Nonlinear Piecewise Caputo Fractional Pantograph System with Respect to Another Function

Abstract

The existence, uniqueness, and various forms of Ulam–Hyers (UH)-type stability results for nonlocal pantograph equations are developed and extended in this study within the frame of novel $psi$-piecewise Caputo fractional derivatives, which generalize the piecewise operators recently presented in the literature. The required results are proven using Banach’s contraction mapping and Krasnoselskii’s fixed-point theorem. Additionally, results pertaining to UH stability are obtained using traditional procedures of nonlinear functional analysis. Additionally, in light of our current findings, a more general challenge for the pantograph system is presented that includes problems similar to the one considered. We provide a pertinent example as an application to support the theoretical findings.

1. Introduction

Fractional calculus (FC) has received great interest from researchers due to its wide range of applications in various scientific fields. The crucial concepts and definitions of FC have been presented [1,2]. In [3,4], the authors introduced some fundamental history of fractional calculus and its applications to engineering and different areas of science.

Many classes of fractional differential equations (FDEs) have been extensively studied and analyzed in the past decades; for example, theories involving the existence of unique solutions have been documented [5,6,7,8,9,10]. Numerical and analytical methods have been developed with the aim of solving such equations [11,12,13,14]. These equations have been tracked as useful in modeling some real-world problems with incredible acheivements.

The qualitative properties of solutions address a vital part of the theory of FDEs. The previously aforementioned region has been studied well for standard differential equations. In any case, for FDEs, there are numerous aspects and perspectives that require further research and surveying. The consideration of the existence and uniqueness has been particularly considered by using Riemann–Liouville (R-L), Caputo, Hilfer, and other FDs (see [15,16,17,18,19,20,21,22,23] and the references therein).

Recently, many generalizations of classical FDs above have been presented by Kilbas et al. [2], Almeida in [24], and Sousa-Oliveira [25]. These derivatives are called the $\psi$-Reiman–Liouville, $\psi$-Caputo, and $\psi$-Hilfer FDs.

Another significant class of FDEs is the pantograph equations (PEs), which have not been as thoroughly investigated in the frame of novel FDs. PEs are an important category of delay equations that give changes in the dependent worth at a past time [26], and they are applied in deterministic situations. A pantograph is basically a tool used for measuring and drawing. This tool is currently used in electric trains and electric cells [27,28,29].

In 1971 Ockendon and Taylor [30] discussed how electric flow is collected by the pantograph of an electric train using the following delay equation

$$\left\{\begin{array}{c}{\upsilon }^{\prime }\left(\varkappa \right)=a\upsilon \left(\varkappa \right)+b\upsilon \left(\lambda \varkappa \right),\varkappa \in [0,T],0<\lambda <1,\\ \upsilon \left(0\right)={\upsilon }_0\end{array}\right.$$

(1)

which is currently called PE. Since that time, many researchers have studied and included it in different mathematical and scientific fields such as number theory, probability, electrodynamics, and medication (see [30,31,32] and the references therein).

The analytical and numerical methods of (1) have been deliberated by several authors [33,34,35]. Derfel and Iserels [36,37], extensively studied the PEs. The following type of nonlinear PE

$$\left\{\begin{array}{c}{\upsilon }^{\prime }\left(\varkappa \right)=f(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\cdots ,\upsilon \left({\lambda }_m\varkappa \right)),\varkappa \in [0,T],0<{\lambda }_1<\cdots <{\lambda }_m<1,\\ \upsilon \left(0\right)={\upsilon }_0\end{array}\right.$$

(2)

was studied by Liu et al. [38], whereas the nonlinear neutral PE

$$\left\{\begin{array}{c}{\upsilon }^{\prime }\left(\varkappa \right)=f(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left(\lambda \varkappa \right),{\upsilon }^{\prime }\left(\lambda \varkappa \right)),\varkappa >0,0<\lambda <1,\\ \upsilon \left(0\right)={\upsilon }_0\end{array}\right.$$

(3)

was considered by Sezer et al. [39].

Many studied have been conducted on fractional PEs because of their significance to numerous areas of exploration. For example, Balachandran et al. [40], discussed the existence of solutions for the following Caputo-type pantograph problem

$$\left\{\begin{array}{c}{}^C{\mathbb{D}}_{0^{+}}^{\vartheta }\upsilon \left(\varkappa \right)=f(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left(\lambda \varkappa \right)),\varkappa \in [0,T],0<\vartheta <1,\\ \upsilon \left(0\right)={\upsilon }_0+g\left(\upsilon \right).\end{array}\right.$$

(4)

In this regard, Atangana and Araz [41] introduced the concept of the piecewise derivative with the aim of modeling real-world problems following multiple processes. Motivated by the above works and by [41], we consider the following piecewise Caputo pantograph problems (PCPPs):

$$\left\{\begin{array}{c}{}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta }\upsilon \left(\varkappa \right)=\phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\cdots ,\upsilon \left({\lambda }_m\varkappa \right)\right),\\ \upsilon \left(0\right)={\upsilon }_0+g\left(\upsilon \right),\end{array}\right.$$

(5)

and

$$\left\{\begin{array}{c}{}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\upsilon \left(\varkappa \right)=\phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\cdots ,\upsilon \left({\lambda }_m\varkappa \right)\right),\\ \upsilon \left(0\right)={\upsilon }_0+g\left(\upsilon \right),\end{array}\right.$$

(6)

where $0<\vartheta \le 1,$ $\varkappa \in \mathbb{J}:=[0,b],$ ${\upsilon }_0\in \mathbb{R},$ $0<{\lambda }_i<1$, for $i=1,2,\cdots ,m$, and ${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta }$, and ${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }$ represent the piecewise and $\psi$-piecewise Caputo FD of order $\vartheta$, respectively, defined by

$${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta }f\left(\varkappa \right)=\left\{\begin{array}{c}\frac{d}{d\varkappa }f\left(\varkappa \right):if\varkappa \in [0,{\varkappa }_1],\\ {}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta }f\left(\varkappa \right):if\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

and

$${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }f\left(\varkappa \right)=\left\{\begin{array}{c}\left(\frac{1}{{\psi }^{\prime }\left(\varkappa \right)}\frac{d}{d\varkappa }\right)f\left(\varkappa \right):if\varkappa \in [0,{\varkappa }_1],\\ {}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta ;\psi }f\left(\varkappa \right):if\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

where ${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta }$ and ${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }$ are classical (or generalized) derivative on $0\le \varkappa \le {\varkappa }_1$ and Caputo (or $\psi$-Caputo) FD on ${\varkappa }_1\le \varkappa \le b$, $\phi :\mathbb{J}\times \mathbb{R}\times {\mathbb{R}}^m\to \mathbb{R}$ and $g:C(\mathbb{J},\mathbb{R})\to \mathbb{R}$ are given functions, ($m\in \mathbb{N}$).

It is essential to note that the utilization of nonlinear condition $\upsilon \left(0\right)={\upsilon }_0+g\left(\upsilon \right)$ in physical issues yields a better impact than the initial condition $\upsilon \left(0\right)={\upsilon }_0$ (see [42]).

We pay attention to the topic of novel piecewise operators. To the best of our knowledge, no results in the literature address the qualitative aspects of the aforesaid problems using the $\psi$-piecewise FC. Consequently, to close this gap and enrich the literature, we developed and extended the existence, uniqueness, and Ulam–Hyers stability results of $\psi$-piecewise Caputo pantograph problems (5) and (6) based on known fixed-point theorems of the Banach type and Krasnoselskii type. Furthermore, we present a more general problem as a system that covers the problems at hand.

Remark 1.

 

(i) If $\psi \left(\varkappa \right)=\varkappa ,$ then problem (6) reduces to problem (5).

(ii) If $\psi \left(\varkappa \right)=\varkappa$ and $m=1,$ then for $\varkappa \in [{\varkappa }_1,b],$ the problem (6) reduces to problem (4) considered in [40].

(iii) If $\psi \left(\varkappa \right)=\varkappa ,$$g\equiv 0$ and $m=2,$ then for $\varkappa \in [0,{\varkappa }_1],$ the problem (6) reduces to problem (3) for an implicit term [39].

(iv) If $\psi \left(\varkappa \right)=\varkappa$ and $g\equiv 0,$ then for $\varkappa \in [0,{\varkappa }_1],$ the problem (6) reduces to (2), see [38].

(v) Our current results for problem (6) stay available for problem (5).

This paper is arranged as follows: Section 2 provides some required results and fundamentals about piecewise FC. Our major outcomes for problems (5) and (6) are proved in Section 3. A comprehensive example verifying the validity of the theories is presented in Section 4. The conclusions of our study are summarized in the final section.

2. Primitive Results

In this section, we provide some notions and basic results of a piecewise FC. Let

$$\mathcal{C}:=\mathcal{C}(\mathbb{J},\mathbb{R})=\left\{\varpi :\mathbb{J}\to \mathbb{R};∥\varpi ∥=\underset{\varkappa \in \mathbb{J}}{max}\left|\varpi \left(\varkappa \right)\right|\right\}$$

$\mathcal{C}$ is a Banach space under the norm $∥·∥$.

