Existence and Hyers–Ulam Stability Analysis of Nonlinear Multi-Term Ψ-Caputo Fractional Differential Equations Incorporating Infinite Delay

Abstract

The aim of the paper is to prove the existence results and Hyers–Ulam stability to nonlinear multi-term $\Psi$-Caputo fractional differential equations with infinite delay. Some specified assumptions are supposed to be satisfied by the nonlinear item and the delayed term. The Leray–Schauder alternative theorem and the Banach contraction principle are utilized to analyze the existence and uniqueness of solutions for infinite delay problems. Some new inequalities are presented in this paper for delayed fractional differential equations as auxiliary results, which are convenient for analyzing Hyers–Ulam stability. Some examples are discussed to illustrate the obtained results.

1. Introduction

Fractional differential equations (Fra-Diff-Equs) [1,2,3,4,5] are powerful tools for describing complex systems. In the early years, research was mainly focused on mathematical theoretical explorations, with less attention to time delays. They gained attention in the late 20th century and began to combine fractional calculus with time delays. Nowadays, fractional time delays have been applied in the fields of physics [6,7], solid mechanics [8], electromechanics [9], and finance [10], and chemistry [11], which help accurately describe the system dynamics, optimize the performance, improve the accuracy of the models, and assist in the decision-making process in these fields.

In order to ensure the solvability of the equations and to provide a prerequisite for numerical calculations, in recent years, scholars have devoted themselves to studying the properties of solutions of Fra-Diff-Equs [12,13,14]. Benchohra M et al. [15] explored the existence of solutions to Caputo Fra-Diff-Equs; Bao et al. [16] explored the existence results of solutions to neutral stochastic Fra-Diff-Equs in Lp$(\Omega ,Ch)$.

However, due to the constraints of the time-delay conditions, the study of the stability of Fra-Diff-Equs [17,18,19,20,21,22] is extremely difficult. It requires scholars attempt to prove new inequalities according to the requirements of the conditions. Over the past years, numerous articles have been dedicated to studying the Hyers–Ulam stability (Hs-Um-St) of Fra-Diff-Equs.

In [23], Dong et al. investigated a type of Riemann–Liouville Fra-Diff-Equs with time delay

$$\left\{\begin{array}{cc}{D}^{\Theta }y\left(v\right)=f(v,\widehat{y_v}),\hfill & v\in (0,♭],\hfill \\ \widehat{y_0}=\aleph \in \breve{\mathfrak{B}},\hfill \end{array}\right.$$

where $\widehat{y}\left(v\right)=v^{1-\Theta }y\left(v\right)$, $0<\Theta \le 1$, $\breve{\mathfrak{B}}$ represents the phase space. For Fra-Diff-Equs with non-zero initial values, the weighted-delay method was employed to investigate the properties of the solutions. Dong et al. proved new Gronwall inequalities, which laid the foundation for the proof of Hs-Um-St.

In [24], Chen et al. investigated the existence, uniqueness, and Hs-Um-St of solutions for Fra-Diff-Equs with infinite delay

$$\left\{\begin{array}{cc}{}_{c}D^{\Theta }y\left(v\right)-\mathfrak{a}{}_{c}D^{\varrho }y\left(v\right)=f(v,y_v),\hfill & v\in J=[0,♭],\hfill \\ y\left(v\right)=\aleph \left(v\right),\hfill & v\in (-\infty ,0],\hfill \end{array}\right.$$

where $y_v\left(s\right)=y(v+s)$, $s\in (-\infty ,0], v\in J$, ${}_{c}D^{\Theta }$ and ${}_{c}D^{\varrho }$ are a Caputo fractional derivative (Fr-De) with $0<\varrho <\Theta \le 1$, $f:J\times \breve{\mathfrak{B}}\to R$ is given that satisfies certain assumptions, the function $\aleph \in \breve{\mathfrak{B}}$.

Inspired by the works in the discussions above, in this paper, we conduct an in-depth study on a class of nonlinear Fra-Diff-Equs with infinite delay and featuring the $\Psi$-Caputo Fr-De

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{}^{c}D^{\Theta ;\Psi }y\left(v\right)-\mathfrak{a}{}^{c}D^{\varrho ;\Psi }y\left(v\right)=f(v,y_v),\hfill & v\in J=[0,♭],\hfill \\ y\left(v\right)=\aleph \left(v\right),\hfill & v\in (-\infty ,0],\hfill \end{array}\right.\end{array}$$

(1)

where ${}^{c}D^{\Theta ;\Psi }$ and ${}^{c}D^{\varrho ;\Psi }$ stand for $\Psi$-Caputo Fr-Des with $0<\varrho <\Theta \le 1$, $\mathfrak{a}\in R$ is a constant, and the function $f:J\times \breve{\mathfrak{B}}\to R$ satisfies the specified hypotheses, $\aleph \in \breve{\mathfrak{B}}$, and $\breve{\mathfrak{B}}$ is a phase space. $y_v$ is a function defined on $(-\infty ,♭]$ as

$$y_v\left(s\right)=y(v+s),s\in (-\infty ,0],v\in J.$$

(2)

Our approach mainly focuses on transforming delayed differential equations into integral equations. Subsequently, these integral equations are properly extended within the phase space. We use the Leray–Schauder alternative and Banach fixed-point theorems to examine the existence and uniqueness of solutions. Furthermore, we initiate an exploration of the Hs-Um-St of the solution. It is worth noting that this exploration is faced with substantial challenges. These challenges mainly originate from the specific constraints caused by the delay conditions, which increase the complexity of analyzing the stability of the solution. Specifically, the difficulty lies in the fact that the original Gronwall inequality is inapplicable. To overcome this difficulty, we need to extend new inequalities.

In Section 2, a comprehensive review of some basic definitions and lemmas will be carried out. Section 3 is devoted to investigating the existence and uniqueness of the solutions. In Section 4, we discuss Hs-Um-St in detail. At the end of the article, two examples are provided to illustrate the conclusions.

2. Preliminaries and Lemmas

We first list several basic definitions and theorems. Subsequently, we proceed to prove a series of inequalities that are used to verify the Hs-Um-St. It should be emphasized that the function $\Psi ([a,♭]\to R_{+})$ is increasing, positive, uniformly integrable, and ${\Psi }^{\prime }\left(v\right)\ne 0$. This makes the subsequent proofs more rigorous.

Definition 1

([25]). Let $\Theta >0$ and $n\in \mathbb{N}$ with $n=[\Theta ]+1$. The Riemann–Liouville Fr-De with respect to Ψ with the order Θ of y is defined by

$${}^{RL}D_{a^{+}}^{\Theta ;\Psi }y\left(v\right):={({\displaystyle \frac{1}{{\Psi }^{\prime }\left(v\right)}}{\displaystyle \frac{d}{dv}})}^n{J}_{a^{+}}^{n-\Theta ;\Psi }y\left(v\right),$$

(3)

where $J_{a^{+}}^{\Theta ;\Psi }y\left(v\right)$ is defined by

$$J_{a^{+}}^{\Theta ;\Psi }y\left(v\right):={\displaystyle \frac{1}{\Gamma (\Theta )}}{\int }_a^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}y\left(s\right)\mathrm{d}s.$$

(4)

Definition 2

([26]). Let $\Theta >0$ and $n\in \mathbb{N}$. The Caputo-type Fr-De with respect to Ψ with the order Θ of y is defined by

$${}^{c}D_{a^{+}}^{\Theta ;\Psi }y\left(v\right):={}^{RL}D_{a^{+}}^{\Theta ;\Psi }[y\left(v\right)-\sum _{i=0}^{n-1}{\displaystyle \frac{y_{\Psi }^{\left[i\right]}\left(a\right)}{i!}}{(\Psi \left(v\right)-\Psi \left(a\right))}^i],$$

(5)

where $y\in C^{n-1}([a,♭],R),$ ${}^{RL}D_{a^{+}}^{\Theta ;\Psi }y\left(v\right)$ exists and

$$y_{\Psi }^{\left[i\right]}\left(v\right):={({\displaystyle \frac{1}{{\Psi }^{\prime }\left(v\right)}}{\displaystyle \frac{d}{dv}})}^{i}y\left(v\right).$$

Theorem 1

([27]). Let $\Theta >0$.

