1 Introduction

Fractional derivatives and integrals find numerous applications in many branches of physics and engineering ranging from quantum optics to astro-physics and cosmology, dynamics of materials to biophysics and medicine, dynamical chaos to control, signal processing to communications, and more. For recent comprehensive reviews on fractional derivatives and their applications, we refer the reader to the monographs [25, 34, 36] and the recent undermentioned papers [1–3, 5, 6, 9, 11, 13, 17, 23, 26–29, 31, 32, 35]. Due to their widespread applications, a system of fractional differential equations subject to boundary conditions has received much attention amongst researchers who accommodate various numerical methods to establish their results; see for instance the papers [18, 22, 33].

Particularly, coupled fractional boundary systems, which study interaction between two quantities, have been under consideration as they provide adequate interpretations for models describing chaotic behavior, anomalous diffusion, ecological effects, and biological models. Many relevant results have been reported in this direction with different boundary conditions; see [4, 7, 8, 14–16, 21, 30, 37–39] and the references therein.

Tripled fractional boundary systems, which are considered as a generalization of coupled fractional systems, are governed by three associated differential equations with three initial or boundary conditions [12, 24]. In [12], Berinde and Borcut introduced the concept of tripled fixed point for nonlinear mappings in partially ordered complete metric spaces and obtained existence and uniqueness theorems for contractive type mappings. Karakaya et al.[24] gave some results concerning the existence of tripled fixed points for a class of condensing operators in Banach spaces.

The cyclic boundary conditions have many applications on channel flow with fully developed flow at inlet as well as outlet using simple foam. In addition, some researchers introduced a railway track coupled dynamics model based on cyclic boundary conditions (see [10] and the references therein).

Unlike coupled fractional systems, the investigations of tripled fractional systems have gained less attention amongst researchers. To the best of authors’ observation, indeed, there is no analytical literature on studying the existence of tripled systems of fractional differential equations.

Motivated by these research works, we investigate in this paper a tripled fractional abstract system with cyclic tripled boundary conditions that has the following form:

$$ \textstyle\begin{cases} ^{c}D_{0}^{\alpha _{k}}x_{k}(t)=f_{k}(t,x ( t ) ),&1< \alpha _{k}\leq 2, \\ x_{k}^{(j)}(0)=a_{k,j}x_{\sigma (k)}^{(j)}(T),& k=1,2,3; j=0,1,\end{cases} $$
(1.1)

where \({}^{c}D_{0}^{\alpha _{k}}\) denotes the Caputo fractional derivative of order \(\alpha _{k}\), \(t\in J= [ 0,T ], f_{k}:J\times \mathbb{R} ^{3}\rightarrow \mathbb{R} \) are continuous functions, \(x=(x_{1},x_{2},x_{3})\in \mathbb{R} ^{3}\), \(\sigma =(1\ 2\ 3)\) is a cycle permutation, and \(a_{k,j}\in \mathbb{R} \), \(k=1,2,3\), \(j=0,1\), such that \(\prod_{k=1}^{3}a_{k,j}\neq 1, j=0,1\). System (1.1) is converted into an equivalent integral form by the help of fractional calculus. The existence and uniqueness of solutions with cyclic permutation of tripled boundary conditions are investigated. We employ the Banach and Krasnoselskii fixed point theorems to prove our main results.

The railway track coupled system investigated in [10] can be modeled as a classical tripled system if it undergoes an external influence, by which many researchers can be prompted to generalize this idea using fractional differential models. We emphasize that the problem considered in the present settings is new and has novel approach that will provide further insight into the analytical study of tripled fractional systems with cyclic boundary conditions.

2 Preliminary assertions

In this section, we recall some basic definitions of fractional calculus [25]. Meanwhile, the integral form of the solution of system (1.1) as well as the definition of permutation group are presented. The notations and terminologies herein will be used in the subsequent section.

Definition 2.1

([25])

The Riemann–Liouville fractional integral of a real-valued function \(f\in C(J)\) is defined by

$$ I_{0}^{\alpha }f ( t ) = \int _{0}^{t} \frac{ ( t-s ) ^{\alpha -1}}{\varGamma ( \alpha ) }f ( s ) \,ds,\quad t\in J, \alpha >0, $$

provided the integral exists, and \(I_{0}^{0}f ( t ):=f(t)\). The Caputo fractional derivative of \(f\in C^{(n)}(J)\) is given by

$$ {}^{c}D_{0}^{\alpha }f ( t ) =I_{0}^{n-\alpha }f^{(n)} ( t ), $$

where \(n=[\alpha ]\) is the greatest integer function.

Lemma 2.2

([25])

Let \([\alpha ]=n\in \mathbb{N} \), and \(f,^{c}D_{0}^{\alpha }f\in {C(J)}\). Then

$$ I_{0}^{\alpha } {}^{c}D_{0}^{\alpha }f ( t ) =f ( t ) +c_{0}+c_{1}t+c_{2}t^{2}+ \cdots+c_{n-1}t^{n-1} $$

for \(c_{i}\in \mathbb{R}, i=0,1,2,\ldots,n-1\).

For convenience, we introduce the following notations:

