Existence of Solutions for Coupled Higher-Order Fractional Integro-Differential Equations with Nonlocal Integral and Multi-Point Boundary Conditions Depending on Lower-Order Fractional Derivatives and Integrals

Abstract

In this article, we investigate the existence and uniqueness of solutions for a nonlinear coupled system of Liouville–Caputo type fractional integro-differential equations supplemented with non-local discrete and integral boundary conditions. The nonlinearity relies both on the unknown functions and their fractional derivatives and integrals in the lower order. The consequence of existence is obtained utilizing the alternative of Leray–Schauder, while the result of uniqueness is based on the concept of Banach contraction mapping. We introduced the concept of unification in the present work with varying parameters of the multi-point and classical integral boundary conditions. With the help of examples, the main results are well demonstrated.

1. Introduction

In the mathematical modeling of many real-world problems, the study of coupled systems of fractional orders of differential equations (FDEs) has gained significant attention; for example, chaotic system synchronization [1,2], anomalous diffusion [3], ecological models [4], etc. We refer to some papers for some recent results on coupled systems with FDEs [5,6,7,8,9,10,11,12,13]. The use of fractional calculus methods is quite prominent in the mathematical modeling of various processes and phenomena. The main reason is that fractional operators, unlike integer operators, are non-local and able to trace the past effects of the phenomena involved; see [14,15,16,17,18,19] for examples and details. Some researchers have addressed the problem of fractional boundary value problems (BVPs), and a significant trend can be seen in the recent literature; for example, see [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37] and the references cited therein. A few authors have recently started investigating coupled fractional BVPs. Ahmad et al. [38] discussed the solvability of the following coupled FDEs with integral boundary conditions:

$$\left\{\begin{array}{c}{{}^C\mathcal{D}}^{q}x\left(t\right)=f(t,x\left(t\right),y\left(t\right)),\hfill \\ {{}^C\mathcal{D}}^{p}y\left(t\right)=h(t,x\left(t\right),y\left(t\right)),\hfill \\ x^{{}^{\prime }}\left(0\right)=\alpha {\int }_0^{\xi }{x}^{{}^{\prime }}\left(s\right)ds,x\left(1\right)=\beta {\int }_0^{1}g\left(x^{{}^{\prime }}\left(s\right)\right)ds,\hfill \\ y^{{}^{\prime }}\left(0\right)={\alpha }_1{\int }_0^{\theta }{y}^{{}^{\prime }}\left(s\right)ds,y\left(1\right)={\beta }_1{\int }_0^{1}g\left(y^{{}^{\prime }}\left(s\right)\right)ds,\hfill \\ t\in [0,1],1<q,p\le 2,0\le \xi ,\theta \le 1,\hfill \end{array}\right.$$

where ${{}^C\mathcal{D}}^q$, ${{}^C\mathcal{D}}^p$ denote the Caputo fractional derivatives (CFDs) of order q, p, f, h: $[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are given continuous functions, and $\alpha$, $\beta$, ${\alpha }_1$, ${\beta }_1$ are real constants. The FDEs with integral and ordinary-fractional flux boundary conditions

$$\left\{\begin{array}{c}{{}^C\mathcal{D}}^{\alpha }x\left(t\right)=f(t,x\left(t\right),y\left(t\right)),\hfill \\ {{}^C\mathcal{D}}^{\beta }y\left(t\right)=h(t,x\left(t\right),y\left(t\right)),\hfill \\ x\left(0\right)+x\left(1\right)=a{\int }_0^{1}x\left(s\right)ds,x^{{}^{\prime }}\left(0\right)=b{{}^C\mathcal{D}}^{\gamma }x\left(1\right),\hfill \\ y\left(0\right)+y\left(1\right)=a_1{\int }_0^{1}y\left(s\right)ds,y^{{}^{\prime }}\left(0\right)=b_1{{}^C\mathcal{D}}^{\delta }y\left(1\right),\hfill \\ t\in [0,1],1<\alpha ,\beta \le 2,0<\gamma ,\delta \le 1,\hfill \end{array}\right.$$

were discussed in [39], where ${{}^C\mathcal{D}}^{\alpha }$, ${{}^C\mathcal{D}}^{\beta }$, ${{}^C\mathcal{D}}^{\gamma }$, ${{}^C\mathcal{D}}^{\delta }$ denote the CFDs of order $\alpha$, $\beta$, $\gamma$, $\delta$, f, h: $[0,1]\times {\mathbb{R}}^2\to \mathbb{R}$, are given continuous functions, and a, b, $a_1$, $b_1$ are real constants. Ahmad et al. analyzed in [40] the existence results for coupled system of FDEs:

$$\left\{\begin{array}{c}{\mathcal{D}}^{\alpha }u\left(t\right)=f(t,v\left(t\right),{\mathcal{D}}^{p}v\left(t\right)),\hfill \\ {\mathcal{D}}^{\beta }v\left(t\right)=g(t,u\left(t\right),{\mathcal{D}}^{q}u\left(t\right)),\hfill \\ u\left(0\right)=0,u\left(1\right)=\gamma u\left(\eta \right),\hfill \\ v\left(0\right)=0,v\left(1\right)=\gamma v\left(\eta \right),\hfill \\ 0<t<1,1<\alpha ,\beta <2,0<\eta <1,\hfill \end{array}\right.$$

where ${\mathcal{D}}^{\alpha }$, ${\mathcal{D}}^{\beta }$, ${\mathcal{D}}^p$, ${\mathcal{D}}^q$ denote the Riemann–Liouville fractional derivatives of order $\alpha$, $\beta$, p, q, f, g : $[0,1]\times {\mathbb{R}}^2\to \mathbb{R}$ are given continuous functions, and $\gamma$ is the real constant. Agarwal et al. [41] analyzed the results with discrete and integral boundary conditions of the existence of coupled fractional-order systems. In fractional BVP involving the Caputo derivatives, Subramanian et al. [42] studied coupled non-local slit-strip conditions.

In this article, we are investigating the existence of solutions for nonlinear coupled Caputo fractional integro-differential equations,

$$\left\{\begin{array}{c}{\displaystyle {{}^C\mathcal{D}}^{\varrho }u\left(\tau \right)=f(\tau ,u\left(\tau \right),v\left(\tau \right),{{}^C\mathcal{D}}^{{\varsigma }_1}v\left(\tau \right),{\mathcal{I}}^{\xi }v\left(\tau \right)),\tau \in [0,T]:=\mathcal{U},}\hfill \\ {{}^C\mathcal{D}}^{\varsigma }v\left(\tau \right)=g(\tau ,u\left(\tau \right),{{}^C\mathcal{D}}^{{\varrho }_1}u\left(\tau \right),{\mathcal{I}}^{\zeta }u\left(\tau \right),v\left(\tau \right)),\tau \in [0,T]:=\mathcal{U},\hfill \end{array}\right.$$

(1)

supplemented by nonlocal integral and multi-point boundary conditions,

$$\left\{\begin{array}{c}u\left(0\right)={\psi }_1\left(v\right),u^{{}^{\prime }}\left(0\right)={ϵ}_1{\int }_0^{{\nu }_1}{v}^{{}^{\prime }}\left(\theta \right)d\theta ,u^{{}^{″}}\left(0\right)=0,···,u^{n-2}\left(0\right)=0,\hfill \\ {\displaystyle u\left(T\right)={\lambda }_1{\int }_0^{{\delta }_1}v\left(\theta \right)d\theta +{\mu }_1\sum _{j=1}^{k-2}{\varpi }_{j}v\left({\vartheta }_j\right),}\hfill \\ v\left(0\right)={\psi }_2\left(u\right),v^{{}^{\prime }}\left(0\right)={ϵ}_2{\int }_0^{{\nu }_2}{u}^{{}^{\prime }}\left(\theta \right)d\theta ,v^{{}^{″}}\left(0\right)=0,···,v^{n-2}\left(0\right)=0,\hfill \\ {\displaystyle v\left(T\right)={\lambda }_2{\int }_0^{{\delta }_2}u\left(\theta \right)d\theta +{\mu }_2\sum _{j=1}^{k-2}{\omega }_{j}u\left({\phi }_j\right),}\hfill \end{array}\right.$$

(2)

where ${{}^C\mathcal{D}}^{\varrho }$, ${{}^C\mathcal{D}}^{\varsigma }$, ${{}^C\mathcal{D}}^{{\varrho }_1}$, ${{}^C\mathcal{D}}^{{\varsigma }_1}$ are the Caputo fractional derivatives of order $n-1<\varrho ,\varsigma <n$, $0<{\varrho }_1,{\varsigma }_1<1$, ${\mathcal{I}}^{\zeta }$, ${\mathcal{I}}^{\xi }$ are the Riemann–Liouville fractional integrals of order $\zeta ,\xi >0$, f, g:$\mathcal{U}\times {\mathbb{R}}^4\to \mathbb{R}$, ${\psi }_1$, ${\psi }_2$ : $\mathcal{C}(\mathcal{U},\mathbb{R})\to \mathbb{R}$ are given continuous functions, $0<{\nu }_1<{\nu }_2<{\delta }_1<{\delta }_2<{\vartheta }_1<{\phi }_1<···<{\vartheta }_{n-2}<{\phi }_{n-2}<T$, and ${ϵ}_i$, ${\lambda }_i$, ${\mu }_i$$(i=1,2)$, ${\varpi }_j$, ${\omega }_j$$(j=1,2,\cdots ,k-2)$ are positive real constants.

The rest of the article is assembled appropriately. In Section 2, we retrieve those concepts for a good reference and prove an auxiliary lemma, which provides the basis for solving the problem. Section 3 presents the primary outcomes, while Section 4, Section 5 and Section 6 provide examples, some important observations, and closing remarks, respectively.

2. Preliminaries

Firstly, we remember some fundamental fractional calculus definitions.

Definition 1.

The fractional integral of order α with the lower limit zero for a function $\mathfrak{f}$ is defined as

$$I^{\varrho }\mathfrak{f}\left(\tau \right)=\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^{\tau }\frac{\mathfrak{f}\left(s\right)}{{(\tau -s)}^{1-\varrho }}ds,\tau >0,\varrho >0,$$

(3)

provided that the right-hand side is point-wise defined on [0.∞), where $\Gamma (·)$ is the gamma function, which is defined by $\Gamma \left(\varrho \right)={\int }_0^{\infty }{\tau }^{\varrho -1}{e}^{-\tau }d\tau .$

Definition 2.

The Riemann–Liouville fractional derivative of order $\varrho >0$ , $n-1<\varrho <n$, $n\in \mathbb{N}$ is defined as

$$D_{0+}^{\varrho }\mathfrak{f}\left(\tau \right)=\frac{1}{\Gamma (n-\varrho )}{\left(\frac{d}{d\tau }\right)}^n{\int }_0^{\tau }{(\tau -s)}^{n-\varrho -1}\mathfrak{f}\left(s\right)ds,\tau >0,$$

(4)

where the function $\mathfrak{f}$ has an absolutely continuous derivative up to order $(n-1)$.

Definition 3.

The Caputo derivative of order $\varrho \in [n-1,n)$ for a function $\mathfrak{f}:[0,\infty )\to \left(\mathbb{R}\right)$ can be written as

$${}^C{\mathcal{D}}_{0+}^{\varrho }\mathfrak{f}\left(\tau \right)={\mathcal{D}}_{0+}^{\varrho }\left(\mathfrak{f}\left(\tau \right)-\sum _{k=0}^{n-1}\frac{{\tau }^k}{k!}{\mathfrak{f}}^{\left(k\right)}\left(0\right)\right),\tau >0,n-1<\varrho <n.$$

(5)

Note that the Caputo fractional derivative of order $\varrho \in [n-1,n)$ exists almost everywhere on $[0,\infty )$ if $\mathfrak{f}\in {\mathcal{AC}}^n([0,\infty ),\left(\mathbb{R}\right))$.

Remark 1.

If $\mathfrak{f}\in {\mathcal{C}}^n[0,\infty ),$ then

$${}^C{D}_{0+}^{\varrho }\widehat{\mathfrak{f}}\left(\tau \right)=\frac{1}{\Gamma (n-\varrho )}{\int }_0^{\tau }\frac{\mathfrak{f}(n)\left(s\right)}{{(\tau -s)}^{\varrho +1-n}}ds=I^{n-\varrho }\mathfrak{f}\left(n\right)\left(\tau \right),\tau >0,n-1<\varrho <n.$$

Lemma 1.

For any $\widehat{f},\widehat{g}\in C[0,T]$, the solution of the linear system of FDEs

$$\left\{\begin{array}{c}{\displaystyle {{}^C\mathcal{D}}^{\varrho }u\left(\tau \right)=\widehat{f}\left(\tau \right),\tau \in \mathcal{U},}\hfill \\ {\displaystyle {{}^C\mathcal{D}}^{\varsigma }v\left(\tau \right)=\widehat{g}\left(\tau \right),\tau \in \mathcal{U},}\hfill \end{array}\right.$$

(6)

supplemented with the boundary conditions (2) is equivalent to the system of integral equations

$$\begin{array}{ccc}\hfill u\left(\tau \right)& =& \frac{1}{\Gamma \left(\varrho \right)}{\int }_0^{\tau }{(\tau -\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta +{\psi }_1\left(v\right)[1+{\kappa }_1{\Lambda }_4\left(\tau \right)-{\Lambda }_3\left(\tau \right)]+{\psi }_2\left(u\right)[{\kappa }_2{\Lambda }_3\left(\tau \right)-{\Lambda }_4\left(\tau \right)]\hfill \\ & & +\frac{{\Lambda }_2\left(\tau \right){ϵ}_2}{\Gamma (\varrho -1)}{\int }_0^{{\nu }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -2}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta +\frac{{\Lambda }_1\left(\tau \right){ϵ}_1}{\Gamma (\varsigma -1)}{\int }_0^{{\nu }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -2}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\Lambda }_4\left(\tau \right)[\frac{{\lambda }_2}{\Gamma \left(\varrho \right)}{\int }_0^{{\delta }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -1}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta +{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j}{\Gamma \left(\varrho \right)}{\int }_0^{{\phi }_j}{({\phi }_j-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta \hfill \\ & & -\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^T{(T-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta ]+{\Lambda }_3\left(\tau \right)[\frac{{\lambda }_1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\delta }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -1}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j}{\Gamma \left(\varsigma \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^T{(T-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta ],\hfill \end{array}$$

