On a Coupled Differential System Involving (k,ψ)-Hilfer Derivative and (k,ψ)-Riemann–Liouville Integral Operators

Abstract

We investigate a nonlinear, nonlocal, and fully coupled boundary value problem containing mixed $(k,\widehat{\psi })$-Hilfer fractional derivative and $(k,\widehat{\psi })$-Riemann–Liouville fractional integral operators. Existence and uniqueness results for the given problem are proved with the aid of standard fixed point theorems. Examples illustrating the main results are presented. The paper concludes with some interesting findings.

1. Introduction

We consider a nonlinear system of $(k,\widehat{\psi })$-Hilfer fractional differential equations:

$$\left\{\begin{array}{c}{\displaystyle {}^{k,H}{\mathcal{D}}^{\tilde{\alpha },\tilde{\beta };\widehat{\psi }}\breve{k}(s)=\check{L}(s,\breve{k}(s),\breve{l}(s)),s\in [l_1,l_2],}\hfill \\ {\displaystyle {}^{k,H}{\mathcal{D}}^{\tilde{p},\tilde{q};\widehat{\psi }}\breve{l}(s)=\breve{L}(s,\breve{k}(s),\breve{l}(s)),s\in [l_1,l_2],}\hfill \end{array}\right.$$

(1)

supplemented with coupled mixed boundary conditions containing $(k,\widehat{\psi })$-derivative and integral operators

$$\left\{\begin{array}{c}{\displaystyle \breve{k}(l_1)=0,\breve{k}(l_2)=\tilde{\lambda }{}^{k,H}{\mathcal{D}}^{\tilde{r},\tilde{s},\widehat{\psi }}\breve{l}(\tilde{\xi })+\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{v},\widehat{\psi }}\breve{l}(\tilde{\sigma }),\tilde{\lambda },\tilde{\mu }\in \mathbb{R},}\hfill \\ {\displaystyle \breve{l}(l_1)=0,\breve{l}(l_2)=\tilde{\nu }{}^{k,H}{\mathcal{D}}^{\tilde{z},\tilde{w},\widehat{\psi }}\breve{k}(\tilde{\eta })+\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{u},\widehat{\psi }}\breve{k}(\tilde{\tau }),\tilde{\nu },\tilde{\theta }\in \mathbb{R},}\hfill \end{array}\right.$$

(2)

where ${}^{k,H}{\mathcal{D}}^{\varrho ,\varpi ;\widehat{\psi }}$ represents the $(k,\widehat{\psi })$-Hilfer fractional derivative operator of order $\varrho$ and parameter $\varpi$ with $\varrho =\{\tilde{\alpha },\tilde{p},\tilde{r},\tilde{z}\}$ and $\varpi =\{\tilde{\beta },\tilde{q},\tilde{s},\tilde{w}\},$ such that $1<\tilde{\alpha }, \tilde{p}<2,$$0<\tilde{r}, \tilde{z}<1,$$0<\varpi <1,$ $0\le l_1<l_2<\infty$, $\check{L},\breve{L}\in C([l_1,l_2]\times \mathbb{R}\times \mathbb{R},\mathbb{R}),$ and ${}^k{\mathcal{I}}^{\widehat{v},\widehat{\psi }},$${}^k{\mathcal{I}}^{\widehat{u},\widehat{\psi }}$ are $(k,\widehat{\psi })$-Riemann–Liouville fractional integrals of order $\widehat{v}>0,\widehat{u}>0$, respectively, and $l_1<\tilde{\xi }, \tilde{\sigma }, \tilde{\eta }, \tilde{\tau }<l_2.$

The objective of the present work is to develop the existence theory for the Problem (1) and (2) via the tools of the fixed point theory. A uniqueness result for the Problem (1) and (2) is proved by means of a fixed point theorem due to Banach, while the Leray–Schauder alternative and Krasnosel’skiĭ’s fixed-point theorem are applied to derive the two existence results for the problem at hand. The results established in this paper will contribute significantly to the literature on coupled $(k,\widehat{\psi })$-Hilfer fractional differential systems, which is indeed scarce and needs to be enriched and extended further in several directions.

Boundary value problems involving different kinds of fractional derivative operators such as Caputo–Liouville, Riemann–Liouville, $\widehat{\psi }$-Riemann–Liouville [1], Hilfer [2], k-Riemann–Liouville, $(k,\widehat{\psi })$-Riemann–Liouville [3], $\widehat{\psi }$-Hilfer [4], etc., have been addressed by several authors. Some recent results on nonlocal multipoint single-valued and multi-valued boundary value problems containing Hilfer and Caputo–Hadamard type fractional derivative operators can be found in the papers [5,6,7]. For preliminary concepts of fractional calculus, for example, see the books [1,8]. Here we mention that the Hilfer fractional derivative unifies the definitions of both Riemann–Liouville and Caputo fractional derivatives. For some applications of Hilfer fractional derivative operator, see [2,9,10,11].

Let us now review some recent works on fractional differential equations and systems equipped with different boundary conditions. In [12], the authors proved the existence and uniqueness of solutions for a boundary value problem involving $(k,\widehat{\psi })$-Hilfer type fractional derivative and integral operators of the form:

$$\left\{\begin{array}{c}{\displaystyle {}^{k,H}{D}^{\alpha ,\beta ;\widehat{\psi }}\breve{k}(s)=\check{L}(s,\breve{k}(s)),s\in [l_1,l_2],}\hfill \\ {\displaystyle \breve{k}(l_1)=0,\breve{k}(l_2)=\tilde{\lambda }{}^{k,H}{D}^{p,q;\widehat{\psi }}\breve{k}(\tilde{\eta })+\tilde{\mu }{}^k{\mathfrak{I}}^{v,\widehat{\psi }}\breve{k}(\tilde{\sigma }),}\hfill \end{array}\right.$$

where ${}^{k,H}{D}^{\alpha ,\beta ;\widehat{\psi }}$ and ${}^{k,H}{D}^{p,q;\widehat{\psi }}$ represent the $(k,\widehat{\psi })$-Hilfer type fractional derivative operators of orders $\alpha \in (1,2)$, and $p\in (0,1)$ with parameters $\beta ,q\in [0,1]$, respectively, $\check{L}\in C([l_1,l_2]\times \mathbb{R},\mathbb{R}),$${}^k{\mathfrak{I}}^{v,\widehat{\psi }}$ is the $(k,\widehat{\psi })$-Riemann–Liouville fractional integral of order $v>0,$$\tilde{\lambda },\tilde{\mu }\in \mathbb{R},$ and $l_1<\tilde{\xi },\tilde{\sigma }<l_2.$ For some recent results on $(k,\widehat{\psi })$-Hilfer fractional differential equations, see [13].

In [14], the authors applied the standard tools of the fixed point theory to establish the existence and uniqueness results for the coupled $(k,\phi )$-Hilfer type fractional differential system (1) equipped with nonlocal multipoint boundary conditions:

$$\breve{k}(l_1)=0,\breve{l}(l_1)=0,\breve{k}(l_2)=\sum _{i=1}^m{\tilde{\lambda }}_i\breve{l}({\tilde{\xi }}_i),\breve{l}(l_2)=\sum _{j=1}^k{\tilde{\mu }}_j\breve{k}({\tilde{\eta }}_j),$$

where ${\tilde{\tilde{\lambda }}}_i,{\tilde{\mu }}_j\in \mathbb{R},$ and $l_1<{\tilde{\xi }}_i,{\tilde{\eta }}_j<l_2,i=1,2,\dots ,m,j=1,2,\dots ,k.$

As far as the authors know, the paper [14] is the only work in the literature dealing with coupled systems of $(k,\widehat{\psi })$-Hilfer fractional derivative operator of the order in $(1,2].$ Our goal in the present paper is to enrich this new research area on coupled $(k,\widehat{\psi })$-Hilfer fractional systems by introducing and investigating the new boundary value Problem (1) and (2).

Concerning the importance of coupled fractional differential systems, it is well-known that such systems appear in the mathematical models of many physical phenomena related to bio-engineering [15], fractional dynamics [16], financial economics [17], etc. In [18,19], some interesting results for $\widehat{\psi }$-Hilfer fractional differential coupled systems were obtained.

The structure of the remaining paper is designed as follows. Section 2 contains basic definitions and an auxiliary lemma. Existence and uniqueness results for the given problem are presented in Section 3, while illustrative examples for these results are discussed in Section 4. In the last section, we indicate some new results arising as special cases of the present work.

2. A Preliminary Result

Let us begin this section with the definitions involved in the Problem (1) and (2).

Definition 1 

([3]). The fractional integral of $(k,\widehat{\psi })$-Riemann–Liouville type of order $\tilde{\alpha }>0$ ($\tilde{\alpha }\in \mathbb{R}$) of a function $\check{L}\in L^1([l_1,l_2],\mathbb{R})$ is defined by

$${}^k{\mathfrak{I}}_{l_1+}^{\tilde{\alpha };\widehat{\psi }}\check{L}(s)=\frac{1}{k{\Gamma }_k(\tilde{\alpha })}{\int }_{l_1}^s{\widehat{\psi }}^{\prime }(v){(\widehat{\psi }(s)-\widehat{\psi }(v))}^{\frac{\tilde{\alpha }}{k}-1}\check{L}(v)dv,k>0,$$

(3)

where $\widehat{\psi }:[l_1,l_2]\to \mathbb{R}$ is an increasing function with ${\widehat{\psi }}^{\prime }(s)\ne 0$ for all $s\in [l_1,l_2]$.

Definition 2 

([13]). For $\tilde{\alpha },k\in {\mathbb{R}}^{+}=(0,+\infty ),$$\tilde{\beta }\in [0,1],$$\widehat{\psi }\in C^n([l_1,l_2],\mathbb{R}),$${\widehat{\psi }}^{\prime }(s)\ne 0,s\in [l_1,l_2],$ the fractional derivative of $(k,\widehat{\psi })$-Hilfer type for the function $\check{L}\in C^n([l_1,l_2],\mathbb{R})$ of order $\tilde{\alpha }$ and type $\tilde{\beta }$ is given by

$${}^{k,H}{D}^{\tilde{\alpha },\tilde{\beta };\widehat{\psi }}\check{L}(s)={}^k{\mathfrak{I}}_{l_1+}^{\beta (nk-\tilde{\alpha });\widehat{\psi }}{\left(\frac{k}{{\widehat{\psi }}^{\prime }(s)}\frac{d}{ds}\right)}^n{}^k{\mathfrak{I}}_{l_1+}^{(1-\tilde{\beta })(nk-\tilde{\alpha });\widehat{\psi }}\check{L}(s),n=⌈\frac{\tilde{\alpha }}{k}⌉.$$

(4)

We solve the linear variant of the nonlinear Problem (1) and (2) in the following lemma.

Lemma 1. 

