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A study of a nonlinear coupled system of three fractional differential equations with nonlocal coupled boundary conditions

Abstract

In this research we introduce and study a new coupled system of three fractional differential equations supplemented with nonlocal multi-point coupled boundary conditions. Existence and uniqueness results are established by using the Leray–Schauder alternative and Banach’s contraction mapping principle. Illustrative examples are also presented.

Introduction

The methods of fractional calculus significantly improved the study of integer-order mathematical models associated with real-world problems appearing in scientific and technical disciplines. A point of central importance for choosing fractional order derivative operators is their nonlocal nature, which accounts for the history of the associated phenomena under investigation. For application details, see financial economics [1], ecology [2], immune systems [3], HIV/AIDS [4], chaotic synchronization [5, 6], etc. Inspired by the great popularity of the subject, many researchers turned to the further development of this branch of mathematical analysis. In particular, fractional order boundary value problems attracted considerable attention, and the literature on the topic was enriched with a huge number of interesting articles, for instance, see [712].

Fractional differential systems have also been studied by many researchers in view of their occurrence in the mathematical modeling of several physical and engineering processes [1315]. One can find the details about the theoretical development of the topic in the articles [1626].

We introduce and study a new class of coupled systems of mixed-order three fractional differential equations equipped with nonlocal multi-point coupled boundary conditions. In precise terms, we consider the following fully coupled system:

$$ \textstyle\begin{cases} {}^{C}{D}_{a^{+}}^{\eta }u(t)= \rho (t,u(t),x(t),y(t)),\quad 1< \eta \leq 2, t \in [a,b], \\ {}^{C}{D}_{a^{+}}^{\xi }x(t)= \varphi (t,u(t),x(t),y(t)),\quad 1< \xi \leq 2, t\in [a,b], \\ {}^{C}{D}_{a^{+}}^{\zeta }y(t)= \psi (t,u(t),x(t),y(t)),\quad 2< \zeta \leq 3, t\in [a,b], \\ u(a)=u_{0}, \qquad u(b)=\sum_{i=1}^{m} p_{i} x(\alpha _{i}), \\ x(a)=0, \qquad x(b)= \sum_{j=1}^{n} q_{j} y(\beta _{j}), \\ y(\xi _{1})=0, \qquad y(\xi _{2})=0,\qquad y(b)=\sum_{k=1}^{l}r_{k} u(\gamma _{k}), \\ a< \xi _{1} < \xi _{2} < \alpha _{1}< \cdots < \alpha _{m}< \beta _{1}< \cdots < \beta _{n}< \gamma _{1}< \cdots < \gamma _{l}< b, \end{cases} $$
(1.1)

where \({}^{C}{D}^{\chi }\) is a Caputo fractional derivative of order \(\chi \in \{\eta ,\xi ,\zeta \}\), \(\rho , \varphi , \psi :[a,b]\times \mathbb{R}\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) are given functions, \(p_{i},q_{j}, r_{k} \in \mathbb{R}\), \(i=1,\ldots ,m\), \(j=1,\ldots ,n\), \(k=1,\ldots ,l\).

Here we emphasize that the novelty of the present work lies in the fact that we consider a coupled system of three fractional differential equations of different orders on an arbitrary domain equipped with coupled nonlocal multi-point boundary conditions. One can observe that the multi-point boundary conditions are of cyclic nature and involve different nonlocal positions. Moreover, it is worthwhile to mention that much of the work on coupled fractional systems involves two fractional differential equations on the fixed domain. Thus our results are more general and contribute significantly to the existing literature on the topic.

The rest of the paper is organized as follows: In Sect. 2 we recall some basic definitions from fractional calculus and present an auxiliary result, which plays a pivotal role in transforming system (1.1) into equivalent integral equations. An existence result for the problem at hand is proved via the Leray–Schauder alternative, while the existence of a unique solution is established via Banach’s contraction mapping principle. These results are presented in Sect. 3. Examples are also discussed for illustration of the obtained results.

Preliminaries

Let us begin this section with some definitions related to our study [27].

Definition 2.1

The Riemann–Liouville fractional integral of order \(\omega \in \mathbb{R}\) (\(\omega >0\)) for a locally integrable real-valued function h defined on \(-\infty \leq a< t< b\leq +\infty \), denoted by \(I_{a^{+}} ^{\omega }h\), is defined by

$$ I_{a^{+}} ^{\omega }h ( t ) = {\mathrm{{ }}} \frac{1}{{\Gamma ( \omega )}} \int _{a}^{t} { ( {t - s} )^{\omega - 1} h ( s )} \,ds, $$

where Γ denotes the Euler gamma function.

Definition 2.2

Let \(h, h^{(m)} \in L^{1}[a,b]\) for \(-\infty \leq a< t< b\leq +\infty \). The Riemann–Liouville fractional derivative \(D_{a^{+}} ^{\omega }h\) of order \(\omega \in (m-1, m]\), \(m\in \mathbb{N}\) is defined as

$$ D_{a^{+}} ^{\omega }h ( t ) ={\mathrm{{}}} \frac{1}{{\Gamma ( m-\omega )}} \frac{d^{m}}{dt^{m}} \int _{a}^{t} { ({t - s} )^{m-1-\omega } h ( s )} \,ds, $$

while the Caputo fractional derivative \({{}^{C}{D}_{a^{+}}^{\omega }h}\) of order \(\omega \in (m-1, m]\), \(m\in \mathbb{N}\) is defined as

$$\begin{aligned} {{}^{C}{D}_{a^{+}}^{\omega }} h ( t ) = D_{a^{+}} ^{\omega } \biggl[ h ( t ) - h ( a ) -h' ( a ) \frac{(t-a)}{1!}-\cdots - h^{(m-1)} ( a ) \frac{(t-a)^{m-1}}{(m-1)!} \biggr]. \end{aligned}$$

Remark 2.3

The Caputo fractional derivative of order \(\omega \in (m-1, m]\), \(m\in \mathbb{N}\) for a continuous function \(h: (0,\infty )\to {\mathbb{R}}\) such that \(h\in C^{m}[a,b]\), existing almost everywhere on \([a, b]\), is defined by

$$\begin{aligned} ^{C}D^{\omega }h(t)=\frac{1}{\Gamma (m-\omega )} \int _{a}^{t}(t-s)^{m- \omega -1} h^{(m)}(s)\,ds. \end{aligned}$$

Now we present an important result to analyze problem (1.1).

Lemma 2.4

Let \(\overline{\rho },\overline{\varphi },\overline{\psi } \in C[a,b]\) and \(\Delta \ne 0\). Then the unique solution of the system

$$ \textstyle\begin{cases} {}^{C}{D}_{a^{+}}^{\eta }u(t)= \overline{\rho }(t),\quad 1< \eta \leq 2, t \in [a,b], \\ {}^{C}{D}_{a^{+}}^{\xi }x(t)= \overline{\varphi }(t),\quad 1< \xi \leq 2, t\in [a,b], \\ {}^{C}{D}_{a^{+}}^{\zeta }y(t)=\overline{\psi }(t),\quad 2< \zeta \leq 3, t\in [a,b], \\ u(a)=u_{0},\qquad u(b)=\sum_{i=1}^{m} p_{i} x(\alpha _{i}), \\ x(a)=0, \qquad x(b)= \sum_{j=1}^{n} q_{j} y(\beta _{j}), \\ y(\xi _{1})=0, \qquad y(\xi _{2})=0, \qquad y(b)=\sum_{k=1}^{l}r_{k} u(\gamma _{k}), \\ a< \xi _{1} < \xi _{2} < \alpha _{1}< \cdots < \alpha _{m}< \beta _{1}< \cdots < \beta _{n}< \gamma _{1}< \cdots < \gamma _{l}< b. \end{cases} $$
(2.1)

is given by the formulas

$$\begin{aligned}& u(t) = \int _{a}^{t}\frac{(t-s)^{\eta -1}}{\Gamma {(\eta )}} \overline{\rho }(s) \,ds+ u_{0} +(t-a) \Biggl\lbrace a_{12} \int _{a}^{b} \frac{(b-s)^{\eta - 1}}{\Gamma {(\eta )}}\overline{\rho }(s) \,ds \\& \hphantom{ u(t) =}{} + \frac{\sum_{i=1}^{m}p_{i}}{b-a} \Biggl(a_{13} \sum _{i=1}^{m}p_{i} (\alpha _{i} - a)+ 1 \Biggr) \int _{a}^{\alpha _{i}} \frac{(\alpha _{i} - a)^{\xi - 1}}{\Gamma {(\xi )}}\overline{\varphi }(s)\,ds \\& \hphantom{ u(t) =}{} + \frac{\sum_{i=1}^{m}p_{i}(\alpha _{i} - a)}{\Delta (b-a)} \Biggl(a_{3} \int _{a}^{\xi _{1}} \frac{(\xi _{1} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\overline{\psi }(s) \,ds + a_{4} \int _{a}^{\xi _{2}} \frac{(\xi _{2} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\overline{\psi }(s) \,ds \\& \hphantom{ u(t) =}{} - A_{3} \int _{a}^{b} \frac{(b - s)^{\zeta -1}}{\Gamma {(\eta )}}\overline{\psi }(s) \,ds - A_{1} \sum_{j=1}^{n}q _{j} \int _{a}^{\beta _{j}} \frac{(\beta _{j} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\overline{\psi }(s) \,ds \\& \hphantom{ u(t) =}{} - A_{1} \int _{a}^{b}\frac{(b - s)^{\xi -1}}{\Gamma {(\xi )}} \overline{\varphi }(s)\,ds \\& \hphantom{ u(t) =}{} + A_{3} \sum_{k=1}^{l}r_{k} \int _{a}^{\gamma _{k}} \frac{(\gamma _{k} - s)^{\eta -1}}{\Gamma {(\eta )}}\overline{\rho }(s) \,ds \Biggr) \Biggr\rbrace + a_{11} (t-a), \end{aligned}$$
(2.2)
$$\begin{aligned}& x(t) = \int _{a}^{t} \frac{(t-s)^{\xi -1}}{\Gamma {(\xi )}}\overline{\varphi }(s)\,ds + \frac{(t-a)}{\Delta } \Biggl\lbrace - a_{5} \int _{a}^{b} \frac{(b-s)^{\eta - 1}}{\Gamma {(\eta )}}\overline{\rho }(s) \,ds \\& \hphantom{x(t) =}{} + a_{5}\sum_{i=1}^{m}p_{i} \int _{a}^{\alpha _{i}} \frac{(\alpha _{i} - a)^{\xi - 1}}{\Gamma {(\xi )}}\overline{\varphi }(s)\,ds + a_{3} \int _{a}^{\xi _{1}} \frac{(\xi _{1} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\overline{\psi }(s) \,ds \\& \hphantom{x(t) =}{} + a_{4} \int _{a}^{\xi _{2}} \frac{(\xi _{2} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\overline{\psi }(s) \,ds \\& \hphantom{x(t) =}{} - A_{3} \int _{a}^{b} \frac{(b - s)^{\zeta -1}}{\Gamma {(\eta )}}\overline{\psi }(s) \,ds - A_{1} \sum_{j=1}^{n}q _{j} \int _{a}^{\beta _{j}} \frac{(\beta _{j} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\overline{\psi }(s) \,ds \\& \hphantom{x(t) =}{} - A_{1} \int _{a}^{b}\frac{(b - s)^{\xi -1}}{\Gamma {(\xi )}} \overline{\varphi }(s)\,ds \\& \hphantom{x(t) =}{} + A_{3} \sum_{k=1}^{l}r_{k} \int _{a}^{\gamma _{k}} \frac{(\gamma _{k} - s)^{\eta -1}}{\Gamma {(\eta )}}\overline{\rho }(s) \,ds +a_{6} \Biggr\rbrace , \end{aligned}$$
(2.3)
$$\begin{aligned}& y(t) = \int _{a}^{t} \frac{(t-s)^{\zeta -1}}{\Gamma {(\zeta )}}\overline{\psi }(s) \,ds +b_{1}(t) \int _{a}^{\xi _{1}} \frac{(\xi _{1} - a)^{\zeta - 1}}{\Gamma {(\zeta )}}\overline{\psi }(s) \,ds \\& \hphantom{y(t) =}{}+ b_{2}(t) \int _{a}^{\xi _{2}} \frac{(\xi _{2} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\overline{\psi }(s) \,ds \\& \hphantom{y(t) =}{} + b_{3}(t) \int _{a}^{b} \frac{(b-s)^{\zeta - 1}}{\Gamma {(\zeta )}}\overline{\psi }(s) \,ds + b_{4}(t) \sum_{j=1}^{n} q_{j} \int _{a}^{b} \frac{(\beta _{j} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\overline{\psi }(s) \,ds \\& \hphantom{y(t) =}{}+b_{5}(t) \int _{a}^{b}\frac{(b-s)^{\eta - 1}}{\Gamma {(\eta )}}\overline{\rho }(s) \,ds \\& \hphantom{y(t) =}{} - b_{3}(t) \sum_{k=1}^{l} r_{k} \int _{a}^{\gamma _{k}} \frac{(\gamma _{k} - s)^{\eta -1}}{\Gamma {(\eta )}}\overline{\rho }(s) \,ds - b_{4}(t) \int _{a}^{b} \frac{(b - s)^{\xi -1}}{\Gamma {(\xi )}} \overline{\varphi }(s)\,ds \\& \hphantom{y(t) =}{} - b_{5}(t)\sum_{i=1}^{m} \alpha _{i} \int _{a}^{ \alpha _{i}} \frac{(\alpha _{i} - s)^{\xi -1}}{\Gamma {(\xi )}} \overline{\varphi }(s)\,ds + b_{6}(t), \end{aligned}$$
(2.4)