Definition 1

([41]). Let $\vartheta >0$ and $\varpi :\mathbb{J}\to \mathbb{R}$ be continuous. Then, the piecewise RL fractional integral is given by

$${}^{PRL}{\mathbb{I}}_{0^{+}}^{\vartheta }\varpi \left(\varkappa \right)=\left\{\begin{array}{c}\mathbb{I}\varpi \left(\varkappa \right),if\varkappa \in [0,{\varkappa }_1],\\ {}^{RL}{\mathbb{I}}_{{\varkappa }_1}^{\vartheta }\varpi \left(\varkappa \right)if\varkappa \in [{\varkappa }_1,b].\end{array}\right.$$

where $\mathbb{I}\varpi \left(\varkappa \right)={\int }_0^{{\varkappa }_1}\varpi \left(t\right)dt$ is the classical integral on $[0,{\varkappa }_1]$ and ${}^{RL}{\mathbb{I}}_{{\varkappa }_1}^{\vartheta }\varpi \left(\varkappa \right)=\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{(\varkappa -t))}^{\vartheta -1}\varpi \left(t\right)dt$ is the RL fractional integral on $[{\varkappa }_1,b].$

Definition 2

([41]). Let $0<\vartheta \le 1$ and $\varpi :\mathbb{J}\to \mathbb{R}$ be continuous. Then, the piecewise Caputo FD is given by

$${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta }\varpi \left(\varkappa \right)=\left\{\begin{array}{c}\mathbb{D}\varpi \left(\varkappa \right),if\varkappa \in [0,{\varkappa }_1],\\ {}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta }\varpi \left(\varkappa \right)if\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

where $\mathbb{D}\varpi \left(\varkappa \right)=\frac{d}{d\varkappa }\varpi \left(\varkappa \right)$ is the classical derivative on $[0,{\varkappa }_1]$ and ${}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta }\varpi \left(\varkappa \right)=\frac{1}{\mathrm{\Gamma }(1-\vartheta )}{\int }_{{\varkappa }_1}^{\varkappa }{(\varkappa -t))}^{-\vartheta }{\varpi }^{\prime }\left(t\right)dt$ is a Caputo FD on $[{\varkappa }_1,b].$

Lemma 1

([41]). For a given function $\varpi :\mathbb{J}\to \mathbb{R},$ and $0<\vartheta \le 1.$ Then, the following PC-FDE

$$\begin{array}{ccc}\hfill {}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta }\varpi \left(\varkappa \right)& =& f\left(\varkappa \right)\hfill \\ \hfill \varpi \left(0\right)& =& {\varkappa }_0\hfill \end{array}$$

has the following solution

$$\varpi \left(\varkappa \right)=\left\{\begin{array}{c}\varpi \left(0\right)+{\int }_0^{{\varkappa }_1}\varpi \left(\varkappa \right)d\varkappa ,if\varkappa \in [0,{\varkappa }_1],\\ \varpi \left({\varkappa }_1\right)+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }(\varkappa -t){)}^{\vartheta -1}\varpi \left(t\right)dtif\varkappa \in [{\varkappa }_1,b].\end{array}\right.$$

The following definition is the mainstay of our results, so we present the piecewise version of the $\psi$-fractional derivative and integral as follows:

Definition 3.

Let $0<\vartheta \le 1,$$\mathbb{I}=[a,b]$ be a finite or infinite interval, $\varpi :\mathbb{I}\to \mathbb{R}$ be an integrable function, and $\psi \in C^1\left(\mathbb{I}\right)$ be an increasing function such that $\psi \left(\varkappa \right)\ne 0$, for all $\varkappa \in \mathbb{I}$. Then, the piecewise version of ψ-Caputo FD is given by

$${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\varpi \left(\varkappa \right)=\left\{\begin{array}{c}{\mathbb{D}}^{1;\psi }\varpi \left(\varkappa \right),if\varkappa \in [0,{\varkappa }_1],\\ {}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta ;\psi }\varpi \left(\varkappa \right),if\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

(7)

where ${\mathbb{D}}^{1;\psi }\varpi \left(\varkappa \right)=\frac{{\varpi }^{\prime }\left(\varkappa \right)}{{\psi }^{\prime }\left(\varkappa \right)},$ and ${}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta ;\psi }\varpi \left(\varkappa \right)$ is a ψ-Caputo FD defined by Almeida [24]), that is

$${}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta ;\psi }\varpi \left(\varkappa \right)=\frac{1}{\mathrm{\Gamma }(1-\vartheta )}{\int }_{{\varkappa }_1}^{\varkappa }\frac{{\psi }^{\prime }\left(t\right)}{{(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta }}{\varpi }_{\psi }^{\prime }\left(t\right)dt.$$

The associated ψ-piecewise fractional integral by

$${}^{PRL}{\mathbb{I}}_{0^{+}}^{\vartheta ;\psi }\varpi \left(\varkappa \right)=\left\{\begin{array}{c}{\mathbb{I}}^{1;\psi }\varpi \left(t\right),if\varkappa \in [0,{\varkappa }_1],\\ {}^{RL}{\mathbb{I}}_{{\varkappa }_1}^{\vartheta ;\psi }\varpi \left(\varkappa \right),if\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

(8)

where ${\mathbb{I}}^{1;\psi }\varpi \left(t\right)={\int }_0^{{\varkappa }_1}{\psi }^{\prime }\left(t\right)\varpi \left(t\right)dt$ represents classical integral with respect to ψ on $[0,{\varkappa }_1]$ and ${}^{PRL}{\mathbb{I}}_{{\varkappa }_1}^{\vartheta ;\psi }\varpi \left(\varkappa \right)=\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }\frac{{\psi }^{\prime }\left(t\right)}{{(\psi \left(\varkappa \right)-\psi \left(t\right))}^{1-\vartheta }}\varpi \left(t\right)dt$ is a ψ-RL fractional integral (see [2]).

We recall the definitions of Ulam–Hyers (UH) stability and generalized Ulam–Hyers (GUH) stability.

Definition 4

([43]). ψ-PCPP (6) is UH stable if there exists a ${\chi }_{\phi }>0$ such that $\forall \epsilon >0$, and for each solution $\omega \in \mathcal{C}$ of the inequality

$$\left|{}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\omega \left(\varkappa \right)-\phi \left(\varkappa ,\omega \left(\varkappa \right),\omega \left({\lambda }_1\varkappa \right),\cdots ,\omega \left({\lambda }_m\varkappa \right)\right)\right|\le \epsilon ,\varkappa \in \mathbb{J},$$

(9)

there exists a solution $\upsilon \in \mathcal{C}$ of ψ-PCPP (6) satisfies

$$\left|\omega \left(\varkappa \right)-\upsilon \left(\varkappa \right)\right|\le {\chi }_{\phi }\epsilon .$$

(10)

Additionally, if there exists a nondecreasing function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\varphi \left(0\right)=0$ such that

$$\left|\omega \left(\varkappa \right)-\upsilon \left(\varkappa \right)\right|\le {\chi }_{\phi }\varphi \left(\epsilon \right),\varkappa \in \mathbb{J},$$

then the concerned solution is GUH stable.

For our forthcoming analysis, we need Banach’s contraction map [44] and Krasnoselskii’s fixed-point theorem [45].

3. Main Results

In this section, we provide some qualitative analyses of the $\psi$-piecewise Caputo pantograph FDE (6). First, certain crucial results are provided for the benefit of the upcoming analysis.

Remark 2.

 

(i) If $\psi \left(\varkappa \right)=\varkappa ,$ then (7) reduces to a piecewise Caputo FD [41].

(ii) If $\psi \left(\varkappa \right)=\varkappa ,$ then (8) reduces to a piecewise RL fractional integral [41].

Lemma 2.

Let $\vartheta \in (0,1]$ and for a given function $\varpi \in \mathcal{C}$. Then,

$${}^{PRL}{\mathbb{I}}_{0^{+}}^{\vartheta ;\psi }{}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\varpi \left(\varkappa \right)=\left\{\begin{array}{c}{\mathbb{I}}^{1;\psi }{\mathbb{D}}^{1;\psi }\varpi \left(\varkappa \right)=\varpi \left({\varkappa }_1\right)-\varpi \left(0\right),if\varkappa \in [0,{\varkappa }_1],\\ {}^{RL}{\mathbb{I}}_{{\varkappa }_1}^{\vartheta ;\psi }{}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta ;\psi }\varpi \left(\varkappa \right)=\varpi \left(\varkappa \right)-\varpi \left({\varkappa }_1\right),if\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

(11)

and

$${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }{}^{PRL}{\mathbb{I}}_{0^{+}}^{\vartheta ;\psi }\varpi \left(\varkappa \right)=\left\{\begin{array}{c}{\mathbb{D}}^{1;\psi }{\mathbb{I}}^{1;\psi }\varpi \left(\varkappa \right)=\varpi \left(\varkappa \right),if\varkappa \in [0,{\varkappa }_1],\\ {{}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta }}^{RL}{\mathbb{I}}_{{\varkappa }_1}^{\vartheta }\varpi \left(\varkappa \right)=\varpi \left(\varkappa \right),if\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

(12)

Proof. 

For $\varkappa \in [{\varkappa }_1,b],$ the proof can be accomplished following similar kinds of steps as in the proof for Theorem 4 presented by Almeida [24].

For $\varkappa \in [0,{\varkappa }_1]$ in (11), we have from (8) and (7) that

$${\mathbb{I}}^{1;\psi }{\mathbb{D}}^{1;\psi }\varpi \left(\varkappa \right)={\int }_0^{{\varkappa }_1}{\psi }^{\prime }\left(t\right){\mathbb{D}}^{1;\psi }\varpi \left(t\right)dt={\int }_0^{{\varkappa }_1}{\psi }^{\prime }\left(t\right)\frac{{\varpi }^{\prime }\left(t\right)}{{\psi }^{\prime }\left(t\right)}dt={\int }_0^{{\varkappa }_1}{\varpi }^{\prime }\left(t\right)dt=\varpi \left({\varkappa }_1\right)-\varpi \left(0\right).$$

For $\varkappa \in [0,{\varkappa }_1]$ in (12), using the Lemma 2.4 [2], for $n-1<\vartheta \le n\in \mathbb{N}$ and $\varpi \in \mathcal{C}$, then ${\mathbb{D}}^{\vartheta ;\psi }$${\mathbb{I}}^{\vartheta ;\psi }\varpi \left(\varkappa \right)=\varpi \left(\varkappa \right).$

As a special case for $\vartheta =1$, then ${\mathbb{D}}^{1;\psi }{\mathbb{I}}^{1;\psi }\varpi \left(\varkappa \right)=\varpi \left(\varkappa \right).$ □

Lemma 3.

Let $0<\vartheta \le 1,0<{\lambda }_1<\cdots <{\lambda }_m<1,$ and $\phi :\mathbb{J}\times \mathbb{R}\times {\mathbb{R}}^m\to \mathbb{R},$$g:\mathcal{C}\to \mathbb{R}$ be continuous. Then, the ψ-PCPP (6) is equivalent to

$$\upsilon \left(\varkappa \right)=\left\{\begin{array}{l}{\upsilon }_0+g\left(\upsilon \right)+{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt,if\varkappa \in [0,{\varkappa }_1],\\ {\upsilon }_{{\varkappa }_1}+g\left(\upsilon \right)+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\\ \times \phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt,if\varkappa \in [{\varkappa }_1,b].\end{array}\right.$$

(13)

Proof. 