Let ${\tilde{\vartheta }}_k=\tilde{\alpha }+\tilde{\beta }(2k-\tilde{\alpha })$, ${\tilde{\eta }}_k=\tilde{p}+\tilde{q}(2k-\tilde{p})$, $\mathcal{B}\ne 0,$ and $\check{L},\breve{L}\in C([l_1,l_2],\mathbb{R})$. Then the pair $(\breve{k},\breve{l})$ is a solution of the linear version of the Problem (1) and (2) given by

$$\left\{\begin{array}{c}{\displaystyle {}^{k,H}{\mathcal{D}}^{\tilde{\alpha },\tilde{\beta };\widehat{\psi }}\breve{k}(s)=\check{L}(s),s\in (l_1,l_2],}\hfill \\ {\displaystyle {}^{k,H}{\mathcal{D}}^{\tilde{p},\tilde{q};\widehat{\psi }}\breve{l}(s)=\breve{L}(s),s\in (l_1,l_2],}\hfill \\ {\displaystyle \breve{k}(l_1)=0,\breve{k}(l_2)=\tilde{\lambda }{}^{k,H}{\mathcal{D}}^{\tilde{r},\tilde{s},\widehat{\psi }}\breve{l}(\tilde{\xi })+\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{v},\widehat{\psi }}\breve{l}(\tilde{\sigma }),}\hfill \\ {\displaystyle \breve{l}(l_1)=0,\breve{l}(l_2)=\tilde{\nu }{}^{k,H}{\mathcal{D}}^{\tilde{z},\tilde{w},\widehat{\psi }}\breve{k}(\tilde{\eta })+\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{u},\widehat{\psi }}\breve{k}(\tilde{\tau }),}\hfill \end{array}\right.$$

(5)

if and only if

$$\begin{array}{ccc}\hfill \breve{k}(s)& =& {\displaystyle {}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(s)}\hfill \\ & & {\displaystyle +\frac{{(\widehat{\psi }(s)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{\mathcal{B}{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4\left(\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{p}+\tilde{v},\widehat{\psi }}\breve{L}(\tilde{\sigma })+\tilde{\lambda }{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}\breve{L}(\tilde{\xi })-{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(l_2)\right)}\hfill \\ & & {\displaystyle +{\mathcal{B}}_2\left(\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}\check{L}(\tilde{\tau })+\tilde{\nu }{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}\check{L}(\tilde{\eta })-{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}\breve{L}(l_2)\right)],}\hfill \end{array}$$

(6)

and

$$\begin{array}{ccc}\hfill \breve{l}(s)& =& {\displaystyle {}^k{\mathcal{I}}^{\breve{p},\widehat{\psi }}\breve{L}(s)}\hfill \\ & & {\displaystyle +\frac{{(\widehat{\psi }(s)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k}{k}-1}}{\mathcal{B}{\Gamma }_k({\tilde{\eta }}_k)}[{\mathcal{B}}_1\left(\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}\check{L}(\tilde{\tau })+\tilde{\nu }{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}\check{L}(\tilde{\eta })-{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}\breve{L}(l_2)\right)}\hfill \\ & & {\displaystyle +{\mathcal{B}}_3\left(\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{p}+\tilde{v},\widehat{\psi }}\breve{L}(\tilde{\sigma })+\tilde{\lambda }{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}\breve{L}(\tilde{\xi })-{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(l_2)\right)],}\hfill \end{array}$$

(7)

where

$$\mathcal{B}:={\mathcal{B}}_1{\mathcal{B}}_4-{\mathcal{B}}_2{\mathcal{B}}_3\ne 0,$$

(8)

$$\begin{array}{ccc}\hfill {\mathcal{B}}_1& =& \frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{{\Gamma }_k({\tilde{\vartheta }}_k)},\hfill \\ \hfill {\mathcal{B}}_2& =& \frac{\tilde{\lambda }{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k-\tilde{r}}{k}-1}}{{\Gamma }_k({\tilde{\eta }}_k-\tilde{r})}+\frac{\tilde{\mu }{(\widehat{\psi }(\tilde{\tilde{\sigma }})-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k+\tilde{v}}{k}-1}}{{\Gamma }_k({\tilde{\eta }}_k+\tilde{v})},\hfill \\ \hfill {\mathcal{B}}_3& =& \frac{\tilde{\nu }{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k-\tilde{z}}{k}-1}}{{\Gamma }_k({\tilde{\vartheta }}_k-\tilde{z})}+\frac{\tilde{\theta }{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k+\tilde{u}}{k}-1}}{{\Gamma }_k({\tilde{\vartheta }}_k+\tilde{u})},\hfill \\ \hfill {\mathcal{B}}_4& =& \frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k}{k}-1}}{{\Gamma }_k({\tilde{\eta }}_k)}.\hfill \end{array}$$

(9)

Proof. 

Assume that the pair $(\breve{k},\breve{l})$ is the solution of the System (5). As argued in [12], operating fractional integrals ${\displaystyle {}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}}$ and ${\displaystyle {}^k{\mathcal{I}}^{\widehat{p},\widehat{\psi }}}$ on the first and second $(k,\widehat{\psi })$-Hilfer fractional differential equations in system (5), respectively, we obtain

$$\begin{array}{ccc}\hfill \breve{k}(s)& =& {\displaystyle {}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(s)+c_0\frac{{(\widehat{\psi }(s)-\widehat{\psi }(a))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{{\Gamma }_k({\tilde{\vartheta }}_k)}+c_1\frac{{(\widehat{\psi }(s)-\widehat{\psi }(l_1))}^{\frac{{\vartheta }_k}{k}-2}}{{\Gamma }_k({\tilde{\vartheta }}_k-k)},}\hfill \\ \hfill \breve{l}(s)& =& {\displaystyle {}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}\breve{L}(s)+d_0\frac{{(\widehat{\psi }(s)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k}{k}-1}}{{\Gamma }_k({\tilde{\eta }}_k)}+d_1\frac{{(\widehat{\psi }(s)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k}{k}-2}}{{\Gamma }_k({\tilde{\eta }}_k-k)},}\hfill \end{array}$$

(10)

where $c_0,c_1,d_0$ and $d_1$ are constants. Making use of the boundary conditions $\breve{k}(l_1)=0$ and $\breve{l}(l_2)=0$ in Equations (10), we find that $c_1=0$ and $d_1=0$ since $\frac{{\tilde{\vartheta }}_k}{k}-2<0$, $\frac{{\tilde{\eta }}_k}{k}-2<0$.

On the other hand, due to the conditions ${\displaystyle \breve{k}(l_2)=\tilde{\lambda }{}^{k,H}{\mathcal{D}}^{\tilde{r},\tilde{s},\widehat{\psi }}\breve{l}(\tilde{\xi })+\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{v},\widehat{\psi }}\breve{l}(\tilde{\sigma })}$ and ${\displaystyle \breve{l}(l_2)=\tilde{\nu }{}^{k,H}{\mathcal{D}}^{\tilde{z},\tilde{w},\widehat{\psi }}\breve{k}(\tilde{\eta })+\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{u},\widehat{\psi }}\breve{k}(\tilde{\tau })}$, we obtain from Equations (10) after inserting $c_1=0$ and $d_1=0$ that

$$\begin{array}{ccc}\hfill {\displaystyle {}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(l_2)}& +& {\displaystyle c_0\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{{\Gamma }_k({\tilde{\vartheta }}_k)}=\tilde{\lambda }{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}\breve{L}(\tilde{\xi })+\tilde{\lambda }{d}_0\frac{{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k-\tilde{r}}{k}-1}}{{\Gamma }_k({\tilde{\eta }}_k-\tilde{r})}}\hfill \\ & & {\displaystyle +\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{p}+\widehat{v},\widehat{\psi }}\breve{L}(\tilde{\sigma })+\tilde{\mu }{d}_0\frac{{(\widehat{\psi }(\tilde{\sigma })-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k+\tilde{v}}{k}-1}}{{\Gamma }_k({\tilde{\eta }}_k+\tilde{v})},}\hfill \end{array}$$

(11)

and

$$\begin{array}{ccc}\hfill {\displaystyle {}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}\breve{L}(l_2)}& +& {\displaystyle d_0\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k}{k}-1}}{{\Gamma }_k({\tilde{\eta }}_k)}=\tilde{\nu }{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}\check{L}(\tilde{\eta })+\tilde{\nu }{c}_0\frac{{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k-\tilde{z}}{k}-1}}{{\Gamma }_k({\tilde{\vartheta }}_k-\tilde{z})}}\hfill \\ & & {\displaystyle +\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}\check{L}(\tilde{\tau })+\tilde{\theta }{c}_0\frac{{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k+\tilde{u}}{k}-1}}{{\Gamma }_k({\tilde{\vartheta }}_k+\tilde{u})}.}\hfill \end{array}$$

(12)

In view of the Notation (9), we can rewrite Equations (11) and (12) as

$$\begin{array}{ccc}\hfill {\mathcal{B}}_1{c}_0-{\mathcal{B}}_2{d}_0& =& {\displaystyle \tilde{\mu }{}^k{\mathcal{I}}^{\tilde{p}+\tilde{v},\widehat{\psi }}\breve{L}(\tilde{\sigma })+\tilde{\lambda }{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}\breve{L}(\tilde{\xi })-{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(l_2),}\hfill \\ \hfill -{\mathcal{B}}_3{c}_0+{\mathcal{B}}_4{d}_0& =& {\displaystyle \tilde{\theta }{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}\check{L}(\tilde{\tau })+\tilde{\nu }{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}\check{L}(\tilde{\eta })-{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}\breve{L}(l_2).}\hfill \end{array}$$

(13)

Solving the System (13) for $c_0$ and $d_0$, we obtain

$$\begin{array}{ccc}\hfill c_0& =& {\displaystyle \frac{1}{\mathcal{B}}[{\mathcal{B}}_4\left(\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{p}+\tilde{v},\widehat{\psi }}\breve{L}(\tilde{\sigma })+\tilde{\lambda }{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}\breve{L}(\tilde{\xi })-{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(l_2)\right)}\hfill \\ & & {\displaystyle +{\mathcal{B}}_2\left(\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}\check{L}(\tilde{\tau })+\tilde{\nu }{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}\check{L}(\tilde{\eta })-{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}\breve{L}(l_2)\right)],}\hfill \\ \hfill d_0& =& {\displaystyle \frac{1}{\mathcal{B}}[{\mathcal{B}}_1\left(\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}\check{L}(\tilde{\tau })+\tilde{\nu }{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}\check{L}(\tilde{\eta })-{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}\breve{L}(l_2)\right)}\hfill \\ & & {\displaystyle +{\mathcal{B}}_3\left(\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{p}+\tilde{v},\widehat{\psi }}\tilde{h}(\tilde{\sigma })+\tilde{\lambda }{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}\breve{L}(\tilde{\xi })-{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(l_2)\right)].}\hfill \end{array}$$

Replacing $c_0$ and $d_0$ in Equation (10) by the above values, we obtain Equations (6) and (7). The converse is obtained by direct calculation. This ends the proof. □

3. Existence and Uniqueness Results

Suppose that $\mathbb{X}=C([l_1,l_2],\mathbb{R})$ is the Banach space consisting of all continuous real-valued functions on $[l_1,l_2]$ to $\mathbb{R}$, equipped with the norm $\parallel \breve{k}\parallel =max\{|\breve{k}(s)|;s\in [l_1,l_2]\}$. Then $(\mathbb{X}\times \mathbb{X},\parallel (\breve{k},\breve{l})\parallel )$ is also a Banach space endowed with the norm $\parallel (\breve{k},\breve{l})\parallel =\parallel \breve{k}\parallel +\parallel \breve{l}\parallel$.