(i) If the function $y:[a,♭]\to R$ is continuous, then

$${}^{c}D_{a^{+}}^{\Theta ;\Psi }{J}_{a^{+}}^{\Theta ;\Psi }y\left(v\right)=y\left(v\right).$$

(ii) If $y\in C^{n-1}([a,♭],R)$ and ${}^{RL}D_{a^{+}}^{\Theta ;\Psi }y\left(v\right)$ exists, then

$$J_{a^{+}}^{\Theta ;\Psi }{}^{c}D_{a^{+}}^{\Theta ;\Psi }y\left(v\right)=y\left(v\right)-\sum _{i=0}^{n-1}{\displaystyle \frac{y_{\Psi }^{\left[i\right]}\left(a\right)}{i!}}{(\Psi \left(v\right)-\Psi \left(a\right))}^i.$$

Definition 3

([28]). Let $\mathbb{X}$ be a Banach space. A linear topological space of functions from $(-\infty ,0]$ into $\mathbb{X}$, with the seminorm ${∥·∥}_{\breve{\mathfrak{B}}}$, is said to be an admissible phase space if $\breve{\mathfrak{B}}$ has the following properties:

(A1) There exists a constant $\mathbb{H}>0$ and functions $\mathbb{K}(·)$, $\mathbb{M}(·):[0,+\infty )\to [0,+\infty ),$ such that $\mathbb{K}$ is a continuous function and $\mathbb{M}$ is locally bounded, such that for any constant $a,♭\in R$ with $♭>a$, if the function $k:(-\infty ,♭]\to \mathbb{X}$, $k_a\in \breve{\mathfrak{B}}$ and function $k(·)$ is continuous on $[a,♭]$, then for every $v\in [a,♭]$, the following conditions (i)-(iii) hold:

(i) $k_v\in \breve{\mathfrak{B}}$;

(ii) $∥k\left(v\right)∥\le \mathbb{H}{∥k_v∥}_{\breve{\mathfrak{B}}}$;

(iii) ${∥k_v∥}_{\breve{\mathfrak{B}}}\le \mathbb{K}(v-a){sup}_{a\le s\le v}∥k\left(s\right)∥+\mathbb{M}(v-a){∥k_a∥}_{\breve{\mathfrak{B}}}$.

(A2) For the function $k(·)$ in (A1), $v\mapsto k_v$ is a $\breve{\mathfrak{B}}$-valued continuous function for $v\in [a,♭]$.

(B1) The space $\breve{\mathfrak{B}}$ is complete.

Definition 4

([29]). The problem (1) is said to be Hs-Um-St if there exists a real number $\mathfrak{x}$ > 0, such that for each $\epsilon >0$ and for every solution $\gimel (·)$ of the inequalities

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\left|{}^{c}D^{\Theta ;\Psi }\gimel \left(v\right)-\mathfrak{a}{}^{c}D^{\varrho ;\Psi }\gimel \left(v\right)-f(v,{\gimel }_v)\right|\le \epsilon ,\hfill & v\in J=[0,♭],\hfill \\ \gimel \left(v\right)=\aleph \left(v\right),\hfill & v\in (-\infty ,0],\hfill \end{array}\right.\end{array}$$

(6)

there exists a solution $\daleth (·)$ of the problem (1) with

$$\left|\gimel \left(v\right)-\daleth \left(v\right)\right|\le \mathfrak{x}\epsilon ,v\in J=[0,♭].$$

Lemma 1

([30]). (Leray–Schauder alternative) Let $\mathbb{X}$ be a Banach space, O is a subset of $\mathbb{X}$, and O is a closed and convex subset. Moreover, assume that $\mathcal{V}$ is an open subset of O with $0\in \mathcal{V}$. Suppose $\tau :\overline{\mathcal{V}}\to O$ is a continuous and compact map. Then, either τ has a fixed point in $\mathcal{V}$, or there is a $s\in \partial \mathcal{V}$ and $\wp \in (0,1)$ with $s=\wp \nu \left(s\right)$.

Next, for conveniently proving Theorem 5, we will extend certain properties of the Caputo derivative to the $\Psi$-Caputo derivative and introduce some integral inequalities that can be regarded as an extended version of the Gronwall inequality.

Lemma 2.

Suppose $\Theta >0$, a function $f\in C[0,♭]$, which is nonnegative and nondecreasing, and Ψ is an increasing function. Then, the function

$$F\left(v\right)=J_{0^{+}}^{\Theta ;\Psi }f\left(v\right)={\displaystyle \frac{1}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}f\left(s\right)\mathrm{d}s$$

is nondecreasing on $[0,♭]$.

Proof. 

Assuming that the function f satisfies the conditions mentioned above, we can obtain that $f^{\prime }\left(v\right)\ge 0$ for all $v\in [0,♭]$. Let $\Psi \left(v\right)-\Psi \left(s\right)=\nu$. Then $s={\Psi }^{-1}(\Psi \left(v\right)-\nu )$. It follows that

$$\begin{array}{cc}\hfill F^{\prime }\left(v\right)& ={\displaystyle \frac{f\left(0\right)}{\Gamma (\Theta )}}{\Psi }^{\prime }\left(v\right){(\Psi \left(v\right)-\Psi \left(0\right))}^{\Theta -1}\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\Theta )}}{\int }_0^{\Psi \left(v\right)-\Psi \left(0\right)}{\nu }^{\Theta -1}{f}^{\prime }\left({\Psi }^{-1}(\Psi \left(v\right)-\nu )\right){\displaystyle \frac{\mathrm{d}\left({\Psi }^{-1}(\Psi \left(v\right)-\nu )\right)}{\mathrm{d}v}}\mathrm{d}\nu .\hfill \end{array}$$

(7)

Let $\theta \left(v\right)={\displaystyle \frac{f\left(0\right)}{\Gamma (\Theta )}}{\Psi }^{\prime }\left(v\right){(\Psi \left(v\right)-\Psi \left(0\right))}^{\Theta -1}$, $\mu \left(v\right)={\Psi }^{-1}(\Psi \left(v\right)-\nu ).$ Then, $\theta \left(v\right)>0$ because Ψ is increasing, and ${\displaystyle \frac{\mathrm{d}\mu \left(v\right)}{\mathrm{d}v}}>0$ due to $\mu (·)$ is nondecreasing. So, we have

$$F^{\prime }\left(v\right)=\theta \left(v\right)+{\displaystyle \frac{1}{\Gamma (\Theta )}}{\int }_0^{\Psi \left(v\right)-\Psi \left(0\right)}{\nu }^{\Theta -1}{f}^{\prime }\left(\mu \left(v\right)\right){\displaystyle \frac{\mathrm{d}\mu \left(v\right)}{\mathrm{d}v}}\mathrm{d}\nu >0$$

for all $v\in [0,♭]$ and, hence, $F(·)$ is nondecreasing on $[0,♭]$. □

To study the problem of infinite delay, we extend Lemma 5 in [24] and prove the following inequality.

Lemma 3.

For any function $\omega \ge 0$ belonging to the space $\mathcal{C}[a,♭]$ and any $v\in [a,♭]$, the following integral inequality

$$\underset{0\le \nu \le v}{sup}{\int }_0^{\nu }{\Psi }^{\prime }\left(s\right){(\Psi \left(\nu \right)-\Psi \left(s\right))}^{\Theta -1}\omega \left(s\right)\mathrm{d}s\le {\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}\underset{0\le \sigma \le s}{sup}\omega \left(\sigma \right)\mathrm{d}s,$$

(8)

can be obtained.

Proof. 

Fix $v\in [a,♭]$. Given the function $\omega (·)$, which is nonnegative, and ${\displaystyle \underset{0\le \sigma \le s}{sup}}\omega \left(\sigma \right)$, which is nondecreasing, this results in $F\left(v\right)={\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}{\displaystyle \underset{0\le \sigma \le s}{sup}}\omega \left(\sigma \right)\mathrm{d}s$ also being nondecreasing according to Lemma 2. Then, for any $\nu \in [0,v]$, we have

$$\begin{array}{cc}\hfill {\int }_0^{\nu }{\Psi }^{\prime }\left(s\right){(\Psi \left(\nu \right)-\Psi \left(s\right))}^{\Theta -1}\omega \left(s\right)\mathrm{d}s& \le {\int }_0^{\nu }{\Psi }^{\prime }\left(s\right){(\Psi \left(\nu \right)-\Psi \left(s\right))}^{\Theta -1}\underset{0\le \sigma \le s}{sup}\omega \left(\sigma \right)\mathrm{d}s\hfill \\ \hfill & \le {\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}\underset{0\le \sigma \le s}{sup}\omega \left(\sigma \right)\mathrm{d}s,\hfill \end{array}$$

therefore,

$$\underset{0\le \nu \le v}{sup}{\int }_0^{\nu }{\Psi }^{\prime }\left(s\right){(\Psi \left(\nu \right)-\Psi \left(s\right))}^{\Theta -1}\omega \left(s\right)\mathrm{d}s\le {\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}\underset{0\le \sigma \le s}{sup}\omega \left(\sigma \right)\mathrm{d}s.$$

Thus, the proof is completed. □

Next, we extend the Lemma 2.6 in [1] to $\Psi$-Caputo calculus.

Lemma 4.

Let $\Theta >0$, $/a>0$. Consider $g(v,s)$ as a non-negative continuous function that is defined on the domain $[0,T]\times [0,T]$, where $g(v,s)\le \mathbb{M}$. The function $g(v,s)$ has the property of being non-decreasing in v and non-increasing in s. Let Ψ be an increasing and uniformly integrable function. Assume that the function $\varpi \left(t\right)$ is non-negative and integrable on the interval $[0,T]$ with

$$\varpi \left(v\right)\le /a+{\int }_0^{v}g(v,s){\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}\varpi \left(s\right)\mathrm{d}s,v\in [0,T].$$

Then, we have

$$\varpi \left(v\right)\le /a+/a{\int }_0^v\sum _{n=1}^{\infty }{\displaystyle \frac{{(g(v,s)\Gamma (\Theta ))}^n}{\Gamma (n\Theta )}}{(\Psi \left(v\right)-\Psi \left(s\right))}^{n\Theta -1}{\Psi }^{\prime }\left(s\right)\mathrm{d}s.$$

(9)

Proof. 