Let us assume we have the $\psi$-PCPP (6) and show that $\upsilon \in \mathcal{C}$ satisfies (13). By Definition 3, we have

$${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\upsilon \left(\varkappa \right)=\left\{\begin{array}{c}\frac{1}{{\psi }^{\prime }\left(\varkappa \right)}{\upsilon }^{\prime }\left(\varkappa \right):if\varkappa \in [0,{\varkappa }_1],\\ {}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta ;\psi }\upsilon \left(\varkappa \right):if\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

In view of Lemma 2, we have

$${{}^{PRL}{\mathbb{I}}_{0^{+}}^{\vartheta ;\psi }}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\upsilon \left(\varkappa \right)=\left\{\begin{array}{c}{\mathbb{I}}^{1;\psi }{\mathbb{D}}^{1;\psi }\upsilon \left(\varkappa \right)=\upsilon \left({\varkappa }_1\right)-\upsilon \left(0\right),if\varkappa \in [0,{\varkappa }_1],\\ {{}^{RL}{\mathbb{I}}_{{\varkappa }_1}^{\vartheta ;\psi }}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta ;\psi }\upsilon \left(\varkappa \right)=\upsilon \left(\varkappa \right)-\upsilon \left({\varkappa }_1\right),if\varkappa \in [{\varkappa }_1,b].\end{array}\right.$$

(14)

Applying ${}^{PRL}{\mathbb{I}}_{0^{+}}^{\vartheta ;\psi }$ on (6), we have

$${{}^{PRL}{\mathbb{I}}_{0^{+}}^{\vartheta ;\psi }}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\upsilon \left(\varkappa \right)=\left\{\begin{array}{c}{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt,if\varkappa \in [0,{\varkappa }_1],\\ {}^{PRL}{\mathbb{I}}_{{\varkappa }_1}^{\vartheta ;\psi }\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right),if\varkappa \in [{\varkappa }_1,b].\end{array}\right.$$

(15)

Comparing (14) and (15), we obtain

Case 1: For $\varkappa \in [0,{\varkappa }_1],$

$$\upsilon \left(\varkappa \right)=\upsilon \left(0\right)+{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt.$$

Case 2: For $\varkappa \in [{\varkappa }_1,b],$

$$\upsilon \left(\varkappa \right)=\upsilon \left({\varkappa }_1\right)+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt.$$

Using the condition in both cases, we obtain

$$\begin{array}{c}\upsilon \left(\varkappa \right)=\left\{\begin{array}{c}{\upsilon }_0+g\left(\upsilon \right)+{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt,if\varkappa \in [0,{\varkappa }_1],\\ {\upsilon }_{{\varkappa }_1}+g\left(\upsilon \right)+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\\ \times \phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt,if\varkappa \in [{\varkappa }_1,b],\end{array}\right.\end{array}$$

which is (13).

Let $\upsilon \in \mathcal{C}$ satisfy (13), and we prove that (6) holds. Applying ${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }$ on (13), we have

$${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\upsilon \left(\varkappa \right)=\left\{\begin{array}{c}\frac{1}{{\psi }^{\prime }\left(\varkappa \right)}\frac{d}{d\varkappa }\left({\upsilon }_0+g\left(\upsilon \right)+{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt\right)if\varkappa \in [0,{\varkappa }_1],\\ {}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta ;\psi }\left({\upsilon }_{{\varkappa }_1}+g\left(\upsilon \right)+{}^{RL}{\mathbb{I}}_{{\varkappa }_1}^{\vartheta ;\psi }\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)\right),if\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

(16)

From the fact that classical derivative and Caputo derivative of any constant function are zero, Lemma 2 shows that

$$\begin{array}{ccc}& & \frac{1}{{\psi }^{\prime }\left(\varkappa \right)}\frac{d}{d\varkappa }{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt\hfill \\ & =& \phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\cdots ,\upsilon \left({\lambda }_m\varkappa \right)\right),\mathrm{on}0\le \varkappa \le {\varkappa }_1,\hfill \end{array}$$

and

$$\begin{array}{ccc}& & {}^C{\mathbb{D}}_{{\varkappa }_1}^{\vartheta ;\psi }{}^{RL}{\mathbb{I}}_{{\varkappa }_1}^{\vartheta ;\psi }\phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left(\lambda \varkappa \right)\right)\hfill \\ & =& \phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\cdots ,\upsilon \left({\lambda }_m\varkappa \right)\right),\mathrm{on}{\varkappa }_1\le \varkappa \le b.\hfill \end{array}$$

Hence

$${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\upsilon \left(\varkappa \right)=\phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\cdots ,\upsilon \left({\lambda }_m\varkappa \right)\right),\mathrm{for}\mathrm{each}\varkappa \in \mathbb{J}.$$

Moreover, $\upsilon \left(0\right)={\upsilon }_0+g\left(\upsilon \right)$ on $[0,{\varkappa }_1],$ and $\upsilon \left({\varkappa }_1\right)={\upsilon }_{{\varkappa }_1}+g\left(\upsilon \right),$ on $[{\varkappa }_1,b].$ □

Remark 3.

Let $m=1$ in Lemma 3. Then, we have the following ψ-piecewise Caputo pantograph problem (ψ-PCPP):

$$\left\{\begin{array}{c}{}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\upsilon \left(\varkappa \right)=\phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left(\lambda \varkappa \right)\right),\varkappa \in \mathbb{J},\\ \upsilon \left(0\right)={\upsilon }_0+g\left(\upsilon \right).\end{array}\right.$$

(17)

In particular, if we replace ${}^C{\mathbb{D}}_{0^{+}}^{\vartheta }$ instead of ${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }$ with $\psi \left(\varkappa \right)=\varkappa$, then (17) reduces to (4), which was considered by Balachandran et al. [40].

Corollary 1.

Let $0<\vartheta \le 1,0<\lambda <1,$ and $\phi :\mathbb{J}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R},$ $g:\mathcal{C}\to \mathbb{R}$ be continuous. Then, the ψ-PCPP (17) is equivalent to

$$\begin{array}{c}\upsilon \left(\varkappa \right)=\left\{\begin{array}{c}{\upsilon }_0+g\left(\upsilon \right)+{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\phi \left(t,\upsilon \left(t\right),\upsilon \left(\lambda t\right))\right)dt,if\varkappa \in [0,{\varkappa }_1],\\ {\upsilon }_{{\varkappa }_1}+g\left(\upsilon \right)+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\\ \times \phi \left(t,\upsilon \left(t\right),\upsilon \left(\lambda t\right)\right)dt,if\varkappa \in [{\varkappa }_1,b].\end{array}\right.\end{array}$$

(18)

As per Lemma 3, we define an operator $\mathcal{K}:\mathcal{C}\to \mathcal{C}$ by

$$\begin{array}{c}\left(\mathcal{K}\upsilon \right)\left(\varkappa \right)=\left\{\begin{array}{c}\left({\mathcal{K}}_1\upsilon \right)\left(\varkappa \right),if\varkappa \in [0,{\varkappa }_1],\\ \left({\mathcal{K}}_2\upsilon \right)\left(\varkappa \right)if\varkappa \in [{\varkappa }_1,b],\end{array}\right.\end{array}$$

where

$$\left({\mathcal{K}}_1\upsilon \right)\left(\varkappa \right)={\upsilon }_0+g\left(\upsilon \right)+{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt,if\varkappa \in [0,{\varkappa }_1],$$

$$\begin{array}{c}\left({\mathcal{K}}_2\upsilon \right)\left(\varkappa \right)={\upsilon }_{{\varkappa }_1}+g\left(\upsilon \right)+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\\ \times \phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt,if\varkappa \in [{\varkappa }_1,b].\end{array}$$

Note that problem (6) has solutions if and only if the operator $\mathcal{K}$ has fixed points, i.e., $\left(\mathcal{K}\upsilon \right)\left(\varkappa \right)=\upsilon \left(\varkappa \right).$

The uniqueness result is based on the Banach contraction map [44].

Theorem 1.

Assume that:

(H1). 

There exists $L_0,L_i>0$ $(i=1,2,\cdots ,m)$ such that

$$\left|\phi (\varkappa ,\upsilon ,{\upsilon }_{{\lambda }_1},\cdots ,{\upsilon }_{{\lambda }_m})-\phi (\varkappa ,\omega ,{\omega }_{{\lambda }_1},\cdots ,{\omega }_{{\lambda }_m})\right|\le L_0\left|\upsilon -\omega \right|+\sum _{i=1}^m{L}_i\left|{\upsilon }_{{\lambda }_i}-{\omega }_{{\lambda }_i}\right|,$$

for each $\varkappa \in \mathbb{J},$$\upsilon ,{\upsilon }_{{\lambda }_i},{\omega }_{{\lambda }_i}\in \mathbb{R};$

(H2). 

There exists $L_g>0$ such that $0<L_g<1$ and $\left|g\left(\upsilon \right)-g\left(\omega \right)\right|\le L_g\left|\upsilon -\omega \right|,$ for $\upsilon ,\omega \in \mathcal{C}.$ If

$$\begin{array}{ccc}\hfill {\aleph }_1& =& L_g+\zeta {\varkappa }_1\left(L_0+\sum _{i=1}^m{L}_i\right)<1,\hfill \\ \hfill {\aleph }_2& =& L_g+\frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\left(L_0+\sum _{i=1}^m{L}_i\right)<1,\hfill \end{array}$$

(19)

then the ψ-PCPP (6) has a unique solution on $\mathbb{J}$.

Proof. 

