Skip to main content

Nonlinear second-order impulsive q-difference Langevin equation with boundary conditions

Abstract

In this paper, we discuss the existence and uniqueness of solutions for Langevinimpulsive q-difference equations with boundary conditions. Our studyrelies on Banach’s and Schaefer’s fixed point theorems. Illustrativeexamples are also presented.

MSC: 26A33, 39A13, 34A37.

1 Introduction and preliminaries

In recent years, the boundary value problems of fractional order differentialequations have emerged as an important area of research, since these problems haveapplications in various disciplines of science and engineering such as mechanics,electricity, chemistry, biology, economics, control theory, signal and imageprocessing, polymer rheology, regular variation in thermodynamics, biophysics,aerodynamics, viscoelasticity and damping, electro-dynamics of complex medium, wavepropagation, blood flow phenomena, etc.[15]. Many researchers have studied the existence theory for nonlinearfractional differential equations with a variety of boundary conditions, forinstance, see the papers [618], and the references therein.

The Langevin equation (first formulated by Langevin in 1908) is found to be aneffective tool to describe the evolution of physical phenomena in fluctuatingenvironments [19]. For some new developments on the fractional Langevin equation, see, forexample, [2027].

Nowadays there is a significant increase of activities in the area ofq-calculus due to its applications in various fields such as mathematics,mechanics, and physics. The book by Kac and Cheung [28] covers many of the fundamental aspects of the quantum calculus.A variety of new results can be found in the papers [2941] and the references cited therein.

Impulsive differential equations serve as basic models to study the dynamics ofprocesses that are subject to sudden changes in their states. Recent development inthis field has been motivated by many applied problems, such as control theory,population dynamics and medicine. For some recent works on the theory of impulsivedifferential equations, we refer the interested reader to the monographs [4244].

Recently in [45] the notions of q k -derivative and q k -integral on finite intervals were introduced. Let usrecall here these notions. 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 lett 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 thefunction 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 be 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 q k -integral is defined as follows.

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 afunction f(t), defined by 0 t f(s) d q s=(1q)t n = 0 q n f( q n t) for t[0,).

For the basic properties of the q k -derivative and q k -integral we refer to [45].

In this paper we combine all the above subjects and investigate the nonlinearsecond-order impulsive q k -difference Langevin equation with boundary conditionsof the form

{ D q k ( D q k + λ ) 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 , α x ( 0 ) + β D q 0 x ( 0 ) = x ( T ) , γ x ( 0 ) + η D q 0 x ( 0 ) = D q m x ( T ) ,
(1.4)

where 0= t 0 < t 1 < t 2 << t k << t m < t m + 1 =T, f:J×RR is a continuous function, λ is a givenconstant, I k , I k C(R,R), Δx( t k )=x( t k + )x( t k ) for k=1,2,,m, x( t k + )= lim h 0 + x( t k +h), 0< q k <1 for k=0,1,2,,m, and α, β,γ, η are given constants.

The rest of this paper is organized as follows. In Section 2, we present apreliminary result which will be used in this paper. In Section 3, we willconsider the existence results for problem (1.4) while in Section 4, we willgive examples to illustrate our main results.

2 An auxiliary lemma

In this section, we present an auxiliary lemma which will be used throughout thispaper. Let J=[0,T], J 0 =[ t 0 , t 1 ], J k =( t k , t k + 1 ] for k=1,2,,m.

Lemma 2.1 LetλT(η+βλα)(α1)(η1)+γ(Tβ). The unique solution of problem (1.4) isgiven by

x ( t ) = δ 1 + δ 2 t Ω { k = 1 m ( t k 1 t k t k 1 s f ( r , x ( r ) ) d q k 1 r d q k 1 s λ t k 1 t k x ( s ) d q k 1 s + I k ( x ( t k ) ) ) + k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) ( T t k ) + t m T t m s f ( r , x ( r ) ) d q m r d q m s λ t m T x ( s ) d q m s } + δ 3 + δ 4 t Ω { k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) + t m T f ( s , x ( s ) ) d q m s } + 0 < t k < t ( t k 1 t k t k 1 s f ( r , x ( r ) ) d q k 1 r d q k 1 s λ t k 1 t k x ( s ) d q k 1 s + I k ( x ( t k ) ) ) + 0 < t k < t ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) ( t t k ) + t k t t k s f ( r , x ( r ) ) d q k r d q k s λ t k t x ( s ) d q k s ,
(2.1)

with 0 < 0 ()=0, where

Ω = ( α 1 ) ( η 1 ) λ T ( η + β λ α ) + γ ( T β ) , δ 1 = η 1 + β λ , δ 2 = λ ( η + β λ 1 α ) + 1 γ , δ 3 = T β , δ 4 = α 1 β λ .

Proof For t J 0 using q 0 -integral for the first equation of (1.4), we get

D q 0 x(t)= D q 0 x(0)+λx(0)+ 0 t f ( s , x ( s ) ) d q 0 sλx(t).