Let $P\varpi \left(v\right)={\int }_0^{v}g(v,s){\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}\varpi \left(s\right)\mathrm{d}s$, it follows that $\varpi \left(v\right)\le /a+P\varpi \left(v\right)$, which implies

$$\varpi \left(v\right)\le \sum _{k=0}^{n-1}{P}^k/a+P^n\varpi \left(v\right)(P^0/a=/a).$$

(10)

We now prove that

$$P^n\varpi \left(v\right)\le {\int }_0^v{\displaystyle \frac{{(g(v,s)\Gamma (\Theta ))}^n}{\Gamma (n\Theta )}}{(\Psi \left(v\right)-\Psi \left(s\right))}^{n\Theta -1}{\Psi }^{\prime }\left(s\right)\varpi \left(s\right)\mathrm{d}s.$$

(11)

For $n=1$, the demonstration is straightforward. Assume that Equation (11) is valid when ($n=k$). Then, when ($n=k+1$), we obtain

$$\begin{array}{cc}\hfill & P^{k+1}\varpi \left(v\right)=P\left(P^k\varpi \left(v\right)\right)\hfill \\ \hfill & ={\int }_0^{v}g(v,s){\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}{P}^k\varpi \left(s\right)\mathrm{d}s\hfill \\ \hfill & \le {\int }_0^{v}g(v,s){\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}{\int }_0^s{\displaystyle \frac{{(g(s,\nu )\Gamma (\Theta ))}^k}{\Gamma (k\Theta )}}{(\Psi \left(s\right)-\Psi \left(\nu \right))}^{k\Theta -1}{\Psi }^{\prime }\left(\nu \right)\varpi \left(\nu \right)\mathrm{d}\nu \mathrm{d}s\hfill \\ \hfill & \le {\int }_0^{v}g(v,\nu ){\Psi }^{\prime }\left(s\right){\int }_0^s{\displaystyle \frac{{(g(v,\nu )\Gamma (\Theta ))}^k}{\Gamma (k\Theta )}}{(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}{(\Psi \left(s\right)-\Psi \left(\nu \right))}^{k\Theta -1}{\Psi }^{\prime }\left(\nu \right)\varpi \left(\nu \right)\mathrm{d}\nu \mathrm{d}s\hfill \\ \hfill & ={\int }_0^{v}g{(v,\nu )}^{k+1}[{\int }_{\nu }^v{\displaystyle \frac{\Gamma {(\Theta )}^k}{\Gamma (k\Theta )}}{(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}{(\Psi \left(s\right)-\Psi \left(\nu \right))}^{k\Theta -1}{\Psi }^{\prime }\left(s\right)\mathrm{d}s]{\Psi }^{\prime }\left(\nu \right)\varpi \left(\nu \right)\mathrm{d}\nu \hfill \\ \hfill & ={\int }_0^v{\displaystyle \frac{{(g(v,\nu )\Gamma (\Theta ))}^{k+1}}{\Gamma \left(\right(k+1)\Theta )}}{(\Psi \left(v\right)-\Psi \left(\nu \right))}^{(k+1)\Theta -1}{\Psi }^{\prime }\left(\nu \right)\varpi \left(\nu \right)\mathrm{d}\nu \hfill \\ \hfill & ={\int }_0^v{\displaystyle \frac{{(g(v,s)\Gamma (\Theta ))}^{k+1}}{\Gamma \left(\right(k+1)\Theta )}}{(\Psi \left(v\right)-\Psi \left(s\right))}^{(k+1)\Theta -1}{\Psi }^{\prime }\left(s\right)\varpi \left(s\right)\mathrm{d}s.\hfill \end{array}$$

Through the method of mathematical induction, inequality (11) holds for all $n\in \mathbb{N}$. Replacing $\varpi \left(v\right)$ with $/a$ in (11), we deduce that $P^k/a\le /a{\int }_0^v{\displaystyle \frac{{(g(v,s)\Gamma (\Theta ))}^k}{\Gamma (k\Theta )}}{(\Psi \left(v\right)-\Psi \left(s\right))}^{k\Theta -1}{\Psi }^{\prime }\left(s\right)\mathrm{d}s$, $k=1,2,\cdots$. Analogous to the proof process of Lemma 3.4 in [23], we are able to confirm that

$$P^n\varpi \left(v\right)\le {\int }_0^v{\displaystyle \frac{{(\mathbb{M}\Gamma (\Theta ))}^n}{\Gamma (n\Theta )}}{(\Psi \left(v\right)-\Psi \left(s\right))}^{n\Theta -1}{\Psi }^{\prime }\left(s\right)\varpi \left(s\right)\mathrm{d}s\to 0$$

as n approaches infinity, uniformly in $v\in [0,T]$. Finally, letting n approach infinity in (10), we obtain

$$\varpi \left(v\right)\le \sum _{n=0}^{\infty }{P}^n/a\le /a+/a{\int }_0^v\sum _{n=1}^{\infty }{\displaystyle \frac{{(g(v,s)\Gamma (\Theta ))}^n}{\Gamma (n\Theta )}}{(\Psi \left(v\right)-\Psi \left(s\right))}^{n\Theta -1}{\Psi }^{\prime }\left(s\right)\mathrm{d}s.$$

The lemma is proved. □

Lemma 5.

The solution of Equation (1) is

$$y\left(v\right)=\mathfrak{a}{J}^{\Theta -\varrho ;\Psi }y\left(v\right)+J^{\Theta ;\Psi }f(v,y_v)+\vartheta \left(v\right),v\in J=[0,♭],$$

(12)

where $\vartheta \left(v\right)=y\left(0\right)(1-{\displaystyle \frac{\mathfrak{a}{(\Psi \left(v\right)-\Psi \left(0\right))}^{\Theta -\varrho }}{\Gamma (1+\Theta -\varrho )}})$ is a function.

Proof. 

The proof is evident from Definition 1 and Theorem 1. □

3. Existence Results

In this part, we mainly use Lemma 1 and the Banach contraction principle to prove the existence results of solutions.

We list the following four assumptions.

(D1) The continuous function $f:[0,♭]\times \breve{\mathfrak{B}}\to R$ has a bounded subset $W_0\subset \breve{\mathfrak{B}}$, such that $f:[0,♭]\times W_0$, on which it is uniformly continuous.

(D2) There exists a constant $\mathbb{L}$, such that $\left|f(v,\hslash )-f(v,ð)\right|\le \mathbb{L}{∥\hslash -ð∥}_{\breve{\mathfrak{B}}}$ for each $v\in J$ and every $\hslash ,ð\in \breve{\mathfrak{B}}$.

(D3) There exist functions $g,l\in C(J,R^{+})$, such that $\left|f(v,\hslash )\right|\le g\left(v\right)+l\left(v\right){∥\hslash ∥}_{\breve{\mathfrak{B}}}$ for each $v\in J$ and every $\hslash \in \breve{\mathfrak{B}}$.

(D4) There exists a nonnegative function $\tilde{\eta }\in {\mathbb{L}}^p[0,♭]$ with $p>{\displaystyle \frac{1}{\Theta }}$ and a continuously non-decreasing function $\Omega :[0,+\infty )\to [0,+\infty )$, such that $\left|f(v,\hslash )\right|\le \tilde{\eta }\left(v\right)\Omega \left({∥\hslash ∥}_{\breve{\mathfrak{B}}}\right)$ for all $v\in J$ and every $\hslash \in \breve{\mathfrak{B}}$.

Before the proof of the main result, let us initiate our analysis by transforming the problem of solutions into a fixed-point problem. According to Lemma 5, it is known that y is a solution to (1), precisely when y satisfies

$$y\left(v\right)=\left\{\begin{array}{cc}\mathfrak{a}{J}^{\Theta -\varrho ;\Psi }y\left(v\right)+J^{\Theta ;\Psi }f(v,y_v)+\vartheta \left(v\right),\hfill & v\in [0,♭],\hfill \\ \aleph \left(v\right),\hfill & v\in (-\infty ,0].\hfill \end{array}\right.$$

For any $\aleph \in \breve{\mathfrak{B}}$, define the function $\widehat{\aleph }$ as

$$\widehat{\aleph }\left(v\right)=\left\{\begin{array}{cc}\aleph \left(0\right),\hfill & v\in [0,♭],\hfill \\ \aleph \left(v\right),\hfill & v\in (-\infty ,0].\hfill \end{array}\right.$$

For $\Xi \in C\left(\right[0,♭],R),$ we define the function $\widehat{\Xi }$ as

$$\widehat{\Xi }\left(v\right)=\left\{\begin{array}{cc}\Xi \left(v\right)-\aleph \left(0\right),\hfill & v\in [0,♭],\hfill \\ 0,\hfill & v\in (-\infty ,0].\hfill \end{array}\right.$$

We can decompose $y(·)$ into the sum of two functions. For $v\in [0,♭],$ we have $y\left(v\right)=\widehat{\aleph }\left(v\right)+\widehat{\Xi }\left(v\right)$. Moreover, for all $v\in [0,♭],$ we can easily obtain that $y_v={\widehat{\aleph }}_v+{\widehat{\Xi }}_v$, and note that $\Xi (·)$ satisfies

$$\Xi \left(v\right)=\mathfrak{a}{J}^{\Theta -\varrho ;\Psi }\Xi \left(v\right)+J^{\Theta ;\Psi }f(v,{\widehat{\aleph }}_v+{\widehat{\Xi }}_v)+\vartheta \left(v\right).$$

Now, we define a set $C_0=\left\{\Xi \in C\left(\right[0,♭],R):\Xi \left(0\right)=\aleph \left(0\right)\right\}.$ It is not difficult to see that the set $C_0$ is closed and, thus, completed. Based on this, we define an operator $T:C_0\to C_0$ by

$$(T\Xi )\left(v\right)=\mathfrak{a}{J}^{\Theta -\varrho ;\Psi }\Xi \left(v\right)+J^{\Theta ;\Psi }f(v,{\widehat{\aleph }}_v+{\widehat{\Xi }}_v)+\vartheta \left(v\right).$$

(13)

Theorem 2.