Let ${sup}_{\varkappa \in \mathbb{J}}\left|\phi (\varkappa ,0,0,\cdots ,0)\right|=M_{\phi }<\infty$, and ${sup}_{\varkappa \in \mathbb{J}}\left|g\left(0\right)\right|=M_g<\infty .$ Becaue $\psi \in {\mathcal{C}}^1,$ there exists a $\zeta$ such that ${sup}_{t\in \mathbb{J}}\left|{\psi }^{\prime }\left(t\right)\right|\le \zeta .$ Choose $r>0$ such that

$$r\ge \left\{\begin{array}{c}\frac{\left|{\upsilon }_0\right|+M_g+M_{\phi }\zeta {\varkappa }_1}{1-{\aleph }_1},\mathrm{on}[0,{\varkappa }_1],\\ \frac{\left|{\upsilon }_{{\varkappa }_1}\right|+M_g+M_{\phi }\frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}}{1-{\aleph }_2},\mathrm{on}[{\varkappa }_1,b].\end{array}\right.$$

(20)

Now, we show that ${\mathcal{KB}}_r\subset {\mathcal{B}}_r$, where ${\mathcal{B}}_r=\left\{\upsilon \in \mathcal{C}:∥\upsilon ∥\le r\right\}.$ For any $\upsilon \in {\mathcal{B}}_r,$ and $\varkappa \in [0,{\varkappa }_1],$ we have

$$\begin{array}{ccc}\hfill \left|\left({\mathcal{K}}_1\upsilon \right)\left(\varkappa \right)\right|& \le & \underset{\varkappa \in [0,{\varkappa }_1]}{sup}\left\{\left|{\upsilon }_0\right|+\left|g\left(\upsilon \right)\right|+{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\left|\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)\right|dt\right\}\hfill \\ & \le & \left|{\upsilon }_0\right|+\left|g\left(\upsilon \right)\right|+{\int }_0^{{\varkappa }_1}{\psi }^{\prime }\left(t\right)\left|\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)\right|dt\hfill \\ & \le & \left|{\upsilon }_0\right|+\left|g\left(\upsilon \right)-g\left(0\right)\right|+M_g+{\int }_0^{{\varkappa }_1}{\psi }^{\prime }\left(t\right)M_{\phi }dt+{\int }_0^{{\varkappa }_1}{\psi }^{\prime }\left(t\right)\hfill \\ & & \times \left|\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)-\phi \left(t,0,0,\cdots ,0\right)\right|dt\hfill \\ & \le & \left|{\upsilon }_0\right|+L_g\left|\upsilon \right|+M_g+M_{\phi }\zeta {\varkappa }_1+{\int }_0^{{\varkappa }_1}{\psi }^{\prime }\left(t\right)\left(L_0\left|\upsilon \right|+\sum _{i=1}^m{L}_i\left|\upsilon \left({\lambda }_{i}t\right)\right|\right)dt\hfill \\ & \le & \left|{\upsilon }_0\right|+L_g∥\upsilon ∥+M_g+M_{\phi }\zeta {\varkappa }_1+\zeta {\varkappa }_1\left(L_0∥\upsilon ∥+\sum _{i=1}^m{L}_i∥\upsilon ∥\right)\hfill \\ & \le & \left|{\upsilon }_0\right|+M_g+M_{\phi }\zeta {\varkappa }_1+\left(L_g+\zeta {\varkappa }_1(L_0+\sum _{i=1}^m{L}_i)\right)r\hfill \\ & \le & \left(1-{\aleph }_1\right)r+{\aleph }_{1}r=r.\hfill \end{array}$$

(21)

For any $\upsilon \in {\mathcal{B}}_r,$ and $\varkappa \in [{\varkappa }_1,b],$ we have

$$\begin{array}{ccc}\hfill \left|\left({\mathcal{K}}_2\upsilon \right)\left(\varkappa \right)\right|& \le & \underset{\varkappa \in [{\varkappa }_1,b]}{sup}\left\{\left|{\upsilon }_{{\varkappa }_1}\right|+\left|g\left(\upsilon \right)\right|+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\right.\hfill \\ & & \left.\times \left|\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)\right|dt\right\}\hfill \\ & \le & \left|{\upsilon }_{{\varkappa }_1}\right|+\left|g\left(\upsilon \right)\right|+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^b{\psi }^{\prime }\left(t\right){(\psi \left(b\right)-\psi \left(t\right))}^{\vartheta -1}\hfill \\ & & \times \left|\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)\right|dt\hfill \\ & \le & \left|{\upsilon }_{{\varkappa }_1}\right|+\left|g\left(\upsilon \right)-g\left(0\right)\right|+M_g+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^b{\psi }^{\prime }\left(t\right){(\psi \left(b\right)-\psi \left(t\right))}^{\vartheta -1}{M}_{\phi }dt\hfill \\ & & +{\int }_{{\varkappa }_1}^b{\psi }^{\prime }\left(t\right){(\psi \left(b\right)-\psi \left(t\right))}^{\vartheta -1}\left|\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)-\phi \left(t,0,0,\cdots ,0\right)\right|dt\hfill \\ & \le & \left|{\upsilon }_{{\varkappa }_1}\right|+L_g∥\upsilon ∥+M_g+M_{\phi }\frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\hfill \\ & & +\frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\left(L_0∥\upsilon ∥+\sum _{i=1}^m{L}_i∥\upsilon ∥\right)\hfill \\ & \le & \left|{\upsilon }_{{\varkappa }_1}\right|+M_g+M_{\phi }\frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}+\left(L_g+\frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}(L_0+\sum _{i=1}^m{L}_i)\right)r\hfill \\ & \le & \left(1-{\aleph }_1\right)r+{\aleph }_{2}r=r.\hfill \end{array}$$

(22)

Equations (21) and (22) show that ${\mathcal{KB}}_r\subset {\mathcal{B}}_r.$

Next, let $\upsilon$,$\overline{\upsilon }$$\in \mathcal{C}$. Then

Case 1: For $\varkappa \in [0,{\varkappa }_1],$

$$\begin{array}{ccc}& & \left|\left({\mathcal{K}}_1\upsilon \right)\left(\varkappa \right)-\left({\mathcal{K}}_1\overline{\upsilon }\right)\left(\varkappa \right)\right|\hfill \\ & & \le \left|g\left(\upsilon \right)-g\left(\overline{\upsilon }\right)\right|+{\int }_0^{{\varkappa }_1}{\psi }^{\prime }\left(t\right)\hfill \\ & & \times \left|\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)-\phi \left(t,\overline{\upsilon }\left(t\right),\overline{\upsilon }\left({\lambda }_{1}t\right),\cdots ,\overline{\upsilon }\left({\lambda }_{m}t\right)\right)\right|dt\hfill \\ & & \le L_g\left|\upsilon -\overline{\upsilon }\right|+{\int }_0^{{\varkappa }_1}{\psi }^{\prime }\left(t\right)\left(L_0\left|\upsilon \left(t\right)-\overline{\upsilon }\left(t\right)\right|+\sum _{i=1}^m{L}_i\left|\upsilon \left({\lambda }_{i}t\right)-\overline{\upsilon }\left({\lambda }_{i}t\right)\right|\right)dt\hfill \\ & & \le \left[L_g+\zeta {\varkappa }_1\left(L_0+\sum _{i=1}^m{L}_i\right)\right]{∥\upsilon -\overline{\upsilon }∥}_{\mathcal{C}}\hfill \\ & & ={\aleph }_1{∥\upsilon -\overline{\upsilon }∥}_{\mathcal{C}}.\hfill \end{array}$$

(23)

Case 2: For $\varkappa \in [{\varkappa }_1,b],$

$$\begin{array}{ccc}& & \left|\left({\mathcal{K}}_2\upsilon \right)\left(\varkappa \right)-\left({\mathcal{K}}_2\overline{\upsilon }\right)\left(\varkappa \right)\right|\hfill \\ & & \le \left|g\left(\upsilon \right)-g\left(\overline{\upsilon }\right)\right|+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\hfill \\ & & \times \left|\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)-\phi \left(t,\overline{\upsilon }\left(t\right),\overline{\upsilon }\left({\lambda }_{1}t\right),\cdots ,\overline{\upsilon }\left({\lambda }_{m}t\right)\right)\right|dt\hfill \\ & & \le L_g\left|\upsilon -\overline{\upsilon }\right|+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\hfill \\ & & \times \left(L_0\left|\upsilon \left(t\right)-\overline{\upsilon }\left(t\right)\right|+\sum _{i=1}^m{L}_i\left|\upsilon \left({\lambda }_{i}t\right)-\overline{\upsilon }\left({\lambda }_{i}t\right)\right|\right)dt\hfill \\ & & \le L_g{∥\upsilon -\overline{\upsilon }∥}_{\mathcal{C}}+\frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\left(L_0+\sum _{i=1}^m{L}_i\right){∥\upsilon -\overline{\upsilon }∥}_{\mathcal{C}}\hfill \\ & & \le \left(L_g+\frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\left(L_0+\sum _{i=1}^m{L}_i\right)\right){∥\upsilon -\overline{\upsilon }∥}_{\mathcal{C}}\hfill \\ & & ={\aleph }_2{∥\upsilon -\overline{\upsilon }∥}_{\mathcal{C}}.\hfill \end{array}$$

(24)

It follows from (23) and (24) that

$${∥\mathcal{K}\upsilon -\mathcal{K}\overline{\upsilon }∥}_{\mathcal{C}}\le \left\{\begin{array}{c}{\aleph }_1{∥\upsilon -\overline{\upsilon }∥}_{\mathcal{C}},on\in [0,{\varkappa }_1],\\ {\aleph }_2{∥\upsilon -\overline{\upsilon }∥}_{\mathcal{C}}on\in [{\varkappa }_1,b].\end{array}\right.$$

Because ${\aleph }_1,{\aleph }_2<1$, $\mathcal{K}$ is a contraction map. Therefore, we deduce from the Banach contraction map that $\psi$-PCPP (6) has a unique solution existing on $\mathbb{J}$. □

Next, we provide existence results based on Krasnoselskii’s fixed-point theorem [45].

Theorem 2.

Let $\phi :\mathbb{J}\times \mathbb{R}\times {\mathbb{R}}^m\to \mathbb{R},$ and $g:\mathcal{C}\to \mathbb{R}$ be continuous, satisfying (H1) and (H2). In addition, we assume that:

(H3) 

$\left|\phi \left(t,\upsilon ,{\upsilon }_{{\lambda }_1},\cdots ,{\upsilon }_{{\lambda }_m}\right)\right|\le {\mu }_{\phi }\left(\varkappa \right)$ for each $\varkappa \in \mathbb{J},$$\upsilon ,{\upsilon }_{{\lambda }_i}\in \mathbb{R},$ and ${\mu }_{\phi }\in \mathcal{C}.$

(H4) 

$\left|g\left(\upsilon \right)\right|\le {\mu }_g\left|\upsilon \right|,$ for $\upsilon \in \mathcal{C},$ ${\mu }_g>0.$

  

If $0\ne {\mu }_g<1,$ then ψ-PCPP (6) has a least one solution on $\mathbb{J}.$

Proof. 