Setting x(0)=A and D q 0 x(0)=B, we have

D q 0 x(t)=λA+B+ 0 t f ( s , x ( s ) ) d q 0 sλx(t),
(2.2)

which leads to

D q 0 x( t 1 )=λA+B+ 0 t 1 f ( s , x ( s ) ) d q 0 sλx( t 1 ).
(2.3)

For t J 0 we obtain by q 0 -integrating (2.2),

x(t)=A+(λA+B)t+ 0 t 0 s f ( r , x ( r ) ) d q 0 r d q 0 sλ 0 t x(s) d q 0 s.

In particular, for t= t 1

x( t 1 )=A+(λA+B) t 1 + 0 t 1 0 s f ( r , x ( r ) ) d q 0 r d q 0 sλ 0 t 1 x(s) d q 0 s.
(2.4)

For t J 1 =( t 1 , t 2 ], q 1 -integrating (1.4), we have

D q 1 x(t)= D q 1 x ( t 1 + ) +λx ( t 1 + ) + t 1 t f ( s , x ( s ) ) d q 1 sλx(t).

From the second impulsive equations of (1.4), we have

D q 1 x ( t ) = λ A + B + 0 t 1 f ( s , x ( s ) ) d q 0 s + I 1 ( x ( t 1 ) ) + λ I 1 ( x ( t 1 ) ) + t 1 t f ( s , x ( s ) ) d q 1 s λ x ( t ) .
(2.5)

Applying q 1 -integral to (2.5) for t J 1 , we obtain

x ( t ) = x ( t 1 + ) + [ λ A + B + 0 t 1 f ( s , x ( s ) ) d q 0 s + I 1 ( x ( t 1 ) ) + λ I 1 ( x ( t 1 ) ) ] ( t t 1 ) + t 1 t t 1 s f ( r , x ( r ) ) d q 1 r d q 1 s λ t 1 t x ( s ) d q 1 s .
(2.6)

Using the second impulsive equation of (1.4) with (2.4) and (2.6), one has

x ( t ) = A + ( λ A + B ) t 1 + 0 t 1 0 s f ( r , x ( r ) ) d q 0 r d q 0 s λ 0 t 1 x ( s ) d q 0 s + I 1 ( x ( t 1 ) ) + [ λ A + B + 0 t 1 f ( s , x ( s ) ) d q 0 s + I 1 ( x ( t 1 ) ) + λ I 1 ( x ( t 1 ) ) ] ( t t 1 ) + t 1 t t 1 s f ( r , x ( r ) ) d q 1 r d q 1 s λ t 1 t x ( s ) d q 1 s = A + ( λ A + B ) t + 0 t 1 0 s f ( r , x ( r ) ) d q 0 r d q 0 s λ 0 t 1 x ( s ) d q 0 s + I 1 ( x ( t 1 ) ) + [ 0 t 1 f ( s , x ( s ) ) d q 0 s + I 1 ( x ( t 1 ) ) + λ I 1 ( x ( t 1 ) ) ] ( t t 1 ) + t 1 t t 1 s f ( r , x ( r ) ) d q 1 r d q 1 s λ t 1 t x ( s ) d q 1 s .

Repeating the above process, for tJ, we get

x ( t ) = A + ( λ A + B ) t + 0 < t k < t ( t k 1 t k t k 1 s f ( r , x ( r ) ) d q k 1 r d q k 1 s λ t k 1 t k x ( s ) d q k 1 s + I k ( x ( t k ) ) ) + 0 < t k < t ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) ( t t k ) + t k t t k s f ( r , x ( r ) ) d q k r d q k s λ t k t x ( s ) d q k s .
(2.7)

For t=T, we get

x ( T ) = ( 1 + λ T ) A + B T + k = 1 m ( t k 1 t k t k 1 s f ( r , x ( r ) ) d q k 1 r d q k 1 s λ t k 1 t k x ( s ) d q k 1 s + I k ( x ( t k ) ) ) + k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) ( T t k ) + t m T t m s f ( r , x ( r ) ) d q m r d q m s λ t m T x ( s ) d q m s .
(2.8)

It is easy to see that

D q k x ( t ) = λ A + B + 0 < t k < t ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) + t k t f ( s , x ( s ) ) d q k s λ x ( t ) .

For t=T and using x(T)=αA+βB, we have

D q m x ( T ) = λ A + B + k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) + t m T f ( s , x ( s ) ) d q m s λ x ( T ) = ( 1 α ) λ A + ( 1 λ β ) B + k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) + t m T f ( s , x ( s ) ) d q m s .
(2.9)

Applying the boundary conditions of (1.4) with (2.8) and (2.9), it follows that

A = η + λ β 1 Ω { k = 1 m ( t k 1 t k t k 1 s f ( r , x ( r ) ) d q k 1 r d q k 1 s λ t k 1 t k x ( s ) d q k 1 s + I k ( x ( t k ) ) ) + k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) ( T t k ) + t m T t m s f ( r , x ( r ) ) d q m r d q m s λ t m T x ( s ) d q m s } β T Ω { k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) + t m T f ( s , x ( s ) ) d q m s }

and

B = α 1 λ T Ω { k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) + t m T f ( s , x ( s ) ) d q m s } γ 1 + α λ Ω { k = 1 m ( t k 1 t k t k 1 s f ( r , x ( r ) ) d q k 1 r d q k 1 s λ t k 1 t k x ( s ) d q k 1 s + I k ( x ( t k ) ) ) + k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) ( T t k ) + t m T t m s f ( r , x ( r ) ) d q m r d q m s λ t m T x ( s ) d q m s } .

