Analytic Study on Φ-Hilfer Fractional Neutral-Type Functional Integro-Differential Equations with Terminal Conditions

Abstract

The current manuscript is concerned with the uniqueness and existence of a solution for a new class of $\Phi$-Hilfer fractional neutral functional integro-differential equations ($\Phi$-HFNFIDEs) with terminal conditions. Firstly, employing Babenko’s approach, we convert the aforesaid equation under consideration into an analogous integral equation. More precisely, using the multivariate Mittag-Leffler function, Banach contraction principle, and Krasnoselskii’s fixed-point theorem, we derive some conditions that guarantee the uniqueness and the existence of the solutions. For an illustration of our results in this manuscript, two examples are provided as well.

1. Introduction

The theory of fractional differential equations (FDEs) is undoubtedly a powerful and versatile tool for modeling and analyzing a wide range of real-world physical processes. In biophysical applications, the fractional relaxation equation can be used to describe the relaxation behavior of ion channels in cell membranes, the mechanical response of biological tissues, and the dynamic behavior of complex biological networks. In physics and engineering, FDEs are widely used to describe anomalous diffusion processes, including particle transport in porous, inhomogeneous, and disordered media. Moreover, FDEs offer a more accurate representation of neutron flux distribution by capturing long-range interactions and memory effects observed during the neutron deceleration process (we refer the interested reader to see [1,2,3,4,5,6,7]). However, Hilfer FDEs, named after mathematician Rainer Hilfer [8], are a specific class of FDEs that incorporate the Hilfer fractional derivative (HFD) operator. These equations provide a powerful tool for modeling and analyzing phenomena that exhibit complex dynamics, viscoelasticity, fractional optics, and control systems. This often leads to the study of the initial and boundary value problems (BVPs) together with integral equations. Such problems form the basis of the mathematical modeling of several dynamic phenomena. We suggest the following papers to see how the HFD has been recently used: [9,10,11,12,13,14,15,16]. Currently, the generalized fractional derivative (FD) introduced by Katugampola [17,18] has been unified with the HFD by Oliveira and Capelas de Oliveira and is now referred to as the Hilfer–Katugampola FD [19]. In fact, the $\psi$-Hilfer FDE is a generalization of the classical FDE, in which the FD is defined using the $\psi$-Hilfer fractional operator. This operator introduces a more flexible kernel compared to the standard Riemann–Liouville FD. The exploration of new physical phenomena and the investigation of chaotic systems have led to the proposal of $\psi$-Hilfer FDEs [20]. Furthermore, the existence and uniqueness of solutions to $\psi$-Hilfer-type fractional BVPs have received considerable attention due to their important qualitative properties. The results based on these settings can be found in [21,22,23,24,25,26,27,28].

Fractional-order delay models are gaining prominence across diverse fields due to their capability to capture complex dynamic behaviors that traditional integer-order models may fail to attain. First and foremost, they can incorporate delays and fractional orders, thus enabling them to capture more meticulous behaviors. In modeling real-world phenomena where the future state of a system depends not only on its current state but also on past states, fractional delay differential equations (FDDEs) play a crucial role. The motivation for solving FDDEs arises from their widespread application in various fields, including science, engineering (for example, control systems, mechanical systems, robotics, aerospace, traffic dynamics, and signal processing), and abundant diverse areas [29,30]. Because fractional derivatives are non-local, solving FDDE is indeed computationally imperative and challenging. Therefore, establishing an efficacious numerical algorithm for solving FDDE is essential. There has been a growing interest in solving FDDE numerically. Moreover, delay terms could show up in the FDs, also known as neutral FDDEs, used in the mathematical modeling of a number of phenomena, such as electric networks with lossless transmission lines (those found in high-speed computers [1], the force of sliding friction in connected pairs [31], and oscillating theory [32]). Neutral-type FDDEs involving different FDs have been studied by many researchers [33,34,35,36,37,38,39,40].

In [24], Almalahi et al. studied the existence and uniqueness of the solutions for the terminal value problem for fractional implicit differential equations characterized by the $\psi$-Hilfer FD:

$$\begin{array}{c}{D}_{a^{+}}^{\alpha ,\beta ,\psi }y\left(t\right)=f(t,y\left(t\right),D_{a^{+}}^{\alpha ,\beta ,\psi }y\left(t\right)),\hspace{1em}t\in (a,T],a>0,\hfill \\ y\left(T\right)=w\in \mathbb{R},\hfill \end{array}$$

where $D_{a^{+}}^{\alpha ,\beta ,\psi }(·)$ symbolizes the $\psi$-HFD of order $\alpha \in (0,1)$, type $\beta \in [0,1]$, and $f:(a,T]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a given function. In [41], the authors considered the terminal value problem (TVP) with neutral functional Hilfer–Katugampola FDEs:

$$\begin{array}{c}{}^{\rho }D_{0^{+}}^{\alpha ,\beta }[y\left(t\right)-H(t,y_t)]=f(t,y_t),\hspace{1em}t\in (0,b],0<b<\infty ,\hfill \\ y\left(b\right)=c\in \mathbb{R},\hfill \\ y\left(t\right)=\phi \left(t\right),\hspace{1em}t\in [-r,0],r>0,\hfill \end{array}$$

where ${}^{\rho }D_{0^{+}}^{\alpha ,\beta }$ and ${}^{\rho }I_{0^{+}}^{1-\gamma }$ are the Hilfer–Katugampola FD of order $\alpha \in (0,1)$ and type $\beta \in [0,1]$, and Katugampola fractional integral of order $1-\gamma ,(\gamma =\alpha +\beta -\alpha \beta )$, and $f,H:(0,b]\times C\left(\right[-r,0],\mathbb{R})\to \mathbb{R}$, respectively, are two given functions, in addition to $\phi \in C\left(\right[-r,0],\mathbb{R})$ with $\phi \left(0\right)=H(0,y_0)$. In [35], Bettayeb et al. studied the implicit neutral Caputo tempered FDEs with delay

$$\begin{array}{c}{}_0{}^{C}D_t^{\zeta ,\omega }(y\left(t\right)-h(t,y_t))=\Psi \left(t,y_t,{}_0{}^{C}D_t^{\zeta ,\omega }(y\left(t\right)-h(t,y_t))\right),\hspace{1em}t\in [0,\varkappa ],\hfill \\ y\left(t\right)=\varphi \left(t\right),\hspace{1em}t\in [-\zeta ,0],\hfill \end{array}$$

where $\zeta \in (0,1)$, $\omega \ge 0$, $\varkappa ,\zeta >0$, and ${}_0{}^{C}D_t^{\zeta ,\omega }(·)$ represent the Caputo tempered FD; $h:[0,\varkappa ]\times C\left(\right[-\zeta ,0],\mathbb{R})\to \mathbb{R}$ and $\Psi :[0,\varkappa ]\times C\left(\right[-\zeta ,0],\mathbb{R})\times \mathbb{R}\to \mathbb{R}$ are given continuous functions, $\varphi \in C\left(\right[-\zeta ,0],\mathbb{R})$. Some existence and Ulam stability results were obtained by employing Krasnoselskii’s fixed-point theorem and Gronwall’s lemma. Recently, Li [42] studied the uniqueness of the initial value problem for integro-differential equations involving an HFD of the following form:

$$\begin{array}{cc}\hfill & D_{a^{+}}^{\alpha ,\beta }u\left(x\right)+\sum _{i=1}^m{\lambda }_i{I}_{a^{+}}^{{\beta }_i}u\left(x\right)=I_{a^{+}}^{\beta }g(x,u\left(x\right)),\hspace{1em}0<\alpha <1,0\le \beta <1,{\beta }_i\ge \beta ,\hfill \\ \hfill & I_{a^{+}}^{1-\gamma }u\left(a\right)=u_a,\hspace{1em}\gamma =\alpha +\beta -\alpha \beta ,\hfill \end{array}$$

where ${\lambda }_i\in \mathbb{R}$, $x\in (a,b]$; $D_{a^{+}}^{\alpha ,\beta }(·)$ is the HFD of order $\alpha \in (0,1)$, type $\beta \in [0,1]$; $I_{a^{+}}^{{\beta }_i}(·)$ is the Riemann–Liouville fractional integral of order ${\beta }_i$; the nonlinear term $g:(a,b]\times R\to R$ is a given function. Vivek et al. [43] studied a new class of $\Phi$-Hilfer fractional-order integro-differential delay system with initial conditions for the existence and Hyers–Ulam–Mittag-Leffler stability results using the multivariate Mittag-Leffler function, fixed-point theory, and generalized Gronwall inequality. The said problem is described by

$$\begin{array}{c}{}^H\mathfrak{D}_{0^{+}}^{\sigma ,\xi ,\Phi }v\left(t\right)+\sum _{i=1}^m{\mu }_i{\mathfrak{I}}_{0^{+}}^{{\xi }_i,\Phi }v\left(t\right)={\mathfrak{I}}_{0^{+}}^{\xi ,\Phi }f(t,v\left(t\right),v\left(g\left(t\right)\right)),\hspace{1em}t\in (0,b]\hfill \\ {\mathfrak{I}}_{0^{+}}^{1-\vartheta ,\Phi }v\left(0^{+}\right)=v_0\in \mathbb{R},\hfill \\ v\left(t\right)=\eta \left(t\right),\hspace{1em}t\in [-\rho ,0],\hfill \end{array}$$

where ${\mu }_i\in \mathbb{R}$ for $i=1,2,\cdots ,m$, ${}^H\mathfrak{D}_{0^{+}}^{\sigma ,\xi ,\Phi }(·)$ is the $\Phi$-HFD of order $\sigma \in (0,1)$ and type $\xi \in [0,1]$, ${\xi }_i\ge \xi$; ${\mathfrak{I}}_{0^{+}}^{1-\vartheta ,\Phi }(·),{\mathfrak{I}}_{0^{+}}^{{\xi }_i,\Phi }(·)$ are $\Phi$-Riemann–Liouville fractional integrals of orders $1-\vartheta (\vartheta =\sigma +\xi -\sigma \xi )$; and ${\xi }_i,f:(0,b]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a given function.