(7)

and

$$\begin{array}{ccc}\hfill v\left(\tau \right)& =& \frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^{\tau }{(\tau -\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta +{\psi }_2\left(u\right)[1+{\kappa }_2{\Lambda }_7\left(\tau \right)-{\Lambda }_8\left(\tau \right)]+{\psi }_1\left(v\right)[{\kappa }_1{\Lambda }_8\left(\tau \right)-{\Lambda }_7\left(\tau \right)]\hfill \\ & & +\frac{{\Lambda }_5\left(\tau \right){ϵ}_1}{\Gamma (\varsigma -1)}{\int }_0^{{\nu }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -2}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta +\frac{{\Lambda }_6\left(\tau \right){ϵ}_2}{\Gamma (\varrho -1)}{\int }_0^{{\nu }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -2}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\Lambda }_7\left(\tau \right)[\frac{{\lambda }_1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\delta }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -1}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta +{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j}{\Gamma \left(\varsigma \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta \hfill \\ & & -\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^T{(T-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta ]+{\Lambda }_8\left(\tau \right)[\frac{{\lambda }_2}{\Gamma \left(\varrho \right)}{\int }_0^{{\delta }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -1}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j}{\Gamma \left(\varrho \right)}{\int }_0^{{\phi }_j}{({\phi }_j-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^T{(T-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta ],\hfill \end{array}$$

(8)

where

$$\begin{array}{ccc}& & {\xi }_1=\frac{{\lambda }_1{\delta }_1^2}{2}+{\mu }_1\sum _{j=1}^{k-2}{\varpi }_j{\vartheta }_j,{\xi }_2=\frac{{\lambda }_1{\delta }_1^n}{n}+{\mu }_1\sum _{j=1}^{k-2}{\varpi }_j{\vartheta }_j^{n-1},{\xi }_3=\frac{{\lambda }_2{\delta }_2^2}{2}+{\mu }_2\sum _{j=1}^{k-2}{\omega }_j{\phi }_j,{\xi }_4=\frac{{\lambda }_2{\delta }_2^n}{n}+{\mu }_2\sum _{j=1}^{k-2}{\omega }_j{\phi }_j^{n-1},\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & {\widehat{\gamma }}_1=1-{\nu }_1{\nu }_2{ϵ}_1{ϵ}_2,{\widehat{\gamma }}_2=T^2-{\xi }_1{\xi }_3,{\widehat{\gamma }}_3=T^n-{\xi }_1{\xi }_4,{\widehat{\gamma }}_4=T^{n-1}{\xi }_3-{\xi }_{4}T,{\widehat{\gamma }}_5={\xi }_1{T}^{n-1}-T{\xi }_2,{\widehat{\gamma }}_6=T^n-{\xi }_2{\xi }_3,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & {\upsilon }_1={\widehat{\gamma }}_2{\nu }_1{\nu }_2^{n-1}{ϵ}_1{ϵ}_2+{\widehat{\gamma }}_1{\widehat{\gamma }}_3,{\upsilon }_2={\widehat{\gamma }}_2{\nu }_2^{n-1}{ϵ}_2+{\widehat{\gamma }}_1{\widehat{\gamma }}_4,{\upsilon }_3={\widehat{\gamma }}_2{\nu }_1^{n-1}{ϵ}_1+{\widehat{\gamma }}_1{\widehat{\gamma }}_5,{\upsilon }_4={\widehat{\gamma }}_2{\nu }_1^{n-1}{\nu }_2{ϵ}_1{ϵ}_2+{\widehat{\gamma }}_1{\widehat{\gamma }}_6,\hfill \\ & & \upsilon ={\upsilon }_2{\upsilon }_3-{\upsilon }_1{\upsilon }_4\ne 0,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & {\eta }_1=1+\frac{({\nu }_1{\nu }_2^{n-1}{ϵ}_2{\beta }_1-{\nu }_1^{n-1}{\beta }_5){ϵ}_1}{\upsilon },{\eta }_2={ϵ}_1{\nu }_1+\frac{({\nu }_1{\nu }_2^{n-1}{ϵ}_2{\beta }_2-{\nu }_1^{n-1}{\beta }_6){ϵ}_1}{\upsilon },{\eta }_3=\frac{({\nu }_1{\nu }_2^{n-1}{ϵ}_2{\beta }_3-{\nu }_1^{n-1}{\beta }_7){ϵ}_1}{\upsilon },\hfill \\ & & {\eta }_4=\frac{({\nu }_1{\nu }_2^{n-1}{ϵ}_2{\beta }_4-{\nu }_1^{n-1}{\beta }_8){ϵ}_1}{\upsilon },{\eta }_5={ϵ}_2{\nu }_2+\frac{({\nu }_2^{n-1}{\beta }_1-{ϵ}_1{\nu }_1^{n-1}{\nu }_2{\beta }_5){ϵ}_2}{\upsilon },{\eta }_6=1+\frac{({\nu }_2^{n-1}{\beta }_2-{ϵ}_1{\nu }_1^{n-1}{\nu }_2{\beta }_6){ϵ}_2}{\upsilon },\hfill \\ & & {\eta }_7=\frac{({\nu }_2^{n-1}{\beta }_3-{ϵ}_1{\nu }_1^{n-1}{\nu }_2{\beta }_7){ϵ}_2}{\upsilon },{\eta }_8=\frac{({\nu }_2^{n-1}{\beta }_4-{ϵ}_1{\nu }_1^{n-1}{\beta }_8{\nu }_2){ϵ}_2}{\upsilon },\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& & {\beta }_1={\widehat{\gamma }}_2({\upsilon }_4-{\upsilon }_3{ϵ}_2{\nu }_2),{\beta }_2=({\upsilon }_4{ϵ}_1{\nu }_1-{\upsilon }_3){\widehat{\gamma }}_2,{\beta }_3=({\upsilon }_3{\xi }_3-{\upsilon }_{4}T){\widehat{\gamma }}_1,{\beta }_4={\widehat{\gamma }}_1({\upsilon }_{3}T-{\upsilon }_4{\xi }_1),\hfill \\ & & {\beta }_5=({\upsilon }_2-{\upsilon }_1{ϵ}_2{\nu }_2){\widehat{\gamma }}_2,{\beta }_6=({\upsilon }_2{ϵ}_1{\nu }_1-{\upsilon }_1){\widehat{\gamma }}_2,{\beta }_7=({\upsilon }_1{\xi }_3-{\upsilon }_{2}T){\widehat{\gamma }}_1,{\beta }_8={\widehat{\gamma }}_1({\upsilon }_{1}T-{\upsilon }_2{\xi }_1),\hfill \end{array}$$

(13)

$$\begin{array}{c}{\kappa }_1={\lambda }_2{\delta }_2+{\mu }_2\sum _{j=1}^{k-2}{\omega }_j,{\kappa }_2={\lambda }_1{\delta }_1+{\mu }_1\sum _{j=1}^{k-2}{\varpi }_j,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}& & {\Lambda }_1\left(\tau \right)=\frac{\tau {\eta }_1}{{\widehat{\gamma }}_1}+\frac{{\tau }^{n-1}{\beta }_1}{\upsilon },{\Lambda }_2\left(\tau \right)=\frac{\tau {\eta }_2}{{\widehat{\gamma }}_1}+\frac{{\tau }^{n-1}{\beta }_2}{\upsilon },{\Lambda }_3\left(\tau \right)=\frac{\tau {\eta }_3}{{\widehat{\gamma }}_1}+\frac{{\tau }^{n-1}{\beta }_3}{\upsilon },{\Lambda }_4\left(\tau \right)=\frac{\tau {\eta }_4}{{\widehat{\gamma }}_1}+\frac{{\tau }^{n-1}{\beta }_4}{\upsilon },\hfill \\ & & {\Lambda }_5\left(\tau \right)=\frac{\tau {\eta }_5}{{\widehat{\gamma }}_1}-\frac{{\tau }^{n-1}{\beta }_5}{\upsilon },{\Lambda }_6\left(\tau \right)=\frac{\tau {\eta }_6}{{\widehat{\gamma }}_1}-\frac{{\tau }^{n-1}{\beta }_6}{\upsilon },{\Lambda }_7\left(\tau \right)=\frac{\tau {\eta }_7}{{\widehat{\gamma }}_1}-\frac{{\tau }^{n-1}{\beta }_7}{\upsilon },{\Lambda }_8\left(\tau \right)=\frac{\tau {\eta }_8}{{\widehat{\gamma }}_1}-\frac{{\tau }^{n-1}{\beta }_8}{\upsilon }.\hfill \end{array}$$

(15)

Proof. 

Solving the FDEs (6) in a standard manner, we get

$$\begin{array}{cc}\hfill y\left(\tau \right)& ={\int }_0^{\tau }\frac{{(\tau -\theta )}^{\varrho -1}}{\Gamma \left(\varrho \right)}\widehat{f}\left(\theta \right)d\theta +a_0+a_1\tau +···+a_{n-1}{\tau }^{n-1},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill z\left(\tau \right)& ={\int }_0^{\tau }\frac{{(\tau -\theta )}^{\varsigma -1}}{\Gamma \left(\varsigma \right)}\widehat{g}\left(\theta \right)d\theta +b_0+b_1\tau +···+b_{n-1}{\tau }^{n-1},\hfill \end{array}$$

(17)

where $a_i,b_i\in \mathbb{R}$, $i=0,1,2,···,n-1$, are arbitrary constants. Using the boundary conditions (2) in (16) and (17) together with notations (9)–(15), we obtain $a_0={\varphi }_1\left(z\right)$, $b_0={\varphi }_2\left(y\right)$, and

$$\begin{array}{cc}\hfill a_1-b_1{ϵ}_1{\nu }_1-b_{n-1}{ϵ}_1{\nu }_1^{n-1}& =\frac{{ϵ}_1}{\Gamma (\varsigma -1)}{\int }_0^{{\nu }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -2}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta ,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill b_1-a_1{\nu }_2{ϵ}_2-a_{n-1}{ϵ}_2{\nu }_2^{n-1}& =\frac{{ϵ}_2}{\Gamma (\varrho -1)}{\int }_0^{{\nu }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -2}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta ,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill a_{1}T+a_{n-1}{T}^{n-1}-b_1{\xi }_1-b_{n-1}{\xi }_2& =\frac{{\lambda }_1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\delta }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -1}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j}{\Gamma \left(\varsigma \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta \hfill \\ & -\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^T{(T-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta \hfill \\ & +{\kappa }_2{\varphi }_2\left(y\right)-{\varphi }_1\left(z\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill b_{1}T+b_{n-1}{T}^{n-1}-a_1{\xi }_3-a_{n-1}{\xi }_4& =\frac{{\lambda }_2}{\Gamma \left(\varrho \right)}{\int }_0^{{\delta }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -1}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j}{\Gamma \left(\varrho \right)}{\int }_0^{{\phi }_j}{({\phi }_j-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta \hfill \\ & -\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^T{(T-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta \hfill \\ & +{\kappa }_1{\varphi }_1\left(z\right)-{\varphi }_2\left(y\right).\hfill \end{array}$$

(21)

Solving the system (18)–(21) for $a_1$, $a_{n-1}$, $b_1$ and $b_{n-1}$, we get

$$\begin{array}{cc}\hfill a_1& =\frac{1}{{\widehat{\gamma }}_1}[\frac{{\eta }_1{ϵ}_1}{\Gamma (\varsigma -1)}{\int }_0^{{\nu }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -2}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +\frac{{\eta }_2{ϵ}_2}{\Gamma (\varrho -1)}{\int }_0^{{\nu }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -2}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\eta }_3(\frac{{\lambda }_1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\delta }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -1}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j}{\Gamma \left(\varsigma \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^T{(T-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta \hfill \\ & +{\kappa }_2{\varphi }_2\left(y\right)-{\varphi }_1\left(z\right))+{\eta }_4(\frac{{\lambda }_2}{\Gamma \left(\varrho \right)}{\int }_0^{{\delta }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -1}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j}{\Gamma \left(\varrho \right)}{\int }_0^{{\phi }_j}{({\phi }_j-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^T{(T-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta \hfill \\ & +{\kappa }_1{\varphi }_1\left(z\right)-{\varphi }_2\left(y\right)\left)\right],\hfill \\ \hfill b_1& =\frac{1}{{\widehat{\gamma }}_1}[\frac{{\eta }_5{ϵ}_1}{\Gamma (\varsigma -1)}{\int }_0^{{\nu }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -2}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +\frac{{\eta }_6{ϵ}_2}{\Gamma (\varrho -1)}{\int }_0^{{\nu }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -2}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\eta }_7(\frac{{\lambda }_1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\delta }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -1}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j}{\Gamma \left(\varsigma \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^T{(T-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta \hfill \\ & +{\kappa }_2{\varphi }_2\left(y\right)-{\varphi }_1\left(z\right))+{\eta }_8(\frac{{\lambda }_2}{\Gamma \left(\varrho \right)}{\int }_0^{{\delta }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -1}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j}{\Gamma \left(\varrho \right)}{\int }_0^{{\phi }_j}{({\phi }_j-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^T{(T-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta \hfill \\ & +{\kappa }_1{\varphi }_1\left(z\right)-{\varphi }_2\left(y\right)\left)\right],\hfill \\ \hfill a_{n-1}& =\frac{1}{\upsilon }[\frac{{\beta }_1{ϵ}_1}{\Gamma (\varsigma -1)}{\int }_0^{{\nu }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -2}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +\frac{{\beta }_2{ϵ}_2}{\Gamma (\varrho -1)}{\int }_0^{{\nu }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -2}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\beta }_3(\frac{{\lambda }_1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\delta }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -1}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j}{\Gamma \left(\varsigma \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^T{(T-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta \hfill \\ & +{\kappa }_2{\varphi }_2\left(y\right)-{\varphi }_1\left(z\right))+{\beta }_4(\frac{{\lambda }_2}{\Gamma \left(\varrho \right)}{\int }_0^{{\delta }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -1}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j}{\Gamma \left(\varrho \right)}{\int }_0^{{\phi }_j}{({\phi }_j-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^T{(T-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta \hfill \\ & +{\kappa }_1{\varphi }_1\left(z\right)-{\varphi }_2\left(y\right)\left)\right],\hfill \\ \hfill b_{n-1}& =\frac{-1}{\upsilon }[\frac{{\beta }_5{ϵ}_1}{\Gamma (\varsigma -1)}{\int }_0^{{\nu }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -2}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +\frac{{\beta }_6{ϵ}_2}{\Gamma (\varrho -1)}{\int }_0^{{\nu }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -2}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\beta }_7(\frac{{\lambda }_1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\delta }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -1}\widehat{g}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j}{\Gamma \left(\varsigma \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^T{(T-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta \hfill \\ & +{\kappa }_2{\varphi }_2\left(y\right)-{\varphi }_1\left(z\right))+{\beta }_8(\frac{{\lambda }_2}{\Gamma \left(\varrho \right)}{\int }_0^{{\delta }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -1}\widehat{f}\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & +{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j}{\Gamma \left(\varrho \right)}{\int }_0^{{\phi }_j}{({\phi }_j-\theta )}^{\varrho -1}\widehat{f}\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^T{(T-\theta )}^{\varsigma -1}\widehat{g}\left(\theta \right)d\theta \hfill \\ & +{\kappa }_1{\varphi }_1\left(z\right)-{\varphi }_2\left(y\right)\left)\right],\hfill \end{array}$$

where ${\widehat{\gamma }}_1$, υ, ${\eta }_i$ and ${\beta }_i$$i=1,2,···,8$, are given by (10)–(13) respectively. Substituting the values of $a_0$, $a_1$, $a_{n-1}$, $b_0$, $b_1$ and $b_{n-1}$, in (16) and (17), we obtain the solutions (7) and (8). □

