Ulam-Type Stability Results for Fractional Integro-Delay Differential and Integral Equations via the ψ-Hilfer Operator

Abstract

In this article, we investigate a nonlinear $\psi$-Hilfer fractional order Volterra integro-delay differential equation ($\psi$-Hilfer FRVIDDE) and a nonlinear $\psi$-Hilfer fractional Volterra delay integral equation ($\psi$-Hilfer FRVDIE), both of which incorporate multiple variable time delays. We establish sufficient conditions for the existence of a unique solution and the Ulam–Hyers stability (U-H stability) of both the $\psi$-Hilfer FRVIDDE and $\psi$-the Hilfer FRVDIE through two new main results. The proof technique relies on the Banach contraction mapping principle, properties of the Hilfer operator, and some additional analytical tools. The considered $\psi$-Hilfer FRVIDDE and $\psi$-Hilfer FRVDIE are new fractional mathematical models in the relevant literature. They extend and improve some available related fractional mathematical models from cases without delay to models incorporating multiple variable time delays, and they also provide new contributions to the qualitative theory of fractional delay differential and fractional delay integral equations. We also give two new examples to verify the applicability of main results of the article. Finally, the article presents substantial and novel results with new examples, contributing to the relevant literature.

1. Introduction

According to the existing body of literature, the first result addressing stability problems in the sense of Ulam was discussed by Ulam [1] in 1940, who posed a fundamental question regarding the stability of the Cauchy functional equation. One year later, in 1941, Hyers provided an affirmative answer to Ulam’s [2] question. Nearly four decades after these initial ideas on what is now known as U–H stability, Rassias proposed a significant generalization of the Hyers theorem [3], establishing the concept of Hyers–Ulam–Rassias stability by proving the existence of unique linear mappings in a neighborhood of approximate additive mappings. Since that time, various forms of Ulam-type stability have been introduced and extensively studied. Following these developments, numerous important books and a large number of research articles have been published, addressing the existence theory as well as Ulam-type stability for a wide range of mathematical models. Today, these qualitative properties constitute an active and attractive area of research, and many authors continue to investigate different classes of mathematical models within this framework.

On the other hand, in recent years, fractional calculus has emerged as one of the most active and rapidly developing research areas in applied mathematics. Although fractional calculus is often viewed as a modern topic, fractional derivatives and fractional integrals date back to the very origins of classical calculus. Since then, numerous definitions of fractional operators have also been formulated, typically by allowing the order of differentiation or integration to be a noninteger parameter. Notably, Kilbas et al. developed the Riemann–Liouville (RL) fractional derivative [4], which extends the classical integer-order derivative through a nonlocal integral formulation. Building on this framework, Almeida proposed a generalized Caputo fractional derivative [5] by incorporating a suitable kernel function, thereby broadening the classical Caputo setting. Besides these information, more recently, the introduction of the $\psi$-Hilfer fractional derivative ($\psi$-HFD), an operator that unifies several well-known fractional derivatives [6], has led to renewed interest in qualitative analysis, particularly in the study of existence theory, Ulam-type stability and some related properties of fractional order mathematical models involving the $\psi$-HFD. Indeed, the $\psi$-HFD contains the Riemann–Liouville and Caputo derivatives as special cases, as detailed in Definition 2. Owing to its high degree of generality and flexibility, this fractional operator has played a significant role as an effective modeling too in a wide range of applications, including finance, physics, biology, and image processing. As a consequence, growing attention has been devoted to the qualitative analysis of mathematical models involving such generalized fractional derivatives.

In this way, fractional-order mathematical models have attracted considerable attention due to their ability to capture memory and hereditary effects that cannot be adequately described by classical integer-order equations. In particular, fractional differential equations, with or without delays, now play an essential role in modeling phenomena arising in biology, physics, engineering, economy, medicine, and the applied sciences [7,8,9,10]. These kinds of mathematical models often lead a more accurate representation of dynamical behaviors than their integer-order counterparts. Comprehensive presentations of the theory and applications of fractional calculus can be found in several standard books, including works on the analysis of fractional differential equations [11], the theory and applications of fractional differential equations [4,12], introduction to fractional calculus and fractional differential equations [13], fractional calculus [14], and studies of fractional derivatives and integrals [15,16].

Motivated by the above developments, numerous authors have obtained interesting results on existence theory, various types of Ulam-type stability, and other qualitative properties of fractional differential equations, with and without delay, involving a wide range of fractional derivative operators, and integral equations. For further (but limited) details, we refer to studies on stability for nonlinear implicit fractional differential equations [17], existence and Ulam-type stability for nonlinear implicit fractional differential equations involving the Hadamard derivative [18], Ulam-type stability of fractional delay differential equations [19], Ulam-type stability of arbitrary-order fractional stochastic differential equations [20], Ulam-type stability of Volterra integral equations [21], global existence and uniqueness of solutions for delay integro-differential equations in Banach spaces [22], Ulam-type stability of zero-point equations [23], Lyapunov stability for Caputo fractional integro-differential equations [24,25], and existence and uniqueness of solutions for systems of fractional differential equations [26], together with the references cited therein.

Specifically, there is a substantial body of recent and noteworthy literature devoted to the qualitative analysis of Hilfer fractional differential equations and their modified types, with or without delay, for instance, existence and Ulam-type stability results for Hilfer and Hilfer–Hadamard fractional differential inclusions and equations are investigated in [27,28], the existence of solutions and U-H stability for a fully coupled system of nonlinear sequential Hilfer fractional differential equations are discussed in [29], the U-H stability and controllability of a coupled system of sequential partial $\sigma$-Hilfer fractional differential equations with a weighted double-phase operator are investigated in [30], the Ulam-type stability of fractional Volterra integro-differential equations without delay is studied in [31,32,33] by means of the $\psi$-Hilfer operator, an analysis of the $\psi$-Hilfer fractional derivative is provided in [6], a theoretical study of $\psi$-Hilfer fractional differential systems in Banach spaces is presented in [34], and the existence and stability of solutions for nonlinear fractional differential equations are discussed in [35], stability of fractional Volterra integro-differential equations via the $\psi$-Hilfer operator is addressed in [36], Ulam–Hyers–Rassias stability for classes of fractional integro-differential equations with single or multiple delays is analyzed in [37,38,39,40,41], Lyapunov stability for delay integro-differential equations involving the Caputo proportional fractional derivative is discussed in [42], and existence and stability results for Hilfer fractional implicit differential equations are presented in [43]. However, we only find the papers of Baleanu et al. and Sousa et al. with respect to the uniqueness of solutions and H-U stability of $\psi$-Hilfer fractional Volterra integral equations ($\psi$-Hilfer FRVIEs) [31,36]. This scarcity in the literature is likely due to the inherent complexity of the subject.

We now outline a brief survey of the main contributions in the literature that motivate the development of the present study.

In 2019, Sousa et al. investigated the uniqueness of solutions and H-U stability of the following $\psi$-Hilfer FRVIDE and $\psi$-Hilfer FRVIE [31], respectively:

$${}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }x\left(t\right)=g\left(t,x\left(t\right)\right)+\underset{0}{\overset{x}{\int }}K\left(x,s,x\left(s\right)\right)ds$$

(1)

and

$$x\left(t\right)=g\left(t,x\left(t\right)\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{x}{\int }}{Q}_{\psi }^{\alpha }(t,s)K\left(x,s,x\left(s\right)\right)ds,$$

(2)

respectively.

Later, in 2021, Baleanu et al. slightly modified the second main result of (Theorem 4 in [31]) concerning $\psi$-Hilfer FRVIE (2) by means of $\psi$-Hilfer Hilfer operator and the Banach contraction mapping principle [36].

In this paper, drawing inspiration from the related works referenced above and in particular from that of [31,36], we address the following $\psi$-Hilfer FRVIDDE and $\psi$-Hilfer FRVDIE, containing N-time-varying delays, respectively:

$$\left\{\begin{array}{c}{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\ell \left(t\right)=g\left(t,\ell \left(t\right),\ell \left(\upsilon \right(t\left)\right)\right)+\underset{0}{\overset{t}{\int }}\left(h\left(t,\ell \left(t\right)\right){\displaystyle \sum _{i=1}^N}{\mathcal{\Im }}_i\left(t,s,\ell \left(s\right),\ell \left({\upsilon }_i\left(s\right)\right)\right)\right)ds,\hfill \\ t\in (0,T],\hfill \\ I_{0^{+}}^{1-\gamma ;\psi }\ell \left(0^{+}\right)=0,\ell \left(t\right)=\theta \left(t\right),t\in [-\upsilon ,0],\hfill \end{array}\right.$$

(3)

and

$$\left\{\begin{array}{c}\ell \left(t\right)=g\left(t,\ell \left(t\right),\ell \left(\upsilon \right(t\left)\right)\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}\left(h\left(t,\ell \left(t\right)\right){\displaystyle \sum _{i=1}^N}{Q}_{\psi }^{\alpha }\left(t,s\right){\mathcal{\Im }}_i\left(t,s,\ell \left(s\right),\ell \left({\upsilon }_i\left(s\right)\right)\right)\right)ds\hfill \\ t\in (0,T],\hfill \\ I_{0^{+}}^{1-\gamma ;\psi }\ell \left(0^{+}\right)=0,\ell \left(t\right)=\theta \left(t\right),t\in [-\upsilon ,0],\hfill \end{array}\right.$$

(4)

with $Q_{\psi }^{\alpha }\left(t,s\right)={\psi }^{\prime }\left(s\right){\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\alpha -1}$, where $t\in [0,T],T>0,T\in R$, ${}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }(.)$ is right-sided HFD, $0<\alpha <1$, $0\le \beta \le 1$, $I_{0^{+}}^{1-\gamma ;\psi }(.)$ is $\psi$-Riemann-Liouville fractional integral ($\psi$-RLFRI) of order $1-\gamma$ such that $\gamma =\alpha +\beta (1-\alpha )$ with respect to $\psi$, ${\upsilon }_i\in C\left(\left[0,T\right],R^{+}\right)$, $0\le {\upsilon }_i\left(t\right)\le {\upsilon }_i<\infty$, ${\upsilon }_i\in R$, ${\upsilon }_i>0$, $\upsilon =max\left\{{\upsilon }_i\right\}$, $\ell \left(t\right)=\theta \left(t\right),t\in [-\upsilon ,0]$, $\theta \in C\left([-\upsilon ,0],\aleph \right)$, $\ell \in C\left(\left[0,T\right],\aleph \right)$, $h\in C\left(\left[0,T\right]\times \aleph ,\aleph \right)$, $g\in C\left(\left[0,T\right]\times \aleph \times \aleph ,\aleph \right)$ and ${\mathcal{\Im }}_i\in C\left(\left[0,T\right]\times \left[0,T\right]\times \aleph \times \aleph ,\aleph \right)$, ℵ is a Banach space. Indeed, $\psi$-Hilfer FRVIDDE (3) and $\psi$-Hilfer FRVDIE (4) are referred to as Hammerstein-type mathematical models of integro-delay differential and delay integrals equations, respectively, incorporating N-time varying delays.