Using Lemma 1, an operator $\mathcal{F}:\mathbb{X}\times \mathbb{X}⟶\mathbb{X}\times \mathbb{X}$ can be defined as

$$\mathcal{F}(\breve{k},\breve{l})(s)=\left(\begin{array}{c}{\mathcal{F}}_1(\breve{k},\breve{l})(s)\\ {\mathcal{F}}_2(\breve{k},\breve{l})(s)\end{array}\right),$$

(14)

where

$$\begin{array}{ccc}\hfill & & {\mathcal{F}}_1(\breve{k},\breve{l})(s)\hfill \\ & =& {}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(s,\breve{k}(s),\breve{l}(s))+\frac{{(\widehat{\psi }(s)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{\mathcal{B}{\Gamma }_k({\tilde{\vartheta }}_k)}\times \hfill \\ & & {\displaystyle [{\mathcal{B}}_4\left(\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{p}+\tilde{v},\widehat{\psi }}\breve{L}(\tilde{\sigma },\breve{k}(\tilde{\sigma }),\breve{l}(\tilde{\sigma }))+\tilde{\lambda }{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}\breve{L}(\tilde{\xi },\breve{k}(\tilde{\xi }),\breve{l}(\tilde{\xi }))-{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(l_2,\breve{k}(l_2),\breve{l}(l_2))\right)}\hfill \\ & & {\displaystyle +{\mathcal{B}}_2\left(\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}\check{L}(\tilde{\tau },\breve{k}(\tilde{\tau }),\breve{l}(\tilde{\tau }))+\tilde{\nu }{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}\check{L}(\tilde{\eta },\breve{r}(\tilde{\eta }),\breve{l}(\tilde{\eta }))-{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}\breve{L}(l_2,\breve{k}(l_2),\breve{l}(l_2)\right)],}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill & & {\mathcal{F}}_2(\breve{k},\breve{l})(s)\hfill \\ & =& {\displaystyle {}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}{\breve{L}}_{\breve{k},\breve{l}}(s)+\frac{{(\widehat{\psi }(s)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k}{k}-1}}{\mathcal{B}{\Gamma }_k({\tilde{\eta }}_k)}\times }\hfill \\ & & {\displaystyle \left[{\mathcal{B}}_1\left(\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}\check{L}(\tilde{\tau },\breve{k}(\tilde{\tau }),\breve{l}(\tilde{\tau }))+\tilde{\nu }{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}\check{L}(\tilde{\eta },\breve{k}(\tilde{\eta }),\breve{l}(\tilde{\eta }))-{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}\breve{L}(l_2,\breve{k}(l_2),\breve{l}(l_2)\right)\right)}\hfill \\ & & {\displaystyle +{\mathcal{B}}_3\left(\tilde{\mu }{\displaystyle {}^k{\mathcal{I}}^{\tilde{p}+\tilde{v},\widehat{\psi }}}\breve{L}(\tilde{\sigma },\breve{k}(\tilde{\sigma }),\breve{l}(\tilde{\sigma }))+\tilde{\lambda }{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}\breve{L}(\tilde{\xi },\breve{k}(\tilde{\xi }),\breve{l}(\tilde{\xi }))-{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(l_2,\breve{k}(l_2),\breve{l}(l_2))\right)].}\hfill \end{array}$$

Here one can notice that the fixed point problem $\mathcal{F}(\breve{k},\breve{l})=(\breve{k},\breve{l})$ is equivalent to the nonlinear Problem (1) and (2).

For the sake of computational convenience, we introduce the notation:

$$\begin{array}{ccc}\hfill {\mathfrak{R}}_1& =& \frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}+\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}\hfill \\ & & +{\mathcal{B}}_2(|\tilde{\theta }|\frac{{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }+\tilde{u}}{k}}}{{\Gamma }_k(\tilde{\alpha }+\tilde{u}+k)}+|\tilde{\nu }|\frac{{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }-\tilde{z}}{k}}}{{\Gamma }_k(\tilde{\alpha }-\tilde{z}+k)})],\hfill \\ \hfill {\mathfrak{R}}_2& =& \frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(|\tilde{\mu }|\frac{{(\widehat{\psi }(\tilde{\sigma })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}+\widehat{v}}{k}}}{{\Gamma }_k(\tilde{p}+\widehat{v}+k)}+|\tilde{\lambda }|\frac{{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}-\tilde{r}}{k}}}{{\Gamma }_k(\tilde{p}-\tilde{r}+k)})\hfill \\ & & +{\mathcal{B}}_2\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}],\hfill \\ \hfill {\mathfrak{R}}_3& =& \frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\eta }}_k)}[{\mathcal{B}}_1(|\tilde{\theta }|\frac{{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }+\tilde{u}}{k}}}{{\Gamma }_k(\tilde{\alpha }+\tilde{u}+k)}+|\tilde{\nu }|\frac{{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }-\tilde{z}}{k}}}{{\Gamma }_k(\tilde{\alpha }-\tilde{z}+k)})\hfill \\ & & +{\mathcal{B}}_3\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}],\hfill \\ \hfill {\mathfrak{R}}_4& =& \frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}+\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_1\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\widehat{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}\hfill \\ & & +{\mathcal{B}}_3(|\tilde{\mu }|\frac{{(\widehat{\psi }(\tilde{\sigma })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}+\tilde{v}}{k}}}{{\Gamma }_k(\tilde{p}+\tilde{v}+k)}+|\tilde{\tilde{\lambda }}|\frac{{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\tilde{r}-\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}-\tilde{r}+k)})]\hfill \end{array}$$

(15)

and

$$\begin{array}{c}\hfill {\mathfrak{R}}_1^{*}={\mathfrak{R}}_1-\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)},{\mathfrak{R}}_4^{*}={\mathfrak{R}}_4-\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}.\end{array}$$

(16)

3.1. Existence of a Unique Solution

In the following result, the Banach’s fixed point theorem is applied to establish the uniqueness of solutions for the System (1) and (2).

Theorem 1. 

Let $\check{L},\breve{L}:[l_1,l_2]\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ satisfy the Lipschitz condition, that is, for all $s\in [l_1,l_2]$ and ${\breve{k}}_i,{\breve{l}}_i\in \mathbb{R}$, $i=1,2$,

$$\begin{array}{ccc}\hfill & & |\check{L}(s,{\breve{k}}_1,{\breve{k}}_2)-\check{L}(s,{\breve{l}}_1,{\breve{l}}_2)|\le {\widehat{m}}_1|{\breve{k}}_1-{\breve{l}}_1|+{\widehat{m}}_2|{\breve{k}}_2-{\breve{l}}_2|,\hfill \\ & & |\breve{f}(s,{\breve{k}}_1,{\breve{k}}_2)-\breve{f}(s,{\breve{l}}_1,{\breve{l}}_2)|\le {\widehat{n}}_1|{\breve{k}}_1-{\breve{l}}_1|+{\widehat{n}}_2{\breve{k}}_2-{\breve{l}}_2|,\hfill \end{array}$$

(17)

where ${\widehat{m}}_i,{\widehat{n}}_i,i=1,2$ are real constants. Moreover, we suppose that

$$({\mathfrak{R}}_1+{\mathfrak{R}}_3)({\widehat{m}}_1+{\widehat{m}}_2)+({\mathfrak{R}}_2+{\mathfrak{R}}_4)({\widehat{n}}_1+{\widehat{n}}_2)<1,$$

(18)

where ${\mathfrak{R}}_i,i=1,2,3,4,$ are defined in Equation (15). Then, the System (1) and (2) has a unique solution on $[l_1,l_2]$.

Proof. 

Let us consider a closed ball ${\mathbb{B}}_r=\{(\breve{k},\breve{l})\in \mathbb{X}\times \mathbb{X}:\parallel (\breve{k},\breve{l})\parallel \le r\},$ where

$$\begin{array}{c}\hfill r\ge \frac{({\mathfrak{R}}_1+{\mathfrak{R}}_3)\mathfrak{D}+({\mathfrak{R}}_2+{\mathfrak{R}}_4){\mathfrak{D}}_1}{1-[({\mathfrak{R}}_1+{\mathfrak{R}}_3)({\widehat{m}}_1+{\widehat{m}}_2)+({\mathfrak{R}}_2+{\mathfrak{R}}_4)({\widehat{n}}_1+{\widehat{n}}_2)]},\end{array}$$

${sup}_{s\in [l_1,l_2]}|\check{L}(s,0,0)|=\mathfrak{D}<\infty$ and ${sup}_{s\in [l_1,l_2]}|\breve{L}(s,0,0)|={\mathfrak{D}}_1<\infty$. Then we show that $\mathcal{F}{\mathbb{B}}_r\subseteq {\mathbb{B}}_r$. For $(\breve{k},\breve{l})\in {\mathbb{B}}_r$, we obtain

$$\begin{array}{ccc}& & |{\mathcal{F}}_1(\breve{k},\breve{l})(s)|\hfill \\ & \le & {\displaystyle {}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}|[\check{L}(s,\breve{k}(s),\breve{l}(s))-\check{L}(s,0,0)|+|\check{L}(s,0,0)|]}\hfill \\ & & {\displaystyle +\frac{{(\widehat{\psi }(s)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(|\tilde{\mu }|{}^k{\mathcal{I}}^{\tilde{p}+\tilde{v},\widehat{\psi }}[|\breve{L}(\tilde{\sigma },\breve{k}(\tilde{\sigma }),\breve{l}(\tilde{\sigma }))-\breve{L}(\tilde{\sigma },0,0)|+|\breve{L}(\tilde{\sigma },0,0)|]}\hfill \\ & & {\displaystyle +|\tilde{\lambda }|{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}[|\check{L}(\tilde{\xi },\breve{k}(\tilde{\xi }),\breve{l}(\tilde{\xi }))-\check{L}(\tilde{\xi },0,0)|+|\check{L}(\tilde{\xi },0,0)|]}\hfill \\ & & {\displaystyle +{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}[|\check{L}(l_2,\breve{k}(l_2),\breve{l}(l_2))-\check{L}(l_2,0,0)|+|\check{L}(l_2,0,0)|])}\hfill \\ & & {\displaystyle +{\mathcal{B}}_2(|\tilde{\theta }|{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}[|\check{L}(\tilde{\tau },\breve{k}(\tilde{\tau }),\breve{l}(\tilde{\tau })-\check{L}(\tilde{\tau },0,0)|+|\check{L}(\tilde{\tau },0,0)|]}\hfill \\ & & {\displaystyle +|\tilde{\nu }|{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}[|\check{L}(\tilde{\eta },\breve{k}(\tilde{\eta }),\breve{l}(\tilde{\eta }))-\check{L}(\tilde{\eta },0,0)|+|\check{L}(\tilde{\eta },0,0)|]}\hfill \\ & & {\displaystyle +{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}[|\breve{L}(l_2,\breve{k}(l_2),\breve{l}(l_2))-\breve{L}(l_2,0,0)|+|\breve{L}(l_2,0,0)|])]}\hfill \\ & \le & \frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}[{\widehat{m}}_1\parallel \breve{k}\parallel +{\widehat{m}}_2\parallel \breve{l}\parallel +\mathfrak{D}]\hfill \\ & & +\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(|\tilde{\mu }|\frac{{(\widehat{\psi }(\tilde{\sigma })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}+\widehat{v}}{k}}}{{\Gamma }_k(\tilde{p}+\tilde{v}+k)}[{\widehat{n}}_1\parallel \breve{k}\parallel +{\widehat{n}}_2\parallel \breve{l}\parallel +{\mathfrak{D}}_1]\hfill \\ & & {\displaystyle +|\tilde{\lambda }|\frac{{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}-\tilde{r}}{k}}}{{\Gamma }_k(\tilde{p}-\tilde{r}+k)}[{\widehat{n}}_1\parallel \breve{k}\parallel +{\widehat{n}}_2\parallel \breve{l}\parallel +{\mathfrak{D}}_1]}\hfill \\ & & +\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}[{\widehat{m}}_1\parallel \breve{k}\parallel +{\widehat{m}}_2\parallel \breve{l}\parallel +\mathfrak{D}])\hfill \\ & & +{\mathcal{B}}_2(|\tilde{\theta }|\frac{{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }+\tilde{u}}{k}}}{{\Gamma }_k(\tilde{\alpha }+\tilde{u}+k)}[{\widehat{m}}_1\parallel \breve{k}\parallel +{\widehat{m}}_2\parallel \breve{l}\parallel +\mathfrak{D}]\hfill \\ & & {\displaystyle +|\tilde{\nu }|\frac{{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }-\tilde{z}}{k}}}{{\Gamma }_k(\tilde{\alpha }-\tilde{z}+k)}[{\widehat{m}}_1\parallel \breve{k}\parallel +{\widehat{m}}_2\parallel \breve{l}\parallel +\mathfrak{D}]}\hfill \\ & & +\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}[{\widehat{n}}_1\parallel \breve{k}\parallel +{\widehat{n}}_2\parallel \breve{l}\parallel +{\mathfrak{D}}_1])]\hfill \\ & =& \{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}+\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}\hfill \\ & & +{\mathcal{B}}_2(|\tilde{\theta }|\frac{{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }+\tilde{u}}{k}}}{{\Gamma }_k(\tilde{\alpha }+\tilde{u}+k)}+|\tilde{\nu }|\frac{{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }-\tilde{z}}{k}}}{{\Gamma }_k(\tilde{\alpha }-\tilde{z}+k)})]\}[{\widehat{m}}_1\parallel \tilde{r}\parallel +{\widehat{m}}_2\parallel \breve{l}\parallel +\mathfrak{D}]\hfill \\ & & +\{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(|\tilde{\mu }|\frac{{(\widehat{\psi }(\tilde{\sigma })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}+\tilde{v}}{k}}}{{\Gamma }_k(\tilde{p}+\tilde{v}+k)}+|\tilde{\lambda }|\frac{{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}-\tilde{r}}{k}}}{{\Gamma }_k(\tilde{p}-\tilde{r}+k)})]\hfill \\ & & +{\mathcal{B}}_2\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}\}[{\widehat{n}}_1\parallel \breve{r}\parallel +{\widehat{n}}_2\parallel \breve{l}\parallel +{\mathfrak{D}}_1]\hfill \\ & =& {\mathfrak{R}}_1[{\widehat{m}}_1\parallel \breve{k}\parallel +{\widehat{m}}_2\parallel \breve{l}\parallel +\mathfrak{D}]+{\mathfrak{R}}_2[{\widehat{n}}_1\parallel \breve{k}\parallel +{\widehat{n}}_2\parallel \breve{l}\parallel +{\mathfrak{D}}_1]\hfill \\ & =& ({\mathfrak{R}}_1{\widehat{m}}_1+{\mathfrak{R}}_2{\widehat{n}}_1)\parallel \breve{k}\parallel +({\mathfrak{R}}_1{\widehat{m}}_2+{\mathfrak{R}}_2{\widehat{n}}_2)\parallel \breve{l}\parallel +{\mathfrak{R}}_1\mathfrak{D}+{\mathfrak{R}}_2{\mathfrak{D}}_1\hfill \\ & \le & {\mathfrak{R}}_1{\widehat{m}}_1+{\mathfrak{R}}_2{\widehat{n}}_1+{\mathfrak{R}}_1{\widehat{m}}_2+{\mathfrak{R}}_2{\widehat{n}}_2)r+{\mathfrak{R}}_1\mathfrak{D}+{\mathfrak{R}}_2{\mathfrak{D}}_1.\hfill \end{array}$$

