RETRACTED ARTICLE: The Existence of Solution for k-Dimensional System of Langevin Hadamard-Type Fractional Differential Inclusions with 2k Different Fractional Orders

Abstract

In this paper, we investigate the existence of solution for k-dimensional system of Langevin Hadamard-type fractional differential inclusions with 2k different fractional orders. Our investigate relies on fixed point theorems and covers the cases when the right-hand side of the inclusion is sum of two multifunctions. Also, we provide an example to illustrate our main results.

1 Introduction

There are many published works about the existence of solutions for many fractional differential equations using fixed point theory (for example, see [2, 14,15,16,17,18,19,20, 39] and the references there in). Also, some researchers have been focused on fractional differential inclusions (for more details, see [1, 3,4,5,6, 12, 13, 21, 22, 24, 25, 27, 28, 30, 32, 34, 36, 37, 40] and the references there in). For finding more details about elementary notions and definitions of fractional differential equations, one can study [31, 35, 38]. The Langevin equation, first introduced by Langevin in 1908. It is well known that a Langevin equation is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments (for example, see [26, 42]). There are many works about the fractional Langevin equation and inclusions (for more information, consider [7,8,9,10, 41]). As you know, the Hadamard fractional integral of order $\alpha >0$ for a function f is defined by

$$\begin{aligned} I^\alpha f(t) = \frac{1}{\Gamma (\alpha )} \int _1^t \left( \ln \frac{t}{s} \right) ^{\alpha -1} \frac{f(s)}{s} \mathrm{d}s. \end{aligned}$$

Also, the Hadamard derivative of fractional order $\alpha$ for function f is defined by

$$\begin{aligned} D^\alpha f(t) = \frac{1}{\Gamma (n-\alpha )} \left( t\frac{\mathrm{d}}{\mathrm{d}t} \right) ^n \int _1^t \left( \ln \frac{t}{s} \right) ^{n-\alpha -1} \frac{f(s)}{s}\mathrm{d}s, \end{aligned}$$

where $n=[\alpha ]+1$. It has been proved that the general solution of the Hadamard fractional differential equation $D^\alpha x(t)=0$ is given by

$$\begin{aligned} x(t) =c_1(\ln t)^{\alpha -1}+ c_2(\ln t)^{\alpha -2}+ \cdots + c_n (\ln t)^{\alpha -n}, \end{aligned}$$

where $c_1, \ldots , c_n$ are real constants and $n=[\alpha ]+1$. Details and properties of the Hadamard fractional derivative and integral can be found in [31]. Let (X, d) be a metric space. Denote by P(X) and $2^X$ the class of all subsets and the class of all nonempty subsets of X, respectively. Thus, $P_{cl}(X)$, $P_{bd}(X)$, $P_{cv}(X)$ and $P_{cp}(X)$ denote the class of all closed, bounded, convex and compact subsets of X, respectively. A mapping $Q: X\rightarrow 2^X$ is called a multifunction on X and $x\in X$ is called a fixed point of Q whenever $x\in Q(x)$. A multifunction $Q : X\rightarrow P(X)$ is lower semi-continuous, if for any open set A of X, the set

$$\begin{aligned} Q^{-1} (A) : = \left\{ x\in X : Q(x)\cap A\ne \emptyset \right\} \end{aligned}$$

is open in X. When for any open set A of X, the set $\{x\in X: Qx\subset A\}$ is open in X, we say that Q is upper semi-continuous. Also, $Q: X\rightarrow P_{cp}(X)$ is called compact if $\overline{Q(S)}$ is a compact set of X for any bounded subsets S of X. A multifunction $G: [1,e]\rightarrow P_{cl}({\mathbb {R}})$ is said to be measurable, whenever the function $t \mapsto d(y, G(t)) = \inf \{|y-z|:z\in G(t)\}$ is measurable for all $y\in {\mathbb {R}}$. Define the Hausdorff metric $H: 2^X\times 2^X\rightarrow [0,\infty )$ by

$$\begin{aligned} H( A, B)= \max \left\{ \sup _{a\in A}d(a,B), \sup _{b\in B}d(A,b) \right\} , \end{aligned}$$

where $d(A,b)=\inf _{a\in A}d(a; b)$. Then, $(P_{b,cl} (X), H)$ is a metric space and $(P_{cl}(X),H)$ is a generalized metric space [23, 32]. A multifunction $N:X\rightarrow P_{cl}(X)$ is called a $\gamma$-contraction whenever there exists $\gamma \in (0,1)$ such that $H(N(x), N(y))\le \gamma d(x,y)$ for all $x, y\in X$. Covitz and Nadler [27] proved that each closed-valued contractive multifunction on a complete metric space has a fixed point. Ahmad and Ntouyas [11] investigated the following boundary value problem of Hadamard-type fractional differential inclusions $D^\alpha x(t)\in F(t,x(t) )$, via the boundary conditions $x(1)=0, x(e)=I^\beta x(\eta ), 1<\eta <e$, where $1<t<e$, $1<\alpha \le 2$, $\beta >0$, $D^{\alpha }$ is Hadamard fractional differential and $I^{\beta }$ is Hadamard fractional integral and $F:[1,e] \times {\mathbb {R}} \rightarrow P({\mathbb {R}})$.

Motivated by the above-mentioned works, in this paper, we investigate the existence of solution for k-dimensional system of Langevin Hadamard-type fractional differential inclusions:

$$\begin{aligned} \left\{ \begin{array}{ll} D^{\beta _1} \left( D^{\alpha _1} + \lambda _1 \right) x_1(t) &{} \in F_1 \big (t, x_1(t), \ldots , x_k(t), I^{\nu _1} x_1(t), \ldots , I^{\nu _k}x_k(t)\big ) \\ &{} \quad + \, G_1\big ( t, x_1(t), \ldots ,x_k(t) \big ), \\ D^{\beta _2} \left( D^{\alpha _2} + \lambda _2 \right) x_2(t)&{} \in F_2 \big (t, x_1(t), \ldots , x_k(t), I^{\nu _1} x_1(t),\ldots , I^{\nu _k}x_k(t) \big ) \\ &{} \quad + \, G_2 \big (t, x_1(t), \ldots , x_k(t) \big ),\\ \quad \quad \vdots &{} \quad \quad \vdots \\ D^{\beta _k} \left( D^{\alpha _k} + \lambda _k \right) x_k(t) &{} \in F_k \big (t, x_1(t), \ldots , x_k(t), I^{\nu _1} x_1(t), \ldots , I^{\nu _k} x_k(t) \big )\\ &{} \quad + \, G_k \big (t, x_1(t), \ldots , x_k(t) \big ), \end{array} \right. \end{aligned}$$

(1.1)

via the boundary conditions $x_i(t)\big |_{t\rightarrow 1^+}=0$, $I^{\gamma _i} x_i(\eta )+ D^{\gamma _i} x_i(\eta )=0$ and $I^{\gamma _i} x_i(e) + D^{\gamma _i} x_i(e) =0$ for $i=1, \ldots , k$, where $1<\beta _i\le 2$, $0< \gamma _i< \alpha _i < 1$, $\nu _i>0$, $1<\eta <e$, $t\in [1,e]$, $D^{(.)}$ is Hadamard derivative of fractional and $I^{(.)}$ is Hadamard fractional integral and $F_i : [1,e] \times {\mathbb {R}}^{2k} \rightarrow 2^{{\mathbb {R}}}$, $G_i : [1,e] \times {\mathbb {R}}^k \rightarrow 2^{{\mathbb {R}}}$ are multifunctions for all $1\le i\le k$. We say that $G: [1,e]\times {\mathbb {R}}^{k} \rightarrow 2^{{\mathbb {R}}}$ is a Carathéodory multifunction whenever $t \mapsto G(t,x_1,\ldots , x_{k})$ is measurable for all $x_1,\ldots , x_{k} \in {\mathbb {R}}$ and $(x_1, \ldots , x_{k}) \mapsto G(t,x_1,\ldots , x_{k})$ is an upper semi-continuous map for almost all $t\in [1,e]$ (for more details, see [13, 28, 32]). Also, a Carathéodory multifunction $G: [1,e] \times {\mathbb {R}}^{k} \rightarrow 2^{{\mathbb {R}}}$ is called $L^{1}$-Carathéodory whenever for each $\rho >0$ there exists $\phi _{\rho } \in L^{1}([1,e], {\mathbb {R}}^+)$ such that

$$\begin{aligned} \parallel G(t,x_1,\ldots , x_{k})\parallel =\sup \{|v|:v\in G(t,x_1,\ldots , x_{k})\}\le \phi _{\rho }(t), \end{aligned}$$

for all $|x_1|,\ldots , |x_{k}|\le \rho$ and for almost all $t\in [1,e]$ (for more information, see [13, 28, 32]). Define the space $X=C([1,e],{\mathbb {R}})$ endowed with the norm $\Vert x \Vert =\sup _{t\in [1,e]} |x(t)|$. In fact, $(X,\Vert .\Vert )$ and the product space

$$\begin{aligned} \bigg ( X^k = \underbrace{ X \times X \times \ldots \times X}_{k}, \Vert . \Vert _{*} \bigg ) \end{aligned}$$

endowed with the norm

$$\begin{aligned} \left\| (x_1, x_2, \ldots , x_k ) \right\| _{*} = \left\| x_1 \right\| + \left\| x_2 \right\| + \cdots + \left\| x_k \right\| \end{aligned}$$

are Banach spaces. Using the idea of another papers (for example, see [6, 12, 36]), define the set of the selections of $F_i$, $G_i$ at $(x_1, \ldots , x_k)$ by

$$\begin{aligned} S_{F_i, (x_1, \ldots , x_k)}&= \left\{ v\in L^1[1,e] : v(t)\in F_i\big (t, x_1(t), \ldots , x_k(t), I^{\nu _1} x_1(t), \right. \\&\quad \left. \ldots , I^{\nu _k} x_k(t) \big ) \right\} ,\\ S_{G_i,(x_1, \ldots , x_k)}&= \left\{ v \in L^1[1,e] : v(t)\in G_i \big (t, x_1(t), \ldots , x_k(t) \big ) \right\} , \end{aligned}$$

for almost all $t\in [1,e]$ and for all $1\le i\le k$. We need the following fixed point lemmas in our main results.