In this work, we generalize the problem considered in [41] to integro-differential systems [42], and we discuss the existence and uniqueness of solutions to the following TVP of $\Phi$-HFNFIDEs:

$$\begin{array}{c}{}^H\mathbb{D}_{0^{+}}^{\xi ,\nu ,\Phi }[v\left(t\right)-F(t,v_t)]+\sum _{i=1}^m{\eta }_i{\mathbb{I}}_{0^{+}}^{{\nu }_i,\Phi }[v\left(t\right)-F(t,v_t)]={\mathbb{I}}_{0^{+}}^{\nu ,\Phi }g(t,v_t),\hspace{1em}t\in \mathbb{J}:=(0,d],\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & v\left(d\right)=\rho \in \mathbb{R},\hfill \end{array}$$

(2)

$$\begin{array}{c}v\left(t\right)=\omega \left(t\right),\hspace{1em}t\in [-\mu ,0],\hspace{1em}\mu >0.\hfill \end{array}$$

(3)

In the considered equations, ${}^H\mathbb{D}_{0^{+}}^{\xi ,\nu ,\Phi }(·)$ is the FD in the $\Phi$-Hilfer sense of order $\xi \in (0,1)$; type $\nu \in [0,1]$, ${\mathbb{I}}_{0^{+}}^{{\nu }_i,\Phi }(·)$, and ${\mathbb{I}}_{0^{+}}^{1-\vartheta ,\Phi }(·)$ are $\Phi$-Riemann–Liouville fractional integrals of orders ${\nu }_i({\nu }_i\ge \nu )$ and $1-\vartheta (\vartheta =\xi +\nu -\xi \nu )$. Moreover, $g,F:\mathbb{J}\times C\left(\right[-\mu ,0],\mathbb{R})$ are two given functions, and $\omega \in C\left(\right[-\mu ,0],\mathbb{R})$ such that $\omega \left(0\right)=F(0,v_0)$.

For any function v defined on $[-\mu ,d]$ and any $t\in {\mathbb{J}}^{\ast }:=[0,d]$, we represent the elements of $C\left(\right[-\mu ,0],\mathbb{R})$ by $v_t$ and define it as

$$\begin{array}{c}\hfill v_t\left(\tau \right)=v(t+\tau ),\hspace{1em}\tau \in [-\mu ,0].\end{array}$$

Initiated by the above, certain novel results on the $\Phi$-HFNFIDEs with terminal conditions have been examined. In this work, we use the concept of $\Phi$-HFD, fixed-point techniques, and multivariate Mittag-Leffler function to derive existence results. The concept is novel, yet there is no literature available for dealing with $\Phi$-HFNFIDEs involving terminal conditions. Thus, researchers are unable to undertake comparative studies.

The novelties and difficulties of this manuscript are described as follows:

(i)

A neutral-type fractional functional integro-differential system with terminal conditions is new in $\Phi$-HFD settings.

(ii)

Under the multivariate Mittag-Leffler function, the uniqueness and existence results are derived through the Banach contraction principle and Krasnoselskii’s fixed-point technique.

This article is arranged as follows. In Section 2, we give some definitions and auxiliary results necessary in this study. In Section 3, we give existence results for Equations (1) and (3) that are based on the fixed-point theorem. In Section 4, as a final point, examples are given to illustrate our results.

2. Auxiliary Facts and Notations

By $C\left(\right[-\mu ,0],\mathbb{R})$, $C({\mathbb{J}}^{\ast },\mathbb{R})$, we denote the Banach spaces of all continuous functions from $[-\mu ,0]$ into $\mathbb{R}$ (from ${\mathbb{J}}^{\ast }$ into $\mathbb{R}$) with the following norms:

$$\begin{array}{c}\hfill {\parallel v\parallel }_C=\underset{t\in [-\mu ,0]}{sup}\left|v\left(t\right)\right|,\end{array}$$

and

$$\begin{array}{c}\hfill {\parallel v\parallel }_{\infty }=\underset{t\in {\mathbb{J}}^{\ast }}{sup}\left|v\left(t\right)\right|,\end{array}$$

respectively.

We consider the weighted spaces of continuous functions

$$\begin{array}{c}\hfill C_{\vartheta ,\Phi }\left({\mathbb{J}}^{\ast }\right)=\{v:\mathbb{J}\to \mathbb{R}:{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta }v\left(t\right)\in C({\mathbb{J}}^{\ast },\mathbb{R})\},\hspace{1em}0\le \vartheta <1,\end{array}$$

and

$$\begin{array}{cc}\hfill & C_{\vartheta ,\Phi }^n\left({\mathbb{J}}^{\ast }\right)=\{v\in C^{n-1}\left({\mathbb{J}}^{\ast }\right):v^{\left(n\right)}\in C_{\vartheta ,\Phi }\left({\mathbb{J}}^{\ast }\right)\},\hspace{1em}n\in \mathbb{N},\hfill \\ \hfill & C_{\vartheta ,\Phi }^0\left({\mathbb{J}}^{\ast }\right)=C_{\vartheta ,\Phi }\left({\mathbb{J}}^{\ast }\right),\hfill \end{array}$$

with the norms

$$\begin{array}{c}\hfill {\parallel v\parallel }_{C_{\vartheta ,\Phi }}=\underset{t\in {\mathbb{J}}^{\ast }}{sup}\left|{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta }v\left(t\right)\right|,\end{array}$$

and

$$\begin{array}{c}\hfill {\parallel v\parallel }_{C_{\vartheta ,\Phi }^n}=\sum _{k=0}^{n-1}{\parallel v^{\left(k\right)}\parallel }_{\infty }+{\parallel v^{\left(n\right)}\parallel }_{C_{\vartheta ,\Phi }}.\end{array}$$

Consider the space ${\mathbb{X}}_b^p(c,d)$, $(b\in \mathbb{R},1\le p\le \infty )$ of those complex-valued Lebesgue measurable functions, g, on $[c,d]$, for which ${\parallel g\parallel }_{{\mathbb{X}}_b^p}<\infty$, where the norm is defined by

$$\begin{array}{c}\hfill {\parallel g\parallel }_{{\mathbb{X}}_b^p}={\left({\int }_c^d{\left|t^{b}g\left(t\right)\right|}^p{\displaystyle \frac{dt}{t}}\right)}^{{\displaystyle \frac{1}{p}}},\hspace{1em}(1\le p<\infty ,b\in \mathbb{R}).\end{array}$$

In the case of $b={\displaystyle \frac{1}{p}}$, the space is ${\mathbb{X}}_b^p(c,d)=L_p(c,d)$.

Definition 1

([20]). A generalized multivariate Mittag-Leffler function ${\mathbb{E}}_{({\xi }_1,\dots ,{\xi }_m),\nu }(u_1,\dots ,u_m)$ is defined by the following series:

$$\begin{array}{c}\hfill {\mathbb{E}}_{({\xi }_1,\dots ,{\xi }_m),\nu }(u_1,\dots ,u_m)=\sum _{r=0}^{\infty }\sum _{r_1+\cdots +r_m=r}\left({\displaystyle \genfrac{}{}{0pt}{}{r}{r_1,\dots ,r_m}}\right){\displaystyle \frac{u_1^{r_1}\cdots u_m^{r_m}}{\Gamma ({\xi }_1{r}_1+\cdots +{\xi }_m{r}_m+\nu )}},\end{array}$$

where ${\xi }_i,\nu >0$ and $u_i\in \mathbb{C}$ for $i=1,2,\dots ,m$ and

$$\begin{array}{c}\hfill \left({\displaystyle \genfrac{}{}{0pt}{}{r}{r_1,\dots ,r_m}}\right)={\displaystyle \frac{r!}{r_1!\cdots r_m!}}.\end{array}$$

Definition 2

([20]). Let ${\Phi }^{\prime }\left(t\right)>0$ and $\xi >0$. The Φ-Riemann–Liouville fractional integral of order ξ of a function $\widehat{g}\in {\mathbb{X}}_b^p(c,d)$ with respect to another function Φ is defined by the following:

$$\begin{array}{c}\hfill {\mathbb{I}}_{c^{+}}^{\xi ,\Phi }\widehat{g}\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\xi \right)}}{\int }_c^t{\Phi }^{\prime }\left(s\right){(\Phi \left(t\right)-\Phi \left(s\right))}^{\xi -1}\widehat{g}\left(s\right)ds,\hspace{1em}t>c,\end{array}$$

where $\Gamma (·)$ indicates the Gamma function.