The main goal of the present study is to undertake a study of qualitative properties, including the uniqueness of solutions and U–H stability, for the Hammerstein-type $\psi$-Hilfer FRVIDDE (3) and the Hammerstein-type $\psi$-Hilfer FRVDIE (4), both involving multiple variable time delays. Hence, this paper presents two new and improved results on the U–H stability and the uniqueness of solutions for two new fractional mathematical models described by $\psi$-Hilfer FRVIDDE (3) and $\psi$-Hilfer FRVDIE (4). Notably, to the best of our knowledge based on a thorough review of the relevant literature databases, the existing literature has not addressed the problems considered in this study for $\psi$-Hilfer FRVIDDE (3) and $\psi$-Hilfer FRVDIE (4). Therefore, we believe that it is worthwhile to investigate these problems for this class of fractional and delay mathematical models. In this framework, we extend and refine the results of (Theorems 3 and 4 in [31]) and that of [36], which were established for the $\psi$-Hilfer FRVIDE and the Hilfer $\psi$-FRVIE without delay, to more general settings incorporating multi time-varying delays. Notably, it is obvious that $\psi$-Hilfer FRVIDE (1) and $\psi$-Hilfer FRVIE (2) are particular cases of the Hammerstein-type $\psi$-fractional delay mathematical models (3) and (4), respectively. Indeed, if we take

$$g\left(t,\ell \left(t\right),\ell \left(\upsilon \right(t\left)\right)\right)=g\left(t,x\left(t\right)\right),h\left(t,\ell \left(t\right)\right)=1$$

and

$$\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\ell \left(s\right),\ell \left({\upsilon }_i\left(s\right)\right)\right)=K\left(t,s,x\left(s\right)\right),$$

in $\psi$-Hilfer FRVIDDE (3) and $\psi$-Hilfer FRVDIE (4), then it follows that $\psi$-Hilfer FRVIDDE (3) and $\psi$-Hilfer FRVDIE (4) reduce to $\psi$-Hilfer FRVIDE (1) of [31] and Hilfer $\psi$-FRVIE (2) considered in [31,36], respectively. Moreover, the proposed Hammerstein-type $\psi$-Hilfer FRVIDDE (3) and $\psi$-Hilfer FRVDIE (4) constitute new fractional mathematical models within the existing literature. It is also evident that $\psi$-Hilfer FRVIDDE (3) and $\psi$-Hilfer FRVDIE (4) incorporate N-time-varying delays. However, $\psi$-Hilfer FRVIDE (1) of [31] and $\psi$-Hilfer FRVIE (2) considered in [31,36] do not involve any delay. This distinction highlights an additional aspect of the novelty and originality of the present study. In light of the above literature review and the information provided, the results of this paper contribute new findings to the qualitative theory of fractional calculus through two novel theoretical results and two illustrative examples. Moreover, these results further extend and improve some existing works by generalizing previously studied delay-free cases to more general fractional models involving multiple variable delays. The organization of the remainder of this paper is as follows. In Section 2, we collect the necessary preliminaries, including essential definitions, a lemma, a proposition, and two theorems. Section 3 is devoted to establishing two new results on the uniqueness of solutions and the U–H stability of $\psi$-Hilfer FRVIDDE (3) and $\psi$-Hilfer FRVDIE (4) in a Banach space, respectively. Section 4 concludes the paper.

2. Preliminary Definitions and Results

This section provides a summary of the key properties of the $\psi$-RLFRI $I_{a^{+}}^{\alpha ;\psi }(.)$ and the $\psi$-HFD ${}^{H}D_{a^{+}}^{\alpha ,\beta ;\psi }(.)$, along with relevant definitions and auxiliary results required for the analysis presented in Section 3.

Let $\left[e,f\right],(0<e<f<\infty )$, be a finite interval on $R^{+}$, $R^{+}=\left[0,\infty \right)$, and let $C\left[e,f\right]$ denote the space of the continuous functions $\vartheta$ on the interval $\left[e,f\right]$ with the norm depicted by

$${∥\vartheta ∥}_{C\left[e,f\right]}=\underset{t\in \left[e,f\right]}{max}\left|\vartheta \left(t\right)\right|.$$

The weight space $C_{1-\zeta ;\psi }\left[e,f\right]$ of a continuous function $\vartheta$ is depicted on the interval $\left(e,f\right]$ by

$$C_{1-\zeta ;\psi }\left[e,f\right]=\left\{\vartheta \in \left(e,f\right]\to R:{\left(\psi \left(t\right)-\psi \left(e\right)\right)}^{1-\zeta }\vartheta \left(t\right)\in C\left[e,f\right]\right\},0\le \zeta <1$$

with the norm depicted by

$${∥\vartheta ∥}_{C_{1-\zeta ;\psi }[e,f]}={∥{\left(\psi \left(t\right)-\psi \left(e\right)\right)}^{1-\zeta }\vartheta \left(t\right)∥}_{C\left[e,f\right]}=\underset{t\in \left[e,f\right]}{max}\left|{\left(\psi \left(t\right)-\psi \left(e\right)\right)}^{1-\zeta }\vartheta \left(t\right)\right|.$$

The weight space $C_{\zeta ;\psi }^n\left[e,f\right]$ of a continuous function $\vartheta$ is depicted on the interval $\left(e,f\right]$ by

$$C_{\zeta ;\psi }^n\left[e,f\right]=\left\{\vartheta :\left(e,f\right]\to R:\vartheta \left(t\right)\in C^{n-1}\left[e,f\right]\mathrm{and}{\vartheta }^{\left(n\right)}\left(t\right)\in {C_{\zeta }}_{;\psi }\left[e,f\right]\right\},$$

$$0\le \zeta <1$$

with the norm depicted by

$${∥\vartheta ∥}_{C_{\zeta ;\psi }^n[e,f]}=\sum _{k=0}^{n-1}{∥{\vartheta }^{\left(k\right)}∥}_{C\left[e,f\right]}+{∥{\vartheta }^{\left(n\right)}∥}_{{C_{\zeta }}_{;\psi }[e,f]}.$$

Definition 1

([6]). Let $\left(e,f\right),\left(-\infty \le e<f\le \infty \right)$, be a finite interval (or infinite interval) of R, $R=\left(-\infty ,\infty \right)$, and let $\alpha >0$. Besides, let $\psi \left(t\right)$ be an increasing and positive monotone function on $\left(e,f\right]$ with the property that $\psi \in C^1\left(e,f\right)$. Then, left-sided RLFRI and right-sided RLFRI of a function ϑ with regard to the function ψ on the interval $\left[e,f\right]$ are depicted by

$$I_{e^{+}}^{\alpha ;\psi }\vartheta \left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{e}{\overset{t}{\int }}{Q}_{\psi }^{\alpha }\left(t,s\right)\vartheta \left(s\right)ds$$

and

$$I_{f^{-}}^{\alpha ;\psi }\vartheta \left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{t}{\overset{f}{\int }}{Q}_{\psi }^{\alpha }\left(t,s\right)\vartheta \left(s\right)ds,$$

respectively, where $Q_{\psi }^{\alpha }\left(t,s\right)={\psi }^{\prime }\left(s\right){\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\alpha -1}$.

Definition 2.

([6]). Let $\alpha \in \left(n-1,n\right)$, $n\in N$, $\left[e,f\right]$ be an interval possessing the property that $-\infty \le e<f\le \infty$ and $\vartheta ,\psi \in C^n\left[e,f\right]$ be two functions with the property that ψ is an increasing function and ${\psi }^{\prime }\left(t\right)\ne 0$, for each $t\in \left[e,f\right]$. Then, the left-sided ψ-HFD of the function ϑ, i.e., ${}^{H}D_{e^{+}}^{\alpha ,\beta ;\psi }\vartheta \left(t\right)$, of order α and type $0\le \beta \le 1,$ is depicted by

$${}^{H}D_{e^{+}}^{\alpha ,\beta ;\psi }\vartheta \left(t\right)=I_{e^{+}}^{\beta (n-\alpha );\psi }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{I}_{e^{+}}^{(1-\beta )(n-\alpha );\psi }\vartheta \left(t\right).$$

The right-sided ψ-HFD of the function ϑ, i.e., ${}^{H}D_{f^{-}}^{\alpha ,\beta ;\psi }\vartheta \left(t\right)$, of order α and type $0\le \beta \le 1,$ is depicted by the same way.

Lemma 1.

