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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 kN{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=(1q)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 kN. 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:TT 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]×RP(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,βR 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 P c l (X)={YP(X):Y is closed}, P c p (X)={YP(X):Y is compact}, and P c p , c (X)={YP(X):Y is compact and convex}.

A multivalued map G:XP(X) is convex (closed) valued if G(x) is convex (closed) for all xX; 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|:yG(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 yC, define the set of selections of F by

S F , y := { v C : v ( t ) F ( t , y ( t ) )  on  [ 0 , T ] } .

Definition 2.1 A multivalued map F:J×RP(R) is said to be Carathéodory (in the sense of q k -calculus) if xF(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|:vF(t,x)} φ α (t) for all xα 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 0U. 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 uU and λ(0,1) with uλF(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:JR: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)|;tJ}.

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 yPC(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 xPC(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×RP(R) is Carathéodory and has nonempty compact and convex values;

(H2) there exist a continuous nondecreasing function ψ:[0,)(0,) and a function pC(J, R + ) such that

F ( t , x ) P :=sup { | y | : y F ( t , x ) } p(t)ψ ( x ) for each (t,x)J×R;

(H3) there exist constants c k such that | I k (y)| c k , k=1,2,,m for each yR;

(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)=hPC(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 xPC(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 ={xC(J,R):xρ} be a bounded ball in C(J,R). Then, for each hH(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 tJ 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 , vu for some u,v{0,1,2,,m} and x B ρ . For each hH(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 tJ,

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 tJ,

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

fΘ(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 UC(J,R) with xH(x) for any λ(0,1) and all xU. Let λ(0,1) and xλH(x). Then there exists v L 1 (J,R) with f S F , x such that, for tJ, 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 xM. 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 xU such that xλH(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)|xy| and | I k (x)|1.

  1. (a)

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

    xF(t,x)= [ | x | | x | + sin 2 x + 1 + t + 1 , e x 2 + 4 5 t 2 + 3 ] .
    (3.4)

For fF, we have

|f|max ( | x | | x | + sin 2 x + 1 + t + 1 , e x 2 + t 2 + 3 ) 5,xR.

Thus,

F ( t , x ) P :=sup { | y | : y F ( t , x ) } 5=p(t)ψ ( x ) ,xR,

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]×RP(R) is a multivalued map given by

    xF(t,x)= [ ( t + 1 ) x 2 x 2 + 1 , t | x | ( cos 2 x + 1 ) 2 ( | x | + 1 ) ] .
    (3.5)

For fF, we have

|f|max ( ( t + 1 ) x 2 x 2 + 1 , t | x | ( cos 2 x + 1 ) 2 ( | x | + 1 ) ) t+1,xR.

Here F ( t , x ) P :=sup{|y|:yF(t,x)}(t+1)=p(t)ψ(x), xR, 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 yC(J,R), then for any tJ, 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 xPC(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,yR;

(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 xPC(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 ={xPC(J,R):xρ} be a bounded ball in PC(J,R). Then, for each hH(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 tJ 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 , uv for some u,v{0,1,2,,m} and x B ρ . For each hH(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 tJ,

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 tJ,

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 UC(J,R) with xH(x) for any λ(0,1) and all xU. Let λ(0,1) and xλH(x). Then there exists f L 1 (J,R) with f S F , x such that, for tJ, 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 xM. 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 xU such that xλH(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)|xy|, | I k (x) I k (y)|(1/57)|xy|, 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]×RP(R) be a multivalued map given by

    xF(t,x)= [ | x | | x | + sin 2 x + 1 + t + 1 , e x 2 + 4 5 t 2 + 3 ] .
    (4.6)

For fF, we have

|f|max ( | x | | x | + sin 2 x + 1 + t + 1 , e x 2 + t 2 + 3 ) 5,xR.

Thus,

F ( t , x ) P :=sup { | y | : y F ( t , x ) } 5=p(t)ψ ( x ) ,xR,

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]×RP(R) is a multivalued map given by

    xF(t,x)= [ ( t + 1 ) x 2 x 2 + 1 , t | x | ( cos 2 x + 1 ) 2 ( | x | + 1 ) ] .
    (4.7)

For fF, we have

|f|max ( ( t + 1 ) x 2 x 2 + 1 , t | x | ( cos 2 x + 1 ) 2 ( | x | + 1 ) ) t+1,xR.

Here F ( t , x ) P :=sup{|y|:yF(t,x)}(t+1)=p(t)ψ(x), xR, 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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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

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Keywords

  • q k -derivative
  • q k -integral
  • impulsive q-difference inclusion