3. Existence and Uniqueness Results

We define space $\mathcal{G}=\left\{u\right|u\in C(\mathcal{U},\mathbb{R})$, ${{}^C\mathcal{D}}^{{\varrho }_1}u\in C(\mathcal{U},\mathbb{R})\}$ equipped with norm ${\displaystyle {\parallel u\parallel }_{\mathcal{G}}=\parallel u\parallel +\parallel {{}^C\mathcal{D}}^{{\varrho }_1}u\parallel =\underset{\tau \in \mathcal{U}}{sup}\left|u\left(\tau \right)\right|+\underset{\tau \in \mathcal{U}}{sup}\left|{{}^C\mathcal{D}}^{{\varrho }_1}u\left(\tau \right)\right|}$. Furthermore, $\mathcal{H}=\left\{v\right|v\in C(\mathcal{U},\mathbb{R})$, ${{}^C\mathcal{D}}^{{\varsigma }_1}v\in C(\mathcal{U},\mathbb{R})\}$ equipped with norm ${\displaystyle {\parallel v\parallel }_{\mathcal{H}}=\parallel v\parallel +\parallel {{}^C\mathcal{D}}^{{\varsigma }_1}v\parallel =\underset{\tau \in \mathcal{U}}{sup}\left|v\left(\tau \right)\right|+\underset{\tau \in \mathcal{U}}{sup}\left|{{}^C\mathcal{D}}^{{\varsigma }_1}v\left(\tau \right)\right|}$. Obviously $(\mathcal{G},\parallel ·{\parallel }_{\mathcal{G}})$ and $(\mathcal{H},\parallel ·{\parallel }_{\mathcal{H}})$ are Banach spaces, and thus the product space $(\mathcal{G}\times \mathcal{H},\parallel ·{\parallel }_{\mathcal{G}\times \mathcal{H}})$ is a Banach space with norm ${\parallel (u,v)\parallel }_{\mathcal{G}\times \mathcal{H}}={\parallel u\parallel }_{\mathcal{G}}+{\parallel v\parallel }_{\mathcal{H}}$ for $(u,v)\in \mathcal{G}\times \mathcal{H}$.

Using Lemma 1, we consider an operator $\Pi :\mathcal{G}\times \mathcal{H}\to \mathcal{G}\times \mathcal{H}$ as

$$\Pi (u,v)\left(\tau \right)=({\Pi }_1(u,v)\left(\tau \right),{\Pi }_2(u,v)\left(\tau \right)),$$

(22)

where

$$\begin{array}{ccc}\hfill {\Pi }_1(u,v)\left(\tau \right)& =& \frac{1}{\Gamma \left(\varrho \right)}{\int }_0^{\tau }{(\tau -\theta )}^{\varrho -1}{\widehat{\mathcal{S}}}_u\left(\theta \right)d\theta +{\psi }_1\left(v\right)[1+{\kappa }_1{\Lambda }_4\left(\tau \right)-{\Lambda }_3\left(\tau \right)]+{\psi }_2\left(u\right)[{\kappa }_2{\Lambda }_3\left(\tau \right)-{\Lambda }_4\left(\tau \right)]\hfill \\ & & +\frac{{\Lambda }_2\left(\tau \right){ϵ}_2}{\Gamma (\varrho -1)}{\int }_0^{{\nu }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -2}{\widehat{\mathcal{S}}}_u\left(\sigma \right)d\sigma \right)d\theta +\frac{{\Lambda }_1\left(\tau \right){ϵ}_1}{\Gamma (\varsigma -1)}{\int }_0^{{\nu }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -2}{\tilde{\mathcal{S}}}_v\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\Lambda }_4\left(\tau \right)[\frac{{\lambda }_2}{\Gamma \left(\varrho \right)}{\int }_0^{{\delta }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -1}{\widehat{\mathcal{S}}}_u\left(\sigma \right)d\sigma \right)d\theta +{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j}{\Gamma \left(\varrho \right)}{\int }_0^{{\phi }_j}{({\phi }_j-\theta )}^{\varrho -1}{\widehat{\mathcal{S}}}_u\left(\theta \right)d\theta \hfill \\ & & -\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^T{(T-\theta )}^{\varsigma -1}{\tilde{\mathcal{S}}}_v\left(\theta \right)d\theta ]+{\Lambda }_3\left(\tau \right)[\frac{{\lambda }_1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\delta }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -1}{\tilde{\mathcal{S}}}_v\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j}{\Gamma \left(\varsigma \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varsigma -1}{\tilde{\mathcal{S}}}_v\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^T{(T-\theta )}^{\varrho -1}{\widehat{\mathcal{S}}}_u\left(\theta \right)d\theta ],\hfill \end{array}$$

(23)

and

$$\begin{array}{ccc}\hfill {\Pi }_2(u,v)\left(\tau \right)& =& \frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^{\tau }{(\tau -\theta )}^{\varsigma -1}{\tilde{\mathcal{S}}}_v\left(\theta \right)d\theta +{\psi }_2\left(u\right)[1+{\kappa }_2{\Lambda }_7\left(\tau \right)-{\Lambda }_8\left(\tau \right)]+{\psi }_1\left(v\right)[{\kappa }_1{\Lambda }_8\left(\tau \right)-{\Lambda }_7\left(\tau \right)]\hfill \\ & & +\frac{{\Lambda }_5\left(\tau \right){ϵ}_1}{\Gamma (\varsigma -1)}{\int }_0^{{\nu }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -2}{\tilde{\mathcal{S}}}_v\left(\sigma \right)d\sigma \right)d\theta +\frac{{\Lambda }_6\left(\tau \right){ϵ}_2}{\Gamma (\varrho -1)}{\int }_0^{{\nu }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -2}{\widehat{\mathcal{S}}}_u\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\Lambda }_7\left(\tau \right)[\frac{{\lambda }_1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\delta }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -1}{\tilde{\mathcal{S}}}_v\left(\sigma \right)d\sigma \right)d\theta +{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j}{\Gamma \left(\varsigma \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varsigma -1}{\tilde{\mathcal{S}}}_v\left(\theta \right)d\theta \hfill \\ & & -\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^T{(T-\theta )}^{\varrho -1}{\widehat{\mathcal{S}}}_u\left(\theta \right)d\theta ]+{\Lambda }_8\left(\tau \right)[\frac{{\lambda }_2}{\Gamma \left(\varrho \right)}{\int }_0^{{\delta }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -1}{\widehat{\mathcal{S}}}_u\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j}{\Gamma \left(\varrho \right)}{\int }_0^{{\phi }_j}{({\phi }_j-\theta )}^{\varrho -1}{\widehat{\mathcal{S}}}_u\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^T{(T-\theta )}^{\varsigma -1}{\tilde{\mathcal{S}}}_v\left(\theta \right)d\theta ].\hfill \end{array}$$

(24)

where

$$\begin{array}{ccc}\hfill {\widehat{\mathcal{S}}}_u\left(\tau \right)& =& f(\tau ,u\left(\tau \right),v\left(\tau \right),{{}^C\mathcal{D}}^{{\varsigma }_1}v\left(\tau \right),{\mathcal{I}}^{\xi }v\left(\tau \right)),\tau \in \mathcal{U},\hfill \\ \hfill {\tilde{\mathcal{S}}}_v\left(\tau \right)& =& g(\tau ,u\left(\tau \right),{{}^C\mathcal{D}}^{{\varrho }_1}u\left(\tau \right),{\mathcal{I}}^{\zeta }u\left(\tau \right),v\left(\tau \right)),\tau \in \mathcal{U},\hfill \end{array}$$

and ${\Lambda }_i(i=1,2,···,8)$ are given by (15). Suitable for computation, we represent

$$\begin{array}{ccc}\hfill {\Delta }_1& =& \frac{1}{\Gamma (\varrho +1)}\left[T^{\varrho }(1+{\overline{\Lambda }}_3)+{\overline{\Lambda }}_2{ϵ}_2{\nu }_2^{\varrho }+{\overline{\Lambda }}_4\left({\lambda }_2\frac{{\delta }_2^{\varrho +1}}{(\varrho +1)}+{\mu }_2\sum _{j=1}^{k-2}{\omega }_j{\phi }_j^{\varrho }\right)\right],\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill {\Delta }_2& =& \frac{1}{\Gamma (\varsigma +1)}\left[T^{\varsigma }{\overline{\Lambda }}_4+{\overline{\Lambda }}_1{ϵ}_1{\nu }_1^{\varsigma }+{\overline{\Lambda }}_3\left({\lambda }_1\frac{{\delta }_1^{\varsigma +1}}{(\varsigma +1)}+{\mu }_1\sum _{j=1}^{k-2}{\varpi }_j{\vartheta }_j^{\varsigma }\right)\right],\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill {\Delta }_3& =& \frac{1}{\Gamma (\varrho +1)}\left[T^{\varrho }{\overline{\Lambda }}_7+{\overline{\Lambda }}_6{ϵ}_2{\nu }_2^{\varrho }+{\overline{\Lambda }}_8\left({\lambda }_2\frac{{\delta }_2^{\varrho +1}}{(\varrho +1)}+{\mu }_2\sum _{j=1}^{k-2}{\omega }_j{\phi }_j^{\varrho }\right)\right],\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill {\Delta }_4& =& \frac{1}{\Gamma (\varsigma +1)}\left[T^{\varsigma }(1+{\overline{\Lambda }}_8)+{\overline{\Lambda }}_5{ϵ}_1{\nu }_1^{\varsigma }+{\overline{\Lambda }}_7\left({\lambda }_1\frac{{\delta }_1^{\varsigma +1}}{(\varsigma +1)}+{\mu }_1\sum _{j=1}^{k-2}{\varpi }_j{\vartheta }_j^{\varsigma }\right)\right],\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill {\widehat{\Delta }}_1& =& \frac{1}{\Gamma (\varrho +1)}\left[\varrho T^{\varrho -1}+{\overline{\Lambda }}_3^{{}^{\prime }}{T}^{\varrho }+{\overline{\Lambda }}_2^{{}^{\prime }}{ϵ}_2{\nu }_2^{\varrho }+{\overline{\Lambda }}_4^{{}^{\prime }}\left({\lambda }_2\frac{{\delta }_2^{\varrho +1}}{(\varrho +1)}+{\mu }_2\sum _{j=1}^{k-2}{\omega }_j{\phi }_j^{\varrho }\right)\right],\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill {\widehat{\Delta }}_2& =& \frac{1}{\Gamma (\varsigma +1)}\left[T^{\varsigma }{\overline{\Lambda }}_4^{{}^{\prime }}+{\overline{\Lambda }}_1^{{}^{\prime }}{ϵ}_1{\nu }_1^{\varsigma }+{\overline{\Lambda }}_3^{{}^{\prime }}\left({\lambda }_1\frac{{\delta }_1^{\varsigma +1}}{(\varsigma +1)}+{\mu }_1\sum _{j=1}^{k-2}{\varpi }_j{\vartheta }_j^{\varsigma }\right)\right],\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill {\widehat{\Delta }}_3& =& \frac{1}{\Gamma (\varrho +1)}\left[T^{\varrho }{\overline{\Lambda }}_7^{{}^{\prime }}+{\overline{\Lambda }}_6^{{}^{\prime }}{ϵ}_2{\nu }_2^{\varrho }+{\overline{\Lambda }}_8^{{}^{\prime }}\left({\lambda }_2\frac{{\delta }_2^{\varrho +1}}{(\varrho +1)}+{\mu }_2\sum _{j=1}^{k-2}{\omega }_j{\phi }_j^{\varrho }\right)\right],\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill {\widehat{\Delta }}_4& =& \frac{1}{\Gamma (\varsigma +1)}\left[\varsigma T^{\varsigma -1}+{\overline{\Lambda }}_8^{{}^{\prime }}{T}^{\varsigma }+{\overline{\Lambda }}_5^{{}^{\prime }}{ϵ}_1{\nu }_1^{\varsigma }+{\overline{\Lambda }}_7^{{}^{\prime }}\left({\lambda }_1\frac{{\delta }_1^{\varsigma +1}}{(\varsigma +1)}+{\mu }_1\sum _{j=1}^{k-2}{\varpi }_j{\vartheta }_j^{\varsigma }\right)\right],\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill {\Phi }_1& =& \left({\Delta }_1+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}{\widehat{\Delta }}_1+{\Delta }_3+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\widehat{\Delta }}_3\right)r_0+\left({\Delta }_2+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}{\widehat{\Delta }}_2+{\Delta }_4+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\widehat{\Delta }}_4\right)s_0,\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill {\Phi }_2& =& \left({\Delta }_1+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}{\widehat{\Delta }}_1+{\Delta }_3+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\widehat{\Delta }}_3\right)r_1\hfill \\ & & +\left({\Delta }_2+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}{\widehat{\Delta }}_2+{\Delta }_4+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\widehat{\Delta }}_4\right)\left(max\{s_1,s_2\}+\frac{s_3{T}^{\zeta }}{\Gamma (\zeta +1)}\right)\hfill \\ & & +\left\{{\kappa }_2{\overline{\Lambda }}_3+{\overline{\Lambda }}_4+\frac{T^{1-{\varrho }_1}({\kappa }_2{\overline{\Lambda }}_3^{{}^{\prime }}+{\overline{\Lambda }}_4^{{}^{\prime }})}{\Gamma (2-{\varrho }_1)}+1+{\kappa }_2{\overline{\Lambda }}_7+{\overline{\Lambda }}_8+\frac{T^{1-{\varsigma }_1}({\kappa }_2{\overline{\Lambda }}_7^{{}^{\prime }}+{\overline{\Lambda }}_8^{{}^{\prime }})}{\Gamma (2-{\varsigma }_1)}\right\}{\mathcal{W}}_2,\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill {\Phi }_3& =& \left({\Delta }_1+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}{\widehat{\Delta }}_1+{\Delta }_3+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\widehat{\Delta }}_3\right)\left(max\{r_2,r_3\}+\frac{r_4{T}^{\xi }}{\Gamma (\xi +1)}\right)\hfill \\ & & +\left({\Delta }_2+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}{\widehat{\Delta }}_2+{\Delta }_4+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\widehat{\Delta }}_4\right)s_4\hfill \\ & & +\left\{1+{\kappa }_1{\overline{\Lambda }}_4+{\overline{\Lambda }}_3+\frac{T^{1-{\varrho }_1}({\kappa }_1{\overline{\Lambda }}_4^{{}^{\prime }}+{\overline{\Lambda }}_3^{{}^{\prime }})}{\Gamma (2-{\varrho }_1)}+{\kappa }_1{\overline{\Lambda }}_8+{\overline{\Lambda }}_7+\frac{T^{1-{\varsigma }_1}({\kappa }_1{\overline{\Lambda }}_8^{{}^{\prime }}+{\overline{\Lambda }}_7^{{}^{\prime }})}{\Gamma (2-{\varsigma }_1)}\right\}{\mathcal{W}}_1,\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill {\Psi }_1& =& {\Delta }_1{\iota }_1{\mathcal{V}}_1+{\Delta }_2{\iota }_2{\mathcal{V}}_2+(1+{\kappa }_1{\overline{\Lambda }}_4+{\overline{\Lambda }}_3){\mathcal{V}}_1+({\kappa }_2{\overline{\Lambda }}_3+{\overline{\Lambda }}_4){\mathcal{V}}_2,\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill {\Psi }_2& =& {\widehat{\Delta }}_1{\iota }_1{\mathcal{V}}_1+{\widehat{\Delta }}_2{\iota }_2{\mathcal{V}}_2+({\kappa }_1{\overline{\Lambda }}_4^{{}^{\prime }}+{\overline{\Lambda }}_3^{{}^{\prime }}){\mathcal{V}}_1+({\kappa }_2{\overline{\Lambda }}_3^{{}^{\prime }}+{\overline{\Lambda }}_4^{{}^{\prime }}){\mathcal{V}}_2,\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill {\Psi }_3& =& {\Delta }_4{\iota }_2{\mathcal{V}}_2+{\Delta }_3{\iota }_1{\mathcal{V}}_1+(1+{\kappa }_2{\overline{\Lambda }}_7+{\overline{\Lambda }}_8){\mathcal{V}}_2+({\kappa }_1{\overline{\Lambda }}_8+{\overline{\Lambda }}_7){\mathcal{V}}_1,\hfill \end{array}$$