([6]). Let $\alpha >0$ and $\beta >0$. Then, we have the semigroup depicted by

$$I_{e^{+}}^{\alpha ;\psi }{I}_{e^{+}}^{\beta ;\psi }\vartheta \left(t\right)=I_{e^{+}}^{\alpha +\beta ;\psi }\vartheta \left(t\right).$$

Definition 3.

Let $\ell =\ell \left(t\right)$. If for each given function $\ell \in C^1\left(\left[0,T\right],R\right)$, satisfying the inequality

$${∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\ell -g\left(t,\ell ,\ell \left(\upsilon \right(t\left)\right)\right)-\underset{0}{\overset{t}{\int }}\left(h\left(t,\ell \right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\ell \left(s\right),\ell \left({\upsilon }_i\left(s\right)\right)\right)\right)ds∥}_{C_{1-\zeta ;\psi }\left[0,T\right]}\le \varphi \left(t\right),$$

(5)

with $\varphi \left(t\right)>0$ and $t\in \left[0,T\right]$, there exists a solution $\eta \left(t\right)$ of ψ-Hilfer FRVIDDE (3) and a constant $C\in R,C>0$, which is independent of $\ell \left(t\right)$ and $\eta \left(t\right)$, such that

$${∥\ell \left(t\right)-\eta \left(t\right)∥}_{C_{1-\zeta ;\psi }\left[0,T\right]}\le C\varphi \left(t\right),$$

for each $t\in \left[0,T\right]$, then we say that ψ-Hilfer FRVIDDE (3) involving N-multiple variable time delays possesses the U-H stability.

Definition 4.

Let $\ell =\ell \left(t\right)$. If for each given function $\ell \in C\left(\left[0,T\right],R\right)$, satisfying the inequality

$${∥\ell -g\left(t,\ell ,\ell \left(\upsilon \right(t\left)\right)\right)-{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}\left(h\left(t,\ell \right)\sum _{i=1}^N{Q}_{\psi }^{\alpha }\left(t,s\right){\mathcal{\Im }}_i\left(t,s,\ell \left(s\right),\ell \left({\upsilon }_i\left(s\right)\right)\right)\right)ds∥}_{C_{1-\zeta ;\psi }\left[0,T\right]}\le \varphi \left(t\right),$$

(6)

with $\varphi \left(t\right)>0$ and $t\in \left[0,T\right]$, there exists a solution $\eta \left(t\right)$ of ψ-Hilfer FRVDIE (4) and a constant $C\in R,C>0$, which is independent of $\ell \left(t\right)$ and $\eta \left(t\right)$, such that

$${∥\ell \left(t\right)-\eta \left(t\right)∥}_{C_{1-\zeta ;\psi }\left[0,T\right]}\le C\varphi \left(t\right),$$

for each $t\in \left[0,T\right]$, then we say that ψ-Hilfer FRVDIE (4) involving multiple variable time delays possesses the U-H stability.

The next Theorem 1 is fundamental to this study, as it forms the basis for our analysis of the unique solution and stability within the U–H stability framework.

Theorem 1.

Let $(\aleph ,d)$ be a complete generalized metric space. Assume that $\Lambda :\aleph \to \aleph$ is a strictly contractive operator with the Lipschitz constant $C_L<1$. If an $n\in Z$, $n\ge 0$, exists with the property that $d({\Lambda }^{n+1}\ell ,{\Lambda }^n\ell )<\infty$ for each $\ell \in \aleph ,$ then the following results are true:

• $(C.1)$ the sequence $\left\{{\Lambda }^n\ell \right\}$ converges to a fixed point ${\ell }^{\ast }$ of Λ;
• $(C.2)$${\ell }^{\ast }$ is the unique fixed point of Λ in ${\aleph }^{\ast }=\left\{\eta \in \aleph :d({\Lambda }^n\ell ,\eta )<\infty \right\};$
• $(C.3)$ if $\eta \in {\aleph }^{\ast }$, then $d\left(\eta ,{\ell }^{\ast }\right)\le {\displaystyle \frac{1}{1-C_L}}d(\Lambda \eta ,\eta )$.

Proof. 

See ref. [44]. □

Theorem 2.

If $\rho \in C^1\left(\left[e,f\right]\right)$, $0<\alpha <1$, $0\le \beta \le 1$, then

$${}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi H}{I}_{0^{+}}^{\alpha ,\psi }\rho \left(t\right)=\rho \left(t\right).$$

Proof. 

See ref. [6]. □

Proposition 1.

Let $\left(\aleph ,{∥.∥}_{C_{1-\zeta ;\psi }}\right)$ be a normed space. Then, the function $d:\aleph \times \aleph \to R$ with $d\left(\ell ,\eta \right)={∥\ell -\eta ∥}_{C_{1-\zeta ;\psi }}$ is a metric on ℵ. This metric is referred to as the metric induced by the norm of the weighted space.

Proof. 

See ref. [31]. □

3. Uniqueness of Solutions and U-H Stability

We now present the first of result this study, concerning the uniqueness of solutions and the U-H stability of $\psi$-Hilfer FRVIDDE (3), in the following theorem.

Remark 1.

Throughout this section, when it is needed to short some formulas, $\ell ,\rho$, ${\rho }_0$ and σ will represent $\ell \left(t\right),\rho \left(t\right)$, ${\rho }_0\left(t\right)$ and $\sigma \left(t\right)$, respectively.

Theorem 3.

Assume that ℵ is a Banach space and $h_0$, $L_g,L_{{\mathcal{\Im }}_1},\dots ,L_{{\mathcal{\Im }}_N},M$ are positive constants such that the following conditions hold, respectively:

• $\left(As.1\right)$ $h\in C\left(\left[0,T\right]\times \aleph ,\aleph \right)$, $g\in C\left(\left[0,T\right]\times \aleph \times \aleph ,\aleph \right)$, ${\mathcal{\Im }}_1\in C\left(\left[0,T\right]\times \left[0,T\right]\times \aleph \times \aleph ,\aleph \right),\dots ,$${\mathcal{\Im }}_N\in C\left(\left[0,T\right]\times \left[0,T\right]\times \aleph \times \aleph ,\aleph \right)$ with properties that

$${∥h\left(t,\ell \right)∥}_{C_{1-\zeta ;\psi }}\le h_0,$$

$${∥g\left(t,\ell ,\ell \left(\upsilon \right(t\left)\right)\right)-g\left(t,\eta ,\eta \left(\upsilon \right(t\left)\right)\right)∥}_{C_{1-\zeta ;\psi }}\le L_g{∥\ell -\eta ∥}_{C_{1-\zeta ;\psi }\left[0,T\right]}$$

$$+L_g{∥\ell \left(\upsilon \right(t\left)\right)-\eta \left(\upsilon \right(t\left)\right)∥}_{C_{1-\zeta ;\psi }\left[0,T\right]},$$

$$\sum _{i=1}^N{∥{\mathcal{\Im }}_i\left(t,s,\ell ,\ell \left({\upsilon }_i\left(s\right)\right)\right)-{\mathcal{\Im }}_i\left(t,s,\eta ,\eta \left({\upsilon }_i\left(s\right)\right)\right)∥}_{C_{1-\zeta ;\psi }}\le \sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{∥\ell -\eta ∥}_{C_{1-\zeta ;\psi }\left[0,T\right]}$$

$$+\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{∥\ell \left({\upsilon }_i\left(s\right)\right)-\eta \left({\upsilon }_i\left(s\right)\right)∥}_{C_{1-\zeta ;\psi \left[0,T\right]}}$$

for each $t,s\in \left[0,T\right]$ and $\ell ,\eta \in \aleph ;$

• $\left(As.2\right)$ $\varphi \in C\left(\left[0,T\right],\left(0,\infty \right)\right)$ is an increasing function with the property that

$${\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}\left(∥h\left(t,\ell \right)∥\sum _{i=1}^N{Q}_{\psi }^{\alpha }\left(t,s\right)\varphi \left(s\right)\right)ds\le h_{0}M\varphi \left(t\right)$$

for each $t,s\in \left[0,T\right]$ and $\ell \in \aleph$;

• $\left(As.3\right)$

$$0<2\left(\left(1+M\right)L_g+h_0\left(1+M\right)M\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\right)<1.$$

If a function $\rho \in C^1\left(\left[0,T\right],\aleph \right)$ satisfies the relation

$${∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\rho -g\left(t,\rho ,\rho \left(\upsilon \right(t\left)\right)\right)-\underset{0}{\overset{t}{\int }}\left(h\left(t,\rho \right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\rho \left(s\right),\rho \left({\upsilon }_i\left(s\right)\right)\right)\right)ds∥}_{C_{1-\zeta ;\psi }\left[0,T\right]}\le \varphi \left(t\right),$$

(7)

for each $t,s\in \left[0,T\right]$, then there exists a unique differentiable function ${\rho }_0\in \left[0,T\right]\to \aleph$ with the property that for each $t\in \left[0,T\right]$,

$${}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }{\rho }_0=g\left(t,{\rho }_0,{\rho }_0\left(\upsilon \left(t\right)\right)\right)+\left[\underset{0}{\overset{t}{\int }}\left(h\left(t,{\rho }_0\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,{\rho }_0\left(s\right),{\rho }_0\left({\upsilon }_i\left(s\right)\right)\right)\right)ds\right]$$

(8)

$${∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\rho \left(t\right)-{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }{\rho }_0\left(t\right)∥}_{C_{1-\zeta ,\psi }}+{∥\rho \left(t\right)-{\rho }_0\left(t\right)∥}_{C_{1-\zeta ,\psi }}$$

$$\le {\displaystyle \frac{1+M}{1-2\left(\left(1+M\right)L_g+h_0\left(1+M\right)M{\displaystyle \sum _{i=1}^N}{L}_{{\mathcal{\Im }}_i}\right)}}.$$

Remark 2.

