# 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 $f: J k :=[ t k , t k + 1 ]→R$, 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 $k∈N∪{0}$ let $J k :=[ t k , t k + 1 ]⊂R$ be an interval and $0< q k <1$ be a constant. We define $q k$-derivative of a function $f: J k →R$ at a point $t∈ J k$ as follows.

Definition 1.1 Assume $f: J k →R$ is a continuous function and let $t∈ J k$. Then the expression

$D q k f(t)= f ( t ) − f ( q k t + ( 1 − q k ) t k ) ( 1 − q k ) ( t − t k ) ,t≠ t k , D q k f( t k )= lim t → t k D q k f(t),$
(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 $t∈ J k$. 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

$D q f(t)= f ( t ) − f ( q t ) ( 1 − q ) t .$
(1.2)

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

Definition 1.2 Let $f: J k →R$ 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 $D q k 2 f= D q k ( D q k f): J k →R$. Similarly, we define higher order $q k$-derivative $D q k n : J k →R$.

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

Definition 1.3 Assume $f: J k →R$ is a continuous function. Then the $q k$-integral is defined by

$∫ t k t f(s) d q k s=(1− q k )(t− t k ) ∑ n = 0 ∞ q k n f ( q k n t + ( 1 − q k n ) t k )$
(1.3)

for $t∈ J k$. Moreover, if $a∈( t k ,t)$ then the definite $q k$-integral is defined by

$∫ a t f ( s ) d q k s = ∫ t k t f ( s ) d q k s − ∫ t k a f ( s ) d q k s = ( 1 − q k ) ( t − t k ) ∑ n = 0 ∞ q k n f ( q k n t + ( 1 − q k n ) t k ) − ( 1 − q k ) ( a − t k ) ∑ n = 0 ∞ q k n f ( q k n a + ( 1 − q k n ) t k ) .$

Note that if $t k =0$ and $q k =q$, then (1.3) reduces to q-integral of a function $f(t)$, defined by $∫ 0 t f(s) d q s=(1−q)t ∑ n = 0 ∞ q n f( q n t)$ for $t∈[0,∞)$.

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 [315] 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 [1618].

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 $t k ∈(qt,t)$ for some $k∈N$. On the other hand, this situation does not arise for impulsive problems on a q-time scale as the points t and $qt=ρ(t)$ are consecutive points, where $ρ:T→T$ 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:

$D q k x ( t ) ∈ F ( t , x ( t ) ) , t ∈ J : = [ 0 , T ] , t ≠ t k , Δ x ( t k ) = I k ( x ( t k ) ) , k = 1 , 2 , … , m , x ( 0 ) = x 0 ,$
(1.4)

where $x 0 ∈R$, $0= t 0 < t 1 < t 2 <⋯< t k <⋯< t m < t m + 1 =T$, $f:[0,T]×R→P(R)$ is a multivalued function, $P(R)$ is the family of all nonempty subjects of , $I k ∈C(R,R)$, $Δx( t k )=x( t k + )−x( t k )$, $k=1,2,…,m$ and $0< q k <1$ for $k=0,1,2,…,m$.

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

$D q k 2 x ( t ) ∈ F ( t , x ( t ) ) , t ∈ J , t ≠ t k , Δ x ( t k ) = I k ( x ( t k ) ) , k = 1 , 2 , … , m , D q k x ( t k + ) − D q k − 1 x ( t k ) = I k ∗ ( x ( t k ) ) , k = 1 , 2 , … , m , x ( 0 ) = α , D q 0 x ( 0 ) = β ,$
(1.5)

where $α,β∈R$ and $I k , I k ∗ ∈C(R,R)$.

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 $(X,∥⋅∥)$, let , , and .

A multivalued map $G:X→P(X)$ is convex (closed) valued if $G(x)$ is convex (closed) for all $x∈X$; is bounded on bounded sets if $G(B)= ⋃ x ∈ B G(x)$ is bounded in X for all $B∈ P b (X)$ (i.e. $sup x ∈ B {sup{|y|:y∈G(x)}}<∞$); is called upper semicontinuous (u.s.c.) on X if for each $x 0 ∈X$, 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 $N 0$ of $x 0$ such that $G( N 0 )⊆N$; is said to be completely continuous if $G(B)$ is relatively compact for every $B∈ P b (X)$.