Lemma 1.1

[28]. If $G: X \rightarrow P_{cl}(Y)$ is upper semi-continuous, then Gr(G) is a closed subset of $X\times Y$. Conversly, if G is completely continuous and has a closed graph, then it is upper semi-continuous.

Lemma 1.2

[33]. Suppose that X be a Banach space, $F: J\times X\rightarrow P_{cp,cv}(X)$ an $L^{1}$-Carathéodory multifunction and $\Theta$ a linear continuous mapping from $L^{1}(J,X)$ to C(J, X). Then, the operator

$$\begin{aligned} \left\{ \begin{array}{l} \Theta o S_{F}:C(J,X)\rightarrow P_{cp,cv}(C(J),X), \\ (\Theta o S_{F})(x)=\Theta (S_{F,x}), \end{array} \right. \end{aligned}$$

is a closed graph operator in $C(J,X) \times C(J,X)$.

Lemma 1.3

[29]. Consider B(0, r) and B[0, r] denote, respectively, the open and closed balls in Banach space X centered at origin and of radius r and let $\Phi _1: X\rightarrow P_{bd, cl, cv}(X)$ and $\Phi _2: B[0,r]\rightarrow P_{ cp, cv}(X)$ be two multivalued operators such that $\Phi _1$ is contraction, $\Phi _2$ is upper semi-continuous and completely continuous. Then, either the operator inclusion $x \in \Phi _1(x) + \Phi _2(x)$ has a solution in B[0, r] or there exists $u\in X$ with $\Vert u\Vert =r$ such that $\lambda u \in \Phi _1(u) + \Phi _2(u)$.

2 Main Results

Lemma 2.1

For $v\in C([1,e], {\mathbb {R}})$, $\lambda \in {\mathbb {R}}$, $\beta \in (1,2]$ and $\alpha \in ( 0,1]$, the unique solution of the fractional problem

$$\begin{aligned} \left\{ \begin{array}{l} D^\beta (D^\alpha +\lambda ) x(t)=v(t), \\ I^\gamma x(\eta ) + D^\gamma x(\eta )=0, \\ I^\gamma x(e) +D^\gamma x(e)=0,\\ x(t)\big |_{t\rightarrow 1^+}=0, \end{array} \right. \end{aligned}$$

(2.1)

where $\eta \in (1,e)$, $D^{(.)}$ is Hadamard fractional differential and $I^{(.)}$ is Hadamard fractional integral, is given by

$$\begin{aligned} x(t)&= \frac{1}{ \Gamma (\alpha )} \int _1^t \frac{1}{s} \left( \ln \frac{t}{s} \right) ^{\alpha -1} \left( I^\beta v(s)- \lambda x(s) \right) \mathrm{d}s\\&\quad - \frac{a_1 (\eta ) a_2(t)}{b_1 a_3(\eta )\left[ a_3(\eta ) b_2 - a_1(\eta ) b_3 \right] } \\&\quad \times \left[ b_3 \int _1^\eta \frac{1}{s} a_4(\eta ) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3(\eta ) \int _1^e \frac{1}{s} a_4(e) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \right] \\&\quad - \frac{a_2(t)}{b_1 a_3(\eta ) } \int _1^\eta \frac{1}{s} a_4(\eta ) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \\&\quad + \frac{a_5(t) }{b_4 \left[ a_3(\eta ) b_2 - a_1(\eta ) b_3 \right] } \\&\quad \times \left[ b_3 \int _1^e \frac{1}{s} a_4(\eta ) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3(\eta ) \int _1^e \frac{1}{s} a_4(e) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \right] , \end{aligned}$$

where

$$\begin{aligned} I^\beta v(s)= & {} \frac{1}{\Gamma (\beta )} \int _1^s \left( \ln \frac{s}{u} \right) ^{\beta -1} \frac{v(u)}{u} \mathrm{d}u, \nonumber \\ a_1(\eta ):= & {} \Gamma (\alpha +\beta +\gamma -1)\left( \ln \eta \right) ^{\alpha +\beta -\gamma -2} \nonumber \\&+ \Gamma ( \alpha + \beta -\gamma - 1) \left( \ln \eta \right) ^{ \alpha + \beta + \gamma - 2}, \nonumber \\ a_2(t):= & {} \Gamma (\alpha +\beta - \gamma ) \Gamma ( \alpha + \beta +\gamma )(\ln t)^{\alpha + \beta -1}, \nonumber \\ a_3(\eta ):= & {} \Gamma (\alpha + \beta + \gamma ) \left( \ln \eta \right) ^{\alpha + \beta - \gamma -1 } \nonumber \\&+ \Gamma ( \alpha + \beta - \gamma ) \left( \ln \eta \right) ^{ \alpha + \beta + \gamma -1}, \nonumber \\ a_4(\eta ):= & {} \frac{1}{\Gamma (\alpha +\gamma )} \left( \ln \frac{\eta }{s} \right) ^{\alpha +\gamma -1} + \frac{1}{\Gamma (\alpha - \gamma )} \left( \ln \frac{\eta }{s} \right) ^{\alpha -\gamma -1}, \nonumber \\ a_4 (e):= & {} \frac{1}{\Gamma (\alpha + \gamma )} \left( \ln \frac{e}{s} \right) ^{\alpha +\gamma - 1} + \frac{1}{ \Gamma ( \alpha -\gamma )} \left( \ln \frac{e}{s} \right) ^{ \alpha - \gamma - 1}, \nonumber \\ a_5(t):= & {} \Gamma (\alpha +\beta -\gamma -1)\Gamma (\alpha + \beta +\gamma -1)(\ln t)^{\alpha +\beta -2}, \nonumber \\ b_1:= & {} \Gamma (\alpha +\beta ), \nonumber \\ b_2:= & {} \Gamma ( \alpha + \beta -\gamma -1) +\Gamma (\alpha +\beta + \gamma -1), \nonumber \\ b_3:= & {} \Gamma (\alpha + \beta -\gamma )+\Gamma ( \alpha + \beta + \gamma ), \nonumber \\ b_4:= & {} \Gamma (\alpha +\beta -1). \end{aligned}$$

(2.2)

Proof

It is known that the general solution of the equation $D^\beta \left( D^\alpha + \lambda \right) x(t) = v(t)$ is

$$\begin{aligned} x(t)&= \frac{1}{\Gamma (\alpha )} \int _1^t \left( \ln \frac{t}{s} \right) ^{\alpha -1} \frac{1}{s} \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \\&\quad + c_1 \frac{\Gamma (\beta )(\ln t)^{\beta +\alpha -1}}{\Gamma (\alpha + \beta )} + c_2\frac{\Gamma (\beta -1)(\ln t)^{\beta +\alpha -1}}{\Gamma (\alpha +\beta -1)}+c_3(\ln t)^{\alpha -1}, \end{aligned}$$

where $c_1, c_2, c_3$ are arbitrary constants and $t \in [1,e]$. Thus,

$$\begin{aligned} I^\gamma x(t)&= \frac{1}{\Gamma (\gamma +\alpha )} \int _1^t \left( \ln \frac{t}{s} \right) ^{\alpha +\gamma - 1} \frac{1}{s} \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \\&\quad + c_1\frac{\Gamma (\beta )(\ln t)^{\beta +\gamma +\alpha -1}}{\Gamma ( \alpha + \beta + \gamma )} + c_2 \frac{\Gamma (\beta -1)(\ln t)^{\beta + \gamma + \alpha - 2}}{\Gamma ( \alpha + \gamma + \beta -1)} \\&\quad + c_3\frac{\Gamma (\alpha )( \ln t)^{\alpha + \gamma -1}}{\Gamma (\alpha +\gamma )},\\ D^\gamma x(t)&= \frac{1}{\Gamma ( \alpha - \gamma )} \int _1^t \left( \ln \frac{t}{s} \right) ^{ \alpha - \gamma -1}\frac{1}{s} \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \\&\quad + c_1\frac{\Gamma (\beta )(\ln t)^{\alpha + \beta - \gamma - 1}}{\Gamma (\alpha + \beta -\gamma )} + c_2\frac{\Gamma (\beta -1)(\ln t)^{\alpha +\beta -\gamma - 2}}{\Gamma (\alpha + \gamma -\beta -1)} \\&\quad + c_3\frac{\Gamma (\alpha )(\ln t)^{\alpha - \gamma - 1}}{\Gamma (\alpha -\gamma )}. \end{aligned}$$