(38)

$$\begin{array}{ccc}\hfill {\Psi }_4& =& {\widehat{\Delta }}_4{\iota }_2{\mathcal{V}}_2+{\widehat{\Delta }}_3{\iota }_1{\mathcal{V}}_1+({\kappa }_2{\overline{\Lambda }}_7^{{}^{\prime }}+{\overline{\Lambda }}_8^{{}^{\prime }}){\mathcal{V}}_2+({\kappa }_1{\overline{\Lambda }}_8^{{}^{\prime }}+{\overline{\Lambda }}_7^{{}^{\prime }}){\mathcal{V}}_1,\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill {\mathcal{P}}_1& =& {\Delta }_1{\mathcal{T}}_1+{\Delta }_2{\mathcal{T}}_2,{\mathcal{P}}_2={\widehat{\Delta }}_1{\mathcal{T}}_1+{\widehat{\Delta }}_2{\mathcal{T}}_2,{\mathcal{P}}_3={\Delta }_4{\mathcal{T}}_2+{\Delta }_3{\mathcal{T}}_1,{\mathcal{P}}_4={\widehat{\Delta }}_4{\mathcal{T}}_2+{\widehat{\Delta }}_3{\mathcal{T}}_1,\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill \overline{\mathcal{P}}& =& {\mathcal{P}}_1+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}{\mathcal{P}}_2,\widehat{\mathcal{P}}={\mathcal{P}}_3+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\mathcal{P}}_4,\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill \Psi & =& {\Psi }_1+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}{\Psi }_2,\widehat{\Psi }={\Psi }_3+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\Psi }_4\hfill \\ \hfill {\mathcal{T}}_1& =& \underset{\tau \in \mathcal{U}}{sup}f(\tau ,0,0,0,0)<\infty ,{\mathcal{T}}_2=\underset{\tau \in \mathcal{U}}{sup}g(\tau ,0,0,0,0)<\infty ,\hfill \end{array}$$

(42)

$$\begin{array}{ccc}\hfill {\iota }_1& =& 1+\frac{T^{\xi }}{\Gamma (\xi +1)},{\iota }_2=1+\frac{T^{\zeta }}{\Gamma (\zeta +1)},\hfill \end{array}$$

(43)

where ${\displaystyle {\overline{\Lambda }}_i=\underset{\tau \in \mathcal{H}}{max}\left|{\Lambda }_i\left(\tau \right)\right|}$ and ${\displaystyle {\overline{\Lambda }}_i^{{}^{\prime }}=\underset{\tau \in \mathcal{H}}{max}\left|{\Lambda }_i^{{}^{\prime }}\left(\tau \right)\right|}$$i=1,2,···,8$. We need the following assumptions in the forthcoming analysis: f, $g:\mathcal{U}\times {\mathbb{R}}^4\to \mathbb{R}$ and ${\psi }_1$, ${\psi }_2:\mathcal{C}(\mathcal{U},\mathbb{R})\to \mathbb{R}$ are continuous functions ${\psi }_1\left(0\right)={\psi }_2\left(0\right)=0$.

$\left({\mathcal{F}}_1\right)$

There exist positive constants $r_i$ and $s_i\ge 0$, $r_0>0$, $s_0>0$, $\forall w_i\in \mathbb{R}$, $i=1,2,3,4$.

$$\begin{array}{ccc}\hfill \left|f\right(\tau ,w_1,w_2,w_3,w_4\left)\right|& \le & r_0+r_1|w_1|+r_2|w_2|+r_3|w_3|+r_4\left|w_4\right|,\hfill \\ \hfill \left|g\right(\tau ,w_1,w_2,w_3,w_4\left)\right|& \le & s_0+s_1|w_1|+s_2|w_2|+s_3|w_3|+s_4\left|w_4\right|.\hfill \end{array}$$

$\left({\mathcal{F}}_2\right)$

There exists positive constants ${\mathcal{W}}_1,{\mathcal{W}}_2>0$,

$$|{\psi }_1\left(v\right)|\le {\mathcal{W}}_1\parallel v\parallel ,|{\psi }_2\left(u\right)|\le {\mathcal{W}}_2\parallel u\parallel ,\forall u,v\in \mathcal{C}(\mathcal{U},\mathbb{R}).$$

$\left({\mathcal{F}}_3\right)$

There exist positive constants ${\mathcal{V}}_i,i=1,2$, ∀$\tau \in \mathcal{U}$ and $r_i,s_i\in \mathbb{R}(i=1,2,3,4)$, we have

$$\begin{array}{ccc}\hfill |f(\tau ,r_1,r_2,r_3,r_4)-f(\tau ,s_1,s_2,s_3,s_4)|& \le & {\mathcal{V}}_1\left(|r_1-s_1|+|r_2-s_2|+|r_3-s_3|+|r_4-s_4|\right),\hfill \\ \hfill |g(\tau ,r_1,r_2,r_3,r_4)-g(\tau ,s_1,s_2,s_3,s_4)|& \le & {\mathcal{V}}_2\left(|r_1-s_1|+|r_2-s_2|+|r_3-s_3|+|r_4-s_4|\right).\hfill \end{array}$$

$\left({\mathcal{F}}_4\right)$

There exist positive constants ${\mathcal{V}}_i(i=1,2)$ such that

$$\begin{array}{c}\hfill |{\psi }_1\left(r_1\right)-{\psi }_1\left(r_2\right)|\le {\mathcal{V}}_1\parallel r_1-r_2\parallel ,|{\psi }_2\left(r_1\right)-{\psi }_2\left(r_2\right)|\le {\mathcal{V}}_2\parallel r_1-r_2\parallel ,\end{array}$$

$\forall r_1,r_2\in \mathbb{R}$.

Theorem 1.

Assume that $\left({\mathcal{F}}_1\right)$ and $\left({\mathcal{F}}_2\right)$ hold. Further, if $\widehat{\Phi }=min\{{\Phi }_2,{\Phi }_3\}<1$, then the problem (1) and (2) has at least one solution on $\mathcal{U}$.

Proof. 

In the first step, we show that operator $\Pi :\mathcal{G}\times \mathcal{H}\to \mathcal{G}\times \mathcal{H}$ is completely continuous. Operator $\Pi$ is continuous by the continuity of f, g, ${\psi }_1$, ${\psi }_2$ functions. Let $\Omega \subset \mathcal{G}\times \mathcal{H}$ be bounded. Then, ∃ positive constants ${\mathcal{L}}_f$ and ${\mathcal{L}}_g$ such that

$$\begin{array}{ccc}\hfill |{\widehat{\mathcal{S}}}_u\left(\tau \right)|& =& |f(\tau ,u\left(\tau \right),v\left(\tau \right),{{}^C\mathcal{D}}^{{\varsigma }_1}v\left(\tau \right),{\mathcal{I}}^{\xi }v\left(\tau \right))|\le {\mathcal{L}}_f\hfill \\ \hfill |{\tilde{\mathcal{S}}}_v\left(\tau \right)|& =& |g(\tau ,u\left(\tau \right),{{}^C\mathcal{D}}^{{\varrho }_1}u\left(\tau \right),{\mathcal{I}}^{\zeta }u\left(\tau \right),v\left(\tau \right))|\le {\mathcal{L}}_g,\hfill \end{array}$$

$\forall (u,v)\in \Omega$, and constants ${\mathcal{L}}_{{\psi }_1}$, ${\mathcal{L}}_{{\psi }_2}>0$ such that $|{\psi }_1\left(v\right)|\le {\mathcal{L}}_{{\psi }_1}$, $|{\psi }_2\left(u\right)|\le {\mathcal{L}}_{{\psi }_2}$, $\forall u$, $v\in \mathcal{C}(\mathcal{U},\mathbb{R})$. Then, for any $(u,v)\in \Omega$, we have

$$\begin{array}{ccc}\hfill |{\Pi }_1(u,v)\left(\tau \right)|& \le & \frac{1}{\Gamma \left(\varrho \right)}{\int }_0^{\tau }{(\tau -\theta )}^{\varrho -1}|{\widehat{\mathcal{S}}}_u\left(\theta \right)|d\theta +|{\psi }_1\left(v\right)|[1+{\kappa }_1|{\Lambda }_4\left(\tau \right)|+|{\Lambda }_3\left(\tau \right)|]+|{\psi }_2\left(u\right)|[{\kappa }_2|{\Lambda }_3\left(\tau \right)|+|{\Lambda }_4\left(\tau \right)|]\hfill \\ & & +\frac{|{\Lambda }_2\left(\tau \right)|{ϵ}_2}{\Gamma (\varrho -1)}{\int }_0^{{\nu }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -2}|{\widehat{\mathcal{S}}}_u\left(\sigma \right)d\sigma \right)d\theta +\frac{|{\Lambda }_1\left(\tau \right)|{ϵ}_1}{\Gamma (\varsigma -1)}{\int }_0^{{\nu }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -2}|{\tilde{\mathcal{S}}}_v\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +|{\Lambda }_4\left(\tau \right)|[\frac{{\lambda }_2}{\Gamma \left(\varrho \right)}{\int }_0^{{\delta }_2}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -1}|{\widehat{\mathcal{S}}}_u\left(\sigma \right)d\sigma \right)d\theta +{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j}{\Gamma \left(\varrho \right)}{\int }_0^{{\phi }_j}{({\phi }_j-\theta )}^{\varrho -1}|{\widehat{\mathcal{S}}}_u\left(\theta \right)|d\theta \hfill \\ & & +\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^T{(T-\theta )}^{\varsigma -1}|{\tilde{\mathcal{S}}}_v\left(\theta \right)|d\theta ]+|{\Lambda }_3\left(\tau \right)|[\frac{{\lambda }_1}{\Gamma \left(\varsigma \right)}{\int }_0^{{\delta }_1}\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -1}|{\tilde{\mathcal{S}}}_v\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j}{\Gamma \left(\varsigma \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varsigma -1}|{\tilde{\mathcal{S}}}_v\left(\theta \right)|d\theta +\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^T{(T-\theta )}^{\varrho -1}|{\widehat{\mathcal{S}}}_u\left(\theta \right)|d\theta ]\hfill \\ & \le & {\Delta }_1{\mathcal{L}}_f+{\Delta }_2{\mathcal{L}}_g+[1+{\kappa }_1\overline{{\Lambda }_4}+\overline{{\Lambda }_3}]{\mathcal{L}}_{{\psi }_1}+[{\kappa }_2\overline{{\Lambda }_3}+\overline{{\Lambda }_4}]{\mathcal{L}}_{{\psi }_2}.\hfill \end{array}$$

Likewise, we get $|{\Pi }_1^{{}^{\prime }}(u,v)\left(\tau \right)|\le {\widehat{\Delta }}_1{\mathcal{L}}_f+{\widehat{\Delta }}_2{\mathcal{L}}_g+[{\kappa }_1{\overline{\Lambda }}_4^{{}^{\prime }}+{\overline{\Lambda }}_3^{{}^{\prime }}]{\mathcal{L}}_{{\psi }_1}+[{\kappa }_2{\overline{\Lambda }}_3^{{}^{\prime }}+{\overline{\Lambda }}_4^{{}^{\prime }}]{\mathcal{L}}_{{\psi }_2}$, which implies that $|{{}^C\mathcal{D}}^{{\varrho }_1}{\Pi }_1(u,v)\left(\tau \right)|\le \frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}({\widehat{\Delta }}_1{\mathcal{L}}_f+{\widehat{\Delta }}_2{\mathcal{L}}_g+[{\kappa }_1{\overline{\Lambda }}_4^{{}^{\prime }}+{\overline{\Lambda }}_3^{{}^{\prime }}]{\mathcal{L}}_{{\psi }_1}+[{\kappa }_2{\overline{\Lambda }}_3^{{}^{\prime }}+$${\overline{\Lambda }}_4^{{}^{\prime }}\left]{\mathcal{L}}_{{\psi }_2}\right)$. Thus, we have

$$\begin{array}{ccc}\hfill \parallel {\Pi }_1{(u,v)\parallel }_{\mathcal{G}}& \le & {\Delta }_1{\mathcal{L}}_f+{\Delta }_2{\mathcal{L}}_g+[1+{\kappa }_1\overline{{\Lambda }_4}+\overline{{\Lambda }_3}]{\mathcal{L}}_{{\psi }_1}+[{\kappa }_2\overline{{\Lambda }_3}+\overline{{\Lambda }_4}]{\mathcal{L}}_{{\psi }_2}\hfill \\ & & +\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}({\widehat{\Delta }}_1{\mathcal{L}}_f+{\widehat{\Delta }}_2{\mathcal{L}}_g+[{\kappa }_1{\overline{\Lambda }}_4^{{}^{\prime }}+{\overline{\Lambda }}_3^{{}^{\prime }}]{\mathcal{L}}_{{\psi }_1}+[{\kappa }_2{\overline{\Lambda }}_3^{{}^{\prime }}+{\overline{\Lambda }}_4^{{}^{\prime }}]{\mathcal{L}}_{{\psi }_2}).\hfill \end{array}$$

(44)