In the sequel, we denote by $C=C([0,T],R)$ the space of all continuous functions from $[0,T]→R$ with norm $∥x∥=sup{|x(t)|:t∈[0,T]}$. By $L 1 ([0,T],R)$ we denote the space of all functions f defined on $[0,T]$ such that $∥ x ∥ L 1 = ∫ 0 T |x(t)|dt<∞$.

For each $y∈C$, define the set of selections of F by

Definition 2.1 A multivalued map $F:J×R→P(R)$ is said to be Carathéodory (in the sense of $q k$-calculus) if $x↦F(t,x)$ is upper semicontinuous on J. Further a Carathéodory function F is called $L 1$-Carathéodory if there exists $φ α ∈ L 1 (J, R + )$ such that $∥F(t,x)∥=sup{|v|:v∈F(t,x)}≤ φ α (t)$ for all $∥x∥≤α$ on J for each $α>0$.

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 $0∈U$. Suppose that $F: U ¯ → P c p , c (C)$ is a upper semicontinuous compact map. Then either

1. (i)

F has a fixed point in $U ¯$, or

2. (ii)

there is a $u∈∂U$ and $λ∈(0,1)$ with $u∈λF(u)$.

Lemma 2.3 ([23, 24])

Let X be a Banach space. Let $F:J×R→ P c p , c (X)$ be an $L 1$-Carathéodory multivalued map and let Θ be a linear continuous mapping from $L 1 (J,R)$ to $C(J,R)$. Then the operator

$Θ∘ S F :C(J,R)→ P c p , c ( C ( J , R ) ) ,x↦(Θ∘ S F )(x)=Θ( S F , x )$

is a closed graph operator in $C(J,R)×C(J,R)$.

Let $J=[0,T]$, $J 0 =[ t 0 , t 1 ]$, $J k =( t k , t k + 1 ]$ for $k=1,2,…,m$. Let $PC(J,R)$ = {$x:J→R:x(t)$ is continuous everywhere except for some $t k$ at which $x( t k + )$ and $x( t k − )$ exist and $x( t k − )=x( t k )$, $k=1,2,…,m$}. $PC(J,R)$ is a Banach space with the norms $∥ x ∥ P C =sup{|x(t)|;t∈J}$.

## 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 $y∈PC(J,R)$, then for any $t∈ J k$, $k=0,1,2,…,m$, the solution of the problem

$D q k x ( t ) = y ( t ) , t ∈ J , t ≠ t k , Δ x ( t k ) = I k ( x ( t k ) ) , k = 1 , 2 , … , m , x ( 0 ) = x 0$
(3.1)

is given by

$x(t)= x 0 + ∑ 0 < t k < t ∫ t k − 1 t k y(s) d q k − 1 s+ ∑ 0 < t k < t I k ( x ( t k ) ) + ∫ t k t y(s) d q k s,$
(3.2)

with $∑ 0 < 0 (⋅)=0$.

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

Definition 3.2 A function $x∈PC(J,R)$ is said to be a solution of (1.4) if $x(0)= x 0$, $Δx( t k )= I k (x( t k ))$, $k=1,2,…,m$, and there exists $f∈ L 1 (J,R)$ such that $f(t)∈F(t,x(t))$ on J and

$x(t)= x 0 + ∑ 0 < t k < t ∫ t k − 1 t k f(s) d q k − 1 s+ ∑ 0 < t k < t I k ( x ( t k ) ) + ∫ t k t f(s) d q k s.$

Theorem 3.3 Assume that:

(H1) $F:J×R→P(R)$ is Carathéodory and has nonempty compact and convex values;

(H2) there exist a continuous nondecreasing function $ψ:[0,∞)→(0,∞)$ and a function $p∈C(J, R + )$ such that

(H3) there exist constants $c k$ such that $| I k (y)|≤ c k$, $k=1,2,…,m$ for each $y∈R$;

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

$M | x 0 | + T ψ ( M ) ∥ p ∥ + ∑ k = 1 m c k >1.$

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

Proof Define the operator $H:PC(J,R)→P(PC(J,R))$ by

$H(x)=h∈PC(J,R):h(t)= x 0 + ∑ 0 < t k < t ∫ t k − 1 t k f(s) d q k − 1 s+ ∑ 0 < t k < t I k ( x ( t k ) ) + ∫ t k t f(s) d q k s,$

for $f∈ S F , x$.