Note that Theorem 3 establishes the existence of a unique solution and the U-H stability of ψ-Hilfer FRVIDDE (3).

Proof. 

Let

$$\Omega =\left\{\ell :\left[0,T\right]\to \aleph ,\ell \mathrm{is}\mathrm{differentiable}\right\}$$

and define a mapping $d:\Omega \times \Omega \to \left[0,\infty \right]$ by

$$d\left(\ell ,\eta \right)=inf\{C\in \left[0,\infty \right]:{∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\ell \left(t\right)-{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\eta \left(t\right)∥}_{C_{1-\zeta ;\psi }}$$

$$+{∥\ell \left(t\right)-\eta \left(t\right)∥}_{C_{1-\zeta ;\psi }}\le C\varphi \left(t\right),t\in \left[0,T\right]\}.$$

Depending on the conditions of Theorem 3, it will be shown that $\left(\Omega ,d\right)$ is a complete generalized metric space. To proceed the proof, assume that $d\left(\ell ,\eta \right)>d\left(\ell ,\sigma \right)+d\left(\sigma ,\eta \right)$ whenever $\ell ,\eta ,\sigma \in \Omega$. Then, there exists a $t_0\in \left[0,T\right]$ with the property that

$${∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\ell \left(t_0\right)-{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\eta \left(t_0\right)∥}_{C_{1-\zeta ;\psi }}+{∥\ell \left(t_0\right)-\eta \left(t_0\right)∥}_{C_{1-\zeta ;\psi }}>\left(d\left(\ell ,\sigma \right)+d\left(\sigma ,\eta \right)\right)\varphi \left(t_0\right).$$

Hence, from the definition of the metric d, we obtain

$${∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\ell \left(t_0\right)-{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\eta \left(t_0\right)∥}_{C_{1-\zeta ;\psi }}+{∥\ell \left(t_0\right)-\eta \left(t_0\right)∥}_{C_{1-\zeta ;\psi }}$$

$$>{∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\ell \left(t_0\right)-{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\sigma \left(t_0\right)∥}_{C_{1-\zeta ;\psi }}+{∥\ell \left(t_0\right)-\sigma \left(t_0\right)∥}_{C_{1-\zeta ;\psi }}$$

$$+{∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\sigma \left(t_0\right)-{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\eta \left(t_0\right)∥}_{C_{1-\zeta ;\psi }}+{∥\sigma \left(t_0\right)-\eta \left(t_0\right)∥}_{C_{1-\zeta ;\psi }},$$

which is a contradiction.

We will now prove that $\left(\Omega ,d\right)$ is a complete generalized metric space. Let $\left({\ell }_n\right)$ be a Cauchy sequence in the space $\left(\Omega ,d\right)$. According to the definition of a Cauchy sequence, from the definition of the metric d, it follows that $\forall \epsilon >0$, $N_{\epsilon }\in N$, $\forall m,n>N_{\epsilon }$, $\forall \in \left[0,T\right]$,

$${∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }{\ell }_n\left(t\right)-{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }{\ell }_m\left(t\right)∥}_{C_{1-\zeta ;\psi }}+{∥{\ell }_n\left(t\right)-{\ell }_m\left(t\right)∥}_{C_{1-\zeta ;\psi }}<\epsilon \varphi \left(t\right).$$

(9)

We note that, in light of the inequality (9) and the continuity of the function $\varphi$ on the compact interval $\left[0,T\right]$, the sequence $\left({\ell }_n\right)$ and $\left({}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }{\ell }_n\right)$ are uniformly convergent on the interval to $\left[0,T\right]$. Thus, there exists a differentiable function ℓ with the property that the sequence $\left({\ell }_n\right)$ and $\left({}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }{\ell }_n\right)$ are uniformly convergent to ℓ and ${}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\ell$, respectively. Thereby, $\ell \in \Omega$ and from the inequality (9), letting $m\to \infty$, we get that $\forall \epsilon >0$, $\exists N_{\epsilon }\in N$, $n\ge N_{\epsilon }$, $t\in \left[0,T\right]$,

$${∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }{\ell }_n\left(t\right)-{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\ell \left(t\right)∥}_{C_{1-\zeta ;\psi }}+{∥{\ell }_n\left(t\right)-\ell \left(t\right)∥}_{C_{1-\zeta ;\psi }}<\epsilon \varphi \left(t\right).$$

Accordingly, we get $\forall \epsilon >0$, $\exists N_{\epsilon }\in N$, $n\ge N_{\epsilon }$, $d\left(x_n,x\right)\le \epsilon$. Hence, it follows that the space $\left(\Omega ,d\right)$ is complete. We now depict an operator $\Lambda :\Omega \to \Omega$ by

$$\Lambda \left(\ell \right)=0,t\in [-\upsilon ,0]$$

and

$$\Lambda \left(\ell \right)=I_{0^{+}}^{\alpha ;\psi }g\left(t,\ell ,\ell \left(\upsilon \right(t\left)\right)\right)+I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{t}{\int }}\left(h\left(t,\ell \right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\ell \left(s\right),\ell \left({\upsilon }_i\left(s\right)\right)\right)\right)ds\right].$$

(10)

Subsequently, we now demonstrate that $\Lambda$ is a strictly contractive operator on $\Omega$. Let $C_{\ell \eta }\in \left[0,\infty \right]$ be a constant such that $d(\ell ,\eta )\le C_{\ell \eta }$ for each $\ell ,\eta \in \aleph$. Thereby, in subsequent, according to this information, we get

$${∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\ell \left(t\right)-{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\eta \left(t\right)∥}_{C_{1-\zeta ,\psi }}+{∥\ell \left(t\right)-\eta \left(t\right)∥}_{C_{1-\zeta ,\psi }}<C_{\ell \eta }\varphi \left(t\right),$$

for reach $t\in \left[0,T\right]$.

Using (10) and the conditions $\left(As.1\right),\left(As.2\right)$, we have

$$∥\ell \left(t\right)-\eta \left(t\right)∥=0\mathrm{for}\mathrm{reach}t\in \left[-\tau ,0\right]$$

and

$${∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\left(\Lambda \ell \left(t\right)-\Lambda \eta \left(t\right)\right)∥}_{C_{1-\zeta ;\psi }}+{∥\Lambda \ell \left(t\right)-\Lambda \eta \left(t\right)∥}_{C_{1-\zeta ;\psi }}$$

$$\le {∥g\left(t,\ell \left(t\right),\ell \left(\upsilon \right(t\left)\right)\right)-g\left(t,\eta \left(t\right),\eta \left(\upsilon \right(t\left)\right)\right)∥}_{C_{1-\zeta ;\psi }}$$

$$+h_0\underset{0}{\overset{t}{\int }}\left(\sum _{i=1}^N{∥{\mathcal{\Im }}_i\left(t,s,\ell \left(s\right),\ell \left({\upsilon }_i\left(s\right)\right)\right)-{\mathcal{\Im }}_i\left(t,s,\eta \left(s\right),\eta \left({\upsilon }_i\left(s\right)\right)\right)∥}_{C_{1-\zeta ;\psi }}\right)ds$$

$$+{∥I_{0^{+}}^{\alpha ;\psi }\left(g\left(t,\ell \left(t\right),\ell \left(\upsilon \right(t\left)\right)\right)-g\left(t,\eta \left(t\right),\eta \left(\upsilon \right(t\left)\right)\right)\right)∥}_{C_{1-\zeta ;\psi }}$$

$$+{∥I_{0^{+}}^{\alpha ;\psi }\left[h_0\underset{0}{\overset{t}{\int }}\left(\sum _{i=1}^N\left({\mathcal{\Im }}_i\left(t,s,\ell \left(s\right),\ell \left({\upsilon }_i\left(s\right)\right)\right)-{\mathcal{\Im }}_i\left(t,s,\eta \left(s\right),\eta \left({\upsilon }_i\left(s\right)\right)\right)\right)\right)ds\right]∥}_{C_{1-\zeta ;\psi }}$$

$$\le L_g{∥\ell \left(t\right)-\eta \left(t\right)∥}_{C_{1-\zeta ;\psi }}+L_g{∥\ell \left(\upsilon \right(t\left)\right)-\eta \left(\upsilon \right(t\left)\right)∥}_{C_{1-\zeta ;\psi }}$$

$$+h_0\underset{0}{\overset{t}{\int }}\left(\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{∥\ell \left(s\right)-\eta \left(s\right)∥}_{C_{1-\zeta ;\psi }}\right)ds+h_0\underset{0}{\overset{t}{\int }}\left(\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{∥\ell \left({\upsilon }_i\left(s\right)\right)-\eta \left({\upsilon }_i\left(s\right)\right)∥}_{C_{1-\zeta ;\psi }}\right)ds$$

$$+I_{0^{+}}^{\alpha ;\psi }\left(L_g{∥\ell \left(t\right)-\eta \left(t\right)∥}_{C_{1-\zeta ;\psi }}\right)+I_{0^{+}}^{\alpha ;\psi }\left(L_g{∥\ell \left(\nu \right(t\left)\right)-\eta \left(\nu \right(t\left)\right)∥}_{C_{1-\zeta ;\psi }}\right)$$

$$+I_{0^{+}}^{\alpha ;\psi }∥\left[\underset{0}{\overset{t}{\int }}\left(h_0\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{∥\ell \left(s\right)-\eta \left(s\right)∥}_{C_{1-\zeta ;\psi }}\right)ds\right]∥$$

$$+I_{0^{+}}^{\alpha ;\psi }∥\left[\underset{0}{\overset{t}{\int }}\left(h_0\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{∥\ell \left({\upsilon }_i\left(s\right)\right)-\eta \left({\upsilon }_i\left(s\right)\right)∥}_{C_{1-\zeta ;\psi }}\right)ds\right]∥$$