Analogously, we have

$$\begin{array}{ccc}& & |{\mathcal{F}}_2(\breve{k},\breve{l})(s)|\hfill \\ \hfill & \le & \{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\eta }}_k)}[{\mathcal{B}}_1(|\tilde{\theta }|\frac{{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }+\tilde{u}}{k}}}{{\Gamma }_k(\tilde{\alpha }+\tilde{u}+k)}+|\tilde{\nu }|\frac{{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }-\tilde{z}}{k}}}{{\Gamma }_k(\tilde{\alpha }-\tilde{z}+k)})\hfill \\ & & +{\mathcal{B}}_3\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}]\}[{\widehat{m}}_1\parallel \breve{k}\parallel +{\widehat{m}}_2\parallel \breve{l}\parallel +\mathfrak{D}]\hfill \\ & & +\{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}+\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_1\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\widehat{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}\hfill \\ & & +{\mathcal{B}}_3(|\tilde{\mu }|\frac{{(\widehat{\psi }(\tilde{\sigma })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}+\widehat{v}}{k}}}{{\Gamma }_k(\tilde{p}+\tilde{v}+k)}+|\tilde{\lambda }|\frac{{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}-\tilde{r}}{k}}}{{\Gamma }_k(\tilde{p}-\tilde{r}+k)})]\}[{\widehat{n}}_1\parallel \breve{k}\parallel +{\widehat{n}}_2\parallel \breve{l}\parallel +{\mathfrak{D}}_1]\hfill \\ & =& ({\mathfrak{R}}_3{\widehat{m}}_1+{\mathfrak{R}}_4{\widehat{n}}_1)\parallel \breve{k}\parallel +({\mathfrak{R}}_3{\widehat{m}}_2+{\mathfrak{R}}_4{\widehat{n}}_2)\parallel \breve{l}\parallel +{\mathfrak{R}}_3\mathfrak{D}+{\mathfrak{R}}_4{\mathfrak{D}}_1\hfill \\ & \le & {\mathfrak{R}}_3{\widehat{m}}_1+{\mathfrak{R}}_4{\widehat{n}}_1+{\mathfrak{R}}_3{\widehat{m}}_2+{\mathfrak{R}}_4{\widehat{n}}_2)r+{\mathfrak{R}}_3\mathfrak{D}+{\mathfrak{R}}_4{\mathfrak{D}}_1.\hfill \end{array}$$

Accordingly, we obtain

$$\begin{array}{ccc}\hfill \parallel \mathcal{F}(\breve{k},\breve{l})\parallel & =& \parallel {\mathcal{F}}_1(\breve{k},\breve{l})\parallel +\parallel {\mathcal{F}}_2(\breve{k},\breve{l})\parallel \hfill \\ & \le & [({\mathfrak{R}}_1+{\mathfrak{R}}_3))({\widehat{m}}_1+{\widehat{m}}_2)+({\mathfrak{R}}_2+{\mathfrak{R}}_4)({\widehat{n}}_1+{\widehat{n}}_2)]r\hfill \\ & & +({\mathfrak{R}}_1+{\mathfrak{R}}_3))\mathfrak{D}+({\mathfrak{R}}_2+{\mathfrak{R}}_4)){\mathfrak{D}}_1\le r,\hfill \end{array}$$

which implies that $\mathcal{F}({\mathbb{B}}_r)\subseteq {\mathbb{B}}_r$ since $(\breve{k},\breve{l})\in {\mathbb{B}}_r$ is an arbitrary element. On the other hand, for $({\breve{k}}_2,{\breve{l}}_2),({\breve{k}}_1,{\breve{l}}_1)\in \mathbb{X}\times \mathbb{X}$ and $s\in [l_1,l_2]$, we obtain

$$\begin{array}{ccc}& & |{\mathcal{F}}_1({\breve{k}}_2,{\breve{l}}_2)(s)-{\mathcal{F}}_1({\breve{k}}_1,{\breve{l}}_1)(s)|\hfill \\ & \le & {\displaystyle {}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}|\check{L}(s,{\breve{k}}_2(s),{\breve{l}}_2(s))-\check{L}(s,{\breve{k}}_1(s),{\breve{l}}_1(s))|}\hfill \\ & & {\displaystyle +\frac{{(\widehat{\psi }(s)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(|\tilde{\mu }|{}^k{\mathcal{I}}^{\tilde{p}+\tilde{v},\widehat{\psi }}|\check{L}(\tilde{\sigma },{\breve{k}}_2(\tilde{\sigma }),{\breve{l}}_2(\tilde{\sigma }))-\check{L}(\tilde{\sigma },{\breve{k}}_1(\tilde{\sigma }),{\breve{l}}_1(\tilde{\sigma }))|}\hfill \\ & & {\displaystyle +|\tilde{\lambda }|{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}|\check{L}(\tilde{\xi },{\breve{k}}_2(\xi ),{\breve{l}}_2(\xi ))-\check{L}(\tilde{\xi },{\breve{k}}_1(\xi ),{\breve{l}}_1(\xi ))|}\hfill \\ & & {\displaystyle +{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}|\check{L}(l_2,{\breve{k}}_2(l_2),{\breve{l}}_2(l_2))-\check{L}(l_2,{\breve{k}}_1(l_2),{\breve{l}}_1(l_2))|)}\hfill \\ & & {\displaystyle +{\mathcal{B}}_2(|\tilde{\theta }|{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}|\check{L}(\tilde{\tau },{\breve{k}}_2(\tilde{\tau }),{\breve{l}}_2(\tilde{\tau })-\check{L}(\tilde{\tau },{\breve{k}}_1(\tilde{\tau }),{\breve{l}}_1(\tilde{\tau })|}\hfill \\ & & {\displaystyle +|\tilde{\nu }|{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}|\check{L}(\tilde{\eta },{\breve{k}}_2(\tilde{\eta }),{\breve{l}}_2(\tilde{\eta }))-\check{L}(\tilde{\eta },{\breve{k}}_1(\tilde{\eta }),{\breve{l}}_1(\tilde{\eta }))|}\hfill \\ & & {\displaystyle +{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}|\breve{f}(l_2,{\breve{k}}_2(l_2),{\breve{l}}_2(l_2))-\breve{L}(l_2,{\breve{k}}_1(l_2),{\breve{l}}_1(l_2))|)]}\hfill \\ & \le & {\displaystyle {}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}[{\widehat{m}}_1\parallel {\breve{k}}_2-{\breve{k}}_1\parallel +{\widehat{m}}_2\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ]}\hfill \\ & & {\displaystyle +\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(|\tilde{\mu }|{}^k{\mathcal{I}}^{\tilde{p}+\tilde{v},\widehat{\psi }}[{\widehat{n}}_1\parallel {\breve{k}}_2-{\breve{k}}_1\parallel +{\widehat{n}}_2\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ]}\hfill \\ & & {\displaystyle +|\tilde{\lambda }|{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}[{\widehat{n}}_1\parallel {\breve{k}}_2-{\breve{k}}_1\parallel +n_2\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ]}\hfill \\ & & {\displaystyle +{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}[{\widehat{m}}_1\parallel {\breve{k}}_2-{\breve{k}}_1\parallel +{\widehat{m}}_2\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ])}\hfill \\ & & {\displaystyle +{\mathcal{B}}_2(|\tilde{\theta }|{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}[{\widehat{m}}_1\parallel {\breve{k}}_2-{\breve{k}}_1\parallel +{\widehat{m}}_2\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ]}\hfill \\ & & {\displaystyle +|\tilde{\nu }|{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}[{\widehat{m}}_1\parallel {\breve{k}}_2-{\breve{k}}_1\parallel +{\widehat{m}}_2\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ]}\hfill \\ & & {\displaystyle +{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}[{\widehat{n}}_1\parallel {\breve{k}}_2-{\breve{k}}_1\parallel +{\widehat{n}}_2\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ])]}\hfill \\ & =& \{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}+\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}\hfill \\ & & +{\mathcal{B}}_2(|\tilde{\theta }|\frac{{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }+\tilde{u}}{k}}}{{\Gamma }_k(\tilde{\alpha }+\tilde{u}+k)}+|\tilde{\nu }|\frac{{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }-\tilde{z}}{k}}}{{\Gamma }_k(\tilde{\alpha }-\tilde{z}+k)})]\}\hfill \\ & & \times [{\widehat{m}}_1\parallel {\breve{k}}_2-{\breve{k}}_1\parallel +{\widehat{m}}_2\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ]\hfill \\ & & +\{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(|\mu |\frac{{(\widehat{\psi }(\tilde{\sigma })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}+\tilde{v}}{k}}}{{\Gamma }_k(\tilde{p}+\tilde{v}+k)}+|\tilde{\lambda }|\frac{{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}-\tilde{r}}{k}}}{{\Gamma }_k(\tilde{p}-\tilde{r}+k)})]\hfill \\ & & +{\mathcal{B}}_2\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}\}[{\widehat{n}}_1\parallel {\breve{k}}_2-{\breve{k}}_1\parallel +{\widehat{n}}_2\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ]\hfill \\ & =& ({\mathfrak{R}}_1{\widehat{m}}_1+{\mathfrak{R}}_2{\widehat{n}}_1)(\parallel {\breve{k}}_2-{\breve{k}}_1\parallel )+({\mathfrak{R}}_1{\widehat{m}}_2+{\mathfrak{R}}_2{\widehat{n}}_2)(\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ).\hfill \end{array}$$

Thus, we obtain

$$\begin{array}{ccc}\hfill & & \parallel {\mathcal{F}}_1({\breve{k}}_2,{\breve{l}}_2)-{\mathcal{F}}_1({\breve{k}}_1,{\breve{l}}_1)\parallel \hfill \\ & & \le ({\mathfrak{R}}_1{\widehat{m}}_1+{\mathfrak{R}}_2{\widehat{n}}_1+{\mathfrak{R}}_1{\widehat{m}}_2+{\mathfrak{R}}_2{\widehat{n}}_2)[\parallel {\breve{k}}_2-{\breve{k}}_1\parallel +\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ].\hfill \end{array}$$

(19)

Similarly, one can find that

$$\begin{array}{ccc}\hfill & & \parallel {\mathcal{F}}_2({\breve{k}}_2,{\breve{l}}_2)-{\mathcal{F}}_2({\breve{k}}_1,{\breve{l}}_1)\parallel \hfill \\ & & \le ({\mathfrak{R}}_3{\widehat{m}}_1+{\mathfrak{R}}_4{\widehat{n}}_1+{\mathfrak{R}}_3{\widehat{m}}_2+{\mathfrak{R}}_4{\widehat{n}}_2)[\parallel {\breve{k}}_2-{\breve{k}}_1\parallel +\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ].\hfill \end{array}$$

(20)

Then, it follows from from Equations (19) and (20) that

$$\begin{array}{ccc}\hfill & & \parallel \mathcal{F}({\breve{k}}_2,{\breve{l}}_2)-\mathcal{F}({\breve{k}}_1,{\breve{k}}_1)\parallel \hfill \\ & & \le [({\mathfrak{R}}_1+{\mathfrak{R}}_3)({\widehat{m}}_1+{\widehat{m}}_2)+({\mathfrak{R}}_2+{\mathfrak{R}}_4)({\widehat{n}}_1+n_2)](\parallel {\breve{k}}_2-{\breve{k}}_1\parallel +\parallel {\breve{l}}_2-{\breve{l}}_1\parallel ),\hfill \end{array}$$

which, in view of the Condition (18), verifies that the operator $\mathcal{F}$ is a contraction. Hence, by Banach’s contraction mapping principle, the operator $\mathcal{F}$ has a unique fixed point. Therefore, the System (1) and (2) has a unique solution on $[l_1,l_2].$ □

3.2. Existence Results

We rely on the Leray–Schauder alternative [20] to establish our first existence result.