At present, using the boundary conditions (2.1), item of $x(t)\big |_{t \rightarrow 0^+}=0$, since $\alpha -1 \le 0$, we obtain $c_3 = 0$,

$$\begin{aligned} c_1 A_1(\eta ) + c_2 A_2(\eta )&= -\int _1^\eta \frac{1}{s} A_3(\eta ) \left( I^\beta v(s)-\lambda x(s) \right) \mathrm{d}s,\\ c_1 B_1 + c_2 B_2&= - \int _1^e \frac{1}{s} A_3(e) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s, \end{aligned}$$

where

$$\begin{aligned} A_1(\eta ):= & {} \frac{\Gamma (\beta )}{\Gamma (\alpha +\beta +\gamma )}\left( \ln \eta \right) ^{\alpha +\beta +\gamma -1} + \frac{\Gamma (\beta )}{ \Gamma (\alpha +\beta -\gamma )}(\ln \eta )^{\alpha + \beta -\gamma -1}, \nonumber \\ A_2(\eta ):= & {} \frac{\Gamma (\beta -1)}{\Gamma (\alpha +\beta +\gamma -1)}(\ln \eta )^{ \alpha + \beta +\gamma -2}+ \frac{\Gamma (\beta -1)}{\Gamma (\alpha +\beta -\gamma -1)}(\ln \eta )^{\alpha +\beta - \gamma -2}, \nonumber \\ A_3(\eta ):= & {} \frac{1}{ \Gamma (\alpha + \gamma )}\left( \ln \frac{ \eta }{s} \right) ^{ \alpha +\gamma -1}+\frac{1}{ \Gamma ( \alpha -\gamma )}\left( \ln \frac{\eta }{s} \right) ^{\alpha - \gamma -1}, \nonumber \\ A_3(e):= & {} \frac{1}{ \Gamma (\alpha + \gamma )} \left( \ln \frac{e}{s} \right) ^{ \alpha +\gamma -1}+\frac{1}{ \Gamma ( \alpha -\gamma )}\left( \ln \frac{e}{s} \right) ^{\alpha -\gamma - 1}, \nonumber \\ B_1:= & {} \frac{\Gamma (\beta )}{\Gamma ( \alpha + \beta +\gamma )}+\frac{\Gamma (\beta )}{ \Gamma (\alpha +\beta -\gamma )}, \nonumber \\ B_2:= & {} \frac{\Gamma (\beta -1)}{\Gamma (\alpha + \beta +\gamma -1)} + \frac{\Gamma (\beta -1)}{\Gamma (\alpha + \beta -\gamma -1)}. \end{aligned}$$

(2.3)

Thus,

$$\begin{aligned} c_1&= - \frac{ a_1(\eta ) b_5}{ \Gamma (\beta ) a_3(\eta ) \left[ a_3(\eta ) b_2 - a_2 (\eta ) b_3 \right] } \\&\quad \times \left[ b_3 \int _1^\eta \frac{1}{s} a_4(\eta ) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3(\eta ) \int _1^e \frac{1}{s} a_4(e) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \right] \\&\quad - \frac{b_5}{\Gamma (\beta ) a_3(\eta )} \int _1^\eta \frac{1}{s} a_4(\eta ) \left( I^\beta v(s) -\lambda x(s) \right) \mathrm{d}s,\\ c_2&= \frac{b_6}{ \Gamma (\beta -1) \left[ a_3(\eta ) b_2 - a_2 (\eta ) b_3 \right] } \\&\quad \times \left[ b_3 \int _1^\eta \frac{1}{s} a_4(\eta ) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3(\eta ) \int _1^e \frac{1}{s} a_4(e) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \right] , \end{aligned}$$

where

$$\begin{aligned} b_5:= & {} \Gamma ( \alpha + \beta - \gamma )\Gamma (\alpha + \beta +\gamma ), \nonumber \\ b_6:= & {} \Gamma (\alpha +\beta -\gamma -1) \Gamma ( \alpha + \beta + \gamma -1). \end{aligned}$$

(2.4)

Hence,

$$\begin{aligned} x(t)&= \frac{1}{ \Gamma (\alpha )} \int _1^t \frac{1}{s} \left( \ln \frac{t}{s} \right) ^{\alpha -1} \left( I^\beta v(s)- \lambda x(s) \right) \mathrm{d}s\\&\quad - \frac{a_1 (\eta ) a_2(t)}{b_1 a_3(\eta )\left[ a_3(\eta ) b_2 - a_1(\eta ) b_3 \right] } \\&\quad \times \left[ b_3 \int _1^\eta \frac{1}{s} a_4(\eta ) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3(\eta ) \int _1^e \frac{1}{s} a_4(e) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \right] \\&\quad - \frac{a_2(t)}{b_1 a_3(\eta ) } \int _1^\eta \frac{1}{s} a_4(\eta ) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \\&\quad + \frac{a_5(t) }{b_4 \left[ a_3(\eta ) b_2 - a_1(\eta ) b_3 \right] } \\&\quad \times \left[ b_3 \int _1^\eta \frac{1}{s} a_4(\eta ) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3(\eta ) \int _1^e \frac{1}{s} a_4(e) \left( I^\beta v(s) - \lambda x(s) \right) \mathrm{d}s \right] , \end{aligned}$$

This completes the proof. $\square$

A function $(x_1, x_2, \ldots , x_k ) \in X^k$ is a solution for the k-dimensional inclusions problem if there exist functions

$$\begin{aligned} \left( v_1, v_2, \ldots , v_k \right) , \left( v'_1, v'_2, \ldots , v'_k \right) \in \underbrace{ L^1[1,e] \times L^1[1,e] \times \cdots \times L^1[1,e]}_k \end{aligned}$$

such that

$$\begin{aligned} v_i(t)\in F_i \left( t, x_1(t), \ldots , x_k(t), I^{\nu _1} x_1(t), \ldots , I^{\nu _k} x_k(t) \right) , \end{aligned}$$

$v'_i(t) \in G_i(t, x_1(t),\ldots ,x_k(t))$, a.e. on [1, e] and $D^{\beta _i} \left( D^{\alpha _i} + \lambda _i \right) x_i(t) = v_i(t) + v'_i(t)$ a.e. on [1, e], $x_i(1)=0$, $I^{\gamma _i} x_i(\eta ) + D^{\gamma _i} x_i(\eta )=0$ and $I^{\gamma _i} x_i(e) + D^{\gamma _i} x_i(e) =0$ for $i =1, \ldots , k$.

Theorem 2.2

Suppose that $G_i: [1,e]\times {\mathbb {R}}^k \rightarrow P_{cp,cv} ({\mathbb {R}})$ are Carathéodory multifunctions and $F_i: [1,e] \times {\mathbb {R}}^{2k} \rightarrow P_{cp,cv}({\mathbb {R}})$ be such that $F_i \left( ., x_1, \ldots , x_{2k} \right) : [1,e] \rightarrow P_{cp,cv}({\mathbb {R}})$ are measurable and there exist continuous functions $p_i, m_i: [1,e] \rightarrow (0 , \infty )$ and continuous nondecreasing functions $\psi _i : [0, \infty ) \rightarrow [0,\infty )$ such that

$$\begin{aligned} \left\| G_i \left( t, x_1, \ldots , x_k \right) \right\|&= \sup \left\{ |v| : v \in G_i \left( t, x_1, \ldots ,x_k \right) \right\} \le p_i(t) \psi _i \bigg ( \sum _{i=1}^k |x_i| \bigg ), \\ \left\| F_i \left( t, x_1, \ldots , x_{2k} \right) \right\|&= \sup \left\{ |v| : v\in F_i \left( t, x_1, \ldots , x_{2k} \right) \right\} \le m_i(t) \end{aligned}$$

and

$$\begin{aligned} H \left( F_i \left( t, x_1, \ldots , x_{2k} \right) , F_i \left( t, y_1, \ldots , y_{2k} \right) \right) \le m_i(t) \sum _{i=1}^{2k} \left| x_i - y_i \right| , \end{aligned}$$

for all $x_1, \ldots , x_{2k}, y_1, \ldots , y_{2k} \in {\mathbb {R}}$ and $1\le i\le k$. If

$$\begin{aligned} \sum _{i=1}^k \bigg ( \Vert m_i \Vert \Lambda _1^i \sum _{j=1}^k \bigg ( 1 + \frac{1}{\Gamma (\nu _j+1)} \bigg ) + \left| \lambda _i \right| \Lambda _2^i \bigg ) < 1, \end{aligned}$$

such that

$$\begin{aligned} \Lambda _1^i&= \frac{1}{\Gamma (\alpha _i + \beta _i+1)} + \left[ \frac{a_1^i(\eta ) b_5^i}{b_1^i a_3^i (\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } + \frac{b_6^i}{ b_4^i \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| }\right] \\&\quad \times \left[ b_3^i a_6^i(\eta ) + a_3^i (\eta ) b_7^i \right] + \frac{b_5^i a_6^i (\eta )}{b_1^i a_3^i(\eta )}, \\ \Lambda _2^i&= \frac{1}{\Gamma (\alpha _i+1)} + \left[ \frac{a_1^i(\eta ) b_5^i}{b_1^i a_3^i(\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } + \frac{b_6^i}{ b_4^i\left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \right] \\&\quad \times \left[ b_3^i a_7^i(\eta ) + a_3^i(\eta ) b_8^i\right] + \frac{b_5^i a_7^i (\eta )}{b_1^i a_3^i(\eta )}, \end{aligned}$$

where

$$\begin{aligned} a_6^i(\eta )&:= \frac{(\ln \eta )^{ \alpha _i + \beta _i + \gamma _i}}{\Gamma ( \alpha _i + \beta _i + \gamma _i+1)}+\frac{( \ln \eta )^{\alpha _i + \beta _i - \gamma _i}}{\Gamma (\alpha _i + \beta _i-\gamma _i+1)}, \\ a_7^i(\eta )&:= \frac{(\ln \eta )^{\alpha _i+\gamma _i}}{\Gamma (\alpha _i+\gamma _i+1)}+\frac{(\ln \eta )^{\alpha _i-\gamma _i}}{\Gamma (\alpha _i-\gamma _i+1)}, \\ b_7^i&:= \frac{1}{\Gamma (\alpha _i+\beta _i+\gamma _i+1)}+\frac{1}{\Gamma (\alpha _i+\beta _i-\gamma _i+1)},\\ b_8^i&:= \frac{1}{\Gamma ( \alpha _i+\gamma _i+1)}+\frac{1}{\Gamma (\alpha _i-\gamma _i+1)}, \end{aligned}$$

for all $1\le i\le k$, then the k-dimensional system of fractional differential inclusions has at least one solution on [1, e].

