\left(x,\nu \left(x\right),\underset{0}{\overset{x}{\int }}K\left(x,\tau ,\nu \left(\tau \right),\nu \left(\delta \right(\tau \left)\right)\right)\mathrm{d}s\right),\hfill \\ I_{0^{+}}^{1-\gamma }\nu \left(a\right)=c.\hfill \end{array}\right.$$

(3)

Applying the contraction mapping theorem, they investigated the U-H, the U-H-R, and semi-U-H-R stability of (3) in the intervals $[a,b]$ and $[a,\infty )$, respectively.

Influenced by the works referenced above, in this paper, we consider the very general $\psi$-Hilfer FrOVIDE with N-time-varying delays

$$\left\{\begin{array}{c}{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\vartheta \left(t\right)=h\left(\vartheta \left(t\right)\right)+f(t,\vartheta \left(t\right))\hfill \\ +\underset{0}{\overset{t}{\int }}\left(g\left(t,\vartheta \left(t\right)\right){\displaystyle \sum _{i=1}^N}{K}_i\left(t,s,\vartheta \left(s\right),\vartheta (s-{\tau }_i\left(s\right))\right)\right)\mathrm{d}s,\hfill & t\in (0,T],\hfill \\ I_{0^{+}}^{1-\gamma }\vartheta \left(0\right)=\sigma ,\hfill \\ \vartheta \left(t\right)=\theta \left(t\right),\hfill & t\in [-\tau ,0].\hfill \end{array}\right.$$

(4)

Here, $t\in I=[0,T]$, $\vartheta \in C\left(\mathbb{R}\right)$, ${}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }(·)$ is the right-hand Hilfer fractional derivative, $\alpha \in (0,1)$, $\beta \in [0,1]$, $I_{0^{+}}^{1-\gamma }(·)$ is a $\psi$-Riemann–Liouville fractional integral, $\sigma \in \mathbb{R}$, $0\le \gamma <1$, and ${\tau }_i\in C(I,{\mathbb{R}}^{+})$ with $0\le {\tau }_i\left(t\right)\le {\tau }_i<\infty$ for $0<{\tau }_i\in \mathbb{R}$, $i=1,2,\dots ,N$. Moreover, $\tau =max\left\{{\tau }_i\right\}$, $\theta \in C\left([-\tau ,0],\mathbb{R}\right)$, $h\in C\left(\mathbb{R},\mathbb{R}\right)$, $f,g\in C\left(I\times \mathbb{R},\mathbb{R}\right)$, and $K_i\in C\left(I\times I\times \mathbb{R}\times \mathbb{R},\mathbb{R}\right)$, $i=1,2,\dots ,N$. Problem (4) can be thought of as a Hammerstein-type $\psi$-Hilfer FrOVIDE.

One primary goal here is to investigate the U-H-R and U-H stability of the $\psi$-Hilfer FrOVIDE (4) by applying fixed-point techniques. As far as we are aware, existing research does not include any studies on the U-H-R and the U-H stability of $\psi$- Hilfer FrOVIDEs with N-time-varying delays using fixed-point methods. Clearly, the $\psi$-Hilfer FrOVIDE (4) is considerably different from those in the references mentioned above and in the other related works referenced in this paper. This highlights the key novelty and significant contribution that this paper makes to the literature. Studying, for the first time, the U-H-R and U-H stability of $\psi$-Hammerstein-type Hilfer FrOVIDE (4) with the incorporation of N-time-varying delays provides a new and meaningful contribution to the qualitative theory of Hammerstein-type Hilfer FrOVIDEs as well. This underscores the second major contribution that this paper makes to the current literature.

The structure of our paper is as follows. Section 2 contains key notions needed in our study. In Section 3, we prove two new results on the U-H-R and U-H stability of our $\psi$-Hilfer FrOVIDE with N-multiple variable time delays.

2. Preliminaries

Let $0<\alpha <1$, $I=[0,T]$ with $T\in \mathbb{R}$ and $T>0$ or $I=[0,\infty )$. Let $f:I\to \mathbb{R}$ be an integrable function and $\psi \in C^1([0,T],\mathbb{R})$ be an increasing function with ${\psi }^{\prime }\left(t\right)\ne 0$ for $t\in I$. Then, the right-hand $\psi$-Hilfer-FrD is given by

$${}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }f\left(t\right)=I_{0^{+}}^{\beta (1-\alpha );\psi }\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)I_{0^{+}}^{(1-\beta )(1-\alpha );\psi }f\left(t\right)$$

(see [12]).

Definition 1.