We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that is convex for each $x∈PC(J,R)$. 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 maps bounded sets (balls) into bounded sets in $PC(J,R)$. For a positive number ρ, let $B ρ ={x∈C(J,R):∥x∥≤ρ}$ be a bounded ball in $C(J,R)$. Then, for each $h∈H(x)$, $x∈ B ρ$, there exists $f∈ S F , x$ such that

$h(t)= x 0 + ∑ 0 < t k < t ∫ t k − 1 t k f(s) d q k − 1 s+ ∑ 0 < t k < t I k ( x ( t k ) ) + ∫ t k t f(s) d q k s.$

Then for $t∈J$ we have

$| h ( t ) | ≤ | x 0 | + ∑ 0 < t k < t ∫ t k − 1 t k | f ( s ) | d q k − 1 s + ∑ 0 < t k < t | I k ( x ( t k ) ) | + ∫ t k t | f ( s ) | d q k s ≤ | x 0 | + ∑ 0 < t k < t ∫ t k − 1 t k p ( s ) ψ ( ∥ x ∥ ) d q k − 1 s + ∑ k = 1 m c k + ∫ t k t p ( s ) ψ ( ∥ x ∥ ) d q k s ≤ | x 0 | + ψ ( ∥ x ∥ ) ∑ 0 < t k < t ∫ t k − 1 t k p ( s ) d q k − 1 s + ∑ k = 1 m c k + ψ ( ∥ x ∥ ) ∫ t k t p ( s ) d q k s ≤ | x 0 | + T ψ ( ∥ x ∥ ) ∥ p ∥ + ∑ k = 1 m c k .$

Consequently,

$∥h∥≤| x 0 |+Tψ(ρ)∥p∥+ ∑ k = 1 m c k .$

Now we show that maps bounded sets into equicontinuous sets of $PC(J,R)$. Let $τ 1 , τ 2 ∈J$, $τ 1 < τ 2$ with $τ 1 ∈ J v$, $τ 2 ∈ J u$, $v≤u$ for some $u,v∈{0,1,2,…,m}$ and $x∈ B ρ$. For each $h∈H(x)$, we obtain

$| h ( τ 2 ) − h ( τ 1 ) | ≤ | ∫ t u τ 2 f ( s ) d q k s − ∫ t v τ 1 f ( s ) d q k s | + | ∑ τ 1 < t k < τ 2 I k ( x ( t k ) ) | + | ∑ τ 1 < t k < τ 2 ∫ t k − 1 t k f ( s ) d q k − 1 s | ≤ | ∫ t u τ 2 f ( s ) d q k s − ∫ t v τ 1 f ( s ) d q k s | + ∑ τ 1 < t k < τ 2 | I k ( x ( t k ) ) | + ∑ τ 1 < t k < τ 2 ∫ t k − 1 t k | f ( s ) | d q k − 1 s .$

Obviously the right hand side of the above inequality tends to zero independently of $x∈ B ρ$ as $τ 2 − τ 1 →0$. Therefore it follows by the Arzelá-Ascoli theorem that $H:PC(J,R)→P(PC(J,R))$ is completely continuous.

Since 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 has a closed graph. Let $x n → x ∗$, $h n ∈H( x n )$ and $h n → h ∗$. Then we need to show that $h ∗ ∈H( x ∗ )$. Associated with $h n ∈H( x n )$, there exists $f n ∈ S F , x n$ such that, for each $t∈J$,

$h n (t)= x 0 + ∑ 0 < t k < t ∫ t k − 1 t k f n (s) d q k − 1 s+ ∑ 0 < t k < t I k ( x n ( t k ) ) + ∫ t k t f n (s) d q k s.$

Thus it suffices to show that there exists $f ∗ ∈ S F , x ∗$ such that, for each $t∈J$,

$h ∗ (t)= x 0 + ∑ 0 < t k < t ∫ t k − 1 t k f ∗ (s) d q k − 1 s+ ∑ 0 < t k < t I k ( x ∗ ( t k ) ) + ∫ t k t f ∗ (s) d q k s.$

Let us consider the linear operator $Θ: L 1 (J,R)→PC(J,R)$ given by

$f↦Θ(f)(t)= x 0 + ∑ 0 < t k < t ∫ t k − 1 t k f(s) d q k − 1 s+ ∑ 0 < t k < t I k ( x ( t k ) ) + ∫ t k t f(s) d q k s.$

Observe that

$∥ h n ( t ) − h ∗ ( t ) ∥ = ∥ ∑ 0 < t k < t ∫ t k − 1 t k ( f n ( u ) − f ∗ ( u ) ) d q k − 1 s + ∑ 0 < t k < t | I k ( x n ( t k ) ) − I k ( x ∗ ( t k ) ) | + ∫ t k t ( f n ( u ) − f ∗ ( u ) ) d q k s ∥ → 0 ,$

as $n→∞$.