Theorem 2. 

Let $\check{L},\breve{L}:[l_1,l_2]\times \mathbb{R}⟶\mathbb{R}$ be two continuous functions such that, for all $s\in [l_1,l_2]$ and ${\breve{k}}_i,{\breve{l}}_i\in \mathbb{R}$, $i=1,2$,

$$\begin{array}{ccc}& & |\check{L}(s,{\breve{k}}_1,{\breve{l}}_1)|\le {\widehat{l}}_0+{\widehat{l}}_1|{\breve{k}}_1|+{\widehat{l}}_2|{\breve{l}}_1|,\hfill \\ & & |\breve{L}(s,{\breve{k}}_2,{\breve{l}}_2)|\le {\widehat{q}}_0+{\widehat{q}}_1|{\breve{k}}_2|+{\widehat{q}}_2|{\breve{l}}_2|,\hfill \end{array}$$

where ${\widehat{l}}_i,{\widehat{q}}_i,i=0,1,2,$ are real constants with ${\widehat{l}}_0,{\widehat{q}}_0>0$. Then, the System (1) and (2) has at least one solution on $[l_1,l_2]$ provided that

$$({\mathfrak{R}}_1+{\mathfrak{R}}_3){\widehat{l}}_1+({\mathfrak{R}}_2+{\mathfrak{R}}_4){\widehat{q}}_1<1and({\mathfrak{R}}_1+{\mathfrak{R}}_3){\widehat{l}}_2+({\mathfrak{R}}_2+{\mathfrak{R}}_4){\widehat{q}}_2<1,$$

(21)

where ${\mathfrak{R}}_i,i=1,2,3,4,$ are defined in Equations (15).

Proof. 

Notice that continuity of the functions $\check{L}$ and $\breve{L}$ implies that of the operator $\mathcal{F}.$ Next, it will be shown that the operator $\mathcal{F}$ is completely continuous. Consider a bounded set $\mathcal{S}$ of $\mathbb{X}\times \mathbb{X}$. Then, there exist positive constants ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ such that $|\check{L}(s,\breve{k}(s),\breve{l}(s)|\le {\mathcal{L}}_1,|\breve{L}(s,\breve{k}(s),\breve{l}(s)|\le {\mathcal{L}}_2,\forall (\breve{k},\breve{l})\in \mathcal{S}.$ In consequence, for all $(\breve{k},\breve{l})\in \mathcal{S}$, we obtain

$$\begin{array}{ccc}\hfill |{\mathcal{F}}_1(\breve{k},\breve{l})(s)|& \le & \{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}+\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}\hfill \\ & & +{\mathcal{B}}_2(|\tilde{\theta }|\frac{{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }+\tilde{u}}{k}}}{{\Gamma }_k(\tilde{\alpha }+\tilde{u}+k)}+|\tilde{\nu }|\frac{{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }-\tilde{z}}{k}}}{{\Gamma }_k(\tilde{\alpha }-\tilde{z}+k)})]\}{\mathcal{L}}_1\hfill \\ & & +\{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(|\tilde{\mu }|\frac{{(\widehat{\psi }(\tilde{\sigma })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}+\tilde{v}}{k}}}{{\Gamma }_k(\tilde{p}+\widehat{v}+k)}\hfill \\ & & +|\tilde{\lambda }|\frac{{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}-\tilde{r}}{k}}}{{\Gamma }_k(\tilde{p}-\tilde{r}+k)})]+{\mathcal{B}}_2\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}\}{\mathcal{L}}_2\hfill \\ & & \le {\mathfrak{R}}_1{\mathcal{L}}_1+{\mathfrak{R}}_2{\mathcal{L}}_2,\hfill \end{array}$$

which yields

$$\begin{array}{c}\hfill \parallel {\mathcal{F}}_1(\breve{k},\breve{l})\parallel \le {\mathfrak{R}}_1{\mathcal{L}}_1+{\mathfrak{R}}_2{\mathcal{L}}_2.\end{array}$$

Analogously, one can obtain

$$\begin{array}{c}\hfill \parallel {\mathcal{F}}_2(\breve{k},\breve{l})\parallel \le {\mathfrak{R}}_3{\mathcal{L}}_1+{\mathfrak{R}}_4{\mathcal{L}}_2.\end{array}$$

Hence, we have

$$\begin{array}{c}\hfill \parallel \mathcal{F}(\breve{k},\breve{l})\parallel =\parallel {\mathcal{F}}_1(\breve{k},\breve{l})\parallel +\parallel {\mathcal{F}}_2(\breve{k},\breve{l}\parallel \le ({\mathfrak{R}}_1+{\mathfrak{R}}_3){\mathcal{L}}_1+({\mathfrak{R}}_2+{\mathfrak{R}}_4){\mathcal{L}}_2.\end{array}$$

Consequently, the operator $\mathcal{F}$ is uniformly bounded. To establish equicontinuity property of the operator $\mathcal{F}$, let $s_1,s_2\in [l_1,l_2]$ with $s_1<s_2$. Then, we have

$$\begin{array}{ccc}& & |{\mathcal{F}}_1(\breve{k}(s_2),\breve{l}(s_2))-{\mathcal{F}}_1(\breve{k}(s_1),\breve{l}(s_1))|\hfill \\ & \le & \frac{1}{{\Gamma }_k(\tilde{\alpha })}|{\int }_{s_1}^{s_2}{\widehat{\psi }}^{\prime }(s)[{(\widehat{\psi }(s_2)-\widehat{\psi }(s))}^{\frac{\tilde{\alpha }}{k}-1}-{(\widehat{\psi }(s_1)-\widehat{\psi }(s))}^{\frac{\tilde{\alpha }}{k}-1}]\check{L}(s,\breve{k}(s),\breve{l}(s))ds\hfill \\ & & +{\int }_{s_1}^{s_2}{\widehat{\psi }}^{\prime }(s){(\widehat{\psi }(s_2)-\widehat{\psi }(s))}^{\frac{\tilde{\alpha }}{k}-1}\check{L}(s,\breve{k}(s),\breve{l}(s))ds|\hfill \\ & & {\displaystyle +\frac{{(\widehat{\psi }(s_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\theta }}_k}{k}-1}-{(\widehat{\psi }(s_1)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\theta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(|\tilde{\mu }|{}^k{\mathcal{I}}^{\tilde{p}+\tilde{v},\widehat{\psi }}|\breve{L}(\tilde{\sigma },\breve{k}(\tilde{\sigma }),\breve{l}(\tilde{\sigma }))|}\hfill \\ & & {\displaystyle +|\tilde{\lambda }|{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}|\breve{L}(\tilde{\sigma },\breve{k}(\tilde{\sigma }),\breve{l}(\tilde{\sigma }))|+{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}|\check{L}(l_2,\breve{k}(l_2),\breve{l}(l_2))|)}\hfill \\ & & {\displaystyle +{\mathcal{B}}_2(|\tilde{\theta }|{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}|\check{L}(\tilde{\tau },\breve{k}(\tilde{\tau }),\breve{l}(\tilde{\tau }))+|\tilde{\nu }|{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}|\check{L}(\tilde{\eta },\breve{k}(\tilde{\eta }),\breve{l}(\tilde{\eta }))|}\hfill \\ & & {\displaystyle +{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}|\breve{L}(l_2,\breve{k}(l_2),\breve{l}(l_2))|)]}\hfill \\ & \le & \frac{{\mathcal{L}}_1}{{\Gamma }_k(\tilde{\alpha }+k)}[2{(\widehat{\psi }(s_2)-\widehat{\psi }(s_1))}^{\frac{\tilde{\alpha }}{k}}+|{(\widehat{\psi }(s_2)-\widehat{\psi }(l_2))}^{\frac{\tilde{\alpha }}{k}}-{(\widehat{\psi }(s_1)-\widehat{\psi }(l_2))}^{\frac{\tilde{\alpha }}{k}}|]\hfill \\ & & +\frac{{(\widehat{\psi }(s_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\theta }}_k}{k}-1}-{(\widehat{\psi }(s_1)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\theta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(|\tilde{\mu }|\frac{{(\widehat{\psi }(\tilde{\sigma })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}+\widehat{v}}{k}}}{{\Gamma }_k(\tilde{p}+\tilde{v}+k)}{\mathcal{L}}_2\hfill \\ & & +|\tilde{\lambda }|\frac{{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}-\tilde{r}}{k}}}{{\Gamma }_k(\tilde{p}-\tilde{r}+k)}{\mathcal{L}}_2+\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}{\mathcal{L}}_1)\hfill \\ & & +{\mathcal{B}}_2(|\tilde{\theta }|\frac{{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }+\tilde{u}}{k}}}{{\Gamma }_k(\tilde{\alpha }+\tilde{\tau }+k)}{\mathcal{L}}_1+|\nu |\frac{{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }-\tilde{z}}{k}}}{{\Gamma }_k(\tilde{\alpha }-\tilde{z}+k)}{\mathcal{L}}_1\hfill \\ & & +\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}{\mathcal{L}}_2)]\to 0\mathrm{as}{s}_2-s_1\to 0,\hfill \end{array}$$

independently of $(\breve{k},\breve{l})\in \mathcal{S}.$ Hence, ${\mathcal{F}}_1(\breve{k},\breve{l})$ is equicontinuous. Similarly, it can be shown that ${\mathcal{F}}_2(\breve{k},\breve{l})$ is equicontinuous. Thus, it follows by the foregoing arguments that the operator $\mathcal{F}(\breve{k},\breve{l})$ is completely continuous.