If, for each function $\vartheta \in C^1\left(I,\mathbb{R}\right)$ satisfying

$$\begin{array}{c}|{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\vartheta \left(t\right)-h\left(\vartheta \left(t\right)\right)-f(t,\vartheta \left(t\right))\hfill \\ \hfill -\underset{0}{\overset{t}{\int }}\left(g\left(t,\vartheta \left(t\right)\right){\displaystyle \sum _{i=1}^N}{K}_i\left(t,s,\vartheta \left(s\right),\vartheta (s-{\tau }_i\left(s\right))\right)\right)\mathrm{d}s|\le \Phi \left(t\right),\end{array}$$

where $\Phi :I\to \mathbb{R}$ with $\Phi \left(t\right)\ge 0$ for $t\in \mathrm{I}$, there is a solution ${\vartheta }_0\left(t\right)$ of the ψ-Hilfer FrOVIDE (4) and a constant $C\in {\mathbb{R}}_{+}$ that does not depend on $\vartheta \left(t\right)$ or ${\vartheta }_0\left(t\right)$, such that, if

$$\left|\vartheta \left(t\right)-{\vartheta }_0\left(t\right)\right|\le C\Phi \left(t\right),$$

then we say that (4) is U-H-R stable. If this holds with $\Phi \left(t\right)$ as a constant function, then (4) is said to be U-H stable.

The following result, which is based on the theorem in [14], will prove to be important in obtaining our main stability results.

Lemma 1

(Theorem 2.1 in [12]). Let $(\mathrm{X},d)$ be a complete generalized metric space and ${\rm Y}:\mathrm{X}\to \mathrm{X}$ be a strictly contractive operator with Lipschitz constant $L_C<1$. If there is an $n\in Z_{+}$ such that $d({{\rm Y}}^{n+1}x,{{\rm Y}}^{n}x)<\infty$ for some $x\in X$, then

(A₁)

The sequence $\left\{{{\rm Y}}^{n}x\right\}$ converges to a fixed point $x^{\ast }$ of Υ;

(A₂)

$x^{\ast }$ is the unique fixed point of Υ in $X^{\ast }=\left\{y\in \mathrm{X}:d({{\rm Y}}^{n}x,y)<\infty \right\}$;

(A₃)

If $y\in X^{\ast }$, then $d(y,y\ast )\le {\displaystyle \frac{1}{1-L_C}}d({\rm Y}y,y)$.

3. H-U-R and H-U Stability of $\mathit{\psi }$-Hilfer FrOVIDEs

Our first result on the H-U-R stability of the $\psi$-Hilfer FrOVIDE (4) is contained in the following theorem.

Theorem 1.

Let $\psi \in C^1(\left[0,T\right],\mathbb{R})$ be an increasing function with ${\psi }^{\prime }\left(t\right)\ne 0$ and let ${\ell }_h$, ${\ell }_f$, $g_0$, ${\ell }_{K_i}$, $i=1,2,\dots ,N$, and M be positive constants such that

(H₁)

$h\in C\left(\mathbb{R},\mathbb{R}\right)$, $\left|h\left({\zeta }_1\right)-h\left({\zeta }_2\right)\right|\le {\ell }_h\left|{\zeta }_1-{\zeta }_2\right|$ for each ${\zeta }_1,{\zeta }_2\in \mathbb{R}$;

(H₂)

$f\in C\left(I\times \mathbb{R},\mathbb{R}\right)$, $\left|f(t,{\zeta }_1)-f(t,{\zeta }_2)\right|\le {\ell }_f\left|{\zeta }_1-{\zeta }_2\right|$ for each $t\in I$ and ${\zeta }_1,{\zeta }_2\in \mathbb{R}$;

(H₃)

$g\in C\left(I\times \mathbb{R},\mathbb{R}\right)$, $\left|g(t,\zeta )\right|\le g_0$ for each $t\in \mathrm{I}$ and $\zeta \in \mathbb{R}$;

(H₄)

$K_i\in C\left(I\times I\times \mathbb{R},\mathbb{R}\right)$, $\left|K_i(t,s,{\zeta }_1,{\aleph }_1)-K_i(t,s,{\zeta }_2,{\aleph }_2)\right|\le {\ell }_{K_i}\left|{\zeta }_1-{\zeta }_2\right|$ for each t, $s\in I$ and ${\zeta }_1$, ${\zeta }_2$, ${\aleph }_1$, ${\aleph }_2\in \mathbb{R}$.