We obtain that, equivalently, $|{\Pi }_2(u,v)\left(\tau \right)|\le {\Delta }_4{\mathcal{L}}_g+{\Delta }_3{\mathcal{L}}_f+[1+{\kappa }_2{\overline{\Lambda }}_7+{\overline{\Lambda }}_8]{\mathcal{L}}_{{\psi }_2}+[{\kappa }_1{\overline{\Lambda }}_8+{\overline{\Lambda }}_7]{\mathcal{L}}_{{\psi }_1}$, and $|{\Pi }_2^{{}^{\prime }}(u,v)\left(\tau \right)|\le {\widehat{\Delta }}_4{\mathcal{L}}_g+{\widehat{\Delta }}_3{\mathcal{L}}_f+[{\kappa }_2{\overline{\Lambda }}_7^{{}^{\prime }}+{\overline{\Lambda }}_8^{{}^{\prime }}]{\mathcal{L}}_{{\psi }_2}+[{\kappa }_1{\overline{\Lambda }}_8^{{}^{\prime }}+{\overline{\Lambda }}_7^{{}^{\prime }}]{\mathcal{L}}_{{\psi }_1}$, which implies that $|{{}^C\mathcal{D}}^{{\varsigma }_1}{\Pi }_2(u,v)\left(\tau \right)|\le \frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}({\widehat{\Delta }}_4{\mathcal{L}}_g+{\widehat{\Delta }}_3{\mathcal{L}}_f+[{\kappa }_2{\overline{\Lambda }}_7^{{}^{\prime }}+{\overline{\Lambda }}_8^{{}^{\prime }}]{\mathcal{L}}_{{\psi }_2}+[{\kappa }_1{\overline{\Lambda }}_8^{{}^{\prime }}+{\overline{\Lambda }}_7^{{}^{\prime }}]{\mathcal{L}}_{{\psi }_1})$. Thus, we have

$$\begin{array}{ccc}\hfill \parallel {\Pi }_2{(u,v)\parallel }_{\mathcal{H}}& \le & {\Delta }_4{\mathcal{L}}_g+{\Delta }_3{\mathcal{L}}_f+[1+{\kappa }_2{\overline{\Lambda }}_7+{\overline{\Lambda }}_8]{\mathcal{L}}_{{\psi }_2}+[{\kappa }_1{\overline{\Lambda }}_8+{\overline{\Lambda }}_7]{\mathcal{L}}_{{\psi }_1}\hfill \\ & & +\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}({\widehat{\Delta }}_4{\mathcal{L}}_g+{\widehat{\Delta }}_3{\mathcal{L}}_f+[{\kappa }_2{\overline{\Lambda }}_7^{{}^{\prime }}+{\overline{\Lambda }}_8^{{}^{\prime }}]{\mathcal{L}}_{{\psi }_2}+[{\kappa }_1{\overline{\Lambda }}_8^{{}^{\prime }}+{\overline{\Lambda }}_7^{{}^{\prime }}]{\mathcal{L}}_{{\psi }_1}).\hfill \end{array}$$

(45)

Thus, $\Pi$ is uniformly bounded by (44) and (45). Operator $\Pi$ must be shown to be equicontinuous.

For ${\tau }_1,{\tau }_2\in \mathcal{U}$ with ${\tau }_1<{\tau }_2$, we have

$$\begin{array}{ccc}& & |{\Pi }_1(u,v)\left({\tau }_2\right)-{\Pi }_1(u,v)\left({\tau }_1\right)|\hfill \\ & \le & \frac{{\mathcal{L}}_f}{\Gamma (\varrho +1)}[{({\tau }_2-{\tau }_1)}^{\varrho }+({\tau }_2^{\varrho }-{\tau }_1^{\varrho })]+\left|{\psi }_1\left(v\right)\right|\left[{\kappa }_1\left(\right|{\Lambda }_4\left({\tau }_2\right)-{\Lambda }_4\left({\tau }_1\right)\left|\right)+\left(\right|{\Lambda }_3\left({\tau }_2\right)-{\Lambda }_3\left({\tau }_1\right)\left|\right)\right]\hfill \\ & & +|{\psi }_2\left(u\right)|\left[{\kappa }_2\left(\right|{\Lambda }_3\left({\tau }_2\right)-{\Lambda }_3\left({\tau }_1\right)\left|\right)+\left(\right|{\Lambda }_4\left({\tau }_2\right)-{\Lambda }_4\left({\tau }_1\right)\left|\right)\right]+\frac{\left(\right|{\Lambda }_2\left({\tau }_2\right)-{\Lambda }_2\left({\tau }_1\right)\left|\right){ϵ}_2{\nu }_2^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}\hfill \\ & & +\frac{\left(\right|{\Lambda }_1\left({\tau }_2\right)-{\Lambda }_1\left({\tau }_1\right)\left|\right){ϵ}_1{\nu }_1^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}+\left(\right|{\Lambda }_4\left({\tau }_2\right)-{\Lambda }_4\left({\tau }_1\right)\left|\right)\left[\frac{{\lambda }_2{\delta }_2^{\varrho +1}{\mathcal{L}}_f}{\Gamma (\varrho +2)}+{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j{\left({\phi }_j\right)}^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}+\frac{T^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}\right]\hfill \\ & & +({\Lambda }_3\left({\tau }_2\right)-{\Lambda }_3\left({\tau }_1\right))\left[\frac{{\lambda }_1{\delta }_1^{\varsigma +1}{\mathcal{L}}_g}{\Gamma (\varsigma +2)}+{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j{\vartheta }_j^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}+\frac{T^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}\right],\hfill \end{array}$$

and

$$\begin{array}{ccc}& & |{\Pi }_1^{{}^{\prime }}(u,v)\left({\tau }_2\right)-{\Pi }_1^{{}^{\prime }}(u,v)\left({\tau }_1\right)|\hfill \\ & \le & \left|{\int }_0^{{\tau }_1}\frac{[{({\tau }_2-\theta )}^{\varrho -2}-{({\tau }_1-\theta )}^{\varrho -2}]}{\Gamma (\varrho -1)}\times f(\theta ,u\left(\theta \right),v\left(\theta \right),{{}^C\mathcal{D}}^{{\varsigma }_1}v\left(\theta \right),{\mathcal{I}}^{\xi }v\left(\theta \right))d\theta \right|\hfill \\ & & +\left|{\int }_{{\tau }_1}^{{\tau }_2}\frac{{({\tau }_2-\theta )}^{\varrho -2}}{\Gamma (\varrho -1)}f(\theta ,u\left(\theta \right),v\left(\theta \right),{{}^C\mathcal{D}}^{{\varsigma }_1}v\left(\theta \right),{\mathcal{I}}^{\xi }v\left(\theta \right))d\theta \right|+\left|{\psi }_1\left(v\right)\right|\left[{\kappa }_1\left(\right|{\Lambda }_4^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_4^{{}^{\prime }}\left({\tau }_1\right)\left|\right)+\left(\right|{\Lambda }_3^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_3^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\right]\hfill \\ & & +|{\psi }_2\left(u\right)|\left[{\kappa }_2\left(\right|{\Lambda }_3^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_3^{{}^{\prime }}\left({\tau }_1\right)\left|\right)+\left(\right|{\Lambda }_4^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_4^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\right]+\frac{\left(\right|{\Lambda }_2^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_2^{{}^{\prime }}\left({\tau }_1\right)\left|\right){ϵ}_2{\nu }_2^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}\hfill \\ & & +\frac{\left(\right|{\Lambda }_1^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_1^{{}^{\prime }}\left({\tau }_1\right)\left|\right){ϵ}_1{\nu }_1^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}+\left(\right|{\Lambda }_4^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_4^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\left[\frac{{\lambda }_2{\delta }_2^{\varrho +1}{\mathcal{L}}_f}{\Gamma (\varrho +2)}+{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j{\left({\phi }_j\right)}^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}+\frac{T^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}\right]\hfill \\ & & +\left(\right|{\Lambda }_3^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_3^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\left[\frac{{\lambda }_1{\delta }_1^{\varsigma +1}{\mathcal{L}}_g}{\Gamma (\varsigma +2)}+{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j{\vartheta }_j^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}+\frac{T^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}\right].\hfill \end{array}$$

Thus, we have

$$\begin{array}{ccc}& & |{{}^C\mathcal{D}}^{{\varrho }_1}{\Pi }_1(u,v)\left({\tau }_2\right)-{{}^C\mathcal{D}}^{{\varrho }_1}{\Pi }_1(u,v)\left({\tau }_1\right)|\hfill \\ & \le & \frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}\{\frac{{\mathcal{L}}_f}{\Gamma \left(\varrho \right)}[{({\tau }_2-{\tau }_1)}^{\varrho -1}+({\tau }_2^{\varrho -1}-{\tau }_1^{\varrho -1})]+\left|{\psi }_1\left(v\right)\right|\left[{\kappa }_1\left(\right|{\Lambda }_4^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_4^{{}^{\prime }}\left({\tau }_1\right)\left|\right)+\left(\right|{\Lambda }_3^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_3^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\right]\hfill \\ & & +|{\psi }_2\left(u\right)|\left[{\kappa }_2\left(\right|{\Lambda }_3^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_3^{{}^{\prime }}\left({\tau }_1\right)\left|\right)+\left(\right|{\Lambda }_4^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_4^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\right]+\frac{\left(\right|{\Lambda }_2^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_2^{{}^{\prime }}\left({\tau }_1\right)\left|\right){ϵ}_2{\nu }_2^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}\hfill \\ & & +\frac{\left(\right|{\Lambda }_1^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_1^{{}^{\prime }}\left({\tau }_1\right)\left|\right){ϵ}_1{\nu }_1^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}+\left(\right|{\Lambda }_4^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_4^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\left[\frac{{\lambda }_2{\delta }_2^{\varrho +1}{\mathcal{L}}_f}{\Gamma (\varrho +2)}+{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j{\left({\phi }_j\right)}^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}+\frac{T^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}\right]\hfill \\ & & +\left(\right|{\Lambda }_3^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_3^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\left[\frac{{\lambda }_1{\delta }_1^{\varsigma +1}{\mathcal{L}}_g}{\Gamma (\varsigma +2)}+{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j{\vartheta }_j^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}+\frac{T^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}\right]\}.\hfill \end{array}$$

Thus, we obtain $\parallel {\Pi }_1(u,v)\left({\tau }_2\right)-{\Pi }_1(u,v)\left({\tau }_1\right){\parallel }_{\mathcal{G}}\to 0$ independent of u and v as ${\tau }_2\to {\tau }_1$. According to the above, we get

$$\begin{array}{ccc}& & |{\Pi }_2(u,v)\left({\tau }_2\right)-{\Pi }_2(u,v)\left({\tau }_1\right)|\hfill \\ & \le & \frac{{\mathcal{L}}_g}{\Gamma (\varsigma +1)}[{({\tau }_2-{\tau }_1)}^{\varsigma }+({\tau }_2^{\varsigma }-{\tau }_1^{\varsigma })]+\left|{\psi }_2\left(u\right)\right|\left[{\kappa }_2\left(\right|{\Lambda }_7\left({\tau }_2\right)-{\Lambda }_7\left({\tau }_1\right)\left|\right)+\left(\right|{\Lambda }_8\left({\tau }_2\right)-{\Lambda }_8\left({\tau }_1\right)\left|\right)\right]\hfill \\ & & +|{\psi }_1\left(v\right)|\left[{\kappa }_1\left(\right|{\Lambda }_8\left({\tau }_2\right)-{\Lambda }_8\left({\tau }_1\right)\left|\right)+\left(\right|{\Lambda }_7\left({\tau }_2\right)-{\Lambda }_7\left({\tau }_1\right)\left|\right)\right]+\frac{\left(\right|{\Lambda }_5\left({\tau }_2\right)-{\Lambda }_5\left({\tau }_1\right)\left|\right){ϵ}_1{\nu }_1^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}\hfill \\ & & +\frac{\left(\right|{\Lambda }_6\left({\tau }_2\right)-{\Lambda }_6\left({\tau }_1\right)\left|\right){ϵ}_2{\nu }_2^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}+\left(\right|{\Lambda }_7\left({\tau }_2\right)-{\Lambda }_7\left({\tau }_1\right)\left|\right)\left[\frac{{\lambda }_1{\delta }_1^{\varsigma +1}{\mathcal{L}}_g}{\Gamma (\varsigma +2)}+{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j{\vartheta }_j^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}+\frac{T^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}\right]\hfill \\ & & +\left(\right|{\Lambda }_8\left({\tau }_2\right)-{\Lambda }_8\left({\tau }_1\right)\left|\right)\left[\frac{{\lambda }_2{\delta }_2^{\varrho +1}{\mathcal{L}}_f}{\Gamma (\varrho +2)}+{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j{\left({\phi }_j\right)}^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}+\frac{T^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}\right],\hfill \end{array}$$

and

$$\begin{array}{ccc}& & |{\Pi }_2^{{}^{\prime }}(u,v)\left({\tau }_2\right)-{\Pi }_2^{{}^{\prime }}(u,v)\left({\tau }_1\right)|\hfill \\ & \le & \left|{\int }_0^{{\tau }_1}\frac{[{({\tau }_2-\theta )}^{\varsigma -2}-{({\tau }_1-\theta )}^{\varsigma -2}]}{\Gamma (\varsigma -1)}\times g(\theta ,u\left(\theta \right),{{}^C\mathcal{D}}^{{\varrho }_1}u\left(\theta \right),{\mathcal{I}}^{\zeta }u\left(\theta \right),v\left(\theta \right))d\theta \right|\hfill \\ & & +\left|{\int }_{{\tau }_1}^{{\tau }_2}\frac{{({\tau }_2-\theta )}^{\varsigma -2}}{\Gamma (\varsigma -1)}g(\theta ,u\left(\theta \right),{{}^C\mathcal{D}}^{{\varrho }_1}u\left(\theta \right),{\mathcal{I}}^{\zeta }u\left(\theta \right),v\left(\theta \right))d\theta \right|+\left|{\psi }_2\left(u\right)\right|\left[{\kappa }_2\left(\right|{\Lambda }_7^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_7^{{}^{\prime }}\left({\tau }_1\right)\left|\right)+\left(\right|{\Lambda }_8^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_8^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\right]\hfill \\ & & +|{\psi }_1\left(v\right)|\left[{\kappa }_1\left(\right|{\Lambda }_8^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_8^{{}^{\prime }}\left({\tau }_1\right)\left|\right)+\left(\right|{\Lambda }_7^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_7^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\right]+\frac{\left(\right|{\Lambda }_5^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_5^{{}^{\prime }}\left({\tau }_1\right)\left|\right){ϵ}_1{\nu }_1^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}\hfill \\ & & +\frac{\left(\right|{\Lambda }_6^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_6^{{}^{\prime }}\left({\tau }_1\right)\left|\right){ϵ}_2{\nu }_2^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}+\left(\right|{\Lambda }_7^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_7^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\left[\frac{{\lambda }_1{\delta }_1^{\varsigma +1}{\mathcal{L}}_g}{\Gamma (\varsigma +2)}+{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j{\vartheta }_j^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}+\frac{T^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}\right]\hfill \\ & & +\left(\right|{\Lambda }_8^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_8^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\left[\frac{{\lambda }_2{\delta }_2^{\varrho +1}{\mathcal{L}}_f}{\Gamma (\varrho +2)}+{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j{\left({\phi }_j\right)}^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}+\frac{T^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}\right].\hfill \end{array}$$