Thus, it follows by Lemma 2.3 that $Θ∘ S F$ is a closed graph operator. Further, we have $h n (t)∈Θ( S F , x n )$. Since $x n → x ∗$, therefore, we have

$h ∗ (t)= x 0 + ∑ 0 < t k < t ∫ t k − 1 t k f ∗ (s) d q k − 1 s+ ∑ 0 < t k < t I k ( x ∗ ( t k ) ) + ∫ t k t f ∗ (s) d q k s,$

for some $f ∗ ∈ S F , x ∗$.

Finally, we show there exists an open set $U⊆C(J,R)$ with $x∉H(x)$ for any $λ∈(0,1)$ and all $x∈∂U$. Let $λ∈(0,1)$ and $x∈λH(x)$. Then there exists $v∈ L 1 (J,R)$ with $f∈ S F , x$ such that, for $t∈J$, we have

$x(t)= x 0 + ∑ 0 < t k < t ∫ t k − 1 t k f(s) d q k − 1 s+ ∑ 0 < t k < t I k ( x ( t k ) ) + ∫ t k t f(s) d q k s.$

Repeating the computations of the second step, we have

$| x ( t ) | ≤| x 0 |+Tψ ( ∥ x ∥ ) ∥p∥+ ∑ k = 1 m c k .$

Consequently, we have

$∥ x ∥ | x 0 | + T ψ ( ∥ x ∥ ) ∥ p ∥ + ∑ k = 1 m c k ≤1.$

In view of (H4), there exists M such that $∥x∥≠M$. Let us set

$U= { x ∈ P C ( J , R ) : ∥ x ∥ < M } .$

Note that the operator $H: U ¯ →P(PC(J,R))$ is upper semicontinuous and completely continuous. From the choice of U, there is no $x∈∂U$ such that $x∈λH(x)$ for some $λ∈(0,1)$. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 2.2), we deduce that has a fixed point $x∈ U ¯$ 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:

$D 1 2 + k x ( t ) ∈ F ( t , x ( t ) ) , t ∈ J = [ 0 , 1 ] , t ≠ t k = k 10 , Δ x ( t k ) = | x ( t k ) | 12 + | x ( t k ) | , k = 1 , 2 , … , 9 , x ( 0 ) = 0 .$
(3.3)

Here $q k =1/(2+k)$, $k=0,1,2,…,9$, $m=9$, $T=1$, and $I k (x)=|x|/(12+|x|)$. We find that $| I k (x)− I k (y)|≤(1/12)|x−y|$ and $| I k (x)|≤1$.

1. (a)

Let $F:[0,1]×R→P(R)$ be a multivalued map given by

$x→F(t,x)= [ | x | | x | + sin 2 x + 1 + t + 1 , e − x 2 + 4 5 t 2 + 3 ] .$
(3.4)

For $f∈F$, we have

$|f|≤max ( | x | | x | + sin 2 x + 1 + t + 1 , e − x 2 + t 2 + 3 ) ≤5,x∈R.$

Thus,

$∥ F ( t , x ) ∥ P :=sup { | y | : y ∈ F ( t , x ) } ≤5=p(t)ψ ( ∥ x ∥ ) ,x∈R,$

with $p(t)=1$, $ψ(∥x∥)=5$. 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 $[0,1]$.

1. (b)

If $F:[0,1]×R→P(R)$ is a multivalued map given by

$x→F(t,x)= [ ( t + 1 ) x 2 x 2 + 1 , t | x | ( cos 2 x + 1 ) 2 ( | x | + 1 ) ] .$
(3.5)

For $f∈F$, we have

$|f|≤max ( ( t + 1 ) x 2 x 2 + 1 , t | x | ( cos 2 x + 1 ) 2 ( | x | + 1 ) ) ≤t+1,x∈R.$

Here $∥ F ( t , x ) ∥ P :=sup{|y|:y∈F(t,x)}≤(t+1)=p(t)ψ(∥x∥)$, $x∈R$, with $p(t)=t+1$, $ψ(∥x∥)=1$. 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 $y∈C(J,R)$, then for any $t∈J$, the solution of the problem