Suppose that (D1) and (D2) are satisfied. If

$$0<{\displaystyle \frac{\mid \mathfrak{a}\mid {(\Psi (♭)-\Psi \left(0\right))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}+{\displaystyle \frac{\mathbb{L}{\mathbb{K}}_m{(\Psi (♭)-\Psi \left(0\right))}^{\Theta }}{\Gamma (\Theta +1)}}<1,$$

(14)

then Equation (1) has a unique solution on $(-\infty ,♭]$.

Proof. 

Let $T:C_0\to C_0$ be given as (13). For any $\hslash ,ð\in C_0$ and each $v\in [0,♭]$, we have

$$\begin{array}{cc}\hfill & \mid (T\hslash )\left(v\right)-\left(Tð\right)\left(v\right)\mid \hfill \\ \hfill & \le {\displaystyle \frac{\mid \mathfrak{a}\mid }{\Gamma (\Theta -\varrho )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -\varrho -1}\mid \hslash \left(s\right)-ð\left(s\right)\mid \mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}\mid f(s,{\widehat{\aleph }}_s+{\widehat{\hslash }}_s)-f(s,{\widehat{\aleph }}_s+{\widehat{ð}}_s)\mid \mathrm{d}s\hfill \\ & \le {\displaystyle \frac{\mid \mathfrak{a}\mid {(\Psi (♭)-\Psi \left(0\right))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}∥\hslash -ð∥+{\displaystyle \frac{\mathbb{L}}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}{∥{\widehat{\hslash }}_s-{\widehat{ð}}_s∥}_{\breve{\mathfrak{B}}}\mathrm{d}s.\hfill \end{array}$$

As

$$\begin{array}{cc}\hfill {∥{\widehat{\hslash }}_s-{\widehat{ð}}_s∥}_{\breve{\mathfrak{B}}}& \le \mathbb{K}\left(s\right)\underset{0\le \nu \le s}{sup}∥\widehat{\hslash }\left(\nu \right)-\widehat{ð}\left(\nu \right)∥+\mathbb{M}\left(s\right){∥{\widehat{\hslash }}_0-{\widehat{ð}}_0∥}_{\breve{\mathfrak{B}}}\hfill \\ \hfill & \le {\mathbb{K}}_m\underset{0\le \nu \le s}{sup}∥\hslash \left(\nu \right)-\aleph \left(0\right)-ð\left(\nu \right)+\aleph \left(0\right)∥\hfill \\ \hfill & \le {\mathbb{K}}_m∥\hslash -ð∥,\hfill \end{array}$$

where ${\mathbb{K}}_m=sup\left\{\left|\mathbb{K}\left(v\right)\right|:v\in [a,♭]\right\}$, we get

$$∥T\hslash -Tð∥\le ({\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (♭)-\Psi \left(0\right))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}+{\displaystyle \frac{\mathbb{L}{\mathbb{K}}_m{(\Psi (♭)-\Psi \left(0\right))}^{\Theta }}{\Gamma (\Theta +1)}})∥\hslash -ð∥.$$

Assumption (14) shows that T is a contraction. Applying the Banach contraction principle, we conclude that T has a unique fixed point. Then Equation (1) has a unique solution on $(-\infty ,♭]$. □

Theorem 3.

Suppose that (D1) and (D3) hold. Further, suppose that

$${\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (♭)-\Psi \left(0\right))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}+{\displaystyle \frac{{\mathbb{K}}_m{(\Psi (♭)-\Psi \left(0\right))}^{\Theta }}{\Gamma (\Theta +1)}}∥l∥<1$$

(15)

holds. Then, Equation (1) has at least one solution on $(-\infty ,♭]$.

Proof. 

To apply Lemma 1, we will demonstrate, through the subsequent steps, that the operator T is completely continuous.

Let $\left\{{\Xi }_n\right\}$ be a sequence, such that ${\Xi }_n\to \Xi$ in $C_0$. Then, for each $v\in [0,♭]$, we have

$$\begin{array}{cc}\hfill & \left|T{\Xi }_n\left(v\right)-T\Xi \left(v\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{\mid \mathfrak{a}\mid }{\Gamma (\Theta -\varrho )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -\varrho -1}\mid {\Xi }_n\left(s\right)-\Xi \left(s\right)\mid \mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}\mid f(s,{\widehat{\aleph }}_s+{\widehat{\left({\Xi }_n\right)}}_s)-f(s,{\widehat{\aleph }}_s+{\widehat{\Xi }}_s)\mid \mathrm{d}s.\hfill \end{array}$$

Set $W_0=\left\{{\left({\Xi }_n\right)}_s:s\in [0,♭],n\ge 1\right\}\subset \breve{\mathfrak{B}}$. From (D1), it can be observed that the function f is uniformly continuous with respect to $s\in [0,v]$. This means that for each $\epsilon >0$, there exists $\delta >0$, such that for each ${\Xi }_1,{\Xi }_2\in W_0$ with $\left|{\Xi }_1-{\Xi }_2\right|<\delta$, we have $\left|f(s,{\Xi }_1)-f(s,{\Xi }_2)\right|<\epsilon$. As ${\Xi }_n\to \Xi$, $N>0$, such that for each $n>N$, we have $\left|{\Xi }_n-\Xi \right|<\delta$. Hence, $\left|f(s,{\Xi }_n)-f(s,\Xi )\right|<\epsilon$ for all $s\in [0,v]$. Based on the previously introduced definition ${\Xi }_v={\widehat{\Xi }}_v+{\widehat{\aleph }}_v$, it follows that $\left|f(s,{\widehat{\aleph }}_s+{\widehat{\left({\Xi }_n\right)}}_s)-f(s,{\widehat{\aleph }}_s+{\widehat{\Xi }}_s)\right|<\epsilon$, so we get

$$\left|T{\Xi }_n\left(v\right)-T\Xi \left(v\right)\right|\le {\displaystyle \frac{\mid \mathfrak{a}\mid {(\Psi (♭)-\Psi \left(0\right))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}∥{\Xi }_n-\Xi ∥+{\displaystyle \frac{{(\Psi (♭)-\Psi \left(0\right))}^{\Theta }\epsilon }{\Gamma (\Theta +1)}}.$$

Hence, $∥{\Xi }_n-\Xi ∥\to 0$ as ${\Xi }_n\to \Xi$. Due to the arbitrariness of $\epsilon$, we obtain that $\left|T{\Xi }_n\left(v\right)-T\Xi \left(v\right)\right|\to 0$, which implies that T is continuous.

We now show that T maps bounded subsets into bounded subsets in $C_0$. In fact, it is enough to prove that for any $r>0$, there exists a positive constant $\zeta$, such that for each $\Xi \in B_r=\left\{\Xi \in C_0:∥\Xi ∥\le r\right\}$, we have $∥T\Xi \left(v\right)∥\le \zeta$. Let $\Xi \in B_r$. As f is a continuous function, for each $v\in [0,♭]$, we have

$$\begin{array}{cc}\hfill \left|T\Xi (v)\right|& \le {\displaystyle \frac{\left|\mathfrak{a}\right|}{\Gamma (\Theta -\varrho )}}{\int }_0^v{\Psi }^{\prime }(s){(\Psi (v)-\Psi (s))}^{\Theta -\varrho -1}\left|\Xi (s)\right|\mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }(s){(\Psi (v)-\Psi (s))}^{\Theta -1}\left|f(s,{\widehat{\Xi }}_s+{\widehat{\aleph }}_s)\right|\mathrm{d}s+\left|\vartheta (♭)\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (♭)-\Psi (0))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}∥\Xi ∥+{\displaystyle \frac{1}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }(s){(\Psi (v)-\Psi (s))}^{\Theta -1}\hfill \\ \hfill & (g(s)+l(s){∥{\widehat{\Xi }}_s+{\widehat{\aleph }}_s∥}_{\breve{\mathfrak{B}}})\mathrm{d}s+\left|\vartheta (♭)\right|.\hfill \end{array}$$