Proof

Define an open ball $B(0,r)\in X^k$, where the real number r satisfies the following inequality

$$\begin{aligned} \frac{\sum _{i=1}^k \left\| p_i\right\| \psi _i(r) \Lambda _1^i}{ 1- \sum _{i=1}^k \left( \left\| m_i \right\| \Lambda _1^i \sum _{j=1}^k \left( 1+ \frac{1}{\Gamma (\nu _j+1)} \right) + 2 |\lambda _i| \Lambda _2^i\right) } < r. \end{aligned}$$

Consider the multivalued operators $A ,B : X^k\rightarrow P(X^k)$ by

$$\begin{aligned} A(x_1, \ldots , x_k )= & {} \left( \begin{array}{c} A_1(x_1, \ldots , x_k)\\ A_2(x_1, \ldots , x_k)\\ \vdots \\ A_k(x_1, \ldots , x_k) \end{array} \right) ,\\ B(x_1, \ldots , x_k)= & {} \left( \begin{array}{c} B_1(x_1, \ldots , x_k)\\ B_2(x_1, \ldots , x_k) \\ \vdots \\ B_k(x_1, \ldots , x_k) \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} A_i(x_1, \ldots , x_k)&:= \left\{ u \in X \left| \right. \exists v \in S_{F_i, (x_1, \ldots , x_k)} ~: ~ u(t)= w_i(v,t) \right\} ,\\ B_i( x_1, \ldots , x_k)&:= \left\{ u\in X \left| \right. \exists v \in S_{G_i, (x_1,\ldots ,x_k)} ~:~ u(t)= w_i(v,t) \right\} , \end{aligned}$$

for all $t\in [1,e]$, and

$$\begin{aligned} w_i(v,t)&= \frac{1}{\Gamma (\alpha _i)} \int _1^t \frac{1}{s} \left( \ln \frac{t}{s} \right) ^{ \alpha _i - 1} \left( I^{\beta _i} v(s) -\lambda _i x_i(s) \right) \mathrm{d}s \\&\quad - \frac{a_1^i(\eta ) a_2^t(i)}{b_1^i a_3^i(\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| }\\&\quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i(\eta ) \left( I^{\beta _i} v(s) -\lambda _i x_i(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3^i(\eta ) \int _1^\eta \frac{1}{s} a_4(e) \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \right] \\&\quad - \frac{a_3^i (\eta ) }{b_1^i a_3^i(\eta )} \int _1^\eta \frac{1}{s} a_4^i(\eta ) \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \\&\quad + \frac{a_1^i(\eta )}{b_4 \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i(\eta ) \left( I^{\beta _i} v(s) - \lambda _ix_i(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3^i(\eta ) \int _1^e \frac{1}{s} a_4^i(e) \left( I^{\beta _i} v(s) - \lambda _i x_i(s)\right) \mathrm{d}s\right] , \end{aligned}$$

which

$$\begin{aligned} I^{\beta _i} v(s)= & {} \frac{1}{\Gamma ( \beta _i)} \int _1^s \left( \ln \frac{s}{u} \right) ^{\beta _i - 1} \frac{v(u)}{u} \mathrm{d}u, \nonumber \\ a_1^i(\eta ):= & {} \Gamma (\alpha _i+ \beta _i+ \gamma _i - 1)(\ln \eta )^{\alpha _i + \beta _i - \gamma _i -2} \nonumber \\&+ \Gamma ( \alpha _i +\beta _i -\gamma _i -1)(\ln \eta )^{ \alpha _i +\beta _i +\gamma _i -2}, \nonumber \\ a_2^i(t):= & {} \Gamma ( \alpha _i +\beta _i -\gamma _i )\Gamma (\alpha _i +\beta _i +\gamma _i )(\ln t)^{\alpha _i +\beta _i -1}, \nonumber \\ a_3^i(\eta ):= & {} \Gamma (\alpha _i+\beta _i +\gamma _i )(\ln \eta )^{ \alpha _i +\beta _i -\gamma _i -1} \nonumber \\&+ \Gamma (\alpha _i +\beta _i -\gamma _i )(\ln \eta )^{\alpha _i +\beta _i +\gamma _i -1}, \nonumber \\ a_4^i(\eta ):= & {} \frac{1}{\Gamma (\alpha _i + \gamma _i )} \left( \ln \frac{\eta }{s} \right) ^{\alpha _i +\gamma _i -1} + \frac{1}{\Gamma ( \alpha _i -\gamma _i )} \left( \ln \frac{\eta }{s} \right) ^{\alpha _i -\gamma _i -1}, \nonumber \\ a_4^i(e):= & {} \frac{1}{\Gamma (\alpha _i +\gamma _i )} \left( \ln \frac{e}{s} \right) ^{\alpha _i +\gamma _i -1} + \frac{1}{\Gamma (\alpha _i -\gamma _i )} \left( \ln \frac{e}{s} \right) ^{\alpha _i -\gamma _i -1}, \nonumber \\ a_5^i(t):= & {} \Gamma (\alpha _i +\beta _i - \gamma _i -1) \Gamma (\alpha _i +\beta _i +\gamma _i -1) \left( \ln t \right) ^{\alpha _i +\beta _i -2}, \nonumber \\ b_1^i:= & {} \Gamma (\alpha _i +\beta _i ), \nonumber \\ b_2^i:= & {} \Gamma (\alpha _i +\beta _i -\gamma _i -1)+\Gamma (\alpha _i +\beta _i +\gamma _i -1), \nonumber \\ b_3^i:= & {} \Gamma (\alpha _i + \beta _i -\gamma _i ) + \Gamma ( \alpha _i +\beta _i +\gamma _i), \nonumber \\ b_4^i:= & {} \Gamma ( \alpha _i +\beta _i -1), \end{aligned}$$

(2.5)

for each $1\le i \le k$. Thus, the k-dimensional system of fractional differential inclusions is equivalent to the inclusion problem $(x_1, \ldots , x_k ) \in A(x_1, \ldots , x_k ) + B(x_1, \ldots , x_k )$. We show that the multifunctions A and B satisfy the conditions of Lemma 1.3. As a first step, we show that $B(x_1, \ldots , x_k) \in P_{cl}(X^k)$ for each $(x_1, \ldots , x_k)\in X^k$. Let $\{( u_1^n, \ldots , u_k^n)\}_{n\ge 1}$ be a sequence in $B(x_1, \ldots , x_k)$ such that $(u_1^n, \ldots , u_k^n)\rightarrow (u_1^0,\ldots ,u_k^0)$. Choose

$$\begin{aligned} \left( v_1^n, \ldots , v^n_k \right) \in S_{G_1, (x_1, \ldots , x_k)} \times S_{G_2, (x_1, \ldots , x_k) } \times \cdots \times S_{G_k, (x_1, \ldots , x_k)} \end{aligned}$$

such that $u^n_i(t)=w_i(v_i^n,t)$ for all $t\in [1,e]$ and $i=1, \ldots , k$. Since $G_i$ is compact valued for all i, $\{v_i^n\}_{n\ge 1}$ has a subsequence which converges to some $v_i^0\in L^{1}([1,e],{\mathbb {R}})$. Denote the subsequence again by $\{v_i^n\}_{n\ge 1}$. It is easy to check that $v_i^0\in S_{G_i(x_1, \ldots , x_k)}$ and $u_i^0(t)=w_i(v_i^0,t)$ for all $t \in [1,e]$. This implies that $u_i^0\in B_i(x_1, \ldots , x_k)$ for all i and so $(u_1^0, \ldots ,u_k^0) \in B(x_1, \ldots , x_k)$. Now, we show that $B(x_1, \ldots , x_k)$ is convex for all $(x_1, \ldots , x_k)\in X^k$. Let $(h_{1}, \ldots , h_k), (h'_1, \ldots , h'_k) \in B(x_1, \ldots , x_k)$. Choose $v_{i}, v'_i \in S_{G_i, (x_1, \ldots , x_k)}$ such that $h_i(t)=w_i(v_i, t)$ and $h'_i(t) = w_i(v'_i, t)$ for almost all $t \in [1,e]$ and $1\le i\le k$. Let $0\le \mu \le 1$. Then, we have

$$\begin{aligned}&\left[ \mu h_i + (1- \mu ) h'_i \right] (t) \\&\quad = \frac{1}{ \Gamma (\alpha _i) } \int _1^t \frac{1}{s} \left( \ln \frac{t}{s} \right) ^{\alpha _i -1} \\&\quad \quad \times \left( I^{\beta _i} \left[ \mu v_i(s) + (1-\mu ) v'_i(s) \right] - \lambda _i x_i(s) \right) \mathrm{d}s \\&\quad \quad - \frac{a_1^i(\eta ) a_2^i(t)}{ b_1^i a_3^i(\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| }\\&\quad \quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i(\eta ) \left( I^{\beta _i} \left[ \mu v_i (s) + ( 1- \mu ) v'_i(s)\right] - \lambda _i x_i(s) \right) \mathrm{d}s \right. \\&\quad \quad \left. - a_3^i(\eta ) \int _1^e \frac{1}{s} a_4^i(e) \left( I^{\beta _i} \left[ \mu v_i(s)+ (1-\mu )v'_i(s) \right] - \lambda _i x_i(s) \right) \mathrm{d}s \right] \\&\quad \quad - \frac{a_2^i(t)}{b_1^i a_3^i(\eta )} \int _1^\eta \frac{1}{s} a_4^i(\eta ) \left( I^{\beta _i} \left[ \mu v_i(s) +(1- \mu ) v'_i(s) \right] - \lambda _i x_i(s) \right) \mathrm{d}s \\&\quad \quad + \frac{a_5^i(t)}{b_4^i \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i(\eta ) \left( I^{\beta _i} \left[ \mu v_i (s) + (1-\mu ) v'_i(s) \right] - \lambda _ix_i(s) \right) \mathrm{d}s \right. \\&\quad \quad \left. - a_3^i(\eta ) \int _1^e \frac{1}{s} a_4^i(e) \left( I^{\beta _i} \left[ \mu v_i(s) +(1-\mu ) v'_i(s) \right] - \lambda _i x_i(s)\right) \mathrm{d}s \right] \\&\quad = w_i \left( \mu v_i + (1- \mu ) v'_i ,t \right) . \end{aligned}$$