If there is a function $\vartheta \in C^1\left(I,\mathbb{R}\right)$ satisfying

$$\left|\vartheta \left(t\right)-\theta \left(t\right)\right|=0,\mathit{for}-\tau \le t\le 0,$$

(5)

and

$$\begin{array}{c}|{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\vartheta \left(t\right)-h\left(\vartheta \left(t\right)\right)-f(t,\vartheta \left(t\right))\hfill \\ \hfill -\underset{0}{\overset{t}{\int }}\left(g\left(t,\vartheta \left(t\right)\right){\displaystyle \sum _{i=1}^N}{K}_i\left(t,s,\vartheta \left(s\right),\vartheta (s-{\tau }_i\left(s\right))\right)\right)\mathrm{d}s|\le \Phi \left(t\right),\end{array}$$

(6)

where $\Phi \in C\left(I,(0,\infty )\right)$ for each $t\in I$ with

$$I_{0^{+}}^{\alpha ;\psi }\Phi \left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(\xi \right)\right)}^{\alpha -1}{\psi }^{\prime }\left(\xi \right)\Phi \left(\xi \right)d\xi \le M\Phi \left(t\right),$$

(7)

then there exists a unique continuous function ${\vartheta }_0\in C\left(I,\mathbb{R}\right)$ such that

$${\vartheta }_0\left(t\right)=\left\{\begin{array}{cc}\theta \left(t\right),\hfill & -\tau \le t\le 0,\hfill \\ {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(0\right))}^{y-1}}{\Gamma \left(y\right)}+I_{0^{+}}^{\alpha ;\psi }h\left({\vartheta }_0\left(t\right)\right)+I_{0^{+}}^{\alpha ;\psi }f\left(t,{\vartheta }_0\left(t\right)\right)}\hfill \\ +I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{\xi }{\int }}\left(g\left(t,{\vartheta }_0\left(t\right)\right){\displaystyle \sum _{i=1}^N}{K}_i\left(t,s,{\vartheta }_0\left(s\right),{\vartheta }_0(s-{\tau }_i\left(s\right))\right)\right)\mathrm{d}s\right],\hfill & t\in (0,T],\hfill \end{array}\right.$$

(8)

with $I_{0^{+}}^{1-\gamma }\vartheta \left(0\right)=\sigma$, $\alpha \in (0,1)$, $\beta \in [0,1]$, and

$$\left|\vartheta \left(t\right)-{\vartheta }_0\left(t\right)\right|\le {\displaystyle \frac{M}{1-M\left[{\ell }_h+{\ell }_f+g_{0}M{\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)\right]}}\Phi \left(t\right)\mathit{for}t\in I,$$

(9)

where $0<M{\ell }_h+M{\ell }_f+g_0{M}^2{\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)<1$.

Proof. 

Let X be the set of all continuous real-valued functions on I and let $\upsilon$, $\omega \in X$. Define

$$d(\upsilon ,\omega )=inf\left\{C\in [0,\infty ]:\left|\upsilon \left(t\right)-\omega \left(t\right)\right|\le C\Phi \left(t\right)\mathrm{for}\mathrm{all}t\in I\right\}.$$

(10)

Define the operator ${\rm Y}:X\to X$ by

$${\rm Y}{\upsilon }_0\left(t\right)=\left\{\begin{array}{cc}\theta \left(t\right),\hfill & t\in [-\tau ,0],\hfill \\ {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(0\right))}^{y-1}}{\Gamma \left(y\right)}+I_{0^{+}}^{\alpha ;\psi }h\left({\upsilon }_0\left(t\right)\right)+I_{0^{+}}^{\alpha ;\psi }f\left(t,{\upsilon }_0\left(t\right)\right)}\hfill \\ +I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{\xi }{\int }}\left(g\left(t,{\upsilon }_0\left(t\right)\right){\displaystyle \sum _{i=1}^N}{K}_i\left(t,s,{\upsilon }_0\left(s\right),{\upsilon }_0(s-{\tau }_i\left(s\right))\right)\right)\mathrm{d}s\right],\hfill & t\in (0,T],\hfill \end{array}\right.$$

(11)

for each ${\nu }_0\in X$.

We will now show that ${\rm Y}$ is a strictly contractive operator on X. Let $\upsilon$, $\omega \in X$ and let $C_{\upsilon \omega }\in \left(0,\infty \right]$ be a constant determined by (10) such that $d(\upsilon ,\omega )\le C_{\upsilon \omega }$, and so

$$\left|\upsilon \left(t\right)-\omega \left(t\right)\right|\le C_{\upsilon \omega }\Phi \left(t\right)\mathrm{for}t\in I.$$

(12)