Substituting the values of A and B into (2.7), we get (2.1) asrequired. The proof is completed. □

3 Main results

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 norm x P C =sup{|x(t)|;tJ}.

From Lemma 2.1, we define an operator S:PC(J,R)PC(J,R) by

( S x ) ( t ) = δ 1 + δ 2 t Ω { k = 1 m ( t k 1 t k t k 1 s f ( r , x ( r ) ) d q k 1 r d q k 1 s λ t k 1 t k x ( s ) d q k 1 s + I k ( x ( t k ) ) ) + k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) ( T t k ) + t m T t m s f ( r , x ( r ) ) d q m r d q m s λ t m T x ( s ) d q m s } + δ 3 + δ 4 t Ω { k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) + t m T f ( s , x ( s ) ) d q m s } + 0 < t k < t ( t k 1 t k t k 1 s f ( r , x ( r ) ) d q k 1 r d q k 1 s λ t k 1 t k x ( s ) d q k 1 s + I k ( x ( t k ) ) ) + 0 < t k < t ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) ( t t k ) + t k t t k s f ( r , x ( r ) ) d q k r d q k s λ t k t x ( s ) d q k s ,
(3.1)

where constants δ 1 , δ 2 , δ 3 , δ 4 , and Ω are defined as in Lemma 2.1. Itshould be noticed that problem (1.4) has solutions if and only if the operator has fixed points.

Our first result is an existence and uniqueness result for the impulsive boundaryvalue problem (1.4) by using the Banach contraction mapping principle.

For convenience, we set

Λ 1 = ( | δ 1 | + | δ 2 | T + | Ω | | Ω | ) [ L 1 k = 1 m + 1 ( t k t k 1 ) 2 1 + q k 1 + | λ | k = 1 m + 1 ( t k t k 1 ) + m L 2 + k = 1 m ( L 1 ( t k t k 1 ) + L 3 + | λ | L 2 ) ( T t k ) ] + ( | δ 3 | + | δ 4 | T | Ω | ) [ L 1 T + m L 3 + m | λ | L 2 ]
(3.2)

and

Λ 2 = ( | δ 1 | + | δ 2 | T + | Ω | | Ω | ) [ K 1 k = 1 m + 1 ( t k t k 1 ) 2 1 + q k 1 + m K 2 + k = 1 m ( K 1 ( t k t k 1 ) + K 3 + | λ | K 2 ) ( T t k ) ] + ( | δ 3 | + | δ 4 | T | Ω | ) [ K 1 T + m K 3 + m | λ | K 2 ] .
(3.3)

Theorem 3.1 Assume that the following conditions hold:

(H1) f:[0,T]×RRis a continuous function and there exists a constant L 1 >0such that

|f(t,x)f(t,y)| L 1 |xy|

for eachtJandx,yR.

(H2) The functions I k , I k :RRare continuous and there exist constants L 2 , L 3 >0such that

| I k (x) I k (y)| L 2 |xy|and| I k (x) I k (y)| L 3 |xy|

for eachx,yR, k=1,2,,m.

If

Λ 1 δ<1,
(3.4)

where Λ 1 is defined by (3.2), then the impulsive q k -difference Langevin boundary value problem(1.4) has a unique solution on J.

Proof Firstly, we transform the impulsive q k -difference Langevin boundary value problem (1.4) intoa fixed point problem, x=Sx, where the operator is defined by (3.1).Applying the Banach contraction mapping principle, we shall show that has a fixed point which is the unique solution of theboundary value problem (1.4).

Let K 1 , K 2 , and K 3 be nonnegative constants such that K 1 = sup t J |f(t,0)|, K 2 =sup{| I k (0)|:k=1,2,,m}, and K 3 =sup{| I k (0)|:k=1,2,,m}. We choose a suitable constant ρ by

ρ Λ 2 1 ε ,

where δε<1 and Λ 2 defined by (3.3). Now, we will show thatS B ρ B ρ , where a set B ρ is defined as B ρ ={xPC(J,R):xρ}. For x B ρ , we have

