Application of the fixed point theorems on the existence of solutions for q-fractional boundary value problems

1.   Introduction

The classical fractional calculus is one of the branches of applied mathematics which has merged in pure mathematics. In fact, the topic of fractional differential equations and inclusions, as one of the subjects of the fractional calculus, is an important and effective field of research which the basic techniques of the functional analysis and fixed point theory were used to proving the existence and uniqueness of solutions for this kind of differential equations and inclusions. On the other hand, the extensiveness and importance of this topic has been caused to publish the many works and papers by other researchers (for example, see ^{[2,3,5,7,9,12,13,14,15,19,32,33,34]}.

Later, $q-$difference calculus or quantum calculus, as a generalization of the classical calculus, has gained considerable attention of researchers and mathematicians. The first work on the subject of $q-$difference calculus dates back to Jackson's works ^[25].

In recent years, many researchers have studied and published various and distinct papers on the existence theory of fractional $q-$difference equations and inclusions (for examples, see ^{[1,4,6,17,18,20,21,22,23,26,29,30,35,37,39,41]}).

In ^[8], Bashir Ahmad et al. studied the existence of solutions for the nonlocal boundary value problem of fractional $q-$difference equation

where, $^cD_q^\alpha$ is the fractional $q-$derivative of the Caputo type and $\alpha_i, \beta_i, \gamma_i \in \mathbb{R}$. Authors in that paper, by applying the Banach contraction principle, Krasnoselskii's fixed point theorem and the Leray-Schauder nonlinear alternative studied the existence results.

In ^[40], Zhao et al. dealt with the following nonlinear fractional $q-$difference equation with the nonlocal $q-$integral boundary value conditions:

where, $D_q^\alpha$ is the fractional $q-$derivative of Riemann-Liouville type and $\mu > 0$. They studied the existence of solutions for the above problem by using the generalized Banach contraction principle, the monotone iterative method and Krasnoselskii's fixed point theorem.

In 2015, Alsaedi, Ntouyas and Ahmad investigated the fractional $q-$difference inclusion with nonlocal and substrip type boundary conditions

where $^{c}D_q^\nu$ denotes the Caputo fractional $q-$derivative of order $\nu$ ^[10].

Motivated by the above papers, in this paper, we discuss the existence of solutions for fractional $q-$difference equation

where, $^cD_q^\alpha$ denotes the fractional $q-$derivative of the Caputo type of order $\alpha$ and $\alpha \in (2, 3]$ and $f:[0, 1]\times \mathbb{R}^2 \to \mathbb{R}$ is a continuous mapping.

Also, we study the existence of solutions for the following fractional $q-$difference inclusion

where $F: [0, 1]\times \mathbb{R}^2 \to \mathcal{P}(\mathbb{R})$ is a compact multivalued map.

The rest of the paper is organized as follows: In section 2, we state some important definitions and lemmas on the fundamental concepts of $q-$fractional calculus and fixed point theory. In section 3, we state main results on the existence of solutions for $q-$fractional boundary value problem (1.1). Section 4 contains some main theorems on the existence of solutions for $q-$fractional boundary value problem (1.2). Finally, in section 5, we give some illustrative examples to show the validity and applicability of our results.

2.   Preliminaries

In this section, we first recall some known definitions and lemmas about $q-$fractional calculus. For more details in this regard, see ^{[22,25,26,27]}.

Let $0 < q < 1$. For each $a\in \mathbb{R}$ we define $[a]_q = \frac{1-{q^a}}{1-q}$. The $q-$analogue of the power function $(a-b)^n$ with $n\in \mathbb{N}_0: = \{ 0, 1, 2, \ldots \}$ is given by

In general, if $\alpha$ is real number then

It is clear that if $b = 0$, then $a^{(\alpha)} = a^\alpha$. The $q-$Gamma function is defined by

where $x\in \mathbb{R}\setminus \{ 0, -1, -2, \ldots \}$. Also, we have $\Gamma_q(x+1) = [x]_q\Gamma_q(x)$. The $q-$derivative of a real-valued function $f$ is defined by

The $q-$derivative of higher order of a function $f$ is given by

The $q-$integral of a function $f$ defined in the interval $[0, b]$ is given by

such that the sum is absolutely convergent. Now, if $a\in [0, b]$, then $q-$integral of $f$ from $a$ to $b$ is

provided that the series converges. Similar to $q-$derivatives, an operator $I_q^n$ is given by

Note that $(D_qI_qf)(x) = f(x)$ and if $f$ is continuous at $x = 0$, then $(I_qD_qf)(x) = f(x)-f(0)$.