where

$$\begin{aligned}& b_{1}(t) = \frac{1}{ \xi _{1} - \xi _{2}} \biggl(\xi _{2} - a + \frac{a_{2} a_{7}}{\Delta }+(t-a) \biggl(\frac{a_{1} a_{7}}{\Delta }-1 \biggr) \biggr)+(t-a)^{2} \frac{a_{7}}{\Delta }, \\& b_{2}(t) = \frac{1}{ \xi _{1} - \xi _{2}} \bigg(- \xi _{1} + a + \frac{a_{2} a_{8}}{\Delta }+(t-a) \biggl(\frac{a_{1} a_{8}}{\Delta }+1 \biggr)\bigg)+(t-a)^{2} \frac{a_{8}}{\Delta }, \\& b_{3}(t) = \frac{b-a }{\Delta (\xi _{1} - \xi _{2})}\bigl(a_{2} + a_{1}(t-a)+( \xi _{1} - \xi _{2}) (t-a)^{2}\bigr), \\& b_{4}(t) = \frac{A_{2}}{\Delta ( \xi _{1} - \xi _{2} )}\bigl(a_{2} + a_{1}(t-a)+( \xi _{1} - \xi _{2}) (t-a)^{2}\bigr), \\& b_{5}(t) = \frac{a_{9}}{\Delta ( \xi _{1} - \xi _{2} )}\bigl(a_{2} + a_{1}(t-a)+( \xi _{1} - \xi _{2}) (t-a)^{2}\bigr), \\& b_{6}(t) = \frac{a_{10}}{\Delta ( \xi _{1} - \xi _{2})}\bigl(a_{2} + a_{1}(t-a)+( \xi _{1} - \xi _{2}) (t-a)^{2}\bigr), \\& a_{1} = (\xi _{2} - a)^{2} - (\xi _{1} - a)^{2},\qquad a_{2}=(\xi _{2} - a) ( \xi _{1} - a) (\xi _{1} - \xi _{2}), \\& a_{3} = \frac{A_{3} (b - \xi _{2})-A_{1} \sum_{j=1}^{n}q_{j} (\xi _{2} - \beta _{j})}{\xi _{1} - \xi _{2}},\qquad a_{4} = \frac{A_{3} (\xi _{1} - b)-A_{1} \sum_{j=1}^{n}q_{j} (\beta _{j} - \xi _{1})}{\xi _{1} - \xi _{2}}, \\& a_{5} = \frac{A_{3} \sum_{k=1}^{l}r_{k} (\gamma _{k} - a)}{b-a},\qquad a_{6} = u_{0} A_{3} \Biggl(\sum_{k=1}^{l}r_{k} - \frac{\sum_{k=1}^{l} r_{k} (\gamma _{k} -a)}{b-a} \Biggr), \\& a_{7} = \frac{-(b-a) (b - \xi _{2})+ A_{2} \sum_{j=1}^{n}q_{j} (\xi _{2} - \beta _{j})}{\xi _{1} - \xi _{2}}, \\& a_{8} = \frac{-(b-a)(\xi _{1} - b)+ A_{2} \sum_{j=1}^{n}q_{j} (\beta _{j} - \xi _{1})}{\xi _{1} - \xi _{2}}, \\& a_{9} = \sum_{k=1}^{l}r_{k} (\gamma _{k} - a),\qquad a_{10}= - u_{0} (b-a) \sum _{k=1}^{l}r_{k} + \sum _{k=1}^{l} r_{k} (\gamma _{k} -a), \\& a_{11} = \frac{1}{b-a} \biggl( \frac{a_{6} \sum_{i=1}^{m}p_{i} (\alpha _{i} - a)}{\Delta } - u_{0} \biggr), \\& a_{12} = \frac{-a_{5} \sum_{i=1}^{m} p_{i} (\alpha _{i} - a)}{\Delta (b-a)}- \frac{1}{b-a},\qquad a_{13}=\frac{a_{5}}{\Delta }, \\& A_{1} = \frac{a_{2} + (b-a)a_{1} + (b-a)^{2} (\xi _{1} - \xi _{2})}{\xi _{1} - \xi _{2}},\qquad A_{2} = \frac{- \sum_{k=1}^{l}r_{k} (r_{k} - a)\sum_{i=1}^{m} p_{i} (\alpha _{i} -a)}{b-a}, \\& A_{3} = \frac{- \sum_{j=1}^{n}q_{j}}{\xi _{1} - \xi _{2}} \bigl(a_{2} +a_{1}( \beta _{j} - a)+(\beta _{j} - a)^{2} (\xi _{1} - \xi _{2} )\bigr), \\& \Delta = A_{2} A_{3} - A_{1} (b-a). \end{aligned}$$
(2.5)

Proof

The solution of system (2.1) can be written as

$$\begin{aligned}& u(t)=I^{\eta }_{a^{+}}\overline{\rho }(t)+c_{1}+c_{2}(t-a), \end{aligned}$$
(2.6)
$$\begin{aligned}& x(t)=I^{\xi }_{a^{+}}\overline{\varphi }(t)+c_{3}+c_{4}(t-a), \end{aligned}$$
(2.7)
$$\begin{aligned}& y(t)=I^{\zeta }_{a^{+}}\overline{\psi }(t) + c_{5} + c_{6} (t-a) + c_{7} (t-a)^{2}, \end{aligned}$$
(2.8)

where \(c_{i} \in \mathbb{R}\) (\(i=1,2,\ldots ,7\)) are unknown constants. Using the condition \(u(a)=u_{0}\) in (2.6) gives \(c_{1}=u_{0} \) and applying the condition \(x(a)=0 \) in (2.7) yields \(c_{3}=0 \), while making use of the conditions \(y(\xi _{1})=0\) and \(y(\xi _{2})=0\) leads to the equations

$$\begin{aligned}& I^{\zeta }_{a^{+}}\overline{\psi }(\xi _{1}) + c_{5} + c_{6} (\xi _{1} -a) + c_{7} (\xi _{1} -a)^{2}=0, \end{aligned}$$
(2.9)
$$\begin{aligned}& I^{\zeta }_{a^{+}}\overline{\psi }(\xi _{2}) + c_{5} + c_{6} (\xi _{2} -a) + c_{7} (\xi _{2} -a)^{2}=0. \end{aligned}$$
(2.10)

Using the conditions \(u(b)=\sum_{i=1}^{m} p_{i} x(\alpha _{i})\), \(x(b)=\sum_{j=1}^{n} q_{j} y( \beta _{j})\), and \(y(b)=\sum_{k=1}^{l} r_{k} u(\gamma _{k})\) with \(c_{1}=u_{0} \) and \(c_{3}=0\) in (2.6)–(2.8) gives

$$\begin{aligned}& I^{\eta }_{a^{+}}\overline{\rho }(b)+u_{0} +c_{2}(b-a)=\sum_{i=1}^{m} p_{i} \bigl(I^{\xi }_{a^{+}}\overline{\varphi } (\alpha _{i})+ c_{4}(\alpha _{i} - a) \bigr), \end{aligned}$$
(2.11)
$$\begin{aligned}& I^{\xi }_{a^{+}}\overline{\varphi }(b)+c_{4}(b-a)=\sum_{j=1}^{n}q_{j} \bigl(I^{\zeta }_{a^{+}}\overline{\psi }(\beta _{j})+c_{5}+c_{6}( \beta _{j} - a)+c_{7}(\beta _{j} - a)^{2} \bigr), \end{aligned}$$
(2.12)
$$\begin{aligned}& I^{\zeta }_{a^{+}}\overline{\psi }(b)+c_{5}+c_{6}(b - a)+c_{7}(b - a)^{2} = \sum_{k=1}^{l}r_{k} \bigl(I^{\eta }_{a^{+}}\overline{\rho }(r_{k})+u_{0}+c_{2}(r_{k} - a) \bigr). \end{aligned}$$
(2.13)

Subtracting (2.10) from (2.9), we find that

$$ c_{6} = \frac{1}{\xi _{1} - \xi _{2}} \bigl(a_{1} c_{7} - I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{1})+I^{\zeta }_{a^{+}}\overline{\psi }(\xi _{2}) \bigr), $$
(2.14)

where \(a_{1}\) is given in (2.5). Substituting the value of \(c_{6}\) into (2.9) yields

$$ c_{5} = \frac{1}{\xi _{1} - \xi _{2}} \bigl(a_{2} c_{7} (\xi _{2} - a) I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{1})- (\xi _{1} - a)I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{2}) \bigr), $$
(2.15)

where \(a_{2}\) is given in (2.5). From (2.11), we have

$$ c_{2}=\frac{1}{b-a} \Biggl(-I^{\eta }_{a^{+}} \overline{\rho }(b)+\sum_{i=1}^{m}p_{i} I^{\xi }_{a^{+}}\overline{\varphi }(\alpha _{i})+\sum _{i=1}^{m}p_{i}( \alpha _{i} -a)c_{4}-u_{0} \Biggr). $$
(2.16)