$$\begin{aligned} &b_{1,1} =b_{2,2}=b_{3,3}= \frac{a_{1,1}a_{2,1}a_{3,1}}{1-a_{1,1}a_{2,1}a_{3,1}},\qquad b_{1,2}= \frac{a_{1,1}}{1-a_{1,1}a_{2,1}a_{3,1}}, \\ &b_{1,3} =\frac{a_{1,1}a_{2,1}}{1-a_{1,1}a_{2,1}a_{3,1}},\qquad b_{2,1}= \frac{a_{2,1}a_{3,1}}{1-a_{1,1}a_{2,1}a_{3,1}},\qquad b_{2,3}= \frac{a_{2,1}}{1-a_{1,1}a_{2,1}a_{3,1}}, \\ &b_{3,1} =\frac{a_{3,1}}{1-a_{1,1}a_{2,1}a_{3,1}},\qquad b_{3,2}= \frac{a_{3,1}a_{1,1}}{1-a_{1,1}a_{2,1}a_{3,1}}, \\ &d_{1,1} =d_{2,2}=d_{3,3}= \frac{a_{1,0}a_{2,0}a_{3,0}}{1-a_{1,0}a_{2,0}a_{3,0}},\qquad d_{1,2}= \frac{a_{1,0}}{1-a_{1,0}a_{2,0}a_{3,0}}, \\ &d_{1,3} =\frac{a_{1,0}a_{2,0}}{1-a_{1,0}a_{2,0}a_{3,0}},\qquad d_{2,1}= \frac{a_{2,0}a_{3,0}}{1-a_{1,0}a_{2,0}a_{3,0}},\qquad d_{2,3}= \frac{a_{2,0}}{1-a_{1,0}a_{2,0}a_{3,0}}, \\ &d_{3,1} =\frac{a_{3,0}}{1-a_{1,0}a_{2,0}a_{3,0}},\qquad d_{3,2}= \frac{a_{3,0}a_{1,0}}{1-a_{1,0}a_{2,0}a_{3,0}}, \\ &e_{1,1} =\frac{a_{1,0}a_{3,1}T ( a_{2,0}a_{3,0}a_{1,1}a_{2,1}+a_{2,0}+a_{2,1} ) }{ ( 1-a_{1,0}a_{2,0}a_{3,0} ) ( 1-a_{1,1}a_{2,1}a_{3,1} ) }, \\ &e_{1,2} =\frac{a_{1,0}a_{1,1}T ( a_{2,0}a_{3,0}+a_{2,0}a_{3,1}+a_{2,1}a_{3,1} ) }{ ( 1-a_{1,0}a_{2,0}a_{3,0} ) ( 1-a_{1,1}a_{2,1}a_{3,1} ) }, \\ &e_{1,3} =\frac{a_{1,0}a_{2,1}T ( a_{2,0}a_{3,0}a_{1,1}+a_{2,0}a_{1,1}a_{3,1}+1 ) }{ ( 1-a_{1,0}a_{2,0}a_{3,0} ) ( 1-a_{1,1}a_{2,1}a_{3,1} ) }, \\ &e_{2,1} =\frac{a_{2,0}a_{3,1}T ( a_{1,0}a_{2,1}a_{3,0}+a_{1,1}a_{2,1}a_{3,0}+1 ) }{ ( 1-a_{1,0}a_{2,0}a_{3,0} ) ( 1-a_{1,1}a_{2,1}a_{3,1} ) }, \\ &e_{2,2} =\frac{a_{2,0}a_{1,1}T ( a_{1,0}a_{3,0}a_{2,1}a_{3,1}+a_{3,0}+a_{3,1} ) }{ ( 1-a_{1,0}a_{2,0}a_{3,0} ) ( 1-a_{1,1}a_{2,1}a_{3,1} ) }, \\ &e_{2,3} =\frac{a_{2,0}a_{2,1}T ( a_{1,0}a_{3,0}+a_{3,0}a_{1,1}+a_{3,1}a_{1,1} ) }{ ( 1-a_{1,0}a_{2,0}a_{3,0} ) ( 1-a_{1,1}a_{2,1}a_{3,1} ) }, \\ &e_{3,1} =\frac{a_{3,0}a_{3,1}T ( a_{1,0}a_{2,0}+a_{1,0}a_{2,1}+a_{1,1}a_{2,1} ) }{ ( 1-a_{1,0}a_{2,0}a_{3,0} ) ( 1-a_{1,1}a_{2,1}a_{3,1} ) }, \\ &e_{3,2} =\frac{a_{3,0}a_{1,1}T ( a_{1,0}a_{2,0}a_{3,1}+a_{1,0}a_{2,1}a_{3,1}+1 ) }{ ( 1-a_{1,0}a_{2,0}a_{3,0} ) ( 1-a_{1,1}a_{2,1}a_{3,1} ) }, \\ &e_{3,3} =\frac{a_{3,0}a_{2,1}T ( a_{1,0}a_{2,0}a_{1,1}a_{3,1}+a_{1,0}+a_{1,1} ) }{ ( 1-a_{1,0}a_{2,0}a_{3,0} ) ( 1-a_{1,1}a_{2,1}a_{3,1} ) }. \end{aligned}$$

Lemma 2.3

Let \(f_{k}\in C(J,\mathbb{R} )\)and \(\prod_{k=1}^{3}a_{k,j}\neq 1, j=0,1\). Then the solution of the linear fractional differential system

$$ {}^{c}D_{0}^{\alpha _{k}}x(t)=f_{k}(t),\quad 1< \alpha _{k} \leq 2, t\in (0,T), $$
(2.1)

subject to the conditions

$$ x_{k}^{(j)}(0)=a_{k,j}x_{\sigma (k)}^{(j)}(T),\quad k=1,2,3, j=0,1, $$
(2.2)

is given by

$$ x_{k}(t)=\sum_{m=1}^{3} \bigl( d_{k,m}I_{0}^{\alpha _{m}}f_{m} ( T ) + ( e_{k,m}+tb_{k,m} ) I_{0}^{\alpha _{m}-1}f_{m} ( T ) \bigr) +I_{0}^{\alpha _{k}}f_{k} ( t ). $$
(2.3)

Proof

Applying the fractional integral to both sides of (2.1) and using Lemma 2.2, we obtain

$$ x_{k}(t)=c_{k,0}+c_{k,1}t+I_{0}^{\alpha _{k}}f_{k} ( t ). $$
(2.4)