Let $\alpha \geq 0$ and $f$ be a function defined on $[0, 1]$. The fractional $q-$integral of the Riemann-Liouville type is $(I_q^0f)(x) = f(x)$ and for $x\in [0, 1]$

The fractional $q-$derivative of the Caputo type of order $\alpha \geq 0$ is defined by

where $[\alpha]$ is the smallest integer greater than or equal to $\alpha$ ^[16]. In the following lemmas, we bring some important properties of these $q-$operators.

Lemma 2.1. ^[17] Let $\alpha, \beta \geq 0$ and $f$ be a function defined on $[0, 1]$. Then

(i) $(I_q^\beta I_q^\alpha f)(x) = (I_q^{\alpha +\beta }f)(x),$

(ii) $(D_q^\alpha I_q^\alpha f)(x) = f(x)$.

Lemma 2.2. ^[17] Let $\alpha > 0$ and $n$ be a positive integer. Then the following equality holds:

Now, we recall some definitions and lemmas on the multifunctions and fixed point theory which are needed in the sequal.

Consider the set $X$ with the metric $d$. Denote by $\mathcal{P}(X)$, $2^X$, $\mathcal{P}_{cl}(X)$, $\mathcal{P}_{cp}(X)$, $\mathcal{P}_{b}(X)$ and $\mathcal{P}_{cv}(X)$, the class of all subsets, the class of all nonempty subsets of $X$, the class of all closed subsets of $X$, the class of all compact subsets of $X$, the class of all bounded subsets of $X$ and the class of all convex subsets of $X$, respectively. Let $F:X\to 2^X$ be a multivalued map. If $u\in Fu$ then we say that $u\in X$ is a fixed point of the multifunction $F$ ^[16,24,38]. The fixed point set of the multivalued operator $F$ will be denoted by ${\it{Fix}} (F)$.

A multifunction $F:[0, 1] \to \mathcal{P}_{cl}(\mathbb{R})$ is said to be measurable whenever the function $(t)\mapsto d(y, F(t)) = \inf \{ \Vert y-v\Vert:\; v\in F(t)\}$ is measurable for all $y\in \mathbb{R}$ ^[16,24]. The Pompeiu-Hausdorff metric $H_d:2^X\times 2^X \to [0, \infty)$ is defined by

where $d(a, B) = \inf_{b\in B}d(a, b),$ $d(A, b) = \inf_{a\in A}d(a, b)$ ^[16,24]. Then the space $(\mathcal{P}_{cl, b}(X), H_d)$ is a metric space and $(\mathcal{P}_{cl, b}(X), H_d)$ is a generalized metric space, where $\mathcal{P}_{cl, b}(X)$ is the set of closed and bounded subsets of $X.$ ^[16,24].

A multi-valued mapping $F:X\to {\mathcal P}_{cl}(X)$ is called a contraction if there exists $\gamma\in (0, 1)$ such that $H_{d}(F(x), F(y))\leq \gamma d(x, y)$ for all $x, y\in X$ ^[16].

$F$ is called upper semi-continuous (u.s.c.) on $X$ if for each $x_0 \in X,$ the set $F(x_0)$ is a nonempty closed subset of $X$, and if for each open set $N$ of $X$ containing $F(x_0),$ there exists an open neighborhood $\mathcal{N}_0$ of $x_0$ such that $F(\mathcal{N}_0) \subseteq N$ ^[16]. The operator $F$ is said to be completely continuous if $F(\mathbb{B})$ is relatively compact for every $\mathbb{B} \in \mathcal P_{b} (X).$

An element $x\in X$ is called an endpoint of a multifunction $F:X\to {\mathcal P}(X)$ whenever $Fx = \{ x\}$ ^[11]. Also, we say that $F$ has an approximate endpoint property whenever $\inf_{x\in X}\sup_{y\in Fx}d(x, y) = 0$ ^[11]. A real-valued function $f: \mathbb{R} \to \mathbb{R}$ is called upper semi-continuous whenever $\lim \sup_{n\to \infty }f(\lambda_n)\leq f(\lambda)$ for all sequence $\{ \lambda_n \}_{n\geq 1}$ with $\lambda_n \to \lambda$.