Substituting the values of \(c_{2}\), \(c_{5}\), and \(c_{6}\) into (2.12) and (2.13), we get the system

$$\begin{aligned}& A_{1} c_{7} + A_{2} c_{4} \\& \quad = \sum_{k=1}^{l}r_{k} I^{\eta }_{a^{+}} \overline{\rho }(\gamma _{k})+\sum _{k=1}^{l}r_{k} u_{0}+ \frac{\sum_{k=1}^{l}r_{k}(\gamma _{k} - a)}{b-a} \Biggl(\sum_{i=1}^{m}p_{i}I^{\xi }_{a^{+}} \overline{\varphi }(\alpha _{i})-u_{0}-I^{\eta }_{a^{+}} \overline{\rho }(b) \Biggr) \\& \qquad {} +\frac{1}{\xi _{1} - \xi _{2}} \bigl( (\xi _{1} - b)I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{2})+(b-\xi _{2})I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{1}) \bigr)-I^{\zeta }_{a^{+}} \overline{\psi }(b), \\& A_{3} c_{7} + (b-a)c_{4} \\& \quad = \sum _{j=1}^{n}q_{j} I^{\zeta }_{a^{+}} \overline{\psi }(\beta _{j})+ \frac{\sum_{j=1}^{n}q_{j}}{\xi _{1} - \xi _{2}} \bigl( (\xi _{2} - \beta _{j})I^{\zeta }_{a^{+}}\overline{ \psi }(\xi _{1})+(\beta _{j} - \xi _{1})I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{2}) \bigr)-I^{\xi }_{a^{+}} \overline{\varphi }(b). \end{aligned}$$

Solving the above system together with the notation in (2.5), we obtain

$$\begin{aligned}& c_{4} = \frac{1}{\Delta } \Biggl\lbrace a_{3} I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{1})+ a_{4} I^{\zeta }_{a^{+}}\overline{\psi }(\xi _{2})-A_{3} I^{\zeta }_{a^{+}}\overline{ \psi }(b)- A_{1}\sum_{j=1}^{n}q_{j}I^{\zeta }_{a^{+}} \overline{\psi }(\beta _{j})-a_{5} I^{\eta }_{a^{+}} \overline{\rho }(b) \\& \hphantom{c_{4} =}{} + A_{3} \sum_{k=1}^{l}r_{k} I^{\eta }_{a^{+}} \overline{\rho }(\gamma _{k})+a_{6} \Biggr\rbrace , \\& c_{7} = \frac{1}{\Delta } \Biggl\lbrace a_{7} I^{\zeta }_{a^{+}}\overline{\psi }(\xi _{1})+ a_{8} I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{2})+ (b-a)I^{\zeta }_{a^{+}}\overline{\psi }(b) \\& \hphantom{c_{7} =}{}+ A_{2} \sum_{j=1}^{n}q_{j}I^{\zeta }_{a^{+}} \overline{\psi }(\beta _{j})-a_{9} \sum _{i=1}^{m}p_{i} I^{\xi }_{a^{+}} \overline{\varphi }(\alpha _{i}) \\& \hphantom{c_{7} =}{} - A_{2} I^{\xi }_{a^{+}}\overline{\varphi }(b) + a_{9} I^{\eta }_{a^{+}}\overline{\rho }(b)-(b-a)\sum _{k=1}^{l}r_{k} I^{\eta }_{a^{+}} \overline{\rho }(\gamma _{k}) + a_{10} \Biggr\rbrace . \end{aligned}$$

Substituting the value of \(c_{4}\) into (2.16) yields

$$\begin{aligned} c_{2} = & \biggl[ \frac{-1}{b-a}-a_{5} \frac{\sum_{i=1}^{m}p_{i}(\alpha _{i} - a)}{\Delta (b-a)} \biggr] I^{\eta }_{a^{+}}\overline{\rho }(b)+\sum _{i=1}^{m} \biggl[\frac{1}{b-a}+ a_{5} \frac{\sum_{i=1}^{m}p_{i}(\alpha _{i} - a)}{\Delta (b-a)}I^{\xi }_{a^{+}} \overline{ \varphi }(\alpha _{i}) \biggr] \\ &{}+ \frac{\sum_{i=1}^{m}p_{i} (\alpha _{i} -a)}{\Delta (b-a)} \Biggl\lbrace a_{3} I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{1}) + a_{4} I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{2})- A_{3} I^{\zeta }_{a^{+}} \overline{\psi }(b) \\ &{}-A_{1} \sum_{j=1}^{n}q_{j} I^{\zeta }_{a^{+}} \overline{\psi }(\beta _{j})-A_{1} I^{\xi }_{a^{+}}\overline{\varphi }(b) \\ &{}+ A_{3} \sum_{k=1}^{l}r_{k} I^{\eta }_{a^{+}} \overline{\rho }(\gamma _{k}) \Biggr\rbrace + a_{11}. \end{aligned}$$

Inserting the value of \(c_{7}\) into (2.14) and (2.15), we get

$$\begin{aligned}& c_{5} = \frac{1}{\xi _{1} - \xi _{2}} \Biggl\lbrace \biggl[ \xi _{2} - a +\frac{a_{2} a_{7}}{\Delta } \biggr] I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{1}) + \biggl[ -\xi _{1} + a + \frac{a_{2} a_{8}}{\Delta } \biggr] I^{\zeta }_{a^{+}}\overline{\psi }( \xi _{2}) \\& \hphantom{c_{5} =}{} +\frac{a_{2}}{\Delta } \Biggl( (b-a)I^{\zeta }_{a^{+}} \overline{\psi }(b) \\& \hphantom{c_{5} =}{} + A_{2} \sum_{j=1}^{n}q_{j} I^{\zeta }_{a^{+}} \overline{\psi }(\beta _{j})-a_{9} \sum_{i=1}^{m}p_{i} I^{\xi }_{a^{+}} \overline{\varphi }(\alpha _{i})- A_{2} I^{\xi }_{a^{+}} \overline{\varphi }(b)+a_{9} I^{\eta }_{a^{+}}\overline{\rho }(b) \\& \hphantom{c_{5} =}{} -(b-a) \sum_{k=1}^{l}r_{k} I^{\eta }_{a^{+}}\overline{\rho }(\gamma _{k})+ a_{10} \Biggr) \Biggr\rbrace , \\& c_{6} = \frac{1}{\xi _{1} - \xi _{2}} \Biggl\lbrace \biggl[ \frac{a_{1} a_{7}-1}{\Delta } \biggr] I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{1}) + \biggl[ \frac{a_{1} a_{8}}{\Delta } +1 \biggr] I^{\zeta }_{a^{+}} \overline{\psi }(\xi _{2})+ \frac{a_{1}}{\Delta } \Biggl( (b-a)I^{\zeta }_{a^{+}}\overline{\psi }(b) \\& \hphantom{c_{6} =}{} + A_{2} \sum_{j=1}^{n}q_{j} I^{\zeta }_{a^{+}} \overline{\psi }(\beta _{j})-a_{9} \sum_{i=1}^{m}p_{i} I^{\xi }_{a^{+}} \overline{\varphi }(\alpha _{i})- A_{2} I^{\xi }_{a^{+}} \overline{\varphi }(b)+a_{9} I^{\eta }_{a^{+}}\overline{\rho }(b) \\& \hphantom{c_{6} =}{}-(b-a) \sum_{k=1}^{l}r_{k} I^{\eta }_{a^{+}}\overline{\rho }(\gamma _{k}) + a_{10} \Biggr) \Biggr\rbrace . \end{aligned}$$

Finally, substituting the values of \(c_{i}\), \(i=1,2,\ldots , 7\), into (2.6), (2.7), and (2.8), we obtain (2.2), (2.3), and (2.4). We can prove the converse of this lemma by direct computation. □

Main results

Let \(X=C([a,b], \mathbb{R})\) be a Banach space endowed with the norm \(\Vert x\Vert =\sup \{\vert x(t)\vert ,t \in [a,b]\}\). Then \((X \times X \times X, \Vert (u,x,y)\Vert _{X})\) is also a Banach space equipped with the norm \(\Vert (u,x,y)\Vert _{X}= \Vert u \Vert + \Vert x \Vert +\Vert y\Vert \), \(u,x,y\in X\).

In view of Lemma 2.4, we define an operator \(T:X \times X \times X \rightarrow X \times X \times X\) by

$$ T\bigl(u(t),x(t),y(t)\bigr)=\bigl(T_{1}\bigl(u(t),x(t),y(t) \bigr),T_{2}\bigl(u(t),x(t),y(t)\bigr),T_{3} \bigl(u(t),x(t),y(t)\bigr)\bigr), $$

where

$$\begin{aligned}& T_{1}\bigl(u(t),x(t),y(t)\bigr) \\& \quad = \int _{a}^{t} \frac{(t-s)^{\eta -1}}{\Gamma {(\eta )}}\rho \bigl(s,u(s),x(s),y(s)\bigr)\,ds + u_{0} \\& \qquad {}+ (t-a) \Biggl\lbrace a_{12} \int _{a}^{b} \frac{(b-s)^{\eta - 1}}{\Gamma {(\eta )}}\rho \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + \frac{\sum_{i=1}^{m}p_{i}}{b-a} \Biggl( a_{13} \sum _{i=1}^{m}p_{i} (\alpha _{i} - a)+ 1 \Biggr) \int _{a}^{\alpha _{i}} \frac{(\alpha _{i} - a)^{\xi - 1}}{\Gamma {(\xi )}}\varphi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + \frac{\sum_{i=1}^{m}p_{i}(\alpha _{i} - a)}{\Delta (b-a)} (a_{3} \int _{a}^{\xi _{1}} \frac{(\xi _{1} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + a_{4} \int _{a}^{\xi _{2}} \frac{(\xi _{2} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} - A_{3} \int _{a}^{b} \frac{(b - s)^{\zeta -1}}{\Gamma {(\eta )}}\psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} - A_{1} \sum_{j=1}^{n}q_{j} \int _{a}^{\beta _{j}} \frac{(\beta _{j} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} - A_{1} \int _{a}^{b} \frac{(b - s)^{\xi -1}}{\Gamma {(\xi )}}\varphi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + A_{3} \sum_{k=1}^{l}r_{k} \int _{a}^{\gamma _{k}} \frac{(\gamma _{k} - s)^{\eta -1}}{\Gamma {(\eta )}}\rho \bigl(s,u(s),x(s),y(s)\bigr)\,ds + a_{12} ) \Biggr\rbrace + a_{11}(t-a), \\& T_{2}\bigl(u(t),x(t),y(t)\bigr) \\& \quad = \int _{a}^{t} \frac{(t-s)^{\xi -1}}{\Gamma {(\xi )}}\varphi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + \frac{(t-a)}{\Delta } \Biggl\lbrace -a_{5} \int _{a}^{b} \frac{(b-s)^{\eta - 1}}{\Gamma {(\eta )}}\rho \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + a_{5}\sum_{i=1}^{m}p_{i} \int _{a}^{\alpha _{i}} \frac{(\alpha _{i} - a)^{\xi - 1}}{\Gamma {(\xi )}}\varphi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + a_{3} \int _{a}^{\xi _{1}} \frac{(\xi _{1} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + a_{4} \int _{a}^{\xi _{2}} \frac{(\xi _{2} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} - A_{3} \int _{a}^{b} \frac{(b - s)^{\zeta -1}}{\Gamma {(\eta )}}\psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} - A_{1} \sum_{j=1}^{n}q _{j} \int _{a}^{\beta _{j}} \frac{(\beta _{j} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} - A_{1} \int _{a}^{b} \frac{(b - s)^{\xi -1}}{\Gamma {(\xi )}}\varphi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + A_{3} \sum_{k=1}^{l}r_{k} \int _{a}^{\gamma _{k}} \frac{(\gamma _{k} - s)^{\eta -1}}{\Gamma {(\eta )}}\rho \bigl(s,u(s),x(s),y(s)\bigr)\,ds +a_{6} \Biggr\rbrace , \\& T_{3}\bigl(u(t),x(t),y(t)\bigr) \\& \quad = \int _{a}^{t} \frac{(t-s)^{\zeta -1}}{\Gamma {(\zeta )}}\psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + b_{1}(t) \int _{a}^{\xi _{1}} \frac{(\xi _{1} - a)^{\zeta - 1}}{\Gamma {(\zeta )}}\psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + b_{2}(t) \int _{a}^{\xi _{2}} \frac{(\xi _{2} - s)^{\zeta -1}}{\Gamma {(\zeta )}} \psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + b_{3}(t) \int _{a}^{b} \frac{(b-s)^{\zeta - 1}}{\Gamma {(\zeta )}}\psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + b_{4}(t)\sum_{j=1}^{n} q_{j} \int _{a}^{b} \frac{(\beta _{j} - s)^{\zeta -1}}{\Gamma {(\zeta )}}\psi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} + b_{5}(t) \int _{a}^{b} \frac{(b-s)^{\eta - 1}}{\Gamma {(\eta )}}\rho \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} - b_{3}(t) \sum_{k=1}^{l} r_{k} \int _{a}^{\gamma _{k}} \frac{(\gamma _{k} - s)^{\eta -1}}{\Gamma {(\eta )}}\rho \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} - b_{4}(t) \int _{a}^{b} \frac{(b - s)^{\xi -1}}{\Gamma {(\xi )}}\varphi \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\& \qquad {} - b_{5}(t)\sum_{i=1}^{m} \alpha _{i} \int _{a}^{ \alpha _{i}} \frac{(\alpha _{i} - s)^{\xi -1}}{\Gamma {(\xi )}} \varphi \bigl(s,u(s),x(s),y(s)\bigr)\,ds + b_{6}(t). \end{aligned}$$

