Fractional Sequential Coupled Systems of Hilfer and Caputo Integro-Differential Equations with Non-Separated Boundary Conditions

Abstract

In studying boundary value problems and coupled systems of fractional order in $(1,2]$, involving Hilfer fractional derivative operators, a zero initial condition is necessary. The consequence of this fact is that boundary value problems and coupled systems of fractional order with non-zero initial conditions cannot be studied. For example, such boundary value problems and coupled systems of fractional order are those including separated, non-separated, or periodic boundary conditions. In this paper, we propose a method for studying a coupled system of fractional order in $(1,2],$ involving fractional derivative operators of Hilfer and Caputo with non-separated boundary conditions. More precisely, a sequential coupled system of fractional differential equations including Hilfer and Caputo fractional derivative operators and non-separated boundary conditions is studied in the present paper. As explained in the concluding section, the opposite combination of Caputo and Hilfer fractional derivative operators requires zero initial conditions. By using Banach’s fixed point theorem, the uniqueness of the solution is established, while by applying the Leray–Schauder alternative, the existence of solution is obtained. Numerical examples are constructed illustrating the main results.

1. Introduction

Fractional calculus and fractional differential equations have received substantial consideration owing to the broad applications of fractional derivative operators in mathematical modeling, describing many real-world processes more accurately than classical-order differential equations. For a systematic development of this topic, see [1,2,3,4,5,6,7], while for an extensive study of fractional boundary value problems, see the monograph in [8]. Usually, fractional derivative operators are defined with the help of fractional integral operators and Euler’s gamma function. There exists a variety of fractional derivative operators, such as Riemann–Liouville, Caputo, Erdélyi–Kober, Hadamard, Hilfer, Katugampola, etc. The k-Riemann–Liouville fractional integral operator was defined in [9], while the k-Riemann–Liouville fractional derivative was introduced in [10]. See [11,12,13,14,15,16] and the references cited therein for recent results regarding k-Riemann–Liouville fractional derivative operators. The $\psi$-Riemann–Liouville integral and derivative fractional operators were introduced in [2], while in [17] and [10], respectively, the $(k,\psi )$-Riemann–Liouville integral and derivative fractional operators were defined. The Hilfer fractional derivative, defined in [18], extends both the Riemann–Liouville and Caputo fractional derivatives. The $\psi$-Hilfer fractional derivative was defined in [19]. For applications of Hilfer fractional derivatives in mathematics, physics, etc., see [20,21,22,23,24,25,26,27,28]. For recent results on boundary value problems for fractional differential equations and inclusions with Hilfer fractional derivatives, see the survey paper by Ntouyas in [29].

In a series of papers, namely in [30,31,32], some authors have studied existence and uniqueness results for Hilfer fractional differential equations subject to a variety of boundary conditions, including multipoint boundary conditions, non-local integral boundary conditions, non-local integro-multipoint boundary conditions, integro-multistrip–multipoint boundary conditions, Riemann–Stieltjes integral multistrip boundary conditions, etc. In all of the above-mentioned boundary value problems, in order for the solution to be well defined, a zero initial condition is required.

In [33], the authors proposed a combination of Hilfer and Caputo fractional derivative operators, making it possible to investigate boundary value problems with a non-zero initial condition. They investigated a sequential fractional boundary value problem that contains a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions of the form

$$\left\{\begin{array}{c}{\displaystyle {}^H\mathbb{D}^{a,b,\varphi }({}^C\mathbb{D}^{\delta ,\varphi }\pi )(\omega )=f\left(\omega ,\pi (\omega ),{\mathbb{I}}^{c,\varphi }\pi (\omega ),{\int }_0^T\pi (s)dw(s)\right),\omega \in [0,T],}\hfill \\ {\displaystyle \pi (0)+{\lambda }_1\pi (T)=0,{}^C\mathbb{D}^{\gamma +\delta -1,\varphi }\pi (0)+{\lambda }_2{}^C\mathbb{D}^{\gamma +\delta -1,\varphi }\pi (T)=0,}\hfill \end{array}\right.$$

(1)

where ${\displaystyle {}^H\mathbb{D}^{a,b,\varphi }}$ and ${\displaystyle {}^C\mathbb{D}^{\delta ,\varphi },}$$0<a,b,\delta <1,$$\gamma =a+b-ab,$$a+\delta >1$ are the $\varphi$-Hilfer fractional derivative and $\varphi$-Caputo fractional derivative, respectively. Moreover, ${\lambda }_1,{\lambda }_2\in \mathbb{R}$, ${\displaystyle {\mathbb{I}}^{c,\varphi }}$ is the Riemann–Liouville fractional integral of order $c>0$, with respect to a function $\varphi$; $f:[0,T]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ is a nonlinear continuous function; ${\int }_0^T\pi (s)dw(s)$ is the Riemann–Stieltjes integral; and $w:[0,T]\to \mathbb{R}$ is a function of bounded variation.

In the present paper, we continue the study this topic by extending the results of [33] to cover coupled sequential fractional systems containing a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions of the form

$$\left\{\begin{array}{c}{\displaystyle {}^H\mathbb{D}^{a_1,b_1,\varphi }({}^C\mathbb{D}^{{\delta }_1,\varphi }{\pi }_1)(\omega )=f_1\left(\omega ,{\pi }_1(\omega ),{\pi }_2(\omega ),{\mathbb{I}}^{c_2,\varphi }{\pi }_2(\omega )\right),\omega \in [0,T],}\hfill \\ {\displaystyle {}^H\mathbb{D}^{a_2,b_2,\varphi }({}^C\mathbb{D}^{{\delta }_2,\varphi }{\pi }_2)(\omega )=f_2\left(\omega ,{\pi }_1(\omega ),{\mathbb{I}}^{c_1,\varphi }{\pi }_1(\omega ),{\pi }_2(\omega )\right),\omega \in [0,T],}\hfill \\ {\displaystyle {\pi }_1(0)+{\lambda }_1{\pi }_2(T)=0,{}^C\mathbb{D}^{{\gamma }_1+{\delta }_1-1,\varphi }{\pi }_1(0)+{\lambda }_2{}^C\mathbb{D}^{{\gamma }_2+{\delta }_2-1,\varphi }{\pi }_2(T)=0,}\hfill \\ {\displaystyle {\pi }_2(0)+{\mu }_1{\pi }_1(T)=0,{}^C\mathbb{D}^{{\gamma }_2+{\delta }_2-1,\varphi }{\pi }_2(0)+{\mu }_2{}^C\mathbb{D}^{{\gamma }_1+{\delta }_1-1,\varphi }{\pi }_1(T)=0,}\hfill \end{array}\right.$$

(2)

where ${\displaystyle {}^H\mathbb{D}^{a_i,b_i,\varphi }}$ and ${\displaystyle {}^C\mathbb{D}^{{\delta }_i,\varphi },}$ $0<a_i,b_i,{\delta }_i<1,$ ${\gamma }_i=a_i+b_i-a_i{b}_i,$ with $a_i+{\delta }_i>1$, $i=1,2$, are the $\varphi$-Hilfer fractional derivative and $\varphi$-Caputo fractional derivative, respectively. Moreover, ${\lambda }_i,{\mu }_i\in \mathbb{R}$ with ${\lambda }_i{\mu }_i\ne 1$, $i=1,2,$ ${\displaystyle {\mathbb{I}}^{c_i,\varphi }}$ are the Riemann–Liouville fractional integral of order $c_i>0$, $i=1,2$ with respect to a function $\varphi$, and $f_i:[0,T]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R},i=1,2$ are nonlinear continuous functions.

We use fixed point theory to obtain our main results. Thus, by using the Banach’s fixed point theorem, we prove the uniqueness of the solution, while via the Leray–Schauder alternative, we establish an existence result. Furthermore, numerical examples are illustrated to support the theoretical analysis.

The novelty of the present study lies in the fact that we consider a combination of fractional derivative operators in the sense of Hilfer and Caputo, subjected to non-separated boundary conditions, which is a new topic of research. The method used is standard, but its configuration in the problem at hand is new. It is imperative to note that the proposed combination of Hilfer and Caputo fractional derivative operators of order $(1,2]$ can be applied in other types of coupled boundary conditions, such as periodic, i.e.,

$$\begin{array}{c}{\pi }_1(0)+{\pi }_2(T)=0,{\pi }_1(0)+{}^{C}D^{a_2+{\delta }_2-1,\varphi }{\pi }_2(T)=0,\\ {\pi }_2(0)+{\pi }_1(T)=0,{\pi }_2(0)+{}^{C}D^{a_1+{\delta }_1-1,\varphi }{\pi }_1(T)=0,\end{array}$$

or separated, i.e.,

$$\begin{array}{c}{\pi }_1(0)+{\lambda }_1{}^C\mathbb{D}^{a_2+{\delta }_2-1,\varphi }{\pi }_2(0)=0,{\pi }_1(T)+{\overline{\lambda }}_1{}^C\mathbb{D}^{a_2+{\delta }_2-1,\varphi }{\pi }_2(T)=0,\\ {\pi }_2(0)+{\lambda }_2{}^C\mathbb{D}^{a_1+{\delta }_1-1,\varphi }{\pi }_1(0)=0,{\pi }_2(T)+{\overline{\lambda }}_2{}^C\mathbb{D}^{a_1+{\delta }_1-1,\varphi }{\pi }_1(T)=0.\end{array}$$

This paper is organized as follows: In Section 2, we recall the basic definitions and lemmas needed to prove our results. We prove also a lemma concerning a linear variant of the problem (2), which allows us to transform the nonlinear system (2) into integral equations. In Section 3, we prove our main results by using the Banach fixed point theorem as well as the Leray–Schauder alternative. In Section 4, some numerical examples are constructed to illustrate the obtained theoretical results. This paper closes with some concluding remarks.

2. Preliminaries

In this section, we recall some definitions, lemmas, and remarks that are used later in this paper.

Definition 1

([2]). Assume that $a_1>0$ and $g\in C([0,T],\mathbb{R})$ and $\varphi \in C^1([0,T],\mathbb{R})$ with ${\varphi }^{\prime }(\omega )>0$ for all $\omega \in [0,T]$. Then, ${\mathbb{I}}^{a_1,\varphi }g$ is defined by

$$\begin{array}{c}\hfill {\displaystyle {\mathbb{I}}^{a_1,\varphi }g(\omega )=\frac{1}{\Gamma (a_1)}{\int }_0^{\omega }{\varphi }^{\prime }(s){(\varphi (\omega )-\varphi (s))}^{a_1-1}g(s)ds,}\end{array}$$

where $\Gamma (a_1)$ is the Euler gamma function, and is called the ϕ-Riemann–Liouville fractional integral of the function g with respect to ϕ of order $a_1.$

Definition 2

([19]). Let $n-1<a_1<n$, $n\in \mathbb{N}$ and $g,\varphi \in C^n([0,T],\mathbb{R})$. Then,

$${\displaystyle {}^H\mathcal{D}^{a_1,b_1,\varphi }g(\omega )={\mathbb{I}}^{b_1(n-a_1),\varphi }{\left(\frac{1}{{\varphi }^{\prime }(\omega )}\frac{d}{d\omega }\right)}^n{\mathbb{I}}^{(1-b_1)(n-a_1),\varphi }g(\omega ),}$$

is called the ϕ-Hilfer fractional derivative of order $a_1$ of the function g with a parameter $b_1\in [0,1].$ In addition, the ϕ-Hilfer operator can be written as

$${}^H\mathcal{D}^{a_1,b_1,\varphi }g(\omega )={\mathbb{I}}^{{\gamma }_1-a_1,\varphi }\left({}^R\mathcal{D}^{{\gamma }_1,\varphi }g\right)(\omega ),$$

(3)

where ${}^R\mathcal{D}^{{\gamma }_1,\varphi }(·)$ is the ϕ-Riemann–Liouville fractional operator and ${\gamma }_1=a_1+b_1(n-a_1)$.