Hence, we deduce that

$$ x_{k}^{\prime }(t)=c_{k,1}+I_{0}^{\alpha _{k}-1}f_{k} ( t ) . $$

The boundary conditions in (2.2) imply that

$$ \textstyle\begin{cases} c_{1,0}=a_{1,0} ( c_{2,0}+c_{2,1}T+I_{0}^{\alpha _{2}}f_{2} ( T ) ), \\ c_{2,0}=a_{2,0} ( c_{3,0}+c_{3,1}T+I_{0}^{\alpha _{3}}f_{3} ( T ) ), \\ c_{3,0}=a_{3,0} ( c_{1,0}+c_{1,1}T+I_{0}^{\alpha _{1}}f_{1} ( T ) ),\end{cases} $$
(2.5)

and

$$ \textstyle\begin{cases} c_{1,1}=a_{1,1} ( c_{2,1}+I_{0}^{\alpha _{2}-1}f_{2} ( T ) ), \\ c_{2,1}=a_{2,1} ( c_{3,1}+I_{0}^{\alpha _{3}-1}f_{3} ( T ) ), \\ c_{3,1}=a_{3,1} ( c_{1,1}+I_{0}^{\alpha _{1}-1}f_{1} ( T ) ).\end{cases} $$
(2.6)

By direct substitutions of the equations in (2.5), we get

$$\begin{aligned} &c_{1,1}=b_{1,1}I_{0}^{\alpha _{1}-1}f_{1} ( T ) +b_{1,2}I_{0}^{ \alpha _{2}-1}f_{2} ( T ) +b_{1,3}I_{0}^{\alpha _{3}-1}f_{3} ( T ), \end{aligned}$$
(2.7)
$$\begin{aligned} &c_{2,1}=b_{2,1}I_{0}^{\alpha _{1}-1}f_{1} ( T ) +b_{2,2}I_{0}^{ \alpha _{2}-1}f_{2} ( T ) +b_{2,3}I_{0}^{\alpha _{3}-1}f_{3} ( T ), \end{aligned}$$
(2.8)

and

$$ c_{3,1}=b_{3,1}I_{0}^{\alpha _{1}-1}f_{1} ( T ) +b_{3,2}I_{0}^{ \alpha _{2}-1}f_{2} ( T ) +b_{3,3}I_{0}^{\alpha _{3}-1}f_{3} ( T ). $$
(2.9)

Similarly, the equations in (2.6) together with the last constants lead to

$$\begin{aligned} c_{1,0}={}&d_{1,1}I_{0}^{\alpha _{1}}f_{1} ( T ) +d_{1,2}I_{0}^{ \alpha _{2}}f_{2} ( T ) +d_{1,3}I_{0}^{\alpha _{3}}f_{3} ( T ) \\ &{}+e_{1,1}I_{0}^{\alpha _{1}-1}f_{1} ( T ) +e_{1,2}I_{0}^{ \alpha _{2}-1}f_{2} ( T ) +e_{1,3}I_{0}^{\alpha _{3}-1}f_{3} ( T ), \end{aligned}$$
(2.10)
$$\begin{aligned} c_{2,0} ={}&d_{2,1}I_{0}^{\alpha _{1}}f_{1} ( T ) +d_{2,2}I_{0}^{ \alpha _{2}}f_{2} ( T ) +d_{2,3}I_{0}^{\alpha _{3}}f_{3} ( T ) \\ &{}+e_{2,1}I_{0}^{\alpha _{1}-1}f_{1} ( T ) +e_{2,2}I_{0}^{ \alpha _{2}-1}f_{2} ( T ) +e_{2,3}I_{0}^{\alpha _{3}-1}f_{3} ( T ), \end{aligned}$$
(2.11)

and

$$\begin{aligned} c_{3,0}={}&d_{3,1}I_{0}^{\alpha _{1}}f_{1} ( T ) +d_{3,2}I_{0}^{ \alpha _{2}}f_{2} ( T ) +d_{3,3}I_{0}^{\alpha _{3}}f_{3} ( T ) \\ &{}+e_{3,1}I_{0}^{\alpha _{1}-1}f_{1} ( T ) +e_{3,2}I_{0}^{ \alpha _{2}-1}f_{2} ( T ) +e_{3,3}I_{0}^{\alpha _{3}-1}f_{3} ( T ). \end{aligned}$$
(2.12)

Substituting the values of \(c_{k,j},k=1,2,3,j=0,1\), in (2.4), we get (2.3). This completes the proof. □

We adopt the following definition of permutation groups.

Definition 2.4

([19])

A permutation of a set A is a function \(\sigma:A\rightarrow A\) that is one to one and onto.

This defines the so-called permutation group \((A,\sigma )\). Let \(A=\{1,2,3\}\), then the cardinality of this group is \(3!=6\) permutations. For instance, one of such permutations is given by \(\sigma (1)=2, \sigma (2)=3, \sigma (3)=1\), and this constitutes a cycle σ= ( 1 2 3 2 3 1 ) =(123). However, we use this cycle in the boundary conditions of system (1.1) such that, for a triple \((x_{1}, x_{2}, x_{3})\), we have \(x^{(j)}_{1}(0)=a_{1,j}x^{(j)}_{2}(T)\), \(x^{(j)}_{2}(0)=a_{2,j}x^{(j)}_{3}(T)\), and \(x^{j}_{3}(0)=a_{3,j}x^{j}_{1}(T)\), \(j=1,2\). The other five permutations can be used and another integral solution can be obtained which is isomorphic to the one in (2.3) with constant differences. To explain this more, we consider permutation \((1 3)(2)\). As a consequence of Lemma 2.3, we find the same solution as (2.3) but with different coefficients. Indeed, we find the following:

