Applications of quantum calculus on finite intervals to impulsive difference inclusions

Abstract

Recently Tariboon and Ntouyas (Adv. Differ. Equ. 2013:282, 2013) introduced the notions of $q_k$-derivative and $q_k$-integral of a function on finite intervals. As applications existence and uniqueness results for initial value problems for first- and second-order impulsive $q_k$-difference equations was proved. In this paper, continuing the study of Tariboon and Ntouyas (Adv. Differ. Equ. 2013:282, 2013), we apply the quantum calculus to initial value problems for impulsive first- and second-order $q_k$-difference inclusions. We establish new existence results, when the right hand side is convex valued, by using the nonlinear alternative of Leray-Schauder type. Some illustrative examples are also presented.

MSC:34A60, 26A33, 39A13, 34A37.

1 Introduction and preliminaries

In [1] the notions of $q_k$-derivative and $q_k$-integral of a function $f:J_k:=[t_k,t_{k+1}]\to \mathbb{R}$, have been introduced and their basic properties was proved. As applications, existence and uniqueness results for initial value problems for first- and second-order impulsive $q_k$-difference equations was proved.

We recall the notions of $q_k$-derivative and $q_k$-integral on finite intervals. For a fixed $k\in \mathbb{N}\cup \{0\}$ let $J_k:=[t_k,t_{k+1}]\subset \mathbb{R}$ be an interval and $0<q_k<1$ be a constant. We define $q_k$-derivative of a function $f:J_k\to \mathbb{R}$ at a point $t\in J_k$ as follows.

Definition 1.1 Assume $f:J_k\to \mathbb{R}$ is a continuous function and let $t\in J_k$. Then the expression

$$D_{q_k}f(t)=\frac{f(t)-f(q_{k}t+(1-q_k)t_k)}{(1-q_k)(t-t_k)},t\ne t_k,D_{q_k}f(t_k)=\underset{t\to t_k}{lim}{D}_{q_k}f(t),$$

(1.1)

is called the $q_k$-derivative of function f at t.

We say that f is $q_k$-differentiable on $J_k$ provided $D_{q_k}f(t)$ exists for all $t\in J_k$. Note that if $t_k=0$ and $q_k=q$ in (1.1), then $D_{q_k}f=D_{q}f$, where $D_q$ is the well-known q-derivative of the function $f(t)$ defined by

$$D_{q}f(t)=\frac{f(t)-f(qt)}{(1-q)t}.$$

(1.2)

In addition, we should define the higher $q_k$-derivative of functions.

Definition 1.2 Let $f:J_k\to \mathbb{R}$ is a continuous function, we call the second-order $q_k$-derivative $D_{q_k}^{2}f$ provided $D_{q_k}f$ is $q_k$-differentiable on $J_k$ with $D_{q_k}^{2}f=D_{q_k}(D_{q_k}f):J_k\to \mathbb{R}$. Similarly, we define higher order $q_k$-derivative $D_{q_k}^n:J_k\to \mathbb{R}$.

The properties of $q_k$-derivative are discussed in [1].

Definition 1.3 Assume $f:J_k\to \mathbb{R}$ is a continuous function. Then the $q_k$-integral is defined by

$${\int }_{t_k}^{t}f(s)d_{q_k}s=(1-q_k)(t-t_k)\sum _{n=0}^{\mathrm{\infty }}{q}_k^{n}f(q_k^{n}t+(1-q_k^n)t_k)$$

(1.3)

for $t\in J_k$. Moreover, if $a\in (t_k,t)$ then the definite $q_k$-integral is defined by

$$\begin{array}{rcl}{\int }_a^{t}f(s)d_{q_k}s& =& {\int }_{t_k}^{t}f(s)d_{q_k}s-{\int }_{t_k}^{a}f(s)d_{q_k}s=(1-q_k)(t-t_k)\sum _{n=0}^{\mathrm{\infty }}{q}_k^{n}f(q_k^{n}t+(1-q_k^n)t_k)\\ -(1-q_k)(a-t_k)\sum _{n=0}^{\mathrm{\infty }}{q}_k^{n}f(q_k^{n}a+(1-q_k^n)t_k).\end{array}$$

Note that if $t_k=0$ and $q_k=q$, then (1.3) reduces to q-integral of a function $f(t)$, defined by ${\int }_0^{t}f(s)d_{q}s=(1-q)t{\sum }_{n=0}^{\mathrm{\infty }}{q}^{n}f(q^{n}t)$ for $t\in [0,\mathrm{\infty })$.

The book by Kac and Cheung [2] covers many of the fundamental aspects of the quantum calculus. In recent years, the topic of q-calculus has attracted the attention of several researchers and a variety of new results can be found in the papers [3–15] and the references cited therein.

Impulsive differential equations, that is, differential equations involving the impulse effect, appear as a natural description of observed evolution phenomena of several real world problems. For some monographs on the impulsive differential equations we refer to [16–18].

Here, we remark that the classical q-calculus cannot be considered in problems with impulses as the definition of q-derivative fails to work when there are impulse points $t_k\in (qt,t)$ for some $k\in \mathbb{N}$. On the other hand, this situation does not arise for impulsive problems on a q-time scale as the points t and $qt=\rho (t)$ are consecutive points, where $\rho :\mathbb{T}\to \mathbb{T}$ is the backward jump operator; see [19]. In [1], quantum calculus on finite intervals, the points t and $q_{k}t+(1-q_k)t_k$ are considered only in an interval $[t_k,t_{k+1}]$. Therefore, the problems with impulses at fixed times can be considered in the framework of $q_k$-calculus.