Choose

$$r\ge \left\{\begin{array}{c}\frac{1}{1-{\mu }_g}\left(\left|{\upsilon }_0\right|+{\mu }_{\phi }^{*}\zeta {\varkappa }_1\right),\mathrm{on}\varkappa \in [0,{\varkappa }_1],\\ \frac{1}{1-{\mu }_g}\left(\left|{\upsilon }_{{\varkappa }_1}\right|+\frac{{\mu }_{\phi }^{*}{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\right),\mathrm{on}\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

where ${\mu }_{\phi }^{*}={sup}_{\varkappa \in \mathbb{J}}\left|{\mu }_{\phi }\left(\varkappa \right)\right|,$ and ${sup}_{t\in \mathbb{J}}\left|{\psi }^{\prime }\left(t\right)\right|\le \zeta .$

Consider the operators $\mathcal{P},\mathcal{O}:\mathcal{C}(\mathbb{J},{\mathcal{B}}_r)\to \mathcal{C}(\mathbb{J},{\mathcal{B}}_r)$ defined by

$$\left(\mathcal{P}\upsilon \right)\left(\varkappa \right)=\left\{\begin{array}{c}{\upsilon }_0+g\left(\upsilon \right),\mathrm{if}\varkappa \in [0,{\varkappa }_1],\\ {\upsilon }_{{\varkappa }_1}+g\left(\upsilon \right),\mathrm{if}\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

and

$$\begin{array}{c}\left(\mathcal{O}\upsilon \right)\left(\varkappa \right)=\left\{\begin{array}{c}{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt,if\varkappa \in [0,{\varkappa }_1],\\ \frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt,if\varkappa \in [{\varkappa }_1,b],\end{array}\right..\end{array}$$

where $\left(\mathcal{P}\upsilon +\mathcal{O}\upsilon \right)\left(\varkappa \right)=\left(\mathcal{K}\upsilon \right)\left(\varkappa \right).$

For any $\upsilon ,\omega \in {\mathcal{B}}_r$, we have

Case 1: For $\varkappa \in [0,{\varkappa }_1],$

$$\begin{array}{ccc}\hfill \left|\left(\mathcal{P}\upsilon +\mathcal{O}\upsilon \right)\left(\varkappa \right)\right|& \le & \left|\left(\mathcal{P}\upsilon \right)\left(\varkappa \right)\right|+\left|\left(\mathcal{O}\upsilon \right)\left(\varkappa \right)\right|\hfill \\ & \le & \left|{\upsilon }_0\right|+\left|g\left(\upsilon \right)\right|+{\int }_0^{\varkappa }\left|{\psi }^{\prime }\left(t\right)\right|\left|\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)\right|dt\hfill \\ & \le & \left|{\upsilon }_0\right|+{\mu }_g\left|\upsilon \right|+{\int }_0^{\varkappa }\left|{\psi }^{\prime }\left(t\right)\right|\left|{\mu }_{\phi }\left(t\right)\right|dt\hfill \\ & \le & \left|{\upsilon }_0\right|+{\mu }_g{∥\upsilon ∥}_{\mathcal{C}}+{\mu }_{\phi }^{*}\zeta {\varkappa }_1\hfill \\ & \le & \left|{\upsilon }_0\right|+{\mu }_{\phi }^{*}\zeta {\varkappa }_1+{\mu }_{g}r\hfill \\ & \le & r.\hfill \end{array}$$

Case 2: For $\varkappa \in [{\varkappa }_1,b],$

$$\begin{array}{ccc}\hfill \left|\left(\mathcal{P}\upsilon +\mathcal{O}\upsilon \right)\left(\varkappa \right)\right|& \le & \left|\left(\mathcal{P}\upsilon \right)\left(\varkappa \right)\right|+\left|\left(\mathcal{O}\upsilon \right)\left(\varkappa \right)\right|\hfill \\ & \le & \left|{\upsilon }_{{\varkappa }_1}\right|+\left|g\left(\upsilon \right)\right|\hfill \\ & & +\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\left|\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)\right|dt\hfill \\ & \le & \left|{\upsilon }_{{\varkappa }_1}\right|+{\mu }_g\left|\upsilon \right|+{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\left|{\mu }_{\phi }\left(t\right)\right|dt\hfill \\ & \le & \left|{\upsilon }_{{\varkappa }_1}\right|+{\mu }_g{∥\upsilon ∥}_{\mathcal{C}}+{\mu }_{\phi }^{*}\frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\hfill \\ & \le & \left|{\upsilon }_{{\varkappa }_1}\right|+{\mu }_{g}r+{\mu }_{\phi }^{*}\frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\hfill \\ & \le & r.\hfill \end{array}$$

Hence, we deduce that ${∥\mathcal{P}\upsilon +\mathcal{O}\upsilon ∥}_{\mathcal{C}}\le r.$

Next, for any $\varkappa \in \mathbb{J}$ and $\upsilon ,\omega \in {\mathcal{B}}_r$, we have

$$\begin{array}{ccc}\hfill \left|\mathcal{P}\upsilon \left(\varkappa \right)-\mathcal{P}\omega \left(\varkappa \right)\right|& \le & \left\{\begin{array}{c}\left|g\left(\upsilon \right)-g\left(\omega \right)\right|,\mathrm{if}\varkappa \in [0,{\varkappa }_1],\\ \left|g\left(\upsilon \right)-g\left(\omega \right)\right|,\mathrm{if}\varkappa \in [{\varkappa }_1,b],\end{array}\right.\hfill \\ & \le & \left\{\begin{array}{c}{L}_g\left|\upsilon -\omega \right|,\mathrm{if}\varkappa \in [0,{\varkappa }_1],\\ L_g\left|\upsilon -\omega \right|,\mathrm{if}\varkappa \in [{\varkappa }_1,b].\end{array}\right.\hfill \end{array}$$

Thus, ${∥\mathcal{P}\upsilon -\mathcal{P}\omega ∥}_{\mathcal{C}}\le L_g{∥\upsilon -\omega ∥}_{\mathcal{C}}.$ As $L_g<1,$$\mathcal{P}$ is a contraction map.

Finally, we show that $\mathcal{O}$ is continuous and compact.

Initially, we show that $\mathcal{O}$ is continuous.

Let ${\left\{{\upsilon }_n\right\}}_{n\ge 1}$ in ${\mathcal{B}}_r$ such that ${\upsilon }_n\to \upsilon$ in ${\mathcal{B}}_r.$ Then, ${\mathrm{\Phi }}_{{\upsilon }_n}\left(\varkappa \right)\to {\mathrm{\Phi }}_{\upsilon }\left(\varkappa \right)$$n\to \infty ,$ where

$${\mathrm{\Phi }}_{{\upsilon }_n}\left(\varkappa \right):=\phi \left(\varkappa ,{\upsilon }_n\left(\varkappa \right),{\upsilon }_n\left({\lambda }_1\varkappa \right),\cdots ,{\upsilon }_n\left({\lambda }_m\varkappa \right)\right)\mathrm{and}{\mathrm{\Phi }}_{\upsilon }\left(\varkappa \right):=\phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\cdots ,\upsilon \left({\lambda }_m\varkappa \right)\right).$$

For each $\varkappa ,$ we have

$$\begin{array}{ccc}\hfill \left|\mathcal{O}{\upsilon }_n\left(\varkappa \right)-\mathcal{O}\upsilon \left(\varkappa \right)\right|& =& \left\{\begin{array}{c}{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\left|{\mathrm{\Phi }}_{{\upsilon }_n}\left(t\right)-{\mathrm{\Phi }}_{\upsilon }\left(t\right)\right|dt,if\varkappa \in [0,{\varkappa }_1],\\ \frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\left|{\mathrm{\Phi }}_{{\upsilon }_n}\left(t\right)-{\mathrm{\Phi }}_{\upsilon }\left(t\right)\right|dt,if\varkappa \in [0,{\varkappa }_1],\end{array}\right.\hfill \\ & \le & \left\{\begin{array}{c}{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)∥{\mathrm{\Phi }}_{{\upsilon }_n}(·)-{\mathrm{\Phi }}_{\upsilon }(·)∥dt,if\varkappa \in [0,{\varkappa }_1],\\ \frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}∥{\mathrm{\Phi }}_{{\upsilon }_n}(·)-{\mathrm{\Phi }}_{\upsilon }(·)∥dt,if\varkappa \in [0,{\varkappa }_1].\end{array}\right.\hfill \end{array}$$

The continuity of $\phi$ and $\psi$ implies that $\left|\mathcal{O}{\upsilon }_n\left(\varkappa \right)-\mathcal{O}\upsilon \left(\varkappa \right)\right|\to 0,$ as $n\to \infty .$

Next, $\mathcal{O}$ is uniformly bounded on ${\mathcal{B}}_r$ as

$$\begin{array}{c}\left|\left(\mathcal{O}\upsilon \right)\left(\varkappa \right)\right|\le \left\{\begin{array}{c}{\mu }_{\phi }^{*}\zeta {\varkappa }_1,if\varkappa \in [0,{\varkappa }_1],\\ {\mu }_{\phi }^{*}\frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)},if\varkappa \in [{\varkappa }_1,b].\end{array}\right.\end{array}$$

At last, we show the compactness of $\mathcal{O}$. Let $\varkappa \in [0,{\varkappa }_1]$ with ${\varkappa }_{ϵ}<$${\varkappa }_{\delta }$$\in [0,{\varkappa }_1]$. Then,

$$\begin{array}{ccc}\hfill \left|\left(\mathcal{O}\upsilon \right)\left({\varkappa }_{ϵ}\right)-\left(\mathcal{O}\upsilon \right)\left({\varkappa }_{\delta }\right)\right|& =& \left|{\int }_{{\varkappa }_{ϵ}}^{{\varkappa }_{\delta }}{\psi }^{\prime }\left(t\right)\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)dt\right|\hfill \\ & \le & {\int }_{{\varkappa }_{ϵ}}^{{\varkappa }_{\delta }}\left|{\psi }^{\prime }\left(t\right)\right|\left|\phi \left(t,\upsilon \left(t\right),\upsilon \left({\lambda }_{1}t\right),\cdots ,\upsilon \left({\lambda }_{m}t\right)\right)\right|dt\hfill \\ & \le & \left({\varkappa }_{\delta }-{\varkappa }_{ϵ}\right)\zeta {\mu }_{\phi }^{*}.\hfill \end{array}$$