As

$$\begin{array}{cc}\hfill {∥{\widehat{\Xi }}_s+{\widehat{\aleph }}_s∥}_{\breve{\mathfrak{B}}}& \le {∥{\widehat{\Xi }}_s∥}_{\breve{\mathfrak{B}}}+{∥{\widehat{\aleph }}_s∥}_{\breve{\mathfrak{B}}}\hfill \\ \hfill & \le \mathbb{K}\left(s\right)\underset{0\le \nu \le s}{sup}∥\widehat{\Xi }\left(\nu \right)∥+\mathbb{M}\left(s\right){∥{\widehat{\Xi }}_0∥}_{\breve{\mathfrak{B}}}+\mathbb{K}\left(s\right)\underset{0\le \nu \le s}{sup}∥\widehat{\aleph }\left(\nu \right)∥+\mathbb{M}\left(s\right){∥{\widehat{\aleph }}_0∥}_{\breve{\mathfrak{B}}}\hfill \\ \hfill & \le {\mathbb{K}}_m\underset{0\le \nu \le s}{sup}∥\Xi \left(\nu \right)-\aleph \left(0\right)∥+{\mathbb{K}}_m∥\aleph \left(0\right)∥+{\mathbb{M}}_m{∥\aleph ∥}_{\breve{\mathfrak{B}}}\hfill \\ \hfill & \le {\mathbb{K}}_{m}r+2{\mathbb{K}}_m∥\aleph \left(0\right)∥+{\mathbb{M}}_m{∥\aleph ∥}_{\breve{\mathfrak{B}}}\hfill \\ \hfill & \le {\mathbb{K}}_{m}r+2{\mathbb{K}}_m\mathbb{H}{∥\aleph ∥}_{\breve{\mathfrak{B}}}+{\mathbb{M}}_m{∥\aleph ∥}_{\breve{\mathfrak{B}}}\hfill \\ \hfill & \le {\mathbb{K}}_{m}r+(2{\mathbb{K}}_m\mathbb{H}+{\mathbb{M}}_m){∥\aleph ∥}_{\breve{\mathfrak{B}}}:=r_0,\hfill \end{array}$$

where $\mathbb{H}>0$, ${\mathbb{M}}_m=sup\left\{\left|\mathbb{M}\left(v\right)\right|:v\in [a,♭]\right\}$, ${\mathbb{K}}_m=sup\left\{\left|\mathbb{K}\left(v\right)\right|:v\in [a,♭]\right\}$, we have

$$\left|T\Xi \left(v\right)\right|\le {\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (♭)-\Psi \left(0\right))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}r+{\displaystyle \frac{{(\Psi (♭)-\Psi \left(0\right))}^{\Theta }}{\Gamma (\Theta +1)}}(∥g∥+∥l∥r_0)+\left|\vartheta (♭)\right|:=\zeta .$$

This can be proven.

Next, we prove that T maps bounded subsets into equicontinuous subsets of $C_0$. For all $\Xi \in B_r$ and $v_1,v_2\in [0,♭]$, $v_1<v_2$, we have

$$\begin{array}{cc}\hfill & \left|(T\Xi )(v_2)-(T\Xi )(v_1)\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left|\mathfrak{a}\right|∥\Xi ∥}{\Gamma (\Theta -\varrho )}}|{\int }_0^{v_1}({(\Psi (v_2)-\Psi (s))}^{\Theta -\varrho -1}-{(\Psi (v_1)-\Psi (s))}^{\Theta -\varrho -1})\mathrm{d}\Psi (s)\hfill \\ \hfill & +{\int }_{v_1}^{v_2}{(\Psi (v_2)-\Psi (s))}^{\Theta -\varrho -1}\mathrm{d}\Psi (s)|\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\Theta )}}|{\int }_0^{v_1}({(\Psi (v_2)-\Psi (s))}^{\Theta -1}-{(\Psi (v_1)-\Psi (s))}^{\Theta -1})f(s,{\widehat{\Xi }}_s+{\widehat{\aleph }}_s)\mathrm{d}\Psi (s)\hfill \\ \hfill & +{\int }_{v_1}^{v_2}{(\Psi (v_2)-\Psi (s))}^{\Theta -1}f(s,{\widehat{\Xi }}_s+{\widehat{\aleph }}_s)\mathrm{d}\Psi (s)|+\left|\vartheta (v_2)-\vartheta (v_1)\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left|\mathfrak{a}\right|∥\Xi ∥}{\Gamma (\Theta -\varrho +1)}}({(\Psi (v_2)-\Psi (0))}^{\Theta -\varrho }-{(\Psi (v_1)-\Psi (0))}^{\Theta -\varrho }+2{(\Psi (v_2)-\Psi (v_1))}^{\Theta -\varrho })\hfill \\ \hfill & +{\displaystyle \frac{∥g∥+∥l∥r_0}{\Gamma (\Theta +1)}}({(\Psi (v_2)-\Psi (0))}^{\Theta }-{(\Psi (v_1)-\Psi (0))}^{\Theta }\hfill \\ \hfill & +2{(\Psi (v_2)-\Psi (v_1))}^{\Theta })+\left|\vartheta (v_2)-\vartheta (v_1)\right|.\hfill \end{array}$$

As $\Theta -\varrho >0$, $\Theta >0$, it is easy to see that $\left|(T\Xi )\left(v_2\right)-(T\Xi )\left(v_1\right)\right|\to 0$ as $v_1-v_2\to 0$. As a consequence, $T\left(B_r\right)$ is equicontinuous. According to the Arzela-Ascolli theorem, we can conclude that $T:C_0\to C_0$ is a completely continuous mapping.

Now, we have to verify that there exists at least one fixed point $\Xi$ of T. We assume that there exists an open set $\mathcal{V}\subseteq C_0$, such that for every $\Xi \in \partial \mathcal{V}$ and for each $\wp \in (0,1)$, the inequality $\Xi \ne \wp T(\Xi )$ holds. Define the set $\chi =\left\{\Xi \in C_0:∥\Xi ∥<N\right\}$. So, the operator $T:\overline{\chi }\to C_0$ satisfies the complete continuity. Assume

$$\Xi =\wp T\Xi$$

holds for some $\Xi \in \overline{\chi }$ and $\wp \in (0,1)$. Then, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|\Xi \left(v\right)\right|& =\left|\wp T\Xi \left(v\right)\right|\le \left|T\Xi \left(v\right)\right|\hfill \\ & \le {\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (♭)-\Psi \left(0\right))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}∥\Xi ∥\hfill \\ & +{\displaystyle \frac{{(\Psi (♭)-\Psi \left(0\right))}^{\Theta }}{\Gamma (\Theta +1)}}(∥g∥+∥l∥({\mathbb{K}}_m∥\Xi ∥+(2{\mathbb{K}}_m\mathbb{H}+{\mathbb{M}}_m){∥\aleph ∥}_{\breve{\mathfrak{B}}}))+\left|\vartheta (♭)\right|.\hfill \end{array}\end{array}$$

Hence, the following inequalities can be obtained based on the conditions (15).

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill ∥\Xi ∥& \le {\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (♭)-\Psi \left(0\right))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}∥\Xi ∥+{\displaystyle \frac{{(\Psi (♭)-\Psi \left(0\right))}^{\Theta }}{\Gamma (\Theta +1)}}{\mathbb{K}}_m∥l∥∥\Xi ∥\hfill \\ & +{\displaystyle \frac{{(\Psi (♭)-\Psi \left(0\right))}^{\Theta }}{\Gamma (\Theta +1)}}(∥g∥+∥l∥(2{\mathbb{K}}_m\mathbb{H}+{\mathbb{M}}_m){∥\aleph ∥}_{\breve{\mathfrak{B}}})+\left|\vartheta (♭)\right|\hfill \\ & <N\hfill \end{array}\end{array}$$

holds, which contradicts with the fact that $N=∥\Xi ∥$. Thus, we obtain

$$\Xi \ne \wp T\Xi$$

for any $\Xi \in \overline{\chi }$ and $\wp \in (0,1)$.

Utilizing Lemma 1, we infer that there exists at least one fixed point $\Xi$ of T. This completes the proof. □

Theorem 4.

Suppose that (D1) and (D4) hold. Additionally, suppose that

$${\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (♭)-\Psi \left(0\right))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}+{\displaystyle \frac{{(\Psi (♭)-\Psi \left(0\right))}^{r_1}{∥\tilde{\eta }∥}_p{\mathbb{K}}_m}{\Gamma (\Theta )r_2}}\underset{r\to \infty }{lim\; sup}{\displaystyle \frac{\Omega \left(r\right)}{r}}<1$$

(16)

holds, where $r_1=[(\Theta -1)q+1]/q$, $r_2={(1+(\Theta -1)q)}^{{\displaystyle \frac{1}{q}}}$. Then, the Equation (1) has at least one solution on $(-\infty ,♭]$.

Proof. 

With the aid of the Lebesgue dominated convergence theorem, it is straightforward to confirm that T is continuous.