S x sup t J { | δ 1 | + | δ 2 | t | Ω | { k = 1 m ( t k 1 t k t k 1 s | f ( r , x ( r ) ) | d q k 1 r d q k 1 s + | λ | t k 1 t k | x ( s ) | d q k 1 s + | I k ( x ( t k ) ) | ) + k = 1 m ( t k 1 t k | f ( s , x ( s ) ) | d q k 1 s + | I k ( x ( t k ) ) | + | λ | | I k ( x ( t k ) ) | ) ( T t k ) + t m T t m s | f ( r , x ( r ) ) | d q m r d q m s + | λ | t m T | x ( s ) | d q m s } + | δ 3 | + | δ 4 | t | Ω | { k = 1 m ( t k 1 t k | f ( s , x ( s ) ) | d q k 1 s + | I k ( x ( t k ) ) | + | λ | | I k ( x ( t k ) ) | ) + t m T | f ( s , x ( s ) ) | d q m s } + 0 < t k < t ( t k 1 t k t k 1 s | f ( r , x ( r ) ) | d q k 1 r d q k 1 s + | λ | t k 1 t k | x ( s ) | d q k 1 s + | I k ( x ( t k ) ) | ) + 0 < t k < t ( t k 1 t k | f ( s , x ( s ) ) | d q k 1 s + | I k ( x ( t k ) ) | + | λ | | I k ( x ( t k ) ) | ) ( t t k ) + t k t t k s | f ( r , x ( r ) ) | d q k r d q k s + | λ | t k t | x ( s ) | d q k s } | δ 1 | + | δ 2 | T | Ω | { k = 1 m ( t k 1 t k t k 1 s ( | f ( r , x ( r ) ) f ( r , 0 ) | + | f ( r , 0 ) | ) d q k 1 r d q k 1 s + | λ | t k 1 t k x d q k 1 s + ( | I k ( x ( t k ) ) I k ( 0 ) | + | I k ( 0 ) | ) ) + k = 1 m ( t k 1 t k ( | f ( s , x ( s ) ) f ( s , 0 ) | + | f ( s , 0 ) | ) d q k 1 s + ( | I k ( x ( t k ) ) I k ( 0 ) | + | I k ( 0 ) | ) + | λ | ( | I k ( x ( t k ) ) I k ( x ( 0 ) ) | + | I k ( x ( 0 ) ) | ) ) ( T t k ) + t m T t m s ( | f ( r , x ( r ) ) f ( r , 0 ) | + | f ( r , 0 ) | ) d q m r d q m s + | λ | t m T x d q m s } + | δ 3 | + | δ 4 | T | Ω | { k = 1 m ( t k 1 t k ( | f ( s , x ( s ) ) f ( s , 0 ) | + | f ( s , 0 ) | ) d q k 1 s + ( | I k ( x ( t k ) ) I k ( 0 ) | + | I k ( 0 ) | ) + | λ | ( | I k ( x ( t k ) ) I k ( x ( 0 ) ) | + | I k ( x ( 0 ) ) | ) ) + t m T ( | f ( s , x ( s ) ) f ( s , 0 ) | + | f ( s , 0 ) | ) d q m s } + k = 1 m ( t k 1 t k t k 1 s ( | f ( r , x ( r ) ) f ( r , 0 ) | + | f ( r , 0 ) | ) d q k 1 r d q k 1 s + | λ | t k 1 t k x d q k 1 s + ( | I k ( x ( t k ) ) I k ( x ( 0 ) ) | + | I k ( x ( 0 ) ) | ) ) + k = 1 m ( t k 1 t k ( | f ( s , x ( s ) ) f ( s , 0 ) | + | f ( s , 0 ) | ) d q k 1 s + ( | I k ( x ( t k ) ) I k ( 0 ) | + | I k ( 0 ) | ) + | λ | ( | I k ( x ( t k ) ) I k ( x ( 0 ) ) | + | I k ( x ( 0 ) ) | ) ) ( T t k ) + t m T t m s ( | f ( r , x ( r ) ) f ( r , 0 ) | + | f ( r , 0 ) | ) d q m r d q m s + | λ | t m T x d q m s | δ 1 | + | δ 2 | T | Ω | { k = 1 m ( t k 1 t k t k 1 s ( L 1 ρ + K 1 ) d q k 1 r d q k 1 s + | λ | t k 1 t k ρ d q k 1 s + ( L 2 ρ + K 2 ) ) + k = 1 m ( t k 1 t k ( L 1 ρ + K 1 ) d q k 1 s + ( L 3 ρ + K 3 ) + | λ | ( L 2 ρ + K 2 ) ) ( T t k ) + t m T t m s ( L 1 ρ + K 1 ) d q m r d q m s + | λ | t m T ρ d q m s } + | δ 3 | + | δ 4 | T | Ω | { k = 1 m ( t k 1 t k ( L 1 ρ + K 1 ) d q k 1 s + ( L 3 ρ + K 3 ) + | λ | ( L 2 ρ + K 2 ) ) + t m T ( L 1 ρ + K 1 ) d q m s } + k = 1 m ( t k 1 t k t k 1 s ( L 1 ρ + K 1 ) d q k 1 r d q k 1 s + | λ | t k 1 t k ρ d q k 1 s + ( L 2 ρ + K 2 ) ) + k = 1 m ( t k 1 t k ( L 1 ρ + K 1 ) d q k 1 s + ( L 3 ρ + K 3 ) + | λ | ( L 2 ρ + K 2 ) ) ( T t k ) + t m T t m s ( L 1 ρ + K 1 ) d q m r d q m s + | λ | t m T ρ d q m s = Λ 1 ρ + Λ 2 ( δ + 1 ε ) ρ ρ ,

which implies that S B ρ B ρ .