In this paper, continuing the study of [1], we apply $q_k$-calculus to establish existence results for initial value problems for impulsive first- and second-order $q_k$-difference inclusions. In Section 3, we consider the following initial value problem for the first-order $q_k$-difference inclusion:

$$\begin{array}{r}{D}_{q_k}x(t)\in F(t,x(t)),t\in J:=[0,T],t\ne t_k,\\ \mathrm{\Delta }x(t_k)=I_k(x(t_k)),k=1,2,\dots ,m,\\ x(0)=x_0,\end{array}$$

(1.4)

where $x_0\in \mathbb{R}$, $0=t_0<t_1<t_2<\cdots <t_k<\cdots <t_m<t_{m+1}=T$, $f:[0,T]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ is a multivalued function, $\mathcal{P}(\mathbb{R})$ is the family of all nonempty subjects of ℝ, $I_k\in C(\mathbb{R},\mathbb{R})$, $\mathrm{\Delta }x(t_k)=x(t_k^{+})-x(t_k)$, $k=1,2,\dots ,m$ and $0<q_k<1$ for $k=0,1,2,\dots ,m$.

In Section 4, we study the existence of solutions for the following initial value problem for second-order impulsive $q_k$-difference inclusion:

$$\begin{array}{r}{D}_{q_k}^{2}x(t)\in F(t,x(t)),t\in J,t\ne t_k,\\ \mathrm{\Delta }x(t_k)=I_k(x(t_k)),k=1,2,\dots ,m,\\ D_{q_k}x\left(t_k^{+}\right)-D_{q_{k-1}}x(t_k)=I_k^{\ast }(x(t_k)),k=1,2,\dots ,m,\\ x(0)=\alpha ,D_{q_0}x(0)=\beta ,\end{array}$$

(1.5)

where $\alpha ,\beta \in \mathbb{R}$ and $I_k,I_k^{\ast }\in C(\mathbb{R},\mathbb{R})$.

We establish new existence results, when the right hand side is convex valued by using the nonlinear alternative of Leray-Schauder type.

The paper is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel. In Section 3 we establish the existence result for first-order $q_k$-difference inclusions, while the existence result for second-order $q_k$-difference inclusions is presented in Section 4. Some illustrative examples are also presented.

2 Preliminaries

In this section we recall some basic concepts of multivalued analysis [20, 21].

For a normed space $(X,\parallel \cdot \parallel )$, let ${\mathcal{P}}_{cl}(X)=\{Y\in \mathcal{P}(X):Y\text{ is closed}\}$, ${\mathcal{P}}_{cp}(X)=\{Y\in \mathcal{P}(X):Y\text{ is compact}\}$, and ${\mathcal{P}}_{cp,c}(X)=\{Y\in \mathcal{P}(X):Y\text{ is compact and convex}\}$.

A multivalued map $G:X\to \mathcal{P}(X)$ is convex (closed) valued if $G(x)$ is convex (closed) for all $x\in X$; is bounded on bounded sets if $G(\mathbb{B})={\bigcup }_{x\in \mathbb{B}}G(x)$ is bounded in X for all $\mathbb{B}\in {\mathcal{P}}_b(X)$ (i.e. ${sup}_{x\in \mathbb{B}}\{sup\{|y|:y\in G(x)\}\}<\mathrm{\infty }$); is called upper semicontinuous (u.s.c.) on X if for each $x_0\in X$, the set $G(x_0)$ is a nonempty closed subset of X, and if for each open set N of X containing $G(x_0)$, there exists an open neighborhood ${\mathcal{N}}_0$ of $x_0$ such that $G({\mathcal{N}}_0)\subseteq N$; is said to be completely continuous if $G(\mathbb{B})$ is relatively compact for every $\mathbb{B}\in {\mathcal{P}}_b(X)$.

In the sequel, we denote by $\mathcal{C}=C([0,T],\mathbb{R})$ the space of all continuous functions from $[0,T]\to \mathbb{R}$ with norm $\parallel x\parallel =sup\{|x(t)|:t\in [0,T]\}$. By $L^1([0,T],\mathbb{R})$ we denote the space of all functions f defined on $[0,T]$ such that ${\parallel x\parallel }_{L^1}={\int }_0^T|x(t)|dt<\mathrm{\infty }$.

For each $y\in \mathcal{C}$, define the set of selections of F by

$$S_{F,y}:=\{v\in \mathcal{C}:v(t)\in F(t,y(t))\text{ on }[0,T]\}.$$

Definition 2.1 A multivalued map $F:J\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ is said to be Carathéodory (in the sense of $q_k$-calculus) if $x\mapsto F(t,x)$ is upper semicontinuous on J. Further a Carathéodory function F is called $L^1$-Carathéodory if there exists ${\phi }_{\alpha }\in L^1(J,{\mathbb{R}}^{+})$ such that $\parallel F(t,x)\parallel =sup\{|v|:v\in F(t,x)\}\le {\phi }_{\alpha }(t)$ for all $\parallel x\parallel \le \alpha$ on J for each $\alpha >0$.

We recall the well-known nonlinear alternative of Leray-Schauder for multivalued maps and a useful result regarding closed graphs.

Lemma 2.2 (Nonlinear alternative for Kakutani maps) [22]

Let E be a Banach space, C a closed convex subset of E, U an open subset of C and $0\in U$. Suppose that $F:\overline{U}\to {\mathcal{P}}_{cp,c}(C)$ is a upper semicontinuous compact map. Then either

• (i) F has a fixed point in $\overline{U}$, or

• (ii) there is a $u\in \partial U$ and $\lambda \in (0,1)$ with $u\in \lambda F(u)$.

Lemma 2.3 ([23, 24])

Let X be a Banach space. Let $F:J\times \mathbb{R}\to {\mathcal{P}}_{cp,c}(X)$ be an $L^1$-Carathéodory multivalued map and let Θ be a linear continuous mapping from $L^1(J,\mathbb{R})$ to $C(J,\mathbb{R})$. Then the operator

$$\mathrm{\Theta }\circ S_F:C(J,\mathbb{R})\to {\mathcal{P}}_{cp,c}(C(J,\mathbb{R})),x\mapsto (\mathrm{\Theta }\circ S_F)(x)=\mathrm{\Theta }(S_{F,x})$$

is a closed graph operator in $C(J,\mathbb{R})\times C(J,\mathbb{R})$.