(25)

Let $\varkappa \in [{\varkappa }_1,b]$ with ${\varkappa }_{ϵ}<$ ${\varkappa }_{\delta }\in [{\varkappa }_1,b]$. Then,

$$\begin{array}{ccc}& & \left|\left(\mathcal{O}\upsilon \right)\left({\varkappa }_{ϵ}\right)-\left(\mathcal{O}\upsilon \right)\left({\varkappa }_{\delta }\right)\right|\hfill \\ & & \le \left|\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{{\varkappa }_{ϵ}}{\psi }^{\prime }\left(t\right){(\psi \left({\varkappa }_{ϵ}\right)-\psi \left(t\right))}^{\vartheta -1}{\mathrm{\Phi }}_{\upsilon }\left(t\right)dt\right.\hfill \\ & & \left.-\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{{\varkappa }_{\delta }}{\psi }^{\prime }\left(t\right){(\psi \left({\varkappa }_{\delta }\right)-\psi \left(t\right))}^{\vartheta -1}{\mathrm{\Phi }}_{\upsilon }\left(t\right)dt\right|\hfill \\ & & \le \frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{{\varkappa }_{ϵ}}{\psi }^{\prime }\left(t\right)\left[{(\psi \left({\varkappa }_{ϵ}\right)-\psi \left(t\right))}^{\vartheta -1}-{(\psi \left({\varkappa }_{\delta }\right)-\psi \left(t\right))}^{\vartheta -1}\right]\left|{\mathrm{\Phi }}_{\upsilon }\left(t\right)\right|dt\hfill \\ & & +\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_{ϵ}}^{{\varkappa }_{\delta }}{\psi }^{\prime }\left(t\right){(\psi \left({\varkappa }_{\delta }\right)-\psi \left(t\right))}^{\vartheta -1}\left|{\mathrm{\Phi }}_{\upsilon }\left(t\right)\right|dt\hfill \\ & & \le \frac{{\mu }_{\phi }^{☆}}{\mathrm{\Gamma }(\vartheta +1)}\left[{(\psi \left({\varkappa }_{\delta }\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }-{(\psi \left({\varkappa }_{ϵ}\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }+2{(\psi \left({\varkappa }_{\delta }\right)-\psi \left({\varkappa }_{ϵ}\right))}^{\vartheta }\right].\hfill \end{array}$$

(26)

From (25) and (26), we obtain

$$\left|\left(\mathcal{O}\upsilon \right)\left({\varkappa }_{ϵ}\right)-\left(\mathcal{O}\upsilon \right)\left({\varkappa }_{\delta }\right)\right|\to 0,\mathrm{as}{\varkappa }_{\delta }-{\varkappa }_{ϵ}\to 0.$$

Thus, $\mathcal{O}$ is equicontinuous. As per the previous steps, $\mathcal{O}$ is relatively compact on ${\mathcal{B}}_r$. Consequently, the Arzela–Ascoli lemma shows that $\mathcal{O}$ is compact on ${\mathcal{B}}_r$. From to Krasnoselskii’s theorem [45], the $\psi$-PCPP (6) has a least one solution on $\mathbb{J}.$ □

UH Stability Analysis

In this part, we provide the UH and GUH stability of $\psi$-PCPP (6).

Remark 4.

$\omega \in \mathcal{C}$ satisfies (9) if there exists $\varsigma \in \mathcal{C}$ with

(i) $\left|\varsigma \left(\varkappa \right)\right|\le \epsilon ,$$\varkappa \in \mathbb{J};$

(ii) For all $\varkappa \in \mathbb{J},$

$${}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\omega \left(\varkappa \right)=\phi \left(\varkappa ,\omega \left(\varkappa \right),\omega \left({\lambda }_1\varkappa \right),\cdots ,\omega \left({\lambda }_m\varkappa \right)\right)+\varsigma \left(\varkappa \right).$$

(27)

Lemma 4.

Let $0<\vartheta \le 1,$ and $\omega \in \mathcal{C}$ is a solution of (9). Then, ω satisfies

$$\left\{\begin{array}{c}\left|\omega \left(\varkappa \right)-{\mathcal{W}}_0-{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right){\mathrm{\Phi }}_{\omega }\left(t\right)dt\right|\le \zeta {\varkappa }_1\epsilon ,if\varkappa \in [0,{\varkappa }_1],\\ \left|\omega \left(\varkappa \right)-{\mathcal{W}}_1-\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}{\mathrm{\Phi }}_{\omega }\left(t\right)dt\right|\le \frac{{(b-{\varkappa }_1)}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\epsilon ,if\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

where ${\mathrm{\Phi }}_{\omega }\left(t\right):=\phi \left(t,\omega \left(t\right),\omega \left({\lambda }_{1}t\right),\cdots ,\omega \left({\lambda }_{m}t\right)\right),{\mathcal{W}}_0={\omega }_0+g\left(\omega \right)$, ζ is a constant with ${sup}_{t\in \mathbb{J}}\left|{\psi }^{\prime }\left(t\right)\right|\le \zeta$, and ${\mathcal{W}}_1={\omega }_{{\varkappa }_1}+g\left(\omega \right).$

Proof. 

Let $\omega$ be a solution of (9). It follows from (ii) of Remark 4 that

$$\left\{\begin{array}{c}{}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\omega \left(\varkappa \right)=\phi \left(\varkappa ,\omega \left(\varkappa \right),\omega \left({\lambda }_1\varkappa \right),\cdots ,\omega \left({\lambda }_m\varkappa \right)\right)+\varsigma \left(\varkappa \right)\\ \omega \left(0\right)={\omega }_0+g\left(\omega \right).\end{array}\right.$$

(28)

Then, the solution of problem (28) is

$$\begin{array}{c}\omega \left(\varkappa \right)=\left\{\begin{array}{c}{\mathcal{W}}_0+{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\left[{\mathrm{\Phi }}_{\omega }\left(t\right)+\varsigma \left(t\right)\right]dt,if\varkappa \in [0,{\varkappa }_1],\\ {\mathcal{W}}_1+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\left[{\mathrm{\Phi }}_{\omega }\left(t\right)+\varsigma \left(t\right)\right]dt,if\varkappa \in [{\varkappa }_1,b].\end{array}\right.\end{array}$$

Once more, by (i) of Remark 4, we obtain

$$\left\{\begin{array}{c}\left|\omega \left(\varkappa \right)-{\mathcal{W}}_0-{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right){\mathrm{\Phi }}_{\omega }\left(t\right)dt\right|\le {\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\left|\varsigma \left(t\right)\right|dt\le \epsilon \zeta {\varkappa }_1,\mathrm{for}\varkappa \in [0,{\varkappa }_1],\\ \left|\omega \left(\varkappa \right)-{\mathcal{W}}_1-\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}{\mathrm{\Phi }}_{\omega }\left(t\right)dt\right|\\ \le \frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\left|\varsigma \left(t\right)\right|dt\le \frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\epsilon ,\mathrm{for}\varkappa \in [{\varkappa }_1,b].\end{array}\right.$$

□

Theorem 3.

Under the hypotheses of Theorem 1. Then, the solution of the ψ-PCPP (6) is UH and GUH stable.

Proof. 

Let $\omega \in \mathcal{C}$ be a solution of (9), and $\upsilon \in \mathcal{C}$ be a unique solution of the following problem

$$\left\{\begin{array}{c}{}^{PC}{\mathbb{D}}_{0^{+}}^{\vartheta ;\psi }\upsilon \left(\varkappa \right)={\mathrm{\Phi }}_{\upsilon }\left(\varkappa \right),\\ \upsilon \left(0\right)=\omega \left(0\right).\end{array}\right.$$

From Lemma 1, we have

$$\begin{array}{c}\upsilon \left(\varkappa \right)=\left\{\begin{array}{c}{\mathcal{V}}_0+{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right){\mathrm{\Phi }}_{\upsilon }\left(t\right)dt,if\varkappa \in [0,{\varkappa }_1],\\ {\mathcal{V}}_1+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}{\mathrm{\Phi }}_{\upsilon }\left(t\right)dt,if\varkappa \in [{\varkappa }_1,b],\end{array}\right.\end{array}$$

(29)

where ${\mathcal{V}}_0={\upsilon }_0+g\left(\upsilon \right)$ and ${\mathcal{V}}_1={\upsilon }_{{\varkappa }_1}+g\left(\upsilon \right).$ Clearly, if $\upsilon \left(0\right)=\omega \left(0\right),$ then ${\upsilon }_0={\upsilon }_{{\varkappa }_1},$${\mathcal{V}}_0={\mathcal{W}}_0,$ and ${\mathcal{V}}_1={\mathcal{W}}_1.$ Hence, (29) becomes

$$\begin{array}{c}\upsilon \left(\varkappa \right)=\left\{\begin{array}{c}{\mathcal{W}}_0+{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right){\mathrm{\Phi }}_{\upsilon }\left(t\right)dt,if\varkappa \in [0,{\varkappa }_1],\\ {\mathcal{W}}_1+\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}{\mathrm{\Phi }}_{\upsilon }\left(t\right)dt,if\varkappa \in [{\varkappa }_1,b].\end{array}\right.\end{array}$$

(30)