We now show that T maps bounded subsets into bounded subsets in $C_0$. Let $B_r=\left\{\Xi \in C_0:∥\Xi ∥\le r\right\}$. Then, for any $\Xi \in B_r$ and $v\in [0,♭]$, we have

$$\begin{array}{cc}\hfill \left|T\Xi (v)\right|& \le {\displaystyle \frac{\left|\mathfrak{a}\right|}{\Gamma (\Theta -\varrho )}}{\int }_0^v{\Psi }^{\prime }(s){(\Psi (v)-\Psi (s))}^{\Theta -\varrho -1}\left|\Xi (s)\right|\mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }(s){(\Psi (v)-\Psi (s))}^{\Theta -1}\left|f(s,{\widehat{\Xi }}_s+{\widehat{\aleph }}_s)\right|\mathrm{d}s+\left|\vartheta (♭)\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (v)-\Psi (0))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}∥\Xi ∥+{\displaystyle \frac{1}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }(s){(\Psi (v)-\Psi (s))}^{\Theta -1}\hfill \\ \hfill & \times \tilde{\eta }(s)\Omega ({∥{\widehat{\Xi }}_s+{\widehat{\aleph }}_s∥}_{\breve{\mathfrak{B}}})\mathrm{d}s+\left|\vartheta (♭)\right|.\hfill \end{array}$$

As

$$\begin{array}{cc}\hfill {∥{\widehat{\Xi }}_s+{\widehat{\aleph }}_s∥}_{\breve{\mathfrak{B}}}& \le {\mathbb{K}}_{m}r+(2{\mathbb{K}}_m\mathbb{H}+{\mathbb{M}}_m){∥\aleph ∥}_{\breve{\mathfrak{B}}}:=r_0,\hfill \end{array}$$

where ${\mathbb{M}}_m=sup\left\{\left|\mathbb{M}\left(v\right)\right|:v\in [a,♭]\right\}$, ${\mathbb{K}}_m=sup\left\{\left|\mathbb{K}\left(v\right)\right|:v\in [a,♭]\right\}$ and $\mathbb{H}>0$ is a constant. By Hölder inequality, we have

$$\begin{array}{cc}\hfill \left|T\Xi (v)\right|& \le {\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (♭)-\Psi (0))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}∥\Xi ∥+{\displaystyle \frac{\Omega (r_0)}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }(s){(\Psi (v)-\Psi (s))}^{\Theta -1}\tilde{\eta }(s)\mathrm{d}s+\left|\vartheta (♭)\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (♭)-\Psi (0))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}∥\Xi ∥+{\displaystyle \frac{\Omega (r_0)}{\Gamma (\Theta )}}{({\int }_0^v{(\Psi (v)-\Psi (s))}^{(\Theta -1)q}\mathrm{d}\Psi (s))}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {({\int }_0^{♭}{\tilde{\eta }}^p(\Psi (s))\mathrm{d}\Psi (s))}^{{\displaystyle \frac{1}{p}}}+\left|\vartheta (♭)\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (♭)-\Psi (0))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}r+{\displaystyle \frac{\Omega (r_0){(\Psi (♭)-\Psi (0))}^{r_1}}{\Gamma (\Theta )r_2}}{∥\tilde{\eta }∥}_p+\left|\vartheta (♭)\right|=\zeta .\hfill \end{array}$$

where $r_1=[(\Theta -1)q+1]/q$, $r_2={(1+(\Theta -1)q)}^{{\displaystyle \frac{1}{q}}}$, ${∥\tilde{\eta }∥}_p={\left({\int }_0^{♭}{\left|\tilde{\eta }\left(s\right)\right|}^p\right)}^{{\displaystyle \frac{1}{p}}}$, ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$ and $(\Theta -1)>-1$.

Next, we prove that T maps bounded subsets into equicontinuous subsets of $C_0$. Consider the set $B_r=\left\{\Xi \in C_0:∥\Xi ∥\le r\right\}$ and take an arbitrary $\Xi \in B_r$. Then, for $v_1,v_2\in [0,♭]$ with $v_1<v_2$, we have

$$\begin{array}{cc}& \left|(T\Xi )(v_2)-(T\Xi )(v_1)\right|\hfill \\ & \le {\displaystyle \frac{\left|\mathfrak{a}\right|∥\Xi ∥}{\Gamma (\Theta -\varrho )}}[{\int }_0^{v_1}({(\Psi (v_2)-\Psi (s))}^{\Theta -\varrho -1}-{(\Psi (v_1)-\Psi (s))}^{\Theta -\varrho -1})\mathrm{d}\Psi (s)\hfill \\ & +{\int }_{v_1}^{v_2}{(\Psi (v_2)-\Psi (s))}^{\Theta -\varrho -1}\mathrm{d}\Psi (s)]\hfill \\ & +{\displaystyle \frac{1}{\Gamma (\Theta )}}\left|{\int }_0^{v_1}({(\Psi (v_2)-\Psi (s))}^{\Theta -1}-{(\Psi (v_1)-\Psi (s))}^{\Theta -1})f(s,{\widehat{\Xi }}_s+{\widehat{\aleph }}_s)\mathrm{d}\Psi (s)\right|\hfill \\ & +\left|{\int }_{v_1}^{v_2}{(\Psi (v_2)-\Psi (s))}^{\Theta -1}f(s,{\widehat{\Xi }}_s+{\widehat{\aleph }}_s)\mathrm{d}\Psi (s)\right|+\left|\vartheta (v_2)-\vartheta (v_1)\right|\hfill \\ & \le {\displaystyle \frac{\left|\mathfrak{a}\right|∥\Xi ∥}{\Gamma (\Theta -\varrho +1)}}({(\Psi (v_2)-\Psi (0))}^{\Theta -\varrho }-{(\Psi (v_1)-\Psi (0))}^{\Theta -\varrho }+2{(\Psi (v_2)-\Psi (v_1))}^{\Theta -\varrho })\hfill \\ & +{\displaystyle \frac{\Omega (r_0)}{\Gamma (\Theta )}}{({\int }_0^{v_1}{[{(\Psi (v_2)-\Psi (s))}^{\Theta -1}-{(\Psi (v_1)-\Psi (s))}^{\Theta -1}]}^q\mathrm{d}\Psi (s))}^{{\displaystyle \frac{1}{q}}}{({\int }_0^{v_1}{\tilde{\eta }}^p(\Psi (s))\mathrm{d}\Psi (s))}^{{\displaystyle \frac{1}{p}}}\hfill \\ & +{\displaystyle \frac{\Omega (r_0)}{\Gamma (\Theta )}}{({\int }_{v_1}^{v_2}{(\Psi (v_2)-\Psi (s))}^{(\Theta -1)q}\mathrm{d}\Psi (s))}^{{\displaystyle \frac{1}{q}}}{({\int }_{v_1}^{v_2}{\tilde{\eta }}^p(\Psi (s))\mathrm{d}\Psi (s))}^{{\displaystyle \frac{1}{p}}}+\left|\vartheta (v_2)-\vartheta (v_1)\right|.\hfill \end{array}$$

As $n>1$, $a>b$, ${(a-b)}^n<a^n-b^n$, we can obtain

$$\begin{array}{cc}\hfill & \left|(T\Xi )(v_2)-(T\Xi )(v_1)\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left|\mathfrak{a}\right|∥\Xi ∥}{\Gamma (\Theta -\varrho +1)}}({(\Psi (v_2)-\Psi (0))}^{\Theta -\varrho }-{(\Psi (v_1)-\Psi (0))}^{\Theta -\varrho }+2{(\Psi (v_2)-\Psi (v_1))}^{\Theta -\varrho })\hfill \\ \hfill & +{\displaystyle \frac{\Omega (r_0)}{\Gamma (\Theta )}}{({\int }_0^{v_1}({(\Psi (v_2)-\Psi (s))}^{(\Theta -1)q}-{(\Psi (v_1)-\Psi (s))}^{(\Theta -1)q})\mathrm{d}\Psi (s))}^{{\displaystyle \frac{1}{q}}}{∥\tilde{\eta }∥}_p\hfill \\ \hfill & +{\displaystyle \frac{{(\Psi (v_2)-\Psi (v_1))}^{r_1}}{r_2}}{∥\tilde{\eta }∥}_p+\left|\vartheta (v_2)-\vartheta (v_1)\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left|\mathfrak{a}\right|∥\Xi ∥}{\Gamma (\Theta -\varrho +1)}}({(\Psi (v_2)-\Psi (0))}^{\Theta -\varrho }-{(\Psi (v_1)-\Psi (0))}^{\Theta -\varrho }+2{(\Psi (v_2)-\Psi (v_1))}^{\Theta -\varrho })\hfill \\ \hfill & +{\displaystyle \frac{\Omega (r_0){∥\tilde{\eta }∥}_p}{\Gamma (\Theta )r_2}}({(\Psi (v_2)-\Psi (0))}^{r_1}-{(\Psi (v_1)-\Psi (0))}^{r_1}+2{(\Psi (v_2)-\Psi (v_1))}^{r_1})\hfill \\ \hfill & +\left|\vartheta (v_2)-\vartheta (v_1)\right|,\hfill \end{array}$$

where $r_1=[(\Theta -1)q+1]/q$, $r_2={(1+(\Theta -1)q)}^{{\displaystyle \frac{1}{q}}}$, $r_0={\mathbb{K}}_{m}r+(2{\mathbb{K}}_m\mathbb{H}+{\mathbb{M}}_m){∥\aleph ∥}_{\breve{\mathfrak{B}}}$. As $\Theta -\varrho >0$, $r_1>0$, it is easy to see that $\left|(T\Xi )\left(v_2\right)-(T\Xi )\left(v_1\right)\right|\to 0$ as $v_1-v_2\to 0$. By the Arzela–Ascolli theorem, we can conclude that $T:C_0\to C_0$ is a completely continuous mapping.