$$\begin{aligned} &b_{1,1} =b_{3,3}=\frac{a_{1,1}a_{3,1}}{1-a_{1,1}a_{3,1}},\qquad b_{2,2}= \frac{a_{2,1}}{1-a_{2,1}}, \\ &b_{1,3} =\frac{a_{1,1}}{1-a_{1,1}a_{3,1}},\qquad b_{3,1}= \frac{a_{3,1}}{1-a_{3,1}a_{1,1}}, \\ &b_{1,2} =b_{2,1}=b_{2,3}=b_{3,2}=0, \\ &d_{1,1} =\frac{1}{1-a_{1,0}a_{3,0}},\qquad d_{2,2}= \frac{a_{2,0}}{1-a_{2,0}},\qquad d_{3,3}= \frac{a_{3,0}a_{1,0}}{1-a_{1,0}a_{3,0}}, \\ &d_{1,3} =\frac{a_{1,0}}{1-a_{1,0}a_{3,0}},\qquad d_{3,1}=a_{3,0} \biggl( \frac{2-a_{1,0}a_{3,0}}{1-a_{1,0}a_{3,0}} \biggr), \\ &d_{1,2} =d_{2,1}=d_{2,3}=d_{3,2}=0, \\ &e_{1,1} =a_{3,1}a_{1,0}T \biggl( \frac{1+a_{1,1}a_{3,0}}{ ( 1-a_{1,0}a_{3,0} ) ( 1-a_{1,1}a_{3,1} ) } \biggr), \\ &e_{1,3} =a_{1,1}a_{1,0}T \biggl( \frac{a_{3,1}+a_{3,0}}{ ( 1-a_{1,0}a_{3,0} ) ( 1-a_{1,1}a_{3,1} ) } \biggr), \\ &e_{2,2} =\frac{a_{2,1}a_{2,0}T}{ ( 1-a_{2,0} ) ( 1-a_{2,1} ) }, \\ &e_{3,1} =a_{3,0}a_{3,1}T \biggl( \frac{a_{1,0}+a_{1,1}}{ ( 1-a_{1,0}a_{3,0} ) ( 1-a_{1,1}a_{3,1} ) } \biggr), \\ &e_{3,3} =a_{3,0}T \biggl( \frac{a_{3,1}a_{1,1}a_{1,0}}{ ( 1-a_{1,0}a_{3,0} ) ( 1-a_{3,1}a_{1,1} ) } \biggr), \\ &e_{2,1} =0=e_{1,2}=e_{2,3}=e_{3,2}=0. \end{aligned}$$

A tripled fixed point of a mapping is given next.

Definition 2.5

([12])

An element \((x_{1},x_{2},x_{3})\in X\times X\times X\) is called a tripled fixed point of a mapping \(\digamma:X\times X\times X\rightarrow X\) if \(\digamma (x_{1},x_{2},x_{3})=x_{1}\), \(\digamma (x_{2},x_{1},x_{3})=x_{2}\), and \(\digamma (x_{3},x_{2},x_{1})=x_{3}\).

Define an operator \(\varPsi:X\times X\times X\rightarrow X\times X\times X\) such that

$$ \varPsi (x_{1},x_{2},x_{3})= \bigl( \digamma (x_{1},x_{2},x_{3}), \digamma (x_{2},x_{1},x_{3}),\digamma (x_{3},x_{2},x_{1}) \bigr). $$

Then \((x_{1},x_{2},x_{3})\) is a tripled fixed point of Ϝ iff \((x_{1},x_{2},x_{3})\) is a fixed point of Ψ, that is, \(\varPsi (x_{1},x_{2},x_{3})=(x_{1},x_{2},x_{3})\).

For completeness, we recall the following tools of fixed point theory.

Theorem 2.6

(Banach fixed point theorem [20])

Let D be a nonempty closed subset of a Banach space E. Then any contraction mapping T from D into itself has a unique fixed point.

Theorem 2.7

(Krasnoselskii fixed point theorem [20])

Let B be a closed convex and nonempty subset of a Banach space X. Let \(\varPsi _{1}\), \(\varPsi _{2}\)be operators defined on B such that

  1. (i)

    \(\varPsi _{1}x+\varPsi _{2}y\in B\)whenever \(x,y\in B\);

  2. (ii)

    \(\varPsi _{1}\)is a contraction mapping;

  3. (iii)

    \(\varPsi _{2}\)is compact and continuous.

Then there exists \(z\in B\)such that \(z=\varPsi _{1}z+\varPsi _{2}z\).

3 Main results

In this section we use the Banach and Krasnoselskii fixed point theorems to ensure the existence of solution for tripled system (1.1).

The Banach space \(X=C ( J,\mathbb{R} ) \) of continuous real-valued functions is defined on J with the usual maximum norm. Hence, we obtain a Banach space \(X^{3}=X\times X\times X\) equipped with the norm \({ \Vert x \Vert }_{X^{3}} ={ \Vert ( x_{1},x_{2},x_{3} ) \Vert }_{X^{3}} = \Vert x_{1} \Vert + \Vert x_{2} \Vert + \Vert x_{3} \Vert \). Using the result of Lemma 2.3, we define the operator \(\varPsi:X^{3}\rightarrow X^{3}\) by

$$ \varPsi x ( t ) = \bigl( \varPsi _{1}x_{1} ( t ), \varPsi _{2}x_{2} ( t ),\varPsi _{3}x_{3} ( t ) \bigr), $$

where

$$\begin{aligned} \varPsi _{k}x_{k} ( t ) ={}&I_{0}^{\alpha _{k}}f_{k} \bigl( t,x(t) \bigr) +\sum_{m=1}^{3} \bigl( d_{k,m}I_{0}^{\alpha _{m}}f_{m} \bigl( T,x(T) \bigr) \\ &{} + ( e_{k,m}+tb_{k,m} ) I_{0}^{ \alpha _{m}-1}f_{m} \bigl( T,x(T) \bigr) \bigr). \end{aligned}$$
(3.1)

If the operator \(\varPsi _{k}:X\rightarrow X\) given by (3.1) has a fixed point in X, then \(\varPsi _{k}x_{k}=x_{k}\), \(k=1,2,3\). Hence in connection with Definition 2.5, we let \(\varPsi _{1}x_{1}=\digamma (x_{1},x_{2},x_{3})\), \(\varPsi _{2}x_{2}=\digamma (x_{2},x_{1},x_{3})\), and \(\varPsi _{3}x_{3}=\digamma (x_{3},x_{2},x_{1})\). This assumption connects the definition of the tripled fixed point introduced in Definition 2.5 with the fixed point of the tripled operator \(\varPsi =(\varPsi _{1}, \varPsi _{2}, \varPsi _{3})\). By this idea, we obtain the main results later.

We make use of the following assumption:

\((\varLambda )\):

Let \(f_{k}:J\times X^{3}\rightarrow X,k=1,2,3\), be a jointly continuous function, and there exists a positive constant \(L_{k}\)such that

$$ \bigl\vert f_{k} ( t,x ) -f_{k} ( t,y ) \bigr\vert \leq L_{k} \Vert x-y \Vert _{X^{3}}$$

for all \(t\in J\)and \(x,y\in X ^{3}\).