$$\le L_g{C}_{\ell \eta }\varphi \left(t\right)+L_g{C}_{\ell \eta }\varphi \left(\upsilon \left(t\right)\right)+h_0{C}_{\ell \eta }\underset{0}{\overset{t}{\int }}\left(\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\varphi \left(s\right)\right)ds$$

$$+h_0{C}_{\ell \eta }\underset{0}{\overset{t}{\int }}\left(\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\varphi \left(\upsilon \left(s\right)\right)\right)ds+L_g{I}_{0^{+}}^{\alpha ;\psi }{C}_{\ell \eta }\varphi \left(t\right)+L_g{I}_{0^{+}}^{\alpha ;\psi }{C}_{\ell \eta }\varphi \left(\upsilon \left(t\right)\right)$$

$$+I_{0^{+}}^{\alpha ;\psi }∥\left[\underset{0}{\overset{t}{\int }}\left(h_0\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{C}_{\ell \eta }\varphi \left(s\right)\right)ds\right]∥+I_{0^{+}}^{\alpha ;\psi }∥\left[\underset{0}{\overset{t}{\int }}\left(h_0\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{C}_{\ell \eta }\varphi \left(\upsilon \left(s\right)\right)\right)ds\right]∥$$

$$\le L_g{C}_{\ell \eta }\varphi \left(t\right)+L_g{C}_{\ell \eta }\varphi \left(\upsilon \left(t\right)\right)+h_0{C}_{\ell \eta }M\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\varphi \left(t\right)+h_0{C}_{\ell \eta }M\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\varphi (\upsilon \left(t\right))$$

$$+L_{g}M{C}_{\ell \eta }\varphi \left(t\right)+L_{g}M{C}_{\ell \eta }\varphi \left(\upsilon \left(t\right)\right)+h_0{C}_{\ell \eta }{M}^2\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\varphi \left(t\right)+h_0{C}_{\ell \eta }{M}^2\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\varphi \left(\upsilon \left(t\right)\right)$$

$$\le \left[\left(1+M\right)L_g+h_0\left(1+M\right)M\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\right]C_{\ell \eta }\varphi \left(t\right)$$

$$+\left[\left(1+M\right)L_g+h_0\left(1+M\right)M\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\right]C_{\ell \eta }\varphi \left(\upsilon \left(t\right)\right)$$

$$\le 2\left[\left(1+M\right)L_g+h_0\left(1+M\right)M\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\right]C_{\ell \eta }\varphi \left(t\right)$$

(11)

since the function $\varphi$ is increasing. Taking inequality (11) into account, we obtain

$$d\left(\Lambda \ell ,\Lambda \eta \right)\le 2\left(\left(1+M\right)L_g+h_0\left(1+M\right)M\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\right)d\left(\ell ,\eta \right),$$

(12)

for each $t\in [0,T]$. Thereby, since

$$0<2\left(\left(1+M\right)L_g+h_0\left(1+M\right)M\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\right)<1,$$

it follows immediately that $\Lambda$ is strictly contractive.

In contrast to the previous result, it is obvious that $\rho \in \Omega$, and we get from (7) that

$${∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\left(\Lambda \rho \left(t\right)-\rho \left(t\right)\right)∥}_{C_{1-\zeta ;\psi }}+{∥\Lambda \rho \left(t\right)-\rho \left(t\right)∥}_{C_{1-\zeta ;\psi }}$$

$$={∥g\left(t,\rho \left(t\right),\rho \left(\upsilon \right(t\left)\right)\right)+\underset{0}{\overset{t}{\int }}\left(h\left(t,\rho \left(t\right)\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\rho \left(s\right),\rho \left({\upsilon }_i\left(s\right)\right)\right)\right)ds-{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\rho \left(t\right)∥}_{C_{1-\zeta ;\psi }}$$

$$+{∥I_{0^{+}}^{\alpha ;\psi }g\left(t,\rho \left(t\right),\rho \left(\upsilon \right(t\left)\right)\right)+I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{t}{\int }}\left(h\left(t,\rho \left(t\right)\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\rho \left(s\right),\rho \left({\upsilon }_i\left(s\right)\right)\right)\right)ds\right]-\rho \left(t\right)∥}_{C_{1-\zeta ;\psi }}$$

$$\le \varphi \left(t\right)+{∥I_{0^{+}}^{\alpha ;\psi }g\left(t,\rho \left(t\right),\rho \left(\upsilon \right(t\left)\right)\right)+I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{t}{\int }}\left(h\left(t,\rho \left(t\right)\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\rho \left(s\right),\rho \left({\upsilon }_i\left(s\right)\right)\right)\right)ds\right]-\rho \left(t\right)∥}_{C_{1-\zeta ;\psi }}$$

$$\le (1+M)\varphi \left(t\right).$$

For this reason, we reach the result as follows:

$$d\left(\Lambda \rho ,\rho \right)\le 1+M<\infty ,\mathrm{where}M<1.$$

(13)

For the condition $(C.2)$ of Theorem 1, it follows that there exists a unique element

$${\rho }_0\in {\aleph }^{\ast }=\left\{\eta \in \aleph :d\left(\Lambda \rho ,\eta \right)<\infty \right\}$$

with the property that $\Lambda {\rho }_0={\rho }_0$ or equivalently

$${\rho }_0=I_{0^{+}}^{\alpha ;\psi }g\left(t,{\rho }_0,{\rho }_0\left(\upsilon \left(t\right)\right)\right)+I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{t}{\int }}\left(h\left(t,{\rho }_0\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,{\rho }_0\left(s\right),{\rho }_0\left({\upsilon }_i\left(s\right)\right)\right)\right)ds\right].$$

(14)

Since ${\rho }_0$ is differentiable, the functions g and ${\mathcal{\Im }}_i,i=1,\dots ,N,$ are continuous, then applying the $\psi$-HFD ${}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }(.)$ on the both sides of (14) and using Theorem 2, we obtain from (14) that

$${}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }{\rho }_0=g\left(t,{\rho }_0,{\rho }_0\left(\upsilon \left(t\right)\right)\right)+\left[\underset{0}{\overset{t}{\int }}\left(h\left(t,{\rho }_0\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,{\rho }_0\left(s\right),{\rho }_0\left({\upsilon }_i\left(s\right)\right)\right)\right)ds\right].$$

Next, according to the condition (C.2) of Theorem 1 and (13), we get

$$d\left(\rho ,{\rho }_0\right)\le {\displaystyle \frac{1}{1-2\left(\left(1+M\right)L_g+h_0\left(1+M\right)M{\displaystyle \sum _{i=1}^N}{L}_{{\mathcal{\Im }}_i}\right)}}d\left(\Lambda \rho ,\rho \right)$$

$$\le {\displaystyle \frac{1+M}{1-2\left(\left(1+M\right)L_g+h_0\left(1+M\right)M{\displaystyle \sum _{i=1}^N}{L}_{{\mathcal{\Im }}_i}\right)}}.$$

By virtue of the definition of the metric d, we reach that the inequality (9) holds, for each $t\in \left[0,T\right]$. Let

$$\xi ={\displaystyle \frac{1+M}{1-2\left(\left(1+M\right)L_g+h_0\left(1+M\right)M{\displaystyle \sum _{i=1}^N}{L}_{{\mathcal{\Im }}_i}\right)}}.$$

Subsequently, assume that $\sigma$ is another differentiable function satisfying $\psi$-FRVIDDE (8) and inequality (9). Then, we have $\rho \in \Omega$, $d\left(\rho ,\sigma \right)<\xi$, and

$${}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\sigma =g\left(t,\sigma ,\sigma \left(\upsilon \right(t\left)\right)\right)+\underset{0}{\overset{t}{\int }}\left(h\left(t,\sigma \right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\sigma \left(s\right),\sigma \left({\upsilon }_i\left(s\right)\right)\right)\right)ds.$$

(15)

To complete the proof, it remains to show the uniqueness of ${\rho }_0$. Hence, it suffices to show that $\sigma$ is a fixed point of the operator $\Lambda$ and $\sigma \in {\Omega }^{\ast }$. Using $\psi$-Hilfer FRVIDDE (15), we can verify that $\Lambda \sigma =\sigma$. We will show that $d\left(\Lambda \rho ,\sigma \right)<\infty$. Taking into consideration $\psi$-Hilfer FRVIDDE (15) and the fact that $d\left(\rho ,\sigma \right)<\xi$, we obtain

$${∥{}^{H}D_{0^{+}}^{\alpha ,\beta ;\psi }\left(\Lambda \rho \left(t\right)-\sigma \left(t\right)\right)∥}_{C_{1-\zeta ;\psi }}+{∥\Lambda \rho \left(t\right)-\sigma \left(t\right)∥}_{C_{1-\zeta ;\psi }}$$

$$=∥g\left(t,\rho \left(t\right),\rho \left(\upsilon \right(t\left)\right)\right)+\underset{0}{\overset{t}{\int }}\left(h\left(t,\rho \left(t\right)\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\rho \left(s\right),\rho \left({\upsilon }_i\left(s\right)\right)\right)\right)ds$$

$$-{g\left(t,\sigma \left(t\right),\sigma \left(\upsilon \right(t\left)\right)\right)-\underset{0}{\overset{t}{\int }}\left(h\left(t,\sigma \left(t\right)\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\sigma \left(s\right),\sigma \left({\upsilon }_i\left(s\right)\right)\right)\right)ds∥}_{C_{1-\zeta ;\psi }}$$

$$+∥I_{0^{+}}^{\alpha ;\psi }g\left(t,\rho \left(t\right),\rho \left(\upsilon \right(t\left)\right)\right)+I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{t}{\int }}\left(h\left(t,\rho \left(t\right)\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\rho \left(s\right),\rho \left({\upsilon }_i\left(s\right)\right)\right)\right)ds\right]$$