Let $J=[0,T]$, $J_0=[t_0,t_1]$, $J_k=(t_k,t_{k+1}]$ for $k=1,2,\dots ,m$. Let $PC(J,\mathbb{R})$ = {$x:J\to \mathbb{R}:x(t)$ is continuous everywhere except for some $t_k$ at which $x(t_k^{+})$ and $x(t_k^{-})$ exist and $x(t_k^{-})=x(t_k)$, $k=1,2,\dots ,m$}. $PC(J,\mathbb{R})$ is a Banach space with the norms ${\parallel x\parallel }_{PC}=sup\{|x(t)|;t\in J\}$.

3 First-order impulsive $q_k$-difference inclusions

In this section, we study the existence of solutions for the first-order impulsive $q_k$-difference inclusion (1.4).

The following lemma was proved in [1].

Lemma 3.1 If $y\in PC(J,\mathbb{R})$, then for any $t\in J_k$, $k=0,1,2,\dots ,m$, the solution of the problem

$$\begin{array}{r}{D}_{q_k}x(t)=y(t),t\in J,t\ne t_k,\\ \mathrm{\Delta }x(t_k)=I_k(x(t_k)),k=1,2,\dots ,m,\\ x(0)=x_0\end{array}$$

(3.1)

is given by

$$x(t)=x_0+\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}y(s)d_{q_{k-1}}s+\sum _{0<t_k<t}{I}_k(x(t_k))+{\int }_{t_k}^{t}y(s)d_{q_k}s,$$

(3.2)

with ${\sum }_{0<0}(\cdot )=0$.

Before studying the boundary value problem (1.4) let us begin by defining its solution.

Definition 3.2 A function $x\in PC(J,\mathbb{R})$ is said to be a solution of (1.4) if $x(0)=x_0$, $\mathrm{\Delta }x(t_k)=I_k(x(t_k))$, $k=1,2,\dots ,m$, and there exists $f\in L^1(J,\mathbb{R})$ such that $f(t)\in F(t,x(t))$ on J and

$$x(t)=x_0+\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+\sum _{0<t_k<t}{I}_k(x(t_k))+{\int }_{t_k}^{t}f(s)d_{q_k}s.$$

Theorem 3.3 Assume that:

(H₁) $F:J\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ is Carathéodory and has nonempty compact and convex values;

(H₂) there exist a continuous nondecreasing function $\psi :[0,\mathrm{\infty })\to (0,\mathrm{\infty })$ and a function $p\in C(J,{\mathbb{R}}^{+})$ such that

$${\parallel F(t,x)\parallel }_{\mathcal{P}}:=sup\{|y|:y\in F(t,x)\}\le p(t)\psi (\parallel x\parallel )\mathit{\text{for each }}(t,x)\in J\times \mathbb{R};$$

(H₃) there exist constants $c_k$ such that $|I_k(y)|\le c_k$, $k=1,2,\dots ,m$ for each $y\in \mathbb{R}$;

(H₄) there exists a constant $M>0$ such that

$$\frac{M}{|x_0|+T\psi (M)\parallel p\parallel +{\sum }_{k=1}^m{c}_k}>1.$$

Then the initial value problem (1.4) has at least one solution on J.

Proof Define the operator $\mathcal{H}:PC(J,\mathbb{R})\to \mathcal{P}(PC(J,\mathbb{R}))$ by

$$\mathcal{H}(x)=h\in PC(J,\mathbb{R}):h(t)=x_0+\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+\sum _{0<t_k<t}{I}_k(x(t_k))+{\int }_{t_k}^{t}f(s)d_{q_k}s,$$

for $f\in S_{F,x}$.

We will show that ℋ satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that ℋ is convex for each $x\in PC(J,\mathbb{R})$. This step is obvious since $S_{F,x}$ is convex (F has convex values), and therefore we omit the proof.

In the second step, we show that ℋ maps bounded sets (balls) into bounded sets in $PC(J,\mathbb{R})$. For a positive number ρ, let $B_{\rho }=\{x\in C(J,\mathbb{R}):\parallel x\parallel \le \rho \}$ be a bounded ball in $C(J,\mathbb{R})$. Then, for each $h\in \mathcal{H}(x)$, $x\in B_{\rho }$, there exists $f\in S_{F,x}$ such that

$$h(t)=x_0+\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+\sum _{0<t_k<t}{I}_k(x(t_k))+{\int }_{t_k}^{t}f(s)d_{q_k}s.$$

Then for $t\in J$ we have

$$\begin{array}{rcl}|h(t)|& \le & |x_0|+\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}|f(s)|d_{q_{k-1}}s+\sum _{0<t_k<t}\left|I_k(x(t_k))\right|+{\int }_{t_k}^t|f(s)|d_{q_k}s\\ \le & |x_0|+\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}p(s)\psi (\parallel x\parallel )d_{q_{k-1}}s+\sum _{k=1}^m{c}_k+{\int }_{t_k}^{t}p(s)\psi (\parallel x\parallel )d_{q_k}s\\ \le & |x_0|+\psi (\parallel x\parallel )\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}p(s)d_{q_{k-1}}s+\sum _{k=1}^m{c}_k+\psi (\parallel x\parallel ){\int }_{t_k}^{t}p(s)d_{q_k}s\\ \le & |x_0|+T\psi (\parallel x\parallel )\parallel p\parallel +\sum _{k=1}^m{c}_k.\end{array}$$

Consequently,

$$\parallel h\parallel \le |x_0|+T\psi (\rho )\parallel p\parallel +\sum _{k=1}^m{c}_k.$$