Lastly, it will be shown that the set $\mathcal{D}=\{(\breve{k},\breve{l})\in \mathbb{X}\times \mathbb{X}:(\breve{k},\breve{l})=\omega \mathcal{F}(\breve{k},\breve{l}),0\le \omega \le 1\}$ is bounded. Let $(\breve{k},\breve{l})\in \mathcal{D}$, then $(\breve{k},\breve{l})=\omega \mathcal{F}(\breve{k},\breve{l})$ for all $s\in [l_1,l_2]$ and that

$$\breve{k}(s)=\omega {\mathcal{F}}_1(\breve{k},\breve{l})(s),\breve{l}(s)=\omega {\mathcal{F}}_2(\breve{k},\breve{l})(s).$$

Thus, we have

$$\begin{array}{ccc}\hfill |\breve{k}(s)|& \le & \{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}+\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}\hfill \\ & & +{\mathcal{B}}_2(|\tilde{\theta }|\frac{{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }+\tilde{u}}{k}}}{{\Gamma }_k(\tilde{\alpha }+\tilde{u}+k)}+|\tilde{\nu }|\frac{{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }-\tilde{z}}{k}}}{{\Gamma }_k(\tilde{\alpha }-\tilde{z}+k)})]\}[{\widehat{l}}_0+{\widehat{l}}_1|\breve{k}|+{\widehat{l}}_2|\breve{l}|]\hfill \\ & & +\{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(|\tilde{\mu }|\frac{{(\widehat{\psi }(\tilde{\sigma })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}+\tilde{v}}{k}}}{{\Gamma }_k(\tilde{p}+\tilde{v}+k)}\hfill \\ & & +|\tilde{\lambda }|\frac{{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}-\tilde{r}}{k}}}{{\Gamma }_k(\tilde{p}-\tilde{r}+k)})]+{\mathcal{B}}_2\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}\}[{\widehat{q}}_0+{\widehat{q}}_1|\breve{k}|+{\widehat{q}}_2|\breve{l}|],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill |\breve{l}(s)|& \le & \{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\eta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\eta }}_k)}[{\mathcal{B}}_1(|\tilde{\theta }|\frac{{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }+\tilde{u}}{k}}}{{\Gamma }_k(\tilde{\alpha }+\tilde{u}+k)}\hfill \\ & & +|\tilde{\nu }|\frac{{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }-\tilde{z}}{k}}}{{\Gamma }_k(\tilde{\alpha }-\tilde{z}+k)})+{\mathcal{B}}_3\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}]\}[{\widehat{l}}_0+{\widehat{l}}_1|\breve{k}|+{\widehat{l}}_2|\breve{l}|]\hfill \\ & & +\{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}+\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_1\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\widehat{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}\hfill \\ & & +{\mathcal{B}}_3(|\tilde{\mu }|\frac{{(\widehat{\psi }(\tilde{\sigma })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}+\widehat{v}}{k}}}{{\Gamma }_k(\tilde{p}+\widehat{v}+k)}\hfill \\ & & +|\tilde{\lambda }|\frac{{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}-\tilde{r}}{k}}}{{\Gamma }_k(\tilde{p}-\tilde{r}+k)})]\}[{\widehat{q}}_0+{\widehat{q}}_1|\breve{k}|+{\widehat{q}}_2|\breve{l}|].\hfill \end{array}$$

Consequently, we obtain

$$\begin{array}{ccc}\hfill \parallel \breve{k}\parallel +\parallel \breve{l}\parallel & \le & ({\mathfrak{R}}_1+{\mathfrak{R}}_3){\widehat{l}}_0+({\mathfrak{R}}_2+{\mathfrak{R}}_4){\widehat{q}}_0+[(({\mathfrak{R}}_1+{\mathfrak{R}}_3){\widehat{l}}_1+({\mathfrak{R}}_2+{\mathfrak{R}}_4){\widehat{q}}_1]\parallel \breve{k}\parallel \hfill \\ & & +[(({\mathfrak{R}}_1+{\mathfrak{R}}_3){\widehat{l}}_2+({\mathfrak{R}}_2+{\mathfrak{R}}_4){\widehat{q}}_2]\parallel \breve{l}\parallel ,\hfill \end{array}$$

which can be expressed as

$$\parallel (\breve{k},\breve{l})\parallel \le \frac{({\mathfrak{R}}_1+{\mathfrak{R}}_3){\widehat{l}}_0+({\mathfrak{R}}_2+{\mathfrak{R}}_4){\widehat{q}}_0}{{\mathcal{M}}_0},$$

where

$${\mathcal{M}}_0=min\{1-[({\mathfrak{R}}_1+{\mathfrak{R}}_3){\widehat{l}}_1+({\mathfrak{R}}_2+{\mathfrak{R}}_4){\widehat{q}}_1],1-[({\mathfrak{R}}_1+{\mathfrak{R}}_3){\widehat{l}}_2+({\mathfrak{R}}_2+{\mathfrak{R}}_4){\widehat{q}}_2]\}.$$

Thus, the Leray–Schauder alternative applies and hence its conclusion implies that the operator $\mathcal{F}$ has at least one fixed point. Hence the System (1) and (2) has at least one solution on $[l_1,l_2].$ □

The proof of the next existence result relies on Krasnosel’skiĭ’s fixed point theorem [21].

Theorem 3. 

Assume that $\check{L},\breve{L}:[l_1,l_2]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are two continuous functions which satisfy Condition (17) of Theorem 1. Moreover, it is assumed that

$(\mathcal{H})$

There exist P and $Q\in C([l_1,l_2],{\mathbb{R}}_{+})$ such that

$$|\check{L}(s,\breve{k},\breve{l})|\le P(s),|\breve{L}(s,\breve{k},\breve{l})|\le Q(s),for each(s,\breve{k},\breve{l})\in [l_1,l_2]\times \mathbb{R}\times \mathbb{R}.$$

Then, the Problem (1) and (2) has at least one solution on $[l_1,l_2]$, provided that

$$\begin{array}{c}\hfill [{\mathfrak{R}}_1^{*}+{\mathfrak{R}}_3]({\widehat{m}}_1+{\widehat{m}}_2)+[{\mathfrak{R}}_2+{\mathfrak{R}}_4^{*}]({\widehat{n}}_1+{\widehat{n}}_2)<1.\end{array}$$

(22)

Proof. 

Let us first decompose the operator $\mathcal{F}$ into four operators ${\mathcal{F}}_{1,1},{\mathcal{F}}_{1,2},{\mathcal{F}}_{2,1}$ and ${\mathcal{F}}_{2,2}$ as

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{1,1}(\breve{k},\breve{l})(s)& =& {\displaystyle {}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(s,\breve{k}(s),\breve{l}(s)),s\in [l_1,l_2],}\hfill \\ \hfill {\mathcal{F}}_{1,2}(\breve{k},\breve{l})(s)& =& {\displaystyle \frac{{(\widehat{\psi }(s)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{\mathcal{B}{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{p}+\widehat{v},\widehat{\psi }}\breve{L}(\tilde{\sigma },\breve{k}(\tilde{\sigma }),\breve{l}(\tilde{\sigma }))}\hfill \\ & & {\displaystyle +\tilde{\lambda }{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}\breve{L}(\tilde{\xi },\breve{k}(\tilde{\xi }),\breve{l}(\tilde{\xi }))-{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(l_2,\breve{k}(l_2),\breve{l}(l_2))}\hfill \\ & & {\displaystyle +{\mathcal{B}}_2(\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}\check{L}(\tilde{\tau },\breve{k}(\tilde{\tau }),\breve{l}(\tilde{\tau })))+\tilde{\nu }{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}\check{L}(\tilde{\eta },\breve{k}(\tilde{\eta }),\breve{l}(\tilde{\eta }))}\hfill \\ & & {\displaystyle -{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}\breve{L}(l_2,\breve{k}(l_2),\breve{l}(l_2)))],s\in [l_1,l_2],}\hfill \\ \hfill {\mathcal{F}}_{2,1}(\breve{k},\breve{l})(s)& =& {\displaystyle {}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}\breve{L}(s,\breve{k}(s),\breve{l}(s)),s\in [l_1,l_2],}\hfill \\ \hfill {\mathcal{F}}_{2,2}(\breve{k},\breve{l})(s)& =& {\displaystyle \frac{{(\widehat{\psi }(s)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\tilde{\eta }}}_k}{k}-1}}{\mathcal{B}{\Gamma }_k({\tilde{\eta }}_k)}[{\mathcal{B}}_1(\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{\alpha }+\tilde{u},\widehat{\psi }}\check{L}(\tilde{\tau },\breve{k}(\tilde{\tau }),\breve{l}(\tilde{\tau }))}\hfill \\ & & {\displaystyle +\tilde{\nu }{}^k{\mathcal{I}}^{\tilde{\alpha }-\tilde{z},\widehat{\psi }}\check{L}(\tilde{\eta },\breve{k}(\tilde{\eta }),\breve{l}(\tilde{\eta }))-{}^k{\mathcal{I}}^{\tilde{p},\widehat{\psi }}\breve{L}(l_2,\breve{k}(l_2),\breve{l}(l_2)))}\hfill \\ & & {\displaystyle +{\mathcal{B}}_3(\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{p}+\tilde{\tilde{v}},\widehat{\psi }}\breve{L}(\tilde{\sigma },\breve{k}(\tilde{\sigma }),\breve{l}(\tilde{\sigma }))+\tilde{\lambda }{}^k{\mathcal{I}}^{\tilde{p}-\tilde{r},\widehat{\psi }}\breve{L}(\tilde{\xi },\breve{k}(\tilde{\xi }),\breve{l}(\tilde{\tilde{\xi }}))}\hfill \\ & & {\displaystyle -{}^k{\mathcal{I}}^{\tilde{\alpha },\widehat{\psi }}\check{L}(l_2,\breve{k}(l_2),\breve{l}(l_2)))],s\in [l_1,l_2].}\hfill \end{array}$$

Observe that ${\mathcal{F}}_1={\mathcal{F}}_{1,1}+{\mathcal{F}}_{1,2}$ and ${\mathcal{F}}_2={\mathcal{F}}_{2,1}+{\mathcal{F}}_{2,2}.$ Consider a closed ball ${\mathcal{B}}_{\widehat{\rho }}=\{(\breve{k},\breve{l})\in \mathbb{X}\times \mathbb{X}:\parallel (\breve{k},\breve{l})\parallel \le \widehat{\rho }\}$ with $\widehat{\rho }\ge ({\mathfrak{R}}_1+{\mathfrak{R}}_3)\parallel P\parallel +({\mathfrak{R}}_2+{\mathfrak{R}}_4)\parallel Q\parallel .$ As in the proof of Theorem 2, one can obtain

$$\begin{array}{c}\hfill |{\mathcal{F}}_{1,1}({\breve{k}}_1,{\breve{k}}_2)(s)+{\mathcal{F}}_{1,2}({\breve{l}}_1,{\breve{l}}_2)(s)|\le {\mathfrak{R}}_1\parallel P\parallel +{\mathfrak{R}}_2\parallel Q\parallel ,\end{array}$$

and

$$\begin{array}{c}\hfill |{\mathcal{F}}_{1,1}({\breve{k}}_1,{\breve{k}}_2)(t)+{\mathcal{F}}_{2,2}({\breve{k}}_1,{\breve{k}}_2)(t)|\le {\mathfrak{R}}_3\parallel P\parallel +{\mathfrak{R}}_4\parallel Q\parallel .\end{array}$$

Therefore, we obtain

$$\begin{array}{c}\hfill \parallel {\mathcal{F}}_1({\breve{k}}_1,{\breve{k}}_2)+{\mathcal{F}}_2({\breve{l}}_1,{\breve{l}}_2)\parallel \le ({\mathfrak{R}}_1+{\mathfrak{R}}_3)\parallel P\parallel +({\mathfrak{R}}_2+{\mathfrak{R}}_4)\parallel Q\parallel <\widehat{\rho }.\end{array}$$