Definition 3

([20]). Let ${\Phi }^{\prime }\left(t\right)>0$ and $\xi >0$, $n\in \mathbb{N}$. The Φ-Riemann–Liouville fractional derivative of order ξ of a function $\widehat{g}$ with respect to another function Φ is defined by

$$\begin{array}{cc}\hfill {\mathbb{D}}_{c^{+}}^{\xi ,\Phi }\widehat{g}\left(t\right)& ={\left({\displaystyle \frac{1}{{\Phi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{\mathbb{I}}_{c^{+}}^{n-\xi ,\Phi }\widehat{g}\left(t\right)\hfill \\ & ={\displaystyle \frac{1}{(n-\xi )}}{\left({\displaystyle \frac{1}{{\Phi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{\int }_c^t{\Phi }^{\prime }\left(s\right){(\Phi \left(t\right)-\Phi \left(s\right))}^{n-\xi -1}\widehat{g}\left(s\right)ds,\hfill \end{array}$$

where $n-1<\xi <n$, $n=\left[\xi \right]+1$, and $\left[\xi \right]$ represents the integer part of the real number ξ.

Definition 4

([20]). Let order ξ and type ν satisfy $n-1<\xi <n$ and $0\le \nu \le 1$, with $n\in \mathbb{N}$. The Φ-Hilfer fractional derivative of a function $\widehat{g}\in C_{1-\vartheta ,\Phi }\left({\mathbb{J}}^{\ast }\right)$ is defined by

$$\begin{array}{cc}\hfill {}^H\mathbb{D}_{c^{+}}^{\xi ,\nu ,\Phi }\widehat{g}\left(t\right)& ={\mathbb{I}}_{c^{+}}^{\nu (n-\xi ),\Phi }{\left({\displaystyle \frac{1}{{\Phi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{\mathbb{I}}_{c^{+}}^{(1-\nu )(n-\xi ),\Phi }\widehat{g}\left(t\right)\hfill \\ & ={\mathbb{I}}_{c^{+}}^{\vartheta -\xi ,\Phi }{\mathbb{D}}_{c^{+}}^{\vartheta ,\Phi }\widehat{g}\left(t\right).\hfill \end{array}$$

In this article, due to $0<\xi <1$, we take $n=1$.

Take the following parameters $\xi ,\nu$, and $\vartheta$ satisfying

$$\begin{array}{c}\hfill \vartheta =\xi +\nu -\xi \nu ,\hspace{1em}0<\xi ,\nu ,\vartheta <1.\end{array}$$

We recall the following weighted spaces:

$$\begin{array}{c}\hfill C_{1-\vartheta ,\Phi }^{\xi ,\nu }\left({\mathbb{J}}^{\ast }\right)=\{v\in C_{1-\vartheta ,\Phi }\left({\mathbb{J}}^{\ast }\right),{}^H\mathbb{D}_{c^{+}}^{\xi ,\nu ,\Phi }v\in C_{1-\vartheta ,\Phi }\left({\mathbb{J}}^{\ast }\right)\}\end{array}$$

and

$$\begin{array}{c}\hfill C_{1-\vartheta ,\Phi }^{\vartheta }\left({\mathbb{J}}^{\ast }\right)=\{v\in C_{1-\vartheta ,\Phi }\left({\mathbb{J}}^{\ast }\right),{\mathbb{D}}_{c^{+}}^{\vartheta ,\Phi }v\in C_{1-\vartheta ,\Phi }\left({\mathbb{J}}^{\ast }\right)\}.\end{array}$$

Since ${}^H\mathbb{D}_{c^{+}}^{\xi ,\nu ,\Phi }v={\mathbb{I}}_{c^{+}}^{\nu (1-\xi ),\Phi }{\mathbb{D}}_{c^{+}}^{\vartheta \Phi }v$, clearly, by Lemma 5 in [20], we have

$$\begin{array}{c}\hfill C_{1-\vartheta ,\Phi }^{\vartheta }\left({\mathbb{J}}^{\ast }\right)\subset C_{1-\vartheta ,\Phi }^{\xi ,\nu }\left({\mathbb{J}}^{\ast }\right)\subset C_{1-\vartheta ,\Phi }\left({\mathbb{J}}^{\ast }\right).\end{array}$$

Theorem 1

(Arzela–Ascoli Theorem [44]). Let $\mathbb{X}$ be a compact space. If $\mathbb{H}$ is an equicontinuous and bounded subset of $C\left(\mathcal{B}\right)$, then the operator $\mathbb{H}$ is relatively compact.

Theorem 2

(Banach Contraction Principle [44]). Let $\mathbb{S}$ be a non-empty closed subset of a Banach space $\mathbb{Y}$, then any contraction mapping $\mathbb{M}$ of $\mathbb{S}$ into itself has a unique fixed point.

Theorem 3

(Krasnoselskii’s Fixed-Point Theorem [44]). Let $\mathbb{S}$ be a closed, convex, and non-empty closed subset of a Banach space $\mathbb{Y}$. Suppose that $\mathcal{A}$ and $\mathcal{B}$ map $\mathbb{S}$ into $\mathbb{Y}$ and that the following is the case:

$\left(i\right)$

$\mathcal{A}y+\mathcal{B}z\in \mathbb{S}$, for all $y,z\in \mathbb{S}$;

$\left(ii\right)$

$\mathcal{A}$ is compact and continuous;

$\left(iii\right)$

$\mathcal{B}$ is a contraction mapping.

Then, there exists w in $\mathbb{S}$ such that $w=\mathcal{A}w+\mathcal{B}w$.

3. Main Results

In this section, we prove the existence of a solution for Equations (1)–(3). To this aim, let us introduce the following set:

$$\begin{array}{c}\hfill \Theta =\{v:[-\mu ,d]\to \mathbb{R}:v{|}_{[-\mu ,0]}\in C([-\mu ,0],\mathbb{R}) \mathrm{and} v{|}_{\mathbb{J}}\in C_{1-\vartheta ,\Phi }^{\vartheta }\left({\mathbb{J}}^{\ast }\right)\},\end{array}$$

$\Theta$ is a Banach space with the following norm:

$$\begin{array}{c}\hfill {\parallel v\parallel }_{\Theta }={\parallel v\parallel }_C+{\parallel v\parallel }_{C_{1-\vartheta ,\Phi }}.\end{array}$$

Babenko’s approach [45] is a powerful technique for solving differential and integral equations with initial conditions. While it is generally similar to the Laplace transform when dealing with equations with constant coefficients, it can also be applied to differential and integral equations with continuous variable coefficients and even to boundary value problems [46,47]. To illustrate the applications of this approach, we will derive an equivalent integral equation in that space $\Theta$ for Equations (1)–(3).

Lemma 1.

Consider the mapping $g,F:\mathbb{J}\times C\left(\right[-\mu ,0],\mathbb{R})\to \mathbb{R}$. Then, Equations (1)–(3) are equivalent to the following implicit integral equation:

$$\begin{array}{c}\hfill v\left(t\right)=\left\{\begin{array}{c}\omega \left(t\right),\hspace{1em}t\in [-\mu ,0]\hfill \\ {\displaystyle [\rho -F(d,v_d)-\sum _{r=0}^{\infty }{(-1)}^r\sum _{r_1+r_2+\cdots +r_m=r}\left(\genfrac{}{}{0pt}{}{r}{r_1,r_2,\dots ,r_m}\right){\eta }_1^{r_1}\cdots {\eta }_m^{r_m}}\hfill \\ \hspace{1em}\hspace{1em}\times {\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }g(d,v_d)]{\left({\displaystyle \frac{\Phi \left(t\right)-\Phi \left(0\right)}{\Phi \left(d\right)-\Phi \left(0\right)}}\right)}^{\vartheta -1}+F(t,v_t)\hfill \\ {\displaystyle \hspace{1em}+\sum _{r=0}^{\infty }{(-1)}^r\sum _{r_1+\cdots +r_m=r}\left(\genfrac{}{}{0pt}{}{r}{r_1,\dots ,r_m}\right){\eta }_1^{r_1}\cdots {\eta }_m^{r_m}}\hfill \\ \hspace{1em}\hspace{1em}\times {\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }g(t,v_t),\hspace{1em}t\in \mathbb{J}.\hfill \end{array}\right.\end{array}$$

(4)

Proof. 

The proof follows from Theorems 3 in [42] and Theorem 3.1 in [41]. □

Suppose that the function $g:\mathbb{J}\times C\left(\right[-\mu ,0],\mathbb{R})\to \mathbb{R}$ is continuous and fulfills the following conditions:

$\left(M_1\right)$

The functions $g,F:\mathbb{J}\times C\left(\right[-\mu ,0],\mathbb{R})\to \mathbb{R}$ such that

$$\begin{array}{c}\hfill g(·,x(·))\in C_{1-\vartheta ,\Phi }^{\nu (1-\xi )}\left({\mathbb{J}}^{\ast }\right)\hspace{1em}\mathrm{and}\hspace{1em}F(·,x(·))\in C_{1-\vartheta ,\Phi }^{\vartheta }\left({\mathbb{J}}^{\ast }\right),\hspace{1em}\mathrm{for}\mathrm{any}\hspace{1em}x\in C_{1-\vartheta ,\Phi }\left({\mathbb{J}}^{\ast }\right).\end{array}$$

$\left(M_2\right)$

There exist $P,Q>0$ such that

$$\begin{array}{c}\hfill |g(t,x)-g(t,\overline{x})|\le P\parallel x-\overline{x}{\parallel }_C\end{array}$$

and

$$\begin{array}{c}\hfill |F(t,y)-F(t,\overline{y})|\le Q\parallel y-\overline{y}{\parallel }_C,\end{array}$$

for any $x,y,\overline{x},\overline{y}\in C([-\mu ,0],\mathbb{R})$ and $t\in \mathbb{J}$.