Theorem 3.1

Let condition \((\varLambda )\)be satisfied. Then tripled system (1.1) has a unique solution whenever

$$ \eta =\sum_{k=1}^{3} \Biggl( \frac{L_{k}T^{\alpha _{k}}}{\varGamma (\alpha _{k}+1)}+\sum_{m=1}^{3}L_{m} \biggl( \biggl( \frac{ \vert d_{k,m} \vert }{\alpha _{m}}+ \vert b_{k,m} \vert \biggr) T+ \vert e_{k,m} \vert \biggr) \frac{T^{\alpha _{m}-1}}{\varGamma (\alpha _{m})} \Biggr) < 1. $$

Proof

Let \(B_{r}= \{ x\in X^{3}: \Vert x \Vert _{X^{3}}\leq r \} \) be a closed subset in \(X^{3}\) such that

$$\begin{aligned} r > ( 1-\eta ) ^{-1} \sum_{k=1}^{3} \Biggl( \frac{N_{k}T^{\alpha _{k}}}{\varGamma (\alpha _{k}+1)}+\sum_{m=1}^{3}N_{m} \biggl( \biggl( \frac{ \vert d_{k,m} \vert }{\alpha _{m}}+ \vert b_{k,m} \vert \biggr) T+ \vert e_{k,m} \vert \biggr) \frac{T^{\alpha _{m}-1}}{\varGamma (\alpha _{m})} \Biggr). \end{aligned}$$

Firstly, we show that \(\varPsi ( B_{r} ) \subset B_{r}\). For this, define \(\sup_{t\in J} \vert f_{k} ( t,0 ) \vert =N_{k}< \infty,k=1,2,3\), then \(\vert f_{k} ( t,x ) \vert \leq L_{k} \Vert x \Vert _{X^{3}}+N_{k}\) for any \(t\in J\). Therefore

$$\begin{aligned} \bigl\vert \varPsi _{k}x_{k} ( t ) \bigr\vert \leq {}& \frac{t^{\alpha _{k}}}{\varGamma (\alpha _{k}+1)} \bigl( L_{k} \Vert x \Vert _{X^{3}}+N_{k} \bigr) \\ &{}+\sum_{m=1}^{3} \biggl( \frac{ \vert d_{k,m} \vert T^{\alpha _{m}}}{\varGamma (\alpha _{m}+1)}+ \bigl( \vert e_{k,m} \vert +t \vert b_{k,m} \vert \bigr) \frac{T^{\alpha _{m}-1}}{\varGamma (\alpha _{m})} \biggr) \bigl( L_{m} \Vert x \Vert _{X^{3}}+N_{m} \bigr) \\ \leq {}& \Biggl( \frac{L_{k}t^{\alpha _{k}}}{\varGamma (\alpha _{k}+1)}+\sum_{m=1}^{3}L_{m} \biggl( \frac{ \vert d_{k,m} \vert T}{\alpha _{m}}+ \bigl( \vert e_{k,m} \vert +t \vert b_{k,m} \vert \bigr) \biggr) \frac{T^{\alpha _{m}-1}}{\varGamma (\alpha _{m})} \Biggr) r \\ &{}\times \Biggl( \frac{N_{k}t^{\alpha _{k}}}{\varGamma (\alpha _{k}+1)}+\sum_{m=1}^{3}N_{m} \biggl( \frac{ \vert d_{k,m} \vert T}{\alpha _{m}}+ \bigl( \vert e_{k,m} \vert +t \vert b_{k,m} \vert \bigr) \biggr) \frac{T^{\alpha _{m}-1}}{\varGamma (\alpha _{m})} \Biggr). \end{aligned}$$

Consequently,

$$\begin{aligned} \Vert \varPsi x \Vert _{X^{3}} \leq{} &\sum _{k=1}^{3} \Biggl( \frac{N_{k}T^{\alpha _{k}}}{\varGamma (\alpha _{k}+1)}+\sum _{m=1}^{3}N_{m} \biggl( \biggl( \frac{ \vert d_{k,m} \vert }{\alpha _{m}}+ \vert b_{k,m} \vert \biggr) T+ \vert e_{k,m} \vert \biggr) \frac{T^{\alpha _{m}-1}}{\varGamma (\alpha _{m})} \Biggr) \\ &{}+\sum_{k=1}^{3} \Biggl( \frac{L_{k}T^{\alpha _{k}}}{\varGamma (\alpha _{k}+1)}+\sum_{m=1}^{3}L_{m} \biggl( \biggl( \frac{ \vert d_{k,m} \vert }{\alpha _{m}}+ \vert b_{k,m} \vert \biggr) T+ \vert e_{k,m} \vert \biggr) \frac{T^{\alpha _{m}-1}}{\varGamma (\alpha _{m})} \Biggr) r \\ \leq {}&r. \end{aligned}$$

Next, we show that the operator Ψ is a contraction. For this, let \(x,y\in X^{3}\), then for any \(t\in J\) we get

$$\begin{aligned} &\bigl\vert \varPsi _{k}x_{k} ( t ) -\varPsi _{k}y_{k} ( t ) \bigr\vert \\ &\quad\leq \Biggl( \frac{L_{k}t^{\alpha _{k}}}{\varGamma (\alpha _{k}+1)}+\sum_{m=1}^{3}L_{m} \biggl( \frac{ \vert d_{k,m} \vert T}{\alpha _{m}}+ \bigl( \vert e_{k,m} \vert +t \vert b_{k,m} \vert \bigr) \biggr) \frac{T^{\alpha _{m}-1}}{\varGamma (\alpha _{m})} \Biggr) \Vert x-y \Vert _{X^{3}}. \end{aligned}$$