Using Lemma 4 and (H1), for $\varkappa \in [0,{\varkappa }_1]$, we have

$$\begin{array}{ccc}& & \left|\omega \left(\varkappa \right)-\upsilon \left(\varkappa \right)\right|\hfill \\ & & =\left|\omega \left(\varkappa \right)-{\mathcal{W}}_0-{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right){\mathrm{\Phi }}_{\upsilon }\left(t\right)dt\right|\hfill \\ & & \le \left|\omega \left(\varkappa \right)-{\mathcal{W}}_0-{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right){\mathrm{\Phi }}_{\omega }\left(t\right)dt\right|+{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\left|{\mathrm{\Phi }}_{\omega }\left(t\right)-{\mathrm{\Phi }}_{\upsilon }\left(t\right)\right|dt\hfill \\ & & \le \zeta {\varkappa }_1\epsilon +\zeta {\int }_0^{\varkappa }\left(L_0\left|\upsilon \left(t\right)-\omega \left(t\right)\right|+\sum _{i=1}^m{L}_i\left|\upsilon \left({\lambda }_{i}t\right)-\omega \left({\lambda }_{i}t\right)\right|\right)dt\hfill \\ & & \le \zeta {\varkappa }_1\epsilon +\zeta {\int }_0^{\varkappa }\left(L_0\left|\upsilon \left(t\right)-\omega \left(t\right)\right|+L_1\left|\upsilon \left({\lambda }_{1}t\right)-\omega \left({\lambda }_{1}t\right)\right|+\cdots +L_m\left|\upsilon \left({\lambda }_{m}t\right)-\omega \left({\lambda }_{m}t\right)\right|\right)dt\hfill \\ & & \le \zeta {\varkappa }_1\epsilon +\zeta \left(L_0+L_1+\cdots +L_m\right){\int }_0^{\varkappa }\left|\upsilon \left(t\right)-\omega \left(t\right)\right|dt.\hfill \end{array}$$

Using classical Gronwall’s Lemma [46], we obtain

$$\begin{array}{ccc}\hfill \left|\omega \left(\varkappa \right)-\upsilon \left(\varkappa \right)\right|& \le & \zeta {\varkappa }_1\epsilon exp\left({\int }_0^{\varkappa }\zeta \left(L_0+L_1+\cdots +L_m\right)\right)dt\hfill \\ & =& \zeta {\varkappa }_1\epsilon exp\left(\zeta \left(L_0+L_1+\cdots +L_m\right){\varkappa }_1\right):=\epsilon {\chi }_{\phi }^0.\hfill \end{array}$$

(31)

For $\varkappa \in [{\varkappa }_1,b]$, we have

$$\begin{array}{ccc}& & \left|\omega \left(\varkappa \right)-\upsilon \left(\varkappa \right)\right|\hfill \\ & & =\left|\omega \left(\varkappa \right)-{\mathcal{W}}_0-\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}{\mathrm{\Phi }}_{\upsilon }\left(t\right)dt\right|\hfill \\ & & \le \left|\omega \left(\varkappa \right)-{\mathcal{W}}_0-\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}{\mathrm{\Phi }}_{\omega }\left(t\right)dt\right|\hfill \\ & & +\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\left|{\mathrm{\Phi }}_{\omega }\left(t\right)-{\mathrm{\Phi }}_{\upsilon }\left(t\right)\right|dt\hfill \\ & & \le \frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\epsilon +\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\hfill \\ & & \times \left(L_0\left|\upsilon \left(t\right)-\omega \left(t\right)\right|+\sum _{i=1}^m{L}_i\left|\upsilon \left({\lambda }_{i}t\right)-\omega \left({\lambda }_{i}t\right)\right|\right)dt\hfill \\ & & \le \frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\epsilon +\frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\hfill \\ & & \times \left(L_0\left|\upsilon \left(t\right)-\omega \left(t\right)\right|+L_1\left|\upsilon \left({\lambda }_{1}t\right)-\omega \left({\lambda }_{1}t\right)\right|+\cdots +L_m\left|\upsilon \left({\lambda }_{m}t\right)-\omega \left({\lambda }_{m}t\right)\right|\right)dt\hfill \\ & & \le \frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\epsilon +\frac{\left(L_0+L_1+\cdots +L_m\right)}{\mathrm{\Gamma }\left(\vartheta \right)}\hfill \\ & & \times {\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}\left|\upsilon \left(t\right)-\omega \left(t\right)\right|dt.\hfill \end{array}$$

Using the generalized fractional Gronwall’s Lemma [47], we obtain

$$\begin{array}{ccc}\hfill \left|\omega \left(\varkappa \right)-\upsilon \left(\varkappa \right)\right|& \le & \frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\epsilon +\frac{\left(L_0+L_1+\cdots +L_m\right)}{\mathrm{\Gamma }(\vartheta +1)}\epsilon \hfill \\ & & \times \frac{1}{\mathrm{\Gamma }\left(\vartheta \right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }\left(t\right){(\psi \left(\varkappa \right)-\psi \left(t\right))}^{\vartheta -1}{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }dt\hfill \\ & \le & \frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\epsilon +\frac{\left(L_0+L_1+\cdots +L_m\right)}{\mathrm{\Gamma }(\vartheta +1)}\epsilon \frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{2\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\hfill \\ & =& \frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\left(1+\frac{\left(L_0+L_1+\cdots +L_m\right)}{\mathrm{\Gamma }(\vartheta +1)}{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }\right)\epsilon \hfill \\ & :& =\epsilon {\chi }_{\phi }^1.\hfill \end{array}$$

(32)

It follows from (31) and (32) that

$$\left|\omega \left(\varkappa \right)-\upsilon \left(\varkappa \right)\right|\le \left\{\begin{array}{c}\epsilon {\chi }_{\phi }^0,for\varkappa \in [0,{\varkappa }_1]\\ \epsilon {\chi }_{\phi }^1,for\varkappa \in [{\varkappa }_1,b]\end{array}\right.,$$

(33)

where

$${\chi }_{\phi }^0=\zeta {\varkappa }_{1}exp\left(\zeta \left(L_0+L_1+\cdots +L_m\right){\varkappa }_1\right),\mathrm{and}$$

$${\chi }_{\phi }^1=\frac{{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }}{\mathrm{\Gamma }(\vartheta +1)}\left(1+\frac{\left(L_0+L_1+\cdots +L_m\right)}{\mathrm{\Gamma }(\vartheta +1)}{(\psi \left(b\right)-\psi \left({\varkappa }_1\right))}^{\vartheta }\right).$$

Hence, the $\psi$-PCPP (6) is UH stable in $\mathcal{C}$.

Moreover, from Definition 4, there exists a nondecreasing function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that $\varphi \left(ϵ\right)=ϵ$. Then, from (33), we have

$$\left|\omega \left(\varkappa \right)-\upsilon \left(\varkappa \right)\right|\le \left\{\begin{array}{c}{\chi }_{\phi }^0\varphi \left(ϵ\right),for\varkappa \in [0,{\varkappa }_1],\\ {\chi }_{\phi }^1\varphi \left(ϵ\right),for\varkappa \in [{\varkappa }_1,b],\end{array}\right.$$

with $\varphi \left(0\right)=0,$ which proves that $\psi$-PCPP (6) is GUH stable in $\mathcal{C}.$ □

Remark 5.

Consider a more general problem as a system that contains a number of problems similar to our current problem (6) as follows:

$$\left\{\begin{array}{c}{}^{PC}{\mathbb{D}}_{0^{+}}^{{\vartheta }_i;\psi }{\upsilon }_i\left(\varkappa \right)={\phi }_i\left(\varkappa ,{\upsilon }_{1,{\lambda }_j}\left(\varkappa \right),{\upsilon }_{2,{\lambda }_j}\left(\varkappa \right),\cdots ,{\upsilon }_{n,{\lambda }_j}\left(\varkappa \right)\right),\varkappa \in [0,b],\\ {\upsilon }_i\left(0\right)={\upsilon }_{0i}+g\left({\upsilon }_i\right),i=1,\cdots ,n,j=1,\cdots ,m,\end{array}\right.$$

(34)

where $0<{\vartheta }_i\le 1$, $0<{\lambda }_j<1$, ${}^{PC}{\mathbb{D}}_{0^{+}}^{{\vartheta }_i;\psi }$ is the generalized piecewise FD of order ${\vartheta }_i$, and

$$\begin{array}{c}{\upsilon }_{1,{\lambda }_j}\left(\varkappa \right)={\upsilon }_1\left(\varkappa \right),{\upsilon }_1\left({\lambda }_1\varkappa \right),\cdots ,{\upsilon }_1\left({\lambda }_m\varkappa \right),\\ {\upsilon }_{2,{\lambda }_j}\left(\varkappa \right)={\upsilon }_2\left(\varkappa \right),{\upsilon }_2\left({\lambda }_1\varkappa \right),\cdots ,{\upsilon }_2\left({\lambda }_m\varkappa \right),\\ ⋮\\ {\upsilon }_{n,{\lambda }_j}\left(\varkappa \right)=\varkappa ,{\upsilon }_n\left(\varkappa \right),{\upsilon }_n\left({\lambda }_1\varkappa \right),\cdots ,{\upsilon }_n\left({\lambda }_m\varkappa \right).\end{array}$$

The pantograph system (34) can be written as:

$$\left\{\begin{array}{c}{}^{PC}{\mathbb{D}}_{0^{+}}^{\mathrm{\Theta };\psi }\mathcal{V}\left(\varkappa \right)=\mathrm{\Phi }\left(\varkappa ,{\mathcal{V}}_{\lambda }\left(\varkappa \right)\right),\varkappa \in [0,b],\\ \mathcal{V}\left(0\right)={\mathcal{V}}_0+g\left(\mathcal{V}\right),\end{array}\right.$$

(35)

where

$$\mathcal{V}\left(\varkappa \right)=\left[\begin{array}{c}{\upsilon }_1\left(\varkappa \right)\\ {\upsilon }_2\left(\varkappa \right)\\ ⋮\\ {\upsilon }_n\left(\varkappa \right)\end{array}\right],\mathrm{\Phi }\left(\varkappa ,{\mathcal{V}}_{\lambda }\left(\varkappa \right)\right)=\left[\begin{array}{c}{\phi }_1(\varkappa ,{\upsilon }_{1,{\lambda }_j}\left(\varkappa \right))\\ {\phi }_2(\varkappa ,{\upsilon }_{2,{\lambda }_j}\left(\varkappa \right))\\ ⋮\\ {\phi }_n(\varkappa ,{\upsilon }_{n,{\lambda }_j}\left(\varkappa \right))\end{array}\right],\mathrm{and}$$

$$\mathcal{V}\left(0\right)=\left[\begin{array}{c}{\upsilon }_1\left(0\right)\\ {\upsilon }_2\left(0\right)\\ ⋮\\ {\upsilon }_n\left(0\right)\end{array}\right],{\mathcal{V}}_0=\left[\begin{array}{c}{\upsilon }_{01}\\ {\upsilon }_{02}\\ ⋮\\ {\upsilon }_{0n}\end{array}\right],\mathrm{\Theta }=\left[\begin{array}{c}{\vartheta }_1\\ {\vartheta }_2\\ ⋮\\ {\vartheta }_n\end{array}\right].$$

Using Lemma 1, system (35) has the following solution:

$$\begin{array}{c}\mathcal{V}\left(\varkappa \right)=\left\{\begin{array}{c}{\mathcal{V}}_0+g\left(\mathcal{V}\right)+{\int }_0^{\varkappa }{\psi }^{\prime }\left(t\right)\mathrm{\Phi }\left({\mathcal{V}}_{\lambda }\left(t\right)\right)dt,if\varkappa \in [0,{\varkappa }_1],\\ {\mathcal{V}}_{{\varkappa }_1}+g\left(\mathcal{V}\right)+\frac{1}{\mathrm{\Gamma }\left(\mathrm{\Theta }\right)}{\int }_{{\varkappa }_1}^{\varkappa }{\psi }^{\prime }{\left(t\right)(\psi \left(\varkappa \right)-\psi \left(t\right)))}^{\mathrm{\Theta }-1}\mathrm{\Phi }\left(t,{\mathcal{V}}_{\lambda }\left(t\right)\right)dt,if\varkappa \in [{\varkappa }_1,b].\end{array}\right.\end{array}$$

(36)

Remark 6.