Now, in view of Lemma 1, we consider an operator $\Lambda :\Theta \to \Theta$ as follows:

$$(\Lambda v)\left(t\right)=\left\{\begin{array}{c}\omega \left(t\right),\hspace{1em}t\in [-\mu ,0]\hfill \\ {\displaystyle [\rho -F(d,v_d)-\sum _{r=0}^{\infty }{(-1)}^r\sum _{r_1+r_2+\cdots +r_m=r}\left(\genfrac{}{}{0pt}{}{r}{r_1,r_2,\dots ,r_m}\right){\eta }_1^{r_1}\cdots {\eta }_m^{r_m}}\hfill \\ \hspace{1em}\hspace{1em}\times {\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }f\left(d\right)]{\left({\displaystyle \frac{\Phi \left(t\right)-\Phi \left(0\right)}{\Phi \left(d\right)-\Phi \left(0\right)}}\right)}^{\vartheta -1}+F(t,v_t)\hfill \\ {\displaystyle \hspace{1em}+\sum _{r=0}^{\infty }{(-1)}^r\sum _{r_1+r_2+\cdots +r_m=r}\left(\genfrac{}{}{0pt}{}{r}{r_1,\dots ,r_m}\right){\eta }_1^{r_1}\cdots {\eta }_m^{r_m}}\hfill \\ \hspace{1em}\hspace{1em}\times {\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }f\left(t\right),\hspace{1em}t\in \mathbb{J},\hfill \end{array}\right.$$

(5)

where $f:\mathbb{J}\to \mathbb{R}$ is a function satisfying functional equation $f\left(t\right)=g(t,v_t)$.

Obviously, the fixed points of $\Lambda$ are the solutions of Equations (1)–(3). For computational convenience, we set

$$\begin{array}{c}\hfill {\displaystyle \Omega =\sum _{r=0}^{\infty }{(-1)}^r\sum _{r_1+r_2+\cdots +r_m=r}\left(\genfrac{}{}{0pt}{}{r}{r_1,\dots ,r_m}\right){\eta }_1^{r_1}\cdots {\eta }_m^{r_m},}\end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle {\Omega }^{\ast }=\sum _{r=0}^{\infty }\sum _{r_1+r_2+\cdots +r_m=r}\left(\genfrac{}{}{0pt}{}{r}{r_1,\dots ,r_m}\right){\left|{\eta }_1\right|}^{r_1}\cdots {\left|{\eta }_m\right|}^{r_m}.}\end{array}$$

Uniqueness and Existence of Solutions

In this subsection, we study the uniqueness and existence of solutions for Equations (1)–(3).

Theorem 4.

Assume that the conditions $\left(M_1\right)-\left(M_2\right)$ hold. Then, if

$$\begin{array}{c}max\{Q+{\displaystyle \frac{P\Gamma \left(\vartheta \right){(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}};\hfill \\ \hspace{1em} \hspace{1em}\hspace{1em}{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +\xi +\nu }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m+1)}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+Q{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }\}<{\displaystyle \frac{1}{2}},\hfill \end{array}$$

(6)

then the solutions of Equations (1)–(3) are unique.

Proof. 

To transform Equations (1)–(3) into a fixed-point problem, we consider an operator $\Lambda$ defined in Equation (5). Let $v,x\in \Theta$; then, for $t\in [-\mu ,0]$, we have

$$\begin{array}{c}\hfill \left|\right(\Lambda v\left)\right(t)-(\Lambda x\left)\right(t\left)\right|=0.\end{array}$$

For $t\in \mathbb{J}$, we have

$$\begin{array}{c}\left|\right(\Lambda v\left)\right(t)-(\Lambda x\left)\right(t\left)\right|\hfill \\ \hspace{1em}\le {\left({\displaystyle \frac{\Phi \left(t\right)-\Phi \left(0\right)}{\Phi \left(d\right)-\Phi \left(0\right)}}\right)}^{\vartheta -1}{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }|f\left(d\right)-h\left(d\right)|+|F(t,v_t)-F(t,x_t)|\hfill \\ \hspace{1em}\hspace{1em}+{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }|f\left(t\right)-h\left(t\right)|+{\left({\displaystyle \frac{\Phi \left(t\right)-\Phi \left(0\right)}{\Phi \left(d\right)-\Phi \left(0\right)}}\right)}^{\vartheta -1}|F(d,v_d)-F(d,x_d)|,\hfill \end{array}$$

where $f,h\in C_{1-\vartheta ,\Phi }\left({\mathbb{J}}^{\ast }\right)$ such that

$$\begin{array}{cc}\hfill & f\left(t\right)=g(t,v_t),\hfill \\ \hfill & h\left(t\right)=g(t,x_t).\hfill \end{array}$$

By $\left(M_2\right)$, we have the following:

$$\begin{array}{cc}\hfill \left|f\right(t)-h(t\left)\right|& =|g(t,v_t)-g(t,x_t)|\hfill \\ \hfill & \le P{\parallel v_t-x_t\parallel }_C.\hfill \end{array}$$

On the other hand, we have ${\displaystyle {\parallel v_t-x_t\parallel }_C=\underset{\tau \in [-\mu ,0]}{sup}|v_t\left(\tau \right)-x_t\left(\tau \right)|}$. Then, there exists at least one ${\tau }^{\ast }\in [-\tau ,0]$ such that

$$\begin{array}{c}\hfill {\parallel v_t-x_t\parallel }_C=|v_t\left({\tau }^{\ast }\right)-x_t\left({\tau }^{\ast }\right)|=|v(t+{\tau }^{\ast })-x(t+{\tau }^{\ast })|.\end{array}$$

If $t+{\tau }^{\ast }\in [-\mu ,0]$, then

$$\begin{array}{c}\hfill {\parallel v_t-x_t\parallel }_C\le {\parallel v-x\parallel }_C={\parallel v-x\parallel }_{\Theta }.\end{array}$$

This gives the following for any $t\in \mathbb{J}$:

$$\begin{array}{c}\left|\right(\Lambda v\left)\right(t)-(\Lambda x\left)\right(t\left)\right|\hfill \\ \le P{\left({\displaystyle \frac{\Phi \left(t\right)-\Phi \left(0\right)}{\Phi \left(d\right)-\Phi \left(0\right)}}\right)}^{\vartheta -1}{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }{\parallel v_d-x_d\parallel }_C\hfill \\ \hspace{1em}+Q{\parallel v_t-x_t\parallel }_C+P{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }{\parallel v_t-x_t\parallel }_C\hfill \\ \hspace{1em}+Q{\left({\displaystyle \frac{\Phi \left(t\right)-\Phi \left(0\right)}{\Phi \left(d\right)-\Phi \left(0\right)}}\right)}^{\vartheta -1}{\parallel v_d-x_d\parallel }_C\hfill \\ \le P{\left({\displaystyle \frac{\Phi \left(t\right)-\Phi \left(0\right)}{\Phi \left(d\right)-\Phi \left(0\right)}}\right)}^{\vartheta -1}{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }{\parallel v_t-x_t\parallel }_{\Theta }\hfill \\ \hspace{1em}+Q{\parallel v_t-x_t\parallel }_C+P{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }{\parallel v_t-x_t\parallel }_{\Theta }\hfill \\ \hspace{1em}+Q{\left({\displaystyle \frac{\Phi \left(t\right)-\Phi \left(0\right)}{\Phi \left(d\right)-\Phi \left(0\right)}}\right)}^{\vartheta -1}{\parallel v_d-x_d\parallel }_{\Theta }.\hfill \end{array}$$

Thus,

$$\begin{array}{c}\left|{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }[(\Lambda v)\left(t\right)-(\Lambda x)\left(t\right)]\right|\hfill \\ \le [{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +(\xi +\nu )}{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m+1)}}\hfill \\ \hspace{1em}+Q{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }+P{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta +(\xi +\nu )}{\Omega }^{\ast }\hfill \\ \hspace{1em}\hspace{1em}\times {\displaystyle \frac{{(\Phi \left(t\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m+1)}}+Q{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }]{\parallel v-x\parallel }_{\Theta }\hfill \\ \le 2[{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +(\xi +\nu )}{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m+1)}}\hfill \\ \hspace{1em}+Q{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }]{\parallel v-x\parallel }_{\Theta }.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill {\parallel \Lambda v-\Lambda x\parallel }_{C_{1-\vartheta ,\Phi }}=& {\parallel \Lambda v-\Lambda x\parallel }_{\Theta }\hfill \\ & \le 2[{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +(\xi +\nu )}{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m+1)}}\hfill \\ & \hspace{1em}+Q{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }]{\parallel v-x\parallel }_{\Theta }.\hfill \end{array}$$

If $t+{\tau }^{\ast }\in {\mathbb{J}}^{\ast }$, then we have

$$\begin{array}{cc}\hfill \parallel v_t-x_t{\parallel }_C& ={(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }{\parallel v_t-x_t\parallel }_C\hfill \\ & \le {(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\parallel v-x\parallel }_{C_{1-\vartheta ,\Phi }}\hfill \\ & \le {(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\parallel v-x\parallel }_{\Theta }.\hfill \end{array}$$