Now we show that ℋ maps bounded sets into equicontinuous sets of $PC(J,\mathbb{R})$. Let ${\tau }_1,{\tau }_2\in J$, ${\tau }_1<{\tau }_2$ with ${\tau }_1\in J_v$, ${\tau }_2\in J_u$, $v\le u$ for some $u,v\in \{0,1,2,\dots ,m\}$ and $x\in B_{\rho }$. For each $h\in \mathcal{H}(x)$, we obtain

$$\begin{array}{rcl}|h({\tau }_2)-h({\tau }_1)|& \le & |{\int }_{t_u}^{{\tau }_2}f(s)d_{q_k}s-{\int }_{t_v}^{{\tau }_1}f(s)d_{q_k}s|+\left|\sum _{{\tau }_1<t_k<{\tau }_2}{I}_k(x(t_k))\right|\\ +|\sum _{{\tau }_1<t_k<{\tau }_2}{\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s|\\ \le & |{\int }_{t_u}^{{\tau }_2}f(s)d_{q_k}s-{\int }_{t_v}^{{\tau }_1}f(s)d_{q_k}s|+\sum _{{\tau }_1<t_k<{\tau }_2}\left|I_k(x(t_k))\right|\\ +\sum _{{\tau }_1<t_k<{\tau }_2}{\int }_{t_{k-1}}^{t_k}|f(s)|d_{q_{k-1}}s.\end{array}$$

Obviously the right hand side of the above inequality tends to zero independently of $x\in B_{\rho }$ as ${\tau }_2-{\tau }_1\to 0$. Therefore it follows by the Arzelá-Ascoli theorem that $\mathcal{H}:PC(J,\mathbb{R})\to \mathcal{P}(PC(J,\mathbb{R}))$ is completely continuous.

Since ℋ is completely continuous, in order to prove that it is upper semicontinuous it is enough to prove that it has a closed graph. Thus, in our next step, we show that ℋ has a closed graph. Let $x_n\to x_{\ast }$, $h_n\in \mathcal{H}(x_n)$ and $h_n\to h_{\ast }$. Then we need to show that $h_{\ast }\in \mathcal{H}(x_{\ast })$. Associated with $h_n\in \mathcal{H}(x_n)$, there exists $f_n\in S_{F,x_n}$ such that, for each $t\in J$,

$$h_n(t)=x_0+\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}{f}_n(s)d_{q_{k-1}}s+\sum _{0<t_k<t}{I}_k(x_n(t_k))+{\int }_{t_k}^t{f}_n(s)d_{q_k}s.$$

Thus it suffices to show that there exists $f_{\ast }\in S_{F,x_{\ast }}$ such that, for each $t\in J$,

$$h_{\ast }(t)=x_0+\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}{f}_{\ast }(s)d_{q_{k-1}}s+\sum _{0<t_k<t}{I}_k(x_{\ast }(t_k))+{\int }_{t_k}^t{f}_{\ast }(s)d_{q_k}s.$$

Let us consider the linear operator $\mathrm{\Theta }:L^1(J,\mathbb{R})\to PC(J,\mathbb{R})$ given by

$$f\mapsto \mathrm{\Theta }(f)(t)=x_0+\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+\sum _{0<t_k<t}{I}_k(x(t_k))+{\int }_{t_k}^{t}f(s)d_{q_k}s.$$

Observe that

$$\begin{array}{rcl}\parallel h_n(t)-h_{\ast }(t)\parallel & =& \parallel \sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}(f_n(u)-f_{\ast }(u))d_{q_{k-1}}s+\sum _{0<t_k<t}|I_k(x_n(t_k))-I_k(x_{\ast }(t_k))|\\ +{\int }_{t_k}^t(f_n(u)-f_{\ast }(u))d_{q_k}s\parallel \to 0,\end{array}$$

as $n\to \mathrm{\infty }$.

Thus, it follows by Lemma 2.3 that $\mathrm{\Theta }\circ S_F$ is a closed graph operator. Further, we have $h_n(t)\in \mathrm{\Theta }(S_{F,x_n})$. Since $x_n\to x_{\ast }$, therefore, we have

$$h_{\ast }(t)=x_0+\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}{f}_{\ast }(s)d_{q_{k-1}}s+\sum _{0<t_k<t}{I}_k(x_{\ast }(t_k))+{\int }_{t_k}^t{f}_{\ast }(s)d_{q_k}s,$$

for some $f_{\ast }\in S_{F,x_{\ast }}$.

Finally, we show there exists an open set $U\subseteq C(J,\mathbb{R})$ with $x\notin \mathcal{H}(x)$ for any $\lambda \in (0,1)$ and all $x\in \partial U$. Let $\lambda \in (0,1)$ and $x\in \lambda \mathcal{H}(x)$. Then there exists $v\in L^1(J,\mathbb{R})$ with $f\in S_{F,x}$ such that, for $t\in J$, we have

$$x(t)=x_0+\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+\sum _{0<t_k<t}{I}_k(x(t_k))+{\int }_{t_k}^{t}f(s)d_{q_k}s.$$

Repeating the computations of the second step, we have

$$|x(t)|\le |x_0|+T\psi (\parallel x\parallel )\parallel p\parallel +\sum _{k=1}^m{c}_k.$$

Consequently, we have

$$\frac{\parallel x\parallel }{|x_0|+T\psi (\parallel x\parallel )\parallel p\parallel +{\sum }_{k=1}^m{c}_k}\le 1.$$

In view of (H₄), there exists M such that $\parallel x\parallel \ne M$. Let us set

$$U=\{x\in PC(J,\mathbb{R}):\parallel x\parallel <M\}.$$

Note that the operator $\mathcal{H}:\overline{U}\to \mathcal{P}(PC(J,\mathbb{R}))$ is upper semicontinuous and completely continuous. From the choice of U, there is no $x\in \partial U$ such that $x\in \lambda \mathcal{H}(x)$ for some $\lambda \in (0,1)$. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 2.2), we deduce that ℋ has a fixed point $x\in \overline{U}$ which is a solution of the problem (1.4). This completes the proof. □

Example 3.4 Let us consider the following first-order initial value problem for impulsive $q_k$-difference inclusions:

$$\begin{array}{r}{D}_{\frac{1}{2+k}}x(t)\in F(t,x(t)),t\in J=[0,1],t\ne t_k=\frac{k}{10},\\ \mathrm{\Delta }x(t_k)=\frac{|x(t_k)|}{12+|x(t_k)|},k=1,2,\dots ,9,\\ x(0)=0.\end{array}$$

(3.3)

Here $q_k=1/(2+k)$, $k=0,1,2,\dots ,9$, $m=9$, $T=1$, and $I_k(x)=|x|/(12+|x|)$. We find that $|I_k(x)-I_k(y)|\le (1/12)|x-y|$ and $|I_k(x)|\le 1$.