We denote by $\Psi$, the family of nondecreasing functions $\psi: [0, \infty)\to [0, \infty)$ such that $\sum_{n = 1}^\infty \psi^n(t) < \infty$ for all $t > 0$ ^[36]. It is known that $\psi(t) < t$ for all $t > 0$ ^[36]. In 2012, Samet, Vetro and Vetro introduced the notion of $\alpha$-$\psi$-contractive type mappings ^[36]. We say that the selfmap $T: X\to X$ is an $\alpha$-$\psi$-contraction whenever $\alpha (u, v)d(Tu, Tv)\leq \psi(d(u, v))$ for all $u, v\in X$ ^[36]. Also, the selfmap $T$ is called $\alpha$-admissible whenever $\alpha(u, v)\geq 1$ implies $\alpha (Tu, Tv)\geq 1$ ^[36]. We say that $X$ has the property $(B)$ whenever for each sequence $\{u_n\}$ in $X$ with $\alpha(u_n, u_{n+1})\geq 1$ for all $n\geq 1$ and $u_n\to u$, we have $\alpha(u_n, u)\geq 1$ for all $n$ ^[36].

In 2013, Mohammadi, Rezapour and Shahzad generalized this notion to multifunctions ^[31]. A multifunction $F: X \to \mathcal{P}_{cl, b}(X)$ is called $\alpha - \psi -$contraction whenever

for each $u, v\in X$ ^[31]. Similarly, the space $X$ has the property $(C_\alpha)$ whenever for each sequence $\{ u_n \}$ in X with $\alpha (u_n, u_{n+1}) \geq 1$ for all $n\in \mathbb{N}$, there exists a subsequence $\{ u_{n_k} \}$ of $\{ u_n \}$ such that $\alpha (u_{n_k}, u)\geq 1$ for all $k\in \mathbb{N}$. The multi-valued map $F$ is $\alpha -$admissible whenever for each $u\in X$ and $v\in Fu$ with $\alpha (u, v)\geq 1$, we have $\alpha (v, w)\geq 1$ for all $w\in Fv$ ^[31].

Our main results based on the following fixed point theorems.

Theorem 2.1. (^[36]) Let $(X, d)$ be a complete metric space, $\psi \in \Psi$, $\alpha : X\times X \to \mathbb{R}$ a map and $T$ an $\alpha -$admissible and $\alpha -\psi -$contractive selfmap on $X$ such that $\alpha (x_0, Tx_0)\geq 1$, for some $x_0 \in X$. If $X$ has the property $(B)$, then $T$ has a fixed point.

Theorem 2.2. (^[28,38], Krasnoselskii) Let $M$ be a closed, bounded, convex and nonempty subset of a Banach space $X$. Let $A$ and $B$ be two operators mapping $M$ to $X$ such that:

(i) $Ax+By\in M$ whenever $x, y\in M$;

(ii) $A$ is compact and continuous;

(iii) $B$ is a contraction mapping.

Then there exists $z\in M$ such that $z = Az+Bz$.

Theorem 2.3. (^[31]) Let $(X, d)$ be a complete metric space, $\alpha : X\times X \to [0, \infty)$ a map, $\psi \in \Psi$ a strictly increasing map, $F: X \to \mathcal{P}_{cl, b}(X)$ an $\alpha$-admissible $\alpha$-$\psi$-contractive multifunction and $\alpha (u_0, u_1)\geq 1$ for some $u_0 \in X$ and $u_1 \in Fu_0$. If the space $X$ has the property $(C_\alpha)$, then $F$ has a fixed point.

Theorem 2.4. (^[11]) Let $(X, d)$ be a complete metric space and $\psi: [0, \infty) \to [0, \infty)$ be an upper semi-continuous function such that $\psi(t) < t$ and $\liminf_{t\to \infty}(t-\psi(t)) > 0$ for all $t > 0$. Suppose that $T : X\to \mathcal{P}_{cl, b}(X)$ is a multifunction such that $H_d(Tx, Ty)\leq \psi(d(x, y))$ for all $x, y \in X$. Then $T$ has a unique endpoint if and only if $T$ has approximate endpoint property.