Hence, for any $t\in \mathbb{J}$, we have the following

$$\begin{array}{c}\left|\right(\Lambda v\left)\right(t)-(\Lambda x\left)\right(t\left)\right|\hfill \\ \le P{\left({\displaystyle \frac{\Phi \left(t\right)-\Phi \left(0\right)}{\Phi \left(d\right)-\Phi \left(0\right)}}\right)}^{\vartheta -1}{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }{\parallel v_d-x_d\parallel }_C\hfill \\ \hspace{1em}+Q\parallel v_t-x_t{\parallel }_C+P{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }{\parallel v_t-x_t\parallel }_C\hfill \\ \hspace{1em}+{\left({\displaystyle \frac{\Phi \left(t\right)-\Phi \left(0\right)}{\Phi \left(d\right)-\Phi \left(0\right)}}\right)}^{\vartheta -1}Q{\parallel v_d-x_d\parallel }_C\hfill \\ \le P{\left({\displaystyle \frac{\Phi \left(t\right)-\Phi \left(0\right)}{\Phi \left(d\right)-\Phi \left(0\right)}}\right)}^{\vartheta -1}{\Omega }^{\ast }{\parallel v_t-x_t\parallel }_{C_{1-\vartheta ,\Phi }}\left({\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }{(\Phi \left(s\right)-\Phi \left(0\right))}^{\vartheta -1}\right)\left(d\right)\hfill \\ \hspace{1em}+2Q{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\parallel v-x\parallel }_{C_{1-\vartheta ,\Phi }}+P{\Omega }^{\ast }\left({\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }{(\Phi \left(s\right)-\Phi \left(0\right))}^{\vartheta -1}\right)\left(t\right)\hfill \\ {\hspace{1em}\hspace{1em}\times \parallel v-x\parallel }_{C_{1-\vartheta ,\Phi }}\hfill \\ \le [{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }\Gamma \left(\vartheta \right){(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\hfill \\ \hspace{1em}+2Q{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}+{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu +\vartheta -1}\Gamma \left(\vartheta \right){\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}]\hfill \\ {\hspace{1em}\hspace{1em}\times \parallel v-x\parallel }_{C_{1-\vartheta ,\Phi }}\hfill \end{array}$$

Therefore, we obtain the following:

$$\begin{array}{c}\left|{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }\left((\Lambda v)\left(t\right)-(\Lambda x)\left(t\right)\right)\right|\hfill \\ \le [{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }\Gamma \left(\vartheta \right){\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\hfill \\ \hspace{1em}+2Q+{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }\Gamma \left(\vartheta \right){\Omega }^{\ast }{(\Phi \left(t\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}]{\parallel v-x\parallel }_{C_{1-\vartheta ,\Phi }}\hfill \\ \le 2\left[{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }\Gamma \left(\vartheta \right){\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}+Q\right]\hfill \\ {\hspace{1em}\hspace{1em}\times \parallel v-x\parallel }_{\Theta }.\hfill \end{array}$$

Then, we obtain

$$\begin{array}{c}{\parallel \Lambda v-\Lambda x\parallel }_{C_{1-\vartheta ,\Phi }}={\parallel \Lambda v-\Lambda x\parallel }_{\Theta }\hfill \\ \hspace{1em}\le 2\left[{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }\Gamma \left(\vartheta \right){\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}+Q\right]\hfill \\ {\hspace{1em}\hspace{1em}\hspace{1em}\times \parallel v-x\parallel }_{\Theta }.\hfill \end{array}$$

By Equation (6), we infer that $\Lambda$ is a contraction. Now, by Theorem 2, the operator $\Lambda$ has a unique fixed point $v^{\ast }\in \Theta$, which is the unique solution of Equations (1)–(3), and this completes the proof. □

Theorem 5.

Suppose that $g:\mathbb{J}\times C\left(\right[-\mu ,0],\mathbb{R})\to \mathbb{R}$ is continuous and satisfies conditions $\left(M_1\right)-\left(M_2\right)$. If

$$\begin{array}{cc}\hfill Q\left[1+{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }\right]& +{\displaystyle \frac{2\Gamma \left(\vartheta \right)\widehat{q}{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\hfill \\ & \hspace{1em}\times \left[{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }+{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu +1-\vartheta }\right]<1,\hfill \end{array}$$

and

$$\begin{array}{c}max\{{\displaystyle \frac{P\Gamma \left(\vartheta \right){(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}+2Q;\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \frac{P\Gamma {\left(\vartheta \right)}^{\xi +\nu +1-\vartheta }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+2Q{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }\}:=max({\Xi }_2;{\Xi }_1)<1,\hfill \end{array}$$

(7)

then there exists at least one solution to Equations (1)–(3).

Proof. 

Fixing ${\displaystyle g^{\ast }=\underset{t\in {\mathbb{J}}^{\ast }}{sup}{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }\left|g(t,0)\right|}$, ${\displaystyle F^{\ast }=\underset{t\in {\mathbb{J}}^{\ast }}{sup}{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }\left|F(t,0)\right|}$, and $\kappa =\left[{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }+{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu +1-\vartheta }\right]$, we define ${\mathbb{B}}_{\widehat{q}}={\{v\in \Theta :\parallel v\parallel }_{\Theta }\le \widehat{q}\}$, where

$$\begin{array}{cc}\hfill \widehat{q}\ge max\{& {\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }\left|\rho \right|+2F^{\ast }+\frac{2\Gamma \left(\vartheta \right){(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}{g}^{\ast }}{1-\left[Q(1+{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta })+\frac{2P\Gamma \left(\vartheta \right)\widehat{q}{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}\kappa }{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}\right]}};\hfill \\ & \underset{t\in [-\mu ,0]}{sup}\left|\omega \left(t\right)\right|\}.\hfill \end{array}$$

We split operator $\Lambda$ into operators ${\Lambda }_1$ and ${\Lambda }_2$ defined on ${\mathbb{B}}_{\widehat{q}}$ as follows:

$$\begin{array}{c}\hfill {\Lambda }_{1}v\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{2}}\omega \left(t\right),\hspace{1em}t\in [-\mu ,0],\hfill \\ \left[\rho -F(d,v_d)-\Omega {\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }f\left(d\right)\right]{\left({\displaystyle \frac{\Phi \left(t\right)-\Phi \left(0\right)}{\Phi \left(d\right)-\Phi \left(0\right)}}\right)}^{\vartheta -1}\hfill \\ \hspace{1em}+F(t,v_t),\hspace{1em}t\in \mathbb{J},\hfill \end{array}\right.\end{array}$$

(8)

and

$$\begin{array}{c}\hfill {\Lambda }_{2}v\left(t\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{2}}\omega \left(t\right),\hspace{1em}t\in [-\mu ,0],\hfill \\ \Omega {\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }f\left(t\right),\hspace{1em}t\in \mathbb{J}.\hfill \end{array}\right.\end{array}$$

(9)

The rest of the proof consists of several claims.

• Claim 1: ${\Lambda }_{1}v+{\Lambda }_{2}x\in {\mathbb{B}}_{\widehat{q}}$, whenever $v,x\in {\mathbb{B}}_{\widehat{q}}$.

Let $v,x\in {\mathbb{B}}_{\widehat{q}}$. Thus, for each $t\in [-\mu ,0]$, we have

$$\begin{array}{c}\left|{\Lambda }_{1}v\left(t\right)\right|\le {\displaystyle \frac{1}{2}}\underset{t\in [-\mu ,0]}{sup}\left|\omega \left(t\right)\right|,\end{array}$$

and

$$\begin{array}{c}\left|{\Lambda }_{2}x\left(t\right)\right|\le {\displaystyle \frac{1}{2}}\underset{t\in [-\mu ,0]}{sup}\left|\omega \left(t\right)\right|,\end{array}$$

which yields

$$\begin{array}{cc}\hfill |{\Lambda }_{1}v+{\Lambda }_2{x|}_{\Theta }& \le {\displaystyle \frac{1}{2}}\underset{t\in [-\mu ,0]}{sup}\left|\omega \left(t\right)\right|+{\displaystyle \frac{1}{2}}\underset{t\in [-\mu ,0]}{sup}\left|\omega \left(t\right)\right|\hfill \\ & =\underset{t\in [-\mu ,0]}{sup}\left|\omega \left(t\right)\right|\hfill \\ & \le \widehat{q}.\hfill \end{array}$$

If $t\in \mathbb{J}$, then by applying ${(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }$ to both sides of Equation (8), we arrive at

$$\begin{array}{cc}\hfill {(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }{\Lambda }_{1}v\left(t\right)=& {(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }\left[\rho -F(d,v_d)-\Omega {\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }f\left(d\right)\right]\hfill \\ & \hspace{1em}+F(t,v_t){(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }.\hfill \end{array}$$

Then,

$$\begin{array}{c}\left|{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }{\Lambda }_{1}v\left(t\right)\right|\hfill \\ \le {(\Phi \left(b\right)-\Phi \left(0\right))}^{1-\vartheta }\left[{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }\left|f\left(d\right)\right|+\left|\rho \right|+\left|F\right(d,v_d\left)\right|\right]\hfill \\ \hspace{1em}+\left[|F(t,v_t)-F(t,0)|+|F(t,0)|\right]{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }\hfill \\ \le {(\Phi \left(b\right)-\Phi \left(0\right))}^{1-\vartheta }\left[{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }\left|f\left(d\right)\right|+\left|\rho \right|\right]+F^{\ast }\hfill \\ \hspace{1em}+\left[Q\parallel v_t{\parallel }_C+F^{\ast }{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}\right]{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }.\hfill \end{array}$$

Thus

$$\begin{array}{c}{|(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }{\Lambda }_{1}v\left(t\right)|\hfill \\ \le {(\Phi \left(b\right)-\Phi \left(0\right))}^{1-\vartheta }\left[{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }\left|f\left(d\right)\right|+\left|\rho \right|\right]+2F^{\ast }\hfill \\ \hspace{1em}+Q\parallel v_t{\parallel }_C{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }.\hfill \end{array}$$