For computational convenience, let us set

$$\begin{aligned}& L_{1} = \frac{1}{\Gamma {(\eta + 1)}} \biggl\lbrace (b-a)^{\eta }+ \vert a_{12} \vert (b-a)^{\eta + 1}+ \frac{ \vert A_{3} \vert \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)\sum_{k=1}^{l} \vert r_{k} \vert (\gamma _{k} - a)^{\eta }}{ \vert \Delta \vert } \biggr\rbrace , \\& M_{1} = \frac{1}{\Gamma {(\xi + 1)}} \Biggl\lbrace \sum _{i=1}^{m} \vert p_{i} \vert \Biggl( \vert a_{13} \vert \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)+1 \Biggr) (\alpha _{i} - a)^{\xi } \\& \hphantom{M_{1} =}{}+ \frac{ \vert A_{1} \vert \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)(b-a)^{\xi }}{ \vert \Delta \vert } \Biggr\rbrace , \\& N_{1} = \frac{1}{\Gamma {(\zeta + 1)}} \Biggl\lbrace \frac{\sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)}{ \vert \Delta \vert } \Biggl( \vert a_{3} \vert (\xi _{1} - a)^{\zeta }+ \vert a_{4} \vert ( \xi _{2} - a)^{\zeta }+ \vert A_{3} \vert (b-a)^{\zeta } \\& \hphantom{N_{1} =}{} + \vert A_{1} \vert \sum_{j=1}^{n} \vert q_{j} \vert ( \beta _{j} - a)^{\zeta } \Biggr) \Biggr\rbrace , \\& L_{2} = \frac{b-a}{ \vert \Delta \vert \Gamma {(\eta + 1)}} \Biggl\lbrace \vert a_{5} \vert (b-a)^{\eta }+ \vert A_{3} \vert \sum _{k=1}^{l} \vert r_{k} \vert (\gamma _{k} - a)^{\eta } \Biggr\rbrace , \\& M_{2} = \frac{1}{\Gamma {(\xi + 1)}} \Biggl\lbrace (b-a)^{\xi }+ \frac{b-a}{ \vert \Delta \vert } \Biggl( \vert a_{5} \vert \sum _{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)^{\xi }+ \vert A_{1} \vert (b-a)^{\xi } \Biggr) \Biggr\rbrace , \\& N_{2} = \frac{b-a}{ \vert \Delta \vert \Gamma {(\zeta + 1)}} \Biggl\lbrace \vert a_{3} \vert (\xi _{1} - a)^{\zeta }+ \vert a_{4} \vert (\xi _{2} - a)^{\zeta }+ \vert A_{3} \vert (b-a)^{\zeta }+ \vert A_{1} \vert \sum _{j=1}^{n} \vert q_{j} \vert (\beta _{j} - a)^{\zeta } \Biggr\rbrace , \\& L_{3} = \frac{1}{\Gamma {(\eta + 1)}} \Biggl\lbrace \delta _{5} (b-a)^{\eta }+ \delta _{3} \sum_{k=1}^{l} \vert r_{k} \vert (\gamma _{k} - a)^{\eta } \Biggr\rbrace , \\& M_{3} = \frac{1}{\Gamma {(\xi + 1)}} \Biggl\lbrace \delta _{4} (b-a)^{\xi }+ \delta _{5} \sum_{i=1}^{m} \vert \alpha _{i} \vert (\alpha _{i} - a)^{\xi } \Biggr\rbrace , \\& N_{3} = \frac{1}{\Gamma {(\zeta + 1)}} \Biggl\lbrace (b-a)^{\zeta }+ \delta _{1} (\xi _{1} - a)^{\zeta }+ \delta _{2} (\xi _{2} - a)^{\zeta } \\& \hphantom{N_{3} =}{}+ \delta _{3} (b-a)^{\zeta }+ \delta _{4} \sum_{j=1}^{n} \vert q _{j} \vert (\beta _{j} - a)^{\zeta } \Biggr\rbrace , \end{aligned}$$
(3.1)

where

$$\begin{aligned}& \delta _{1} = \frac{1}{ \vert \xi _{1} - \xi _{2} \vert } \biggl( \vert \xi _{2} \vert + \vert a \vert + \frac{ \vert a_{2} a_{7} \vert }{ \vert \Delta \vert }+(b-a) \biggl\vert \frac{ a_{1} a_{7} }{ \Delta }-1 \biggr\vert \biggr)+(b-a)^{2} \frac{ \vert a_{7} \vert }{ \vert \Delta \vert }, \\& \delta _{2} = \frac{1}{ \vert \xi _{1} - \xi _{2} \vert } \biggl( \vert a \vert + \vert \xi _{1} \vert + \frac{ \vert a_{2} a_{8} \vert }{ \vert \Delta \vert }+(b-a) \biggl\vert \frac{ a_{1} a_{8} }{ \Delta }+1 \biggr\vert \biggr)+(b-a)^{2} \frac{ \vert a_{8} \vert }{ \vert \Delta \vert }, \\& \delta _{3} = \frac{b-a }{ \vert \Delta \vert ( \vert \xi _{1} - \xi _{2} \vert )}\bigl( \vert a_{2} \vert + \vert a_{1} \vert (b-a)+\bigl( \vert \xi _{1} - \xi _{2} \vert \bigr) (b-a)^{2}\bigr), \\& \delta _{4} = \frac{ \vert A_{2} \vert }{ \vert \Delta \vert ( \vert \xi _{1} - \xi _{2} \vert )}\bigl( \vert a_{2} \vert + \vert a_{1} \vert (b-a)+\bigl( \vert \xi _{1} - \xi _{2} \vert \bigr) (b-a)^{2}\bigr), \\& \delta _{5} = \frac{ \vert a_{9} \vert }{ \vert \Delta \vert ( \vert \xi _{1} - \xi _{2} \vert )}\bigl( \vert a_{2} \vert + \vert a_{1} \vert (b-a)+\bigl( \vert \xi _{1} - \xi _{2} \vert \bigr) (b-a)^{2}\bigr), \\& \delta _{6} = \frac{ \vert a_{10} \vert }{ \vert \Delta \vert ( \vert \xi _{1} - \xi _{2} \vert )}\bigl( \vert a_{2} \vert + \vert a_{1} \vert (b-a)+\bigl( \vert \xi _{1} - \xi _{2} \vert \bigr) (b-a)^{2}\bigr). \end{aligned}$$

In our first result, we establish the existence of solutions for system (1.1) by applying the Leray–Schauder alternative [28].

Lemma 3.1

(Leray–Schauder alternative)

Let \(\mathfrak{J}:{\mathcal{U}}\longrightarrow \mathcal{U} \) be a completely continuous operator (i.e., a map restricted to any bounded set in \(\mathcal{U}\) is compact). Let \(\mathcal{Q}(\mathfrak{J})=\{x\in \mathcal{U}:x=\eta \mathfrak{J}(x)\textit{ for some }0<\eta <1\}\). Then either the set \(\mathcal{Q}(\mathfrak{J})\) is unbounded, or \(\mathfrak{J}\) has at least one fixed point.

Theorem 3.2

Let \(\Delta \ne 0\), where Δ is defined by (2.5). In addition, we assume that:

\((H_{2})\):

\(\rho , \varphi ,\psi :[a,b] \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) are continuous functions and there exist real constants \(k_{i},\sigma _{i}, \mu _{i} \geq 0\) (\(i=1,2,3\)) and \(k_{0}>0\), \(\sigma _{0}>0\), \(\mu _{0} > 0\) such that, for all \(t\in [a,b]\) and \(u,x,y \in \mathbb{R}\),

$$\begin{aligned}& \bigl\vert \rho (t,u,x,y) \bigr\vert \leq k_{0}+ k_{1} \vert u \vert + k_{2} \vert x \vert + k_{3} \vert y \vert , \\& \bigl\vert \varphi (t,u,x,y) \bigr\vert \leq \sigma _{0}+ \sigma _{1} \vert u \vert + \sigma _{2} \vert x \vert + \sigma _{3} \vert y \vert , \\& \bigl\vert \psi (t,u,x,y) \bigr\vert \leq \mu _{0}+ \mu _{1} \vert u \vert + \mu _{2} \vert x \vert + \mu _{3} \vert y \vert . \end{aligned}$$

Then system (1.1) has at least one solution on \([a,b]\) provided that

$$\begin{aligned}& ( L_{1} + L_{2} + L_{3})k_{1} +(M_{1}+ M_{2} + M_{3})\sigma _{1} + (N_{1}+N_{2}+N_{3}) \mu _{1}< 1, \\& ( L_{1}+ L_{2} + L_{3})k_{2}+(M_{1} +M_{2} +M_{3}) \sigma _{2}+(N_{1}+N_{2}+N_{3}) \mu _{2} < 1, \\& ( L_{1}+ L_{2} + L_{3})k_{3}+(M_{1} +M_{2} +M_{3}) \sigma _{3}+(N_{1}+N_{2}+N_{3}) \mu _{3} < 1, \end{aligned}$$
(3.2)

where \(L_{i}\), \(M_{i}\), \(N_{i}\), \(i=1,2,3\), are given in (3.1).