$D q k 2 x ( t ) = y ( t ) , t ∈ J , t ≠ t k , Δ x ( t k ) = I k ( x ( t k ) ) , k = 1 , 2 , … , m , D q k x ( t k + ) − D q k − 1 x ( t k ) = I k ∗ ( x ( t k ) ) , k = 1 , 2 , … , m , x ( 0 ) = α , D q 0 x ( 0 ) = β ,$
(4.1)

is given by

$x ( t ) = α + β t + ∑ 0 < t k < t ( ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) y ( s ) d q k − 1 s + I k ( x ( t k ) ) ) + t [ ∑ 0 < t k < t ( ∫ t k − 1 t k f y ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) ] − ∑ 0 < t k < t t k ( ∫ t k − 1 t k y ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) + ∫ t k t ( t − q k s − ( 1 − q k ) t k ) y ( s ) d q k s ,$
(4.2)

with $∑ 0 < 0 (⋅)=0$.

Definition 4.2 A function $x∈PC(J,R)$ is said to be a solution of (1.5) if $x(0)= x 0$, $D q 0 x(0)=β$, $Δx( t k )= I k (x( t k ))$, $D q k x( t k + )− D q k − 1 x( t k )= I k ∗ (x( t k ))$, $k=1,2,…,m$ and there exists $f∈ L 1 (J,R)$ such that $f(t)∈F(t,x(t))$ on J and

$x ( t ) = α + β t + ∑ 0 < t k < t ( ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) f ( s ) d q k − 1 s + I k ( x ( t k ) ) ) + t [ ∑ 0 < t k < t ( ∫ t k − 1 t k f ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) ] − ∑ 0 < t k < t t k ( ∫ t k − 1 t k f ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) + ∫ t k t ( t − q k s − ( 1 − q k ) t k ) f ( s ) d q k s ,$
(4.3)

with $∑ 0 < 0 (⋅)=0$.

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

(A1) there exist constants $c k$, $c k ∗$ such that $| I k (x)|≤ c k$, $| I k ∗ (y)|≤ c k ∗$, $k=1,2,…,m$ for each $x,y∈R$;

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

$M | α | + | β | T + ∥ p ∥ ψ ( M ) Λ 1 + ∑ k = 1 m [ c k + c k ∗ ( T + t k ) ] >1,$

where

$Λ 1 = ∑ k = 1 m + 1 ( t k − t k − 1 ) 2 1 + q k − 1 + ∑ k = 1 m (T+ t k )( t k − t k − 1 ).$
(4.4)

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

Proof Define the operator $H:PC(J,R)→P(PC(J,R))$ by

$H ( x ) = h ∈ P C ( J , R ) : h ( t ) = α + β t + ∑ 0 < t k < t ( ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) f ( s ) d q k − 1 s + I k ( x ( t k ) ) ) + t [ ∑ 0 < t k < t ( ∫ t k − 1 t k f ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) ] − ∑ 0 < t k < t t k ( ∫ t k − 1 t k f ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) + ∫ t k t ( t − q k s − ( 1 − q k ) t k ) f ( s ) d q k s ,$

for $f∈ S F , x$.

We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that is convex for each $x∈PC(J,R)$. 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 maps bounded sets (balls) into bounded sets in $PC(J,R)$. For a positive number ρ, let $B ρ ={x∈PC(J,R):∥x∥≤ρ}$ be a bounded ball in $PC(J,R)$. Then, for each $h∈H(x)$, $x∈ B ρ$, there exists $f∈ S F , x$ such that

$h ( t ) = α + β t + ∑ 0 < t k < t ( ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) f ( s ) d q k − 1 s + I k ( x ( t k ) ) ) + t [ ∑ 0 < t k < t ( ∫ t k − 1 t k f ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) ] − ∑ 0 < t k < t t k ( ∫ t k − 1 t k f ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) + ∫ t k t ( t − q k s − ( 1 − q k ) t k ) f ( s ) d q k s .$