It follows that

$$\begin{aligned} \Vert \varPsi x-\varPsi y \Vert _{X^{3}} &\leq \sum _{k=1}^{3} \Biggl( \frac{L_{k}T^{\alpha _{k}}}{\varGamma (\alpha _{k}+1)}+\sum _{m=1}^{3}L_{m} \biggl( \biggl( \frac{ \vert d_{k,m} \vert }{\alpha _{m}}+ \vert b_{k,m} \vert \biggr) T+ \vert e_{k,m} \vert \biggr) \frac{T^{\alpha _{m}-1}}{\varGamma (\alpha _{m})} \Biggr) \\ &\leq \eta \Vert x-y \Vert _{X^{3}}. \end{aligned}$$

Since \(\eta <1\), therefore Ψ is a contraction operator. Then, by the Banach fixed point theorem, the operator Ψ has a unique fixed point which is the unique solution of problem (1.1). This completes the proof. □

In the next result, we apply the Krasnoselskii fixed point theorem (Theorem 2.7) to prove the existence of at least one solution of the tripled fractional system (1.1). For this purpose, we decompose the triple operator \(\varPsi:X^{3}\rightarrow X^{3}\) into two triple operators \(\varPsi _{1}\) and \(\varPsi _{2}\) such that

$$ \varPsi x ( t ) =\varPsi _{1}x ( t ) +\varPsi _{2}x ( t ), $$

where \(\varPsi _{i}x ( t ) = ( \varPsi _{1,i}x_{1} ( t ),\varPsi _{2,i}x_{2} ( t ),\varPsi _{3,i}x_{3} ( t ) ) \), \(i=1,2\), and

$$ \textstyle\begin{cases} \varPsi _{k,1}x_{k} ( t ) =I_{0}^{\alpha _{k}}f_{k} ( t,x(t) ),\quad k=1,2,3, \\ \varPsi _{k,2}x_{k} ( t ) =\sum_{m=1}^{3} ( d_{k,m}I_{0}^{ \alpha _{m}}f_{m} ( T,x(T) ) + ( e_{k,m}+tb_{k,m} ) I_{0}^{\alpha _{m}-1}f_{m} ( T,x(T) ) ).\end{cases} $$

Theorem 3.2

Let \(f_{k}:J\times X ^{3}\rightarrow X,k=1,2,3\), be a jointly continuous function, and there exist nonnegative fractional integrable real-valued functions \(\varphi _{k}\)and \(\mu _{k}\)such that

$$ \textstyle\begin{cases} \vert f_{k} ( t,x ) -f_{k} ( t,y ) \vert \leq \varphi _{k}(t) \Vert x-y \Vert _{X ^{3}}, \\ \vert f_{k} ( t,0 ) \vert \leq \mu _{k}(t),\quad t \in J,k=1,2,3,\end{cases} $$

where \(x,y\in X ^{3}\). Then tripled system (1.1) has a solution provided that

$$ \sum_{k=1}^{3}m_{k}\max _{t\in J}I_{0}^{\alpha _{k}}\varphi _{k}(t)+n_{k} \max_{t\in J}I_{0}^{\alpha _{k}-1}\varphi _{k}(t)< 1, $$

where \(m_{k}=1+\sum_{m=1}^{3} \vert d_{m,k} \vert \)and \(n_{k}=\sum_{m=1}^{3} \vert e_{m,k} \vert +T \vert b_{m,k} \vert \).

Proof

Let \(B_{r}= \{ x\in X^{3}: \Vert x \Vert _{X^{3}}\leq r \} \) be a closed convex nonempty subset in \(X^{3}\) such that

$$ r\geq \frac{\sum_{k=1}^{3}m_{k}\max_{t\in J}I_{0}^{\alpha _{k}}\mu _{k}(t)+n_{k}\max_{t\in J}I_{0}^{\alpha _{k}-1}\mu _{k}(t)}{1-\sum_{k=1}^{3}m_{k}\max_{t\in J}I_{0}^{\alpha _{k}}\varphi _{k}(t)+n_{k}\max_{t\in J}I_{0}^{\alpha _{k}-1}\varphi _{k}(t)}. $$

We show that \(\varPsi _{1}\) is a contraction and \(\varPsi _{2}\) is compact on \(B_{r}\). Before doing these two steps, we show that \(\varPsi _{1}x+\varPsi _{2}y\in B_{r}\) whenever \(x,y\in B_{r}\). Let \(x=(x_{1},x_{2},x_{3})\) and \(y=(y_{1},y_{2},y_{3})\) be any elements of \(B_{r}\), then for \(t\in J\) we have

$$ \bigl\vert \varPsi _{k,1}x_{k} ( t ) \bigr\vert \leq I_{0}^{ \alpha _{k}} \bigl\vert f_{k} \bigl( t,x(t) \bigr) \bigr\vert \leq I_{0}^{\alpha _{k}}\varphi _{k}(t) \Vert x \Vert _{X^{3}}+I_{0}^{ \alpha _{k}}\mu _{k}(t), $$

and

$$\begin{aligned} \bigl\vert \varPsi _{k,2}y_{k} ( t ) \bigr\vert \leq {}& \sum_{m=1}^{3} \bigl( \vert d_{k,m} \vert I_{0}^{ \alpha _{m}} \bigl\vert f_{m} \bigl( T,y(T) \bigr) \bigr\vert + \bigl( \vert e_{k,m} \vert +t \vert b_{k,m} \vert \bigr) I_{0}^{\alpha _{m}-1} \bigl\vert f_{m} \bigl( T,y(T) \bigr) \bigr\vert \bigr) \\ \leq{} & \Vert y \Vert _{X^{3}}\sum_{m=1}^{3} \vert d_{k,m} \vert I_{0}^{\alpha _{m}}\varphi _{m}(t)+ \bigl( \vert e_{k,m} \vert +t \vert b_{k,m} \vert \bigr) I_{0}^{\alpha _{m}-1}\varphi _{m}(t) \\ &{}+\sum_{m=1}^{3} \vert d_{k,m} \vert I_{0}^{ \alpha _{m}}\mu _{m}(t)+ \bigl( \vert e_{k,m} \vert +t \vert b_{k,m} \vert \bigr) I_{0}^{\alpha _{m}-1}\mu _{m}(t). \end{aligned}$$