Proof

Observe that the continuity of the operator \(T: X \times X \times X \rightarrow X \times X \times X\) follows from that of the functions ρ, φ, and ψ. Next, let \(\Omega \subset X \times X \times X\) be bounded such that

$$\begin{aligned}& \bigl\vert \rho \bigl(t,u(t),x(t),y(t)\bigr) \bigr\vert \leq K_{1}, \\& \bigl\vert \varphi \bigl(t,u(t),x(t),y(t)\bigr) \bigr\vert \leq K_{2}, \\& \bigl\vert \psi \bigl(t,u(t),x(t),y(t)\bigr) \bigr\vert \leq K_{3},\quad \forall (u,x,y) \in \Omega , \end{aligned}$$

for positive constants \(K_{1}\), \(K_{2}\), and \(K_{3} \). Then, for any \((u,x,y)\in \Omega \), we have

$$\begin{aligned}& \bigl\vert T_{1}\bigl(u(t),x(t),y(t)\bigr) \bigr\vert \\& \quad \leq \int _{a}^{t}\frac{(t-s)^{\eta -1}}{\Gamma {(\eta )}} \bigl\vert \rho \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds + \vert u_{0} \vert \\& \qquad {} + (b-a) \Biggl\lbrace \vert a_{12} \vert \int _{a}^{b} \frac{(b-s)^{\eta - 1}}{\Gamma {(\eta )}} \bigl\vert \rho \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\& \qquad {} + \frac{\sum_{i=1}^{m} \vert p_{i} \vert }{b-a} \Biggl( \vert a_{13} \vert \sum _{i=1}^{m} \vert p_{i} \vert ( \alpha _{i} - a)+ 1 \Biggr) \int _{a}^{\alpha _{i}} \frac{(\alpha _{i} - a)^{\xi - 1}}{\Gamma {(\xi )}} \bigl\vert \varphi \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\& \qquad {} + \frac{\sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)}{ \vert \Delta \vert (b-a)} ( \vert a_{3} \vert \int _{a}^{\xi _{1}} \frac{(\xi _{1} - s)^{\zeta -1}}{\Gamma {(\zeta )}} \bigl\vert \psi \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\& \qquad {} + \vert a_{4} \vert \int _{a}^{\xi _{2}} \frac{(\xi _{2} - s)^{\zeta -1}}{\Gamma {(\zeta )}} \bigl\vert \psi \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\& \qquad {} + \vert A_{3} \vert \int _{a}^{b} \frac{(b - s)^{\zeta -1}}{\Gamma {(\eta )}} \bigl\vert \psi \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\& \qquad {} + \vert A_{1} \vert \sum _{j=1}^{n} \vert q_{j} \vert \int _{a}^{\beta _{j}} \frac{(\beta _{j} - s)^{\zeta -1}}{\Gamma {(\zeta )}} \bigl\vert \psi \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\& \qquad {} + \vert A_{1} \vert \int _{a}^{b} \frac{(b - s)^{\xi -1}}{\Gamma {(\xi )}} \bigl\vert \varphi \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\& \qquad {} + \vert A_{3} \vert \sum _{k=1}^{l} \vert r_{k} \vert \int _{a}^{\gamma _{k}} \frac{(\gamma _{k} - s)^{\eta -1}}{\Gamma {(\eta )}} \bigl\vert \rho \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds + \vert a_{12} \vert ) \Biggr\rbrace + \vert a_{11} \vert (b-a) \\& \quad \leq \vert u_{0} \vert + \vert a_{11} \vert (b-a)+ \biggl\lbrace \frac{1}{\Gamma {(\eta + 1)}} \biggl\lbrace (b-a)^{\eta }+ \vert a_{12} \vert (b-a)^{\eta + 1} \\& \qquad {} + \frac{ \vert A_{3} \vert \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)\sum_{k=1}^{l} \vert r_{k} \vert (\gamma _{k} - a)^{\eta }}{ \vert \Delta \vert } \biggr\rbrace \biggr\rbrace \Vert \rho \Vert \\& \qquad {} + \Biggl\lbrace \frac{1}{\Gamma {(\xi + 1)}} \Biggl\lbrace \sum _{i=1}^{m} \vert p_{i} \vert \Biggl( \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)+1\Biggr) (\alpha _{i} - a)^{\xi } \\& \qquad {} + \frac{ \vert A_{1} \vert \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)(b-a)^{\xi }}{ \vert \Delta \vert } \Biggr\rbrace \Biggr\rbrace \Vert \varphi \Vert \\& \qquad {}+ \Biggl\lbrace \frac{1}{\Gamma {(\zeta + 1)}} \Biggl\lbrace \frac{\sum_{i=1}^{m} \vert p_{i} (\alpha _{i} - a) \vert }{ \vert \Delta \vert } \Biggl( \vert a_{3} \vert (\xi _{1} - a)^{\zeta }+ \vert a_{4} \vert ( \xi _{2} - a)^{\zeta } \\& \qquad {} + \vert A_{3} \vert (b-a)^{\zeta }+ \vert A_{1} \vert \sum_{j=1}^{n} \vert q_{j} \vert (\beta _{j} - a)^{\zeta } \Biggr) \Biggr\rbrace \Biggr\rbrace \Vert \psi \Vert \\& \quad \leq \vert u_{0} \vert + \vert a_{11} \vert (b-a)+L_{1} K_{1} + M_{1}K_{2} +N_{1}K_{3}, \end{aligned}$$

which implies that

$$ \bigl\Vert T_{1}(u,x,y) \bigr\Vert _{X} \leq \vert u_{0} \vert + \vert a_{11} \vert (b-a)+L_{1} K_{1} + M_{1}K_{2} +N_{1}K_{3}. $$

In a similar way, we can find that

$$ \bigl\Vert T_{2}(u,x,y) \bigr\Vert _{X} \leq \frac{ \vert a_{6} \vert (b-a)}{ \vert \Delta \vert }+L_{2} K_{1} + M_{2} K_{2} +N_{2} K_{3} $$

and

$$\begin{aligned} \bigl\vert T_{3}(u,x,y) (t) \bigr\vert \leq & \int _{a}^{t} \frac{(t-s)^{\zeta -1}}{\Gamma {(\zeta )}} \bigl\vert \psi \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\ &{}+ \max_{t \in [a,b]} \bigl\lbrace b_{1}(t)\bigr\rbrace \int _{a}^{ \xi _{1}}\frac{(\xi _{1} - a)^{\zeta - 1}}{\Gamma {(\zeta )}} \bigl\vert \psi \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\ &{}+ \max_{t \in [a,b]} \bigl\lbrace b_{2}(t)\bigr\rbrace \int _{a}^{ \xi _{2}}\frac{(\xi _{2} - s)^{\zeta -1}}{\Gamma {(\zeta )}} \bigl\vert \psi \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\ &{}+ \max_{t \in [a,b]} \bigl\lbrace b_{3}(t)\bigr\rbrace \int _{a}^{b} \frac{(b-s)^{\zeta - 1}}{\Gamma {(\zeta )}} \bigl\vert \psi \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\ &{}+ \max_{t \in [a,b]} \bigl\lbrace b_{4}(t)\bigr\rbrace \sum_{j=1}^{n} \vert q_{j} \vert \int _{a}^{b} \frac{(\beta _{j} - s)^{\zeta -1}}{\Gamma {(\zeta )}} \bigl\vert \psi \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\ &{}+ \max_{t \in [a,b]} \bigl\lbrace b_{5}(t)\bigr\rbrace \int _{a}^{b} \frac{(b-s)^{\eta - 1}}{\Gamma {(\eta )}} \bigl\vert \rho \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\ &{}+ \max_{t \in [a,b]} \bigl\lbrace b_{3}(t)\bigr\rbrace \sum_{k=1}^{l} \vert r_{k} \vert \int _{a}^{\gamma _{k}} \frac{(\gamma _{k} - s)^{\eta -1}}{\Gamma {(\eta )}} \bigl\vert \rho \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\ &{}+ \max_{t \in [a,b]} \bigl\lbrace b_{4}(t) \bigr\rbrace \int _{a}^{b} \frac{(b - s)^{\xi -1}}{\Gamma {(\xi )}} \bigl\vert \varphi \bigl(s,u(s),x(s),y(s)\bigr) \bigr\vert \,ds \\ &{}+ \max_{t \in [a,b]} \bigl\lbrace b_{5}(t)\bigr\rbrace \sum_{i=1}^{m} \biggl\vert p_{i} \int _{a}^{\alpha _{i}} \frac{(\alpha _{i} - s)^{\xi -1}}{\Gamma {(\xi )}} \biggr\vert \varphi \vert \bigl(s,u(s),x(s),y(s)\bigr)\,ds \\ &{} + \max_{t \in [a,b]} \bigl\lbrace b_{6}(t)\bigr\rbrace \\ \leq & \delta _{6} + L_{3} K_{1} + M_{3} K_{2} +N_{3} K_{3}, \end{aligned}$$

which implies that

$$ \bigl\Vert T_{3}(u,x,y) \bigr\Vert _{X} \leq \delta _{6} + L_{3} K_{1} + M_{3} K_{2} +N_{3} K_{3}. $$

From the above argument, we deduce that the operator T is uniformly bounded, that is,

$$\begin{aligned} \bigl\Vert T(u,x,y) \bigr\Vert _{X} \leq& \vert u_{0} \vert + \vert a_{11} \vert (b-a)+ \frac{ \vert a_{6} (b-a) \vert }{ \vert \Delta \vert }+ \delta _{6} \\ &{} +(L_{1}+ L_{2}+L_{3})K_{1}+ (M_{1}+ M_{2}+M_{3})K_{2}+(N_{1}+N_{2}+N_{3})K_{3}. \end{aligned}$$

Next, we show that T is equicontinuous. Let \(t_{1},t_{2}\in [a,b]\) with \(t_{1}< t_{2}\). Then we have

$$\begin{aligned}& \bigl\vert T_{1}\bigl(u(t_{2}),x(t_{2}),y(t_{2}) \bigr)- T_{1}\bigl(u(t_{1}),x(t_{1}),y(t_{1}) \bigr) \bigr\vert \\& \quad \leq \frac{K_{1}}{\Gamma (\eta +1)}\bigl[2(t_{2}-t_{1})^{\eta }+ \bigl\vert t_{2}^{\eta }-t_{1}^{\eta } \bigr\vert \bigr]+(t_{2} -t_{1}) \vert a_{11} \vert + \biggl\lbrace \frac{t_{2} - t_{1}}{\Gamma {(\eta + 1)}} \biggl\lbrace ( \vert a_{12} \vert (b-a)^{\eta + 1} \\& \qquad {}+ \frac{ \vert A_{3} \vert \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)\sum_{k=1}^{l} \vert r_{k} \vert (\gamma _{k} - a)^{\eta }}{ \vert \Delta \vert (b-a)} \biggr\rbrace \biggr\rbrace K_{1} \\& \qquad {}+ \biggl\lbrace \frac{t_{2} - t_{1}}{\Gamma {(\xi + 1)}} \biggl\lbrace \frac{\sum_{i=1}^{m} \vert p_{i} \vert (\sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)+1)(\alpha _{i} - a)^{\xi }}{ \vert \Delta \vert (b-a)} \\& \qquad {}+ \frac{ \vert A_{1} \vert \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)(b-a)^{\xi }}{ \vert \Delta \vert (b-a)} \biggr\rbrace \biggr\rbrace K_{2} \\& \qquad {}+ \Biggl\lbrace \frac{t_{2} - t_{1}}{\Gamma {(\zeta + 1)}} \Biggl\lbrace \frac{\sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)}{ \vert \Delta \vert (b-a)} \Biggl( \vert a_{3} \vert (\xi _{1} - a)^{\zeta }+ \vert a_{4} \vert ( \xi _{2} - a)^{\zeta }+ \vert A_{3} \vert (b-a)^{\zeta } \\& \qquad {}+ \vert A_{1} \vert \sum_{j=1}^{n} \vert q_{j} \vert (\beta _{j} - a)^{\zeta } \Biggr) \Biggr\rbrace \Biggr\rbrace K_{3}. \end{aligned}$$