Definition 3

([34]). The ϕ-Caputo fractional derivative ${\displaystyle {}^C\mathbb{D}^{a_1,\varphi }(·)}$ of order $a_1$ of a function g is presented as

$${}^C\mathcal{D}^{a_1,\varphi }g(\omega )={\mathbb{I}}^{n-a_1,\varphi }{\left({\displaystyle \frac{1}{{\varphi }^{\prime }(\omega )}}{\displaystyle \frac{d}{d\omega }}\right)}^{n}g(\omega ),$$

(4)

where $n-1<a_1<n$, $n\in \mathbb{N}$ and $g,\varphi \in C^n([0,T],\mathbb{R}),$ while the ϕ-Riemann–Liouville fractional derivative ${\displaystyle {}^R\mathbb{D}^{a_1,\varphi }(·)}$ is defined by interchanging between two operators in (4) as

$${}^R\mathcal{D}^{a_1,\varphi }g(\omega )={\left({\displaystyle \frac{1}{{\varphi }^{\prime }(\omega )}}{\displaystyle \frac{d}{d\omega }}\right)}^n{\mathbb{I}}^{n-a_1,\varphi }g(\omega ).$$

Remark 1

([35]). The parameters $a_1,b_1,{\gamma }_1$ satisfy

$$\begin{array}{c}\hfill {\gamma }_1=a_1+b_1(n-a_1),n-1<a_1,{\gamma }_1<n,0\le b_1\le 1,\end{array}$$

and

$$\begin{array}{c}\hfill {\gamma }_1\ge a_1,{\gamma }_1>b_1,n-{\gamma }_1<n-b_1(n-a_1),\end{array}$$

where $a_1$ is an order of Hilfer operator and $b_1$ is an interpolated value between the Riemann–Liouville and Caputo differential operators.

Lemma 1

([19]). For the constants $a_1,a_2,a>0$, $0\le b_1\le 1$ and $\delta >1$, we have

(i)

${\displaystyle {\mathbb{I}}^{a_1,\varphi }\left({\mathbb{I}}^{a_2,\varphi }g\right)(\omega )={\mathbb{I}}^{a_1+a_2,\varphi }g(\omega );}$

(ii)

${\displaystyle {\mathbb{I}}^{a,\varphi }{(\varphi (\omega )-\varphi (0))}^{\delta -1}=\frac{\Gamma (\delta )}{\Gamma (a+\delta )}{(\varphi (\omega )-\varphi (0))}^{a+\delta -1};}$

(iii)

${\displaystyle {}^H\mathcal{D}^{a_1,b_1,\varphi }\left({\mathbb{I}}^{a_1,\varphi }g\right)(\omega )=g(\omega ).}$

In the following lemma, we denote by $A{C}^k[0,T]$ the k-times absolutely continuous functions on $[0,T]$.

Lemma 2

([19]). Let $g\in C^n[0,T]$, $n-1<a_1\le n$, $n\in \mathbb{N}$, $0\le b_1\le 1$, ${\gamma }_1=a_1+n{b}_1-a_1{b}_1$, ${\displaystyle \left({\mathbb{I}}^{(n-a_1)(1-b_1),\varphi }g\right)\in A{C}^k[0,T]}$. Then, we have

$$\begin{array}{c}\hfill {\displaystyle \left({\mathbb{I}}^{a_1,\varphi }{}^H\mathcal{D}^{a_1,b_1,\varphi }g\right)(\omega )=g(\omega )-\sum _{i=1}^n\frac{{(\varphi (\omega )-\varphi (0))}^{\gamma -i}}{\Gamma (\gamma -i+1)}{\Delta }_{\varphi }^{[n-i]}\left({\mathbb{I}}^{(1-b_1)(n-a_1),\varphi }g\right)(0),}\end{array}$$

where ${\displaystyle {\Delta }_{\varphi }^{[n-i]}={\left(\frac{1}{{\varphi }^{\prime }(\omega )}\frac{d}{d\omega }\right)}^{n-i}}$ and

$$\begin{array}{c}\hfill {\displaystyle {\mathbb{I}}^{a_1,\varphi }\left({}^C\mathcal{D}^{a_1,\varphi }g\right)(\omega )=g(\omega )-\sum _{i=0}^{n-1}\frac{{\Delta }_{\varphi }^{\left[i\right]}g(0)}{i!}{(\varphi (\omega )-\varphi (0))}^i.}\end{array}$$

(5)

In the following lemma, a linear variant of the sequential fractional Hilfer–Caputo system (2) is studied. We will use this lemma in defining the integral operators used in our theorems.

Lemma 3.

Let $g_i\in C^1[0,T],i=1,2,$ and ${\pi }_i\in C^1[0,T],i=1,2,$ be such that

$${\mathbb{I}}^{(1-a_i)(1-b_i),\varphi }{}^{c}D^{{\delta }_i,\varphi }{\pi }_i\in A{C}^1[0,T],{\mathbb{I}}^{1+b_i(a_i-1)}{g}_i\in A{C}^1[0,T],{\mathbb{I}}^{a_i+{\delta }_i}{g}_i\in A{C}^1[0,T],i=1,2,$$

and ${\lambda }_i{\mu }_i\ne 1,i=1,2.$ Then, ${\pi }_1,{\pi }_2$ are solutions of the Hilfer–Caputo fractional linear system

$$\left\{\begin{array}{c}{\displaystyle {}^H\mathbb{D}^{a_1,b_1,\varphi }({}^C\mathbb{D}^{{\delta }_1,\varphi }{\pi }_1)(\omega )=g_1(\omega ),\omega \in [0,T],}\hfill \\ {\displaystyle {}^H\mathbb{D}^{a_2,b_2,\varphi }({}^C\mathbb{D}^{{\delta }_2,\varphi }{\pi }_2)(\omega )=g_2(\omega ),\omega \in [0,T],}\hfill \\ {\displaystyle {\pi }_1(0)+{\lambda }_1{\pi }_2(T)=0,{}^C\mathbb{D}^{{\gamma }_1+{\delta }_1-1,\varphi }{\pi }_1(0)+{\lambda }_2{}^C\mathbb{D}^{{\gamma }_2+{\delta }_2-1,\varphi }{\pi }_2(T)=0,}\hfill \\ {\displaystyle {\pi }_2(0)+{\mu }_1{\pi }_1(T)=0,{}^C\mathbb{D}^{{\gamma }_2+{\delta }_2-1,\varphi }{\pi }_2(0)+{\mu }_2{}^C\mathbb{D}^{{\gamma }_1+{\delta }_1-1,\varphi }{\pi }_1(T)=0,}\hfill \end{array}\right.$$

(6)

if and only if ${\pi }_1,{\pi }_2$ satisfy the integral equations

$$\begin{array}{ccc}\hfill {\pi }_1(\omega )& =& {\displaystyle \frac{{\lambda }_1}{1-{\lambda }_1{\mu }_1}}\left\{\right[-{\mathbb{I}}^{a_2+{\delta }_2,\varphi }{g}_2(T)-{\displaystyle \frac{{(\varphi (T)-\varphi (0))}^{{\gamma }_2+{\delta }_2-1}}{\Gamma ({\gamma }_2+{\delta }_2)}}\hfill \\ & & \times {\displaystyle \frac{{\mu }_2}{1-{\lambda }_2{\mu }_2}}\left[-{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)+{\lambda }_2{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)\right]]\hfill \\ & & -{\lambda }_1[-{\mathbb{I}}^{a_1+{\delta }_1,\varphi }{g}_1(T)-{\mu }_1{\displaystyle \frac{{(\varphi (T)-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}}{\Gamma ({\gamma }_1+{\delta }_1)}}\hfill \\ & & \times {\displaystyle \frac{{\lambda }_2}{1-{\lambda }_2{\mu }_2}}\left[-{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)+{\mu }_2{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)\right]\left]\right\}\hfill \\ & & +{\displaystyle \frac{{(\varphi (\omega )-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}}{\Gamma ({\gamma }_1+{\delta }_1)}}{\displaystyle \frac{{\lambda }_2}{1-{\lambda }_2{\mu }_2}}[-{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)\hfill \\ & & +{\mu }_2{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)]+{\mathbb{I}}^{a_1+{\delta }_1,\varphi }{g}_1(\omega ),\omega \in [0,T],\hfill \end{array}$$

(7)

and

$$\begin{array}{ccc}\hfill {\pi }_2(\omega )& =& {\displaystyle \frac{{\lambda }_1}{1-{\lambda }_1{\mu }_1}}\left\{\right[-{\mathbb{I}}^{a_1+{\delta }_1,\varphi }{g}_1(T)-{\mu }_1{\displaystyle \frac{{(\varphi (T)-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}}{\Gamma ({\gamma }_1+{\delta }_1)}}\hfill \\ & & \times {\displaystyle \frac{{\lambda }_2}{1-{\lambda }_2{\mu }_2}}\left[-{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)+{\mu }_2{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)\right]]\hfill \\ & & -{\mu }_1{\lambda }_1[-{\mathbb{I}}^{a_2+{\delta }_2,\varphi }{g}_2(T)-{\displaystyle \frac{{(\varphi (T)-\varphi (0))}^{{\gamma }_2+{\delta }_2-1}}{\Gamma ({\gamma }_2+{\delta }_2)}}\hfill \\ & & \times {\displaystyle \frac{{\mu }_2}{1-{\lambda }_2{\mu }_2}}\left[-{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)+{\lambda }_2{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)\right]\left]\right\}\hfill \\ & & +{\displaystyle \frac{{(\varphi (\omega )-\varphi (0))}^{{\gamma }_2+{\delta }_2-1}}{\Gamma ({\gamma }_2+{\delta }_2)}}{\displaystyle \frac{{\mu }_2}{1-{\lambda }_2{\mu }_2}}[-{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)\hfill \\ & & +{\lambda }_2{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)]+{\mathbb{I}}^{a_2+{\delta }_2,\varphi }{g}_2(\omega ),\omega \in [0,T].\hfill \end{array}$$

(8)

Proof. 