3.   Existence results for fractional $q-$difference equation

Let $X = \{ u: u, D_qu\in C([0, 1], \mathbb{R})\}$. Then $X$ is a Banach space via the norm

Put

Lemma 3.1. Let $y\in C([0, 1], {\mathbb R}).$ Then the integral solution of the $q-$fractional boundary value problem

is given by

Proof. Choose the constants $c_0$, $c_1$ and $c_2 \in \mathbb{R}$ such that

Thus, we have

By using the boundary value conditions, we find that $c_0 = 0$ and

and

By substituting the values of $c_i$'s in Eq (3.3), we obtain the $q-$integral equation (3.2). The converse follows by direct computation. The proof is completed.

In view of the above lemma, we define an operator $S:X\to X$ as follows:

It is evident that the solution of the problem (1.1) is a fixed point of an operator $S$; that is $Su = u$.

Now, we are ready to prove our main results.

Theorem 3.1. Let $\psi \in \Psi,$ $\xi: \mathbb{R}^2 \times \mathbb{R}^2\to \mathbb{R}$ be a map and $f: [0, 1] \times {\mathbb R}^2 \to {\mathbb R}$ a continuous function. Suppose that:

$(H1)$ For all $u_1, u_2, v_1, v_2 \in {\mathbb R}$,

with $\xi \Big((u_1(t), v_1(t)), (u_2(t), v_2(t))\Big) \geq 0$ for $t\in [0, 1]$.

$(H2)$ There exists $u_0 \in {\mathbb R}$ such that $\xi \Big((u_0(t), D_qu_0(t)), (Fu_0(t), D_q(Fu_0(t)))\Big) \geq 0$ for all $t\in [0, 1]$ and $\xi \Big((u(t), D_qu(t)), (v(t), D_qv(t))\Big) \geq 0$ implies

for all $t\in [0, 1]$ and $u, v\in {\mathbb R}$.

$(H3)$ For each convergent sequence $\{ u_n \}_{n\geq 1}$ in ${\mathbb R}$ with $u_n \to u$ and

for all $n$ and $t\in [0, 1]$, we have $\xi \Big((u_n(t), D_qu_n(t)), (u(t), D_qu(t))\Big) \geq 0$.

Then the fractional $q-$difference Eq (1.1) has at least one solution.

Proof. Let $u, v\in {\mathbb R}$ with $\xi ((u(t), D_qu(t)), ((v(t), D_q v(t))) \geq 0$ for all $t\in [0, 1]$. Then, we get

and

Hence

Now, define the function $\alpha : {\mathbb R}\times {\mathbb R} \to [0, \infty)$ as follows

By definition of the above function, it is clear that

for each $u, v\in {\mathbb R}$. This means that $S$ is an $\alpha -\psi -$contractive operator. Also, it is easy to see that $S$ is an $\alpha -$admissible and $\alpha (u_0, Su_0)\geq 1$. Suppose that $\{ u_n \}_{n\geq 1}$ is a sequence in ${\mathbb R}$ with $u_n \to u$ and $\alpha (u_n, u_{n+1}) \geq 1$ for all $n$. By definition of the function $\alpha$, we have

Thus by the hypothesis, $\xi \Big((u_n(t), D_qu_n(t)), (u(t), D_qu(t))\Big) \geq 0$. This shows that for all $n$, $\alpha (u_n, u) \geq 1$. So ${\mathbb R}$ has the property $(B)$. Finally, Theorem 2.1 implies that the operator $S$ has fixed point $u^*\in {\mathbb R}$ which is the solution for the $q-$fractional problem (1.1). This completes the proof.

Theorem 3.2. Let $f:[0, 1]\times \mathbb{R}^2 \to \mathbb{R}$ be a continuous function. Suppose that:

$(H4)$ There exists a continuous function $L: [0, 1] \to \mathbb{R}$ such that for each $t\in [0, 1]$ and for all $u_i, v_i\in \mathbb{R}$, $i = 1, 2,$ we have

$(H5)$ There exist a continuous function $\mu : [0, 1] \to \mathbb{R}^+$ and a non-decreasing continuous function $\psi : [0, 1] \to \mathbb{R}^+$ such that

Then, the fractional $q-$difference equation (1.1) has at least one solution on $[0, 1]$ if

where $\Vert L \Vert = \sup\limits_{t\in [0, 1]} \vert L (t)\vert$ and $\Lambda_1, \Lambda_2$ are given by Eq (3.1).