Analogously,we can obtain

$$\begin{aligned}& \bigl\vert T_{2}\bigl(u(t_{2}),x(t_{2}),y(t_{2}) \bigr)- T_{2}\bigl(u(t_{1}),x(t_{1}),y(t_{1}) \bigr) \bigr\vert \\& \quad \leq \frac{K_{2}}{\Gamma (\xi +1)}\bigl[2(t_{2}-t_{1})^{\xi }+ \bigl\vert t_{2}^{\xi }- t_{1}^{\xi } \bigr\vert \bigr]+ (t_{2} - t_{1}) \frac{ \vert a_{6} \vert }{ \vert \Delta \vert } + \frac{ t_{2} - t_{1}}{\Delta } \Biggl\lbrace \frac{1}{ \Gamma {(\eta + 1)}} \Biggl\lbrace \vert a_{5} \vert (b-a)^{\eta } \\& \qquad {}+ \vert A_{3} \vert \sum _{k=1}^{l} \vert r_{k} \vert ( \gamma _{k} - a)^{\eta } \Biggr\rbrace \Biggr\rbrace K_{1} \\& \qquad {} + \frac{t_{2} - t_{1}}{\Delta \Gamma {(\xi + 1)}} \Biggl\lbrace \vert a_{5} \vert \sum _{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)^{\xi }+ \vert A_{1} \vert (b-a)^{\xi } \Biggr\rbrace K_{2} \\& \qquad {}+ \frac{t_{2} - t_{1}}{ \vert \Delta \vert \Gamma {(\zeta + 1)}} \Biggl\lbrace \vert a_{3} \vert (\xi _{1} - a)^{\zeta }+ \vert a_{4} \vert (\xi _{2} - a)^{\zeta }+ \vert A_{3} \vert (b-a)^{\zeta } \\& \qquad {}+ \vert A_{1} \vert \sum _{j=1}^{n} \vert q_{j} \vert (\beta _{j} - a)^{\zeta } \Biggr\rbrace K_{3}, \end{aligned}$$

and

$$\begin{aligned}& \bigl\vert T_{3}(t_{2}),x(t_{2}),y(t_{2}))- T_{3}u(t_{1}),x(t_{1}),y(t_{1})) \bigr\vert \\& \quad \leq \frac{K_{3}}{\Gamma {(\zeta +1)}}\bigl[2(t_{2}-t_{1})^{\zeta }+ \bigl\vert t_{2}^{\zeta }- t_{1}^{\zeta } \bigr\vert \bigr] \\& \qquad {}+ (t_{2} - t_{1}) \frac{ \vert a_{1} a_{10} \vert }{ \vert \Delta \vert (\xi _{1} - \xi _{2})} \\& \qquad {} + \frac{(t_{2} - t_{1}) \vert a_{1} \vert }{ \vert \Delta \vert ( \vert \xi _{1} - \xi _{2} \vert ) \Gamma {(\eta + 1)}} \Biggl\lbrace \vert a_{9} \vert (b-a)^{\eta }+ (b-a)\sum_{k=1}^{l} \vert r_{k} \vert (\gamma _{k} - a)^{\eta } \Biggr\rbrace K_{1} \\& \qquad {}+ \frac{(t_{2} - t_{1}) \vert a_{1} \vert }{ \vert \Delta \vert ( \vert \xi _{1} - \xi _{2} \vert ) \Gamma {(\xi + 1)}} \Biggl\lbrace \vert A_{2} \vert (b-a)^{\xi }+ \vert a_{9} \vert \sum _{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)^{\xi } \Biggr\rbrace K_{2} \\& \qquad {}+ \frac{(t_{2} - t_{1}}{( \vert \xi _{1} - \xi _{2} \vert ) \Gamma {(\zeta + 1)}} \Biggl\lbrace \biggl\vert \frac{ a_{1} a_{7} }{ \Delta }-1 \biggr\vert ( \xi _{1} - a)^{\zeta }+ \biggl\vert \frac{ a_{1} a_{8} }{ \Delta }+ 1 \biggr\vert (\xi _{2} -a)^{\zeta } \\& \qquad {}+ \frac{ \vert a_{1} \vert }{ \vert \Delta \vert }(b-a)^{\zeta }+ \frac{ \vert A_{2} a_{1} \vert }{ \vert \Delta \vert } \sum _{j=1}^{n}q_{j} (\beta _{j} - a)^{\zeta } \Biggr\rbrace K_{3} + \bigl\lbrace \bigl(t_{2}^{2} - t_{1}^{2}\bigr)-2a (t_{2} - t_{1}) \bigr\rbrace \Biggl\lbrace \frac{ \vert a_{10 } \vert }{ \vert \Delta \vert } \\& \qquad {}+ \Biggl\lbrace \frac{1}{ \vert \Delta \vert \Gamma {(\eta + 1)}}\Biggl( \vert a_{9} \vert (b-a)^{\eta }+ (b-a)\sum_{k=1}^{l} \vert r_{k} \vert (\gamma _{k} - a)^{\eta } \Biggr) \Biggr\rbrace K_{1} \\& \qquad {}+ \Biggl\lbrace \frac{1}{ \vert \Delta \vert \Gamma {(\xi +1)}}\Biggl( \vert A_{2} \vert (b-a)^{\xi }+ \vert a_{9} \vert \sum _{i=1}^{m}p_{i} (\alpha _{i} - a)^{\xi }\Biggr) \Biggr\rbrace K_{2} \\& \qquad {}+ \Biggl\lbrace \frac{1}{ \vert \Delta \vert \Gamma {(\zeta +1)}} \Biggl( \vert a_{7} \vert ( \xi _{1} - a)^{\zeta }+ \vert a_{8} \vert (\xi _{2} - a)^{\zeta }+(b-a)^{ \zeta +1} \\& \qquad {}+ \vert A_{2} \vert \sum_{j=1}^{n}q_{j}( \beta _{j} - a)^{\zeta }\Biggr) \Biggr\rbrace K_{3} \Biggr\rbrace . \end{aligned}$$

The above inequalities are independent of u, x, y and tend to zero as \(t_{1}\to t_{2}\). This shows that the operator \(T(u,x,y) \) is equicontinuous. In consequence, we deduce that the operator \(T(u,x,y) \) is completely continuous.

Finally, we consider the set \(\mathcal{P} = \lbrace (u,x,y) \in X \times X \times X:(u,x,y)= \nu T(u,x,y), 0 \leq \nu \leq 1\rbrace \) and show that it is bounded.

Let \((u,x,y) \in \mathcal{P} \) with \((u,x,y)= \nu T(u,x,y)\). For any \(t \in [a,b]\), we have \(u(t)= \nu T_{1}(u,x,y)(t)\), \(x(t)= \nu T_{2}(u,x,y)(t)\), \(y(t)= \nu T_{3}(u,x,y)(t)\). Then, by \((H_{2})\), we have

$$\begin{aligned}& \bigl\vert u(t) \bigr\vert \leq \vert u_{0} \vert + \vert a_{11} \vert (b-a) + L_{1} \bigl(k_{0}+ k_{1} \vert u \vert + k_{2} \vert x \vert + k_{3} \vert y \vert \bigr) \\& \hphantom{\bigl\vert u(t) \bigr\vert \leq} {} + M_{1} \bigl(\sigma _{0}+ \sigma _{1} \vert u \vert + \sigma _{2} \vert x \vert + \sigma _{3} \vert y \vert \bigr) \\& \hphantom{\bigl\vert u(t) \bigr\vert \leq} {}+ N_{1}\bigl(\mu _{0} + \mu _{1} \vert u \vert + \mu _{2} \vert x \vert + \mu _{3} \vert y \vert \bigr) \\& \hphantom{\bigl\vert u(t) \bigr\vert } = \vert u_{0} \vert + \vert a_{11} \vert (b-a)+ L_{1}k_{0}+ M_{1} \sigma _{0} + N_{1} \mu _{0} + (L_{1}k_{1}+ M_{1} \sigma _{1} + N_{1} \mu _{1}) \vert u \vert _{2} \\& \hphantom{\bigl\vert u(t) \bigr\vert \leq} {}+ (L_{1}k_{2}+ M_{1}\sigma+ N_{1} \mu _{2}) \vert x \vert + (L_{1}k_{3}+ M_{1} \sigma _{3} + N_{1} \mu _{3}) \vert y \vert , \\& \bigl\vert x(t) \bigr\vert \leq \frac{ \vert a_{6} \vert (b-a)}{ \vert \Delta \vert } + L_{2}k_{0}+ M_{2} \sigma _{0} + N_{2} \mu _{0} + (L_{2}k_{1}+ M_{2} \sigma _{1}+ N_{2} \mu _{1}) \vert u \vert \\& \hphantom{\bigl\vert x(t) \bigr\vert \leq} {}+ (L_{2}k_{2}+ M_{2}\sigma _{2} + N_{2} \mu _{2}) \vert x \vert \\& \hphantom{\bigl\vert x(t) \bigr\vert \leq} {}+ (L_{2}k_{3}+ M_{2}\sigma _{3} + N_{2} \mu _{3}) \vert y \vert , \end{aligned}$$

and

$$\begin{aligned} \bigl\vert y(t) \bigr\vert \leq & \delta _{6} + L_{3}k_{0}+ M_{3} \sigma _{0} + N_{3} \mu _{0} + (L_{3}k_{1}+ M_{3} \sigma _{1}+ N_{3} \mu _{1}) \vert u \vert \\ &{}+ (L_{3} k_{2}+ M_{3}\sigma _{2} + N_{3} \mu _{2}) \vert x \vert \\ &{}+ (L_{3}k_{3}+ M_{3}\sigma _{3} + N_{3} \mu _{3}) \vert y \vert . \end{aligned}$$

It follows from the foregoing arguments that

$$\begin{aligned}& \Vert u \Vert \leq \vert u_{0} \vert + \vert a_{11} \vert (b-a)+ L_{1}k_{0}+ M_{1} \sigma _{0} + N_{1} \mu _{0} + (L_{1}k_{1}+ M_{1} \sigma _{1} + N_{1} \mu _{1}) \Vert u \Vert \\& \hphantom{\Vert u \Vert \leq}{}+ (L_{1}k_{2}+ M_{1}\sigma _{2} + N_{1} \mu _{2}) \Vert x \Vert + (L_{1}k_{3}+ M_{1}\sigma _{3} + N_{1} \mu _{3}) \Vert y \Vert , \\& \Vert x \Vert \leq \frac{ \vert a_{6} \vert (b-a)}{ \vert \Delta \vert } + L_{2}k_{0}+ M_{2} \sigma _{0} + N_{2} \mu _{0} + (L_{2}k_{1}+ M_{2} \sigma _{1}+ N_{2} \mu _{1}) \Vert u \Vert \\& \hphantom{\Vert x \Vert \leq} {}+ (L_{2}k_{2}+ M_{2}\sigma _{2} + N_{2} \mu _{2}) \Vert x \Vert +(L_{2}k_{3}+ M_{2}\sigma _{3} + N_{2} \mu _{3}) \Vert y \Vert , \\& \Vert y \Vert \leq \delta _{6} + L_{3}k_{0}+ M_{3} \sigma _{0} + N_{3} \mu _{0} + (L_{3}k_{1}+ M_{3} \sigma _{1}+ N_{3} \mu _{1}) \Vert u \Vert \\& \hphantom{\Vert y \Vert \leq} {}+ (L_{3} k_{2}+ M_{3}\sigma _{2} + N_{3} \mu _{2}) \Vert x \Vert \\& \hphantom{\Vert y \Vert \leq} {}+ (L_{3}k_{3}+ M_{3}\sigma _{3} + N_{3} \mu _{3}) \Vert y \Vert . \end{aligned}$$

Adding the above three inequalities, we have

$$\begin{aligned}& \Vert u \Vert + \Vert x \Vert + \Vert y \Vert \\& \quad \leq \vert u_{0} \vert + \vert a_{11} \vert (b-a)+ \frac{ \vert a_{6} \vert (b-a)}{ \vert \Delta \vert }+ \delta _{6} + (L_{1}+ L_{2}+L_{3})k_{0} +(M_{1}+ M_{2}+M_{3})\sigma _{0} \\& \qquad {}+ (N_{1}+N_{2}+N_{3})\mu _{0} \\& \qquad {}+\bigl[(L_{1}+ L_{2}+L_{3})k_{1}+(M_{1}+ M_{2}+M_{3}) \sigma _{1} + (N_{1}+N_{2}+N_{3}) \mu _{1}\bigr] \Vert u \Vert \\& \qquad {}+ \bigl[(L_{1}+ L_{2}+L_{3})k_{2} +(M_{1}+ M_{2}+M_{3}) \sigma _{2} +(N_{1}+N_{2}+N_{3})\mu _{2}\bigr] \Vert x \Vert \\& \qquad {}+ \bigl[(L_{1}+ L_{2}+L_{3})k_{3} +(M_{1}+ M_{2}+M_{3}) \sigma _{3} +(N_{1}+N_{2}+N_{3})\mu _{3}\bigr] \Vert y \Vert , \end{aligned}$$

which implies that

$$\begin{aligned}& \bigl\Vert (u,x,y) \bigr\Vert _{X} \\& \quad \leq \frac{1}{M_{0}} \biggl[ \vert u_{0} \vert + \vert a_{11} \vert (b-a)+ \frac{ \vert a_{6} \vert (b-a)}{ \vert \Delta \vert } \\& \qquad {}+ \delta _{6} +(L_{1}+ L_{2}+L_{3})k_{0}+(M_{1}+ M_{2}+M_{3})\sigma _{0} \\& \qquad {}+(N_{1}+N_{2}+N_{3})\mu _{0} \biggr], \end{aligned}$$

where \(M_{0}=\min \lbrace 1-[(L_{1}+L_{2}+L_{3})k_{i}+(M_{1}+M_{2}+M_{3}) \sigma _{i} +(N_{1}+N_{2}+N_{3})\mu _{i}],i=1,2,3 \rbrace \). Hence the set \(\mathcal{P}\) is bounded. Thus, by the Leray–Schauder alternative, we deduce that the operator T has at least one fixed point, which implies that problem (1.1) has at least one solution on \([a,b]\). This completes the proof. □

Our next existence and uniqueness result is based on the contraction mapping principle due to Banach.