In consequence, we obtain

$$ \Vert \varPsi _{1}x \Vert _{X^{3}}\leq \Vert x \Vert _{X^{3}}\sum_{k=1}^{3}\max _{t\in J}I_{0}^{\alpha _{k}} \varphi _{k}(t)+ \sum_{k=1}^{3}\max_{t\in J}I_{0}^{\alpha _{k}} \mu _{k}(t) $$

and

$$\begin{aligned} \Vert \varPsi _{2}y \Vert _{X^{3}} \leq & \Vert y \Vert _{X^{3}}\sum_{m=1}^{3}\sum _{k=1}^{3} \vert d_{k,m} \vert \max_{t\in J}I_{0}^{\alpha _{m}} \varphi _{m}(t)+ \bigl( \vert e_{k,m} \vert +T \vert b_{k,m} \vert \bigr) \max_{t\in J}I_{0}^{\alpha _{m}-1} \varphi _{m}(t) \\ &{}+\sum_{m=1}^{3} \vert d_{k,m} \vert \max_{t \in J}I_{0}^{\alpha _{m}}\mu _{m}(t)+ \bigl( \vert e_{k,m} \vert +t \vert b_{k,m} \vert \bigr) \max_{t\in J}I_{0}^{ \alpha _{m}-1} \mu _{m}(t). \end{aligned}$$
(3.2)

Hence

$$\begin{aligned} \Vert \varPsi _{1}x+\varPsi _{2}y \Vert _{X^{3}} \leq{} &r\sum_{k=1}^{3}m_{k} \max_{t\in J}I_{0}^{\alpha _{k}}\varphi _{k}(t)+n_{k} \max_{t\in J}I_{0}^{\alpha _{k}-1} \varphi _{k}(t) \\ &{}+\sum_{k=1}^{3}m_{k}\max _{t\in J}I_{0}^{\alpha _{k}}\mu _{k}(t)+n_{k} \max_{t\in J}I_{0}^{\alpha _{k}-1}\mu _{k}(t). \end{aligned}$$

In accordance with the previous estimates and the value of r, we deduce that \(\varPsi _{1}x+\varPsi _{2}y\in B_{r}\).

Next we show the contraction of \(\varPsi _{1}\). Let \(x,y\in X^{3}\), then

$$\begin{aligned} \bigl\vert \varPsi _{k,1}x_{k} ( t ) -\varPsi _{k,1}y_{k} ( t ) \bigr\vert &\leq I_{0}^{\alpha _{k}} \bigl\vert f_{k} \bigl( t,x(t) \bigr) -f_{k} \bigl( t,y(t) \bigr) \bigr\vert \\ &\leq I_{0}^{\alpha _{k}}\varphi _{k}(t) \Vert x-y \Vert _{X^{3}}\\ & \leq \max_{t\in J}I_{0}^{\alpha _{k}} \varphi _{k}(t) \Vert x-y \Vert _{X^{3}}. \end{aligned}$$

Hence

$$ \Vert \varPsi _{1}x-\varPsi _{1}y \Vert _{X^{3}}\leq \Biggl( \sum_{k=1}^{3} \max_{t\in J}I_{0}^{\alpha _{k}}\varphi _{k}(t) \Biggr) \Vert x-y \Vert _{X^{3}}. $$

Since \(\max_{t\in J}I_{0}^{\alpha _{k}}\varphi _{k}(t)\leq m_{k}\max_{t \in J}I_{0}^{\alpha _{k}}\varphi _{k}(t)<1\), we deduce the contraction.

The last step shows the compactness of \(\varPsi _{2}\). It is obvious by (3.2) that \(\varPsi _{2}\) maps bounded sets into bounded sets. On the other hand, the continuity of \(f_{k}\) and its fractional integral would imply the continuity of the operator \(\varPsi _{2}\). The only thing we add is the equicontinuity of the family \(\varPsi _{2}B_{r}\). Let \(t_{1},t_{2}\in J\) with \(t_{1}< t_{2}\), then we have

$$ \bigl\vert \varPsi _{k,2}x_{k} ( t_{2} ) - \varPsi _{k,2}x_{k} ( t_{1} ) \bigr\vert \leq ( t_{2}-t_{1} ) \sum_{m=1}^{3} \vert b_{k,m} \vert \bigl( I_{0}^{ \alpha _{m}-1} \bigl( r \varphi _{m}(T)+\mu _{m}(T) \bigr) \bigr). $$

Accordingly, we find that

$$ \Vert \varPsi _{2}x \Vert _{X^{3}}\leq ( t_{2}-t_{1} ) \sum_{m=1}^{3} \Biggl( \bigl( I_{0}^{\alpha _{m}-1} \bigl( r\varphi _{m}(T)+ \mu _{m}(T) \bigr) \bigr) \sum_{m=1}^{3} \vert b_{k,m} \vert \Biggr), $$

which tends to zero as \(t_{1}\rightarrow t_{2}\) independently of x. Hence, by the Arzelà–Ascoli theorem, the operator \(\varPsi _{2}\) is compact. Using Krasnoselskii Theorem 2.7, there exists a fixed point \(x\in B_{r}\subset X^{3}\) satisfying the operator equation \(x=\varPsi _{1}x+\varPsi _{2}x\), which is the solution of tripled system (1.1). This completes the proof. □