(10)

By $\left(M_2\right)$, for each $t\in \mathbb{J}$, we obtain

$$\begin{array}{cc}\hfill \left|f\right(t\left)\right|& =|g(t,v_t)-g(t,0)+g(t,0)|\hfill \\ & \le |g(t,v_t)-g(t,0)|+|g(t,0)|\hfill \\ & \le P{\parallel v_t\parallel }_C+{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{g}^{\ast },\hfill \end{array}$$

which can be expressed as

$$\begin{array}{c}\hfill {|(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }f\left(t\right)|\le g^{\ast }+P{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }{\parallel v_t\parallel }_C.\end{array}$$

If $t+{\tau }^{\ast }\in [-\mu ,0]$, then ${\parallel v_t\parallel }_C={\parallel v\parallel }_C={\parallel v\parallel }_{\Theta }$, which means that for each $t\in \mathbb{J}$, we obtain the following:

$$\begin{array}{cc}\hfill {|(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }f\left(t\right)|& \le g^{\ast }+P{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }{\parallel v_t\parallel }_{\Theta }\hfill \\ & \le g^{\ast }+P{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }\widehat{q}:=\mathbb{L}.\hfill \end{array}$$

(11)

Using inequalities (10) and (11), we obtain

$$\begin{array}{c}\left|{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }{\Lambda }_{1}v\left(t\right)\right|\hfill \\ \le {(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }[\left|\rho \right|+\mathbb{L}{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu +\vartheta -1}\Gamma \left(\vartheta \right){\Omega }^{\ast }\hfill \\ \hspace{1em}\hspace{1em}\times {\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}+Q\widehat{q}]+2F^{\ast }.\hfill \end{array}$$

which gives

$$\begin{array}{c}{\parallel {\Lambda }_{1}v\parallel }_{C_{1-\vartheta ,\Phi }}\hfill \\ \le {(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }[\left|\rho \right|+\mathbb{L}{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu +\vartheta -1}\Gamma \left(\vartheta \right){\Omega }^{\ast }\hfill \\ \hspace{1em}\hspace{1em}\times {\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}+Q\widehat{q}]+2F^{\ast }:={\mathbb{L}}_1.\hfill \end{array}$$

If $t+{\tau }^{\ast }\in {\mathbb{J}}^{\ast }$, then

$$\begin{array}{c}{\parallel v_t\parallel }_C={(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }\parallel v_t{\parallel }_C\le {(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\parallel v\parallel }_{\Theta }.\end{array}$$

Then, for each $t\in \mathbb{J}$, we have

$$\begin{array}{c}\hfill {|(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }f\left(t\right)|\le g^{\ast }+P\widehat{q}={\mathbb{L}}_1.\end{array}$$

(12)

Using inequalities (10) and (12), we obtain

$$\begin{array}{c}\left|{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }{\Lambda }_{1}v\left(t\right)\right|\hfill \\ \le {(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }[\left|\rho \right|+{\mathbb{L}}_1{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu +\vartheta -1}\Gamma \left(\vartheta \right){\Omega }^{\ast }\hfill \\ \hspace{1em}\hspace{1em}\times {\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}]+Q\widehat{q}+2F^{\ast }.\hfill \end{array}$$

This gives

$$\begin{array}{cc}\hfill {\parallel {\Lambda }_{1}v\parallel }_{C_{1-\vartheta ,\Phi }}& \le {(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }[\left|\rho \right|+{\mathbb{L}}_1{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu +\vartheta -1}\Gamma \left(\vartheta \right){\Omega }^{\ast }\hfill \\ & \hspace{1em}\times {\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}]+Q\widehat{q}+2F^{\ast }={\mathbb{L}}_2.\hfill \end{array}$$

For operator ${\Lambda }_2$, if $t+{\tau }^{\ast }\in [-\mu ,0]$ with $t\in \mathbb{J}$, then using Equation (11), we have

$$\begin{array}{cc}\hfill |{\Lambda }_{2}x\left(t\right)|& \le [{\displaystyle \frac{\Gamma \left(\vartheta \right){\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}{g}^{\ast }\hfill \\ & \hspace{1em}+{\displaystyle \frac{P\Gamma \left(\vartheta \right){\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\widehat{q}]{(\Phi \left(t\right)-\Phi \left(0\right))}^{\xi +\nu +\vartheta -1}.\hfill \end{array}$$

This results in

$$\begin{array}{c}\left|{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }{\Lambda }_{2}x\left(t\right)\right|\hfill \\ \le [{\displaystyle \frac{\Gamma \left(\vartheta \right){\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}{g}^{\ast }\hfill \\ \hspace{1em}+{\displaystyle \frac{P\Gamma \left(\vartheta \right){\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\widehat{q}]{(\Phi \left(t\right)-\Phi \left(0\right))}^{\xi +\nu }\hfill \\ \le [{\displaystyle \frac{\Gamma \left(\vartheta \right){\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}{g}^{\ast }\hfill \\ \hspace{1em}+{\displaystyle \frac{P\Gamma \left(\vartheta \right){\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\widehat{q}]{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }.\hfill \end{array}$$

Hence,

$$\begin{array}{c}{\parallel {\Lambda }_{2}x\parallel }_{C_{1-\vartheta ,\Phi }}\hfill \\ \le [\Gamma \left(\vartheta \right){\Omega }^{\ast }{\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}{g}^{\ast }\hfill \\ \hspace{1em}+P\Gamma \left(\vartheta \right){\Omega }^{\ast }{\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\widehat{q}]{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }:={\mathbb{L}}_3.\hfill \end{array}$$

If $t+{\tau }^{\ast }\in {\mathbb{J}}^{\ast }$, then for each $t\in \mathbb{J}$ and by using Equation (12), we have

$$\begin{array}{cc}\hfill \left|{\Lambda }_{2}x\left(t\right)\right|& \le [\Gamma \left(\vartheta \right){\Omega }^{\ast }{\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}{g}^{\ast }\hfill \\ & \hspace{1em}+P\Gamma \left(\vartheta \right){\Omega }^{\ast }{\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\widehat{q}]{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu +\vartheta -1}.\hfill \end{array}$$

Therefore gives

$$\begin{array}{c}\left|{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }{\Lambda }_{2}x\left(t\right)\right|\hfill \\ \le [\Gamma \left(\vartheta \right){\Omega }^{\ast }{\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}{g}^{\ast }\hfill \\ \hspace{1em}+P\Gamma \left(\vartheta \right){\Omega }^{\ast }{\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\widehat{q}]{(\Phi \left(t\right)-\Phi \left(0\right))}^{\xi +\nu }\hfill \\ \le [\Gamma \left(\vartheta \right){\Omega }^{\ast }{\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}{g}^{\ast }\hfill \\ \hspace{1em}+P\Gamma \left(\vartheta \right){\Omega }^{\ast }{\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\widehat{q}]{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }.\hfill \end{array}$$

Hence,

$$\begin{array}{c}{\parallel {\Lambda }_{2}x\parallel }_{C_{1-\vartheta ,\Phi }}\hfill \\ \le [\Gamma \left(\vartheta \right){\Omega }^{\ast }{\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}{g}^{\ast }\hfill \\ \hspace{1em}+P\Gamma \left(\vartheta \right){\Omega }^{\ast }{\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\widehat{q}]{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }:={\mathbb{L}}_4.\hfill \end{array}$$

Which implies that for each $v,x\in {\mathbb{B}}_{\widehat{q}}$, we obtain

$$\begin{array}{c}{\parallel {\Lambda }_{1}v+{\Lambda }_{2}x\parallel }_{C_{1-\vartheta ,\Phi }}\hfill \\ \le \parallel {\Lambda }_1{v\parallel }_{C_{1-\vartheta ,\Phi }}+{\parallel {\Lambda }_{2}x\parallel }_{C_{1-\vartheta ,\Phi }}\hfill \\ \le max\{{\mathbb{L}}_1;{\mathbb{L}}_2\}+max\{{\mathbb{L}}_3;{\mathbb{L}}_4\}\hfill \\ \le {\displaystyle \frac{2\Gamma \left(\vartheta \right)(\Phi \left(d\right))-\Phi {\left(0\right)}^{\xi +\vartheta }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}{g}^{\ast }\hfill \\ \hspace{1em}+{\displaystyle \frac{2P\Gamma \left(\vartheta \right)\widehat{q}{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\hfill \\ \hspace{1em}\hspace{1em}\times [{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }+{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu +1-\vartheta }]\hfill \\ \hspace{1em}+{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }\left|\rho \right|+2F^{\ast }+P\widehat{q}[1+{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }].\hfill \end{array}$$

Since

$$\begin{array}{cc}\hfill \widehat{q}\ge & {\displaystyle \frac{{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }\left|\rho \right|+2F^{\ast }+\frac{2\Gamma \left(\vartheta \right){(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}{g}^{\ast }}{1-\left[Q(1+{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta })+\frac{2P\Gamma \left(\vartheta \right)\widehat{q}{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}\kappa }{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}\right]}},\hfill \end{array}$$

where $\kappa =\left[{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }+{(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu +1-\vartheta }\right]$, then

$$\begin{array}{c}\hfill {\parallel {\Lambda }_{1}v+{\Lambda }_{2}x\parallel }_{C_{1-\vartheta ,\Phi }}={\parallel {\Lambda }_{1}v+{\Lambda }_{2}x\parallel }_{\Theta }\le \widehat{q},\end{array}$$

which implies that, for each $t\in [-\mu ,d]$, we have

$$\begin{array}{c}\hfill {\parallel {\Lambda }_{1}v+{\Lambda }_{2}x\parallel }_{\Theta }\le \widehat{q},\end{array}$$

which infers that, ${\parallel {\Lambda }_{1}v+{\Lambda }_{2}x\parallel }_{\Theta }\le \widehat{q}$.