$$-{I_{0^{+}}^{\alpha ;\psi }g\left(t,\sigma \left(t\right),\sigma \left(\upsilon \right(t\left)\right)\right)-I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{t}{\int }}\left(h\left(t,\sigma \left(t\right)\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\sigma \left(s\right),\sigma \left({\upsilon }_i\left(s\right)\right)\right)\right)ds\right]∥}_{C_{1-\zeta ;\psi }}$$

$$\le {∥g\left(t,\rho \left(t\right),\rho \left(\upsilon \right(t\left)\right)\right)-g\left(t,\sigma \left(t\right),\sigma \left(\upsilon \right(t\left)\right)\right)∥}_{C_{1-\zeta ;\psi }}$$

$$+\underset{0}{\overset{t}{\int }}{∥h\left(t,\rho \left(t\right)\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\rho \left(s\right),\rho \left({\upsilon }_i\left(s\right)\right)\right)-h\left(t,\sigma \left(t\right)\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\sigma \left(s\right),\sigma \left({\upsilon }_i\left(s\right)\right)\right)∥}_{C_{1-\zeta ;\psi }}ds$$

$$+{∥I_{0^{+}}^{\alpha ;\psi }\left[g\left(t,\rho \left(t\right),\rho \left(\upsilon \right(t\left)\right)\right)-g\left(t,\sigma \left(t\right),\sigma \left(\upsilon \right(t\left)\right)\right)\right]∥}_{C_{1-\zeta ;\psi }}$$

$$+\underset{0}{\overset{t}{\int }}{∥I_{0^{+}}^{\alpha ;\psi }\left[h\left(t,\rho \left(t\right)\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\rho \left(s\right),\rho \left({\upsilon }_i\left(s\right)\right)\right)-h\left(t,\sigma \left(t\right)\right)\sum _{i=1}^N{\mathcal{\Im }}_i\left(t,s,\sigma \left(s\right),\sigma \left({\upsilon }_i\left(s\right)\right)\right)\right]ds∥}_{C_{1-\zeta ;\psi }}$$

$$\le L_g{∥\rho \left(t\right)-\sigma \left(t\right)∥}_{C_{1-\zeta ;\psi }}+L_g{∥\rho \left(\upsilon \right(t\left)\right)-\sigma \left(\upsilon \right(t\left)\right)∥}_{C_{1-\zeta ;\psi }}$$

$$+h_0\underset{0}{\overset{t}{\int }}\left(\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{∥\rho \left(s\right)-\sigma \left(s\right)∥}_{C_{1-\zeta ;\psi }}\right)ds+h_0\underset{0}{\overset{t}{\int }}\left(\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{∥\rho \left({\upsilon }_i\left(s\right)\right)-\sigma \left({\upsilon }_i\left(s\right)\right)∥}_{C_{1-\zeta ;\psi }}\right)ds$$

$$+I_{0^{+}}^{\alpha ;\psi }\left(\left[L_g{∥\rho \left(t\right)-\sigma \left(t\right)∥}_{C_{1-\zeta ;\psi }}\right]\right)+I_{0^{+}}^{\alpha ;\psi }\left(\left[L_g{∥\rho \left(\nu \right(t\left)\right)-\sigma \left(\nu \right(t\left)\right)∥}_{C_{1-\zeta ;\psi }}\right]\right)$$

$$+I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{t}{\int }}\left(h_0\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{∥\rho \left(s\right)-\sigma \left(s\right)∥}_{C_{1-\zeta ;\psi }}\right)ds\right]$$

$$+I_{0^{+}}^{\alpha ;\psi }\left[\underset{0}{\overset{t}{\int }}\left(h_0\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{∥\rho \left({\upsilon }_i\left(s\right)\right)-\sigma \left({\upsilon }_i\left(s\right)\right)∥}_{C_{1-\zeta ;\psi }}\right)ds\right]$$

$$\le 2\left[\left(1+M\right)L_g+h_0\left(1+M\right)M\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\right]\xi .$$

(16)

Hence, it follows from (16) that

$$d\left(\Lambda \rho ,\sigma \right)\le 2\left(\left(1+M\right)L_g+h_0\left(1+M\right)M\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\right)\xi <\infty .$$

(17)

Thereby we completed the proof of Theorem 2. □

We now give the second result of this study with respect to the uniqueness solution and the U-H stability of $\psi$-Hilfer FRVDIE (4).

Theorem 4.

Assume that ℵ is a Banach space and $h_0$, $L_g,L_{{\mathcal{\Im }}_1},\dots ,L_{{\mathcal{\Im }}_N},M$ are positive constants such that the following conditions hold, respectively:

• $\left(As.4\right)$ $h\in C\left(\left[0,T\right]\times \aleph ,\aleph \right)$, $g\in C\left(\left[0,T\right]\times \aleph \times \aleph ,\aleph \right)$, ${\mathcal{\Im }}_1\in C\left(\left[0,T\right]\times \left[0,T\right]\times \aleph \times \aleph ,\aleph \right),\dots ,$${\mathcal{\Im }}_N\in C\left(\left[0,T\right]\times \left[0,T\right]\times \aleph \times \aleph ,\aleph \right)$ with

$${∥h\left(t,\ell \right)∥}_{C_{1-\zeta ;\psi }}\le h_0,$$

$${∥g\left(t,\ell ,\ell \left(\upsilon \right(t\left)\right)\right)-g\left(t,\eta ,\eta \left(\upsilon \right(t\left)\right)\right)∥}_{C_{1-\zeta ;\psi }}\le L_g{∥\ell -\eta ∥}_{C_{1-\zeta ;\psi }}$$

$$+L_g{∥\ell \left(\upsilon \right(t\left)\right)-\eta \left(\upsilon \right(t\left)\right)∥}_{C_{1-\zeta ;\psi }},$$

$$\sum _{i=1}^N{∥{\mathcal{\Im }}_i\left(t,s,\ell ,\ell \left({\upsilon }_i\left(s\right)\right)\right)-{\mathcal{\Im }}_i\left(t,s,\eta ,\eta \left({\upsilon }_i\left(s\right)\right)\right)∥}_{C_{1-\zeta ;\psi }}$$

$$\le \sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{∥\ell -\eta ∥}_{C_{1-\zeta ;\psi }}+\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{∥\ell \left({\upsilon }_i\left(s\right)\right)-\eta \left({\upsilon }_i\left(s\right)\right)∥}_{C_{1-\zeta ;\psi }}$$

for each for each $t\in \left[0,T\right]$ and $\ell ,\eta \in \aleph$;

• $\left(As.5\right)$ $\varphi \in C\left(\left[0,T\right],\left(0,\infty \right)\right)$ is a nondecreasing function with the property that

$${\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}\left(∥h\left(t,\ell \right)∥\sum _{i=1}^N{Q}_{\psi }^{\alpha }\left(t,s\right)\varphi \left(s\right)\right)ds\le h_{0}M\varphi \left(t\right)$$

for each for each $t,s\in \left[0,T\right]$ and $\ell ,\eta \in R$, where $Q_{\psi }^{\alpha }\left(t,s\right)$ has been defined previously;

• $(As.6)$

$$0<2\left(L_g+h_0\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\right)M<1.$$

If the function $\rho \in C\left(\left[0,T\right],\aleph \right)$ satisfies the relation

$${∥\rho -g\left(t,\rho ,\rho \left(\upsilon \right(t\left)\right)\right)-{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}\left(h\left(t,\rho \right)\sum _{i=1}^N{Q}_{\psi }^{\alpha }\left(t,s\right){\mathcal{\Im }}_i\left(t,s,\rho \left(s\right),\rho \left({\upsilon }_i\left(s\right)\right)\right)\right)ds∥}_{C_{1-\zeta ;\psi }}\le \varphi \left(t\right)$$

(18)

for each $t\in \left[0,T\right]$, then there exists a unique function ${\rho }_0\in C\left(\left[0,T\right],\aleph \right)$ with the properties that

$${\rho }_0=g\left(t,{\rho }_0,{\rho }_0\left(\upsilon \left(t\right)\right)\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}\left(h\left(t,{\rho }_0\right)\sum _{i=1}^N{Q}_{\psi }^{\alpha }\left(t,s\right){\mathcal{\Im }}_i\left(t,s,{\rho }_0\left(s\right),{\rho }_0\left({\upsilon }_i\left(s\right)\right)\right)\right)ds$$

(19)

and

$${∥\rho \left(t\right)-{\rho }_0\left(t\right)∥}_{C_{1-\zeta ,\psi }}\le {\displaystyle \frac{1}{1-2\left(L_g+h_0{\displaystyle \sum _{i=1}^N}{L}_{{\mathcal{\Im }}_i}\right)M}}\varphi \left(t\right).$$

(20)

Remark 3.

It is worth to note that Theorem 4 allows a unique solution and the U-H stability of ψ-Hilfer FRVDIE (4).

Proof. 