Since $g_i\in C^1[0,T],i=1,2$, by taking the fractional integral operator ${\displaystyle {\mathbb{I}}^{a_i,\varphi }(·),i=1,2}$ on both sides of the first and second equations in (6) and using Lemma 2, we obtain

$${\displaystyle {}^C\mathbb{D}^{{\delta }_1,\varphi }{\pi }_1(\omega )=c_1{(\varphi (\omega )-\varphi (0))}^{{\gamma }_1-1}+{\mathbb{I}}^{a_1,\varphi }{g}_1(\omega ),}$$

(9)

and

$${\displaystyle {}^C\mathbb{D}^{{\delta }_2,\varphi }{\pi }_2(\omega )=c_2{(\varphi (\omega )-\varphi (0))}^{{\gamma }_2-1}+{\mathbb{I}}^{a_2,\varphi }{g}_2(\omega ),}$$

(10)

where ${\gamma }_i=a_i+b_i(1-a_i)$ and $c_i\in \mathbb{R},i=1,2$. Now, by taking the fractional integral ${\displaystyle {\mathbb{I}}^{{\delta }_i,\varphi },i=1,2}$ on both sides of the equations (9), (10) and applying Lemma 1, since ${\pi }_i\in C^1[0,T],{\mathbb{I}}^{a_i+{\delta }_i}{g}_i\in A{C}^1[0,T],i=1,2,$ we obtain

$${\displaystyle {\pi }_1(\omega )=d_1+\frac{\Gamma ({\gamma }_1)}{\Gamma ({\gamma }_1+{\delta }_1)}{c}_1{(\varphi (\omega )-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}+{\mathbb{I}}^{a_1+{\delta }_1,\varphi }{g}_1(\omega ),}$$

(11)

and

$${\displaystyle {\pi }_2(\omega )=d_2+\frac{\Gamma ({\gamma }_2)}{\Gamma ({\gamma }_2+{\delta }_2)}{c}_2{(\varphi (\omega )-\varphi (0))}^{{\gamma }_2+{\delta }_2-1}+{\mathbb{I}}^{a_2+{\delta }_2,\varphi }{g}_2(\omega ).}$$

(12)

Via Lemma 1, we have

$$\begin{array}{c}\hfill {}^C\mathbb{D}^{{\gamma }_i+{\delta }_i-1,\varphi }{\pi }_i(\omega )=\Gamma ({\gamma }_i)c_i+{\mathbb{I}}^{a_i-{\gamma }_i+1,\varphi }{g}_i(\omega ),i=1,2.\end{array}$$

Now, combining the boundary conditions

$${\displaystyle {\pi }_1(0)+{\lambda }_1{\pi }_2(T)=0,\mathrm{and}{}^C\mathbb{D}^{{\gamma }_1+{\delta }_1-1,\varphi }{\pi }_1(0)+{\lambda }_2{}^C\mathbb{D}^{{\gamma }_2+{\delta }_2-1,\varphi }{\pi }_2(T)=0,}$$

with (11) and (12), we obtain

$$\begin{array}{ccc}& & {\displaystyle d_1+{\lambda }_1\left[d_2+\frac{\Gamma ({\gamma }_2)}{\Gamma ({\gamma }_2+{\delta }_2)}{c}_2{(\varphi (T)-\varphi (0))}^{{\gamma }_2+{\delta }_2-1}+{\mathbb{I}}^{a_2+{\delta }_2,\varphi }{g}_2(T)\right]=0,}\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& & \Gamma ({\gamma }_1)c_1+{\lambda }_2\left[\Gamma ({\gamma }_2)c_2+{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)\right]=0.\hfill \end{array}$$

(14)

By ${\pi }_2(0)+{\mu }_1{\pi }_1(T)=0$ and ${\displaystyle {}^C\mathbb{D}^{{\gamma }_2+{\delta }_2-1,\varphi }{\pi }_2(0)+{\mu }_2{}^C\mathbb{D}^{{\gamma }_1+{\delta }_1-1,\varphi }{\pi }_1(T)=0,}$ we obtain

$$\begin{array}{ccc}& & d_2+{\mu }_1\left[d_1+{\displaystyle \frac{\Gamma ({\gamma }_1)}{\Gamma ({\gamma }_1+{\delta }_1)}}{c}_1{(\varphi (T)-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}+{\mathbb{I}}^{a_1+{\delta }_1,\varphi }{g}_1(T)\right]=0,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & \Gamma ({\gamma }_2)c_2+{\mu }_2\left[\Gamma ({\gamma }_1)c_1+{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)\right]=0.\hfill \end{array}$$

(16)

From Equations (14) and (16), we have

$$\begin{array}{ccc}\hfill c_1& =& {\displaystyle \frac{{\lambda }_2}{\Gamma ({\gamma }_1)(1-{\lambda }_2{\mu }_2)}}\left[-{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)+{\mu }_2{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)\right],\hfill \\ \hfill c_2& =& {\displaystyle \frac{{\mu }_2}{\Gamma ({\gamma }_2)(1-{\lambda }_2{\mu }_2)}}\left[-{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)+{\lambda }_2{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)\right].\hfill \end{array}$$

From Equations (13) and (15), we have

$$\begin{array}{ccc}\hfill d_1& =& {\displaystyle \frac{{\lambda }_1}{1-{\lambda }_1{\mu }_1}}\left\{\right[-{\mathbb{I}}^{a_2+{\delta }_2,\varphi }{g}_2(T)-{\displaystyle \frac{1}{\Gamma ({\gamma }_2+{\delta }_2)}}{(\varphi (T)-\varphi (0))}^{{\gamma }_2+{\delta }_2-1}\hfill \\ & & \times {\displaystyle \frac{{\mu }_2}{1-{\lambda }_2{\mu }_2}}\left[-{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)+{\lambda }_2{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)\right]]\hfill \\ & & -{\lambda }_1[-{\mathbb{I}}^{a_1{\delta }_1,\varphi }{g}_1(T)-{\mu }_1{\displaystyle \frac{1}{\Gamma ({\gamma }_1+{\delta }_1)}}{(\varphi (T)-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}\hfill \\ & & \times {\displaystyle \frac{{\lambda }_2}{1-{\lambda }_2{\mu }_2}}\left[-{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)+{\mu }_2{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)\right]\left]\right\},\hfill \\ \hfill d_2& =& {\displaystyle \frac{{\lambda }_1}{1-{\lambda }_1{\mu }_1}}\left\{\right[-{\mathbb{I}}^{a_1+{\delta }_1,\varphi }{g}_1(T)-{\mu }_1{\displaystyle \frac{1}{\Gamma ({\gamma }_1+{\delta }_1)}}{(\varphi (T)-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}\hfill \\ & & \times {\displaystyle \frac{{\lambda }_2}{1-{\lambda }_2{\mu }_2}}\left[-{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)+{\mu }_2{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)\right]]\hfill \\ & & -{\mu }_1{\lambda }_1[-{\mathbb{I}}^{a_2+{\delta }_2,\varphi }{g}_2(T)-{\displaystyle \frac{1}{\Gamma ({\gamma }_2+{\delta }_2)}}{(\varphi (T)-\varphi (0))}^{{\gamma }_2+{\delta }_2-1}\hfill \\ & & \times {\displaystyle \frac{{\mu }_2}{1-{\lambda }_2{\mu }_2}}\left[-{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)+{\lambda }_2{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)\right]\left]\right\}.\hfill \end{array}$$

Replacing the values $c_1,c_2,d_1$, and $d_2$ in (11) and (12), we obtain the solutions (7) and (8).

Conversely, we will show that ${\pi }_1(\omega )$ and ${\pi }_2(\omega )$ expressed in (7) and (8) satisfy the non-separated boundary value problem (6). Firstly, from ${\pi }_i\in C^1[0,T]$, $i=1,2,{\mathbb{I}}^{a_i+{\delta }_i,\varphi }{g}_i\in C^1[0,T]$, $i=1,2$, and by applying the operators ${}^C\mathbb{D}^{{\delta }_1,\varphi }$ and ${}^C\mathbb{D}^{{\delta }_2,\varphi }$ to (7) and (8), respectively, we obtain

$$\begin{array}{ccc}\hfill {}^C\mathbb{D}^{{\delta }_1,\varphi }{\pi }_1(\omega )& =& {\displaystyle \frac{{\lambda }_2{(\varphi (\omega )-\varphi (0))}^{{\gamma }_1-1}}{\Gamma ({\gamma }_1)(1-{\lambda }_2{\mu }_2)}}\left[-{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)+{\mu }_2{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)\right]\hfill \\ & & +{\mathbb{I}}^{a_1,\varphi }{g}_1(\omega ),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {}^C\mathbb{D}^{{\delta }_2,\varphi }{\pi }_2(\omega )& =& {\displaystyle \frac{{\mu }_2{(\varphi (\omega )-\varphi (0))}^{{\gamma }_2-1}}{\Gamma ({\gamma }_2)(1-{\lambda }_2{\mu }_2)}}\left[-{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)+{\lambda }_2{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)\right]\hfill \\ & & {\displaystyle +{\mathbb{I}}^{a_2,\varphi }{g}_2(\omega ),}\hfill \end{array}$$

using the fact that the Caputo derivative of a constant is zero. In the next step, since ${\mathbb{I}}^{(1-a_i)(1-b_i),\varphi }{}^{c}D^{{\delta }_i,\varphi }{\pi }_i\in A{C}^1[0,T],{\mathbb{I}}^{1+b_i(a_i-1)}{g}_i\in A{C}^1[0,T],i=1,2,$ we take the operators ${}^H\mathbb{D}^{a_1,b_1,\varphi }(·)$ and ${}^H\mathbb{D}^{a_2,b_2,\varphi }(·)$ into the above two equations, respectively. We consider the term

$$\begin{array}{ccc}\hfill {}^H\mathbb{D}^{a_1,b_1,\varphi }{(\varphi (\omega )-\varphi (0))}^{{\gamma }_1-1}& =& {\mathbb{I}}^{{\gamma }_1-a_1,\varphi }\left({}^R\mathbb{D}^{{\gamma }_1,\varphi }{(\varphi (\omega )-\varphi (0))}^{{\gamma }_1-1}\right)\hfill \\ & =& {\mathbb{I}}^{{\gamma }_1-a_1,\varphi }\left({\displaystyle \frac{\Gamma ({\gamma }_1)}{\Gamma ({\gamma }_1-{\gamma }_1)}}{(\varphi (\omega )-\varphi (0))}^{-1}\right)\hfill \\ & \to & 0,\hfill \end{array}$$

and, similarly, ${}^H\mathbb{D}^{a_2,b_2,\varphi }{(\varphi (\omega )-\varphi (0))}^{{\gamma }_2-1}\to 0$. According to Lemma 1$(iii)$, ${\pi }_1(\omega )$ and ${\pi }_2(\omega )$ satisfy the first two fractional differential equations in the boundary value problem (6). In the final step, we show that ${\pi }_1(\omega )$ and ${\pi }_2(\omega )$ satisfy the boundary conditions of (6). Taking the $\varphi$-Caputo fractional derivative of orders ${\gamma }_1+{\delta }_1-1$ and ${\gamma }_2+{\delta }_2-1$ into (7) and (8), respectively, since ${\pi }_i\in C^1[0,T]$, we obtain

$${}^C\mathbb{D}^{{\gamma }_1+{\delta }_1-1,\varphi }{\pi }_1(\omega )={\displaystyle \frac{{\lambda }_2}{(1-{\lambda }_2{\mu }_2)}}\left[-{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)+{\mu }_2{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)\right]+{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(\omega )$$

and

$${}^C\mathbb{D}^{{\gamma }_2+{\delta }_2-1,\varphi }{\pi }_2(\omega )={\displaystyle \frac{{\mu }_2}{(1-{\lambda }_2{\mu }_2)}}\left[-{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1(T)+{\lambda }_2{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(T)\right]+{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2(\omega ).$$

Putting the values $\omega =0$ and $\omega =T$ in (7) and (8) and in the above two equations, we can find that ${\pi }_1(\omega )$ and ${\pi }_2(\omega )$ satisfy the boundary conditions in (6). Therefore, the proof is completed. □