• Claim 2: ${\Lambda }_1$ is a contradiction.

Let $v,x\in \Theta$. If $t\in [-\mu ,0]$, then

$$\begin{array}{c}\hfill |{\Lambda }_{1}v\left(t\right)-{\Lambda }_{1}x\left(t\right)|=0.\end{array}$$

For $t\in \mathbb{J}$, we have

$$\begin{array}{c}|{\Lambda }_{1}v\left(t\right)-{\Lambda }_{1}x\left(t\right)|\hfill \\ \le {(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }|f\left(d\right)-h\left(d\right)|\hfill \\ \hspace{1em}+|F(t,v_t)-F(t,x_t)|+{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}|F(d,v_d)-F(d,x_d)|\hfill \\ \le {(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }|f\left(d\right)-h\left(d\right)|\hfill \\ \hspace{1em}+Q\parallel v_t-x_t{\parallel }_C+{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}Q{\parallel v_d-x_d\parallel }_C,\hfill \end{array}$$

where $f,h\in C_{1-\vartheta ,\Phi }$ such that

$$\begin{array}{c}f\left(t\right)=g(t,v_t),\hfill \\ h\left(t\right)=g(t,x_t).\hfill \end{array}$$

By using $\left(M_2\right)$, we have

$$\begin{array}{cc}\hfill \left|f\right(t)-h(t\left)\right|& =|g(t,v_t)-g(t,x_t)|\hfill \\ & \le P\parallel v_t-x_t{\parallel }_C.\hfill \end{array}$$

If $t+{\tau }^{\ast }\in [-\mu ,0]$, then

$$\begin{array}{c}\hfill \parallel v_t-x_t{\parallel }_C\le {\parallel v-x\parallel }_C={\parallel v-x\parallel }_{\Theta },\end{array}$$

which implies that for each $t\in \mathbb{J}$,

$$\begin{array}{c}|{\Lambda }_{1}v\left(t\right)-{\Lambda }_{2}x\left(t\right)|\hfill \\ \le P{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }{\parallel v_d-x_d\parallel }_C\hfill \\ \hspace{1em}+Q\parallel v_t-x_t{\parallel }_C+{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}Q{\parallel v_d-x_d\parallel }_C\hfill \\ \le {\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +(\xi +\nu )}{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m+1)}}\hfill \\ \hspace{1em}\hspace{1em}\times {(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\parallel v-x\parallel }_{\Theta }\hfill \\ {\hspace{1em}+Q\parallel v-x\parallel }_{\Theta }+{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}Q{\parallel v-x\parallel }_{\Theta }.\hfill \end{array}$$

Thus,

$$\begin{array}{c}\left|{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }({\Lambda }_{1}v\left(t\right)-{\Lambda }_{2}x\left(t\right))\right|\hfill \\ \le [{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +(\xi +\nu )}{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m+1)}}\hfill \\ \hspace{1em}+Q{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }+{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }Q]{\parallel v-x\parallel }_{\Theta }.\hfill \end{array}$$

Consequently,

$$\begin{array}{c}{\parallel {\Lambda }_{1}v-{\Lambda }_{2}x\parallel }_{C_{1-\vartheta ,\Phi }}\hfill \\ \le [{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +(\xi +\nu )}{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m+1)}}\hfill \\ \hspace{1em}+2Q{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }{]\parallel v-x\parallel }_{\Theta }:={\Xi }_1{\parallel v-x\parallel }_{\Theta }.\hfill \end{array}$$

If $t+{\tau }^{\ast }\in {\mathbb{J}}^{\ast }$, then

$$\begin{array}{c}\hfill {\parallel v_t-x_t\parallel }_C\le {(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\parallel v-x\parallel }_{C_{1-\vartheta ,\Phi }}={(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\parallel v-x\parallel }_{\Theta }.\end{array}$$

Which implies that, for each $t\in \mathbb{J}$,

$$\begin{array}{c}|{\Lambda }_{1}v\left(t\right)-{\Lambda }_{1}x\left(t\right)|\hfill \\ \le P{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\Omega }^{\ast }{\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }{\parallel v_d-x_d\parallel }_C\hfill \\ \hspace{1em}+Q\parallel v_t-x_t{\parallel }_C+{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}Q{\parallel v_d-x_d\parallel }_C\hfill \\ \le P{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\parallel v-x\parallel }_{\Theta }{\Omega }^{\ast }\hfill \\ \hspace{1em}\times \left({\mathbb{I}}_{0^{+}}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m,\Phi }(\Phi \left(s\right)-\Phi {\left(0\right)}^{\vartheta -1})\right)\left(d\right)+2Q{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\parallel v-x\parallel }_{\Theta }\hfill \\ \le {\displaystyle \frac{P\Gamma \left(\vartheta \right){(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\hfill \\ { \times \parallel v-x\parallel }_{\Theta }+2Q{(\Phi \left(t\right)-\Phi \left(0\right))}^{\vartheta -1}{\parallel v-x\parallel }_{\Theta }.\hfill \end{array}$$

Hence,

$$\begin{array}{c}\left|{(\Phi \left(t\right)-\Phi \left(0\right))}^{1-\vartheta }({\Lambda }_{1}v\left(t\right)-{\Lambda }_{1}x\left(t\right))\right|\hfill \\ \le [{\displaystyle \frac{P\Gamma \left(\vartheta \right){(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\hfill \\ {\hspace{1em}+2Q]\parallel v-x\parallel }_{\Theta },\hfill \end{array}$$

which implies that

$$\begin{array}{c}{\parallel {\Lambda }_{1}v-{\Lambda }_{1}x\parallel }_{C_{1-\vartheta ,\Phi }}\hfill \\ \le [{\displaystyle \frac{P\Gamma \left(\vartheta \right){(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}\hfill \\ {\hspace{1em}+2Q]\parallel v-x\parallel }_{\Theta }:={\Xi }_2{\parallel v-x\parallel }_{\Theta }.\hfill \end{array}$$

Consequently,

$$\begin{array}{c}\hfill {\parallel {\Lambda }_{1}v-{\Lambda }_{1}x\parallel }_{C_{1-\vartheta ,\Phi }}=\parallel {\Lambda }_{1}v-{\Lambda }_1{x\parallel }_{\Theta }\le max({\Xi }_1,{\Xi }_2){\parallel v-x\parallel }_{\Theta }.\end{array}$$

By Equation (7), we infer that $\Lambda$ is a contraction.

• Claim 3: ${\Lambda }_2$ is compact and continuous.

The continuity of g implies that the operator ${\Lambda }_2$ is continuous. Also, ${\Lambda }_2$ is uniformly bounded on ${\mathbb{B}}_{\widehat{q}}$. Let $t\in [-\mu ,0]$ and $x\in {\mathbb{B}}_{\widehat{q}}$. Then, we have

$$\begin{array}{c}\hfill \left|{\Lambda }_{2}x\left(t\right)\right|={\displaystyle \frac{1}{2}}\left|\omega \left(t\right)\right|\le \left|\omega \left(t\right)\right|\le \underset{t\in [-\mu ,0]}{sup}\left|\omega \left(t\right)\right|\le \widehat{q},\end{array}$$

which leads to

$$\begin{array}{c}\hfill {\parallel {\Lambda }_{2}x\parallel }_{\Theta }\le \widehat{q}.\end{array}$$

If $t\in \mathbb{J}$, then we have

$$\begin{array}{c}\hfill {\parallel {\Lambda }_{2}x\parallel }_{\Theta }={\parallel {\Lambda }_{2}x\parallel }_{C_{1-\vartheta ,\Phi }}\le max\{{\mathbb{L}}_3,{\mathbb{L}}_4\}.\end{array}$$

Which means that ${\Lambda }_2$ is uniformly bounded on ${\mathbb{B}}_{\widehat{q}}$. Next, it remains to show that ${\Lambda }_2{\mathbb{B}}_{\widehat{q}}$ is equicontinuous. For $t_1,t_2\in \mathbb{J}$ and $x\in {\mathbb{B}}_{\widehat{q}}$, we have

$$\begin{array}{c}|{(\Phi \left(t_2\right)-\Phi \left(0\right))}^{1-\vartheta }{\Lambda }_{2}x\left(t_2\right)-{(\Phi \left(t_1\right)-\Phi \left(0\right))}^{1-\vartheta }{\Lambda }_{2}x\left(t_1\right)|\hfill \\ \le {\displaystyle \frac{{\Omega }^{\ast }{(\Phi \left(t_2\right)-\Phi \left(0\right))}^{1-\vartheta }}{\Gamma (\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}{\int }_{t_1}^{t_2}{\Phi }^{\prime }\left(s\right){(\Phi \left(t_2\right)-\Phi \left(s\right))}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m-1}\hfill \\ \hspace{1em}\hspace{1em}\times \left|f\left(s\right)\right|ds+{\displaystyle \frac{{\Omega }^{\ast }}{\Gamma (\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}}{\int }_0^{t_1}{\Phi }^{\prime }\left(s\right)|[{(\Phi \left(t_2\right)-\Phi \left(s\right))}^{1-\vartheta }\hfill \\ \hspace{1em}\hspace{1em}\times {(\Phi \left(t_2\right)-\Phi \left(0\right))}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m-1}-{(\Phi \left(t_1\right)-\Phi \left(0\right))}^{1-\vartheta }\hfill \\ \hspace{1em}\hspace{1em}\times {(\Phi \left(t_1\right)-\Phi \left(s\right))}^{\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m-1}]|\left|f\left(s\right)\right|ds\hfill \\ ⟶0 \mathrm{as} t_2-t_1\to 0.\hfill \end{array}$$

This proves the equicontinuity of the set ${\Lambda }_2{\mathbb{B}}_{\widehat{q}}$, and by Theorem 1, it is relatively compact. Thus, as a consequence of Theorem 3, we conclude that $\Lambda$ has at least a fixed point $v^{\ast }\in \Theta$. □

4. Examples

Example 1.