Proof. Define $\Vert \mu \Vert = \sup\limits_{t\in [0, 1]} \vert \mu (t)\vert$ and choose a suitable constant $r$ such that

where $\Delta_i$'s are given by Eq (3.1). We consider the set $B_r = \{ u\in X: \; \Vert u\Vert \leq r \}$, where $r$ is given in Eq (3.6). It is clear that $B_r$ is a closed, bounded, convex and nonempty subset of the Banach space $X$. Now, define two operators $S_1$ and $S_2$ on $B_r$ as follows:

and

for each $t\in [0, 1]$. Put $a = \sup_{u\in X} \psi (\Vert u\Vert)$. For $u, v\in B_r$, we have

Also,

Hence $\Vert S_1u+S_2v\Vert \leq r$ and so, $S_1u+S_2v\in B_r$.

Clearly, the continuity of $S_1$ is follows from the function $f$. Also, for each $u\in B_r$, we have

and

Thus $\Vert S_1u\Vert \leq \Big\{ \frac{1}{\Gamma_q (\alpha +1)} + \frac{1}{\Gamma_q (\alpha)} \Big\} \Vert \mu \Vert \psi(\Vert u\Vert).$ This proves that the operator $S_1$ is uniformly bounded on $B_r$.

Now, we show that the operator $S_1$ is compact on $B_r$. For each $t_1, t_2 \in [0, 1]$ with $t_1 < t_2$, one can write

It is seen that $\vert (S_1u)(t_2)-(S_1u)(t_1)\vert \to 0$ as $t_2 \to t_1$. Also, we have

Again, we see that $\vert (D_qS_1u)(t_2)-(D_qS_1u)(t_1)\vert \to 0$ as $t_2 \to t_1$. Hence $\Vert (S_1u)(t_2)-(S_1u)(t_1) \Vert$ tends to zero as $t_2 \to t_1$. Thus, $S_1$ is equicontinuous and so $S_1$ is relatively compact on $B_r$. Consequently, the Arzelá-Ascoli theorem implies that $S_1$ is a compact operator on $B_r$.

Finally, we prove that $S_2$ is a contraction mapping. For each $u, v\in B_r$, we have

Also,

Hence, we obtain

Thus, $\Vert S_2u-S_2v\Vert \leq \Vert L\Vert (\Delta_1 + \Delta_2)\Vert u-v\Vert$ or $\Vert S_2u -S_2v \Vert \leq k \Vert u-v\Vert$. Thus $S_2$ is contraction on $B_r$ as $k < 1$. Therefore, all the assumptions of Theorem 2.2 are satisfied. Hence, Theorem 2.2 implies that the $q-$fractional boundary value problem (1.1) has at least one solution on $[0, 1]$.

4.   Existence results for fractional $q-$difference inclusion

In this section, we prove our main results about the existence of solutions for fractional $q-$difference inclusion (1.2).

Definition 4.1. A function $u\in C([0, 1], {\mathbb R})$ is called a solution for the fractional $q-$difference inclusion (1.2) whenever it satisfies the boundary value conditions and there exists a function $v\in L^1([0, 1])$ such that $v(t)\in F(t, u(t), D_qu(t))$ for almost all $t\in [0, 1]$ and

for all $t\in [0, 1]$.

Let $X$ be a Banach space with the norm defined in the last section. For each $u\in X$, the set of selections of the operator $F$ is defined by

Also, we define the operator $\mathcal{N}: X\to \mathcal{P}(X)$ by

where

Theorem 4.1. Let $F: [0, 1] \times \mathbb{R}^2 \to \mathcal{P}_{cp}(\mathbb{R})$ be a multifunction. Suppose that:

$(H6)$ The operator $F$ is integrably bounded and $F(\cdot, u, v): [0, 1] \to \mathcal{P}_{cp}(\mathbb{R})$ is measurable for all $u, v\in \mathbb{R}$.

$(H7)$ Assume that there exists $m\in C([0, 1], [0, \infty))$ such that

for all $t\in [0, 1]$ and $u_i, u'_i \in \mathbb{R} \; (i = 1, 2)$ and $\psi \in \Psi$. The constants $\Lambda_1$ and $\Lambda_2$ are given by Eq (3.1).