• (a) Let $F:[0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ be a multivalued map given by

  $$x\to F(t,x)=[\frac{|x|}{|x|+{sin}^{2}x+1}+t+1,e^{-x^2}+\frac{4}{5}{t}^2+3].$$

  (3.4)

For $f\in F$, we have

$$|f|\le max(\frac{|x|}{|x|+{sin}^{2}x+1}+t+1,e^{-x^2}+t^2+3)\le 5,x\in \mathbb{R}.$$

Thus,

$${\parallel F(t,x)\parallel }_{\mathcal{P}}:=sup\{|y|:y\in F(t,x)\}\le 5=p(t)\psi (\parallel x\parallel ),x\in \mathbb{R},$$

with $p(t)=1$, $\psi (\parallel x\parallel )=5$. Further, using the condition (H₄) we find that $M>14$. Therefore, all the conditions of Theorem 3.3 are satisfied. So, problem (3.3) with $F(t,x)$ given by (3.4) has at least one solution on $[0,1]$.

• (b) If $F:[0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ is a multivalued map given by

  $$x\to F(t,x)=[\frac{(t+1)x^2}{x^2+1},\frac{t|x|({cos}^{2}x+1)}{2(|x|+1)}].$$

  (3.5)

For $f\in F$, we have

$$|f|\le max(\frac{(t+1)x^2}{x^2+1},\frac{t|x|({cos}^{2}x+1)}{2(|x|+1)})\le t+1,x\in \mathbb{R}.$$

Here ${\parallel F(t,x)\parallel }_{\mathcal{P}}:=sup\{|y|:y\in F(t,x)\}\le (t+1)=p(t)\psi (\parallel x\parallel )$, $x\in \mathbb{R}$, with $p(t)=t+1$, $\psi (\parallel x\parallel )=1$. It is easy to verify that $M>10.5$. Then, by Theorem 3.3, the problem (3.3) with $F(t,x)$ given by (3.5) has at least one solution on $[0,1]$.

4 Second-order impulsive $q_k$-difference inclusions

In this section, we study the existence of solutions for the second-order impulsive $q_k$-difference inclusion (1.5).

We recall the following lemma from [1].

Lemma 4.1 If $y\in C(J,\mathbb{R})$, then for any $t\in J$, the solution of the problem

$$\begin{array}{r}{D}_{q_k}^{2}x(t)=y(t),t\in J,t\ne t_k,\\ \mathrm{\Delta }x(t_k)=I_k(x(t_k)),k=1,2,\dots ,m,\\ D_{q_k}x\left(t_k^{+}\right)-D_{q_{k-1}}x(t_k)=I_k^{\ast }(x(t_k)),k=1,2,\dots ,m,\\ x(0)=\alpha ,D_{q_0}x(0)=\beta ,\end{array}$$

(4.1)

is given by

$$\begin{array}{rcl}x(t)& =& \alpha +\beta t\\ +\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})y(s)d_{q_{k-1}}s+I_k(x(t_k)))\\ +t\left[\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}fy(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\right]\\ -\sum _{0<t_k<t}{t}_k({\int }_{t_{k-1}}^{t_k}y(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\\ +{\int }_{t_k}^t(t-q_{k}s-(1-q_k)t_k)y(s)d_{q_k}s,\end{array}$$

(4.2)

with ${\sum }_{0<0}(\cdot )=0$.

Definition 4.2 A function $x\in PC(J,\mathbb{R})$ is said to be a solution of (1.5) if $x(0)=x_0$, $D_{q_0}x(0)=\beta$, $\mathrm{\Delta }x(t_k)=I_k(x(t_k))$, $D_{q_k}x(t_k^{+})-D_{q_{k-1}}x(t_k)=I_k^{\ast }(x(t_k))$, $k=1,2,\dots ,m$ and there exists $f\in L^1(J,\mathbb{R})$ such that $f(t)\in F(t,x(t))$ on J and

$$\begin{array}{rcl}x(t)& =& \alpha +\beta t\\ +\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})f(s)d_{q_{k-1}}s+I_k(x(t_k)))\\ +t\left[\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\right]\\ -\sum _{0<t_k<t}{t}_k({\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\\ +{\int }_{t_k}^t(t-q_{k}s-(1-q_k)t_k)f(s)d_{q_k}s,\end{array}$$

(4.3)

with ${\sum }_{0<0}(\cdot )=0$.

Theorem 4.3 Assume that (H₁), (H₂) hold. In addition we suppose that:

(A₁) there exist constants $c_k$, $c_k^{\ast }$ such that $|I_k(x)|\le c_k$, $|I_k^{\ast }(y)|\le c_k^{\ast }$, $k=1,2,\dots ,m$ for each $x,y\in \mathbb{R}$;

(A₂) there exists a constant $M>0$ such that

$$\frac{M}{|\alpha |+|\beta |T+\parallel p\parallel \psi (M){\mathrm{\Lambda }}_1+{\sum }_{k=1}^m[c_k+c_k^{\ast }(T+t_k)]}>1,$$

where

$${\mathrm{\Lambda }}_1=\sum _{k=1}^{m+1}\frac{{(t_k-t_{k-1})}^2}{1+q_{k-1}}+\sum _{k=1}^m(T+t_k)(t_k-t_{k-1}).$$

(4.4)

Then the initial value problem (1.5) has at least one solution on J.