Consequently, we have

$$\begin{array}{ccc}& & |{{}^C\mathcal{D}}^{{\varsigma }_1}{\Pi }_2(u,v)\left({\tau }_2\right)-{{}^C\mathcal{D}}^{{\varsigma }_1}{\Pi }_2(u,v)\left({\tau }_1\right)|\hfill \\ & \le & \frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}\{\frac{{\mathcal{L}}_g}{\Gamma (\varsigma +1)}[{({\tau }_2-{\tau }_1)}^{\varsigma }+({\tau }_2^{\varsigma }-{\tau }_1^{\varsigma })]+\left|{\psi }_2\left(u\right)\right|\left[{\kappa }_2\left(\right|{\Lambda }_7^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_7^{{}^{\prime }}\left({\tau }_1\right)\left|\right)+\left(\right|{\Lambda }_8^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_8^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\right]\hfill \\ & & +|{\psi }_1\left(v\right)|\left[{\kappa }_1\left(\right|{\Lambda }_8^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_8^{{}^{\prime }}\left({\tau }_1\right)\left|\right)+\left(\right|{\Lambda }_7^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_7^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\right]+\frac{\left(\right|{\Lambda }_5^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_5^{{}^{\prime }}\left({\tau }_1\right)\left|\right){ϵ}_1{\nu }_1^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}\hfill \\ & & +\frac{\left(\right|{\Lambda }_6^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_6^{{}^{\prime }}\left({\tau }_1\right)\left|\right){ϵ}_2{\nu }_2^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}+\left(\right|{\Lambda }_7^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_7^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\left[\frac{{\lambda }_1{\delta }_1^{\varsigma +1}{\mathcal{L}}_g}{\Gamma (\varsigma +2)}+{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j{\vartheta }_j^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}+\frac{T^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}\right]\hfill \\ & & +\left(\right|{\Lambda }_8^{{}^{\prime }}\left({\tau }_2\right)-{\Lambda }_8^{{}^{\prime }}\left({\tau }_1\right)\left|\right)\left[\frac{{\lambda }_2{\delta }_2^{\varrho +1}{\mathcal{L}}_f}{\Gamma (\varrho +2)}+{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j{\left({\phi }_j\right)}^{\varrho }{\mathcal{L}}_f}{\Gamma (\varrho +1)}+\frac{T^{\varsigma }{\mathcal{L}}_g}{\Gamma (\varsigma +1)}\right]\},\hfill \end{array}$$

which means that $\parallel {\Pi }_2(u,v)\left({\tau }_2\right)-{\Pi }_2(u,v)\left({\tau }_1\right){\parallel }_{\mathcal{H}}\to 0$ independent of u and v as ${\tau }_2\to {\tau }_1$. Hence, the operator $\Pi (u,v)$ is equicontinuous, and thus it is completely continuous by Lemma (see Lemma 1.2 [15]). Next, we demonstrate that the set $\Phi =\left\{\right(u,v)\in \mathcal{G}\times \mathcal{H}|(u,v)=\eta \Pi (u,v),0<\eta <1\}$ is bounded. Let $(u,v)\in \Phi$; then, $(u,v)=\eta \Pi (u,v)$, and for any $\tau \in \mathcal{U}$, we have $u\left(\tau \right)=\eta {\Pi }_1(u,v)\left(\tau \right)$, $v\left(\tau \right)=\eta {\Pi }_2(u,v)\left(\tau \right)$. Thus,

$$\begin{array}{ccc}\hfill {\left|u\left(\tau \right)\right|}_{\mathcal{G}}& \le & {\Delta }_1\left(r_0+r_1{\parallel u\parallel }_{\mathcal{G}}+\left(max\{r_2,r_3\}+\frac{r_4{T}^{\xi }}{\Gamma (\xi +1)}\right){\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +{\Delta }_2\left(s_0+\left(max\{s_1,s_2\}+\frac{s_3{T}^{\zeta }}{\Gamma (\zeta +1)}\right){\parallel u\parallel }_{\mathcal{G}}+s_4{\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +\left(1+{\kappa }_1|{\overline{\Lambda }}_4|+|{\overline{\Lambda }}_3|\right){\mathcal{W}}_1{\parallel v\parallel }_{\mathcal{H}}+\left({\kappa }_2|{\overline{\Lambda }}_3|+|{\overline{\Lambda }}_4|\right){\mathcal{W}}_2{\parallel u\parallel }_{\mathcal{G}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill |u^{{}^{\prime }}\left(\tau \right)|& \le & {\widehat{\Delta }}_1\left(r_0+r_1{\parallel u\parallel }_{\mathcal{G}}+\left(max\{r_2,r_3\}+\frac{r_4{T}^{\xi }}{\Gamma (\xi +1)}\right){\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +{\widehat{\Delta }}_2\left(s_0+\left(max\{s_1,s_2\}+\frac{s_3{T}^{\zeta }}{\Gamma (\zeta +1)}\right){\parallel u\parallel }_{\mathcal{G}}+s_4{\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +\left({\kappa }_1|{\overline{\Lambda }}_4^{{}^{\prime }}|+|{\overline{\Lambda }}_3^{{}^{\prime }}|\right){\mathcal{W}}_1{\parallel v\parallel }_{\mathcal{H}}+\left({\kappa }_2|{\overline{\Lambda }}_3^{{}^{\prime }}|+|{\overline{\Lambda }}_4^{{}^{\prime }}|\right){\mathcal{W}}_2{\parallel u\parallel }_{\mathcal{G}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill |{{}^C\mathcal{D}}^{{\varrho }_1}u\left(\tau \right)|& \le & \frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}\{{\widehat{\Delta }}_1\left(r_0+r_1{\parallel u\parallel }_{\mathcal{G}}+\left(max\{r_2,r_3\}+\frac{r_4{T}^{\xi }}{\Gamma (\xi +1)}\right){\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +{\widehat{\Delta }}_2\left(s_0+\left(max\{s_1,s_2\}+\frac{s_3{T}^{\zeta }}{\Gamma (\zeta +1)}\right){\parallel u\parallel }_{\mathcal{G}}+s_4{\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +\left({\kappa }_1|{\overline{\Lambda }}_4^{{}^{\prime }}|+|{\overline{\Lambda }}_3^{{}^{\prime }}|\right){\mathcal{W}}_1{\parallel v\parallel }_{\mathcal{H}}+\left({\kappa }_2|{\overline{\Lambda }}_3^{{}^{\prime }}|+|{\overline{\Lambda }}_4^{{}^{\prime }}|\right){\mathcal{W}}_2{\parallel u\parallel }_{\mathcal{G}}\}.\hfill \end{array}$$

Hence, we have

$$\begin{array}{ccc}\hfill \parallel u\parallel & \le & {\Delta }_1\left(r_0+r_1{\parallel u\parallel }_{\mathcal{G}}+\left(max\{r_2,r_3\}+\frac{r_4{T}^{\xi }}{\Gamma (\xi +1)}\right){\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +{\Delta }_2\left(s_0+\left(max\{s_1,s_2\}+\frac{s_3{T}^{\zeta }}{\Gamma (\zeta +1)}\right){\parallel u\parallel }_{\mathcal{G}}+s_4{\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +\left(1+{\kappa }_1|{\overline{\Lambda }}_4|+|{\overline{\Lambda }}_3|\right){\mathcal{W}}_1{\parallel v\parallel }_{\mathcal{H}}+\left({\kappa }_2|{\overline{\Lambda }}_3|+|{\overline{\Lambda }}_4|\right){\mathcal{W}}_2{\parallel u\parallel }_{\mathcal{G}}\hfill \\ & & +\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}\{{\widehat{\Delta }}_1\left(r_0+r_1{\parallel u\parallel }_{\mathcal{G}}+\left(max\{r_2,r_3\}+\frac{r_4{T}^{\xi }}{\Gamma (\xi +1)}\right){\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +{\widehat{\Delta }}_2\left(s_0+\left(max\{s_1,s_2\}+\frac{s_3{T}^{\zeta }}{\Gamma (\zeta +1)}\right){\parallel u\parallel }_{\mathcal{G}}+s_4{\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +\left({\kappa }_1|{\overline{\Lambda }}_4^{{}^{\prime }}|+|{\overline{\Lambda }}_3^{{}^{\prime }}|\right){\mathcal{W}}_1{\parallel v\parallel }_{\mathcal{H}}+\left({\kappa }_2|{\overline{\Lambda }}_3^{{}^{\prime }}|+|{\overline{\Lambda }}_4^{{}^{\prime }}|\right){\mathcal{W}}_2{\parallel u\parallel }_{\mathcal{G}}\}.\hfill \end{array}$$

(46)

According to the above, we get

$$\begin{array}{ccc}\hfill \parallel v\parallel & \le & {\Delta }_4\left(s_0+\left(max\{s_1,s_2\}+\frac{s_3{T}^{\zeta }}{\Gamma (\zeta +1)}\right){\parallel u\parallel }_{\mathcal{G}}+s_4{\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +{\Delta }_3\left(r_0+r_1{\parallel u\parallel }_{\mathcal{G}}+\left(max\{r_2,r_3\}+\frac{r_4{T}^{\xi }}{\Gamma (\xi +1)}\right){\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +\left(1+{\kappa }_2|{\overline{\Lambda }}_7|+|{\overline{\Lambda }}_8|\right){\mathcal{W}}_2{\parallel u\parallel }_{\mathcal{G}}+\left({\kappa }_1|{\overline{\Lambda }}_8|+|{\overline{\Lambda }}_7|\right){\mathcal{W}}_1{\parallel v\parallel }_{\mathcal{H}}\hfill \\ & & +\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}\{{\widehat{\Delta }}_4\left(s_0+\left(max\{s_1,s_2\}+\frac{s_3{T}^{\zeta }}{\Gamma (\zeta +1)}\right){\parallel u\parallel }_{\mathcal{G}}+s_4{\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +{\widehat{\Delta }}_3\left(r_0+r_1{\parallel u\parallel }_{\mathcal{G}}+\left(max\{r_2,r_3\}+\frac{r_4{T}^{\xi }}{\Gamma (\xi +1)}\right){\parallel v\parallel }_{\mathcal{H}}\right)\hfill \\ & & +\left({\kappa }_2|{\overline{\Lambda }}_7^{{}^{\prime }}|+|{\overline{\Lambda }}_8^{{}^{\prime }}|\right){\mathcal{W}}_2{\parallel u\parallel }_{\mathcal{G}}+\left({\kappa }_1|{\overline{\Lambda }}_8^{{}^{\prime }}|+|{\overline{\Lambda }}_7^{{}^{\prime }}|\right){\mathcal{W}}_1{\parallel v\parallel }_{\mathcal{H}}\}.\hfill \end{array}$$

(47)

Using the above inequalities in combination with the notations (46) and (47), we deduce the result below. $\parallel u\parallel +\parallel v\parallel \le {\Phi }_1+min\{{\Phi }_2,{\Phi }_3\}{\parallel (u,v)\parallel }_{\mathcal{G}\times \mathcal{H}}$, which leads to ${\displaystyle {\parallel (u,v)\parallel }_{\mathcal{G}\times \mathcal{H}}\le \frac{{\displaystyle {\Phi }_1}}{{\displaystyle 1-min\{{\Phi }_2,{\Phi }_3\}}}}$. This concludes that the set $min\{{\Phi }_2$, ${\Phi }_3\}$ is bounded. Therefore, operator $\Pi$ has at least one fixed point by Theorem (see Theorem 1.9 [15]), which means the system (1)–(2) has at least one solutions on $\mathcal{U}$. □

Theorem 2.

Assume that $\left({\mathcal{F}}_3\right)$ and $\left({\mathcal{F}}_4\right)$ holds. Additionally, if

$$\Psi +\widehat{\Psi }<1,$$

(48)

where $\Psi ,\widehat{\Psi }$ are defined by (42), then on $\mathcal{U}$ there is a unique solution to the problems (1) and (2).

Proof. 

Let us fix this $\widehat{\epsilon }\ge max\left\{\frac{\overline{\mathcal{P}}+\widehat{\mathcal{P}}}{1-(\Psi +\widehat{\Psi })}\right\}$, where $\Psi$, $\widehat{\Psi }$ and $\overline{\mathcal{P}}$, $\widehat{\mathcal{P}}$ are respectively given by (41) and (42), and show that $\Pi {\mathcal{B}}_{\widehat{\epsilon }}\subset {\mathcal{B}}_{\widehat{\epsilon }}$, where the operator $\Pi$ is given by (22) and ${\displaystyle {\mathcal{B}}_{\widehat{\epsilon }}=\{(u,v)\in \mathcal{G}\times \mathcal{H}:\parallel (u,v)\parallel \le \widehat{\epsilon }\}}$. For $(u,v)\in {\mathcal{B}}_{\widehat{\epsilon }}$, $\tau \in \mathcal{H}$, we have

$$\begin{array}{ccc}\hfill |{\widehat{\mathcal{S}}}_u\left(\tau \right)|& =& |f(\tau ,u\left(\tau \right),v\left(\tau \right),{{}^C\mathcal{D}}^{{\varsigma }_1}v\left(\tau \right),{\mathcal{I}}^{\xi }v\left(\tau \right))|\le {\mathcal{V}}_1{\iota }_1\left({\parallel (u,v)\parallel }_{\mathcal{G}\times \mathcal{H}}\right)+{\mathcal{T}}_1\le {\mathcal{V}}_1{\iota }_1\widehat{\epsilon }+{\mathcal{T}}_1,\hfill \\ \hfill |{\tilde{\mathcal{S}}}_v\left(\tau \right)|& =& |g(\tau ,u\left(\tau \right),{{}^C\mathcal{D}}^{{\varrho }_1}u\left(\tau \right),{\mathcal{I}}^{\zeta }u\left(\tau \right),v\left(\tau \right))|\le {\mathcal{V}}_2{\iota }_2\left({\parallel (u,v)\parallel }_{\mathcal{G}\times \mathcal{H}}\right)+{\mathcal{T}}_2\le {\mathcal{V}}_2{\iota }_2\widehat{\epsilon }+{\mathcal{T}}_2,\hfill \\ \hfill |{\psi }_1\left(v\right)|& \le & {\mathcal{V}}_1\widehat{\epsilon },\left|{\psi }_2\left(u\right)\right|\le {\mathcal{V}}_2\widehat{\epsilon },\hfill \end{array}$$

which lead to

$$|{\Pi }_1(u,v)\left(\tau \right)|\le {\Psi }_1\widehat{\epsilon }+{\mathcal{P}}_1,$$

where ${\Psi }_1$ and ${\mathcal{P}}_1$ are given by (36) and (40). With the above notes, we get

$$|{\Pi }_1^{{}^{\prime }}(u,v)\left(\tau \right)|\le {\Psi }_2\widehat{\epsilon }+{\mathcal{P}}_2,$$

which means that

$$|{{}^C\mathcal{D}}^{{\varrho }_1}{\Pi }_1(u,v)\left(\tau \right)|\le \frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}\left({\Psi }_2\widehat{\epsilon }+{\mathcal{P}}_2\right).$$

Thus, we get

$${\displaystyle \parallel {\Pi }_1{(u,v)\parallel }_{\mathcal{G}}\le \left({\Psi }_1+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}{\Psi }_2\right)\widehat{\epsilon }+\left({\mathcal{P}}_1+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}{\mathcal{P}}_2\right).}$$

(49)

Similarly, we get

$$|{\Pi }_2(u,v)\left(\tau \right)|\le {\Psi }_3\widehat{\epsilon }+{\mathcal{P}}_3,$$

where ${\Psi }_3$ and ${\mathcal{P}}_3$ are given by (38) and (40). With the above notes, we get

$$|{\Pi }_2^{{}^{\prime }}(u,v)\left(\tau \right)|\le {\Psi }_4\widehat{\epsilon }+{\mathcal{P}}_4,$$

which means that

$$|{{}^C\mathcal{D}}^{{\varsigma }_1}{\Pi }_2(u,v)\left(\tau \right)|\le \frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}\left({\Psi }_4\widehat{\epsilon }+{\mathcal{P}}_4\right).$$

Hence, we have

$${\displaystyle \parallel {\Pi }_2{(u,v)\parallel }_{\mathcal{H}}\le \left({\Psi }_3+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\Psi }_4\right)\widehat{\epsilon }+\left({\mathcal{P}}_3+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\mathcal{P}}_4\right).}$$

(50)

So, (49) and (50) follow ${\parallel \Pi (u,v)\parallel }_{\mathcal{G}\times \mathcal{H}}\le \widehat{\epsilon }$, and thus, $\Pi {\mathcal{B}}_{\widehat{\epsilon }}\subset {\mathcal{B}}_{\widehat{\epsilon }}$.