From (11) and conditions (H₁)–(H₄), we have $\left|{\rm Y}\upsilon \left(t\right)-{\rm Y}\omega \left(t\right)\right|=0$ for $-\tau \le t\le 0$ and

$$\begin{array}{cc}\hfill \left|{\rm Y}\upsilon \left(t\right)-{\rm Y}\omega \left(t\right)\right|& \le {\displaystyle \frac{{\ell }_h}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(\xi \right)\right)}^{\alpha -1}{\psi }^{\prime }\left(\xi \right)\left|\upsilon \left(\xi \right)-\omega \left(\xi \right)\right|d\xi \hfill \\ \hfill & +{\displaystyle \frac{{\ell }_f}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(\xi \right)\right)}^{\alpha -1}{\psi }^{\prime }\left(\xi \right)\left|\upsilon \left(\xi \right)-\omega \left(\xi \right)\right|d\xi \hfill \\ \hfill & +{\displaystyle \frac{g_0}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(\xi \right)\right)}^{\alpha -1}{\psi }^{\prime }\left(\xi \right)\left[\underset{0}{\overset{\xi }{\int }}\left({\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)\left|\upsilon \left(s\right)-\omega \left(s\right)\right|\right)\right]\mathrm{d}s\mathrm{d}\xi \hfill \\ \hfill & \le {\ell }_{h}M{C}_{\upsilon \omega }\Phi \left(t\right)+{\ell }_{f}M{C}_{\upsilon \omega }\Phi \left(t\right)+g_{0}M{C}_{\upsilon \omega }{I}_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{\xi }{\int }}\left({\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)\Phi \left(s\right)\right)\right]\mathrm{d}s\hfill \\ \hfill \le & M{C}_{\upsilon \omega }\left[{\ell }_h+{\ell }_f+g_{0}M{\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)\right]\Phi \left(t\right)\hfill \end{array}$$

(13)

for $t\in (0,T]$. In view of (13), we see that

$$d\left({\rm Y}\upsilon ,{\rm Y}\omega \right)\le M{C}_{\upsilon \omega }\left[{\ell }_h+{\ell }_f+g_{0}M{\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)\right]\Phi \left(t\right)\mathrm{for}\mathrm{each}t\in I$$

Moreover, in view of (13), we have

$$d\left({\rm Y}\upsilon ,{\rm Y}\omega \right)\le M\left[{\ell }_h+{\ell }_f+g_{0}M{\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)\right]d\left(\upsilon ,\omega \right)\mathrm{for}\mathrm{each}\upsilon ,\omega \in X.$$

Based on the operator (11), we also have that

$$\left|{\rm Y}\omega \left(t\right)-{\rm Y}{\omega }_0\left(t\right)\right|=0\mathrm{for}\mathrm{each}-\tau \le t\le 0$$

and there is a constant $0<C<\infty$ such that

$$\begin{array}{cc}\hfill \left|{\rm Y}{\omega }_0\left(t\right)\right.& -\left.{\omega }_0\left(t\right)\right|=\left|{\displaystyle \frac{{\left(\psi \left(t\right)-\psi \left(0\right)\right)}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\sigma +\right.I_{0^{+}}^{\alpha ;\psi }h\left({\omega }_0\left(t\right)\right)+I_{0^{+}}^{\alpha ;\psi }f\left(t,{\omega }_0\left(t\right)\right)\hfill \\ \hfill & +\left.I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{\xi }{\int }}\left(g\left(t,{\omega }_0\left(t\right)\right){\displaystyle \sum _{i=1}^N}{K}_i\left(t,s,{\omega }_0\left(s\right),{\omega }_0(s-{\tau }_i\left(s\right))\right)\right)\mathrm{d}s\right]-{\omega }_0\left(t\right)\right|\le C\Phi \left(t\right)\hfill \end{array}$$

for any ${\omega }_0\in X$ and each $t\in (0,T]$. From (10), we conclude that $d\left({\rm Y}{\omega }_0,{\omega }_0\right)<\infty$. In view of Lemma 1, there is a function ${\vartheta }_0\in (I,\mathbb{R})$ such that ${{\rm Y}}^n{\vartheta }_0\to {\vartheta }_0$ in $(X,d)$ and ${\rm Y}{\vartheta }_0={\vartheta }_0$, i.e., ${\vartheta }_0$ is the function given in (8).

Since $\omega$ and ${\omega }_0$ are bounded on the interval I and ${\displaystyle \underset{t\in I}{min}}\Phi \left(t\right)>0$, there is a constant $C_{\omega }\in (0,\infty )$ such that

$$\left|\omega \left(t\right)-{\omega }_0\left(t\right)\right|\le C_{\omega }\Phi \left(t\right),t\in I.$$

Hence, $d\left(\omega ,{\omega }_0\right)<\infty$ for each $\omega \in X$, and so the set

$$\left\{\omega \in X\left|d(\omega ,{\omega }_0)<\infty \right.\right\}$$

is in fact the set X. Therefore, by Lemma 1, the function ${\vartheta }_0$ given in (8) is unique.