Proof Define the operator $\mathcal{H}:PC(J,\mathbb{R})\to \mathcal{P}(PC(J,\mathbb{R}))$ by

$$\begin{array}{rl}\mathcal{H}(x)=h\in PC(J,\mathbb{R}):h(t)=& \alpha +\beta t+\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})f(s)d_{q_{k-1}}s\\ +I_k(x(t_k)))+t\left[\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\right]\\ -\sum _{0<t_k<t}{t}_k({\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\\ +{\int }_{t_k}^t(t-q_{k}s-(1-q_k)t_k)f(s)d_{q_k}s,\end{array}$$

for $f\in S_{F,x}$.

We will show that ℋ satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that ℋ is convex for each $x\in PC(J,\mathbb{R})$. This step is obvious since $S_{F,x}$ is convex (F has convex values), and therefore we omit the proof.

In the second step, we show that ℋ maps bounded sets (balls) into bounded sets in $PC(J,\mathbb{R})$. For a positive number ρ, let $B_{\rho }=\{x\in PC(J,\mathbb{R}):\parallel x\parallel \le \rho \}$ be a bounded ball in $PC(J,\mathbb{R})$. Then, for each $h\in \mathcal{H}(x)$, $x\in B_{\rho }$, there exists $f\in S_{F,x}$ such that

$$\begin{array}{rcl}h(t)& =& \alpha +\beta t\\ +\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})f(s)d_{q_{k-1}}s+I_k(x(t_k)))\\ +t\left[\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\right]\\ -\sum _{0<t_k<t}{t}_k({\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\\ +{\int }_{t_k}^t(t-q_{k}s-(1-q_k)t_k)f(s)d_{q_k}s.\end{array}$$

Then for $t\in J$ we have

$$\begin{array}{rcl}|h(t)|& \le & |\alpha |+|\beta |t\\ +\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})|f(s)|d_{q_{k-1}}s+\left|I_k(x(t_k))\right|)\\ +t\left[\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}|f(s)|d_{q_{k-1}}s+\left|I_k^{\ast }(x(t_k))\right|)\right]\\ +\sum _{0<t_k<t}{t}_k({\int }_{t_{k-1}}^{t_k}|f(s)|d_{q_{k-1}}s+\left|I_k^{\ast }(x(t_k))\right|)\\ +{\int }_{t_k}^t(t-q_{k}s-(1-q_k)t_k)|f(s)|d_{q_k}s\\ \le & |\alpha |+|\beta |T\\ +\sum _{0<t_k<T}({\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})p(s)\psi (\parallel x\parallel )d_{q_{k-1}}s+\left|I_k(x(t_k))\right|)\\ +T\left[\sum _{0<t_k<T}({\int }_{t_{k-1}}^{t_k}p(s)\psi (\parallel x\parallel )d_{q_{k-1}}s+\left|I_k^{\ast }(x(t_k))\right|)\right]\\ +\sum _{0<t_k<T}{t}_k({\int }_{t_{k-1}}^{t_k}p(s)\psi (\parallel x\parallel )d_{q_{k-1}}s+\left|I_k^{\ast }(x(t_k))\right|)\\ +{\int }_{t_m}^T(T-q_{m}s-(1-q_m)t_m)p(s)\psi (\parallel x\parallel )d_{q_m}s\\ =& |\alpha |+|\beta |T+\sum _{k=1}^m(\frac{{(t_k-t_{k-1})}^2}{1+q_{k-1}}\parallel p\parallel \psi (\parallel x\parallel )+c_k)\\ +T\left[\sum _{k=1}^m(\parallel p\parallel \psi (\parallel x\parallel )(t_k-t_{k-1})+c_k^{\ast })\right]\\ +\sum _{k=1}^m{t}_k(\parallel p\parallel \psi (\parallel x\parallel )(t_k-t_{k-1})+c_k^{\ast })+\frac{{(T-t_m)}^2}{1+q_m}\parallel p\parallel \psi (\parallel x\parallel )\\ =& |\alpha |+|\beta |T+\parallel p\parallel \psi (\parallel x\parallel )\{\sum _{k=1}^{m+1}\frac{{(t_k-t_{k-1})}^2}{1+q_{k-1}}+\sum _{k=1}^m(T+t_k)(t_k-t_{k-1})\}\\ +\sum _{k=1}^m[c_k+c_k^{\ast }(T+t_k)].\end{array}$$

Consequently,

$$\begin{array}{rcl}\parallel h\parallel & \le & |\alpha |+|\beta |T+\parallel p\parallel \psi (\rho )\{\sum _{k=1}^{m+1}\frac{{(t_k-t_{k-1})}^2}{1+q_{k-1}}+\sum _{k=1}^m(T+t_k)(t_k-t_{k-1})\}\\ +\sum _{k=1}^m[c_k+c_k^{\ast }(T+t_k)].\end{array}$$

Now we show that ℋ maps bounded sets into equicontinuous sets of $PC(J,\mathbb{R})$. Let ${\tau }_1,{\tau }_2\in J$, ${\tau }_1<{\tau }_2$ with ${\tau }_1\in J_u$, ${\tau }_2\in J_v$, $u\le v$ for some $u,v\in \{0,1,2,\dots ,m\}$ and $x\in B_{\rho }$. For each $h\in \mathcal{H}(x)$, we obtain

$$\begin{array}{rcl}|h({\tau }_2)-h({\tau }_1)|& \le & |\beta ||{\tau }_2-{\tau }_1|\\ +\sum _{{\tau }_1<t_k<{\tau }_2}({\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})|f(s)|d_{q_{k-1}}s+\left|I_k(x(t_k))\right|)\\ +|{\tau }_2-{\tau }_1|\left[\sum _{0<t_k<{\tau }_1}({\int }_{t_{k-1}}^{t_k}|f(s)|d_{q_{k-1}}s+\left|I_k^{\ast }(x(t_k))\right|)\right]\\ +{\tau }_2\left[\sum _{{\tau }_1<t_k<{\tau }_2}({\int }_{t_{k-1}}^{t_k}|f(s)|d_{q_{k-1}}s+\left|I_k^{\ast }(x(t_k))\right|)\right]\\ +\sum _{{\tau }_1<t_k<{\tau }_2}{t}_k({\int }_{t_{k-1}}^{t_k}|f(s)|d_{q_{k-1}}s+\left|I_k^{\ast }(x(t_k))\right|)\\ +|{\int }_{t_v}^{{\tau }_2}({\tau }_2-q_{k}s-(1-q_k)t_k)|f(s)|d_{q_k}s\\ -{\int }_{t_u}^{{\tau }_1}({\tau }_1-q_{k}s-(1-q_k)t_k)|f(s)|d_{q_k}s|.\end{array}$$