3. Main Results

We denote that the space of all continuous functions ${\pi }_1$ from $[0,T]$ to $\mathbb{R}$ by $X.$X endowed with the sup-norm $\parallel {\pi }_1{\parallel }_X=sup\{|{\pi }_1(\omega ):\omega \in [0,T]\}$ is a Banach space. The product space $(X\times Y,\parallel ·{\parallel }_{X\times Y})$ is a Banach space with norm $\parallel ({\pi }_1,{\pi }_2){\parallel }_{X\times Y}=\parallel {\pi }_1{\parallel }_X+{\parallel {\pi }_2\parallel }_Y$ for $({\pi }_1,{\pi }_2)\in X\times Y.$ In view of Lemma 3, we define an operator $\mathbb{Z}:X\times Y⟶X\times Y$ by

$$\mathbb{Z}({\pi }_1,{\pi }_2)(\omega )=\left(\begin{array}{c}{\mathbb{Z}}_1({\pi }_1,{\pi }_2)(\omega )\\ {\mathbb{Z}}_2({\pi }_1,{\pi }_2)(\omega )\end{array}\right),$$

where

$$\begin{array}{ccc}\hfill {\mathbb{Z}}_1({\pi }_1,{\pi }_2)(\omega )& =& {\displaystyle \frac{{\lambda }_1}{1-{\lambda }_1{\mu }_1}}\left\{\right[-{\mathbb{I}}^{a_2+{\delta }_2,\varphi }{g}_2^{\pi }(T)-{\displaystyle \frac{{(\varphi (T)-\varphi (0))}^{{\gamma }_2+{\delta }_2-1}}{\Gamma ({\gamma }_2+{\delta }_2)}}\hfill \\ & & \times {\displaystyle \frac{{\mu }_2}{1-{\lambda }_2{\mu }_2}}\left[-{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1^{\pi }(T)+{\lambda }_2{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2^{\pi }(T)\right]]\hfill \\ & & -{\lambda }_1[-{\mathbb{I}}^{a_1+{\delta }_1,\varphi }{g}_1^{\pi }(T)-{\mu }_1{\displaystyle \frac{{(\varphi (T)-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}}{\Gamma ({\gamma }_1+{\delta }_1)}}\hfill \\ & & \times {\displaystyle \frac{{\lambda }_2}{1-{\lambda }_2{\mu }_2}}\left[-{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2^{\pi }(T)+{\mu }_2{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1^{\pi }(T)\right]\left]\right\}\hfill \\ & & +{\displaystyle \frac{{(\varphi (\omega )-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}}{\Gamma ({\gamma }_1+{\delta }_1)}}{\displaystyle \frac{{\lambda }_2}{1-{\lambda }_2{\mu }_2}}[-{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2^{\pi }(T)\hfill \\ & & +{\mu }_2{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1^{\pi }(T)]+{\mathbb{I}}^{a_1+{\delta }_1,\varphi }{g}_1^{\pi }(\omega ),\omega \in [0,T],\hfill \end{array}$$

(17)

and

$$\begin{array}{ccc}\hfill {\mathbb{Z}}_2({\pi }_1,{\pi }_2)(\omega )& =& {\displaystyle \frac{{\lambda }_1}{1-{\lambda }_1{\mu }_1}}\left\{\right[-{\mathbb{I}}^{a_1+{\delta }_1,\varphi }{g}_1^{\pi }(T)-{\mu }_1{\displaystyle \frac{{(\varphi (T)-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}}{\Gamma ({\gamma }_1+{\delta }_1)}}\hfill \\ & & \times {\displaystyle \frac{{\lambda }_2}{1-{\lambda }_2{\mu }_2}}\left[-{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2^{\pi }(T)+{\mu }_2{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1^{\pi }(T)\right]]\hfill \\ & & -{\mu }_1{\lambda }_1[-{\mathbb{I}}^{a_2+{\delta }_2,\varphi }{g}_2^{\pi }(T)-{\displaystyle \frac{{(\varphi (T)-\varphi (0))}^{{\gamma }_2+{\delta }_2-1}}{\Gamma ({\gamma }_2+{\delta }_2)}}\hfill \\ & & \times {\displaystyle \frac{{\mu }_2}{1-{\lambda }_2{\mu }_2}}\left[-{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1^{\pi }(T)+{\lambda }_2{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2^{\pi }(T)\right]\left]\right\}\hfill \\ & & +{\displaystyle \frac{{(\varphi (\omega )-\varphi (0))}^{{\gamma }_2+{\delta }_2-1}}{\Gamma ({\gamma }_2+{\delta }_2)}}{\displaystyle \frac{{\mu }_2}{1-{\lambda }_2{\mu }_2}}[-{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }{g}_1^{\pi }(T)\hfill \\ & & +{\lambda }_2{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }{g}_2^{\pi }(T)]+{\mathbb{I}}^{a_2+{\delta }_2,\varphi }{g}_2^{\pi }(\omega ),\omega \in [0,T],\hfill \end{array}$$

(18)

where we used the notations

$$g_1^{\pi }(y)=f_1\left(y,{\pi }_1(y),{\pi }_2(y),{\mathbb{I}}^{c_2,\varphi }{\pi }_2(y)\right),y=T,\omega ,$$

and

$$g_2^{\pi }(y)=f_2\left(y,{\pi }_1(y),{\mathbb{I}}^{c_1,\varphi }{\pi }_1(y),{\pi }_2(y)\right),y=T,\omega .$$

For computational convenience, we set some notations of constants

$$\begin{array}{ccc}\hfill {\mathsf{\Phi }}_{\eta }(\omega )& =& {\displaystyle \frac{{(\varphi (\omega )-\varphi (0))}^{\eta }}{\Gamma (\eta +1)}},\hfill \\ \hfill {\mathbb{Q}}_{0i}& =& 1+{\mathsf{\Phi }}_{c_i}(T),i=1,2,\hfill \\ \hfill {\mathbb{Q}}_1& =& {\displaystyle \frac{|{\lambda }_1|}{|1-{\lambda }_1{\mu }_1||1-{\lambda }_2{\mu }_2|}}\{|{\mu }_2|{\mathsf{\Phi }}_{{\gamma }_2+{\delta }_2-1}(T){\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)\hfill \\ & & +|{\lambda }_1|\left[{\mathsf{\Phi }}_{a_1+{\delta }_1}(T)+|{\lambda }_2||{\mu }_1|{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)\right]\}\hfill \\ & & +{\displaystyle \frac{|{\lambda }_2||{\mu }_2|}{|1-{\lambda }_2{\mu }_2|}}{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)+{\mathsf{\Phi }}_{a_1+{\delta }_1}(T),\hfill \\ \hfill {\mathbb{Q}}_2& =& {\displaystyle \frac{|{\lambda }_1|}{|1-{\lambda }_1{\mu }_1||1-{\lambda }_2{\mu }_2|}}\{\left[{\mathsf{\Phi }}_{a_2+{\delta }_2}(T)+|{\lambda }_2||{\mu }_2|{\mathsf{\Phi }}_{{\gamma }_2+{\delta }_2-1}(T){\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)\right]\hfill \\ & & +|{\lambda }_1||{\lambda }_2||{\mu }_1|{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)\}\hfill \\ & & +{\displaystyle \frac{|{\lambda }_2|}{|1-{\lambda }_2{\mu }_2|}}{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T),\hfill \\ \hfill {\mathbb{Q}}_3& =& {\displaystyle \frac{|{\lambda }_1|}{|1-{\lambda }_1{\mu }_1||1-{\lambda }_2{\mu }_2|}}\{{\mathsf{\Phi }}_{a_1+{\delta }_1}(T)+|{\lambda }_2||{\mu }_1||{\mu }_2|{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)\hfill \\ & & +|{\lambda }_1||{\mu }_1||{\mu }_2|{\mathsf{\Phi }}_{{\gamma }_2+{\delta }_2-1}(T){\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)\}\hfill \\ & & +{\displaystyle \frac{|{\mu }_2|}{|1-{\lambda }_2{\mu }_2|}}{\mathsf{\Phi }}_{{\gamma }_2+{\delta }_2-1}(T){\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T),\hfill \\ \hfill {\mathbb{Q}}_4& =& {\displaystyle \frac{|{\lambda }_1|}{|1-{\lambda }_1{\mu }_1||1-{\lambda }_2{\mu }_2|}}\{|{\lambda }_2||{\mu }_1|{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)\hfill \\ & & +|{\lambda }_1||{\mu }_1|\left[{\mathsf{\Phi }}_{a_2+{\delta }_2}(T)+|{\lambda }_2||{\mu }_2|{\mathsf{\Phi }}_{{\gamma }_2+{\delta }_2-1}(T){\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)\right]\}\hfill \\ & & +{\displaystyle \frac{|{\lambda }_2||{\mu }_2|}{|1-{\lambda }_2{\mu }_2|}}{\mathsf{\Phi }}_{{\gamma }_2+{\delta }_2-1}(T){\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)+{\mathsf{\Phi }}_{a_2+{\delta }_2}(T).\hfill \end{array}$$

(19)

The uniqueness of the solution of the system (2) is proved in the next theorem by using the classical contraction fixed point theorem according to Banach [36].

Theorem 1.

Let $f_1,f_2:[0,T]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ be such that for

$({\mathbb{G}}_1)$,

there exists ${\mathbb{K}}_i>0,i=1,2$, such that

$$\begin{array}{ccc}& & |f_1(\omega ,x_1,x_2,x_3)-f_1(\omega ,y_1,y_2,y_3)|\le {\mathbb{K}}_1(|x_1-y_1|+|x_2-y_2|+|x_3-y_3|),\hfill \\ & & |f_2(\omega ,x_1,x_2,x_3)-f_2(\omega ,y_1,y_2,y_3)|\le {\mathbb{K}}_2(|x_1-y_1|+|x_2-y_2|+|x_3-y_3|),\hfill \end{array}$$

for all $\omega \in [0,T]$ and $x_i,y_i\in \mathbb{R}$, $i=1,2,3$.

If

$$({\mathbb{Q}}_1+{\mathbb{Q}}_3){\mathbb{K}}_1(1+{\mathbb{Q}}_{02})+({\mathbb{Q}}_2+{\mathbb{Q}}_4){\mathbb{K}}_2(1+{\mathbb{Q}}_{01})<1,$$

(20)

where ${\mathbb{Q}}_{0i},i=1,2,{\mathbb{Q}}_j,j=1,2,3,4$ are defined by (19), then there exists a unique solution to the system (2) over the interval $[0,T]$.

Proof. 