For any x,yPC(J,R) and for each tJ, we have

| S x ( t ) S y ( t ) | | δ 1 | + | δ 2 | t | Ω | { k = 1 m ( t k 1 t k t k 1 s | f ( r , x ( r ) ) f ( r , y ( r ) ) | d q k 1 r d q k 1 s + | λ | t k 1 t k | x ( s ) y ( s ) | d q k 1 s + | I k ( x ( t k ) ) I k ( y ( t k ) ) | ) + k = 1 m ( t k 1 t k | f ( s , x ( s ) ) f ( s , y ( s ) ) | d q k 1 s + | I k ( x ( t k ) ) I k ( y ( t k ) ) | + | λ | | I k ( x ( t k ) ) I k ( y ( t k ) ) | ) ( T t k ) + t m T t m s | f ( r , x ( r ) ) f ( r , y ( r ) ) | d q m r d q m s + | λ | t m T | x ( s ) y ( s ) | d q m s } + | δ 3 | + | δ 4 | t | Ω | { k = 1 m ( t k 1 t k | f ( s , x ( s ) ) f ( s , y ( s ) ) | d q k 1 s + | I k ( x ( t k ) ) I k ( y ( t k ) ) | + | λ | | I k ( x ( t k ) ) I k ( y ( t k ) ) | ) + t m T | f ( s , x ( s ) ) f ( s , y ( s ) ) | d q m s } + 0 < t k < t ( t k 1 t k t k 1 s | f ( r , x ( r ) ) f ( r , y ( r ) ) | d q k 1 r d q k 1 s + | λ | t k 1 t k | x ( s ) y ( s ) | d q k 1 s + | I k ( x ( t k ) ) I k ( y ( t k ) ) | ) + 0 < t k < t ( t k 1 t k | f ( s , x ( s ) ) f ( s , y ( s ) ) | d q k 1 s + | I k ( x ( t k ) ) I k ( y ( t k ) ) | + | λ | | I k ( x ( t k ) ) I k ( y ( t k ) ) | ) ( t t k ) + t k t t k s | f ( r , x ( r ) ) f ( r , y ( r ) ) | d q k r d q k s + | λ | t k t | x ( s ) y ( s ) | d q k s | δ 1 | + | δ 2 | T | Ω | x y { k = 1 m ( L 1 ( t k t k 1 ) 2 1 + q k 1 + | λ | ( t k t k 1 ) + L 2 ) + k = 1 m ( L 1 ( t k t k 1 ) + L 3 + | λ | L 2 ) ( T t k ) + L 1 ( T t m ) 2 1 + q m + | λ | ( T t m ) } + | δ 3 | + | δ 4 | T | Ω | x y { k = 1 m ( L 1 ( t k t k 1 ) + L 3 + | λ | L 2 ) + L 1 ( T t m ) } + x y k = 1 m ( L 1 ( t k t k 1 ) 2 1 + q k 1 + | λ | ( t k t k 1 ) + L 2 ) + x y k = 1 m ( L 1 ( t k t k 1 ) + L 3 + | λ | L 2 ) ( T t k ) + L 1 ( T t m ) 2 1 + q m x y + | λ | ( T t m ) x y = Λ 1 x y ,

which implies that SxSy Λ 1 xy. As Λ 1 <1, is a contraction.Therefore, by the Banach contraction mapping principle, we find that has a fixed point which is the unique solution of problem(1.4). □

The second existence result is based on Schaefer’s fixed point theorem.

Theorem 3.2 Assume that the following conditions hold:

(H3) f:J×RRis a continuous function and there exists a constant M 1 >0such that

|f(t,x)| M 1

for eachtJand allxR.

(H4) The functions I k , I k :RRare continuous and there exist constants M 2 , M 3 >0such that

| I k (x)| M 2 and| I k (x)| M 3

for allxR, k=1,2,,m.

If

| δ 1 | + | δ 2 | T + | Ω | | Ω | |λ|T<1,
(3.5)

then the impulsive q k -difference Langevin boundary value problem(1.4) has at least one solution on J.

Proof We shall use Schaefer’s fixed point theorem to prove that theoperator defined by (3.1) has a fixed point. We divide the proof intofour steps.

Step 1: Continuity of.

Let { x n } be a sequence such that x n x in PC(J,R). Since f is a continuous function onJ×R and I k , I k are continuous functions on fork=1,2,,m, we have

f ( t , x n ( t ) ) f ( t , x ( t ) ) , I k ( x n ( t k ) ) I k ( x ( t k ) ) and I k ( x n ( t k ) ) I k ( x ( t k ) )

for k=1,2,,m, as n.