Consequently, ${\mathcal{F}}_1({\breve{k}}_1,{\breve{k}}_2)+{\mathcal{F}}_2({\breve{l}}_1,{\breve{l}}_2)\in {\mathcal{B}}_{\widehat{\rho }}.$ Next, it will be accomplished that the $(F_{1,2},F_{2,2})$ is a contraction. As argued in proving Theorem 1, for $({\breve{k}}_1,{\breve{l}}_1),({\breve{k}}_2,{\breve{l}}_2)\in {\mathcal{B}}_{\widehat{\rho }}$, one can find that

$$\begin{array}{ccc}\hfill & & |{\mathcal{F}}_{1,2}({\breve{k}}_1,{\breve{k}}_2)(s)-{\mathcal{F}}_{1,2}({\breve{l}}_1,{\breve{l}}_2)(s)|\hfill \\ & \le & \{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}+{\mathcal{B}}_2(|\tilde{\theta }|\frac{{(\widehat{\psi }(\tilde{\tau })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }+\tilde{u}}{k}}}{{\Gamma }_k(\tilde{\alpha }+\tilde{u}+k)}\hfill \\ & & +|\tilde{\nu }|\frac{{(\widehat{\psi }(\tilde{\eta })-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }-\tilde{z}}{k}}}{{\Gamma }_k(\tilde{\alpha }-\tilde{z}+k)})]\}[{\widehat{m}}_1\parallel {\breve{k}}_1-{\breve{l}}_1\parallel +{\widehat{m}}_2\parallel {\breve{k}}_2-{\breve{l}}_2\parallel ]\hfill \\ & & +\{\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{{\tilde{\vartheta }}_k}{k}-1}}{|\mathcal{B}|{\Gamma }_k({\tilde{\vartheta }}_k)}[{\mathcal{B}}_4(|\tilde{\mu }|\frac{{(\widehat{\psi }(\tilde{\sigma })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}+\tilde{v}}{k}}}{{\Gamma }_k(\tilde{p}+\tilde{v}+k)}+|\tilde{\lambda }|\frac{{(\widehat{\psi }(\tilde{\xi })-\widehat{\psi }(l_1))}^{\frac{\tilde{p}-\tilde{r}}{k}}}{{\Gamma }_k(\tilde{p}-\tilde{r}+k)})]\hfill \\ & & +{\mathcal{B}}_2\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}\}[{\widehat{n}}_1\parallel {\breve{k}}_1-{\breve{l}}_1\parallel +{\widehat{n}}_2\parallel {\breve{k}}_2-{\breve{l}}_2\parallel ]\hfill \\ & =& {\mathfrak{R}}_1^{*}({\widehat{m}}_1\parallel {\breve{k}}_1-{\breve{l}}_1\parallel +{\widehat{m}}_2\parallel {\breve{k}}_2-{\breve{l}}_2\parallel )\hfill \\ & & +{\mathfrak{R}}_2({\widehat{n}}_1\parallel {\breve{k}}_1-{\breve{l}}_1\parallel +{\widehat{n}}_2\parallel {\breve{k}}_2-{\breve{l}}_2\parallel )\hfill \\ & =& \left[{\mathfrak{R}}_1^{*}{\widehat{m}}_1+{\mathfrak{R}}_2{\widehat{n}}_1\right]\parallel {\breve{k}}_1-{\breve{l}}_1\parallel +\left[{\mathfrak{R}}_1^{*}{\widehat{m}}_2+{\mathfrak{R}}_2{\widehat{n}}_2\right]\parallel {\breve{k}}_2-{\breve{l}}_2\parallel ,\hfill \end{array}$$

(23)

and

$$\begin{array}{ccc}\hfill & & |{\mathcal{F}}_{2,2}({\breve{k}}_1,{\breve{k}}_2)(s)-{\mathcal{F}}_{2,2}({\breve{l}}_1,{\breve{l}}_2)(s)|\hfill \\ & \le & \left[{\mathfrak{R}}_3{\widehat{m}}_1+{\mathfrak{R}}_4^{*}{\widehat{n}}_1\right]\parallel {\breve{k}}_1-{\breve{l}}_1\parallel +\left[{\mathfrak{R}}_3{\widehat{m}}_2+{\mathfrak{R}}_4^{*}{\widehat{n}}_2\right]\parallel {\breve{k}}_2-{\breve{l}}_2\parallel .\hfill \end{array}$$

(24)

From Equations (23) and (24), we obtain

$$\begin{array}{ccc}\hfill & & \parallel ({\mathcal{F}}_{1,2},{\mathcal{F}}_{2,2})({\breve{k}}_1,{\breve{k}}_2)-({\mathcal{F}}_{1,2},{\mathcal{F}}_{2,2})({\breve{l}}_1,{\breve{l}}_2)|\hfill \\ & \le & \left\{\left[{\mathfrak{R}}_1^{*}+{\mathfrak{R}}_3\right]({\widehat{m}}_1+{\widehat{m}}_2)+\left[{\mathfrak{R}}_2+{\mathfrak{R}}_4^{*}\right]({\widehat{n}}_1+{\widehat{n}}_2)\right\}(\parallel {\breve{k}}_1-{\breve{l}}_1\parallel +\parallel {\breve{k}}_2-{\breve{l}}_2\parallel ),\hfill \end{array}$$

which, owing to the Condition (22), shows that the operator $({\mathcal{F}}_{1,2},{\mathcal{F}}_{2,1})$ is a contraction. In view of the continuity property of $\check{L}$ and $\breve{L}$, the operator $({\mathcal{F}}_{1,1},{\mathcal{F}}_{2,1})$ is continuous. Moreover,

$$\begin{array}{c}\hfill \parallel ({\mathcal{F}}_{1,1},{\mathcal{F}}_{2,1})(\breve{k},\breve{l})\parallel \le \frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}\parallel P\parallel +\frac{{(\widehat{\psi }(l_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}\parallel Q\parallel ,\end{array}$$

as ${\displaystyle \parallel {\mathcal{F}}_{1,1}(\breve{k},\breve{l})\parallel \le \frac{\widehat{\psi }(l_2)-\widehat{\psi }(l_1){)}^{\frac{\tilde{\alpha }}{k}}}{{\Gamma }_k(\tilde{\alpha }+k)}\parallel P\parallel }$ and ${\displaystyle \parallel {\mathcal{F}}_{2,1}(\breve{k},\breve{l})\parallel \le \frac{\widehat{\psi }(l_2)-\widehat{\psi }(l_1){)}^{\frac{\tilde{p}}{k}}}{{\Gamma }_k(\tilde{p}+k)}\parallel Q\parallel .}$

Thus, $({\mathcal{F}}_{1,1},{\mathcal{F}}_{2,1}){\mathcal{B}}_{\rho }$ is uniformly bounded.

In the next step, we establish that the set $({\mathcal{F}}_{1,1},{\mathcal{F}}_{2,1}){\mathcal{B}}_{\rho }$ is equicontinuous. For $s_1,s_2\in [l_1,l_2]$, $s_1<s_2$ and for all $(\breve{k},\breve{l})\in {\mathcal{B}}_{\tilde{\rho }}$, we have

$$\begin{array}{ccc}& & |{\mathcal{F}}_{1,1}(\breve{k},\breve{l})(s_2)-{\mathcal{F}}_{1,1}(\breve{k},\breve{l})(s_1)\hfill \\ & \le & \frac{1}{{\Gamma }_k(\widehat{\tilde{\alpha }})}|{\int }_{s_1}^{s_2}{\widehat{\psi }}^{\prime }(s)[{(\widehat{\psi }(s_2)-\widehat{\psi }(s))}^{\frac{\tilde{\alpha }-k}{k}}-{(\widehat{\psi }(s_1)-\widehat{\psi }(s))}^{\frac{\tilde{\alpha }-k}{k}}]\check{L}(s,\breve{k}(s),\breve{l}(s))ds\hfill \\ & & +{\int }_{s_1}^{s_2}{\widehat{\psi }}^{\prime }(s){(\widehat{\psi }(s_2)-\widehat{\psi }(s))}^{\frac{\tilde{\alpha }-k}{k}}\check{f}(s,\breve{k}(s),\breve{l}(s))ds|\hfill \\ & \le & \frac{\parallel P\parallel }{{\Gamma }_k(\tilde{\alpha }+k)}[2{(\widehat{\psi }(s_2)-\widehat{\psi }(s_1))}^{\frac{\tilde{\alpha }}{k}}+|{(\widehat{\psi }(s_2)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}-{(\widehat{\psi }(s_1)-\widehat{\psi }(l_1))}^{\frac{\tilde{\alpha }}{k}}|]\hfill \\ & & ⟶0\mathrm{as}{s}_1⟶s_2,\hfill \end{array}$$

independently of $(\breve{k},\breve{l})\in {\mathcal{B}}_{\tilde{\rho }}.$ Analogously, one can obtain that

$$|({\mathcal{F}}_{2,1}(\breve{k},\breve{l})(s_2)-{\mathcal{F}}_{2,1}(\breve{k},\breve{l})(s_1)|\to 0\mathrm{as}{s}_1⟶s_2.$$

Thus, $|({\mathcal{F}}_{1,1},{\mathcal{F}}_{2,1})(\breve{k},\breve{l})(s_2)-({\mathcal{F}}_{1,1},{\mathcal{F}}_{2,1})(\breve{k},\breve{l})(s_1)|\to 0$ as $s_1⟶s_2$. So, $({\mathcal{F}}_{1,1},{\mathcal{F}}_{2,1})$ is equicontinuous. Hence, we deduce by the Arzelá–Ascoli theorem that the operator $({\mathcal{F}}_{1,1},{\mathcal{F}}_{2,1})$ is compact on ${\mathcal{B}}_{\widehat{\rho }}.$ Thus, the hypotheses of Krasnosel’skiĭ fixed point theorem is verified. Therefore, the System (1) and (2) has at least one solution on $[l_1,l_2].$ □

4. Examples

Consider the following boundary value problem after fixing the parameters in the System (1) and (2):

$$\left\{\begin{array}{c}{\displaystyle {}^{\frac{6}{7},H}{\mathfrak{D}}^{\frac{9}{7},\frac{4}{5};s^2+1}\breve{k}(s)=\check{L}(s,\breve{k}(s),\breve{l}(s)),\frac{2}{5}<s<\frac{8}{5},}\hfill \\ {\displaystyle {}^{\frac{6}{7},H}{\mathcal{D}}^{\frac{11}{7},\frac{2}{5};s^2+1}\breve{l}(s)=\breve{L}(s,\breve{k}(s),\breve{l}(s)),\frac{2}{5}<s<\frac{8}{5},}\hfill \\ {\displaystyle \breve{k}\left(\frac{2}{5}\right)=0,\breve{k}\left(\frac{8}{5}\right)=\frac{1}{\sqrt{\pi }}{}^{\frac{6}{7},H}{\mathcal{D}}^{\frac{6}{7},\frac{3}{5};s^2+1}\breve{l}\left(\frac{4}{5}\right)+\frac{2}{59}{}^{\frac{6}{7}}{\mathcal{I}}^{\frac{1}{4};s^2+1}\breve{l}\left(\frac{7}{5}\right),}\hfill \\ {\displaystyle \breve{l}\left(\frac{2}{5}\right)=0,\breve{l}\left(\frac{8}{5}\right)=\frac{4}{79}{}^{\frac{6}{7},H}{\mathcal{D}}^{\frac{5}{7},\frac{1}{5};s^2+1}\breve{k}\left(\frac{3}{5}\right)+\frac{1}{\sqrt{e}}{}^{\frac{6}{7}}{\mathcal{I}}^{\frac{3}{4};s^2+1}\breve{k}\left(\frac{6}{5}\right).}\hfill \end{array}\right.$$

(25)

Here, $k=6/7$, $\tilde{\alpha }=9/7$, $\tilde{p}=11/7$, $\tilde{r}=6/7$, $\tilde{z}=5/7$, $\tilde{\beta }=4/5$, $\tilde{q}=2/5$, $\tilde{s}=3/5$, $\tilde{w}=1/5$, $\tilde{v}=1/4$, $\tilde{u}=3/4$, $\widehat{\psi }(s)=s^2+1$, $\tilde{\lambda }=1/\sqrt{\pi }$, $\tilde{\mu }=2/59$, $\tilde{\nu }=4/79$, $\tilde{\theta }=1/\sqrt{e}$ and $l_1=2/5$, $l_2=8/5$, $\tilde{\xi }=4/5$, $\tilde{\sigma }=7/5$, $\tilde{\eta }=3/5$, $\tilde{\tau }=6/5$. Using the given values, we find that ${\tilde{\vartheta }}_k={\tilde{\eta }}_k=57/35$, ${\Gamma }_k({\tilde{\vartheta }}_k)={\Gamma }_k({\tilde{\eta }}_k)\approx 0.8371768940$, ${\Gamma }_k({\tilde{\vartheta }}_k+\tilde{u})\approx 1.248828596$, ${\Gamma }_k({\tilde{\vartheta }}_k-\tilde{z})\approx 0.9557910248$, ${\Gamma }_k({\tilde{\eta }}_k+\tilde{v})\approx 0.9127761461$, ${\Gamma }_k({\tilde{\eta }}_k-\tilde{r})\approx 1.085229307$, ${\Gamma }_k(\tilde{\alpha }+k)\approx 1.054911472$, ${\Gamma }_k(\tilde{p}+k)\approx 1.299979244$, ${\Gamma }_k(\tilde{\alpha }+\tilde{u}+k)\approx 2.012923279$, ${\Gamma }_k(\tilde{\alpha }-\tilde{z}+k)\approx 2.968888877$, ${\Gamma }_k(\tilde{p}+\tilde{v}+k)\approx 1.622489113$, ${\Gamma }_k(\tilde{p}-\tilde{r}+k)\approx 6.329317026$, ${\mathcal{B}}_1\approx 2.626472658$, ${\mathcal{B}}_2\approx 0.6342926434$, ${\mathcal{B}}_3\approx 0.8003297566$, ${\mathcal{B}}_4\approx 2.626472658$, $\mathcal{B}\approx 6.390715346$ (${\mathcal{B}}_i,i=1,2,3,4,$ and $\mathcal{B}$ are, respectively, given in Equations (9) and (8)), ${\mathfrak{R}}_1\approx 7.483199257$, ${\mathfrak{R}}_2\approx 1.254247333$, ${\mathfrak{R}}_3\approx 1.797703986$, ${\mathfrak{R}}_4\approx 8.040757033$ (${\mathfrak{R}}_i,i=1,2,3,4,$ are defined in (15)), ${\mathfrak{R}}_1^{*}\approx 3.958672213$, ${\mathfrak{R}}_4^{*}\approx 4.211473493$ (${\mathfrak{R}}_1^{*}$ and ${\mathfrak{R}}_2^{*}$ are defined in (16)).