Then for $t∈J$ we have

$| h ( t ) | ≤ | α | + | β | t + ∑ 0 < t k < t ( ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) | f ( s ) | d q k − 1 s + | I k ( x ( t k ) ) | ) + t [ ∑ 0 < t k < t ( ∫ t k − 1 t k | f ( s ) | d q k − 1 s + | I k ∗ ( x ( t k ) ) | ) ] + ∑ 0 < t k < t t k ( ∫ t k − 1 t k | f ( s ) | d q k − 1 s + | I k ∗ ( x ( t k ) ) | ) + ∫ t k t ( t − q k s − ( 1 − q k ) t k ) | f ( s ) | d q k s ≤ | α | + | β | T + ∑ 0 < t k < T ( ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) p ( s ) ψ ( ∥ x ∥ ) d q k − 1 s + | I k ( x ( t k ) ) | ) + T [ ∑ 0 < t k < T ( ∫ t k − 1 t k p ( s ) ψ ( ∥ x ∥ ) d q k − 1 s + | I k ∗ ( x ( t k ) ) | ) ] + ∑ 0 < t k < T t k ( ∫ t k − 1 t k p ( s ) ψ ( ∥ x ∥ ) d q k − 1 s + | I k ∗ ( x ( t k ) ) | ) + ∫ t m T ( T − q m s − ( 1 − q m ) t m ) p ( s ) ψ ( ∥ x ∥ ) d q m s = | α | + | β | T + ∑ k = 1 m ( ( t k − t k − 1 ) 2 1 + q k − 1 ∥ p ∥ ψ ( ∥ x ∥ ) + c k ) + T [ ∑ k = 1 m ( ∥ p ∥ ψ ( ∥ x ∥ ) ( t k − t k − 1 ) + c k ∗ ) ] + ∑ k = 1 m t k ( ∥ p ∥ ψ ( ∥ x ∥ ) ( t k − t k − 1 ) + c k ∗ ) + ( T − t m ) 2 1 + q m ∥ p ∥ ψ ( ∥ x ∥ ) = | α | + | β | T + ∥ p ∥ ψ ( ∥ x ∥ ) { ∑ k = 1 m + 1 ( t k − t k − 1 ) 2 1 + q k − 1 + ∑ k = 1 m ( T + t k ) ( t k − t k − 1 ) } + ∑ k = 1 m [ c k + c k ∗ ( T + t k ) ] .$

Consequently,

$∥ h ∥ ≤ | α | + | β | T + ∥ p ∥ ψ ( ρ ) { ∑ k = 1 m + 1 ( t k − t k − 1 ) 2 1 + q k − 1 + ∑ k = 1 m ( T + t k ) ( t k − t k − 1 ) } + ∑ k = 1 m [ c k + c k ∗ ( T + t k ) ] .$

Now we show that maps bounded sets into equicontinuous sets of $PC(J,R)$. Let $τ 1 , τ 2 ∈J$, $τ 1 < τ 2$ with $τ 1 ∈ J u$, $τ 2 ∈ J v$, $u≤v$ for some $u,v∈{0,1,2,…,m}$ and $x∈ B ρ$. For each $h∈H(x)$, we obtain

$| h ( τ 2 ) − h ( τ 1 ) | ≤ | β | | τ 2 − τ 1 | + ∑ τ 1 < t k < τ 2 ( ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) | f ( s ) | d q k − 1 s + | I k ( x ( t k ) ) | ) + | τ 2 − τ 1 | [ ∑ 0 < t k < τ 1 ( ∫ t k − 1 t k | f ( s ) | d q k − 1 s + | I k ∗ ( x ( t k ) ) | ) ] + τ 2 [ ∑ τ 1 < t k < τ 2 ( ∫ t k − 1 t k | f ( s ) | d q k − 1 s + | I k ∗ ( x ( t k ) ) | ) ] + ∑ τ 1 < t k < τ 2 t k ( ∫ t k − 1 t k | f ( s ) | d q k − 1 s + | I k ∗ ( x ( t k ) ) | ) + | ∫ t v τ 2 ( τ 2 − q k s − ( 1 − q k ) t k ) | f ( s ) | d q k s − ∫ t u τ 1 ( τ 1 − q k s − ( 1 − q k ) t k ) | f ( s ) | d q k s | .$

Obviously the right hand side of the above inequality tends to zero independently of $x∈ B ρ$ as $τ 2 − τ 1 →0$. Therefore it follows by the Arzelá-Ascoli theorem that $H:PC(J,R)→P(PC(J,R))$ is completely continuous.