Since $S_{G_i,(x_1, \ldots ,x_k)}$ ($G_i$ has convex values) is convex for all $1\le i \le k$,

$$\begin{aligned} \left[ \mu h_i+ (1-\mu ) h_i' \right] \in B_i(x_1, x_2, \ldots , x_k). \end{aligned}$$

Thus,

$$\begin{aligned} \mu (h_{1},\ldots ,h_k)+(1-\mu )(h'_1, \ldots , h'_k )&= (\mu h_1 +(1-\mu ) h'_1, \ldots , \mu h_k + (1-\mu )h'_k)\\&\in B(x_1, \ldots ,x_k). \end{aligned}$$

In this step, we show that B maps bounded sets of $X^k$ into bounded sets. Suppose that $\rho >0$ and

$$\begin{aligned} B_\rho = \left\{ (x_1, \ldots , x_k) \in X^k: \left\| (x_1, \ldots , x_k) \right\| _* \le \rho \right\} . \end{aligned}$$

For $(x_1, \ldots , x_k) \in B_\rho$ and $(h_1, \ldots , h_k) \in B(x_1, \ldots , x_k)$ choose

$$\begin{aligned} (v_1, \ldots , v_k) \in S_{G_1, (x_1, \ldots , x_k ) } \times \ldots \times S_{G_k, (x_1, \ldots , x_k)} \end{aligned}$$

such that $h_i(t)=w_i(v_i,t)$ for almost all $t\in [1,e]$ and $1\le i\le k$. Hence,

$$\begin{aligned} \left| h_i(t) \right|&\le \frac{1}{\Gamma (\alpha _i)} \int _1^t \frac{1}{s} \left( \ln \frac{t}{s} \right) ^{\alpha _i-1} \left( I^{\beta _i} \left| v_i(s) \right| + |\lambda _i| |x_i(s)| \right) \mathrm{d}s\\&\quad + \frac{a_1^i(\eta ) a_2^i(t)}{b_1^i a_3^i(\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| }\\&\quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i (\eta ) \left( I^{\beta _i} \left| v_i(s) \right| + |\lambda _i| |x_i(s)| \right) \mathrm{d}s \right. \\&\quad \left. + a_3^i(\eta ) \int _1^e \frac{1}{s} a_4^i (e) \left( I^{\beta _i} \left| v_i(s) \right| + |\lambda _i| |x_i(s)| \right) \mathrm{d}s \right] \\&\quad + \frac{ a_5^i(t)}{ b_1^i a_3^i(\eta )} \int _1^\eta \frac{1}{s} a_4^i (\eta ) \left( I^{\beta _i} \left| v_i(s) \right| + |\lambda _i| |x_i(s)| \right) \mathrm{d}s \\&\quad + \frac{a_5^i(t)}{b_4^i \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i(\eta ) \left( I^{\beta _i} \left| v_i(s) \right| + |\lambda _i| |x_i(s)| \right) ds \right. \\&\quad \left. + a_3^i(\eta ) \int _1^e \frac{1}{s} a_4^i(e) \left( I^{\beta _i} \left| v_i(u) \right| + |\lambda _i| |x_i(s)|\right) ds \right] \\&\le \frac{ \left\| p_i \right\| \psi \left( \Vert (x_1, \ldots , x_k ) \Vert _* \right) }{ \Gamma (\alpha _i +\beta _i+1)} + \frac{ |\lambda _i| \left\| (x_1, \ldots , x_k) \right\| _*}{\Gamma (\alpha _i + 1)} \\&\quad + \frac{a_1^i (\eta ) b_5^i}{b_1^i a_3^i(\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \times \left[ b_3^i \left( \frac{ \Vert p_i\Vert \psi ( \Vert ( x_1, \ldots , x_k )\Vert _*)(\ln \eta )^{\alpha _i +\beta _i +\gamma _i}}{\Gamma (\alpha _i +\beta _i +\gamma _i+1)} \right. \right. \\&\quad + \frac{|\lambda _i|\Vert (x_1, \ldots , x_k)\Vert _*(\ln \eta )^{ \alpha _i + \gamma _i}}{ \Gamma (\alpha _i +\gamma _i+1)}\\&\quad + \frac{\Vert p_i\Vert \psi (\Vert (x_1, \ldots , x_k)\Vert _*)(\ln \eta )^{\alpha _i +\beta _i- \gamma _i}}{\Gamma (\alpha _i+\beta _i-\gamma _i+1)} \\&\quad \left. + \frac{|\lambda _i|\Vert (x_1, \ldots , x_k)\Vert _*(\ln \eta )^{\alpha _i- \gamma _i}}{ \Gamma (\alpha _i -\gamma _i+1)}\right) \\&\quad + a_3^i (\eta ) \left( \frac{\Vert p_i\Vert \psi (\Vert (x_1, \ldots , x_k)\Vert _*)}{\Gamma (\alpha _i+ \beta _i + \gamma _i+1)} \right. \\&\quad + \frac{|\lambda _i|\Vert (x_1, \ldots ,x_k)\Vert _*}{\Gamma (\alpha _i+\gamma _i+1)} + \frac{\Vert p_i\Vert \psi (\Vert (x_1, \ldots , x_k)\Vert _*)}{\Gamma (\alpha _i+\beta _i-\gamma _i+1)} \\&\quad \left. \left. +\frac{|\lambda _i|\Vert (x_1, \ldots , x_k)\Vert _*}{\Gamma (\alpha _i-\gamma _i+1)} \right) \right] \\&\quad + \frac{b_5^i}{b_1^i a_3^i(\eta )} \left( \frac{\Vert p_i\Vert \psi (\Vert (x_1, \ldots , x_k)\Vert _*)(\ln \eta )^{\alpha _i + \beta _i+\gamma _i }}{\Gamma (\alpha _i +\beta _i +\gamma _i+1)} \right. \\&\quad \left. +\frac{|\lambda _i|\Vert (x_1, \ldots , x_k)\Vert _*(\ln \eta )^{\alpha _i+\gamma _i}}{\Gamma (\alpha _i+\gamma _i+1)} \right. \\&\quad + \frac{\Vert p_i\Vert \psi (\Vert (x_1, \ldots , x_k)\Vert _*)(\ln \eta )^{\alpha _i+\beta _i-\gamma _i}}{ \Gamma (\alpha _i +\beta _i-\gamma _i+1)} \\&\quad \left. +\frac{|\lambda _i|\Vert (x_1, \ldots ,x_k)\Vert _*(\ln \eta )^{ \alpha _i - \gamma _i}}{ \Gamma ( \alpha _i -\gamma _i + 1)}\right) \\&\quad + \frac{b_6^i}{b_4^i \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \times \left[ b_3^i \left( \frac{\Vert p_i\Vert \psi (\Vert (x_1, \ldots , x_k)\Vert _*)(\ln \eta )^{\alpha _i + \beta _i + \gamma _i}}{\Gamma (\alpha _i + \beta _i +\gamma _i + 1)} \right. \right. \\&\quad +\frac{|\lambda _i|\Vert (x_1, \ldots , x_k)\Vert _*( \ln \eta )^{\alpha _i + \gamma _i}}{\Gamma (\alpha _i + \gamma _i+1)} \\&\quad + \frac{\Vert p_i\Vert \psi (\Vert (x_1, \ldots ,x_k)\Vert _*)(\ln \eta )^{\alpha _i +\beta _i -\gamma _i}}{\Gamma (\alpha _i + \beta _i-\gamma _i+1)} \\&\quad \left. + \frac{|\lambda _i|\Vert (x_1, \ldots , x_k)\Vert _*(\ln \eta )^{\alpha _i -\gamma _i}}{ \Gamma (\alpha _i - \gamma _i+1)} \right) \\&\quad + a_3^i(\eta ) \left( \frac{\Vert p_i\Vert \psi (\Vert (x_1, \ldots ,x_k)\Vert _*)}{\Gamma (\alpha _i +\beta _i + \gamma _i+1)} + \frac{|\lambda _i|\Vert (x_1, \ldots ,x_k)\Vert _*}{\Gamma (\alpha _i +\gamma _i + 1)} \right. \\&\quad \left. \left. + \frac{\Vert p_i\Vert \psi (\Vert (x_1, \ldots , x_k)\Vert _*)}{ \Gamma (\alpha _i + \beta _i-\gamma _i+1)} +\frac{|\lambda _i|\Vert (x_1, \ldots , x_k)\Vert _*}{\Gamma (\alpha _i - \gamma _i+1)}\right) \right] \\&= \Vert p_i\Vert \psi _i (\rho ) \left[ \frac{1}{ \Gamma (\alpha _i+\beta _i+1)} \right. \\&\quad + \bigg ( \frac{a_1^i (\eta ) b_5^i}{b_1^i a_3^i(\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } + \frac{b_6^i}{b_4^i \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| }\bigg ) \\&\quad \times \left( b_3^i a_6^i(\eta ) + a_3^i(\eta ) b_7^i \right) \left. + \frac{b_5^i a_6^i(\eta )}{b_1^i a_3^i(\eta ) } \right] \\&\quad + \rho |\lambda _i| \left[ \frac{1}{\Gamma (\alpha _i+1)} \right. \\&\quad + \bigg ( \frac{a_1^i(\eta ) b_5^i}{b_1^i a_3^i(\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } + \frac{b_6^i}{b_4^i \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \bigg ) \\&\quad \times \left( b_3^i a_7^i (\eta ) + a_3^i(\eta ) b_8^i \right) \left. + \frac{b_5^i a_7^(\eta )}{b_1^i a_3(\eta ) } \right] \\&= \Vert p_i\Vert \psi _i(\rho ) \Lambda _1^i+\rho |\lambda _i|\Lambda _2^i, \end{aligned}$$

for all $t\in [1,e]$ and $1\le i \le k$. Thus, $\Vert h_i\Vert \le \Vert p_i\Vert \psi _i(\rho ) \Lambda _1^i + \rho |\lambda _i|\Lambda _2^i$ and so

$$\begin{aligned} \left\| (h_1,\ldots , h_k) \right\| = \sum _{i=1}^k \Vert h_i\Vert \le \sum _{i=1}^k \left( \Vert p_i\Vert \psi _i(\rho ) \Lambda _1^i + \rho | \lambda _i| \Lambda _2^i \right) . \end{aligned}$$