Theorem 3.3

Let \(\Delta \ne 0\), where Δ is defined by (2.5). In addition, we assume that:

\((H_{2})\):

\(\rho , \varphi ,\psi :[a,b]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}\) are continuous functions and there exist positive constants \(l_{1}\), \(l_{2}\), and \(l_{3}\) such that, for all \(t\in [a,b]\) and \(u_{i},x_{i},y_{i}\in \mathbb{R}\), \(i=1,2,3\), we have

$$\begin{aligned}& \bigl\vert \rho (t, x_{1}, x_{2}, x_{3}) - \rho (t, y_{1} , y_{2}, y_{3}) \bigr\vert \leq l_{1} \bigl( \vert x_{1}- y_{1} \vert + \vert x_{2} -y_{2} \vert + \vert x_{3} - y_{3} \vert \bigr), \\& \bigl\vert \varphi (t, x_{1}, x_{2},x_{3}) - \varphi (t, y_{1} , y_{2},y_{3}) \bigr\vert \leq l_{2} \bigl( \vert x_{1}- y_{1} \vert + \vert x_{2} -y_{2} \vert + \vert x_{3} - y_{3} \vert \bigr), \\& \bigl\vert \psi (t,x_{1},x_{2},x_{3})- \psi (t,y_{1},y_{2},y_{3}) \bigr\vert \leq l_{3}\bigl( \vert x_{1}- y_{1} \vert + \vert x_{2}- y_{2} \vert + \vert x_{3} - y_{3} \vert \bigr). \end{aligned}$$

If

$$ (L_{1}+L_{2}+L_{3})l_{1}+(M_{1}+M_{2}+M_{3})l_{2}+(N_{1}+N_{2}+N_{3})l_{3}< 1, $$
(3.3)

where \(L_{i}\), \(M_{i}\), \(N_{i}\) are given in (3.1), then system (1.1) has a unique solution on \([a,b]\).

Proof

Define \(\sup_{t\in [a,b]}\rho (t,0,0,0)=Q_{1}<\infty \), \(\sup_{t\in [a,b]} \varphi (t,0,0,0)=Q_{2}<\infty \), \(\sup_{t\in [a,b]}\psi (t,0, 0,0) =Q_{3}<\infty \), and \(r>0\) such that

$$ r> \frac{ \vert u_{0} \vert + (b-a) \vert a_{11} \vert + \frac{ \vert a_{6} \vert (b-a)}{ \vert \Delta \vert }+ \delta _{6} + E}{1-(L_{1}+L_{2}+L_{3})l_{1}-(M_{1}+M_{2}+M_{3})l_{2}-(N_{1}+N_{2}+N_{3})l_{3}}, $$

where \(E=(L_{1}+L_{2}+L_{3})Q_{1}+(M_{1}+M_{2}+M_{3})Q_{2} +(N_{1}+N_{2}+N_{3})Q_{3}\).

In the first step, we show that \(T B_{r} \subset B_{r} \), where \(B_{r} = \lbrace (u,x,y)\in X \times X \times X : \Vert (u,x,y) \Vert \leq r\rbrace \). By assumption \((H_{2})\), for \((u,x,y) \in B_{r} \), \(t \in [a,b] \), we have

$$\begin{aligned} \bigl\vert \rho \bigl(t,u(t),x(t),y(t)\bigr) \bigr\vert \leq & \bigl\vert \rho \bigl(t,u(t),x(t),y(t)\bigr)- \rho (t,0,0,0) \bigr\vert \\ \leq & l_{1}\bigl( \vert u \vert + \bigl\vert x(t) \bigr\vert + \bigl\vert y(t) \bigr\vert \bigr)+ Q_{1} \\ \leq & l_{1}\bigl( \Vert u \Vert + \Vert x \Vert + \Vert y \Vert \bigr)+ Q_{1} \leq l_{1}r+ Q_{1}. \end{aligned}$$
(3.4)

Similarly, we can get

$$\begin{aligned} \bigl\vert \varphi \bigl(t,u(t),x(t),y(t)\bigr) \bigr\vert \leq & \bigl\vert \varphi \bigl(t,u(t),x(t),y(t)\bigr)- \varphi (t,0,0,0) \bigr\vert \\ \leq & l_{2}\bigl( \vert u \vert + \bigl\vert x(t) \bigr\vert + \bigl\vert y(t) \bigr\vert \bigr)+ Q_{2} \\ \leq & l_{2}\bigl( \Vert u \Vert + \Vert x \Vert + \Vert y \Vert \bigr)+ Q_{2} \leq l_{2}r+ Q_{2}, \end{aligned}$$
(3.5)

and

$$ \bigl\vert \psi \bigl(t,u(t),x(t),y(t)\bigr) \bigr\vert \leq l_{3} \bigl( \Vert u \Vert + \Vert x \Vert + \Vert y \Vert \bigr)+ Q_{3} \leq l_{3}r+ Q_{3}. $$
(3.6)

Using (3.4), (3.5), and (3.6), we obtain

$$\begin{aligned} \bigl\vert T_{1}(u,x,y) (t)) \bigr\vert \leq & \vert u_{0} \vert + \vert a _{11} \vert (b-a) + \biggl\lbrace \frac{1}{\Gamma {(\eta + 1)}} \biggl\lbrace (b-a)^{\eta }+ \vert a_{12} \vert (b-a)^{\eta + 1} \\ &{}+ \frac{ \vert A_{3} \vert \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)\sum_{k=1}^{l} \vert r_{k} \vert (\gamma _{k} - a)^{\eta }}{ \vert \Delta \vert } \biggr\rbrace \biggr\rbrace \Vert \rho \Vert \\ &{}+ \Biggl\lbrace \frac{1}{\Gamma {(\xi + 1)}} \Biggl\lbrace \sum _{i=1}^{m} \vert p_{i} \vert \sum _{i=1}^{m} \vert p_{i} \vert ( \alpha _{i} - a)+1) (\alpha _{i} - a)^{\xi } \\ &{}+ \frac{ \vert A_{1} \vert \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)(b-a)^{\xi }}{ \vert \Delta \vert } \Biggr\rbrace \Biggr\rbrace \Vert \varphi \Vert \\ &{}+ \Biggl\lbrace \frac{1}{\Gamma {(\zeta + 1)}} \Biggl\lbrace \frac{\sum_{i=1}^{m} \vert p_{i} (\alpha _{i} - a) \vert }{ \vert \Delta \vert } \Biggl( \vert a_{3} \vert (\xi _{1} - a)^{\zeta }+ \vert a_{4} \vert ( \xi _{2} - a)^{\zeta } \\ &{}+ \vert A_{3} \vert (b-a)^{\zeta }+ \vert A_{1} \vert \sum_{j=1}^{n} \vert q_{j} \vert (\beta _{j} - a)^{\zeta } \Biggr) \Biggr\rbrace \Biggr\rbrace \Vert \psi \Vert \\ \leq & \vert u_{0} \vert + \vert a _{11} \vert (b-a) + L_{1} (l_{1} r+ Q_{1}) + M_{1}(l_{2} r+Q_{2}) +N_{1}(l_{3} r+Q_{3}) \\ \leq & \vert u_{0} \vert + \vert a _{11} \vert (b-a) + (L_{1} l_{1} + M_{1} l_{2} + N_{1} l_{3})r + L_{1} Q_{1} + M_{1} Q_{2} + N_{1} Q_{3}, \end{aligned}$$

which, on taking the norm for \(t \in [a, b]\), yields

$$ \bigl\Vert T_{1}(u,x,y) \bigr\Vert _{X} \leq \vert u_{0} \vert + \vert a_{11} \vert (b-a)+(L_{1}l_{1}+ M_{1}l_{2} + N_{1} l_{3})r+ L_{1}Q_{1}+ M_{1}Q_{2}+N_{1} Q_{3}. $$

Likewise, we can find that

$$ \bigl\Vert T_{2}(u,x,y) \bigr\Vert _{X} \leq \frac{ \vert a_{6} \vert (b-a)}{ \vert \Delta \vert }+ (L_{2}l_{1}+ M_{2}l_{2}+N_{2}l_{3})r+ L_{2}Q_{1}+ M_{2}Q_{2}+N_{2} Q_{3} $$

and

$$ \bigl\Vert T_{3}(u,x,y) \bigr\Vert _{X} \leq \delta _{6} +(L_{3}l_{1}+ M_{3}l_{2}+N_{3}l_{3})r+ L_{3}Q_{1}+ M_{3}Q_{2}+N_{3} Q_{3}. $$

Consequently,

$$\begin{aligned}& \bigl\Vert T(u,x,y) \bigr\Vert _{X} \\& \quad \leq \vert u_{0} \vert + \vert a_{11} \vert (b-a)+ \frac{ \vert a_{6} \vert (b-a)}{ \vert \Delta \vert }+ \delta _{6} + \bigl[(L_{1}+L_{2}+L_{3})l_{1}+ (M_{1}+M_{2}+M_{3})l_{2} \\& \qquad {}+(N_{1}+N_{2}+N_{3})l_{3} \bigr]r + (L_{1}+L_{2}+L_{3})Q_{1}+(M_{1}+M_{2}+M_{3})Q_{2} \\& \qquad {}+ (N_{1}+N_{2}+N_{3})Q_{3} \leq r. \end{aligned}$$