Consider the Φ-HFNFIDE with terminal condition as follows:

$$\begin{array}{c}{}^H\mathbb{D}_{0^{+}}^{0.6,0.4,t}\left[v\left(t\right)-F(t,v_t)\right]+0.1{\mathbb{I}}_{0^{+}}^{0.5,t}\left[v\left(t\right)-F(t,v_t)\right]-0.7{\mathbb{I}}_{0^{+}}^{0.7,t}\left[v\left(t\right)-F(t,v_t)\right]\hfill \\ \hspace{1em}\hspace{1em}={\mathbb{I}}_{0^{+}}^{0.5,t}g(t,v_t),\hspace{1em}t\in (0,1],\hfill \end{array}$$

(13)

$$\begin{array}{c}v\left(1\right)=\rho \in \mathbb{R},\hfill \end{array}$$

(14)

$$\begin{array}{c}v\left(t\right)=\omega \left(t\right),\hspace{1em}t\in [-\mu ,0],\hspace{1em}\mu >0,\hfill \end{array}$$

(15)

where $\omega \in C\left(\right[-\mu ,0],\mathbb{R})$. Set

$$\begin{array}{c}\hfill g(t,x)={\displaystyle \frac{ln(\sqrt{t}+1)}{\sqrt{2\sqrt{t}}}}+{\displaystyle \frac{2+\left|x\right|}{36(1+|x\left|\right)}},\hspace{1em}t\in (0,1],\hspace{1em}x\in C([-\mu ,0],\mathbb{R}),\end{array}$$

(16)

and

$$\begin{array}{c}\hfill F(t,y)={\displaystyle \frac{\left|y\right|}{54(1+|y\left|\right)}}+{\displaystyle \frac{1}{\sqrt{2\sqrt{t}}}},\hspace{1em}t\in (0,1],\hspace{1em}y\in C([-\mu ,0],\mathbb{R}).\end{array}$$

(17)

Then,

$$\begin{array}{c}\hfill C_{1-\vartheta ,\Phi }^{\nu (1-\xi )}\left([0,1]\right)=C_{0.24,t}^{0.16}\left([0,1]\right),\end{array}$$

with $\xi =0.6$, $\nu =0.4$, $\vartheta =0.76$, and $\Phi \left(t\right)=t$. Clearly, we have the functions $g\in C_{0.24,t}^{0.16}[0,1]$ and $F\in C_{0.24,t}^{0.76}[0,1]$. Thus, $\left(M_1\right)$ is fulfilled.

Let $x,\overline{x}\in C([-\mu ,0],\mathbb{R})$ and $t\in (0,4]$. Then, we have

$$\begin{array}{cc}\hfill |g(t,x)-g(t,\overline{x})|& \le {\displaystyle \frac{1}{36}}{\parallel x-\overline{x}\parallel }_C,\hfill \end{array}$$

and

$$\begin{array}{c}\hfill |F(t,x)-F(t,\overline{x})|\le {\displaystyle \frac{1}{54}}{\parallel x-\overline{x}\parallel }_C.\end{array}$$

Therefore, Condition $\left(M_2\right)$ holds with $P={\displaystyle \frac{1}{36}}$ and $Q={\displaystyle \frac{1}{54}}$. We shall check that Condition (6) is fulfilled with $d=1$, and

$$\begin{array}{cc}\hfill {\Omega }^{\ast }& =\sum _{r=0}^{\infty }\sum _{r_1+r_2=r}\left({\displaystyle \genfrac{}{}{0pt}{}{r}{r_1,r_2}}\right){\left|{\eta }_1\right|}^{r_1}{\left|{\eta }_2\right|}^{r_2}\hfill \\ & =\sum _{r=0}^{\infty }\sum _{r_1+r_2=r}\left({\displaystyle \genfrac{}{}{0pt}{}{r}{r_1,r_2}}\right){\left|0.1\right|}^{r_1}{\left|0.7\right|}^{r_2}\hfill \\ & =\sum _{r=0}^{\infty }{\left(0.8\right)}^r={\displaystyle \frac{1}{1-0.8}}=5.\hfill \end{array}$$

Indeed, we have

$$\begin{array}{c}max\{Q+{\displaystyle \frac{P\Gamma \left(\vartheta \right){(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}};\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +\xi +\nu }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m+1)}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+Q{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }\}\approx 0.20120<{\displaystyle \frac{1}{2}}.\hfill \end{array}$$

Hence, by Theorem 4, we conclude that Equations (13)–(15) with g and F mentioned in Equations (16) and (17), respectively, have a unique solution.

Example 2.

Consider the following problem

$$\begin{array}{c}{}^H\mathbb{D}_{0^{+}}^{0.6,0,t}\left[v\left(t\right)-F(t,v_t)\right]+0.1{\mathbb{I}}_{0^{+}}^{0.4,t}\left[v\left(t\right)-F(t,v_t)\right]-0.7{\mathbb{I}}_{0^{+}}^{0.5,t}\left[v\left(t\right)-F(t,v_t)\right]\hfill \\ \hspace{1em}\hspace{1em}={\mathbb{I}}_{0^{+}}^{0.4,t}g(t,v_t),\hspace{1em}t\in (1,2],\hfill \end{array}$$

(18)

$$\begin{array}{c}v\left(2\right)=\rho \in \mathbb{R},\hfill \end{array}$$

(19)

$$\begin{array}{c}v\left(t\right)=\omega \left(t\right),\hspace{1em}t\in [-\mu ,0],\hspace{1em}\mu >0,\hfill \end{array}$$

(20)

where $\omega \in C\left(\right[-\mu ,0],\mathbb{R})$. Set

$$\begin{array}{c}\hfill g(t,x)={\displaystyle \frac{\left|x\right|}{2(1+|x\left|\right)}},\hspace{1em}t\in (1,2],x\in C([-\mu ,0],\mathbb{R})\end{array}$$

and

$$\begin{array}{c}\hfill F(t,y)={\displaystyle \frac{t}{1+\left|y\right|}}\left(e^{-2}+{\displaystyle \frac{1}{e^{t^2+1}}}\right),\hspace{1em}t\in (1,2],y\in C([-\mu ,0],\mathbb{R}),\end{array}$$

Thus, we have

$$\begin{array}{c}\hfill C_{1-\vartheta ,\Phi }^{\nu (1-\xi )}[1,2]=C_{0.5,t}^0[1,2]=C_{0.5,t}[1,2],\end{array}$$

with $\xi =0.5,\nu =0,\vartheta =0.5$, and $\Phi \left(t\right)=t$. Clearly, the functions are $g\in C_{0.5,t}[1,2]$ and $F\in C_{0.5,t}^{0.5}[1,2]$. Thus, Condition $\left(M_1\right)$ is satisfied. For $x,\overline{x}\in C([-\mu ,0],\mathbb{R})$ and $t\in (1,2]$, we have

$$\begin{array}{c}\hfill |g(t,x)-g(t,\overline{x})|\le {\displaystyle \frac{1}{2}}{\parallel x-\overline{x}\parallel }_C,\end{array}$$

and

$$\begin{array}{c}\hfill |F(t,x)-F(t,\overline{x})|\le 2e^{-1}{\parallel x-\overline{x}\parallel }_C.\end{array}$$

Hence, Condition $\left(M_2\right)$ is satisfied with $P={\displaystyle \frac{1}{2}}$ and $Q=2e^{-1}$. We shall check that Condition (6) is fulfilled with $d=2$ and ${\Omega }^{\ast }=5$. Indeed,

$$\begin{array}{c}max\{Q+{\displaystyle \frac{P\Gamma \left(\vartheta \right){(\Phi \left(d\right)-\Phi \left(0\right))}^{\xi +\nu }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (2(\xi +\nu )-\xi \nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m)}};\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \frac{P{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta +\xi +\nu }{\Omega }^{\ast }{(\Phi \left(d\right)-\Phi \left(0\right))}^{(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m}}{\Gamma (\xi +\nu +(\xi +{\nu }_1)r_1+\cdots +(\xi +{\nu }_m)r_m+1)}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+Q{(\Phi \left(d\right)-\Phi \left(0\right))}^{1-\vartheta }\}\approx 0.48<{\displaystyle \frac{1}{2}},\hfill \end{array}$$

which implies that all the assumptions of Theorem 4 are satisfied. Thus, Equations (18)–(20) have a unique solution.

5. Conclusions

In this article, the uniqueness and existence of the solution for $\Phi$-HFNFIDEs with terminal conditions are discussed. The multivariate Mittag-Leffler function, Banach contraction principle, and Krasnoselskii’s fixed-point theorem are utilized to obtain the results. The respective results are novel and interesting to the best of our knowledge. As a conclusion, the $\Phi$-Hilfer fractional derivative can be used as a powerful tool for studying the dynamical behavior of many real-world problems in Banach spaces.