Then, for each tJ, we get

| ( S x n ) ( t ) ( S x ) ( t ) | | δ 1 | + | δ 2 | t | Ω | { k = 1 m ( t k 1 t k t k 1 s | f ( r , x n ( r ) ) f ( r , x ( r ) ) | d q k 1 r d q k 1 s + λ t k 1 t k | x n ( s ) x ( s ) | d q k 1 s + | I k ( x n ( t k ) ) I k ( x ( t k ) ) | ) + k = 1 m ( t k 1 t k | f ( s , x n ( s ) ) f ( s , x ( s ) ) | d q k 1 s + | I k ( x n ( t k ) ) I k ( x ( t k ) ) | + | λ | | I k ( x n ( t k ) ) I k ( x ( t k ) ) | ) ( T t k ) + t m T t m s | f ( r , x n ( r ) ) f ( r , x ( r ) ) | d q m r d q m s + | λ | t m T | x n ( s ) x ( s ) | d q m s } + | δ 3 | + | δ 4 | t Ω { k = 1 m ( t k 1 t k | f ( s , x n ( s ) ) f ( s , x ( s ) ) | d q k 1 s + | I k ( x n ( t k ) ) I k ( x ( t k ) ) | + | λ | | I k ( x n ( t k ) ) I k ( x ( t k ) ) | ) + t m T | f ( s , x n ( s ) ) f ( s , x ( s ) ) | d q m s } + 0 < t k < t ( t k 1 t k t k 1 s | f ( r , x n ( r ) ) f ( r , x ( r ) ) | d q k 1 r d q k 1 s + | λ | t k 1 t k | x n ( s ) x ( s ) | d q k 1 s + | I k ( x n ( t k ) ) I k ( x ( t k ) ) | ) + 0 < t k < t ( t k 1 t k | f ( s , x n ( s ) ) f ( s , x ( s ) ) | d q k 1 s + | I k ( x n ( t k ) ) I k ( x ( t k ) ) | + | λ | | I k ( x n ( t k ) ) I k ( x ( t k ) ) | ) ( t t k ) + t k t t k s | f ( r , x n ( r ) ) f ( r , x ( r ) ) | d q k r d q k s + | λ | t k t | x n ( s ) x ( s ) | d q k s ,

which gives S x n Sx0 as n. This means that is continuous.

Step 2: maps bounded sets into bounded sets inPC(J,R).

Let us prove that for any ρ >0, there exists a positive constant σsuch that for each x B ρ ={xPC(J,R):x ρ }, we have Sxσ. For any x B ρ , we have

| ( S x ) ( t ) | | δ 1 | + | δ 2 | T | Ω | { k = 1 m ( t k 1 t k t k 1 s | f ( r , x ( r ) ) | d q k 1 r d q k 1 s + | λ | t k 1 t k | x ( s ) | d q k 1 s + | I k ( x ( t k ) ) | ) + k = 1 m ( t k 1 t k | f ( s , x ( s ) ) | d q k 1 s + | I k ( x ( t k ) ) | + | λ | | I k ( x ( t k ) ) | ) ( T t k ) + t m T t m s | f ( r , x ( r ) ) | d q m r d q m s + | λ | t m T | x ( s ) | d q m s } + | δ 3 | + | δ 4 | T | Ω | { k = 1 m ( t k 1 t k | f ( s , x ( s ) ) | d q k 1 s + | I k ( x ( t k ) ) | + | λ | | I k ( x ( t k ) ) | ) + t m T | f ( s , x ( s ) ) | d q m s } + k = 1 m ( t k 1 t k t k 1 s | f ( r , x ( r ) ) | d q k 1 r d q k 1 s + | λ | t k 1 t k | x ( s ) | d q k 1 s + | I k ( x ( t k ) ) | ) + k = 1 m ( t k 1 t k | f ( s , x ( s ) ) | d q k 1 s + | I k ( x ( t k ) ) | + | λ | | I k ( x ( t k ) ) | ) ( T t k ) + t m T t m s | f ( r , x ( r ) ) | d q m r d q m s + | λ | t m T | x ( s ) | d q m s | δ 1 | + | δ 2 | T | Ω | { k = 1 m ( M 1 ( t k t k 1 ) 2 1 + q k 1 + ρ | λ | ( t k t k 1 ) + M 2 ) + k = 1 m ( M 1 ( t k t k 1 ) + M 3 + | λ | M 2 ) ( T t k ) + M 1 ( T t m ) 2 1 + q m + ρ | λ | ( T t m ) } + | δ 3 | + | δ 4 | T | Ω | { k = 1 m ( M 1 ( t k t k 1 ) + M 3 + | λ | M 2 ) + M 1 ( T t m ) } + k = 1 m ( M 1 ( t k t k 1 ) 2 1 + q k 1 + ρ | λ | ( t k t k 1 ) + M 2 ) + ρ | λ | ( T t m ) + k = 1 m ( M 1 ( t k t k 1 ) + M 3 + | λ | M 2 ) ( T t k ) + M 1 ( T t m ) 2 1 + q m : = σ .

Hence, we deduce that Sxσ.

Step 3: maps bounded sets into equicontinuous sets ofPC(J,R).

Let τ 1 , τ 2 J i =( t i , t i + 1 ] for some i{0,1,2,,m}, τ 1 < τ 2 , B ρ be a bounded set of PC(J,R) as in Step 2, and let x B ρ . Then we have