Now, by (5), (6), and Theorem 5 in [6], we have that

$$\left|\vartheta \left(t\right)-\theta \left(t\right)\right|=0\mathrm{for}-\tau \le t\le 0,$$

and

$$\begin{array}{c}|\vartheta \left(t\right)-{\displaystyle \frac{{\left(\psi \left(t\right)-\psi \left(0\right)\right)}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\sigma -I_{0^{+}}^{\alpha ;\psi }h\left(\vartheta \left(t\right)\right)-I_{0^{+}}^{\alpha ;\psi }f\left(t,\vartheta \left(t\right)\right)\hfill \\ -I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{\xi }{\int }}\left(g\left(t,\vartheta \left(t\right)\right){\displaystyle \sum _{i=1}^N}{K}_i\left(t,s,\vartheta \left(s\right),\vartheta (s-{\tau }_i\left(s\right))\right)\right)\mathrm{d}s\right]|\\ \hfill \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(\xi \right)\right)}^{\alpha -1}{\psi }^{\prime }\left(\xi \right)\Phi \left(\xi \right)\mathrm{d}\xi \end{array}$$

for $t\in (0,T]$. From (7) and (11),

$$\left|\vartheta \left(t\right)-{\rm Y}\vartheta \left(t\right)\right|\le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(\psi \left(t\right)-\psi \left(\xi \right)\right)}^{\alpha -1}{\psi }^{\prime }\left(\xi \right)\Phi \left(\xi \right)d\xi \le M\Phi \left(t\right)\mathrm{for}\mathrm{each}t\in I.$$

This implies that

$$d\left(\vartheta ,{\rm Y}\vartheta \right)<M,$$

and so, from Lemma 1 and (11), we see that

$$\begin{array}{cc}\hfill d\left(\vartheta ,{\vartheta }_0\right)\le & {\displaystyle \frac{1}{1-M\left[{\ell }_h+{\ell }_f+g_{0}M{\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)\right]}}d\left({\rm Y}\vartheta ,\vartheta \right)\le {\displaystyle \frac{M}{1-M\left[{\ell }_h+{\ell }_f+g_{0}M{\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)\right]}},\hfill \end{array}$$

so (9) holds. This completes the proof of the theorem. □

Remark 1.

Theorem 1 shows that the ψ-Hilfer FrOVIDE (4) is, in fact, U-H-R stable.

In our next result, we prove that (4) is U-H stable.

Theorem 2.

In addition to conditions (H₁)–(H₄), assume that $\alpha \in (0,1)$, $\beta \in [0,1]$, $\psi \in C^1[0,T]$ is an increasing function with ${\psi }^{\prime }\left(t\right)\ne 0$ and there are positive constants ${\ell }_h$, ${\ell }_f$, $g_0$, and ${\ell }_{K_i}$, $i=1,2,\dots ,N$, such that

$$0<T{\ell }_h+T{\ell }_f+2^{-1}{g}_0{T}^2{\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)<1.$$

If, for each $\epsilon >0$, there is a function $\vartheta \in C^1\left(I,\mathbb{R}\right)$ satisfying

$${}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\vartheta \left(t\right)-h\left(\vartheta \left(t\right)\right)-f(t,\vartheta \left(t\right))-\underset{0}{\overset{t}{\int }}\left(g\left(t,\vartheta \left(t\right)\right){\displaystyle \sum _{i=1}^N}{K}_i\left(t,s,\vartheta \left(s\right),\vartheta (s-{\tau }_i\left(s\right))\right)\right)\mathrm{d}s\le \epsilon$$

for $t\in I$, then there is a unique function ${\vartheta }_0\in C\left(I,\mathbb{R}\right)$ that satisfies (8) and

$$\left|\vartheta \left(t\right)-{\vartheta }_0\left(t\right)\right|\le {\displaystyle \frac{\epsilon {\left(\psi \left(T\right)-\psi \left(0\right)\right)}^{\alpha }}{\Gamma (\alpha +1)-{\left(\psi \left(T\right)-\psi \left(0\right)\right)}^{\alpha }\left[{\ell }_h+{\ell }_f+2^{-1}{g}_{0}T{\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)\right]}}$$

for each $t\in I$.

Proof. 