Consider the set defined as follows,

$$\Omega :=\left\{\ell :\left[0,T\right]\to X:\ell \mathrm{is}\mathrm{continuous}\right\}$$

and depict the generalized metric $d:\Omega \times \Omega \to \left[0,\infty \right]$ by

$$d(\ell ,\eta )=inf\left\{C\in [0,\infty ]\left|{∥\ell \left(t\right)-\eta \left(t\right)∥}_{C_{1-\zeta ;\psi }}\le C\varphi \left(t\right),t\in \left[0,T\right]\right.\right\}.$$

(21)

As in the proof of Theorem 3, it can be readily shown that $\left(\Omega ,d\right)$ is a generalized complete metric space. Indeed, following similar calculations as in Theorem 3, one can show that for any $\ell ,\eta \in \Omega$,

$$d\left(\Lambda \ell ,\Lambda \eta \right)\le 2\left(L_g+h_0\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\right)Md\left(\ell ,\eta \right).$$

Then, since $0<2\left(L_g+h_0{\displaystyle \sum _{i=1}^N}{L}_{{\mathcal{\Im }}_i}\right)M<1$, it can be deduced that the operator $\Lambda$ is strictly contractive. Next, in the light of the inequality (18), we obtain $d\left(\Lambda \rho ,\rho \right)\le 1<\infty$. Therefore, according to Theorem 1, the operator $\Lambda$ has a unique fixed point ${\rho }_0$ in the set

$${\Omega }^{\ast }=\left\{\eta \in \Omega :d\left(\Lambda \rho ,\eta \right)<\infty \right\}.$$

Let $\sigma$ be another function satisfying $\psi$-FRVDIE (19) and the inequality (20). Hence, we have $\rho \in \aleph$,

$$d\left(\rho ,\sigma \right)\le {\displaystyle \frac{1}{1-2\left(L_g+h_0{\displaystyle \sum _{i=1}^N}{L}_{{\mathcal{\Im }}_i}\right)M}}$$

and

$$\sigma =g\left(t,\sigma ,\sigma \left(\upsilon \right(t\left)\right)\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}\left(h\left(t,\sigma \right)\sum _{i=1}^N{Q}_{\psi }^{\alpha }(t,s){\mathcal{\Im }}_i\left(t,s,\sigma \left(s\right),\sigma \left({\upsilon }_i\left(s\right)\right)\right)\right)ds.$$

The uniqueness of $\sigma$ follows after showing that $\sigma$ is a fixed point of the operator $\Lambda$ and that $\sigma \in {\aleph }^{\ast }$. From $\psi$-FRVDIE (19), it follows that $\Lambda \sigma =\sigma$. $\psi$-FRVDIE (19) further shows that

$$d\left(\rho ,\sigma \right)\le {\displaystyle \frac{1}{1-2\left(L_g+h_0{\displaystyle \sum _{i=1}^N}{L}_{{\mathcal{\Im }}_i}\right)M}}.$$

By using the conditions (As.4) and (As.5) of Theorem 4, we get

$${∥\Lambda \rho \left(t\right)-\sigma \left(t\right)∥}_{C_{1-\zeta ;\psi }}$$

$$=∥g\left(t,\rho \left(t\right),\rho \left(\upsilon \right(t\left)\right)\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}\left(h\left(t,\rho \left(t\right)\right)\sum _{i=1}^N{Q}_{\psi }^{\alpha }(t,s){\mathcal{\Im }}_i\left(t,s,\rho \left(s\right),\rho \left({\upsilon }_i\left(s\right)\right)\right)\right)ds$$

$$-{g\left(t,\sigma \left(t\right),\sigma \left(\upsilon \right(t\left)\right)\right)-{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}\left(h\left(t,\sigma \left(t\right)\right)\sum _{i=1}^N{Q}_{\psi }^{\alpha }(t,s){\mathcal{\Im }}_i\left(t,s,\sigma \left(s\right),\sigma \left({\upsilon }_i\left(s\right)\right)\right)\right)ds∥}_{C_{1-\zeta ;\psi }}$$

$$\le {∥g\left(t,\rho \left(t\right),\rho \left(\upsilon \right(t\left)\right)\right)-g\left(t,\sigma \left(t\right),\sigma \left(\upsilon \right(t\left)\right)\right)∥}_{C_{1-\zeta ;\psi }}$$

$$+{\displaystyle \frac{h_0}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{Q}_{\psi }^{\alpha }(t,s)\sum _{i=1}^N{∥{\mathcal{\Im }}_i\left(t,s,\rho \left(s\right),\rho \left({\upsilon }_i\left(s\right)\right)\right)-{\mathcal{\Im }}_i\left(t,s,\rho \left(s\right),\rho \left({\upsilon }_i\left(s\right)\right)\right)∥}_{C_{1-\zeta ;\psi }}ds$$

$$\le L_g{∥\rho \left(t\right)-\sigma \left(t\right)∥}_{C_{1-\zeta ;\psi }}+L_g{∥\rho \left(\upsilon \right(t\left)\right)-\sigma \left(\upsilon \right(t\left)\right)∥}_{C_{1-\zeta ;\psi }}$$

$$+{\displaystyle \frac{h_0}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}\left(\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{Q}_{\psi }^{\alpha }(t,s){∥\rho \left(s\right)-\sigma \left(s\right)∥}_{C_{1-\zeta ;\psi }}\right)ds$$

$$+{\displaystyle \frac{h_0}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}\left(\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}{Q}_{\psi }^{\alpha }(t,s){∥\rho \left({\upsilon }_i\left(s\right)\right)-\sigma \left({\upsilon }_i\left(s\right)\right)∥}_{C_{1-\zeta ;\psi }}\right)ds$$

$$\le 2L_g{C}_{\rho \sigma }\varphi \left(t\right)+2h_{0}M{C}_{\rho \sigma }\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\varphi \left(t\right)$$

$$\le 2\left(L_g+h_{0}M\sum _{i=1}^N{L}_{{\mathcal{\Im }}_i}\right)d\left(\rho ,\sigma \right)\varphi \left(t\right)$$

$$\le {\displaystyle \frac{2L_g+2h_0{\displaystyle \sum _{i=1}^N}{L}_{{\mathcal{\Im }}_i}}{1-2\left(L_g+h_0{\displaystyle \sum _{i=1}^N}{L}_{{\mathcal{\Im }}_i}\right)M}}\varphi \left(t\right),$$

which yields that $d\left(\Lambda \rho ,\sigma \right)<\infty$. Thereby, we completed the proof of Theorem 4. □

Remark 4.

It is important to note that the proofs of Theorems 3 and 4 are established using the Banach contraction mapping principle, the definition of U–H stability of ψ-Hilfer FRVIDDE (3) and ψ-Hilfer FRVDIE (4), the properties of ψ-Hilfer operator, and other techniques. Indeed, these fundamental tools necessitate the sufficient conditions (As.1), (As.2) and (As.3) formulated in Theorem 3 and the sufficient conditions (As.4), (As.5) and (As.6) in Theorem 4, respectively.

4. Applications

In this section, we present two illustrative examples that verify the conditions of Theorems 3 and 4.

Example 1.

We analyze the following ψ-Hilfer FRVIDDE incorporating a variable delay:

$$\left\{\begin{array}{ll}{}^{H}D_{0^{+}}^{{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}};t^2}\ell \left(t\right)=& {\displaystyle \frac{1}{500exp\left(1+t^2\right)}}tanh\left(\ell \left(t\right)\right)+{\displaystyle \frac{1}{500exp\left(1+t^2\right)}}tanh\left(\ell \left(2^{-1}t\right)\right)\\ & +{\displaystyle \frac{1}{500}}\left({\displaystyle \frac{tanh\left(\ell \left(t\right)\right)}{1+exp\left(t\right)}}\right)\underset{0}{\overset{t}{\int }}{\displaystyle \frac{tanh\left(\ell \left(s\right)\right)+tanh\left(\ell \left(2^{-1}s\right)\right)}{1+exp\left(s^2+t^2\right)}}ds,t\in \left(0,1\right],\\ & I_{0^{+}}^{1-\gamma ;\psi }\ell \left(0^{+}\right)=0,\ell \left(t\right)=\theta \left(t\right)=0,t\in \left[-2^{-1},0\right].\end{array}\right.$$

(22)

According to the ψ-Hilfer FRVIDDE (22), we have $\alpha =\beta ={\displaystyle \frac{1}{4}}$, $\psi \left(t\right)=t^2$ and ${\upsilon }_1\left(t\right)={\displaystyle \frac{t}{2}}$. Moreover, $\gamma =\alpha +\beta (1-\alpha )={\displaystyle \frac{7}{16}}$, $\upsilon =max\left\{2^{-1}t\right\}=2^{-1},t\in \left[0,1\right]$. Through the comparison of ψ-Hilfer FRVIDDE (22) and ψ-Hilfer FRVIDDE (3), we formulate the relevant relations as follows:

$$(0,T]=(0,1],T=1,$$

$$\Gamma \left(\alpha \right)=\Gamma \left({\displaystyle \frac{1}{4}}\right)\cong 3.6256,$$

$$\Gamma \left(\alpha +1\right)=\Gamma \left({\displaystyle \frac{5}{4}}\right)\cong 0.9064,$$

$${\upsilon }_1\in C\left(\left[0,T\right],R^{+}\right),0\le {\upsilon }_1\left(t\right)={\displaystyle \frac{1}{2}}t\le {\displaystyle \frac{1}{2}}=\upsilon ,t\in \left[0,1\right];$$

$$g\left(t,\ell ,\ell \left(\upsilon \right(t\left)\right)\right)={\displaystyle \frac{1}{500exp\left(1+t^2\right)}}tanh\left(\ell \right)+{\displaystyle \frac{1}{500exp\left(1+t^2\right)}}tanh\left(\ell \left(2^{-1}t\right)\right),$$