Since 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 has a closed graph. Let $x n → x ∗$, $h n ∈H( x n )$ and $h n → h ∗$. Then we need to show that $h ∗ ∈H( x ∗ )$. Associated with $h n ∈H( x n )$, there exists $f n ∈ S F , x n$ such that, for each $t∈J$,

$h n ( t ) = α + β t + ∑ 0 < t k < t ( ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) f n ( s ) d q k − 1 s + I k ( x ( t k ) ) ) + t [ ∑ 0 < t k < t ( ∫ t k − 1 t k f n ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) ] − ∑ 0 < t k < t t k ( ∫ t k − 1 t k f n ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) + ∫ t k t ( t − q k s − ( 1 − q k ) t k ) f n ( s ) d q k s .$

Thus it suffices to show that there exists $f ∗ ∈ S F , x ∗$ such that, for each $t∈J$,

$h ∗ ( t ) = α + β t + ∑ 0 < t k < t ( ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) f ∗ ( s ) d q k − 1 s + I k ( x ( t k ) ) ) + t [ ∑ 0 < t k < t ( ∫ t k − 1 t k f ∗ ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) ] − ∑ 0 < t k < t t k ( ∫ t k − 1 t k f ∗ ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) + ∫ t k t ( t − q k s − ( 1 − q k ) t k ) f ∗ ( s ) d q k s .$

Let us consider the linear operator $Θ: L 1 (J,R)→PC(J,R)$ given by

$f ↦ Θ ( f ) ( t ) = α + β t + ∑ 0 < t k < t ( ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) f ( s ) d q k − 1 s + I k ( x ( t k ) ) ) + t [ ∑ 0 < t k < t ( ∫ t k − 1 t k f ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) ] − ∑ 0 < t k < t t k ( ∫ t k − 1 t k f ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) + ∫ t k t ( t − q k s − ( 1 − q k ) t k ) f ( s ) d q k s .$

Observe that

$∥ h n ( t ) − h ∗ ( t ) ∥ = ∥ ∑ 0 < t k < t ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) ( f n ( u ) − f ∗ ( u ) ) d q k − 1 s + ∑ 0 < t k < t | I k ( x n ( t k ) ) − I k ( x ∗ ( t k ) ) | + T ∑ 0 < t k < t ∫ t k − 1 t k ( f n ( u ) − f ∗ ( u ) ) d q k − 1 s + T ∑ 0 < t k < t | I k ∗ ( x n ( t k ) ) − I k ∗ ( x ∗ ( t k ) ) | + ∑ 0 < t k < t t k ∫ t k − 1 t k ( f n ( u ) − f ∗ ( u ) ) d q k − 1 s + ∑ 0 < t k < t | I k ∗ ( x n ( t k ) ) − I k ∗ ( x ∗ ( t k ) ) | + ∫ t k t ( t − q k s − ( 1 − q k ) t k ) ( f n ( u ) − f ∗ ( u ) ) d q k s ∥ → 0 ,$

as $n→∞$.

Thus, it follows by Lemma 2.3 that $Θ∘ S F$ is a closed graph operator. Further, we have $h n (t)∈Θ( S F , x n )$. Since $x n → x ∗$, therefore, we have

$h ∗ ( t ) = α + β t + ∑ 0 < t k < t ( ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) f ∗ ( s ) d q k − 1 s + I k ( x ( t k ) ) ) + t [ ∑ 0 < t k < t ( ∫ t k − 1 t k f ∗ ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) ] − ∑ 0 < t k < t t k ( ∫ t k − 1 t k f ∗ ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) + ∫ t k t ( t − q k s − ( 1 − q k ) t k ) f ∗ ( s ) d q k s ,$

for some $f ∗ ∈ S F , x ∗$.

Finally, we show there exists an open set $U⊆C(J,R)$ with $x∉H(x)$ for any $λ∈(0,1)$ and all $x∈∂U$. Let $λ∈(0,1)$ and $x∈λH(x)$. Then there exists $f∈ L 1 (J,R)$ with $f∈ S F , x$ such that, for $t∈J$, we have

$x ( t ) = α + β t + ∑ 0 < t k < t ( ∫ t k − 1 t k ( t k − q k − 1 s − ( 1 − q k − 1 ) t k − 1 ) f ( s ) d q k − 1 s + I k ( x ( t k ) ) ) + t [ ∑ 0 < t k < t ( ∫ t k − 1 t k f ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) ] − ∑ 0 < t k < t t k ( ∫ t k − 1 t k f ( s ) d q k − 1 s + I k ∗ ( x ( t k ) ) ) + ∫ t k t ( t − q k s − ( 1 − q k ) t k ) f ( s ) d q k s .$