Similar to the proof of Theorem 1, we let

$$d(\upsilon ,\omega )=inf\left\{C\in [0,\infty ]:\left|\upsilon \left(t\right)-\omega \left(t\right)\right|\le C\mathrm{for}\mathrm{all}t\in I\right\},$$

(14)

define the operator ${\rm Y}:X\to X$ as in (11) for $\upsilon \in X$. Proceeding as above, we again arrive at (12). It then follows that

$$\left|{\rm Y}\upsilon \left(t\right)-{\rm Y}\omega \left(t\right)\right|=0,-\tau \le t\le 0,$$

and

$$\begin{array}{cc}\hfill \left|{\rm Y}\upsilon \left(t\right)-{\rm Y}\omega \left(t\right)\right|& \le {\ell }_h{C}_{\upsilon \omega }\left({\displaystyle \frac{{\left(\psi \left(T\right)-\psi \left(0\right)\right)}^{\alpha }}{\Gamma (\alpha +1)}}\right)+{\ell }_f{C}_{\upsilon \omega }\left({\displaystyle \frac{{\left(\psi \left(T\right)-\psi \left(0\right)\right)}^{\alpha }}{\Gamma (\alpha +1)}}\right)\hfill \\ \hfill & +g_0{C}_{\upsilon \omega }{\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)\left({\displaystyle \frac{T{\left(\psi \left(T\right)-\psi \left(0\right)\right)}^{\alpha }}{\Gamma (\alpha +1)}}\right)\hfill \end{array}$$

(15)

for $t\in I$. In view of (15), we obtain

$$d\left({\rm Y}\upsilon ,{\rm Y}\omega \right)\le C_{\upsilon \omega }\left[\left({\ell }_h+{\ell }_f+2^{-1}T{g}_0{\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)\right)\left({\displaystyle \frac{{\left(\psi \left(T\right)-\psi \left(0\right)\right)}^{\alpha }}{\Gamma (\alpha +1)}}\right)\right]d\left(\upsilon ,\omega \right)$$

for each $\upsilon$, $\omega \in X$.

Let ${\omega }_0\in X$ be given; then,

$$\left|{\rm Y}{\omega }_0\left(t\right)-{\omega }_0\left(t\right)\right|=0,\mathrm{for}-\tau \le t\le 0,$$

and there is a constant $0<C<\infty$ such that

$$\begin{array}{c}\left|{\rm Y}{\omega }_0\left(t\right)-{\omega }_0\left(t\right)\right|=\left|{\displaystyle \frac{{\left(\psi \left(t\right)-\psi \left(0\right)\right)}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\sigma +\right.I_{0^{+}}^{\alpha ;\psi }h\left({\omega }_0\left(t\right)\right)+I_{0^{+}}^{\alpha ;\psi }f\left(t,{\omega }_0\left(t\right)\right)\hfill \\ \hfill +\left.I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{\xi }{\int }}\left(g\left(t,{\omega }_0\left(t\right)\right){\displaystyle \sum _{i=1}^N}{K}_i\left(t,s,{\omega }_0\left(s\right),{\omega }_0(s-{\tau }_i\left(s\right))\right)\right)\mathrm{d}s\right]-{\omega }_0\left(t\right)\right|\le C\end{array}$$

for $t\in (0,T]$. Since the functions f, g, h, $K_i$, and ${\omega }_0$ are bounded on $[0,T]$, from (14), we have $d\left({\rm Y}{\omega }_0,{\omega }_0\right)<\infty$. By Lemma 1, there is function ${\vartheta }_0\in C(I,\mathbb{R})$ such that ${{\rm Y}}^n{\vartheta }_0\to {\vartheta }_0$ in $(X,d)$ as $n\to \infty$ and ${\rm Y}{\vartheta }_0={\vartheta }_0$, i.e., ${\vartheta }_0$ satisfies (11) for each $t\in I$.

Since $\omega$ and ${\omega }_0$ are bounded on the interval I, for each $\omega \in X$, there is a constant $C_{\omega }\in \left(0,\infty \right)$ such that

$$\left|\omega \left(t\right)-{\omega }_0\left(t\right)\right|\le C_{\omega }$$

for each $t\in I$. Hence, $d\left(\omega ,{\omega }_0\right)<\infty$ for each $\omega \in X$. Therefore, the set $\left\{\omega \in X\left|d(\omega ,{\omega }_0)<\infty \right.\right\}$ is equal to X. Thus, by Lemma 1, we conclude that ${\vartheta }_0$ is the unique continuous function given in (8).