Now, for \((u_{1},x_{1},y_{1}),(u_{2},x_{2},y_{2}) \in X \times X \times X \) and for any \(t \in [a,b] \), we get

$$\begin{aligned}& \bigl\vert T_{1}(u_{2}, x_{2}, y_{2}) (t)- T_{1}(u_{1}, x_{1}, y_{1}) (t) \bigr\vert \\& \quad \leq \biggl\lbrace \frac{1}{\Gamma {(\eta + 1)}} \biggl\lbrace (b-a)^{\eta }+ \vert a_{12} \vert (b-a)^{\eta + 1} \\& \qquad {}+ \frac{ \vert A_{3} \vert \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)\sum_{k=1}^{l} \vert r_{k} \vert (\gamma _{k} - a)^{\eta }}{ \vert \Delta \vert } \biggr\rbrace \biggr\rbrace \\& \qquad {}\times l_{1} \bigl( \Vert u_{2} - u_{1} \Vert + \Vert x_{2} - x_{1} \Vert + \Vert y_{2} - y_{1} \Vert \bigr) \\& \qquad {}+ \Biggl\lbrace \frac{1}{\Gamma {(\xi + 1)}} \Biggl\lbrace \sum _{i=1}^{m} \vert p_{i} \vert \Biggl( \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)+1\Biggr) (\alpha _{i} - a)^{\xi } \\& \qquad {}+ \frac{ \vert A_{1} \vert \sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)(b-a)^{\xi }}{ \vert \Delta \vert } \Biggr\rbrace \Biggr\rbrace l_{2} \bigl( \Vert u_{2} - u_{1} \Vert + \Vert x_{2} - x_{1} \Vert + \Vert y_{2} - y_{1} \Vert \bigr) \\& \qquad {}+ \Biggl\lbrace \frac{1}{\Gamma {(\zeta + 1)}} \Biggl\lbrace \frac{\sum_{i=1}^{m} \vert p_{i} \vert (\alpha _{i} - a)}{ \vert \Delta \vert } \Biggl( \vert a_{3} \vert (\xi _{1} - a)^{\zeta }+ \vert a_{4} \vert ( \xi _{2} - a)^{\zeta } \\& \qquad {}+ \vert A_{3} \vert (b-a)^{\zeta }+ \vert A_{1} \vert \sum_{j=1}^{n} \vert q_{j} \vert (\beta _{j} - a)^{\zeta } \Biggr) \Biggr\rbrace \Biggr\rbrace l_{3} \bigl( \Vert u_{2} - u_{1} \Vert + \Vert x_{2} - x_{1} \Vert + \Vert y_{2} - y_{1} \Vert \bigr) \\& \quad \leq (L_{1} l_{1}+ M_{1} l_{2}+N_{1}l_{3} ) \bigl( \Vert u_{2} - u_{1} \Vert + \Vert x_{2} - x_{1} \Vert + \Vert y_{2} - y_{1} \Vert \bigr), \end{aligned}$$

which implies that

$$\begin{aligned}& \bigl\Vert T_{1}(u_{2},x_{2},y_{2})- T_{1}(u_{1},x_{1},y_{1}) \bigr\Vert _{X} \\& \quad \leq (L_{1}l_{1} + M_{1}l_{2}+N_{1}l_{3}) \bigl( \Vert u_{2} - u_{1} + \Vert \Vert x_{2} - x_{1} \Vert + \Vert y_{2} -y_{1} \Vert \bigr). \end{aligned}$$
(3.7)

Similarly, we find that

$$\begin{aligned}& \bigl\Vert T_{2}(u_{2},x_{2},y_{2})- T_{2}(u_{1},x_{1},y_{1}) \bigr\Vert _{X} \\& \quad \leq (L_{2}l_{1}+ M_{2}l_{2}+N_{2}l_{3}) \bigl( \Vert u_{2} - u_{1} \Vert + \Vert x_{2} - x_{1} \Vert + \Vert y_{2} -y_{1} \Vert \bigr) \end{aligned}$$
(3.8)

and

$$\begin{aligned}& \bigl\Vert T_{3}(u_{2},x_{2},y_{2})- T_{3}(u_{1},x_{1},y_{1}) \bigr\Vert _{X} \\& \quad \leq (L_{3}l_{1}+ M_{3}l_{2}+N_{3}l_{3}) \bigl( \Vert u_{2} - u_{1} \Vert + \Vert x_{2} - x_{1} \Vert + \Vert y_{2} -y_{1} \Vert \bigr). \end{aligned}$$
(3.9)

It follows from (3.7), (3.8), and (3.9) that

$$\begin{aligned}& \bigl\Vert T(u_{2},x_{2},y_{2})- T(u_{1},x_{1},y_{1}) \bigr\Vert _{X} \\& \quad \leq \bigl[(L_{1}+ L_{2}+L_{3})l_{1} + (M_{1}+ M_{2}+M_{3})l_{2}+(N_{1}+N_{2}+N_{3})l_{3} \bigr] \\& \qquad {} \times \bigl( \Vert u_{2} - u_{1} \Vert + \Vert x_{2} - x_{1} \Vert + \Vert y_{2} - y_{1} \Vert \bigr). \end{aligned}$$

The above inequality together with (3.3) implies that T is a contraction. Hence it follows by Banach’s fixed point theorem that there exists a unique fixed point for the operator T, which corresponds to a unique solution of problem (1.1) on \([a,b]\). The proof is completed. □

Examples

Let us consider the following mixed-type coupled fractional differential system:

$$ \textstyle\begin{cases} D_{a^{+}}^{\frac{3}{2}}u(t)=\rho (t,u(t),x(t),y(t)), \quad t \in [1,2], \\ D_{a^{+}}^{\frac{7}{4}}x(t)=\varphi (t,u(t),x(t),y(t)), \quad t \in [1,2], \\ D_{a^{+}}^{\frac{5}{2}}x(t)=\psi (t,u(t),x(t),y(t)), \quad t \in [1,2], \\ u(1)=1/400, \qquad u(2)=\frac{1}{20} x (\frac{13}{10} ), \\ x(1)=0, \qquad x(2)=\sum_{j=1}^{2}q_{j} y(\beta -j), \\ y (\frac{11}{10} )=0,\qquad y (\frac{6}{5} )=0,\qquad y(b)= \sum_{k=1}^{2}r_{k} u(\gamma _{k}). \end{cases} $$
(4.1)

Here, \(\eta =3/4\), \(\xi =7/4\), \(\zeta =5/2\), \(a=1\), \(b=2\), \(u_{0}=1/400\), \(m=1\), \(n=2\), \(l=2\), \(p_{1}=1/20\), \(q_{1}=1/100\), \(q_{2}=1/50\), \(r_{1}=1/1000\), \(r_{2}=1/500\), \(\alpha _{1}=13/10\), \(\beta _{1}=7/5\), \(\beta _{2}=3/2\), \(\gamma _{1}=8/5\), \(\gamma _{2}=17/10\). With the given data, it is found that \(L_{1}\simeq 0.75223\), \(L_{2}\simeq eq1397 \times 10^{-5}\), \(L_{3}\simeq 5.0146 \times 10^{-3}\), \(M_{1}\simeq 4.0414 \times 10^{-2}\), \(M_{2}\simeq 1.2435\), \(M_{3} \simeq 4.8058 \times 10^{-5}\), \(N_{1} \simeq 1.8944 \times 10^{-3}\), \(N_{2} \simeq 3.0567 \times 10^{-3}\), \(N_{3} \simeq 1.0942\).

(1) In order to illustrate Theorem 3.2, we take

$$\begin{aligned}& \rho (t,u,x,y) =e^{-2t}+ \frac{1}{8} u \cos x + \frac{e^{-t}}{3}x \sin y + \frac{e^{-t}}{4}y \cos u, \\& \varphi (t,u,x,y) = t \sqrt{t^{2}+3} + \frac{e^{-t}}{3\Pi }u \tan ^{-1} x + \frac{1}{\sqrt{48+t^{2}}}x+ \frac{1}{4}y \sin u, \\& \psi (t,u,x,y) =\frac{e^{-t}}{10}+\frac{e^{-t}}{3}u+ \frac{1}{4+t}x+\frac{e^{-t}}{4}y \cos x. \end{aligned}$$
(4.2)

It is easy to check that condition \((H_{1})\) is satisfied with \(k_{0}= 1/e^{2}\), \(k_{1}= 1/8\), \(k_{2}= 1/(3e)\), \(k_{3}=1/(4e)\), \(\sigma _{0}=2 \sqrt{7}\), \(\sigma _{1}= 1/(6e)\), \(\sigma _{2}= 1/7\), \(\sigma _{3}=1/4\), \(\mu _{0}=1/(10e)\), \(\mu _{1}=1/(3e)\), \(\mu _{2}=1/5\), \(\mu _{3}=1/(4e)\). Furthermore,

$$\begin{aligned}& (L_{1} + L_{2}+L_{3})k_{1} + (M_{1} + M_{2}+M_{3})\sigma _{1}+(N_{1}+N_{2}+N_{3}) \mu _{1} \simeq 0.30801< 1, \\& (L_{1} + L_{2}+L_{3})k_{2} + (M_{1} + M_{2}+M_{3})\sigma _{2}+(N_{1}+N_{2}+N_{3}) \mu _{2} \simeq 0.49596< 1, \\& (L_{1} + L_{2}+L_{3})k_{3} + (M_{1} + M_{2}+M_{3})\sigma _{3}+(N_{1}+N_{2}+N_{3}) \mu _{3} \simeq 0.49161< 1 . \end{aligned}$$

Clearly, the hypotheses of Theorem 3.2 are satisfied, and hence the conclusion of Theorem 3.2 applies to problem (4.1) with ρ, φ, ψ given by (4.2).

(2) In order to illustrate Theorem 3.3, we take

$$\begin{aligned}& \rho (t,u,x,y) = \frac{e^{-t}}{\sqrt{3+t^{2}}} \cos u + \cos t, \\& \varphi (t,u,x,y) = \frac{1}{5+t^{4}}\bigl(\sin u+ \vert x \vert \bigr)+ e^{-t}, \\& \psi (t,u,x,u) = \frac{e^{-t}}{3}\sin y + \tan ^{-1}t, \end{aligned}$$
(4.3)

which clearly satisfies condition \((H_{2})\) with \(l_{1}= 1/(2e)\), \(l_{2}=1/6 \), and \(l_{3}= 1/(3e)\). Moreover, \((L_{1} +L_{2}+L_{3})l_{1} + (M_{1} + M_{2}+M_{3})l_{2}+(N_{1}+N_{2}+N_{3})l_{3} \simeq 0.49596<1\). Thus the hypothesis of Theorem 3.3 holds true, and consequently there exists a unique solution for problem (4.1) on \([1,2]\) with ρ, φ, ψ given by (4.3).

Conclusions

This paper studies a tripled system of nonlinear fractional differential equations of different orders on an arbitrary domain complemented with the multi-point boundary conditions of cyclic nature involving different nonlocal positions. Applying the standard fixed point theorems, we have proved the existence and uniqueness results for the given problem, which are well illustrated with the aid of examples. By taking all \(p_{i}=0\), \(i=1, \dots , m\), \(q_{j}=0\), \(j=1, \dots \), and \(r_{k}=0\), \(k=1, \dots , l\), we obtain the new results for the given tripled system of nonlinear fractional differential equations equipped with the conditions: \(u(a)=u_{0}\), \(u(b)=0\), \(x(a)=0\), \(x(b)=0\), \(y(\xi _{1})=0\), \(y(\xi _{2})=0\), \(y(b)=0\). To the best of our knowledge, it is the first paper dealing with a nonlocal multi-point boundary value problem involving a tripled system of nonlinear fractional differential equations of different orders on an arbitrary domain.

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Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (KEP-PhD-41-130-41). The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors also thank the reviewers for their constructive remarks on our work.

Funding

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (KEP-PhD-41-130-41).

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Each of the authors, BA, SH, AA, and SKN, contributed equally to each part of this work. All authors read and approved the final manuscript.

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Correspondence to Bashir Ahmad.

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Ahmad, B., Hamdan, S., Alsaedi, A. et al. A study of a nonlinear coupled system of three fractional differential equations with nonlocal coupled boundary conditions. Adv Differ Equ 2021, 278 (2021). https://doi.org/10.1186/s13662-021-03440-7

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MSC

  • 34A08
  • 34B15

Keywords

  • Fractional differential equations
  • Caputo fractional derivative
  • System
  • Existence
  • Fixed point theorems