Denote ${\mathbb{M}}_i=sup\{|f_i(\omega ,0,0,0)|:\omega \in [0,T]\},i=1,2$ and ${\mathbb{B}}_r=\{({\pi }_1,{\pi }_2)\in X\times Y:\parallel ({\pi }_1,{\pi }_2)\parallel \le r\}$ with

$$\begin{array}{c}\hfill r\ge {\displaystyle \frac{({\mathbb{Q}}_1+{\mathbb{Q}}_3){\mathbb{M}}_1+({\mathbb{Q}}_2+{\mathbb{Q}}_4){\mathbb{M}}_2}{1-[{\mathbb{K}}_1(1+{\mathbb{Q}}_{02})({\mathbb{Q}}_1+{\mathbb{Q}}_3)+{\mathbb{K}}_2(1+{\mathbb{Q}}_{01})({\mathbb{Q}}_2+{\mathbb{Q}}_4)]}}.\end{array}$$

Using $({\mathbb{G}}_1),$ for all $({\pi }_1,{\pi }_2)\in X\times Y,$ we have

$$\begin{array}{ccc}\hfill |g_1^{\pi }(\omega )|& =& |f_1\left(\omega ,{\pi }_1(\omega ),{\pi }_2(\omega ),{\mathbb{I}}^{c_2,\varphi }{\pi }_2(\omega )\right)|\hfill \\ & \le & |f\left(\omega ,{\pi }_1(\omega ),{\pi }_2(\omega ),{\mathbb{I}}^{c_2,\varphi }{\pi }_2(\omega )\right)-f(\omega ,0,0,0)|+|f(\omega ,0,0,0)|\hfill \\ & \le & {\mathbb{K}}_1\left(|{\pi }_1(\omega )|+|{\pi }_2(\omega )|+{\mathbb{I}}^{c_2,\varphi }|{\pi }_2(\omega )|\right)+{\mathbb{M}}_1\hfill \\ & \le & {\mathbb{K}}_1\left(\parallel {\pi }_1{\parallel }_X+\parallel {\pi }_2{\parallel }_Y+{\displaystyle \frac{{(\varphi (T)-\varphi (0))}^{c_2}}{\Gamma (c_2+1)}}{\parallel {\pi }_2\parallel }_Y\right)+{\mathbb{M}}_1\hfill \\ & =& {\mathbb{K}}_1\left(\parallel {\pi }_1{\parallel }_X+{\parallel {\pi }_2\parallel }_Y\left[1+{\displaystyle \frac{{(\varphi (T)-\varphi (0))}^{c_2}}{\Gamma (c_2+1)}}\right]\right)+\mathbb{M}\hfill \\ & =& {\mathbb{K}}_1\left(\parallel {\pi }_1{\parallel }_X+[1+{\mathsf{\Phi }}_{c_2}(T)]{\parallel {\pi }_2\parallel }_Y\right)+{\mathbb{M}}_1\hfill \\ & \le & {\mathbb{K}}_1\left(\parallel {\pi }_1{\parallel }_X+\parallel {\pi }_2{\parallel }_Y+{\mathbb{Q}}_{02}(\parallel {\pi }_1{\parallel }_X+{\parallel {\pi }_2\parallel }_Y\right)+{\mathbb{M}}_1\hfill \\ & =& {\mathbb{K}}_1\left(1+{\mathbb{Q}}_{02})(\parallel {\pi }_1{\parallel }_X+{\parallel {\pi }_2\parallel }_Y\right)+{\mathbb{M}}_1\hfill \\ & =& {\mathbb{K}}_1(1+{\mathbb{Q}}_{02})\parallel ({\pi }_1,{\pi }_2){\parallel }_{X\times Y}+{\mathbb{M}}_1\hfill \\ & \le & {\mathbb{K}}_1(1+{\mathbb{Q}}_{02})r+{\mathbb{M}}_1,\hfill \end{array}$$

and, similarly,

$$\begin{array}{ccc}\hfill |g_2^{\pi }(\omega )|& =& |f_2\left(\omega ,{\pi }_1(\omega ),{\mathbb{I}}^{c_1,\varphi }{\pi }_1(\omega ),{\pi }_2(\omega )\right)|\hfill \\ & \le & {\mathbb{K}}_2\left(\parallel {\pi }_1{\parallel }_X{\mathbb{Q}}_{01}+{\parallel {\pi }_2\parallel }_Y\right)+{\mathbb{M}}_2\hfill \\ & \le & {\mathbb{K}}_2(1+{\mathbb{Q}}_{01})r+{\mathbb{M}}_2.\hfill \end{array}$$

We will show that $\mathbb{Z}{\mathbb{B}}_r\subseteq {\mathbb{B}}_r.$ For all ${\pi }_1,{\pi }_2\in X\times Y$, we have

$$\begin{array}{ccc}\hfill |{\mathbb{Z}}_1({\pi }_1,{\pi }_2)(\omega )|& \le & {\displaystyle \frac{|{\lambda }_1|}{|1-{\lambda }_1{\mu }_1|}}\{[{\mathbb{I}}^{a_2+{\delta }_2,\varphi }|g_2^{\pi }(T)|+{\mathsf{\Phi }}_{{\gamma }_2+{\delta }_2-1}(T)\hfill \\ & & \times {\displaystyle \frac{|{\mu }_2|}{|1-{\lambda }_2{\mu }_2|}}\left[{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }|g_1^{\pi }(T)|+|{\lambda }_2|{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }|g_2^{\pi }(T)|\right]]\hfill \\ & & +{\lambda }_1|[{\mathbb{I}}^{a_1+{\delta }_1,\varphi }|g_1^{\pi }(T)|+|{\mu }_1|{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T)\hfill \\ & & \times {\displaystyle \frac{|{\lambda }_2|}{|1-{\lambda }_2{\mu }_2|}}\left[{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }|g_2^{\pi }(T)|+|{\mu }_2|{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }|g_1^{\pi }(T)|\right]]\}\hfill \\ & & +{\displaystyle \frac{1}{\Gamma ({\gamma }_1+{\delta }_1)}}{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(\omega ){\displaystyle \frac{|{\lambda }_2|}{|1-{\lambda }_2{\mu }_2|}}[{\mathbb{I}}^{a_2-{\gamma }_2+1,\varphi }|g_2^{\pi }(T)|\hfill \\ & & +|{\mu }_2|{\mathbb{I}}^{a_1-{\gamma }_1+1,\varphi }|g_1^{\pi }(T)|]+{\mathbb{I}}^{a_1+{\delta }_1,\varphi }|g_1^{\pi }(\omega )|\hfill \\ & \le & \{{\displaystyle \frac{|{\lambda }_1|}{|1-{\lambda }_1{\mu }_1||1-{\lambda }_2{\mu }_2|}}\{|{\mu }_2|{\mathsf{\Phi }}_{{\gamma }_2+{\delta }_2-1}(T){\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)\hfill \\ & & +|{\lambda }_1|\left[{\mathsf{\Phi }}_{a_1+{\delta }_1}(T)+|{\lambda }_2||{\mu }_1|{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)\right]\}\hfill \\ & & +{\displaystyle \frac{|{\lambda }_2||{\mu }_2|}{|1-{\lambda }_2{\mu }_2|}}{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)+{\mathsf{\Phi }}_{a_1+{\delta }_1}(T)\}|g_1^{\pi }|(T)\hfill \\ & & +\{{\displaystyle \frac{|{\lambda }_1|}{|1-{\lambda }_1{\mu }_1||1-{\lambda }_2{\mu }_2|}}\{[{\mathsf{\Phi }}_{a_2+{\delta }_2}(T)\hfill \\ & & +|{\lambda }_2||{\mu }_2|{\mathsf{\Phi }}_{{\gamma }_2+{\delta }_2-1}(T){\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)]\hfill \\ & & +|{\lambda }_1||{\lambda }_2||{\mu }_1|{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)\}\hfill \\ & & +{\displaystyle \frac{|{\lambda }_2|}{|1-{\lambda }_2{\mu }_2|}}{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)\}|g_2^{\pi }|(T)\hfill \\ & \le & {\mathbb{Q}}_1[{\mathbb{K}}_1\left(1+{\mathbb{Q}}_{02})r+{\mathbb{M}}_1\right]+{\mathbb{Q}}_2[{\mathbb{K}}_2\left(1+{\mathbb{Q}}_{01})r+{\mathbb{M}}_2\right].\hfill \end{array}$$

Hence,

$$\parallel {\mathbb{Z}}_1({\pi }_1,{\pi }_2){\parallel }_X\le {\mathbb{Q}}_1[{\mathbb{K}}_1\left(1+{\mathbb{Q}}_{02})r+{\mathbb{M}}_1\right]+{\mathbb{Q}}_2[{\mathbb{K}}_2\left(1+{\mathbb{Q}}_{01})r+{\mathbb{M}}_2\right].$$

Similarly, we obtain

$$\parallel {\mathbb{Z}}_2({\pi }_1,{\pi }_2){\parallel }_Y\le {\mathbb{Q}}_3[{\mathbb{K}}_1\left(1+{\mathbb{Q}}_{02})r+{\mathbb{M}}_1\right]+{\mathbb{Q}}_4[{\mathbb{K}}_2\left(1+{\mathbb{Q}}_{01})r+{\mathbb{M}}_2\right].$$

Consequently,

$$\begin{array}{ccc}\hfill \parallel \mathbb{Z}({\pi }_1,{\pi }_2){\parallel }_{X\times Y}& \le & \parallel {\mathbb{Z}}_1({\pi }_1,{\pi }_2){\parallel }_X+{\parallel {\mathbb{Z}}_2({\pi }_1,{\pi }_2)\parallel }_Y\hfill \\ & \le & {\mathbb{K}}_1(1+{\mathbb{Q}}_{02})({\mathbb{Q}}_1+{\mathbb{Q}}_2)+{\mathbb{K}}_2(1+{\mathbb{Q}}_{01})({\mathbb{Q}}_2+{\mathbb{Q}}_4)\hfill \\ & & +({\mathbb{Q}}_1+{\mathbb{Q}}_3){\mathbb{M}}_1+({\mathbb{Q}}_2+{\mathbb{Q}}_4){\mathbb{M}}_2\le r,\hfill \end{array}$$

and, therefore, $\mathbb{Z}{\mathbb{B}}_r\subseteq {\mathbb{B}}_r$.

Again, making use of $({\mathbb{G}}_1),$ we have

$$\begin{array}{ccc}& & |f_1\left(\omega ,x_1,x_2,{\mathbb{I}}^{c_2,\varphi }{x}_2\right)-f_1\left(\omega ,y_1,y_2,{\mathbb{I}}^{c_2,\varphi }{y}_2\right)|\hfill \\ & \le & {\mathbb{K}}_1(|x_1-y_1|+|x_2-y_2|+|{\mathbb{I}}^{c_2,\varphi }{x}_1-{\mathbb{I}}^{c_2,\varphi }{y}_1|)\hfill \\ & \le & {\mathbb{K}}_1\left(\parallel x_1-y_1{\parallel }_X+\parallel x_2-y_2{\parallel }_Y+{\displaystyle \frac{{(\varphi (T)-\varphi (0))}^{c_2}}{\Gamma (c_2+1)}}{\parallel x_2-y_2\parallel }_Y\right)\hfill \\ & \le & {\mathbb{K}}_1(\parallel x_1-y_1{\parallel }_X+{\mathbb{Q}}_{02}\parallel x_2-y_2{\parallel }_Y)\hfill \\ & \le & {\mathbb{K}}_1(1+{\mathbb{Q}}_{02})[\parallel x_1-y_1{\parallel }_X+\parallel x_2-y_2{\parallel }_Y)].\hfill \end{array}$$

Similarly, we obtain

$$\begin{array}{ccc}& & |f_2\left(\omega ,x_1,{\mathbb{I}}^{c_1,\varphi }{x}_1,x_2\right)-f_2\left(\omega ,y_1,{\mathbb{I}}^{c_1,\varphi }{y}_1,y_2\right)|\hfill \\ & \le & {\mathbb{K}}_2({\mathbb{Q}}_{01}\parallel x_1-y_1{\parallel }_X+\parallel x_2-y_2{\parallel }_Y)\hfill \\ & \le & {\mathbb{K}}_2(1+{\mathbb{Q}}_{01})[\parallel x_1-y_1{\parallel }_X+\parallel x_2-y_2{\parallel }_Y)].\hfill \end{array}$$