4 Application

Corresponding to system (1.1), we consider the following tripled fractional system:

$$ \textstyle\begin{cases} {}^{c}D_{0}^{\frac{6}{5}}x_{1} ( t ) =\frac{t}{20}+ \frac{1}{\sqrt{169+t^{2}}} ( \frac{ \vert x_{1} ( t ) \vert }{1+ \vert x ( t ) \vert }+ \frac{ \vert x_{2} ( t ) \vert }{1+ \vert y ( t ) \vert }+ \frac{ \vert x_{3} ( t ) \vert }{1+ \vert z ( t ) \vert } ), \\ {}^{c}D_{0}^{\frac{3}{2}}x_{2} ( t ) = \frac{1}{\sqrt{64+t^{2}}}+ \frac{t \vert x_{1} ( t ) \vert }{20}+ \frac{t \vert x_{2} ( t ) \vert }{15 ( t^{3}+1 ) }+ \frac{t \vert x_{3} ( t ) \vert }{15 ( 2+t ) }, \\ {}^{c}D_{0}^{\frac{9}{8}}x_{3} ( t ) =\frac{e^{-t}}{10+t}+ \frac{ \vert x_{1} ( t ) \vert }{\sqrt{169+t^{2}}}+ \frac{ \vert x_{2} ( t ) \vert }{12+t}+ \frac{ \vert x_{3} ( t ) \vert }{15\sqrt{1+t^{2}}},\end{cases} $$
(4.1)

where \(t\in [ 0,1 ]\) with

$$ \textstyle\begin{cases} x_{1} ( 0 ) =x_{2} ( 1 ),\qquad 5x_{1}^{\prime } ( 0 ) =2x_{2}^{\prime } ( 1 ), \\ 2x_{2} ( 0 ) =x_{3} ( 1 ),\qquad 7x_{2}^{\prime } ( 0 ) =2x_{3}^{\prime } ( 1 ), \\ 3x_{3} ( 0 ) =x_{1} ( 1 ),\qquad 4x_{3}^{\prime } ( 0 ) =3x_{1}^{\prime } ( 1 ).\end{cases} $$
(4.2)

Using the given data, we find the following constants:

$$\begin{aligned} &a_{1,0}=1,\qquad a_{1,1}=\frac{2}{5}, \qquad a_{2,0}= \frac{1}{2},\qquad a_{2,1}= \frac{2}{7}, \\ &a_{3,0}=\frac{1}{3},\qquad a_{3,1}=\frac{3}{4}, \\ &b_{1,1} =b_{2,2}=b_{3,3}=0.09,\qquad b_{1,2}=0.4375,\qquad b_{1,3}=0.125, \\ &b_{2,1} =0.2344, \qquad b_{2,3}=0.3125,\qquad b_{3,1}=0.8203,\qquad b_{3,2}=0.3281 , \\ &d_{1,1} =d_{2,1}=d_{2,2}=d_{3,3}=0.2,\qquad d_{1,2}=1.2,\qquad d_{1,3}=d_{2,3}=0.6, \\ &d_{3,1} =0.4=d_{3,2},\qquad e_{1,1}=0.7921,\qquad e_{1,2}=0.397,\qquad e_{1,3}=0.4562, \\ &e_{2,1} =0.5578,\qquad e_{2,2}=0.3031,\qquad e_{2,3}=0.1437, \\ &e_{3,1} =0.2953,\qquad e_{3,2}=0.2781,\qquad e_{3,3}=0.1937, \\ &L_{1} =\frac{1}{13},\qquad L_{2}=\frac{1}{15},\qquad L_{3}=\frac{1}{12}. \end{aligned}$$
(4.3)

Then we deduce that \(\eta \approx 0.95<1\). Hence by Theorem 3.1 there is a unique solution for system (4.1). Furthermore, we have

$$\begin{aligned} &\varphi _{1}(t) =\frac{1}{\sqrt{169+t^{2}}}, \qquad \varphi _{2}(t)= \frac{t}{15}, \qquad \varphi _{3}(t)=\frac{1}{12}, \\ &\mu _{1}(t) =\frac{t}{20},\qquad \mu _{2}(t)= \frac{1}{\sqrt{64+t^{2}}}, \qquad\mu _{3}(t)=\frac{e^{-t}}{10+t}, \end{aligned}$$

and

$$\begin{aligned} m_{1} &=1.8,\qquad m_{2}=2.8,\qquad m_{3}=2.4, \\ n_{1} &=2.79, \qquad n_{2}=1.834,\qquad n_{3}=1.3211. \end{aligned}$$

Hence

$$ \sum_{k=1}^{3}m_{k}\max _{t\in J}I_{0}^{\alpha _{k}}\varphi _{k}(t)+n_{k} \max_{t\in J}I_{0}^{\alpha _{k}-1}\varphi _{k}(t)=0.812< 1. $$

Therefore, using Theorem 3.2, there exists a solution of system (4.1). The reduction of the condition value from 0.95 to 0.812 is substantial. However, we lose the uniqueness property of the solution.

The used permutation in the boundary condition (4.2) has the form \((1\ 2\ 3)\). Let us use another permutation of the boundary conditions for system (4.1) that has the form \((1\ 3)(2)\) such that

$$ \textstyle\begin{cases} x_{1} ( 0 ) =x_{3} ( 1 ),\qquad 5x_{1}^{\prime } ( 0 ) =2x_{3}^{\prime } ( 1 ), \\ 2x_{2} ( 0 ) =x_{2} ( 1 ),\qquad 7x_{2}^{\prime } ( 0 ) =2x_{2}^{\prime } ( 1 ), \\ 3x_{3} ( 0 ) =x_{1} ( 1 ),\qquad 4x_{3}^{\prime } ( 0 ) =3x_{1}^{\prime } ( 1 ) \end{cases} $$

with the same constants as (4.3). Hence we deduce the same results as in the previous example. Furthermore, one can use four other permutations, namely \((1\ 2)(3)\), \((1)(2\ 3)\), \((1\ 3\ 2)\), and identity \((1)(2)(3)\).

5 Conclusion

In this paper, we investigate a tripled system of three fractional differential equations of order \(\alpha \in (1,2]\). The existence and uniqueness of solutions of the proposed system associated with cyclic permutation boundary conditions are established. The Banach and Krasnoselskii fixed point theorems are used as tools to prove our main results. We present examples to illustrate the applicability of the main results.

We study a fractional system consisting of three associated equations together with a new type of boundary conditions that is related to permutation groups. This might be a novel approach that will provide substantial potential for developing more new ideas in this field.

The results of this paper can be extended to a tripled system of fractional equations with impulsive effects and nonlocal conditions. Indeed, a tripled fractional system along with different boundary conditions can be considered and discussed. Finally, the results of this paper can be extended to m-tuple fractional systems. We leave investigation of these topics as future work for interested readers.