$(H8)$ There exists a function $\xi : \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$ with $\xi \Big((u_1, u'_1), (u_2, u'_2)\Big) \geq 0$ for $u_i, u'_i \in \mathbb{R}\; (i = 1, 2)$.

$(H9)$ If $\{ u_n \}_{n\geq 1}$ is a sequence in $X$ such that $u_n \to u$ and $\xi \Big((u_n(t), D_qu_n (t)), (u(t), D_qu(t){)}\Big) \geq 0$ for all $t\in [0, 1]$, then there exists a subsequence $\{ u_{n_j}\}_{j\geq 1}$ of $\{ u_n \}$ such that

for all $t\in [0, 1]$.

$(H10)$ There exist $u_0\in X$ and $h\in \mathcal{N}(u_0)$ such that $\xi \Big((u_0(t), D_qu_0 (t)), (h(t), D_qh(t){)}\Big) \geq 0$, for all $t\in [0, 1]$, where the operator $\mathcal{N}:X \to \mathcal{P}(X)$ is given by Eq (4.1).

$(H11)$ For each $u\in X$ and $h\in \mathcal{N}(u)$ with $\xi \Big((u(t), D_qu (t)), (h(t), D_qh(t))\Big) \geq 0$, there exists $w\in \mathcal{N}(u)$ such that $\xi \Big((h(t), D_qh (t)), (w(t), D_qw(t){)}\Big) \geq 0$.

Then the fractional $q-$difference inclusion (1.2) has a solution.

Proof. It is evident that the fixed point of the operator $\mathcal{N}: X \to \mathcal{P}(X)$ is a solution of the inclusion problem (1.2). Since the multivalued map $t\mapsto F(t, u(t), D_qu(t))$ is measurable and it closed-valued for all $u\in X$, so $F$ has measurable selection and the set $S_{F, u}$ is not empty.

We prove that $\mathcal{N}(u)$ is a closed subset of $X$ for all $u\in X$, i.e., $\mathcal{N}(u)\in \mathcal{P}_{cl}(X)$. For this, let $\{ u_n \}_{n\geq 1}$ be a sequence in $\mathcal{N}(u)$ which converges to $u$. We should prove that $u\in \mathcal{N}(u)$. For each natural number $n$, there exists $v_n \in S_{F, u}$ such that

for almost all $t\in[0, 1]$.

That $F$ has compact values, we pass into a subsequence (if necessary) to obtain that a subsequence $\{ v_n \}_{n\geq 1}$ converges to some $v\in L^1([0, 1])$. Thus $v\in S_{F, u}$ and we get

for each $t\in [0, 1]$. This shows that $u\in \mathcal{N}(u)$; that is, the operator $\mathcal{N}$ is closed-valued. Now, since $F$ is a multifunction with compact values, it is easy to check that $\mathcal{N}(u)$ is bounded set in $X$ for all $u\in X$. In the next step, we show that the operator $\mathcal{N}$ is an $\alpha -\psi -$contractive multivalued map. For this purpose, we define a function $\alpha : X\times X \to [0, \infty)$ by

for all $u, u'\in X$. Let $u, u'\in X$ and $h_1 \in \mathcal{N}(u')$. We choose $v_1 \in S_{F, u'}$ such that

for all $t\in [0, 1]$. Since by (4.2), we have

for all $u, u'\in X$ with $\xi \Big((u(t), D_qu(t)), (u'(t), D_qu'(t))\Big) \geq 0$ for almost all $t\in [0, 1]$, so there exists $w\in F(t, u(t), D_qu(t))$ such that