Now, we show that B maps bounded sets to equi-continuous subsets of $X^k$. Let $t_{1},t_{2}\in [1, e]$ with $t_{1}<t_{2}$, $(x_1,\ldots ,x_k)\in B_{\rho }$ and $(h_1, \ldots , h_k) \in B(x_1, \ldots ,x_k)$. Then, we have

$$\begin{aligned} \left| h_i(t_2) - h_i(t_1) \right|&=\left| \frac{1}{\Gamma (\alpha _i)}\int _1^{t_2} \frac{1}{s} \left( \ln \frac{t_2}{s} \right) ^{\alpha _i-1} \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \right. \\&\quad -\frac{1}{\Gamma (\alpha _i)}\int _1^{t_1} \frac{1}{s} \left( \ln \frac{t_1}{s} \right) ^{\alpha _i-1} \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s\\&\quad - \frac{a_1^i(\eta ) a_2^i(t)}{b_1^i a_3^i(\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i (\eta ) \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3^i(\eta ) \int _1^e \frac{1}{s} a_4^i (\eta ) \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \right] \\&\quad + \frac{a_1^i (\eta ) a_2^i(t)}{ b_1^i a_3^i(\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i (\eta ) \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3^i(\eta ) \int _1^e \frac{1}{s} a_4^i(e) \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \right] \\&\quad - \frac{a_2^i(t_2)}{b_1^i a_3^i(\eta )} \int _1^\eta \frac{1}{s} a_4^i(\eta ) \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \\&\quad + \frac{a_2^i(t_1)}{b_1^i a_3^i (\eta )} \int _1^\eta \frac{1}{s} a_4^i(\eta ) \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \\&\quad + \frac{a_5^i(t_2)}{b_4^i \left[ a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i\right] } \\&\quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i (\eta ) \left( I^{\beta _i} v(s) - \lambda _ix_i(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3^i(\eta ) \int _1^e \frac{1}{s} a_4^i(e) \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \right] \\&\quad - \frac{ a_2^i (t_1) }{b_4^i \left[ a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i\right] } \\&\quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i(\eta ) \left( I^{\beta _i} v(s) - \lambda _ix_i(s) \right) \mathrm{d}s \right. \\&\quad \left. \left. - a_3^i (\eta ) \int _1^e \frac{1}{s} a_4^i (e) \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \right] \right| \\&\le \frac{1}{\Gamma (\alpha _i)} \int _1^{t_1} \frac{1}{s} \bigg ( \bigg ( \ln \frac{t_2}{s} \bigg )^{\alpha _i-1} - \bigg ( \ln \frac{t_1}{s} \bigg )^{\alpha _i-1} \bigg ) \\&\quad \times \left( \frac{\Vert p_i\Vert \psi _i(\rho )}{ \Gamma (\beta _i+1)} (\ln (s))^{\beta _i} + \rho | \lambda _i| \right) \mathrm{d}s \\&\quad + \frac{1}{ \Gamma ( \alpha _i)}\int _{t_1}^{t_2} \frac{1}{s} \left( \ln \frac{t_2}{s} \right) ^{\alpha _i-1} \\&\quad \times \left( \frac{\Vert p_i\Vert \psi _i(\rho )}{ \Gamma (\beta _i+1)} (\ln (s))^{\beta _i} + \rho |\lambda _i| \right) \mathrm{d}s \\&\quad + \frac{a_1^i (\eta ) b_5^i \left( (\ln t_2)^{\alpha _i+\beta _i-1}-(\ln t_1)^{\alpha _i+\beta _i-1} \right) }{b_1^i a_3^i(\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i (\eta ) \left( I^{\beta _i} v(s) -\lambda _i x_i(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3^i (\eta ) \int _1^e \frac{1}{s} a_4^i(e) \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \right] \\&\quad + \frac{b_5^i \left( \left( \ln t_2 \right) ^{\alpha _i+\beta _i-1}-(\ln t_1)^{\alpha _i+\beta _i-1}\right) }{b_1^i a_3^i(\eta )} \\&\quad \times \int _1^\eta \frac{1}{s} a_4^i(\eta ) \left( I^{\beta _i} v(s) - \lambda _i x_i(s) \right) \mathrm{d}s \\&\quad + \frac{ b_2^i\left( (\ln t_2)^{\alpha _i + \beta _i-1}-(\ln t_1)^{\alpha _i +\beta _i-1}\right) }{b_4^i \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i(\eta ) \left( I^{\beta _i} v(s) - \lambda _ix_i(s) \right) \mathrm{d}s \right. \\&\quad \left. - a_3^i (\eta ) \int _1^e \frac{1}{s} a_4^i(e) \left( I^{\beta _i} v(s) - \lambda _i x_i(s)\right) \mathrm{d}s \right] , \end{aligned}$$

for each $1\le i \le k$. Obviously, the right-hand side of the above inequality tends to zero independent of $(x_1, \ldots , x_k) \in B_\rho$ as $t_2\rightarrow t_1$. This implies that

$$\begin{aligned} \lim _{t_2\rightarrow t_1} \left| (h_1(t_2)-h_1(t_1), \ldots ,h_k(t_2)-h_k(t_1)) \right| =0. \end{aligned}$$

Hence, using the Arzela–Ascoli theorem, B is completely continuous and since $B(x_1, \ldots , x_k)$ is closed-valued, $B(x_1, \ldots , x_k)\in P_{cp,cv}(X^k)$. Similar as B, $A(x_1,\ldots , x_k)\in P_{cl,bd,cv}(X^k)$ too. Here, we show that B has a closed graph. Let $(u_1^n,\ldots ,u_k^n)\in B(x_1^n,\ldots ,x_k^n)$ for all n such that $(x_1^n,\ldots ,x_k^n)\rightarrow (x_1^0,\ldots ,x_k^0)$ and $(u_1^n,\ldots ,u_k^n)\rightarrow (u_1^0,\ldots ,u_k^0)$. We show that $(u_1^0,\ldots ,u_k^0)\in B(x_1^0,\ldots ,x_k^0)$. For each natural number n, choose

$$\begin{aligned} (v_1^n,\ldots , v^n_k) \in S_{G_1, (x^n_1, \ldots , x^n_k)} \times \ldots \times S_{G_k,(x^n_1, \ldots , x^n_k)} \end{aligned}$$

such that $u^n_i(t)=w_i(v_i^n,t)$ for all $t\in [1,e]$ and $1\le i\le k$. Consider the continuous linear operator

$$\begin{aligned} \left\{ \begin{array}{l} \theta _i:L^{1}([1,e],{\mathbb {R}})\rightarrow X, \\ \theta _i(v)(t)=w_i(v,t). \end{array} \right. \end{aligned}$$

Using Lemma 1.2, $\theta _i o S_{G_i}$ is a closed graph operator. Since $u_i^{n}\in \theta _i(S_{G_i,(x^n_1,\ldots ,x^n_k)})$ for all n, $1\le i\le k$ and $(x_1^n,\ldots ,x_k^n) \rightarrow (x_1^0, \ldots , x_k^0)$, there exists $v_i^{0} \in S_{G_i,(x_1^0,\ldots ,x_k^0)}$ such that $u_i^0(t)=w_i(v_i^0,t)$. Hence, $u_i^0\in B_i(x_1^0, \ldots ,x_k^0)$ for all $1\le i\le k$. This implies that $B_i$ has a closed graph for all $1\le i\le k$ and so B has a closed graph and this show that the operator B is upper semi-continuous. Now, we show that A is a contraction multifunction. Let $(x_1,\ldots ,x_k),(y_1,\ldots ,y_k)\in X^k$ and $(h_1,\ldots ,h_k)\in A(y_1,\ldots ,y_k)$ be given. Then, we can choose

$$\begin{aligned} (v_1, \ldots ,v_k)\in S_{F_1,(y_1,\ldots ,y_k)}\times S_{F_2, (y_1, \ldots ,y_k)} \times \cdots \times S_{F_k,(y_1, \ldots , y_k)} \end{aligned}$$

such that $h_i(t)=w_i(v_i,t)$ for all $t\in [1,e]$ and $i=1,\ldots ,k$. Put

$$\begin{aligned} F_{i,x}&= F_i \big (t,x_1(t), \ldots , x_k(t),\quad I^{\nu _1}x_1(t), \ldots , \quad I^{\nu _k}x_k(t) \big ), \\ F_{i,y}&= F_i \big (t, y_1(t), \ldots , y_k(t), \quad I^{\nu _1}y_1(t), \ldots , I^{\nu _k}y_k(t) \big ). \end{aligned}$$