Now, for $(u_1,v_1)$, $(u_2,v_2)$$\in \mathcal{G}\times \mathcal{H}$ and any $\tau \in \mathcal{U}$, we have

$$|{\Pi }_1(u_1,v_1)\left(\tau \right)-{\Pi }_1(u_2,v_2)\left(\tau \right)|\le {\Psi }_1(\parallel u_1-u_2{\parallel }_{\mathcal{G}}+\parallel v_1-v_2{\parallel }_{\mathcal{H}}).$$

Next, we find that

$$|{\Pi }_1^{{}^{\prime }}(u_1,v_1)\left(\tau \right)-{\Pi }_1^{{}^{\prime }}(u_2,v_2)\left(\tau \right)|\le {\Psi }_2(\parallel u_1-u_2{\parallel }_{\mathcal{G}}+\parallel v_1-v_2{\parallel }_{\mathcal{H}}).$$

Thus, we have

$$|{{}^C\mathcal{D}}^{{\varrho }_1}{\Pi }_1(u_1,v_1)\left(\tau \right)-{{}^C\mathcal{D}}^{{\varrho }_1}{\Pi }_1(u_2,v_2)\left(\tau \right)|\le \frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}\left({\Psi }_2(\parallel u_1-u_2{\parallel }_{\mathcal{G}}+\parallel v_1-v_2{\parallel }_{\mathcal{H}})\right),$$

Hence, we get

$$\parallel {\Pi }_1(u_1,v_1)-{\Pi }_1(u_2,v_2){\parallel }_{\mathcal{G}}\le \left({\Psi }_1+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}{\Psi }_2\right)(\parallel u_1-u_2{\parallel }_{\mathcal{G}}+\parallel v_1-v_2{\parallel }_{\mathcal{H}}).$$

(51)

Similarly, we have

$$\parallel {\Pi }_2(u_1,v_1)-{\Pi }_2(u_2,v_2){\parallel }_{\mathcal{H}}\le \left({\Psi }_3+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\Psi }_4\right)(\parallel u_1-u_2{\parallel }_{\mathcal{G}}+\parallel v_1-v_2{\parallel }_{\mathcal{H}}).$$

(52)

So, (51) and (52) follow

$$\begin{array}{ccc}& & \parallel \Pi (u_1,v_1)-\Pi (u_2,v_2){\parallel }_{\mathcal{G}\times \mathcal{H}}\hfill \\ & \le & \left({\Psi }_1+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}{\Psi }_2+{\Psi }_3+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\Psi }_4\right)(\parallel u_1-u_2{\parallel }_{\mathcal{G}}+\parallel v_1-v_2{\parallel }_{\mathcal{H}}).\hfill \end{array}$$

It follows that, in view of the condition (48), the operator $\Pi$ is a contraction. Thus, by Theorem (see Theorem 1.2.2 [14]), the system (1) and (2) has a unique solution on $\mathcal{U}$. □

4. Examples

Example 1.

Consider the system of Caputo type FIDEs given by

$$\left\{\begin{array}{c}{\displaystyle {{}^C\mathcal{D}}^{\frac{68}{25}}u\left(\tau \right)=f(\tau ,u\left(\tau \right),v\left(\tau \right),{{}^C\mathcal{D}}^{\frac{44}{25}}v\left(\tau \right),{\mathcal{I}}^{\frac{46}{25}}v\left(\tau \right)),}\hfill \\ {\displaystyle {{}^C\mathcal{D}}^{\frac{62}{25}}v\left(\tau \right)=g(\tau ,u\left(\tau \right),{{}^C\mathcal{D}}^{\frac{73}{50}}u\left(\tau \right),{\mathcal{I}}^{\frac{39}{25}}u\left(\tau \right),v\left(\tau \right)),}\hfill \end{array}\right.$$

(53)

subject to the boundary conditions

$$\left\{\begin{array}{c}{\displaystyle u\left(0\right)={\psi }_1\left(v\right),u^{{}^{\prime }}\left(0\right)={ϵ}_1{\int }_0^{{\nu }_1}{v}^{{}^{\prime }}\left(\theta \right)d\theta ,u\left(T\right)={\lambda }_1{\int }_0^{{\delta }_1}v\left(\theta \right)d\theta +{\mu }_1\sum _{j=1}^{k-2}{\varpi }_{j}v\left({\vartheta }_j\right),}\hfill \\ {\displaystyle v\left(0\right)={\psi }_2\left(u\right),v^{{}^{\prime }}\left(0\right)={ϵ}_2{\int }_0^{{\nu }_2}{u}^{{}^{\prime }}\left(\theta \right)d\theta ,v\left(T\right)={\lambda }_2{\int }_0^{{\delta }_2}u\left(\theta \right)d\theta +{\mu }_2\sum _{j=1}^{k-2}{\omega }_{j}u\left({\phi }_j\right).}\hfill \end{array}\right.$$

(54)

Here, $\varrho =\frac{68}{25}$, $\varsigma =\frac{62}{25}$, ${\varrho }_1=\frac{73}{50}$, ${\varsigma }_1=\frac{44}{25}$, $\xi =\frac{46}{25}$, $\zeta =\frac{39}{25}$, $T=1$, ${\delta }_1=\frac{1}{8}$, ${\delta }_2=\frac{37}{200}$, ${\vartheta }_1=\frac{37}{250}$, ${\vartheta }_2=\frac{79}{500}$, ${\vartheta }_3=\frac{47}{250}$, ${\vartheta }_4=\frac{11}{50}$, ${\phi }_1=\frac{113}{500}$, ${\phi }_2=\frac{59}{250}$, ${\phi }_3=\frac{6}{25}$, ${\displaystyle {\phi }_4=\frac{49}{200}}$, ${\varpi }_1=\frac{1}{160}$, ${\varpi }_2=\frac{58}{625}$, ${\varpi }_3=\frac{29}{400}$, ${\varpi }_4=\frac{21}{250}$, ${\omega }_1=\frac{17}{200}$, ${\omega }_2=\frac{12}{125}$, ${\omega }_3=\frac{19}{250}$, ${\omega }_4=\frac{7}{125}$, ${\lambda }_1=\frac{1}{80}$, ${\lambda }_2=\frac{7}{200}$, ${\mu }_1=\frac{9}{400}$, ${\mu }_2=\frac{23}{500}$,${ϵ}_1=\frac{11}{400}$, ${ϵ}_2=\frac{19}{500}$, ${\nu }_1=\frac{53}{200}$, ${\nu }_2=\frac{133}{500}$. We consider the functions,

$$\begin{array}{ccc}\hfill \left|f\right(\tau ,u_1,u_2,u_3,u_4\left)\right|& \le & \frac{1}{20({\tau }^2+1)}+\frac{1}{70{(2+\tau )}^2}\left(2u_2+\frac{|u_1|}{1+|u_1|}\right)+\frac{sin{u}_3}{700}+\frac{arctan{u}_4}{140(3+\tau )},\hfill \\ \hfill \left|g\right(\tau ,u_1,u_2,u_3,u_4\left)\right|& \le & \frac{1}{({\tau }^4+1)2}+\frac{1}{150(1+{\tau }^2)}\left(\frac{u_2}{3}+3u_1\right)+\frac{cos{u}_3}{800}+\frac{|u_4|}{400(1+|u_4\left|\right)},\hfill \\ \hfill |{\psi }_1\left(v\right)|& \le & \frac{1}{110}\parallel v\parallel ,|{\psi }_2\left(u\right)|\le \frac{1}{600}\parallel u\parallel .\hfill \end{array}$$

Clearly

$$\begin{array}{ccc}\hfill |f(\tau ,u_1,u_2,u_3,u_4)|& \le & \frac{1}{20}+\frac{1}{140}|u_1|+\frac{1}{70}|u_2|+\frac{1}{700}|u_3|+\frac{1}{420}|u_4|,\hfill \\ \hfill |g(\tau ,u_1,u_2,u_3,u_4)|& \le & \frac{1}{2}+\frac{1}{50}|u_1|+\frac{1}{450}|u_2|+\frac{1}{800}|u_3|+\frac{1}{400}|u_4|,\hfill \\ \hfill |{\psi }_1\left(v\right)|& \le & \frac{1}{110}\parallel v\parallel ,|{\psi }_2\left(u\right)|\le \frac{1}{600}\parallel u\parallel .\hfill \end{array}$$

With the given data, we find that $r_0=\frac{1}{20}$, $r_1=\frac{1}{140}$, $r_2=\frac{1}{70}$, $r_3=\frac{1}{700}$, $r_4=\frac{1}{420}$, $s_0=\frac{1}{2}$, $s_1=\frac{1}{50}$, $s_2=\frac{1}{450}$, $s_3=\frac{1}{800}$, $s_4=\frac{1}{400}$, ${\Delta }_1=0.46845200402823617$, ${\Delta }_2=0.0007227881649342847$, ${\Delta }_3=0.0011251030301272463$, ${\Delta }_4=0.6151822290610394$, ${\widehat{\Delta }}_1=0.8713184743902467$, ${\widehat{\Delta }}_2=0.0014641295646385377$, ${\widehat{\Delta }}_3=0.00186307333192$, ${\widehat{\Delta }}_4=1.0704190784503973$, we find that $\widehat{\Phi }=min\{{\Phi }_1,{\Phi }_2\}<1$. Thus, the assumption of Theorem 1 holds and the problem (53) and (54) has at least one solution on $[0,1]$.

Example 2.

We consider the functions

$$\begin{array}{ccc}\hfill \left|f\right(\tau ,u_1,u_2,u_3,u_4\left)\right|& \le & \frac{1}{9\sqrt{{\tau }^2+64}}\left(\frac{3}{\tau +1}+cos(u_1+u_2)+\frac{|u_3|}{|u_3|+1}+arctan{u}_4\right),\hfill \\ \hfill \left|g\right(\tau ,u_1,u_2,u_3,u_4\left)\right|& \le & \frac{1}{24\sqrt{36+{\tau }^2}}\left(\frac{{\tau }^2}{\tau +2}+\left(u_2+\frac{|u_1|}{1+|u_1|}\right)+sin{u}_3+\left|u_4\right|\right),\hfill \\ \hfill |{\psi }_1\left(v\right)|& \le & \frac{1}{110}\parallel v\parallel ,|{\psi }_2\left(u\right)|\le \frac{1}{60}\parallel u\parallel .\hfill \end{array}$$

Using the given data, it is found that ${\mathcal{V}}_1=\frac{1}{72}$, ${\mathcal{V}}_2=\frac{1}{144}$, ${\mathcal{V}}_1=\frac{1}{110}$, ${\mathcal{V}}_2=\frac{1}{60}$, ${\Delta }_1=0.46845200402823617$, ${\Delta }_2=0.0007227881649342847$, ${\Delta }_3=0.0011251030301272463$, ${\Delta }_4=0.6151822290610394$, ${\widehat{\Delta }}_1=0.8713184743902467$, ${\widehat{\Delta }}_2=0.0014641295646385377$, ${\widehat{\Delta }}_3=0.00186307333192$, ${\widehat{\Delta }}_4=1.0704190784503973$, with ${\Psi }_1+\frac{T^{1-{\varrho }_1}}{\Gamma (2-{\varrho }_1)}$${\Psi }_2\approx$$0.05140181873251667$, and ${\Psi }_3+\frac{T^{1-{\varsigma }_1}}{\Gamma (2-{\varsigma }_1)}{\Psi }_4\approx 0.05324949686716335$; hence, the Theorem 2 is satisfied, and here the problem (53)–(54) has a unique solution on $[0,1]$.