Now, we have to verify that there exists at least one fixed point $\Xi$ of T. We assume that there exists an open set $\mathcal{V}\subseteq C_0$, such that for every $\Xi \in \partial \mathcal{V}$ and for each $\wp \in (0,1)$, the inequality $\Xi \ne \wp T(\Xi )$ holds. Define the set $\chi =\left\{\Xi \in C_0:∥\Xi ∥<N\right\}$. So, the operator $T:\overline{\chi }\to C_0$ satisfies the complete continuity. Assume

$$\Xi =\wp T\Xi$$

holds for some $\Xi \in \overline{\chi }$ and $\wp \in (0,1)$. Then, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|\Xi (v)\right|& =\left|\wp T\Xi (v)\right|\le \left|T\Xi (v)\right|\hfill \\ & \le {\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (♭)-\Psi (0))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}∥\Xi ∥\hfill \\ & +{\displaystyle \frac{\Omega ({\mathbb{K}}_{m}r+(2{\mathbb{K}}_m\mathbb{H}+{\mathbb{M}}_m){∥\aleph ∥}_{\breve{\mathfrak{B}}}){(\Psi (♭)-\Psi (0))}^{r_1}}{\Gamma (\Theta )r_2}}{∥\tilde{\eta }∥}_p+\left|\vartheta (♭)\right|.\hfill \end{array}\end{array}$$

Hence, the following inequalities can be obtained based on the conditions (16).

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill ∥\Xi ∥& <N\hfill \end{array}\end{array}$$

holds, which contradicts $N=∥\Xi ∥$. Thus,

$$\Xi \ne \wp T\Xi$$

for any $\Xi \in \overline{\chi }$ and $\wp \in (0,1)$.

As a consequence of Lemma 1, we deduce that there exists at least one fixed point $\Xi$ of T. The proof is complete. □

4. Stability Analysis

In this section, we conduct an analysis of the Hs-Um-St of the Fra-Diff-Equs (1) with infinite delay.

Theorem 5.

Assume that all the conditions stated in Theorem 2 are met, and inequality (6) admits at least one solution. Under such circumstances, the problem (1) exhibits Hs-Um-St.

Proof. 

For each $\epsilon >0$, and each function ℷ that satisfies the following inequalities

$$\left|{}^{c}D^{\Theta ;\Psi }\gimel \left(v\right)-\mathfrak{a}{}^{c}D^{\varrho ;\Psi }\gimel \left(v\right)-f(v,{\gimel }_v)\right|\le \epsilon ,v\in [0,♭],$$

let $g\left(v\right)={}^{c}D^{\Theta ;\Psi }\gimel \left(v\right)-\mathfrak{a}{}^{c}D^{\varrho ;\Psi }\gimel \left(v\right)-f(v,{\gimel }_v)$. Then we have

$$\gimel \left(v\right)=\mathfrak{a}{J}^{\Theta -\varrho ;\Psi }\gimel \left(v\right)+J^{\Theta ;\Psi }f(v,{\gimel }_v)+J^{\Theta ;\Psi }g\left(v\right)+\vartheta \left(v\right).$$

According to Theorem 2, there is a unique solution $\daleth \left(v\right)$ of problem (1), and the function ℸ can be expressed as

$$\daleth \left(v\right)=\mathfrak{a}{J}^{\Theta -\varrho ;\Psi }\daleth \left(v\right)+J^{\Theta ;\Psi }f(v,{\daleth }_v)+\vartheta \left(v\right).$$

So, we have

$$\begin{array}{cc}\hfill \left|\gimel \left(v\right)-\daleth \left(v\right)\right|& \le {\displaystyle \frac{\left|\mathfrak{a}\right|}{\Gamma (\Theta -\varrho )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -\varrho -1}\left|\gimel \left(v\right)-\daleth \left(v\right)\right|\mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}\left|f(s,{\gimel }_s)-f(s,{\daleth }_s)\right|\mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}\left|g\left(s\right)\right|\mathrm{d}s.\hfill \end{array}$$

As

$$\left|f(s,{\gimel }_s)-f(s,{\daleth }_s)\right|\le \mathbb{L}{∥{\gimel }_s-{\daleth }_s∥}_{\breve{\mathfrak{B}}},$$

together with Definition 3, we obtain

$$\begin{array}{cc}\hfill {∥{\gimel }_s-{\daleth }_s∥}_{\breve{\mathfrak{B}}}& ={∥({\widehat{\gimel }}_s+{\widehat{\aleph }}_s)-({\widehat{\daleth }}_s+{\widehat{\aleph }}_s)∥}_{\breve{\mathfrak{B}}}={∥{\widehat{\gimel }}_s-{\widehat{\daleth }}_s∥}_{\breve{\mathfrak{B}}}\hfill \\ \hfill & \le \mathbb{K}(s)\underset{0\le \nu \le s}{sup}∥\widehat{\gimel }(\nu )-\widehat{\daleth }(\nu )∥+\mathbb{M}(s){∥{\widehat{\gimel }}_0-{\widehat{\daleth }}_0∥}_{\breve{\mathfrak{B}}}\hfill \\ \hfill & \le {\mathbb{K}}_m\underset{0\le \nu \le s}{sup}∥\gimel (\nu )-\aleph (0)-\daleth (\nu )+\aleph (0)∥\hfill \\ \hfill & ={\mathbb{K}}_m\underset{0\le \nu \le s}{sup}\left|\gimel (\nu )-\daleth (\nu )\right|,\hfill \end{array}$$

which indicates that

$$\begin{array}{cc}\hfill \left|\gimel \left(v\right)-\daleth \left(v\right)\right|& \le {\displaystyle \frac{\left|\mathfrak{a}\right|}{\Gamma (\Theta -\varrho )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -\varrho -1}\left|\gimel \left(s\right)-\daleth \left(s\right)\right|\mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{\mathbb{L}{\mathbb{K}}_m}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}\underset{0\le \sigma \le s}{sup}\left|\gimel \left(\sigma \right)-\daleth \left(\sigma \right)\right|\mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{{(\Psi (♭)-\Psi \left(0\right))}^{\Theta }}{\Gamma (\Theta +1)}}\epsilon .\hfill \end{array}$$

According to Lemma 3,

$$\begin{array}{cc}\hfill & \underset{0\le \nu \le v}{sup}\left|\gimel \left(\nu \right)-\daleth \left(\nu \right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left|\mathfrak{a}\right|}{\Gamma (\Theta -\varrho )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -\varrho -1}\underset{0\le \sigma \le s}{sup}\left|\gimel \left(\sigma \right)-\daleth \left(\sigma \right)\right|\mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{\mathbb{L}{\mathbb{K}}_m}{\Gamma (\Theta )}}{\int }_0^v{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}\underset{0\le \sigma \le s}{sup}\left|\gimel \left(\sigma \right)-\daleth \left(\sigma \right)\right|\mathrm{d}s+{\displaystyle \frac{{(\Psi (♭)-\Psi \left(0\right))}^{\Theta }}{\Gamma (\Theta +1)}}\epsilon \hfill \\ \hfill & ={\int }_0^v\left[\left|\mathfrak{a}\right|{\displaystyle \frac{{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -\varrho -1}}{\Gamma (\Theta -\varrho )}}+{\displaystyle \frac{\mathbb{L}{\mathbb{K}}_m{\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -1}}{\Gamma (\Theta )}}\right]\underset{0\le \sigma \le s}{sup}\left|\gimel \left(\sigma \right)-\daleth \left(\sigma \right)\right|\mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{{(\Psi (♭)-\Psi \left(0\right))}^{\Theta }}{\Gamma (\Theta +1)}}\epsilon .\hfill \end{array}$$

Let $\phi \left(v\right):={\displaystyle \underset{0\le \nu \le v}{sup}}\left|\gimel \left(\nu \right)-\daleth \left(\nu \right)\right|$, $\mathbb{M}={\displaystyle \frac{{(\Psi \left(b\right)-\Psi \left(0\right))}^{\Theta }}{\Gamma (\Theta +1)}},$ and $g(v,s)={\displaystyle \frac{\left|\mathfrak{a}\right|}{\Gamma (\Theta -\varrho )}}+{\displaystyle \frac{\mathbb{L}{\mathbb{K}}_b{(\Psi \left(v\right)-\Psi \left(s\right))}^{\varrho }}{\Gamma (\Theta )}}\le {\mathbb{M}}_0.$ We can obtain

$$\phi \left(v\right)\le \mathbb{M}\epsilon +{\int }_0^{v}g(v,s){\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -\varrho -1}\phi \left(s\right)\mathrm{d}s.$$

It is not difficult to show that

$$\begin{array}{cc}\hfill \phi (v)& \le \mathbb{M}\epsilon +\mathbb{M}\epsilon {\int }_0^v\sum _{n=1}^{\infty }{\displaystyle \frac{{({\mathbb{M}}_0\Gamma (\Theta -\varrho ))}^n}{\Gamma (n(\Theta -\varrho ))}}{(\Psi (v)-\Psi (s))}^{n(\Theta -\varrho )-1}{\Psi }^{\prime }(s)\mathrm{d}s\hfill \\ \hfill & \le \mathbb{M}\epsilon +\mathbb{M}\epsilon \sum _{n=1}^{\infty }{\displaystyle \frac{{({\mathbb{M}}_0\Gamma (\Theta -\varrho ))}^n}{\Gamma (n(\Theta -\varrho )+1)}}{(\Psi (b)-\Psi (0))}^{n(\Theta -\varrho )}\hfill \\ \hfill & =\epsilon \mathbb{M}(1+\sum _{n=1}^{\infty }{\displaystyle \frac{{({\mathbb{M}}_0\Gamma (\Theta -\varrho ))}^n}{\Gamma (n(\Theta -\varrho )+1)}}{(\Psi (b)-\Psi (0))}^{n(\Theta -\varrho )})\hfill \\ \hfill & =\epsilon \mathbb{M}{E}_{\Theta -\varrho }({\mathbb{M}}_0\Gamma (\Theta -\varrho ){(\Psi (b)-\Psi (0))}^{\Theta -\varrho }).\hfill \end{array}$$