Now, using (14) and Theorem 5 in [6], we obtain

$$\left|{\vartheta }_0\left(t\right)-\theta \left(t\right)\right|=0,-\tau \le t\le 0,$$

and

$$\begin{array}{cc}\hfill |& {\vartheta }_0\left(t\right)-{\displaystyle \frac{{\left(\psi \left(t\right)-\psi \left(0\right)\right)}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\sigma -I_{0^{+}}^{\alpha ;\psi }h\left({\vartheta }_0\left(t\right)\right)-I_{0^{+}}^{\alpha ;\psi }f\left(t,{\vartheta }_0\left(t\right)\right)\hfill \\ \hfill & -I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{\xi }{\int }}\left(g\left(t,{\vartheta }_0\left(t\right)\right){\displaystyle \sum _{i=1}^N}{K}_i\left(t,s,{\vartheta }_0\left(s\right),{\vartheta }_0(s-{\tau }_i\left(s\right))\right)\right)\mathrm{d}s\right]|\hfill \\ \hfill & \le {\displaystyle \frac{\epsilon {\left(\psi \left(T\right)-\psi \left(0\right)\right)}^{\alpha }}{\Gamma (\alpha +1)}}\mathrm{for}\mathrm{each}t\in I.\hfill \end{array}$$

Therefore,

$$d(\vartheta ,{\rm Y}\vartheta )\le {\displaystyle \frac{\epsilon {\left(\psi \left(T\right)-\psi \left(0\right)\right)}^{\alpha }}{\Gamma (\alpha +1)}}.$$

From Lemma 1 and inequality (14), we can conclude that

$$\left|\vartheta \left(t\right)-{\vartheta }_0\left(t\right)\right|\le {\displaystyle \frac{{\left(\psi \left(T\right)-\psi \left(0\right)\right)}^{\alpha }\epsilon }{\Gamma (\alpha +1)-{\left(\psi \left(T\right)-\psi \left(0\right)\right)}^{\alpha }\left[{\ell }_h+{\ell }_f+2^{-1}T{\displaystyle \sum _{i=1}^N}\left({\ell }_{K_i}\right)\right]}}.$$

This proves the U-H stability and completes the proof of the theorem. □

4. Discussion and an Example

The results for the Hilfer FrOVIDE (4) considered here improve and extend the results of Sousa and Oliveira [12] from a fractional model without delay to a more general fractional model involving multiple variable delays. It can be seen that Equation (2) is in fact a particular case of (4). In addition, the Hilfer FrOVIDE (3) is also particular case of (4). Our results extend the results of Sousa and Oliveira [10] from the fractional model with one delay to a more general fractional model with multiple variable delays. The results presented in this paper are new and contribute to the current literature on these problems.

To illustrate our results, we provide the following example.

Example 1.

We consider the ψ-Hilfer FrOVIDE containing a constant delay

$$\left\{\begin{array}{c}{}^{H}D_{0^{+}}^{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}};exp\left(t\right)}\vartheta \left(t\right)={\displaystyle \frac{1}{200}}arctan\left(\vartheta \left(t\right)\right)+{\displaystyle \frac{\vartheta \left(t\right)}{600\left(1+t+{\vartheta }^2\left(t\right)\right)}}\hfill \\ +{\displaystyle \frac{1}{1+t+{\vartheta }^6\left(t\right)}}\underset{0}{\overset{t}{\int }}{\displaystyle \frac{sin\left(\vartheta \left(s\right)\right)}{400\left(1+s^2+t^2+{\vartheta }^2(s-1)\right)}}ds,t\in I=(0,1],\hfill \\ I_{0^{+}}^{1-\gamma }\vartheta \left(0\right)=\sigma ,\hfill \\ \vartheta \left(t\right)=0,t\in [-1,0]\hfill \end{array}\right.$$

(16)

together with the inequality

$$\begin{array}{cc}\hfill |& {}^{H}D_{0^{+}}^{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}};exp\left(t\right)}\vartheta \left(t\right)-{\displaystyle \frac{1}{200}}arctan\left(\vartheta \left(t\right)\right)-{\displaystyle \frac{\vartheta \left(t\right)}{600\left(1+t+{\vartheta }^2\left(t\right)\right)}}\hfill \\ \hfill & -{\displaystyle \frac{1}{1+t+{\vartheta }^6\left(t\right)}}\underset{0}{\overset{t}{\int }}{\displaystyle \frac{sin\left(\vartheta \left(s\right)\right)}{400\left(1+s^2+t^2+{\vartheta }^2(s-1)\right)}}ds|\hfill \\ \hfill & \le \Phi \left(t\right).\hfill \end{array}$$