Now, consider the multivalued map $A: [0, 1] \to \mathcal{P}(\mathbb{R})$ which is given by

for all $t\in [0, 1]$. Clearly, the multifunction $A(\cdot) \cap F(\cdot, u(\cdot), D_qu(\cdot))$ is measurable, because $v_1$ and $\varphi = m \psi (\vert u-u' \vert + \vert D_qu - D_qu' \vert) \Big(\frac{1}{\Lambda_1 + \Lambda_2 }\Big)$ are measurable. In this step, we can choose $v_2 \in F(t, u(t), D_qu(t))$ such that

for all $t\in [0, 1]$. Define the element $h_2 \in \mathcal{N}(u)$ by follows:

for all $t\in [0, 1]$. Letting $\sup_{t\in [0, 1]}\vert m(t)\vert = \Vert m\Vert$, we have

and

for all $t\in [0, 1]$. Hence, we obtain

Therefore $\alpha (u, u') H_d (\mathcal{N}(u), \mathcal{N}(u')) \leq \psi (\Vert u-u' \Vert)$ for all $u, u'\in X$. This means that $\mathcal{N}$ is an $\alpha -\psi -$contractive multivalued mapping. Now, let $u\in X$ and $u'\in \mathcal{N}(u)$ be such that $\alpha (u, u')\geq 1$. Thus, by definition of $\alpha$, we have $\xi \Big((u(t), D_qu(t)), (u'(t), D_qu'(t))\Big) \geq 0$ and by the hypothesis there exists $w\in \mathcal{N}(u')$ such that $\xi \Big((u'(t), D_qu'(t)), (w(t), D_qw(t))\Big) \geq 0$. This implies that $\alpha (u', w) \geq 1$ and so, this proves that the operator $\mathcal{N}$ is an $\alpha -$admissible.

Finally, we choose $u_0 \in X$ and $u' \in \mathcal{N}(u_0)$ such that

Hence $\alpha (u_0, u') \geq 1$. Consequently, Theorem 2.3 implies that $\mathcal{N}$ has a fixed point. In the other words, there exists $u^* \in X$ such that $u^* \in \mathcal{N}(u^*)$ which $u^*$ is the solution of the fractional $q-$difference inclusion {(1.2)} and the proof is completed.

Theorem 4.2. Let $F: [0, 1] \times \mathbb{R}^2 \to \mathcal{P}_{cp}(\mathbb{R})$ be a multifunction. Suppose that:

$(H12)$ The function $\psi : [0, \infty)\to [0, \infty)$ is a nondecreasing upper semi-continuous mapping such that $\lim \inf_{t\to \infty }(t-\psi (t)) > 0$ and $\psi (t) < t$ for all $t > 0$.

$(H13)$ The operator $F: [0, 1]\times \mathbb{R}^{2} \to {\mathcal P}_{cp}(\mathbb{R})$ be an integrably bounded multifunction such that $F(\cdot, u_1, u_2) : [0, 1] \to {\mathcal P}_{cp}(\mathbb{R})$ is measurable for all $u_1, u_2 \in \mathbb{R}$.

$(H14)$ There exists a function $m\in C([0, 1], [0, \infty))$ such that

for all $t\in [0, 1]$ and $u_i, u'_i \in \mathbb{R}$ $(i = 1, 2)$, where $\Phi_i$'s are given in Eq (3.1).

$(H15)$ The operator $\mathcal{N}$ has the approximate endpoint property where $\mathcal{N}$ is defined in Eq (4.1).

Then the $q-$fractional inclusion problem (1.2) has a solution.

Proof. We show that the multifunction $\mathcal{N}: X\to {\mathcal P}(X)$ has an endpoint. For this, we prove that $\mathcal{N}(u)$ is a closed subset of ${\mathcal P}(X)$ for all $u\in X$. First of all, since the multivalued map $t\mapsto F(t, u(t), D_qu(t))$ is measurable and has closed values for all $u\in X$, so it has measurable selection and thus $S_{F, u}$ is nonempty for all $u\in X$.

Similar to the first part of the proof of Theorem 4.1, one can see that the operator $\mathcal{N}(u)$ is closed-valued. Also, $\mathcal{N}(u)$ is a bounded set for all $u\in X$, because $F$ is a compact multivalued map.

Finally, we show that $H_d (\mathcal{N}(u), \mathcal{N}(w))\leq \psi (\Vert u-w\Vert)$. Let $u, w\in X$ and $h_1 \in \mathcal{N}(w)$. Choose $v_1 \in S_{F, w}$ such that

for almost all $t\in [0, 1]$. Since

for all $t\in [0, 1]$, there exist $z\in F(t, u(t), D_qu(t))$ such that

for all $t\in [0, 1]$. Now, consider the multivalued map $U: [0, 1]\to {\mathcal P}(\mathbb{R})$ which is defined by