| ( S x ) ( τ 2 ) ( S x ) ( τ 1 ) | | δ 2 | | τ 2 τ 1 | | Ω | { k = 1 m ( t k 1 t k t k 1 s | f ( r , x ( r ) ) | d q k 1 r d q k 1 s + | λ | t k 1 t k | x ( s ) | d q k 1 s + | I k ( x ( t k ) ) | ) + k = 1 m ( t k 1 t k | f ( s , x ( s ) ) | d q k 1 s + | I k ( x ( t k ) ) | + | λ | | I k ( x ( t k ) ) | ) ( T t k ) + t m T t m s | f ( r , x ( r ) ) | d q m r d q m s + | λ | t m T | x ( s ) | d q m s } + | δ 4 | | τ 2 τ 1 | | Ω | { k = 1 m ( t k 1 t k | f ( s , x ( s ) ) | d q k 1 s + | I k ( x ( t k ) ) | + | λ | | I k ( x ( t k ) ) | ) + t m T | f ( s , x ( s ) ) | d q m s } + | λ | | t i τ 2 x ( s ) d q i s t i τ 1 x ( s ) d q i s | + | τ 2 τ 1 | k = 1 i ( t k 1 t k | f ( s , x ( s ) ) | d q k 1 s + | I k ( x ( t k ) ) | + | λ | | I k ( x ( t k ) ) | ) + | t i τ 2 t i s f ( r , x ( r ) ) d q i r d q i s t i τ 1 t i s f ( r , x ( r ) ) d q i r d q i s | | δ 2 | | τ 2 τ 1 | | Ω | { k = 1 m ( M 1 ( t k t k 1 ) 2 1 + q k 1 + ρ | λ | ( t k t k 1 ) + M 2 ) + k = 1 m ( M 1 ( t k t k 1 ) + M 3 + | λ | M 2 ) ( T t k ) + M 1 ( T t m ) 2 1 + q m + ρ | λ | ( T t m ) } + | δ 4 | | τ 2 τ 1 | | Ω | { k = 1 m ( M 1 ( t k t k 1 ) + M 3 + | λ | M 2 ) + M 1 ( T t m ) } + | τ 2 τ 1 | ρ | λ | + | τ 2 τ 1 | k = 1 i ( M 1 ( t k t k 1 ) + M 3 + | λ | M 2 ) + | τ 2 τ 1 | M 1 ( τ 2 + τ 1 + 2 t i ) 1 + q i .

The right-hand side of the above inequality is independent of x and tendsto zero as τ 1 τ 2 . As a consequence of Steps 1 to 3, together with theArzelá-Ascoli theorem, we deduce that S:PC(J,R)PC(J,R) is completely continuous.

Step 4: We show that the set

E= { x P C ( J , R ) : x = κ S x  for some  0 < κ < 1 }

is bounded.

Let xE. Then x(t)=κ(Sx)(t) for some 0<κ<1. Thus, for each tJ, we have

x ( t ) = κ ( S x ) ( t ) = κ ( δ 1 + δ 2 t ) Ω { k = 1 m ( t k 1 t k t k 1 s f ( r , x ( r ) ) d q k 1 r d q k 1 s λ t k 1 t k x ( s ) d q k 1 s + I k ( x ( t k ) ) ) + k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) ( T t k ) + t m T t m s f ( r , x ( r ) ) d q m r d q m s λ t m T x ( s ) d q m s } + κ ( δ 3 + δ 4 t ) Ω { k = 1 m ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) + t m T f ( s , x ( s ) ) d q m s } + κ 0 < t k < t ( t k 1 t k t k 1 s f ( r , x ( r ) ) d q k 1 r d q k 1 s λ t k 1 t k x ( s ) d q k 1 s + I k ( x ( t k ) ) ) + κ 0 < t k < t ( t k 1 t k f ( s , x ( s ) ) d q k 1 s + I k ( x ( t k ) ) + λ I k ( x ( t k ) ) ) ( t t k ) + κ t k t t k s f ( r , x ( r ) ) d q k r d q k s κ λ t k t x ( s ) d q k s .

This implies by (H3) and (H4) that for eachtJ, we have

x | δ 1 | + | δ 2 | T | Ω | { k = 1 m ( t k 1 t k t k 1 s M 1 d q k 1 r d q k 1 s + | λ | t k 1 t k | x ( s ) | d q k 1 s + M 2 ) + k = 1 m ( t k 1 t k M 1 d q k 1 s + M 3 + | λ | M 2 ) ( T t k ) + t m T t m s M 1 d q m r d q m s + | λ | t m T | x ( s ) | d q m s } + | δ 3 | + | δ 4 | T | Ω | { k = 1 m ( t k 1 t k M 1 d q k 1 s + M 3 + | λ | M 2 ) + t m T M 1 d q m s } + k = 1 m ( t k 1 t k t k 1 s M 1 d q k 1 r d q k 1 s + | λ | t k 1 t k | x ( s ) | d q k 1 s + M 2 ) + k = 1 m ( t k 1 t k M 1 d q k 1 s + M 3 + | λ | M 2 ) ( T t k ) + t m T t m s M 1 d q m r d q m s + | λ | t m T | x ( s ) | d q m s | δ 1 | + | δ 2 | T + | Ω | | Ω | { M 1 k = 1 m + 1 ( t k t k 1 ) 2 1 + q k 1 + | λ | x T + m M 2 + k = 1 m ( M 1 ( t k t k 1 ) + M 3 + | λ | M 2 ) ( T t k ) } + | δ 3 | + | δ 4 | T | Ω | { M 1 T + m M 3 + m | λ | M 2 } .