Repeating the computations of the second step, we have

$| x ( t ) | ≤ | α | + | β | T + ∥ p ∥ ψ ( ∥ x ∥ ) { ∑ k = 1 m + 1 ( t k − t k − 1 ) 2 1 + q k − 1 + ∑ k = 1 m ( T + t k ) ( t k − t k − 1 ) } + ∑ k = 1 m [ c k + c k ∗ ( T + t k ) ] .$

Consequently, we have

$∥ x ∥ | α | + | β | T + ∥ p ∥ ψ ( ∥ x ∥ ) Λ 1 + ∑ k = 1 m [ c k + c k ∗ ( T + t k ) ] ≤1.$

In view of (A2), there exists M such that $∥x∥≠M$. Let us set

$U= { x ∈ P C ( J , R ) : ∥ x ∥ < M } .$

Note that the operator $H: U ¯ →P(PC(J,R))$ is upper semicontinuous and completely continuous. From the choice of U, there is no $x∈∂U$ such that $x∈λH(x)$ for some $λ∈(0,1)$. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 2.2), we deduce that has a fixed point $x∈ U ¯$ 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:

${ D 2 3 + k 2 x ( t ) ∈ F ( t , x ( t ) ) , t ∈ J = [ 0 , 1 ] , t ≠ t k = k 10 , Δ x ( t k ) = | x ( t k ) | 15 ( 6 + | x ( t k ) | ) , k = 1 , 2 , … , 9 , D 2 3 + k x ( t k + ) − D 2 3 + k − 1 x ( t k ) = | x ( t k ) | 19 ( 3 + | x ( t k ) | ) , k = 1 , 2 , … , 9 , x ( 0 ) = 0 , D 2 3 x ( 0 ) = 0 .$
(4.5)

Here $q k =2/(3+k)$, $k=0,1,2,…,9$, $m=9$, $T=1$, $α=0$, $β=0$, $I k (x)=|x|/(15(6+|x|))$, and $I k ∗ (x)=|x|/(19(3+|x|))$. We find that $| I k (x)− I k (y)|≤(1/90)|x−y|$, $| I k ∗ (x)− I k ∗ (y)|≤(1/57)|x−y|$, and $I k (x)≤1/15$, $I k ∗ (x)≤1/19$; and we have

$Λ 1 = ∑ k = 1 m + 1 ( t k − t k − 1 ) 2 1 + q k − 1 + ∑ k = 1 m (T+ t k )( t k − t k − 1 )≈1.42663542.$
1. (a)

Let $F:[0,1]×R→P(R)$ be a multivalued map given by

$x→F(t,x)= [ | x | | x | + sin 2 x + 1 + t + 1 , e − x 2 + 4 5 t 2 + 3 ] .$
(4.6)

For $f∈F$, we have

$|f|≤max ( | x | | x | + sin 2 x + 1 + t + 1 , e − x 2 + t 2 + 3 ) ≤5,x∈R.$

Thus,

$∥ F ( t , x ) ∥ P :=sup { | y | : y ∈ F ( t , x ) } ≤5=p(t)ψ ( ∥ x ∥ ) ,x∈R,$

with $p(t)=1$, $ψ(∥x∥)=5$. Further, using the condition (A2) we find

$M 5 Λ 1 + ∑ k = 1 9 [ 1 15 + 1 19 ( 1 + t k ) ] >1,$

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 $[0,1]$.

1. (b)

If $F:[0,1]×R→P(R)$ is a multivalued map given by

$x→F(t,x)= [ ( t + 1 ) x 2 x 2 + 1 , t | x | ( cos 2 x + 1 ) 2 ( | x | + 1 ) ] .$
(4.7)

For $f∈F$, we have

$|f|≤max ( ( t + 1 ) x 2 x 2 + 1 , t | x | ( cos 2 x + 1 ) 2 ( | x | + 1 ) ) ≤t+1,x∈R.$

Here $∥ F ( t , x ) ∥ P :=sup{|y|:y∈F(t,x)}≤(t+1)=p(t)ψ(∥x∥)$, $x∈R$, with $p(t)=t+1$, $ψ(∥x∥)=1$. 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.

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## Acknowledgements

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

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### Competing interests

The authors declare that they have no competing interests.

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Ntouyas, S.K., Tariboon, J. Applications of quantum calculus on finite intervals to impulsive difference inclusions. Adv Differ Equ 2014, 262 (2014). https://doi.org/10.1186/1687-1847-2014-262

• $q k$-derivative
• $q k$-integral