5. A Variant of a Problem

Note that the boundary conditions (1) include the strips of different lengths when modifying the strips in boundary conditions to the same lengths (1); then, the problem reduces to the form

$$\left\{\begin{array}{c}u\left(0\right)={\psi }_1\left(v\right),u^{{}^{\prime }}\left(0\right)={ϵ}_1{\int }_0^{\nu }{v}^{{}^{\prime }}\left(\theta \right)d\theta ,u^{{}^{″}}\left(0\right)=0,···,u^{n-2}\left(0\right)=0,\hfill \\ u\left(T\right)={\lambda }_1{\int }_0^{\delta }v\left(\theta \right)d\theta +{\mu }_1\sum _{j=1}^{k-2}{\varpi }_{j}v\left({\vartheta }_j\right),\hfill \\ v\left(0\right)={\psi }_2\left(u\right),v^{{}^{\prime }}\left(0\right)={ϵ}_2{\int }_0^{\nu }{u}^{{}^{\prime }}\left(\theta \right)d\theta ,v^{{}^{″}}\left(0\right)=0,···,v^{n-2}\left(0\right)=0,\hfill \\ v\left(T\right)={\lambda }_2{\int }_0^{\delta }u\left(\theta \right)d\theta +{\mu }_2\sum _{j=1}^{k-2}{\omega }_{j}u\left({\vartheta }_j\right).\hfill \end{array}\right.$$

(55)

Concerning the problem (1) with (55) instead of (2), we obtained the operator $\widehat{\Pi }:\mathcal{G}\times \mathcal{H}\to \mathcal{G}\times \mathcal{H}$ defined by

$$\widehat{\Pi }(u,v)\left(\tau \right)=({\widehat{\Pi }}_1(u,v)\left(\tau \right),{\widehat{\Pi }}_2(u,v)\left(\tau \right)),$$

where

$$\begin{array}{ccc}\hfill {\widehat{\Pi }}_1(u,v)\left(\tau \right)& =& \frac{1}{\Gamma \left(\varrho \right)}{\int }_0^{\tau }{(\tau -\theta )}^{\varrho -1}{\widehat{\mathcal{Q}}}_u\left(\theta \right)d\theta +{\psi }_1\left(v\right)[1+{\kappa }_1{\Lambda }_4\left(\tau \right)-{\Lambda }_3\left(\tau \right)]+{\psi }_2\left(u\right)[{\kappa }_2{\Lambda }_3\left(\tau \right)-{\Lambda }_4\left(\tau \right)]\hfill \\ & & +\frac{{\Lambda }_2\left(\tau \right){ϵ}_2}{\Gamma (\varrho -1)}{\int }_0^{\nu }\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -2}{\widehat{\mathcal{Q}}}_u\left(\sigma \right)d\sigma \right)d\theta +\frac{{\Lambda }_1\left(\tau \right){ϵ}_1}{\Gamma (\varsigma -1)}{\int }_0^{\nu }\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -2}{\tilde{\mathcal{Q}}}_v\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\Lambda }_4\left(\tau \right)[\frac{{\lambda }_2}{\Gamma \left(\varrho \right)}{\int }_0^{\delta }\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -1}{\widehat{\mathcal{Q}}}_u\left(\sigma \right)d\sigma \right)d\theta +{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j}{\Gamma \left(\varrho \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varrho -1}{\widehat{\mathcal{Q}}}_u\left(\theta \right)d\theta \hfill \\ & & -\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^T{(T-\theta )}^{\varsigma -1}{\tilde{\mathcal{Q}}}_v\left(\theta \right)d\theta ]+{\Lambda }_3\left(\tau \right)[\frac{{\lambda }_1}{\Gamma \left(\varsigma \right)}{\int }_0^{\delta }\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -1}{\tilde{\mathcal{Q}}}_v\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j}{\Gamma \left(\varsigma \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varsigma -1}{\tilde{\mathcal{Q}}}_v\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^T{(T-\theta )}^{\varrho -1}{\widehat{\mathcal{Q}}}_u\left(\theta \right)d\theta ],\hfill \end{array}$$

(56)

and

$$\begin{array}{ccc}\hfill {\widehat{\Pi }}_2(u,v)\left(\tau \right)& =& \frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^{\tau }{(\tau -\theta )}^{\varsigma -1}{\tilde{\mathcal{Q}}}_v\left(\theta \right)d\theta +{\psi }_2\left(u\right)[1+{\kappa }_2{\Lambda }_7\left(\tau \right)-{\Lambda }_8\left(\tau \right)]+{\psi }_1\left(v\right)[{\kappa }_1{\Lambda }_8\left(\tau \right)-{\Lambda }_7\left(\tau \right)]\hfill \\ & & +\frac{{\Lambda }_5\left(\tau \right){ϵ}_1}{\Gamma (\varsigma -1)}{\int }_0^{\nu }\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -2}{\tilde{\mathcal{Q}}}_v\left(\sigma \right)d\sigma \right)d\theta +\frac{{\Lambda }_6\left(\tau \right){ϵ}_2}{\Gamma (\varrho -1)}{\int }_0^{\nu }\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -2}{\widehat{\mathcal{Q}}}_u\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\Lambda }_7\left(\tau \right)[\frac{{\lambda }_1}{\Gamma \left(\varsigma \right)}{\int }_0^{\delta }\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varsigma -1}{\tilde{\mathcal{Q}}}_v\left(\sigma \right)d\sigma \right)d\theta +{\mu }_1\sum _{j=1}^{k-2}\frac{{\varpi }_j}{\Gamma \left(\varsigma \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varsigma -1}{\tilde{\mathcal{Q}}}_v\left(\theta \right)d\theta \hfill \\ & & -\frac{1}{\Gamma \left(\varrho \right)}{\int }_0^T{(T-\theta )}^{\varrho -1}{\widehat{\mathcal{Q}}}_u\left(\theta \right)d\theta ]+{\Lambda }_8\left(\tau \right)[\frac{{\lambda }_2}{\Gamma \left(\varrho \right)}{\int }_0^{\delta }\left({\int }_0^{\theta }{(\theta -\sigma )}^{\varrho -1}{\widehat{\mathcal{Q}}}_u\left(\sigma \right)d\sigma \right)d\theta \hfill \\ & & +{\mu }_2\sum _{j=1}^{k-2}\frac{{\omega }_j}{\Gamma \left(\varrho \right)}{\int }_0^{{\vartheta }_j}{({\vartheta }_j-\theta )}^{\varrho -1}{\widehat{\mathcal{Q}}}_u\left(\theta \right)d\theta -\frac{1}{\Gamma \left(\varsigma \right)}{\int }_0^T{(T-\theta )}^{\varsigma -1}{\tilde{\mathcal{Q}}}_v\left(\theta \right)d\theta ].\hfill \end{array}$$

(57)

where

$$\begin{array}{ccc}\hfill {\widehat{\mathcal{Q}}}_u\left(\tau \right)& =& f(\tau ,u\left(\tau \right),v\left(\tau \right),{{}^C\mathcal{D}}^{{\varsigma }_1}v\left(\tau \right),{\mathcal{I}}^{\xi }v\left(\tau \right)),\tau \in \mathcal{U},\hfill \\ \hfill {\tilde{\mathcal{Q}}}_v\left(\tau \right)& =& g(\tau ,u\left(\tau \right),{{}^C\mathcal{D}}^{{\varrho }_1}u\left(\tau \right),{\mathcal{I}}^{\zeta }u\left(\tau \right),v\left(\tau \right)),\tau \in \mathcal{U}.\hfill \end{array}$$

and

$$\begin{array}{ccc}& & {\xi }_1=\frac{{\lambda }_1{\delta }^2}{2}+{\mu }_1\sum _{j=1}^{k-2}{\varpi }_j{\vartheta }_j,{\xi }_2=\frac{{\lambda }_1{\delta }^n}{n}+{\mu }_1\sum _{j=1}^{k-2}{\varpi }_j{\vartheta }_j^{n-1},{\xi }_3=\frac{{\lambda }_2{\delta }^2}{2}+{\mu }_2\sum _{j=1}^{k-2}{\omega }_j{\vartheta }_j,{\xi }_4=\frac{{\lambda }_2{\delta }^n}{n}+{\mu }_2\sum _{j=1}^{k-2}{\omega }_j{\vartheta }_j^{n-1},\hfill \\ & & {\widehat{\gamma }}_1=1-{\nu }^2{ϵ}_1{ϵ}_2,{\widehat{\gamma }}_2=T^2-{\xi }_1{\xi }_3,{\widehat{\gamma }}_3=T^n-{\xi }_1{\xi }_4,{\widehat{\gamma }}_4=T^{n-1}{\xi }_3-{\xi }_{4}T,{\widehat{\gamma }}_5={\xi }_1{T}^{n-1}-T{\xi }_2,{\widehat{\gamma }}_6=T^n-{\xi }_2{\xi }_3,\hfill \\ & & {\upsilon }_1={\widehat{\gamma }}_2{\nu }^n{ϵ}_1{ϵ}_2+{\widehat{\gamma }}_1{\widehat{\gamma }}_3,{\upsilon }_2={\widehat{\gamma }}_2{\nu }^{n-1}{ϵ}_2+{\widehat{\gamma }}_1{\widehat{\gamma }}_4,{\upsilon }_3={\widehat{\gamma }}_2{\nu }^{n-1}{ϵ}_1+{\widehat{\gamma }}_1{\widehat{\gamma }}_5,{\upsilon }_4={\widehat{\gamma }}_2{\nu }^n{ϵ}_1{ϵ}_2+{\widehat{\gamma }}_1{\widehat{\gamma }}_6,\upsilon ={\upsilon }_2{\upsilon }_3-{\upsilon }_1{\upsilon }_4\ne 0,\hfill \\ & & {\eta }_1=1+\frac{({\nu }^n{ϵ}_2{\beta }_1-{\nu }^{n-1}{\beta }_5){ϵ}_1}{\upsilon },{\eta }_2={ϵ}_1\nu +\frac{({\nu }^n{ϵ}_2{\beta }_2-{\nu }^{n-1}{\beta }_6){ϵ}_1}{\upsilon },{\eta }_3=\frac{({\nu }^n{ϵ}_2{\beta }_3-{\nu }^{n-1}{\beta }_7){ϵ}_1}{\upsilon },\hfill \\ & & {\eta }_4=\frac{({\nu }^n{ϵ}_2{\beta }_4-{\nu }^{n-1}{\beta }_8){ϵ}_1}{\upsilon },{\eta }_5={ϵ}_2\nu +\frac{({\nu }^{n-1}{\beta }_1-{ϵ}_1{\nu }^n{\beta }_5){ϵ}_2}{\upsilon },{\eta }_6=1+\frac{({\nu }^{n-1}{\beta }_2-{ϵ}_1{\nu }^n{\beta }_6){ϵ}_2}{\upsilon },\hfill \\ & & {\eta }_7=\frac{({\nu }^{n-1}{\beta }_3-{ϵ}_1{\nu }^n{\beta }_7){ϵ}_2}{\upsilon },{\eta }_8=\frac{({\nu }^{n-1}{\beta }_4-{ϵ}_1{\nu }^n{\beta }_8){ϵ}_2}{\upsilon },\hfill \\ & & {\beta }_1={\widehat{\gamma }}_2({\upsilon }_4-{\upsilon }_3{ϵ}_2\nu ),{\beta }_2=({\upsilon }_4{ϵ}_1\nu -{\upsilon }_3){\widehat{\gamma }}_2,{\beta }_3=({\upsilon }_3{\xi }_3-{\upsilon }_{4}T){\widehat{\gamma }}_1,{\beta }_4={\widehat{\gamma }}_1({\upsilon }_{3}T-{\upsilon }_4{\xi }_1),\hfill \\ & & {\beta }_5=({\upsilon }_2-{\upsilon }_1{ϵ}_2\nu ){\widehat{\gamma }}_2,{\beta }_6=({\upsilon }_2{ϵ}_1\nu -{\upsilon }_1){\widehat{\gamma }}_2,{\beta }_7=({\upsilon }_1{\xi }_3-{\upsilon }_{2}T){\widehat{\gamma }}_1,{\beta }_8={\widehat{\gamma }}_1({\upsilon }_{1}T-{\upsilon }_2{\xi }_1),\hfill \\ & & {\kappa }_1={\lambda }_2\delta +{\mu }_2\sum _{j=1}^{k-2}{\omega }_j,{\kappa }_2={\lambda }_1\delta +{\mu }_1\sum _{j=1}^{k-2}{\varpi }_j,\hfill \\ & & {\Lambda }_1\left(\tau \right)=\frac{\tau {\eta }_1}{{\widehat{\gamma }}_1}+\frac{{\tau }^{n-1}{\beta }_1}{\upsilon },{\Lambda }_2\left(\tau \right)=\frac{\tau {\eta }_2}{{\widehat{\gamma }}_1}+\frac{{\tau }^{n-1}{\beta }_2}{\upsilon },{\Lambda }_3\left(\tau \right)=\frac{\tau {\eta }_3}{{\widehat{\gamma }}_1}+\frac{{\tau }^{n-1}{\beta }_3}{\upsilon },{\Lambda }_4\left(\tau \right)=\frac{\tau {\eta }_4}{{\widehat{\gamma }}_1}+\frac{{\tau }^{n-1}{\beta }_4}{\upsilon },\hfill \\ & & {\Lambda }_5\left(\tau \right)=\frac{\tau {\eta }_5}{{\widehat{\gamma }}_1}-\frac{{\tau }^{n-1}{\beta }_5}{\upsilon },{\Lambda }_6\left(\tau \right)=\frac{\tau {\eta }_6}{{\widehat{\gamma }}_1}-\frac{{\tau }^{n-1}{\beta }_6}{\upsilon },{\Lambda }_7\left(\tau \right)=\frac{\tau {\eta }_7}{{\widehat{\gamma }}_1}-\frac{{\tau }^{n-1}{\beta }_7}{\upsilon },{\Lambda }_8\left(\tau \right)=\frac{\tau {\eta }_8}{{\widehat{\gamma }}_1}-\frac{{\tau }^{n-1}{\beta }_8}{\upsilon }.\hfill \end{array}$$

6. Discussion

For Caputo form FIDEs, we examined the consequences of existence and uniqueness supplemented by non-local multi-point and integral boundary conditions by Leray–Schauder’s alternative and Banach’s fixed-point theorem. By fixing the parameters $({ϵ}_1$, ${ϵ}_2$, ${\lambda }_1$, ${\lambda }_2$, ${\mu }_1$, ${\mu }_2)$ involved in the problem (1) and (2), our results correspond to certain specific problems. Suppose that taking ${\lambda }_1={\lambda }_2={\mu }_1={\mu }_2=0$ in the results provided, we are given the problems (1) with the form

$$\left\{\begin{array}{c}u\left(0\right)={\psi }_1\left(v\right),u^{{}^{\prime }}\left(0\right)={ϵ}_1{\int }_0^{{\nu }_1}{v}^{{}^{\prime }}\left(\theta \right)d\theta ,u^{{}^{″}}\left(0\right)=0,···,u^{n-2}\left(0\right)=0,u\left(T\right)=0,\hfill \\ v\left(0\right)={\psi }_2\left(u\right),v^{{}^{\prime }}\left(0\right)={ϵ}_2{\int }_0^{{\nu }_2}{u}^{{}^{\prime }}\left(\theta \right)d\theta ,v^{{}^{″}}\left(0\right)=0,···,v^{n-2}\left(0\right)=0,v\left(T\right)=0,\hfill \end{array}\right.$$

while the results are

$$\left\{\begin{array}{c}{\displaystyle u\left(0\right)={\psi }_1\left(v\right),u^{{}^{\prime }}\left(0\right)=0,···,u^{n-2}\left(0\right)=0,u\left(T\right)={\lambda }_1{\int }_0^{{\delta }_1}v\left(\theta \right)d\theta +{\mu }_1\sum _{j=1}^{k-2}{\varpi }_{j}v\left({\vartheta }_j\right),}\hfill \\ {\displaystyle v\left(0\right)={\psi }_2\left(u\right),v^{{}^{\prime }}\left(0\right)=0,···,v^{n-2}\left(0\right)=0v\left(T\right)={\lambda }_2{\int }_0^{{\delta }_2}u\left(\theta \right)d\theta +{\mu }_2\sum _{j=1}^{k-2}{\omega }_{j}u\left({\phi }_j\right),}\hfill \end{array}\right.$$

followed by ${ϵ}_1={ϵ}_2=0$. Using the methods used in the previous section, we can solve the above-related problems (1) and (2).