Next, we will show that the operator $\mathbb{Z}$ is a contraction. For $(x_1,y_1),(x_2,y_2)\in {\mathbb{B}}_r,$ we have

$$\begin{array}{ccc}& & |{\mathbb{Z}}_1(x_1,y_1)(\omega )-{\mathbb{Z}}_1(x_2,y_2)(\omega )|\hfill \\ & \le & \left\{{\displaystyle \frac{|{\lambda }_1|}{|1-{\lambda }_1{\mu }_1||1-{\lambda }_2{\mu }_2|}}\right\{|{\mu }_2|{\mathsf{\Phi }}_{{\gamma }_2+{\delta }_2-1}(T){\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)\hfill \\ & & +|{\lambda }_1|\left[{\mathsf{\Phi }}_{a_1+{\delta }_1}(T)+|{\lambda }_2||{\mu }_1|{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)\right]\}\hfill \\ & & +{\displaystyle \frac{|{\lambda }_2||{\mu }_2|}{|1-{\lambda }_2{\mu }_2|}}{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)+{\mathsf{\Phi }}_{a_1+{\delta }_1}(T)\}\hfill \\ & & \times |f_1\left(\omega ,x_1,x_2,{\mathbb{I}}^{c_2,\varphi }{\pi }_2\right)-f_1\left(\omega ,y_1,y_2,{\mathbb{I}}^{c_2,\varphi }{y}_2\right)|\hfill \\ & & +\left\{{\displaystyle \frac{|{\lambda }_1|}{|1-{\lambda }_1{\mu }_1||1-{\lambda }_2{\mu }_2|}}\right\{\left[{\mathsf{\Phi }}_{a_2+{\delta }_2}(T)+|{\lambda }_2||{\mu }_2|{\mathsf{\Phi }}_{{\gamma }_2+{\delta }_2-1}(T){\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)\right]\hfill \\ & & +|{\lambda }_1||{\lambda }_2||{\mu }_1|{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)\}+{\displaystyle \frac{|{\lambda }_2|}{|1-{\lambda }_2{\mu }_2|}}{\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T){\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)\}\hfill \\ & & \times |f_2\left(\omega ,x_1,{\mathbb{I}}^{c_1,\varphi }{x}_1,x_2\right)-f_2\left(\omega ,y_1,{\mathbb{I}}^{c_1,\varphi }{y}_1,y_2\right)|\hfill \\ & \le & [{\mathbb{Q}}_1{\mathbb{K}}_1(1+{\mathbb{Q}}_{02})+{\mathbb{Q}}_2{\mathbb{K}}_2(1+{\mathbb{Q}}_{01})][\parallel x_1-y_1\parallel +\parallel x_2-y_2\parallel ],\hfill \end{array}$$

which implies

$$\begin{array}{ccc}& & \parallel {\mathbb{Z}}_1(x_1,y_1)-{\mathbb{Z}}_1(x_2,y_2){\parallel }_X\hfill \\ & \le & [{\mathbb{Q}}_3{\mathbb{K}}_1(1+{\mathbb{Q}}_{02})+{\mathbb{Q}}_4{\mathbb{K}}_2(1+{\mathbb{Q}}_{01})][\parallel x_1-y_1{\parallel }_X+\parallel x_2-y_2{\parallel }_Y].\hfill \end{array}$$

In a similar way, we obtain

$$\begin{array}{ccc}& & \parallel {\mathbb{Z}}_2(x_1,y_1)-{\mathbb{Z}}_2(x_2,y_2){\parallel }_Y\hfill \\ & \le & [{\mathbb{Q}}_3{\mathbb{K}}_1(1+{\mathbb{Q}}_{02})+{\mathbb{Q}}_4{\mathbb{K}}_2(1+{\mathbb{Q}}_{01})][\parallel x_1-y_1{\parallel }_X+\parallel x_2-y_2{\parallel }_Y].\hfill \end{array}$$

Consequently, it follows that

$$\begin{array}{ccc}& & \parallel \mathbb{Z}(x_1,y_1)-\mathbb{Z}(x_2,y_2){\parallel }_{X\times Y}\hfill \\ & \le & [({\mathbb{Q}}_1+{\mathbb{Q}}_3){\mathbb{K}}_1(1+{\mathbb{Q}}_{02})+({\mathbb{Q}}_2+{\mathbb{Q}}_4){\mathbb{K}}_2(1+{\mathbb{Q}}_{01})][\parallel x_1-y_1{\parallel }_X+\parallel x_2-y_2{\parallel }_Y].\hfill \end{array}$$

According to the given assumption in (20), $({\mathbb{Q}}_1+{\mathbb{Q}}_3){\mathbb{K}}_1(1+{\mathbb{Q}}_{02})+({\mathbb{Q}}_2+{\mathbb{Q}}_4){\mathbb{K}}_2(1+{\mathbb{Q}}_{01})<1$, and thus $\mathbb{Z}$ is a contraction. Therefore, using Banach’s fixed point theorem, the fractional Hilfer–Caputo sequential system (2) has a unique solution on $[0,T]$. □

Next, we prove an existence result for the fractional Hilfer–Caputo sequential system (2) via the Leray–Schauder alternative [37].

Theorem 2.

Let $f_1,f_2:[0,T]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ be continuous. Moreover, we assume that for

$({\mathbb{G}}_2)$,

there exists real constants $p_i,q_i\ge 0,i=1,2$ and $p_0>0,q_0>0$, such that for all $x_i\in \mathbb{R}$, $i=1,2,3$, we have

$$\begin{array}{c}\hfill |f_1(\omega ,x_1,x_2,x_3)|\le p_0+p_1|x_1|+p_2|x_2|+p_3|x_3|,\\ \hfill |f_2(\omega ,x_1,x_2,x_3)|\le q_0+q_1|x_1|+q_2|x_2|+q_3|x_3|.\end{array}$$

Then, the fractional Hilfer–Caputo sequential system (2) has at least one solution on $[0,T],$ provided that

$$\begin{array}{ccc}& & ({\mathbb{Q}}_1+{\mathbb{Q}}_3)p_1+({\mathbb{Q}}_2+{\mathbb{Q}}_4)max\left\{q_1,q_2{\mathsf{\Phi }}_{c_1}(T)\right\}<1,\hfill \\ & & ({\mathbb{Q}}_2+{\mathbb{Q}}_4)q_3+({\mathbb{Q}}_1+{\mathbb{Q}}_3)max\left\{p_2,p_3{\mathsf{\Phi }}_{c_2}(T)\right\}<1.\hfill \end{array}$$

Proof. 

First of all, we show that the operator $\mathbb{Z}:X\times Y\to X\times Y$ is completely continuous. Notice that, since the functions $f_1,f_2$ are continuous, the operator $\mathbb{Z}$ is continuous.

Let ${\mathbb{B}}_{\rho }=\{({\pi }_1,{\pi }_2)\in X\times Y:\parallel ({\pi }_1,{\pi }_2)\parallel \le \rho \}.$ For any ${\pi }_1,{\pi }_2\in {\mathbb{B}}_{\rho },$ we have

$$\begin{array}{ccc}\hfill |g_1^{\pi }(\omega )|& =& |f_1\left(\omega ,{\pi }_1(\omega ),{\pi }_2(\omega ),{\mathbb{I}}^{c_2,\varphi }{\pi }_2(\omega )\right)|\hfill \\ & \le & p_0+p_1|{\pi }_1(\omega )|+p_2|{\pi }_2(\omega )|+p_3{\mathbb{I}}^{c_2,\varphi }|{\pi }_2(\omega )|\hfill \\ & \le & p_0+p_1\parallel {\pi }_1{\parallel }_X+p_2\parallel {\pi }_2{\parallel }_Y+p_3{\displaystyle \frac{{(\varphi (T)-\varphi (0))}^{c_2}}{\Gamma (c_2+1)}}{\parallel {\pi }_2\parallel }_Y\hfill \\ & =& p_0+p_1\parallel {\pi }_1{\parallel }_X+{\parallel {\pi }_2\parallel }_Y\left[p_2+p_3{\mathsf{\Phi }}_{c_2}(T)\right]\hfill \\ & \le & p_0+p_1\rho +\left[p_2+p_3{\mathsf{\Phi }}_{c_2}(T)\right]\rho :=L_1.\hfill \end{array}$$

Similarly, we obtain

$$|g_2^{\pi }(\omega )|=|f_2\left(\omega ,{\pi }_1(\omega ),{\mathbb{I}}^{c_1,\varphi }{\pi }_1(\omega ),{\pi }_2(\omega )\right)|\le q_0+\left[q_1+q_2{\mathsf{\Phi }}_{c_1}(T)\right]\rho +q_3\rho :=L_2.$$

For any ${\pi }_1,{\pi }_2\in {\mathbb{B}}_{\rho }$ we have, as in Theorem 1,

$$|{\mathbb{Z}}_1({\pi }_1,{\pi }_2)(\omega )|\le {\mathbb{Q}}_1{L}_1+{\mathbb{Q}}_2{L}_2,$$

and

$$|{\mathbb{Z}}_2({\pi }_1,{\pi }_2)(\omega )|\le {\mathbb{Q}}_3{L}_1+{\mathbb{Q}}_4{L}_2,$$

and hence

$$\parallel \mathbb{Z}({\pi }_1,{\pi }_2){\parallel }_{X\times Y}=\parallel {\mathbb{Z}}_1({\pi }_1,{\pi }_2){\parallel }_X+{\parallel {\mathbb{Z}}_2({\pi }_1,{\pi }_2)\parallel }_Y\le ({\mathbb{Q}}_1+{\mathbb{Q}}_3)L_1+({\mathbb{Q}}_2+{\mathbb{Q}}_4)L_2,$$

which means that the operator $\mathbb{Z}$ is uniformly bounded.

Now, we want to show that $\mathbb{Z}$ is equicontinuous. Let ${\omega }_1,{\omega }_2\in [0,T]$ with ${\omega }_1<{\omega }_2$. Then, for all ${\pi }_1,{\pi }_2\in {\mathbb{B}}_x$, we have

$$\begin{array}{ccc}& & |{\mathbb{Z}}_1({\pi }_1,{\pi }_2)({\omega }_2)-{\mathbb{Z}}_1({\pi }_1,{\pi }_2)({\omega }_1)|\hfill \\ & \le & {\displaystyle \frac{|{\lambda }_2|}{|1-{\lambda }_2{\mu }_2|\Gamma ({\gamma }_1+{\delta }_1)}}|{(\varphi ({\omega }_2)-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}-{(\varphi ({\omega }_1)-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}|\hfill \\ & & \times \left[L_2{\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)+|{\mu }_2|L_1{\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)\right]\hfill \\ & & +L_1{\displaystyle \frac{1}{\Gamma (a+\delta )}}|{\int }_0^{{\omega }_1}{\varphi }^{\prime }(s)[{(\varphi ({\omega }_2)-\varphi (s))}^{a_1+{\delta }_1-1}-{(\varphi ({\omega }_1)-\varphi (s))}^{a_1+{\delta }_1-1}]\hfill \\ & & +{\int }_{{\omega }_1}^{{\omega }_2}{\varphi }^{\prime }(s){(\varphi ({\omega }_2)-\varphi (s))}^{a_1+{\delta }_1-1}|ds\hfill \\ & \le & {\displaystyle \frac{|{\lambda }_2|}{|1-{\lambda }_2{\mu }_2|\Gamma ({\gamma }_1+{\delta }_1)}}|{(\varphi ({\omega }_2)-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}-{(\varphi ({\omega }_1)-\varphi (0))}^{{\gamma }_1+{\delta }_1-1}|\hfill \\ & & \times \left[L_2{\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)+|{\mu }_2|L_1{\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)\right]\hfill \\ & & +{\displaystyle \frac{L_1}{\Gamma (a_1+{\delta }_1+1)}}[2{(\varphi ({\omega }_2)-\varphi ({\omega }_1))}^{a_1+{\delta }_1}\hfill \\ & & +|{(\varphi ({\omega }_2)-\varphi (0))}^{a_1+{\delta }_1}-{(\varphi ({\omega }_1)-\varphi (0))}^{a_1+{\delta }_1}|].\hfill \end{array}$$

This implies that $|{\mathbb{Z}}_1({\pi }_1,{\pi }_2)({\omega }_2)-{\mathbb{Z}}_1({\pi }_1,{\pi }_2)({\omega }_1)|\to 0$ as ${\omega }_1\to {\omega }_2,$ independently of ${\pi }_1,{\pi }_2$. In a similar way, $|{\mathbb{Z}}_2({\pi }_1,{\pi }_2)({\omega }_2)-{\mathbb{Z}}_2({\pi }_1,{\pi }_2)({\omega }_1)|\to 0,$ ${\omega }_1\to {\omega }_2,$ independently of ${\pi }_1,{\pi }_2$. Hence, using the Arzelá–Ascoli theorem, the equicontinuity of the operator $\mathbb{Z}$ is proved.