Since $v_1$ and $\varphi = m\psi (\vert u-w\vert + \vert D_qu - D_qw \vert) \Big(\frac{1}{\Phi_1 +\Phi_2 } \Big)$ are measurable, the multifunction $U(\cdot)\cap F(\cdot, u(\cdot), D_qu(\cdot))$ is measurable. Choose $v_2(t)\in F(t, u(t), D_qu(t))$ such that

for all $t\in [0, 1]$. We define the element $h_2 \in \mathcal{N}(u)$ as follows:

for all $t\in [0, 1]$. Therefore, similar to the proof of Theorem 4.1, we get

Hence $H_d (\mathcal{N}(u), \mathcal{N}(w))\leq \psi (\Vert u-w\Vert)$ for all $u, w\in X$. By hypothesis $(H15)$, since the multifunction $\mathcal{N}$ has approximate endpoint property, by Theorem 2.4, there exists $u^*\in X$ such that $\mathcal{N}(u^*) = \{ u^* \}$. Consequently, the $q-$fractional inclusion (1.2) has the solution $u^*$ and the proof is completed.

5.   Examples

Now, in this section, we present some illustrative examples to show the validity of our main results.

Example 5.1. Consider the fractional $q-$difference equation

via the boundary value conditions

where $^{c}D_{1/2}^{5/2}$ denotes the Caputo $q-$fractional derivative of order $5/2$. Clearly, $\alpha = 5/2$ and $q = 1/2$. We define $f: [0, 1] \times \mathbb{R}^2 \to \mathbb{R}$ by

In this case, for each $x_i, y_i \in \mathbb{R}\; (i = 1, 2)$, we have

Hence $L(t) = t/100$ and so $\Vert L \Vert = \sup_{t\in [0, 1]} \vert L(t) \vert = 1/100.$ On the other hand, we define continuous and nondecreasing function $\psi : \mathbb{R}^+ \to \mathbb{R}$ by $\psi (x) = x$ for all $x\in \mathbb{R}^+$. We have

Clearly, $\mu : [0, 1] \to \mathbb{R}$ is given by $\mu (t) = t/100$ which is continuous function. Then, we have $\Lambda_1 + \Lambda_2 \approx 6.0085$ and so $k\approx 0.06 < 1$. Since, all assumptions of Theorem 3.2 hold, thus the fractional $q-$difference equation (5.1)–(5.2) has at least one solution on $[0, 1]$.

Example 5.2. We consider the fractional $q-$difference inclusion

via the boundary value conditions

Put $\alpha = 5/2$ and $q = 1/2$. By these values, we get $\Lambda_1 \approx 2.5596$ and $\Lambda_2 \approx 3.4489$. We define multifunction $F: [0, 1]\times \mathbb{R}^2 \to \mathcal{P}(\mathbb{R})$ by follows:

for each $t\in [0, 1]$. By above definition, there exists a continuous function $m: [0, 1] \to [0, \infty)$ by $m(t) = 5t/200$ for all $t$. Then $\Vert m\Vert = 5/200.$ Also, we define upper semi-continuous and nondecreasing function $\psi : (0, \infty) \to [0, \infty)$ by $\psi (t) = t/2$ for all $t > 0$. It is clear that $\lim \inf_{t\to \infty } (t-\psi (t)) > 0$ and $\psi (t) < t$ for all $t > 0$. On the other hand, we have $\Phi_1 \approx 0.06399$ and $\Phi_2 \approx 0.08622$ and $\frac{1}{\Phi_1 + \Phi_2 } \approx 6.6577 > 0$. For every $x_i, y_i \in \mathbb{R}\; (i = 1, 2)$, we have

Now, put $X = \{ u:\; u, D_{1/2}u \in C([0, 1], \mathbb{R}) \}$. Define $\mathcal{N} : X\to \mathcal{P}(\mathbb{R})$ as follows:

where

Also, the operator $\mathcal{N}$ has the approximate endpoint property, because $\sup_{u\in \mathcal{N}(0)} \Vert u\Vert = 0$ and so $\inf_{u\in X}\sup_{z\in \mathcal{N}(u)}\Vert u-z\Vert = 0$. All assumptions of Theorem 4.2 hold. Therefore, by Theorem 4.2, the fractional $q-$difference inclusion (5.3)–(5.4) has a solution.

Conflict of interest

All authors declare no conflicts of interest in this paper.