Since

$$\begin{aligned} H\left( F_{i,x}, F_{i,y} \right) \le m_i(t) \sum _{i=1}^k (|x_i(t)-y_i(t)|+|I^{\nu _i}x_i(t)-I^{\nu _i}y_i(t)|) \end{aligned}$$

for almost all $t\in [1,e]$ and $i=1,\ldots ,k$, there exists

$$\begin{aligned} u_i \in F_i \left( t,x_1(t), \ldots , x_k(t), I^{\nu _1}x_1(t), \ldots , I^{\nu _k}x_k(t) \right) \end{aligned}$$

such that

$$\begin{aligned} \left| v_{i}(t)-u_i \right| \le m_i(t) \sum _{i=1}^k \left( \left| x_i(t) - y_i(t) \right| + \left| I^{\nu _i} x_i(t) - I^{\nu _i} y_i(t) \right| \right) \end{aligned}$$

for almost all $t\in [1,e]$ and $i=1, \ldots , k$. Consider the multifunction $U_i:[1,e] \rightarrow 2^{{\mathbb {R}}}$ defined by

$$\begin{aligned} U_i(t) = \bigg \{ w\in {\mathbb {R}} : \left| v_{i}(t) - w \right| \le m_i(t) \sum _{i=1}^k \left( |x_i(t)-y_i(t)| + |I^{\nu _i}x_i(t)-I^{\nu _i}y_i(t)| \right) \bigg \}, \end{aligned}$$

for almost all $t\in [1,e]$. Since $U_i(t)\cap F_i(t,x_1(t),\ldots ,x_k(t),I^{\nu _1}x_1(t),\ldots ,I^{\nu _k}x_k(t))$ is a measurable multifunction. Thus, we can choose

$$\begin{aligned} v'_i(t) \in F_i \left( t, x_1(t), \ldots , x_k(t), I^{\nu _1} x_1(t), \ldots , I^{\nu _k} x_k(t) \right) \end{aligned}$$

such that

$$\begin{aligned} \left| v_i(t)- v'_i(t) \right| \le m_i(t) \sum _{i=1}^k \left( |x_i(t)- y_i(t)| + |I^{\nu _i} x_i(t) - I^{\nu _i} y_i(t)| \right) . \end{aligned}$$

For each $t\in [1.e]$ and $i=1, \ldots , k$, let us define $h^\prime _i(t)= w_i(v'_i,t)$. Since

$$\begin{aligned} \left| h_i(t)- h'_i(t) \right|&\le \frac{1}{\Gamma (\alpha _i)} \int _1^t \frac{1}{s} \bigg (\ln \frac{t}{s}\bigg )^{\alpha _i-1} \\&\quad \times \left( I^{\beta _i} \left( |v_i(s)- v'_i(s)|\right) + |\lambda _i| |x_i(s)-y_i(s)| \right) \mathrm{d}s \\&\quad + \frac{a_1^i(\eta ) a_2^i(t)}{b_1^i a_3^i(\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \times \left[ b_3^i \int _1^\eta \frac{1}{s} a_4^i (\eta ) \left( I^{\beta _i} \left( |v_i(s) - v'_i(s)|\right) + |\lambda _i| |x_i(s)-y_i(s)| \right) \mathrm{d}s \right. \\&\quad \left. + \, a_3^i (\eta ) \int _1^e \frac{1}{s} a_4^i (e) \left( I^{\beta _i} \left( |v_i(s) - v'_i(s)| \right) + |\lambda _i| |x_i(s)-y_i(s)| \right) \mathrm{d}s\right] \\&\quad + \frac{a_2^o(t) }{b_1^i a_3^i (\eta )} \int _1^\eta \frac{1}{s} a_4^i (\eta ) \left( I^{\beta _i} \left( |v_i(s) -v'_i(s)|\right) + | \lambda _i | |x_i(s)-y_i(s)| \right) \mathrm{d}s\\&\quad +\frac{a_5^i (t)}{b_4^i\left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \times \left[ b_3 \int _1^\eta \frac{1}{s} a_4^i (\eta ) \left( Ii^{\beta _i} \left( |v_i(s) -v'_i(s)| \right) + |\lambda _i| |x_i(s) - y_i(s)| \right) ds \right. \\&\quad \left. + \, a_3^i (\eta ) \int _1^e \frac{1}{s} a_4^i(e) \left( I^{\beta _i} \left( |v_i(s) -v'_i(s)| \right) + |\lambda _i| |x_i(s)-y_i(s)|\right) \mathrm{d}s \right] \\&\le \frac{ \Vert m_i\Vert \Vert (x_1-y_1, \ldots , x_k-y_k)\Vert _* }{ \Gamma (\alpha _i +\beta _i +1)} \sum _{j=1}^k \bigg (1+ \frac{1}{\Gamma (\nu _j+1)} \bigg )\\&\quad + \frac{ |\lambda _i| \Vert (x_1-y_1,\ldots ,x_k - y_k)\Vert _*}{\Gamma (\alpha _i +1)} \\&\quad + \frac{a_1^i (\eta ) b_5^i}{b_1^i a_3^i (\eta ) \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \times \left[ b_3^i \left( a_8^i (\eta ) + a_9^i (\eta ) \right) + a_3^i(\eta ) \left( b_9^i + b_{10}^i\right) \right] \\&\quad + \frac{b_5^i}{b_1^i a_3^i (\eta )} \left[ a_8^i (\eta ) + a_9^i(\eta )\right] \\&\quad + \frac{b_6^i}{b_4^i \left| a_3^i (\eta ) b_2^i - a_1^i(\eta ) b_3^i \right| } \\&\quad \times \left[ b_3^i \left( a_8^i (\eta ) + a_9^i(\eta ) \right) + a_3^i(\eta ) \left( b_9^i + b_{10}^i \right) \right] \\&= \left\| (x_1-y_1, \ldots ,x_k-y_k) \right\| _*\Vert m_i\Vert \Lambda _1^i \sum _{j=1}^k \bigg (1 + \frac{1}{\Gamma (\nu _j+1)} \bigg )\\&\quad + \left\| (x_1-y_1, \ldots , x_k-y_k) \right\| _* |\lambda _i|\Lambda _2^i \\&= \bigg ( \left\| m_i \right\| \Lambda _1^i \sum _{j=1}^k \bigg (1 + \frac{1}{\Gamma (\nu _j + 1)} \bigg ) + |\lambda _i| \Lambda _2^i \bigg ) \left\| (x_1-y_1, \ldots , x_k-y_k) \right\| _*, \end{aligned}$$

where

$$\begin{aligned} a_8^i (\eta ):= & {} \frac{ \ |m_i \Vert \Vert (x_1-y_1,\ldots , x_k-y_k)\Vert _* \sum _{j=1}^k (1 + \frac{1}{\Gamma (\nu _j +1)})(\ln \eta )^{\alpha _i +\beta _i +\gamma _i}}{ \Gamma (\alpha _i + \beta _i + \gamma _i + 1)} \nonumber \\&+ \frac{ \Vert m_i \Vert \Vert (x_1-y_1,\ldots ,x_k-y_k) \Vert _* \sum _{j=1}^k (1 + \frac{1}{\Gamma (\nu _j +1)}))(\ln \eta )^{\alpha _i +\beta _i -\gamma _i}}{\Gamma (\alpha _i +\beta _i - \gamma _i+1)}, \nonumber \\ a_9^i (\eta ):= & {} \frac{ |\lambda _i| \Vert (x_1-y_1, \ldots , x_k-y_k) \Vert _*( \ln \eta )^{\alpha _i + \gamma _i}}{\Gamma (\alpha _i+\gamma _i+1)}\nonumber \\&+ \frac{ |\lambda _i| \Vert (x_1-y_1, \ldots , x_k-y_k) \Vert _*(\ln \eta )^{\alpha _i -\gamma _i}}{\Gamma (\alpha _i-\gamma _i+1)}, \nonumber \\ b_9^i:= & {} \frac{ \Vert m_i \Vert \Vert (x_1-y_1,\ldots , x_k-y_k)\Vert _* \sum _{j=1}^k (1 + \frac{1}{\Gamma (\nu _j + 1)})}{\Gamma (\alpha _i + \beta _i + \gamma _i + 1)} \nonumber \\&+ \frac{ \Vert m_i \Vert \Vert ( x_1 -y_1, \ldots , x_k-y_k) \Vert _* \sum _{j=1}^k(1 +\frac{1}{\Gamma (\nu _j + 1)})}{\Gamma (\alpha _i + \beta _i-\gamma _i + 1)}, \nonumber \\ b_{10}^i:= & {} \frac{ |\lambda _i| \Vert (x_1, \ldots , x_k)\Vert _*}{\Gamma (\alpha _i + \gamma _i + 1)} + \frac{ |\lambda _i| \Vert (x_1-y_1, \ldots , x_k-y_k)\Vert _*}{\Gamma (\alpha _i -\gamma _i+1)}, \end{aligned}$$

(2.6)

we get

$$\begin{aligned} \left\| h_i- h_i' \right\|&\le \bigg ( \Vert m_i\Vert \Lambda _1^i \sum _{j=1}^k \bigg (1 + \frac{1}{\Gamma (\nu _j + 1)} \bigg ) + |\lambda _i| \Lambda _2^i \bigg ) \\&\quad \times \left\| (x_1-y_1, \ldots , x_k-y_k) \right\| _* \end{aligned}$$

for all $i=1, \ldots ,k$. Hence,

$$\begin{aligned} \left\| (h_1, \ldots , h_k) (h'_1, \ldots , h'_k) \right\| _*&= \sum _{i=1}^k \Vert h_i-h'_i \\&\le \sum _{i=1}^k \bigg ( \Vert m_i\Vert \Lambda _1^i \sum _{j=1}^k \bigg (1 + \frac{1}{\Gamma (\nu _j+1)}\bigg ) + |\lambda _i| \Lambda _2^i\bigg ) \\&\quad \times \left\| (x_1-y_1, \ldots , x_k-y_k)\right\| _*. \end{aligned}$$

This implies that

$$\begin{aligned} H\left( A_X, A_Y \right)&\le \sum _{i=1}^k \bigg ( \Vert m_i\Vert \Lambda _1^i\sum _{j=1}^k \bigg (1 + \frac{1}{ \Gamma (\nu _j + 1)}\bigg ) + |\lambda _i|\Lambda _2^i \bigg ) \\&\quad \times \Vert (x_1-y_1, \ldots , x_k-y_k)\Vert _*. \end{aligned}$$

where $A_X=A(x_1,\ldots ,x_k)$ and $A_Y= A(y_1, \ldots ,y_k)$. Since

$$\begin{aligned} \sum _{i=1}^k \bigg ( \Vert m_i\Vert \Lambda _1^i \sum _{j=1}^k \bigg (1 + \frac{1}{ \Gamma (\nu _j + 1) } \bigg ) + |\lambda _i| \Lambda _2^i \bigg ) <1, \end{aligned}$$