Obviously the right hand side of the above inequality tends to zero independently of $x\in B_{\rho }$ as ${\tau }_2-{\tau }_1\to 0$. Therefore it follows by the Arzelá-Ascoli theorem that $\mathcal{H}:PC(J,\mathbb{R})\to \mathcal{P}(PC(J,\mathbb{R}))$ is completely continuous.

Since ℋ is completely continuous, in order to prove that it is upper semicontinuous it is enough to prove that it has a closed graph. Thus, in our next step, we show that ℋ has a closed graph. Let $x_n\to x_{\ast }$, $h_n\in \mathcal{H}(x_n)$ and $h_n\to h_{\ast }$. Then we need to show that $h_{\ast }\in \mathcal{H}(x_{\ast })$. Associated with $h_n\in \mathcal{H}(x_n)$, there exists $f_n\in S_{F,x_n}$ such that, for each $t\in J$,

$$\begin{array}{rcl}{h}_n(t)& =& \alpha +\beta t+\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})f_n(s)d_{q_{k-1}}s+I_k(x(t_k)))\\ +t\left[\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}{f}_n(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\right]\\ -\sum _{0<t_k<t}{t}_k({\int }_{t_{k-1}}^{t_k}{f}_n(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\\ +{\int }_{t_k}^t(t-q_{k}s-(1-q_k)t_k)f_n(s)d_{q_k}s.\end{array}$$

Thus it suffices to show that there exists $f_{\ast }\in S_{F,x_{\ast }}$ such that, for each $t\in J$,

$$\begin{array}{rcl}{h}_{\ast }(t)& =& \alpha +\beta t+\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})f_{\ast }(s)d_{q_{k-1}}s+I_k(x(t_k)))\\ +t\left[\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}{f}_{\ast }(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\right]\\ -\sum _{0<t_k<t}{t}_k({\int }_{t_{k-1}}^{t_k}{f}_{\ast }(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\\ +{\int }_{t_k}^t(t-q_{k}s-(1-q_k)t_k)f_{\ast }(s)d_{q_k}s.\end{array}$$

Let us consider the linear operator $\mathrm{\Theta }:L^1(J,\mathbb{R})\to PC(J,\mathbb{R})$ given by

$$\begin{array}{rcl}f\mapsto \mathrm{\Theta }(f)(t)& =& \alpha +\beta t+\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})f(s)d_{q_{k-1}}s+I_k(x(t_k)))\\ +t\left[\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\right]\\ -\sum _{0<t_k<t}{t}_k({\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\\ +{\int }_{t_k}^t(t-q_{k}s-(1-q_k)t_k)f(s)d_{q_k}s.\end{array}$$

Observe that

$$\begin{array}{rcl}\parallel h_n(t)-h_{\ast }(t)\parallel & =& \parallel \sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})(f_n(u)-f_{\ast }(u))d_{q_{k-1}}s\\ +\sum _{0<t_k<t}|I_k(x_n(t_k))-I_k(x_{\ast }(t_k))|\\ +T\sum _{0<t_k<t}{\int }_{t_{k-1}}^{t_k}(f_n(u)-f_{\ast }(u))d_{q_{k-1}}s\\ +T\sum _{0<t_k<t}|I_k^{\ast }(x_n(t_k))-I_k^{\ast }(x_{\ast }(t_k))|\\ +\sum _{0<t_k<t}{t}_k{\int }_{t_{k-1}}^{t_k}(f_n(u)-f_{\ast }(u))d_{q_{k-1}}s\\ +\sum _{0<t_k<t}|I_k^{\ast }(x_n(t_k))-I_k^{\ast }(x_{\ast }(t_k))|\\ +{\int }_{t_k}^t(t-q_{k}s-(1-q_k)t_k)(f_n(u)-f_{\ast }(u))d_{q_k}s\parallel \to 0,\end{array}$$

as $n\to \mathrm{\infty }$.

Thus, it follows by Lemma 2.3 that $\mathrm{\Theta }\circ S_F$ is a closed graph operator. Further, we have $h_n(t)\in \mathrm{\Theta }(S_{F,x_n})$. Since $x_n\to x_{\ast }$, therefore, we have

$$\begin{array}{rcl}{h}_{\ast }(t)& =& \alpha +\beta t+\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})f_{\ast }(s)d_{q_{k-1}}s+I_k(x(t_k)))\\ +t\left[\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}{f}_{\ast }(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\right]\\ -\sum _{0<t_k<t}{t}_k({\int }_{t_{k-1}}^{t_k}{f}_{\ast }(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\\ +{\int }_{t_k}^t(t-q_{k}s-(1-q_k)t_k)f_{\ast }(s)d_{q_k}s,\end{array}$$

for some $f_{\ast }\in S_{F,x_{\ast }}$.