It remains to show that the set

$$\Xi =\{({\pi }_1,{\pi }_2)\in X\times Y:({\pi }_1,{\pi }_2)=\lambda \mathbb{Z}({\pi }_1,{\pi }_2)0<\lambda <1\},$$

is bounded. Let $({\pi }_1,{\pi }_2)\in X\times Y$; then, $({\pi }_1,{\pi }_2)=\lambda \mathbb{Z}({\pi }_1,{\pi }_2)$ for some $\lambda \in (0,1)$, and for any $\omega \in [0,T]$, we have

$${\pi }_1(\omega )=\lambda {\mathbb{Z}}_1({\pi }_1,{\pi }_2)(\omega ),{\pi }_2(\omega )=\lambda {\mathbb{Z}}_2({\pi }_1,{\pi }_2)(\omega ).$$

As in the first step, for all $\omega \in [0,T]$, we obtain

$$\begin{array}{ccc}\hfill |{\pi }_1(\omega )|& \le & {\mathbb{Q}}_1\left[p_0+p_1\parallel {\pi }_1{\parallel }_X+p_2\parallel {\pi }_2{\parallel }_Y+p_3{\mathsf{\Phi }}_{c_2}(T){\parallel {\pi }_2\parallel }_Y\right]\hfill \\ & & +{\mathbb{Q}}_2\left[q_0+q_1\parallel {\pi }_1{\parallel }_X+q_2{\mathsf{\Phi }}_{c_1}(T)\parallel {\pi }_1{\parallel }_Y+q_3{\parallel {\pi }_2\parallel }_Y\right]\hfill \\ & \le & {\mathbb{Q}}_1\left[p_0+p_1\parallel {\pi }_1{\parallel }_X+max\left\{p_2,p_3{\mathsf{\Phi }}_{c_2}(T)\right\}{\parallel {\pi }_2\parallel }_Y\right]\hfill \\ & & +{\mathbb{Q}}_2\left[q_0+max\left\{q_1,q_2{\mathsf{\Phi }}_{c_1}(T)\right\}\parallel {\pi }_1{\parallel }_X+q_3{\parallel {\pi }_2\parallel }_Y\right].\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill \parallel {\pi }_1{\parallel }_X& \le & {\mathbb{Q}}_1\left[p_0+p_1\parallel {\pi }_1{\parallel }_X+max\left\{p_2,p_3{\mathsf{\Phi }}_{c_2}(T)\right\}{\parallel {\pi }_2\parallel }_Y\right]\hfill \\ & & +{\mathbb{Q}}_2\left[q_0+max\left\{q_1,q_2{\mathsf{\Phi }}_{c_1}(T)\right\}\parallel {\pi }_1{\parallel }_X+q_3{\parallel {\pi }_2\parallel }_Y\right].\hfill \end{array}$$

Similarly, we can obtain

$$\begin{array}{ccc}\hfill \parallel {\pi }_2{\parallel }_Y& \le & {\mathbb{Q}}_3\left[p_0+p_1\parallel {\pi }_1{\parallel }_X+max\left\{p_2,p_3{\mathsf{\Phi }}_{c_2}(T)\right\}{\parallel {\pi }_2\parallel }_Y\right]\hfill \\ & & +{\mathbb{Q}}_4\left[q_0+max\left\{q_1,q_2{\mathsf{\Phi }}_{c_1}(T)\right\}\parallel {\pi }_1{\parallel }_X+q_3{\parallel {\pi }_2\parallel }_Y\right].\hfill \end{array}$$

From the above inequalities, we have

$$\begin{array}{ccc}\hfill \parallel {\pi }_1{\parallel }_X+{\parallel {\pi }_2\parallel }_Y& \le & ({\mathbb{Q}}_1+{\mathbb{Q}}_3)p_0+({\mathbb{Q}}_2+{\mathbb{Q}}_4)q_0\hfill \\ & & +\left[({\mathbb{Q}}_1+{\mathbb{Q}}_3)p_1+({\mathbb{Q}}_2+{\mathbb{Q}}_4)max\left\{q_1,q_2{\mathsf{\Phi }}_{c_1}(T)\right\}\right]{\parallel {\pi }_1\parallel }_X\hfill \\ & & +\left[({\mathbb{Q}}_2+{\mathbb{Q}}_4)q_3+({\mathbb{Q}}_1+{\mathbb{Q}}_3)max\left\{p_2,p_3{\mathsf{\Phi }}_{c_2}(T)\right\}\right]{\parallel {\pi }_2\parallel }_Y.\hfill \end{array}$$

Consequently,

$$\parallel ({\pi }_1,{\pi }_2){\parallel }_{X\times Y}\le {\displaystyle \frac{({\mathbb{Q}}_1+{\mathbb{Q}}_3)p_0+({\mathbb{Q}}_2+{\mathbb{Q}}_4)q_0}{M_0}},$$

where

$$\begin{array}{ccc}\hfill M_0& =& min\{1-\left[({\mathbb{Q}}_1+{\mathbb{Q}}_3)p_1+({\mathbb{Q}}_2+{\mathbb{Q}}_4)max\left\{q_1,q_2{\mathsf{\Phi }}_{c_1}(T)\right\}\right],\hfill \\ & & 1-\left[({\mathbb{Q}}_2+{\mathbb{Q}}_4)q_3+({\mathbb{Q}}_1+{\mathbb{Q}}_3)max\left\{p_2,p_3{\mathsf{\Phi }}_{c_2}(T)\right\}\right]\}.\hfill \end{array}$$

This implies that the set $\Xi$ is bounded. Therefore, according to the Leray–Schauder alternative, the fractional Hilfer–Caputo sequential system (2) has on $[0,T]$ at least one solution. □

4. Examples

In this section, a coupled system of mixed fractional Hilfer and Caputo integro-differential equations with non-separated boundary conditions, by varying the nonlinear functions containing fractional integrals of unknown functions, can be considered to be of the following form:

$$\left\{\begin{array}{c}{\displaystyle {}^H\mathbb{D}^{\frac{14}{25},\frac{2}{5},{\omega }^2+\sqrt{\omega }}({}^C\mathbb{D}^{\frac{13}{25},{\omega }^2+\sqrt{\omega }}{\pi }_1)(\omega )=f_1\left(\omega ,{\pi }_1(\omega ),{\pi }_2(\omega ),{\mathbb{I}}^{c_2,\varphi }{\pi }_2(\omega )\right),\omega \in \left[0,\frac{11}{7}\right],}\hfill \\ {\displaystyle {}^H\mathbb{D}^{\frac{18}{25},\frac{4}{5},{\omega }^2+\sqrt{\omega }}({}^C\mathbb{D}^{\frac{11}{25},{\omega }^2+\sqrt{\omega }}{\pi }_2)(\omega )=f_2\left(\omega ,{\pi }_1(\omega ),{\mathbb{I}}^{c_1,\varphi }{\pi }_1(\omega ),{\pi }_2(\omega )\right),\omega \in \left[0,\frac{11}{7}\right],}\hfill \\ {\displaystyle {\pi }_1(0)+\frac{17}{71}{\pi }_2\left(\frac{11}{7}\right)=0,{}^C\mathbb{D}^{\frac{32}{125},{\omega }^2+\sqrt{\omega }}{\pi }_1(0)+\frac{19}{73}{}^C\mathbb{D}^{\frac{9}{25},{\omega }^2+\sqrt{\omega }}{\pi }_2\left(\frac{11}{7}\right)=0,}\hfill \\ {\displaystyle {\pi }_2(0)+\frac{31}{79}{\pi }_1\left(\frac{11}{7}\right)=0,{}^C\mathbb{D}^{\frac{48}{125},{\omega }^2+\sqrt{\omega }}{\pi }_2(0)+\frac{23}{77}{}^C\mathbb{D}^{\frac{1}{25},{\omega }^2+\sqrt{\omega }}{\pi }_1\left(\frac{11}{7}\right)=0.}\hfill \end{array}\right.$$

(21)

From this information, we set $a_1=14/25$, $a_2=18/25$, $b_1=2/5$, $b_2=4/5$, $\varphi ={\omega }^2+\sqrt{\omega }$, ${\delta }_1=13/25$, ${\delta }_2=11/25$, $T=11/7$, ${\lambda }_1=17/71$, ${\lambda }_2=19/73$, ${\mu }_1=31/79$, and ${\mu }_2=23/77$. Then, we can show that ${\gamma }_1=92/125$, ${\gamma }_2=118/125$, ${\gamma }_1+{\delta }_1-1=32/125$, ${\gamma }_2+{\delta }_2-1=48/125$, ${\lambda }_1{\mu }_1=527/5609\ne 1$, and ${\lambda }_2{\mu }_2=437/5621\ne 1$. In addition, by using the Maple program, we find that ${\mathsf{\Phi }}_{a_1+{\delta }_1}(T)\approx 3.990138697$, ${\mathsf{\Phi }}_{a_2+{\delta }_2}(T)\approx 4.259716999$, ${\mathsf{\Phi }}_{{\gamma }_1+{\delta }_1-1}(T)\approx 1.546712212$, ${\mathsf{\Phi }}_{{\gamma }_2+{\delta }_2-1}(T)\approx 1.865023293$, ${\mathsf{\Phi }}_{a_1-{\gamma }_1+1}(T)\approx 3.149347909$, ${\mathsf{\Phi }}_{a_2-{\gamma }_2+1}(T)\approx 2.997491446$, ${\mathbb{Q}}_1\approx 5.211383734$, ${\mathbb{Q}}_2\approx 2.686036097$, ${\mathbb{Q}}_3\approx 3.135513393$, ${\mathbb{Q}}_4\approx 4.993040295$.