Here, we have $\alpha ={\displaystyle \frac{1}{2}}=\beta$ so that $\gamma =\alpha +\beta (1-\alpha )={\displaystyle \frac{3}{4}}$. Moreover, $N=1$, ${\tau }_1\left(t\right)=1$, $\psi \left(t\right)=exp\left(t\right)$, ${\ell }_h={\displaystyle \frac{1}{200}}$, ${\ell }_f={\displaystyle \frac{1}{200}}$, $g_0=1$, ${\ell }_{K_1}={\displaystyle \frac{1}{400}}$, and $M\in (0,1)$ (see Theorems 3.1 and 3.2 in [10]). By comparing (16) and (4), we see that

$$t\in (0,1],T=1,$$

$$h\left(\vartheta \right)={\displaystyle \frac{1}{200}}arctan\left(\vartheta \right),$$

$$f\left(t,\vartheta \right)={\displaystyle \frac{\vartheta }{600\left(1+t+{\vartheta }^2\right)}},$$

$$g\left(t,\vartheta \right)={\displaystyle \frac{1}{1+t+{\vartheta }^6}},$$

$$K_1\left(t,s,\vartheta ,\vartheta (s-{\tau }_1\left(s\right))\right)={\displaystyle \frac{sin\left(\vartheta \right)}{400\left(1+s^2+t^2+{\vartheta }^2(s-1)\right)}},$$

$$h\in C\left(R,R\right),$$

$$f,g\in C\left((0,T]\times R,R\right),$$

$$K_1\in C\left((0,T]\times (0,T]\times R\times R,R\right),$$

$$\left|h\left({\zeta }_1\right)-h\left({\zeta }_2\right)\right|={\displaystyle \frac{1}{200}}\left|arctan\left({\zeta }_1\right)-arctan\left({\zeta }_2\right)\right|\le {\displaystyle \frac{1}{200}}\left|{\zeta }_1-{\zeta }_2\right|,{\ell }_h={\displaystyle \frac{1}{200}},$$

$$\left|f(t,{\zeta }_1)-f(t,{\zeta }_2)\right|={\displaystyle \frac{1}{600}}\left|{\displaystyle \frac{{\zeta }_1}{1+t+{\zeta }_1^2}}-{\displaystyle \frac{{\zeta }_2}{1+t+{\zeta }_2^2}}\right|$$

$$={\displaystyle \frac{1}{600}}{\displaystyle \frac{\left|{\zeta }_1\left(1+t+{\zeta }_2^2\right)-{\zeta }_2\left(1+t+{\zeta }_1^2\right)\right|}{\left(1+t+{\zeta }_1^2\right)\left(1+t+{\zeta }_2^2\right)}}$$

$$\le {\displaystyle \frac{1}{300}}{\displaystyle \frac{\left|{\zeta }_1-{\zeta }_2\right|}{\left(1+t+{\zeta }_1^2\right)\left(1+t+{\zeta }_2^2\right)}}+{\displaystyle \frac{\left|{\zeta }_1-{\zeta }_2\right|}{600}}{\displaystyle \frac{\left|{\zeta }_1{\zeta }_2\right|}{\left(1+t+{\zeta }_1^2\right)\left(1+t+{\zeta }_2^2\right)}}$$

$$\le {\displaystyle \frac{1}{200}}\left|{\zeta }_1-{\zeta }_2\right|,{\ell }_f={\displaystyle \frac{1}{200}},$$

$$\left|g(t,\vartheta )\right|={\displaystyle \frac{1}{1+t+{\vartheta }^6}}\le 1=g_0,$$

$$\left|K_1\left(t,s,{\zeta }_1,{\aleph }_1\right)-K_1\left(t,s,{\zeta }_2,{\aleph }_2\right)\right|\le {\displaystyle \frac{1}{400}}\left|sin\left({\zeta }_1\right)-sin\left({\zeta }_2\right)\right|$$

$$\le {\displaystyle \frac{1}{400}}\left|{\zeta }_1-{\zeta }_2\right|,{\ell }_{K_1}={\displaystyle \frac{1}{400}}.$$

We then see that conditions (H₁)–(H₄) hold, and the hypotheses of Theorem 1 hold with

$$0<M{\ell }_h+M{\ell }_f+g_0{M}^2{\ell }_{K_1}={\displaystyle \frac{1}{200}}M+{\displaystyle \frac{1}{200}}M+{\displaystyle \frac{1}{400}}{M}^2<1,M\in (0,1).$$

Moreover, the conditions of Theorem 2 hold with

$$0<T{\ell }_h+T{\ell }_f+2^{-1}{g}_0{T}^2{\ell }_{K_1}={\displaystyle \frac{1}{200}}+{\displaystyle \frac{1}{200}}+{\displaystyle \frac{1}{800}}={\displaystyle \frac{9}{800}}<1.$$

Thus, FrOVIDE (16) is Ulam–Hyers–Rassias stable.