Finally, we show there exists an open set $U\subseteq C(J,\mathbb{R})$ with $x\notin \mathcal{H}(x)$ for any $\lambda \in (0,1)$ and all $x\in \partial U$. Let $\lambda \in (0,1)$ and $x\in \lambda \mathcal{H}(x)$. Then there exists $f\in L^1(J,\mathbb{R})$ with $f\in S_{F,x}$ such that, for $t\in J$, we have

$$\begin{array}{rcl}x(t)& =& \alpha +\beta t+\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}(t_k-q_{k-1}s-(1-q_{k-1})t_{k-1})f(s)d_{q_{k-1}}s+I_k(x(t_k)))\\ +t\left[\sum _{0<t_k<t}({\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\right]\\ -\sum _{0<t_k<t}{t}_k({\int }_{t_{k-1}}^{t_k}f(s)d_{q_{k-1}}s+I_k^{\ast }(x(t_k)))\\ +{\int }_{t_k}^t(t-q_{k}s-(1-q_k)t_k)f(s)d_{q_k}s.\end{array}$$

Repeating the computations of the second step, we have

$$\begin{array}{rl}|x(t)|\le & |\alpha |+|\beta |T+\parallel p\parallel \psi (\parallel x\parallel )\{\sum _{k=1}^{m+1}\frac{{(t_k-t_{k-1})}^2}{1+q_{k-1}}+\sum _{k=1}^m(T+t_k)(t_k-t_{k-1})\}\\ +\sum _{k=1}^m[c_k+c_k^{\ast }(T+t_k)].\end{array}$$

Consequently, we have

$$\frac{\parallel x\parallel }{|\alpha |+|\beta |T+\parallel p\parallel \psi (\parallel x\parallel ){\mathrm{\Lambda }}_1+{\sum }_{k=1}^m[c_k+c_k^{\ast }(T+t_k)]}\le 1.$$

In view of (A₂), there exists M such that $\parallel x\parallel \ne M$. Let us set

$$U=\{x\in PC(J,\mathbb{R}):\parallel x\parallel <M\}.$$

Note that the operator $\mathcal{H}:\overline{U}\to \mathcal{P}(PC(J,\mathbb{R}))$ is upper semicontinuous and completely continuous. From the choice of U, there is no $x\in \partial U$ such that $x\in \lambda \mathcal{H}(x)$ for some $\lambda \in (0,1)$. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 2.2), we deduce that ℋ has a fixed point $x\in \overline{U}$ which is a solution of the problem (1.4). This completes the proof. □

Example 4.4 Let us consider the following second-order impulsive $q_k$-difference inclusion with initial conditions:

$$\{\begin{array}{l}{D}_{\frac{2}{3+k}}^{2}x(t)\in F(t,x(t)),t\in J=[0,1],t\ne t_k=\frac{k}{10},\\ \mathrm{\Delta }x(t_k)=\frac{|x(t_k)|}{15(6+|x(t_k)|)},k=1,2,\dots ,9,\\ D_{\frac{2}{3+k}}x(t_k^{+})-D_{\frac{2}{3+k-1}}x(t_k)=\frac{|x(t_k)|}{19(3+|x(t_k)|)},k=1,2,\dots ,9,\\ x(0)=0,D_{\frac{2}{3}}x(0)=0.\end{array}$$

(4.5)

Here $q_k=2/(3+k)$, $k=0,1,2,\dots ,9$, $m=9$, $T=1$, $\alpha =0$, $\beta =0$, $I_k(x)=|x|/(15(6+|x|))$, and $I_k^{\ast }(x)=|x|/(19(3+|x|))$. We find that $|I_k(x)-I_k(y)|\le (1/90)|x-y|$, $|I_k^{\ast }(x)-I_k^{\ast }(y)|\le (1/57)|x-y|$, and $I_k(x)\le 1/15$, $I_k^{\ast }(x)\le 1/19$; and we have

$${\mathrm{\Lambda }}_1=\sum _{k=1}^{m+1}\frac{{(t_k-t_{k-1})}^2}{1+q_{k-1}}+\sum _{k=1}^m(T+t_k)(t_k-t_{k-1})\approx 1.42663542.$$

• (a) Let $F:[0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ be a multivalued map given by

  $$x\to F(t,x)=[\frac{|x|}{|x|+{sin}^{2}x+1}+t+1,e^{-x^2}+\frac{4}{5}{t}^2+3].$$

  (4.6)

For $f\in F$, we have

$$|f|\le max(\frac{|x|}{|x|+{sin}^{2}x+1}+t+1,e^{-x^2}+t^2+3)\le 5,x\in \mathbb{R}.$$

Thus,

$${\parallel F(t,x)\parallel }_{\mathcal{P}}:=sup\{|y|:y\in F(t,x)\}\le 5=p(t)\psi (\parallel x\parallel ),x\in \mathbb{R},$$

with $p(t)=1$, $\psi (\parallel x\parallel )=5$. Further, using the condition (A₂) we find

$$\frac{M}{5{\mathrm{\Lambda }}_1+{\sum }_{k=1}^9[\frac{1}{15}+\frac{1}{19}(1+t_k)]}>1,$$

which implies $M>8.44370316$. Therefore, all the conditions of Theorem 4.3 are satisfied. So, problem (4.5) with $F(t,x)$ given by (4.6) has at least one solution on $[0,1]$.

• (b) If $F:[0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ is a multivalued map given by

  $$x\to F(t,x)=[\frac{(t+1)x^2}{x^2+1},\frac{t|x|({cos}^{2}x+1)}{2(|x|+1)}].$$

  (4.7)

For $f\in F$, we have

$$|f|\le max(\frac{(t+1)x^2}{x^2+1},\frac{t|x|({cos}^{2}x+1)}{2(|x|+1)})\le t+1,x\in \mathbb{R}.$$

Here ${\parallel F(t,x)\parallel }_{\mathcal{P}}:=sup\{|y|:y\in F(t,x)\}\le (t+1)=p(t)\psi (\parallel x\parallel )$, $x\in \mathbb{R}$, with $p(t)=t+1$, $\psi (\parallel x\parallel )=1$. It is easy to verify that $M>3.45047945$. Then, by Theorem 4.3, the problem (4.5) with $F(t,x)$ given by (4.7) has at least one solution on $[0,1]$.

Authors’ information

Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.