A is contraction mapping. Suppose that $(x_1, \ldots , x_k)$ be a possible solution of $\lambda (x_1, \ldots , x_k)\in A(x_1, \ldots ,x_k)+B(x_1, \ldots ,x_k)$ for some real number $\lambda >1$ with $\Vert (x_1, \ldots , x_k)\Vert _*=1$. Then, there exist

$$\begin{aligned} (v_1, \ldots ,v_k) \in S_{F_1,(x_1,\ldots ,x_k)}\times S_{F_2,(x_1,\ldots ,x_k)}\times \cdots \times S_{F_k,(x_1,\ldots ,x_k)} \end{aligned}$$

and

$$\begin{aligned} (v'_1,\ldots ,v'_k)\in S_{G_1,(x_1,\ldots ,x_k)}\times S_{G_2,(x_1,\ldots ,x_k)}\times \cdots \times S_{G_k,(x_1,\ldots ,x_k)} \end{aligned}$$

such that $x_i(t)=\lambda ^{-1} (w_i(v_i,t) + w_i(v'_i,t))$ for each $t\in [1,e]$ and $1\le i\le k$. Clearly, we have

$$\begin{aligned} \Vert x_i\Vert&\le \bigg ( \Vert m_i\Vert \Lambda _1^i \sum _{j=1}^k \bigg ( 1 + \frac{1}{\Gamma (\nu _j+1)} \bigg ) + |\lambda _i|\Lambda _2^i \bigg ) \Vert (x_1, \ldots , x_k)\Vert _*\\&\quad + \Vert p_i \Vert \psi _i(\Vert (x_1, \ldots , x_k)\Vert _*) \Lambda _1^i + |\lambda _i| \Lambda _2^i \Vert (x_1, \ldots , x_k)\Vert _*. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert (x_1, \ldots , x_k)\Vert _*\le \frac{ \sum _{i=1}^k \Vert p_i\Vert \psi _i( \Vert (x_1, \ldots , x_k)\Vert _*)\Lambda _1^i}{1 - \sum _{i=1}^k \left( \Vert m_i\Vert \Lambda _1^i \sum _{j=1}^k (1+\frac{1}{ \Gamma (\nu _j + 1)})+2|\lambda _i|\Lambda _2^i\right) }. \end{aligned}$$

Substituting $\Vert (x_1, \ldots ,x_k)\Vert _*=r$ in the above inequality, we have

$$\begin{aligned} r \le \frac{ \sum _{i=1}^k \Vert p_i\Vert \psi _i(r) \Lambda _1^i}{1-\sum _{i=1}^k \left( \Vert m_i\Vert \Lambda _1^i\sum _{j=1}^k (1 + \frac{1}{\Gamma (\nu _j + 1)})+2| \lambda _i|\Lambda _2^i\right) }, \end{aligned}$$

which is a contradiction. Consequently, by the Lemma 1.3, there exsist $(x_1, \ldots ,x_k)\in B[0,r]$ such that $(x_1, \ldots ,x_k)\in A (x_1, \ldots ,x_k) + B(x_1,\ldots ,x_k)$ which is a solution of $k-$dimensional system of fractional differential inclusions. This completes the proof. $\square$

3 Example

Here, we give an example to illustrate our results.

Example 3.1

Consider the system of Langevin Hadamard-type fractional differential inclusions similar to (1.1) with 2-dimensional

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle D^{\frac{3}{2}} \left( D^\frac{1}{2} + \pi ^{-4} \right) u(t) &{} \in F_1\left( \displaystyle t, u(t), v(t), I^{\frac{1}{4}}u(t), I^{\frac{1}{3}} v(t) \right) \\ &{} \quad + \, G_1 \left( t, u(t), v(t) \right) , \\ \displaystyle D^{\frac{5}{4}} \left( D^\frac{3}{4} + \frac{1}{75} \right) v(t) &{} \in F_1 \left( \displaystyle t, u(t), v(t), I^{\frac{1}{4} } u(t), I^{\frac{1}{3} } v(t) \right) \\ &{} \quad + \, G_1 \left( t, u(t), v(t) \right) , \end{array} \right. \end{aligned}$$

(3.1)

with condition

$$\begin{aligned} \left\{ \begin{array}{l} u(t)\big |_{t \rightarrow 1^+} = v(t)\big |_{t \rightarrow 1^+}=0,\\ I^{\frac{1}{3}} u(2) + D^{\frac{1}{3}} u(2)=0, \qquad I^{\frac{1}{2}} v(2) + D^{\frac{1}{2}} v(2)=0,\\ I^{\frac{1}{3}} u(e) + D^{\frac{1}{3}} u(e)=0, \qquad I^{\frac{1}{2}} u(e) + D^{\frac{1}{2}} u(e)=0, \end{array}\right. \end{aligned}$$

where $F_1, F_2: [1,e]\times {\mathbb {R}}^4\rightarrow P({\mathbb {R}})$ and $G_1, G_2: [1,e]\times {\mathbb {R}}^2\rightarrow P({\mathbb {R}})$ are multivalued maps given by

$$\begin{aligned} F_1 \left( t, x_1, x_2, x_3, x_4 \right)&= \left[ -1, \frac{e^{t-e} \sin x_1}{150 \pi } + \frac{t}{e^8} \cos x_2 \right. \\&\quad \left. + \frac{|x_3|}{ \cosh 7(1+|x_3|)} + \frac{2x_4^2}{10^3(1 + x_4^2)}\right] , \\ G_1 \left( t, x_1, x_2, x_3, x_4 \right)&= \left[ e^{-|x_1|} -\frac{x_2^2}{1 + x_2^2} + \cos t, 2t + \frac{|x_1|}{ 1 +|x_1|} + \sin y + t^2\right] \end{aligned}$$

and

$$\begin{aligned} F_2 \left( t, x_1, x_2, x_3, x_4 \right)&= \left[ 0, \frac{e^t}{25 \pi ^5} \bigg ( \frac{ |x_1| + |x_2| + |x_3| + |x_4|}{1 + |x_1| + |x_2| + |x_3| + |x_4|}\bigg )\right] , \\ G_2 \left( t , x_1, x_2, x_3, x_4 \right)&= \left[ \frac{x_1}{4(1 + x_1)} + \frac{x_2}{1 + x_2}+ 2 + t, \sin x_1 + \cos x_2 + 4t\right] . \end{aligned}$$

Here, $k=2$, $\alpha _1=\frac{1}{2}$, $\alpha _2= \frac{3}{4}$, $\beta _1 = \frac{3}{2}$, $\beta _2 = \frac{5}{4}$, $\gamma _1 = \frac{1}{3}$, $\gamma _2 = \frac{1}{2}$, $\nu _1 =\frac{1}{4}$, $\nu _2=\frac{1}{3}$, $\eta = 2$, $\lambda _1=\pi ^{-4}$ and $\lambda _2 = \frac{1}{75}$. Clearly, we have

$$\begin{aligned} \left\| G_1 \left( t, x_1, x_2 \right) \right\|&=\sup \left\{ |v| : v\in G_1\left( t, x_1, x_2 \right) \right\} \le 17,\\ \left\| G_2 \left( t, x_1, x_2 \right) \right\|&= \sup \left\{ |v| : v \in G_2 \left( t, x_1, x_2 \right) \right\} \le 14, \\ \left\| F_2 \left( t, x_1, x_2, x_3, x_4 \right) \right\|&= \sup \left\{ |v| : v\in F_2 \left( t, x_1, x_2, x_3, x_4 \right) \right\} \\&\le \frac{e^t}{25 \pi ^5}, \\ \left\| F_1 \left( t, x_1, x_2, x_3, x_4 \right) \right\|&=\sup \left\{ |v| : v\in F_1 \left( t, x_1, x_2, x_3, x_4 \right) \right\} \\&\le \frac{e^{t-e}}{150\pi } + \frac{t}{e^8} + \frac{1}{\cosh 7} + \frac{2}{10^3}. \end{aligned}$$

Consider, $p_1(t)=1$, $p_2(t)=1$, $\psi _1(t)=17$, $\psi _2(t)=14$,

$$\begin{aligned} m_1(t) = \frac{e^{t-e}}{ 150\pi } +\frac{t}{e^8}+\frac{1}{\cosh 7}+\frac{2}{10^3}, \end{aligned}$$

$m_2(t)=\frac{e^t}{25\pi ^5}$. Using the given data, it is found that $\Lambda _1^1\approx 6.799$, $\Lambda _1^2\approx 6.93$, $\Lambda _2^1\approx 17.93$, $\Lambda _2^2\approx 15.8$, $\Delta _1\approx 1.54$, $\Delta _2\approx 2.05$ and

$$\begin{aligned} \sum _{i=1}^2 \bigg ( \Vert m_i\Vert \Lambda _1^i \sum _{j=1}^2 \bigg ( 1 +\frac{1}{\Gamma (\nu _j+1)} \bigg ) + |\lambda _i| \Lambda _2^i \bigg )&= \frac{5}{1000} \times 6.799 \times 4.22 + \frac{17.93}{\pi ^{4}}\\&\quad + \frac{1}{1000} \times 6.93 \times 4.22 + \frac{15.8}{75}\\&= 0.66 <1 \end{aligned}$$

and

$$\begin{aligned}&\frac{ \sum _{i=1}^2 \Vert p_i\Vert \psi _i(r) \Lambda _1^i}{1 -\sum _{i=1}^2 \left( \Vert m_i\Vert \Lambda _1^i \sum _{j=1}^2 ( 1 + \frac{1}{\Gamma (\nu _j+1)}) + 2 |\lambda _i|\Lambda _2^i\right) } \\&\quad = \frac{17 \times 6.97 + 14 \times 6.93}{1 - 0.66}=633.852. \end{aligned}$$

Thus, by the Theorem 2.2, the 2-dimensional system of fractional differential inclusions 3.1 has a solution on B[0, 633.852].