Following the approaches of proof used in the preceding, we can obtain the same results (Theorems 1–3) for the nonlinear pantograph system (35) in view of formula (36).

4. Example

In this portion, we present an exhaustive example to illustrate the reported results.

Consider the following $\psi$-PCPP

$$\left\{\begin{array}{c}{}^{PC}{\mathbb{D}}_{0^{+}}^{0.25;e^{\frac{\varkappa }{2}}}\upsilon \left(\varkappa \right)=\phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\upsilon \left({\lambda }_2\varkappa \right)\right),\varkappa \in [0,1],\\ \upsilon \left(0\right)=0.7+\sum _{i=1}^n{c}_i\upsilon \left({\varkappa }_i\right),\end{array}\right.$$

(37)

or

$$\left\{\begin{array}{c}{\mathbb{D}}^{1;e^{\frac{\varkappa }{2}}}\upsilon \left(\varkappa \right)=\phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\upsilon \left({\lambda }_2\varkappa \right)\right),\mathrm{if}\varkappa \in [0,0.5],\\ \upsilon \left(0\right)=0.7+\sum _{i=1}^n{c}_i\upsilon \left({\varkappa }_i\right),\\ \\ {}^C{\mathbb{D}}_{0.5^{+}}^{0.25;e^{\frac{\varkappa }{2}}}\upsilon \left(\varkappa \right)=\phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\upsilon \left({\lambda }_2\varkappa \right)\right),if\varkappa \in [0.5,1],\\ \upsilon \left(0.5\right)=0.8+\sum _{i=1}^n{c}_i\upsilon \left({\varkappa }_i\right),\end{array}\right.$$

where $\vartheta =0.25,$$\psi \left(\varkappa \right)=e^{\frac{\varkappa }{2}},{\upsilon }_0=0.7,$${\upsilon }_{{\varkappa }_1}=0.8,$$0<{\varkappa }_1=0.5<\cdots <{\varkappa }_n<1=b$, and $c_i$ are positive constants with ${\sum }_{i=1}^n{c}_i<\frac{3}{5}.$ Set $m=2$$\left({\lambda }_1=\frac{1}{3},{\lambda }_2=\frac{1}{4}\right)$,

$$\phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\upsilon \left({\lambda }_2\varkappa \right)\right)=\frac{sin\left|\upsilon \left(\varkappa \right)+\upsilon \left(\frac{\varkappa }{3}\right)\right|}{9+2\varkappa }+\frac{cos\left|\upsilon \left(\frac{\varkappa }{4}\right)\right|}{4+2\varkappa },$$

for $\varkappa \in [0,1],$$\upsilon \in [0,\infty ),$ and

$$g\left(\upsilon \right)=\sum _{i=1}^n{c}_i\upsilon \left({\varkappa }_i\right),\upsilon \in [0,\infty ).$$

(I) Let $\upsilon ,\omega \in [0,\infty )$, $\varkappa \in [0,1]$. Then

$$\begin{array}{ccc}& & \left|\phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\upsilon \left({\lambda }_2\varkappa \right)\right)-\phi \left(\varkappa ,\omega \left(\varkappa \right),\omega \left({\lambda }_1\varkappa \right),\omega \left({\lambda }_2\varkappa \right)\right)\right|\hfill \\ & & =\left|\frac{sin\left|\upsilon \left(\varkappa \right)+\upsilon \left(\frac{\varkappa }{3}\right)\right|}{9+2\varkappa }+\frac{cos\left|\upsilon \left(\frac{\varkappa }{4}\right)\right|}{4+2\varkappa }-\frac{sin\left|\omega \left(\varkappa \right)+\omega \left(\frac{\varkappa }{3}\right)\right|}{9+2\varkappa }-\frac{cos\left|\omega \left(\frac{\varkappa }{4}\right)\right|}{4+2\varkappa }\right|\hfill \\ & & \le \frac{1}{9+2\varkappa }\left|\left(\left|\upsilon \left(\varkappa \right)+\upsilon \left(\frac{\varkappa }{3}\right)\right|-\left|\omega \left(\varkappa \right)+\omega \left(\frac{\varkappa }{3}\right)\right|\right)+\frac{1}{4+2\varkappa }\left(\left|\upsilon \left(\frac{\varkappa }{4}\right)\right|-\left|\omega \left(\frac{\varkappa }{4}\right)\right|\right)\right|\hfill \\ & & \le \frac{1}{9+2\varkappa }\left(\left|\upsilon \left(\varkappa \right)-\omega \left(\varkappa \right)\right|+\left|\upsilon \left(\frac{\varkappa }{3}\right)-\omega \left(\frac{\varkappa }{3}\right)\right|\right)+\frac{1}{4+2\varkappa }\left|\upsilon \left(\frac{\varkappa }{4}\right)-\omega \left(\frac{\varkappa }{4}\right)\right|\hfill \\ & & \le \frac{1}{9}\left|\upsilon \left(\varkappa \right)-\omega \left(\varkappa \right)\right|+\frac{1}{9}\left|\upsilon \left(\frac{\varkappa }{3}\right)-\omega \left(\frac{\varkappa }{3}\right)\right|+\frac{1}{4}\left|\upsilon \left(\frac{\varkappa }{4}\right)-\omega \left(\frac{\varkappa }{4}\right)\right|.\hfill \end{array}$$

Thus, (Hy₁) holds with $L_0=L_1=\frac{1}{9}$ and $L_2=\frac{1}{4}.$ Additionally, for $\upsilon ,\omega \in [0,\infty )$, we have

$$\left|g\left(\upsilon \right)-g\left(\omega \right)\right|=\left|\sum _{i=1}^n{c}_i\upsilon \left({\varkappa }_i\right)-\sum _{i=1}^n{c}_i\omega \left({\varkappa }_i\right)\right|\le \sum _{i=1}^n{c}_i\left|\upsilon -\omega \right|\le \frac{3}{5}\left|\upsilon -\omega \right|.$$

Therefore, (Hy₂) holds with $L_g=\frac{3}{5}$. To fulfill condition (19), we have ${sup}_{\varkappa \in [0,1]}$$\left|{\psi }^{\prime }\left(\varkappa \right)\right|={sup}_{\varkappa \in [0,1]}\left|\frac{1}{2}{e}^{\frac{\varkappa }{2}}\right|=\frac{\sqrt{e}}{2}<1:=\zeta .$ Hence, ${\aleph }_1=\frac{301}{360}<1,$ and ${\aleph }_2\approx 0.64<1.$ Thus, Theorem 1 shows that $\psi$-PCPP (37) has a unique solution $[0,1]$.

(II) For $\epsilon >0$ with ${\chi }_{\phi }^0=\frac{1}{2}{e}^{\frac{17}{72}}>0,$ and ${\chi }_{\phi }^1\approx 1.20>0.$ It follows from Theorem 3 that the $\psi$-PCPP (37) is HU and GUH stable.

(III) For $\varkappa \in [0,1]$ and $\upsilon \in [0,\infty ),$ we obtain

$$\left|\phi \left(\varkappa ,\upsilon \left(\varkappa \right),\upsilon \left({\lambda }_1\varkappa \right),\upsilon \left({\lambda }_2\varkappa \right)\right)\right|\le \frac{1}{9+2\varkappa }+\frac{1}{4+2\varkappa }$$

and

$$\left|g\left(\upsilon \right)\right|=\left|\sum _{i=1}^n{c}_i\upsilon \left({\varkappa }_i\right)\right|\le \sum _{i=1}^n{c}_i\left|\upsilon \right|\le \frac{3}{5}\left|\upsilon \right|.$$

Consequently, (Hy₃) and ((Hy₄) hold with ${\mu }_{\phi }\left(\varkappa \right)=\frac{1}{9+2\varkappa }+\frac{1}{4+2\varkappa },$ ${\mu }_{\phi }^{*}=\frac{49}{36},$ and ${\mu }_g=\frac{3}{5}<1.$ Thus, all the assumptions of Theorem 2 are satisfied. Hence, $\psi$-PCPP (37) has a solution on $[0,1]$.

5. Conclusions

Atangana and Araz [41] suggested the idea of piecewise derivatives. In this regard, we created and expanded the existence, uniqueness, and UH–GUH stability results for nonlocal pantograph equations under $\psi$-piecewise Caputo FDs as an additional contribution to this subject. Based on the fixed-point theorems of Banach and Krasnoselskii, we offered numerous new results of existence and uniqueness. Moreover, results pertaining to UH/GUH stability were obtained utilizing traditional methodologies of nonlinear functional analysis. An example to validate the theoretical findings was provided. In light of our recent discoveries, a more general problem for the pantograph system that includes problems related to the study’s subject was presented. In the future, it will be interesting to study the current pantograph systems under piecewise FDs in the Caputo Fabrizio, and Atangana–Baleanu sense [41,48,49].