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

$$\left(R,∥.∥\right)=\left(R,\left|.\right|\right)\mathit{is}a\mathit{Banach}\mathit{space};$$

$$∥g\left(t,\ell ,\ell \left(\upsilon \right(t\left)\right)\right)-g\left(t,\eta ,\eta \left(\upsilon \right(t\left)\right)\right)∥$$

$$\le {\displaystyle \frac{1}{500}}∥tanh\left(\ell \right)-tanh\left(\eta \right)∥+{\displaystyle \frac{1}{500}}∥tanh\left(\ell \left(2^{-1}t\right)\right)-tanh\left(\eta \left(2^{-1}t\right)\right)∥$$

$$\le {\displaystyle \frac{1}{500}}∥\ell -\eta ∥+{\displaystyle \frac{1}{500}}∥\ell \left(2^{-1}t\right)-\eta \left(2^{-1}t\right)∥,$$

where $L_g={\displaystyle \frac{1}{500}};$

$$h\left(t,\ell \right)={\displaystyle \frac{1}{500exp\left(1+t^2\right)}}tanh\left(\ell \right),$$

$$h\in C\left(\left[0,1\right]\times R,R\right),$$

$$∥h\left(t,\ell \right)∥\le {\displaystyle \frac{1}{500exp\left(1+t^2\right)}}tanh\left(\ell \right)\le {\displaystyle \frac{1}{500}}=h_0;$$

$${\mathcal{\Im }}_1\left(t,s,\ell ,\ell \left({\upsilon }_i\left(s\right)\right)\right)={\displaystyle \frac{tanh\left(\ell \right)}{1+exp\left(s^2+t^2\right)}}+{\displaystyle \frac{tanh\left(\ell \left(2^{-1}s\right)\right)}{1+exp\left(s^2+t^2\right)}},$$

$${\mathcal{\Im }}_1\in C\left(\left[0,1\right]\times \left[0,1\right]\times R\times R,R\right),$$

$$∥{\mathcal{\Im }}_1\left(t,s,\ell ,\ell \left({\upsilon }_i\left(s\right)\right)\right)-{\mathcal{\Im }}_1\left(t,s,\eta ,\eta \left({\upsilon }_i\left(s\right)\right)\right)∥\le {\displaystyle \frac{1}{500}}∥tanh\left(\ell \right)-tanh\left(\eta \right)∥$$

$$+{\displaystyle \frac{1}{500}}∥tanh\left(\ell \left(2^{-1}s\right)\right)-tanh\left(\eta \left(2^{-1}s\right)\right)∥$$

$$\le {\displaystyle \frac{1}{500}}∥\ell -\eta ∥+{\displaystyle \frac{1}{500}}∥\ell \left(2^{-1}s\right)-\eta \left(2^{-1}s\right)∥,$$

where $L_{{\mathcal{\Im }}_1}={\displaystyle \frac{1}{500}}$. Thus, it follows that the condition $(As.1)$ is satisfied.

We now define a continuous function $\varphi :\left[0,1\right]\to \left(0,\infty \right)$ by

$$\varphi \left(t\right)=0.1exp\left(2.2t\right).$$

This function is differentiable and non-decreasing on the interval $\left[0,t\right]$ with $t\ge 0$, and ϕ also satisfies the inequality, as follows: $\underset{0}{\overset{t}{\int }}0.1exp\left(2.2s\right)ds\le 0.1Kexp\left(2.2t\right)$, for each $K\ge 0.404$, and $exp\left(2.2s\right)\le exp\left(2.2t\right)$ for $0\le s\le t$. Let $K=0.404$. In the light of these and the above data, we get

$${\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}\left(∥h\left(t,\ell \right)∥Q_{\psi }^{\alpha }\left(t,s\right)\varphi \left(s\right)\right)ds\le {\displaystyle \frac{1}{5000\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\psi }^{\prime }\left(s\right){\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\alpha -1}exp\left(2.2s\right)ds$$

$$\le {\displaystyle \frac{{\left(\psi \left(t\right)-\psi \left(0\right)\right)}^{\alpha }}{5000\Gamma (\alpha +1)}}\times 0.404exp\left(2.2t\right)$$

$$={\displaystyle \frac{{\left(\psi \left(1\right)-\psi \left(0\right)\right)}^{\frac{1}{4}}}{5000\Gamma \left(\frac{5}{4}\right)}}\times 0.404exp\left(2.2t\right)$$

$$={\displaystyle \frac{1}{5000\Gamma \left(\frac{5}{4}\right)}}\times 0.404exp\left(2.2t\right)$$

$$\cong {\displaystyle \frac{1}{5000\times 0.9064}}\times 0.404exp\left(2.2t\right)$$

$$\cong {\displaystyle \frac{0.1h_0}{0.9064}}\times 0.404exp\left(2.2t\right).$$

Let

$$M={\displaystyle \frac{1}{0.9064}}\times 0.404\cong 0.4458.$$

Then, it follows that the condition (As.2) is satisfied. Finally, we obtain

$$2\left(\left(1+M\right)L_g+h_0\left(1+M\right)M{L}_{{\mathcal{\Im }}_1}\right)$$

$$=2\left(1.4458\times {\displaystyle \frac{1}{500}}+{\displaystyle \frac{1}{500}}\times 1.4458\times 0.4458\times {\displaystyle \frac{1}{500}}\right)\cong 0.002897<1.$$

Thereby, the condition (As.3) of Theorem 3 is satisfied. Hence, ψ-FRVIDDE (22) has a unique solution and possesses the U-H stability.

Example 2.

We address the following ψ-Hilfer FRVDIE incorporating a variable delay:

$$\left\{\begin{array}{l}\ell \left(t\right)={\displaystyle \frac{1}{500exp\left(1+t^2\right)}}tanh\left(\ell \left(t\right)\right)+{\displaystyle \frac{1}{500exp\left(1+t^2\right)}}tanh\left(\ell \left(2^{-1}t\right)\right)\\ +{\displaystyle \frac{1}{500}}\left({\displaystyle \frac{tanh\left(\ell \left(t\right)\right)}{1+exp\left(t\right)}}\right)\underset{0}{\overset{t}{\int }}{Q}_{\psi }^{\alpha }\left(t,s\right){\displaystyle \frac{tanh\left(\ell \left(s\right)\right)+tanh\left(\ell \left(2^{-1}s\right)\right)}{1+exp\left(s^2+t^2\right)}}ds,t\in \left(0,1\right],\\ I_{0^{+}}^{1-\gamma ;\psi }\ell \left(0^{+}\right)=0,\ell \left(t\right)=\theta \left(t\right)=t,t\in \left[-2^{-1},0\right]\end{array}\right.$$

(23)

with $Q_{\psi }^{\alpha }\left(t,s\right)={\psi }^{\prime }\left(s\right){\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\alpha -1}=2s{\left(t^2-s^2\right)}^{-{\displaystyle \frac{3}{4}}}$. Since the verification of conditions (As.4) and (As.5) follows exactly as in Example 1, we restrict our attention to condition (As.6) of Theorem 4.

$$2\left(L_g+h_0{L}_{{\mathcal{\Im }}_1}\right)M=2\left({\displaystyle \frac{1}{500}}+{\displaystyle \frac{1}{250000}}\right)\times 0.4458$$

$$={\displaystyle \frac{501}{12500}}\times 0.4458\cong 0.01787<1.$$

Thereby, the condition (As.6) of Theorem 3 is satisfied. Hence, ψ-FRVDIE (23) has a uniqueness of solution and possesses the U-H stability.

Remark 5.

Technically, according to Example 1 and Example 2, ψ-Hilfer FRVIDDE (22) and ψ-Hilfer FRVDIE (23) are particular cases of ψ-Hilfer FRVIDDE (3) and ψ-Hilfer FRVDIE (4), respectively. Both ψ-Hilfer fractional Volterra integro-differential and ψ-Hilfer integral equations are nonlinear and involve a variable delay. From the above discussion, it is clear that the functions appearing in ψ-Hilfer FRVIDDE (22) and ψ-Hilfer FRVDIE (23) satisfy the sufficient conditions (As.1)–(As.3) of Theorems 3 and (As.4)–(As.6) of Theorem 4, respectively. Consequently, these results illustrate and support the applicability of the theoretical stability results presented in this paper. Next, from a conceptual perspective, the U–H stability of ψ-Hilfer FRVIDDE (22) and ψ-Hilfer FRVDIE (23) illustrates the robustness of these mathematical models, in the sense that functions which approximately satisfy the governing equations lie close to their exact solutions. This property ensures that small perturbations or approximation errors do not significantly affect the qualitative behavior of the solutions of these models.

5. Conclusions

Analytical solutions of ordinary, functional, integro-differential, and integral equations, as well as their fractional and delay-fractional generalizations, are rarely obtainable except in very special cases. For most models, particularly nonlinear fractional integro-delay differential and nonlinear fractional delay integral equations, closed-form solutions are unattainable and numerical approximations must instead be employed on finite intervals. These difficulties underscore the importance of qualitative methods that provide information about solutions without requiring explicit representations. Next, fractional calculus has become an essential modeling framework in many scientific and engineering disciplines on account of its ability to capture memory and hereditary effects. As a result, the analysis of qualitative properties such as uniqueness and Ulam-type stability of fractional differential and integral equations has attracted substantial attention. Fixed point techniques, especially the Banach contraction mapping principle, have proven to be effective tools for investigating such properties. In this work, we established uniqueness and U–H stability for a nonlinear $\psi$-Hilfer FRVIDDE and a nonlinear $\psi$-Hilfer FRVDIE with multiple variable time delays. The analysis leverages the Banach contraction mapping principle together with structural properties of the Hilfer operator and some analytical tools. The results of this study improve and extend existing some works on Hilfer-type fractional models and provide new qualitative insights into $\psi$-Hilfer FRVIDDEs and $\psi$-Hilfer FRVDIEs. Finally, several important questions remain open. These include the study of uniqueness of solutions and Ulam-type stabilities for impulsive $\psi$-Hilfer fractional differential equations with or without delays, $\psi$-Hilfer fractional partial differential equations, $\psi$-Hilfer–Hadamard-type integro-differential equations with multiple variable delays, and neutral or stochastic $\psi$-Hilfer FDEs. Progress in these directions will further advance the theoretical foundation of fractional models and broaden their applicability.