Example 1. 

Let $\check{L},\breve{L}:[(2/5),(8/5)]\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ be the nonlinear Lipschitzian unbounded functions given by

$$\begin{array}{ccc}\hfill \check{L}(s,\breve{k},\breve{l})& =& \frac{e^{-(5s-2)}}{(40s+21)}\left(\frac{|\breve{k}|}{1+|\breve{k}|}\right)+\frac{{cos}^2\pi s({\breve{l}}^2+2|\breve{l}|)}{(2{(5s+4)}^2+6)(1+|\breve{l}|)}+\frac{1}{3}s+1,\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill \breve{L}(s,\breve{k},\breve{l})& =& \frac{{sin}^2\pi t({\breve{k}}^2+2|\breve{k}|)}{2{(5s+4)}^2(1+|\breve{k}|)}+\frac{{tan}^{-1}(\breve{l})}{2(35t+5)}+\frac{1}{4}s+2,\hfill \end{array}$$

(27)

which satisfy the Lipschitz condition:

$$|\check{L}(s,{\breve{k}}_1,{\breve{l}}_1)-\check{L}(s,{\breve{k}}_2,{\breve{l}}_2)|\le \frac{1}{37}|{\breve{k}}_1-{\breve{k}}_2|+\frac{1}{39}|{\breve{l}}_1-{\breve{l}}_2|,$$

$$|\breve{L}(s,{\breve{k}}_1,{\breve{l}}_1)-\breve{L}(s,{\breve{k}}_2,{\breve{l}}_2)|\le \frac{1}{36}|{\tilde{r}}_1-{\tilde{r}}_2|+\frac{1}{38}|{\widehat{z}}_1-{\widehat{z}}_2|,$$

with Lipschitz constants ${\widehat{m}}_1=1/37$, ${\widehat{m}}_2=1/39$, ${\widehat{n}}_1=1/36$ and ${\widehat{n}}_2=1/38$. Furthermore, $({\mathfrak{R}}_1+{\mathfrak{R}}_3)({\widehat{m}}_1+{\widehat{m}}_2)+({\mathfrak{R}}_2+{\mathfrak{R}}_4)({\widehat{n}}_1+{\widehat{n}}_2)\approx 0.9916070446<1.$ Thus, the hypotheses of Theorem 1 are satisfied and hence its conclusion implies that the Problem (25) with functions $\check{L}$ and $\breve{L}$ given by Equations (26) and (27), respectively, has a unique solution on the interval $[(2/5),(8/5)]$.

Example 2. 

Consider the functions $\check{L}$, $\breve{L}:[(2/5),(8/5)]\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ as

$$\begin{array}{ccc}\hfill \check{L}(s,\breve{k},\breve{l})& =& \frac{1+{cos}^2(s\breve{k}\breve{l})}{2\pi s}+\frac{e^{-|s\breve{l}|}{|\breve{k}|}^{33}}{20(1+{\breve{k}}^{32})}+\frac{sin|\breve{l}|}{(5s+19)},\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill \breve{L}(s,\breve{k},\breve{l})& =& \frac{1+{sin}^2(s\breve{k}\breve{l})}{4\pi s}+\frac{\breve{k}(1+{cos}^4\breve{l})}{(5s+36)}+\frac{e^{-|s\breve{k}|}{\breve{l}}^{38}}{22(1+|\breve{l}{|}^{37})}.\hfill \end{array}$$

(29)

Clearly $|\tilde{f}(s,\breve{k},\breve{l})|\le (5/2\pi )+(1/20)|\breve{k}|+(1/21)|\breve{l}|$ and $|\breve{L}(s,\breve{k},\breve{l})|\le (5/4\pi )+(1/19)|\breve{k}|+(1/22)|\breve{l}|$, with ${\widehat{l}}_0=5/2\pi$, ${\widehat{l}}_1=1/20$, ${\widehat{l}}_2=1/21$, ${\widehat{q}}_0=5/4\pi$, ${\widehat{q}}_1=1/19$, ${\widehat{q}}_2=1/22$. Moreover, $({\mathfrak{R}}_1+{\mathfrak{R}}_3){\widehat{l}}_1+({\mathfrak{R}}_2+{\mathfrak{R}}_4){\widehat{q}}_1\approx 0.9532559183<1$ and $({\mathfrak{R}}_1+{\mathfrak{R}}_3){\widehat{l}}_2+({\mathfrak{R}}_2+{\mathfrak{R}}_4){\widehat{q}}_2\approx 0.8644479720<1$. Therefore, by the conclusion of Theorem 2, the Problem (25) with functions $\check{L}$, $\breve{L}$ given by Equations (28) and (29), respectively, has at least one solution on $[(2/5),(8/5)]$.

Example 3. 

Let the nonlinear Lipschitzian functions $\check{L},\breve{L}:[(2/5),(8/5)]\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R}$ be defined by

$$\begin{array}{ccc}\hfill \check{L}(s,\breve{k},\breve{l})& =& \frac{1}{2\pi }{sin}^4\pi s+\frac{|\breve{k}|}{24(1+|\breve{k}|)}+\frac{1}{22}{e}^{-(5s-2)}{tan}^{-1}\breve{l},\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill \breve{L}(s,\breve{k},\breve{l})& =& \frac{1}{4\pi }{cos}^4\pi s+\frac{sin\breve{k}}{(10s+19)}+\frac{2|\breve{l}|}{105s(1+|\breve{l}|)}.\hfill \end{array}$$

(31)

Then, we have

$$|\check{L}(s,\breve{k},\breve{l})|\le \frac{1}{2\pi }{sin}^4\pi s+\frac{\pi }{44}{e}^{-(5s-2)}+\frac{1}{24},$$

$$|\breve{L}(s,\breve{k},\breve{l})|\le \frac{1}{4\pi }{cos}^4\pi s+\frac{1}{10s+19}+\frac{2}{105s},$$

and

$$|\check{L}(s,{\breve{k}}_1,{\breve{l}}_1)-\check{L}(s,{\breve{k}}_2,{\breve{l}}_2)|\le \frac{1}{24}|{\breve{k}}_1-{\breve{k}}_2|+\frac{1}{22}|{\breve{l}}_1-{\breve{l}}_2|,$$

$$|\breve{L}(s,{\breve{k}}_1,{\breve{l}}_1)-\breve{L}(s,{\breve{k}}_2,{\breve{l}}_2)|\le \frac{1}{23}|{\breve{k}}_1-{\breve{k}}_2|+\frac{1}{21}|{\breve{l}}_1-{\breve{l}}_2|.$$

Setting ${\widehat{m}}_1=1/24$, ${\widehat{m}}_2=1/22$, ${\widehat{n}}_1=1/23$, ${\widehat{n}}_2=1/21$, we find that $[{\mathfrak{R}}_1^{*}+{\mathfrak{R}}_3]({\widehat{m}}_1+{\widehat{m}}_2)+[{\mathfrak{R}}_2+{\mathfrak{R}}_4^{*}]({\widehat{n}}_1+{\widehat{n}}_2)\approx 0.9994149281<1.$ Therefore, by Theorem 3, the Problem (25) with the functions $\check{L}$, $\breve{L}$ given by Equations (30) and (31), respectively, has at least one solution.

It is interesting to note that the functions given in Equations (30) and (31) satisfy the Lipschitz condition. However, the uniqueness of the solution to the problem at hand does not follow since $({\mathfrak{R}}_1+{\mathfrak{R}}_3)({\widehat{m}}_1+{\widehat{m}}_2)+({\mathfrak{R}}_2+{\mathfrak{R}}_4)({\widehat{n}}_1+{\widehat{n}}_2)\approx 1.655313420>1.$

5. Conclusions

In this work, we have established the existence and uniqueness results for a nonlinear nonlocal boundary value problem involving $(k,\widehat{\psi })$-Hilfer fractional derivative and $(k,\widehat{\psi })$-Riemann–Liouville fractional integral operators. In order to apply the fixed-point technique to the given problem, we first transform it into a fixed-point problem, which facilitates the application of the fixed point theorems chosen for the present analysis. Our problem is novel in the given configuration and the results obtained for it are of more general form. Some new results arising as special cases from our work are listed below.

• By letting $\tilde{\mu }=0=\tilde{\theta }$ in the present results, we obtain the ones for coupled boundary conditions involving only $(k,\widehat{\psi })$-Hilfer derivative operators:

  $${\displaystyle \breve{k}(l_1)=0,\breve{l}(l_1)=0,\breve{k}(l_2)=\tilde{\lambda }{}^{k,H}{\mathcal{D}}^{\tilde{r},\tilde{s},\widehat{\psi }}\breve{l}(\tilde{\xi }),\breve{l}(l_2)=\tilde{\nu }{}^{k,H}{\mathcal{D}}^{\tilde{z},\tilde{w},\widehat{\psi }}\breve{k}(\tilde{\eta }).}$$
• For $\tilde{\lambda }=0=\tilde{\nu }$, our results correspond to the $(k,\widehat{\psi })$-Riemann–Liouville fractional type integral boundary conditions:

  $${\displaystyle \breve{k}(l_1)=0,\breve{l}(l_1)=0,\breve{k}(l_2)=\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{v},\widehat{\psi }}\breve{l}(\tilde{\sigma }),\breve{l}(l_2)=\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{u},\widehat{\psi }}\breve{k}(\tilde{\tau }).}$$
• Fixing $\tilde{\mu }=0$ and $\tilde{\nu }=0$ in the present results, we obtain the ones for the mixed boundary conditions of the form:

  $${\displaystyle \breve{k}(l_1)=0,\breve{l}(l_1)=0,\breve{k}(l_2)=\tilde{\lambda }{}^{k,H}{\mathcal{D}}^{\tilde{r},\widehat{s},\widehat{\psi }}\breve{l}(\xi ),\breve{l}(l_2)=\tilde{\theta }{}^k{\mathcal{I}}^{\tilde{u},\widehat{\psi }}\breve{k}(\tilde{\tau }).}$$
• Letting $\tilde{\lambda }=0$ and $\tilde{\theta }=0$ in the present results, we obtain the ones for the mixed boundary condition:

  $${\displaystyle \breve{k}(l_1)=0,\breve{l}(l_1)=0,\breve{k}(l_2)=\tilde{\mu }{}^k{\mathcal{I}}^{\tilde{v},\widehat{\psi }}\breve{l}(\tilde{\sigma }),\breve{l}(l_2)=\tilde{\nu }{}^{k,H}{\mathcal{D}}^{\tilde{z},\tilde{w},\widehat{\psi }}\breve{k}(\tilde{\eta }).}$$