Then, the inequality

$$\phi \left(v\right)\le \mathfrak{x}\epsilon$$

holds for $\mathfrak{x}=\mathbb{M}{E}_{\Theta -\varrho }({\mathbb{M}}_0\Gamma (\Theta -\varrho ){(\Psi \left(b\right)-\Psi \left(0\right))}^{\Theta -\varrho }),$ which implies that the Hs-Um-St of problem (1) is proved. □

5. Examples

For any real constant $\mathrm{Y}>0$, we set

$$U_{\mathrm{Y}}=\left\{y\in C((-\infty ,0],R):\underset{\theta \to -\infty }{lim}{e}^{\mathrm{Y}\theta }y\left(\theta \right)existinR\right\},$$

and set

$${\left|y\right|}_{\mathrm{Y}}=sup\left\{e^{\mathrm{Y}\theta }\left|y\left(\theta \right)\right|:-\infty <\theta \le 0\right\}.$$

By [31], $U_{\mathrm{Y}}$ satisfies the conditions $\mathfrak{K}=\mathfrak{M}=\mathfrak{H}=1$, and, hence, $U_{\mathrm{Y}}$ is a phase space.

Example 1.

Consider the following nonlinear Ψ-Caputo-type Fra-Diff-Equs with infinite delay of the form

$$\left\{\begin{array}{c}{}^{c}D^{0.7;\Psi }y\left(v\right)-{\displaystyle \frac{1}{4}}{}^{c}D^{0.4;\Psi }y\left(v\right)=e^{-\gamma v}(y_v+cosv),v\in J=[0,1],\hfill \\ y\left(v\right)=\aleph \left(v\right)\in U_{\gamma },v\in (-\infty ,0],\hfill \end{array}\right.$$

(17)

where $\Psi \left(v\right)=\sqrt{1+v}$.

In this case, it is readily observable that (D1) and (D3) are satisfied with the function $l\left(v\right)=e^{-\mathrm{Y}v}$. Due to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\left|\mathfrak{a}\right|{(\Psi (♭)-\Psi \left(0\right))}^{\Theta -\varrho }}{\Gamma (\Theta -\varrho +1)}}+{\displaystyle \frac{{\mathbb{K}}_m{(\Psi (♭)-\Psi \left(0\right))}^{\Theta }}{\Gamma (\Theta +1)}}∥l∥\hfill \\ \hfill & <{\displaystyle \frac{{(\sqrt{2}-1)}^{0.3}}{4\Gamma \left(1.3\right)}}+{\displaystyle \frac{{(\sqrt{2}-1)}^{0.7}}{\Gamma \left(1.7\right)}}\approx 0.7658<1.\hfill \end{array}$$

Thus, all the conditions of Theorem 3 hold. Therefore, the problem (17) has at least one solution on $(-\infty ,1].$

Example 2.

Consider the following problem

$$\left\{\begin{array}{c}{}^{c}D^{0.8;\Psi }y\left(v\right)-{\displaystyle \frac{1}{5}}{}^{c}D^{0.4;\Psi }y\left(v\right)={\displaystyle \frac{c^{\prime }{e}^{-\mathrm{Y}v+v}{\left|y\right|}_{\mathrm{Y}}}{1+{\left|y\right|}_{\mathrm{Y}}}},v\in J=[0,1],\hfill \\ y\left(v\right)=\aleph \left(v\right)\in U_{\mathrm{Y}},v\in (-\infty ,0],\hfill \end{array}\right.$$

(18)

where $\Psi \left(v\right)=2^v$, $c^{\prime }>0$. Set

$$f(v,x)={\displaystyle \frac{c^{\prime }{e}^{-\mathrm{Y}v+v}x}{1+x}},(v,x)\in [0,1]\times R^{+}.$$

Then, for any x, $y\in U_{\mathrm{Y}},$ we have

$$\begin{array}{cc}\hfill \left|f(v,x)-f(v,y)\right|& =c^{\prime }{e}^{-\mathrm{Y}v+v}\left|{\displaystyle \frac{x}{1+x}}-{\displaystyle \frac{y}{1+y}}\right|\hfill \\ \hfill & \le {\displaystyle \frac{c^{\prime }{e}^{-\mathrm{Y}v+v}\left|x-y\right|}{(1+x)(1+y)}}\le c^{\prime }{e}^v\left|x-y\right|\le c^{\prime }e\left|x-y\right|.\hfill \end{array}$$

Hence, $0<c^{\prime }<{\displaystyle \frac{\Gamma \left(1.8\right)(5\Gamma (1.4)-1)}{5e\Gamma \left(1.4\right)}}\approx 0.2674$, condition (D2) holds. Thus, problem (18) has a unique solution to $(-\infty ,1]$ by applying Theorem 2.

Next, we discuss the Hs-Um-St of problem (18). For any $\epsilon >0$, and each function y that satisfies the following inequalities

$$\left|{}^{c}D^{0.8;\Psi }y\left(v\right)-{\displaystyle \frac{1}{5}}{}^{c}D^{0.4;\Psi }y\left(v\right)-{\displaystyle \frac{c^{\prime }{e}^{-\mathrm{Y}v+v}{\left|y\right|}_{\mathrm{Y}}}{1+{\left|y\right|}_{\mathrm{Y}}}}\right|\le \epsilon ,v\in J=[0,1],$$

let $g\left(v\right)$ represent the left side of the inequality above and let $\daleth \left(t\right)$ be the unique solution of problem (18); then, we have

$$\underset{0\le \nu \le v}{sup}\left|y\left(\nu \right)-\daleth \left(\nu \right)\right|\le {\int }_0^v\left[{\displaystyle \frac{2^{s}ln2{(2^v-2^s)}^{-0.6}}{5\Gamma \left(0.4\right)}}+{\displaystyle \frac{c^{\prime }{2}^{s}ln2{(2^v-2^s)}^{-0.2}}{\Gamma \left(0.8\right)}}\right]\underset{0\le \sigma \le s}{sup}\left|y\left(\sigma \right)-\daleth \left(\sigma \right)\right|\mathrm{d}s+{\displaystyle \frac{\epsilon }{\Gamma \left(1.8\right)}}$$

let $\phi \left(v\right):={\displaystyle \underset{0\le \nu \le v}{sup}}\left|y\left(\nu \right)-\daleth \left(\nu \right)\right|$, $\mathbb{M}={\displaystyle \frac{1}{\Gamma \left(1.8\right)}},$ and $g(v,s)={\displaystyle \frac{1}{5\Gamma \left(0.4\right)}}+{\displaystyle \frac{c^{\prime }{(2^v-2^s)}^{0.4}}{\Gamma \left(0.8\right)}}\le {\displaystyle \frac{1}{5\Gamma \left(0.4\right)}}+{\displaystyle \frac{c^{\prime }}{\Gamma \left(0.8\right)}}={\mathbb{M}}_0.$ By Lemma 4,

$$\begin{array}{cc}\hfill \phi \left(v\right)& \le \mathbb{M}\epsilon +{\int }_0^{v}g(v,s){\Psi }^{\prime }\left(s\right){(\Psi \left(v\right)-\Psi \left(s\right))}^{\Theta -\varrho -1}\phi \left(s\right)\mathrm{d}s\hfill \\ \hfill & \le \mathbb{M}\epsilon +\mathbb{M}\epsilon \sum _{n=1}^{\infty }{\displaystyle \frac{{({\mathbb{M}}_0\Gamma \left(0.4\right))}^n}{\Gamma (0.4n+1)}}\hfill \\ \hfill & \le \mathbb{M}\epsilon E_{0.4}({\mathbb{M}}_0\Gamma \left(0.4\right)),\hfill \end{array}$$

(19)

let $\mathfrak{x}=\mathbb{M}{E}_{0.4}({\mathbb{M}}_0\Gamma \left(0.4\right))={\displaystyle \frac{1}{\Gamma \left(1.8\right)}}{E}_{0.4}({\displaystyle \frac{1}{5}}+{\displaystyle \frac{c^{\prime }\Gamma \left(0.4\right)}{\Gamma \left(0.8\right)}})$, it follows that $\phi \left(v\right)\le \mathfrak{x}\epsilon$, which implies that the problem (18) is Hs-Um-St.

6. Conclusions

In this paper, we mainly discuss a class of nonlinear Fra-Diff-Equs with finite time delay. We first use the Leray–Schauder alternation theorem and the Banach fixed-point theorem to study the existence and uniqueness of solutions for this class of $\Psi$-Caputo Fra-Diff-Equs. Then, we study the Hs-Um-St of the Fra-Diff-Equs (1). With the aim of overcoming the difficulty in proving Hs-Um-St, some inequalities are obtained for delayed Fra-Diff-Equs. In addition, there are two examples listed to illustrate the conclusions. The findings of our study enhance and extend the results presented in [23,24].