Case I. Consider the nonlinear functions $f_1,f_2$, defined by

$$\left\{\begin{array}{c}{\displaystyle f_1(\omega ,{\pi }_1,{\pi }_2,{\mathbb{I}}^{\frac{5}{7},{\omega }^2+\sqrt{\omega }}{\pi }_2)={\omega }^2+\frac{1}{2}+\frac{1}{2(\omega +72)}\left(\frac{{\pi }_1^2+2|{\pi }_1|}{1+|{\pi }_1|}\right)+\frac{sin{\pi }_2}{{\omega }^2+75}}\hfill \\ {\displaystyle +\frac{\left|{\mathbb{I}}^{\frac{5}{7},{\omega }^2+\sqrt{\omega }}{\pi }_2\right|}{15{(\omega +3)}^2\left(1+\left|{\mathbb{I}}^{\frac{5}{7},{\omega }^2+\sqrt{\omega }}{\pi }_2\right|\right)},}\hfill \\ {\displaystyle f_2(\omega ,{\pi }_1,{\mathbb{I}}^{\frac{3}{7},{\omega }^2+\sqrt{\omega }}{\pi }_1,{\pi }_2)={\omega }^3+\frac{1}{3}+\frac{1}{2(\omega +36)}\left(\frac{|{\pi }_1|}{1+|{\pi }_1|}\right)+\frac{{tan}^{-1}{\pi }_2}{{(\omega +\sqrt{73})}^2}}\hfill \\ {\displaystyle +\frac{{\left({\mathbb{I}}^{\frac{3}{7},{\omega }^2+\sqrt{\omega }}{\pi }_1\right)}^2+2\left|{\mathbb{I}}^{\frac{3}{7},{\omega }^2+\sqrt{\omega }}{\pi }_1\right|}{(\sqrt{\omega }+140)\left(1+\left|{\mathbb{I}}^{\frac{3}{7},{\omega }^2+\sqrt{\omega }}{\pi }_1\right|\right)}.}\hfill \end{array}\right.$$

(22)

Let us see that for all $\omega \in [0,11/7]$,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial }{\partial \omega }}\left[{\mathbb{I}}^{a_i+{\delta }_i,{\omega }^2+\sqrt{\omega }}{f}_i(·)\right]& =& {\displaystyle \frac{\partial }{\partial \omega }}[{\displaystyle \frac{1}{\Gamma (a_i+{\delta }_i)}}{\int }_0^{\omega }\left(2\omega +{\displaystyle \frac{1}{2\sqrt{\omega }}}\right)\hfill \\ & & \times {\left({\omega }^2+\sqrt{\omega }-s^2-\sqrt{s}\right)}^{a_i+{\delta }_1-1}{f}_i(·)ds],i=1,2.\hfill \end{array}$$

Since $a_i+{\delta }_i>1$, for $i=1,2,$ the above equation exists by the Leibniz fundamental rule of calculus, and then ${\mathbb{I}}^{a_i+{\delta }_i,{\omega }^2+\sqrt{\omega }}{f}_i(·)\in C^1[0,11/7]$, $i=1,2$. Next, for all $\omega \in (0,11/7]$, we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial }{\partial \omega }}\left[{\mathbb{I}}^{(1-b_i)(1-a_i),\varphi }{\mathbb{I}}^{a_i,\varphi }{f}_i(·)\right]& =& {\displaystyle \frac{\partial }{\partial \omega }}\left[{\mathbb{I}}^{(1-b_i)(1-a_i)+a_i,{\omega }^2+\sqrt{\omega }}{f}_i(·)\right]\hfill \\ & =& {\displaystyle \frac{\partial }{\partial \omega }}\left[{\mathbb{I}}^{1-{\gamma }_i+a_i,{\omega }^2+\sqrt{\omega }}{f}_i(·)\right],\hfill \end{array}$$

which exists on $(0,11/7]$ and it is also integrable on $[0,11/7]$.

In this case, we see that $c_1=3/7$, $c_2=5/7$ and then we have ${\mathbb{Q}}_{01}\approx 2.982416725$ and ${\mathbb{Q}}_{02}\approx 3.805800381$. In addition, we can show that the $f_1,f_2$ in (22) satisfy

$$|f_1(\omega ,x_1,x_2,x_3)-f_1(\omega ,y_1,y_2,y_3)|\le {\displaystyle \frac{1}{72}}(|x_1-y_1|+|x_2-y_2|+|x_3-y_3|)$$

and

$$|f_2(\omega ,x_1,x_2,x_3)-f_2(\omega ,y_1,y_2,y_3)|\le {\displaystyle \frac{1}{70}}(|x_1-y_1|+|x_2-y_2|+|x_3-y_3|),$$

with ${\mathbb{K}}_1=1/72$ and ${\mathbb{K}}_2=1/70$. From all of these details, we can find that

$$({\mathbb{Q}}_1+{\mathbb{Q}}_3){\mathbb{K}}_1(1+{\mathbb{Q}}_{02})+({\mathbb{Q}}_2+{\mathbb{Q}}_4){\mathbb{K}}_2(1+{\mathbb{Q}}_{01})\approx 0.9940077024<1,$$

and thus the inequality in (20) in Theorem 1 is fulfilled. This means that, according to Theorem 1, the coupled system of mixed fractional Hilfer and Caputo integro-differential equations with non-separated boundary conditions (21) with $f_1,f_2$ defined in (22) has a unique solution $({\pi }_1,{\pi }_2)(\omega )$ for $\omega \in [0,11/7]$.

Case $II$. Now, we consider the functions $f_1,f_2$, expressed by

$$\left\{\begin{array}{c}{\displaystyle f_1(\omega ,{\pi }_1,{\pi }_2,{\mathbb{I}}^{\frac{9}{7},{\omega }^2+\sqrt{\omega }}{\pi }_2)=\omega +\frac{3}{7}+\frac{1}{2(\omega +19)}\left(\frac{{\pi }_1^2{e}^{-|{\pi }_2|}}{1+|{\pi }_1|}\right)+\frac{{\pi }_2{cos}^8{\pi }_1}{{\omega }^2+41}}\hfill \\ {\displaystyle +\frac{{sin}^4{\pi }_2}{3{(\omega +4)}^2}{\mathbb{I}}^{\frac{9}{7},{\omega }^2+\sqrt{\omega }}{\pi }_2,}\hfill \\ {\displaystyle f_2(\omega ,{\pi }_1,{\mathbb{I}}^{\frac{8}{7},{\omega }^2+\sqrt{\omega }}{\pi }_1,{\pi }_2)=\omega +\frac{10}{7}+\frac{1}{4(\omega +10)}\left(\frac{{\pi }_1^{2024}}{1+|{\pi }_1{|}^{2023}}\right)}\hfill \\ {\displaystyle +\frac{{\pi }_2{sin}^{12}{\pi }_1}{5(\omega +9)}+\frac{e^{-{\pi }_2^2}}{6(\omega +7)}{\mathbb{I}}^{\frac{8}{7},{\omega }^2+\sqrt{\omega }}{\pi }_1.}\hfill \end{array}\right.$$

(23)

Using a method similar to that used in $(I)$, we can show that ${\mathbb{I}}^{a_i+{\delta }_i,{\omega }^2+\sqrt{\omega }}{f}_i(·)\in C^1[0,11/7]$, $i=1,2$ and ${\mathbb{I}}^{(1-b_i)(1-a_i),\varphi }{\mathbb{I}}^{a_i,\varphi }{f}_i(·)\in C^1[0,11/7]$, $i=1,2$. From (23), we set $c_1=8/7$ and $c_2=9/7$, which lead to ${\mathsf{\Phi }}_{c_1}(T)\approx 4.201809053$ and ${\mathsf{\Phi }}_{c_2}(T)\approx 4.685257726$. Note that $f_1,f_2$ do not satisfy the Lipschitz condition, but

$$|f_1(\omega ,x_1,x_2,x_3)|\le 2+{\displaystyle \frac{1}{38}}|x_1|+{\displaystyle \frac{1}{41}}|x_2|+{\displaystyle \frac{1}{48}}|x_3|,$$

and

$$|f_2(\omega ,x_1,x_2,x_3)|\le 3+{\displaystyle \frac{1}{40}}|x_1|+{\displaystyle \frac{1}{42}}|x_2|+{\displaystyle \frac{1}{45}}|x_3|.$$

Thus, we choose $p_0=2$, $p_1=1/38$, $p_2=1/41$, $p_3=1/48$, $q_0=3$, $q_1=1/40$, $q_2=1/42$, $q_3=1/45$. Then, we obtain

$$max\{q_1,q_2{\mathsf{\Phi }}_{c_1}(T)\}\approx 0.1235826192andmax\{p_2,p_3{\mathsf{\Phi }}_{c_2}(T)\}\approx 0.1301460479.$$

Therefore, from all setting constants, we obtain

$$\begin{array}{ccc}& & ({\mathbb{Q}}_1+{\mathbb{Q}}_3)p_1+({\mathbb{Q}}_2+{\mathbb{Q}}_4)max\left\{q_1,q_2{\mathsf{\Phi }}_{c_1}(T)\right\}\approx 0.9878935853<1,\hfill \\ & & ({\mathbb{Q}}_2+{\mathbb{Q}}_4)q_3+({\mathbb{Q}}_1+{\mathbb{Q}}_3)max\left\{p_2,p_3{\mathsf{\Phi }}_{c_2}(T)\right\}\approx 0.9853828970<1.\hfill \end{array}$$

Hence, by applying Theorem 2, the separated boundary value problem (21) with $f_1,f_2$ defined in (23) has at least one solution $({\pi }_1,{\pi }_2)$ on the interval $[0,11/7]$.

5. Conclusions

In the present research, a sequential coupled system of fractional differential equations combining Hilfer and Caputo fractional derivative operators, supplemented with non-separated boundary conditions, was investigated. We applied the method proposed recently by the authors in [33], where a sequential fractional boundary value problem with non-separated boundary conditions was studied. The proposed combination gives us the possibility to discuss sequential coupled systems of fractional differential equations combining Hilfer and Caputo fractional derivative operators subjected to non-zero initial conditions. We emphasize that the combination of Hilfer and Caputo fractional derivative operators allows us to study non-separated boundary conditions with non-zero initial conditions, while the combination of Caputo and Hilfer fractional derivative operators requires zero initial conditions. Indeed, if the sequential fractional differential equation in (6) is interchanged as

$${\displaystyle {}^C\mathbb{D}^{\delta ,\varphi }({}^H\mathbb{D}^{a_1,b_1,\varphi }{\pi }_1)(\omega )=g_1(\omega ),}$$

(24)

then we have

$${\pi }_1(\omega )=c_2{(\varphi (\omega )-\varphi (0))}^{{\gamma }_1-1}+{\displaystyle \frac{c_1}{\Gamma (a_1+1)}}{(\varphi (\omega )-\varphi (0))}^{a_1}+{\mathbb{I}}^{\delta +a_1,\varphi }{g}_1(\omega ),$$

where $c_1,c_2\in \mathbb{R}$. Since ${\gamma }_1=a_1+b_1(1-a_1)\in (0,1)$, we have $c_2=0$ when $\omega \to 0$. This means that the initial condition ${\pi }_1(0)=0$ is necessary for the fractional differential Equation (24). The main existence and uniqueness results have been proven in this paper by using the Banach fixed point theorem and the Leray–Schauder alternative. Furthermore, some examples have been illustrated to support our theoretical analysis.

To obtain the main results, fixed point theorems were used, namely the Banach’s fixed point theorem to establish the uniqueness of the solution and the Leray–Schauder alternative to establish an existence result. Furthermore, numerical examples have been illustrated to support the theoretical analysis.

In future studies, we plan to enrich the literature in this new subject by studying other types of sequential coupled systems of fractional differential equations combining Hilfer and Caputo fractional derivative operators and other types of non-zero boundary conditions.