Setting

Γ = | δ 1 | + | δ 2 | T + | Ω | | Ω | { M 1 k = 1 m + 1 ( t k t k 1 ) 2 1 + q k 1 + m M 2 + k = 1 m ( M 1 ( t k t k 1 ) + M 3 + | λ | M 2 ) ( T t k ) } + | δ 3 | + | δ 4 | T | Ω | { M 1 T + m M 3 + m | λ | M 2 } ,

we have

x | δ 1 | + | δ 2 | T + | Ω | | Ω | |λ|xT+Γ,

which yields

x Γ 1 | δ 1 | + | δ 2 | T + | Ω | | Ω | | λ | T :=M.

This shows that the set E is bounded. As a consequence of Schaefer’sfixed point theorem, we conclude that has a fixed point whichis a solution of the impulsive q k -difference Langevin boundary value problem(1.4). □

4 Examples

Example 4.1 Consider the following boundary value problem for the second-orderimpulsive q k -difference Langevin equation:

{ D ( 2 k + 1 5 k + 2 ) 1 2 ( D ( 2 k + 1 5 k + 2 ) 1 2 + 1 10 ) x ( t ) = t e 2 t ( 10 + t ) 2 | x ( t ) | ( 1 + | x ( t ) | ) , t J = [ 0 , 1 ] , t t k , Δ x ( t k ) = | x ( t k ) | 9 ( 9 + | x ( t k ) | ) , t k = k 10 , k = 1 , 2 , , 9 , D ( 2 k + 1 5 k + 2 ) 1 2 x ( t k + ) D ( 2 k 1 5 k 3 ) 1 2 x ( t k ) = 1 8 tan 1 ( 1 10 x ( t k ) ) , t k = k 10 , k = 1 , 2 , , 9 , 1 7 x ( 0 ) + 2 9 D 1 2 x ( 0 ) = x ( 1 ) , 2 7 x ( 0 ) + 1 9 D 1 2 x ( 0 ) = D 19 47 x ( 1 ) .
(4.1)

Here q k = ( 2 k + 1 ) / ( 5 k + 2 ) , k=0,1,2,,9, m=9, T=1, λ=1/10, α=1/7, β=2/9, γ=2/7, η=1/9, f(t,x)=(t|x(t)|)/( e 2 t ( 10 + t ) 2 (1+|x(t)|)), I k (x)=|x|/(9(9+|x|)), and I k (x)=(1/8) tan 1 (x/10). Since

| f ( t , x ) f ( t , y ) | ( 1 / 100 ) | x y | , | I k ( x ) I k ( y ) | ( 1 / 81 ) | x y | and | I k ( x ) I k ( y ) | ( 1 / 80 ) | x y | ,

then (H1) and (H2) are satisfied with L 1 =(1/100), L 2 =(1/81), L 3 =(1/80). We can find that Ω=3,103/3,150, δ 1 =(13)/15, δ 2 =46/75, δ 3 =7/9, δ 4 =(277)/315 and thus

Λ 1 0.920497882<1.

Hence, by Theorem 3.1, the boundary value problem (4.1) has a unique solutionon [0,1].

Example 4.2 Consider the following boundary value problem for the second-orderimpulsive q k -difference Langevin equation:

{ D ( k + 1 3 k + 4 ) 2 ( D ( k + 1 3 k + 4 ) 2 + 1 5 ) x ( t ) = 3 t 3 ( 4 + x 2 ) 1 2 , t J = [ 0 , 1 ] , t t k , Δ x ( t k ) = 2 k cos 2 π t k + t 2 | x ( t k ) | , t k = k 10 , k = 1 , 2 , , 9 , D ( k + 1 3 k + 4 ) 2 x ( t k + ) D ( k 3 k + 1 ) 2 x ( t k ) = 4 sin ( ( π t ) / 2 ) 3 k + | x ( t k ) | cos 2 2 t , t k = k 10 , k = 1 , 2 , , 9 , 1 4 x ( 0 ) + 1 5 D 1 16 x ( 0 ) = x ( 1 ) , 2 9 x ( 0 ) + 1 7 D 1 16 x ( 0 ) = D 100 961 x ( 1 ) .
(4.2)

Here q k = ( ( k + 1 ) / ( 3 k + 4 ) ) 2 , k=0,1,2,,9, m=9, T=1, λ=1/5, α=1/4, β=1/5, γ=2/9, η=1/7, f(t,x)=((3 t 3 )/ ( 4 + x 2 ) 1 / 2 ), I k (x)=((2k cos 2 πt)/(k+ t 2 |x|)), and I k (x)=((4sin((πt)/2))/(3k+|x| cos 2 2t)). Clearly,

|f(t,x)|=| 3 t 3 ( 4 + x 2 ) 1 2 | 3 2 ,| I k (x)|=| 2 k cos 2 π t k + t 2 | x | |2

and

| I k (x)|=| 4 sin ( ( π t ) / 2 ) 3 k + | x | cos 2 2 t | 4 3 .

We can find that

| δ 1 | + | δ 2 | T + | Ω | | Ω | |λ|T= 13 , 958 26 , 273 <1,

where Ω=(α1)(η1)λT(η+βλα)+γ(Tβ)=26,273/31,500, δ 1 =η1+βλ=(143)/175 and δ 2 =λ(η+βλ1α)+1γ=17,777/31,500.

Hence, by Theorem 3.2, the boundary value problem (4.2) has at least onesolution 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.

References

  1. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon; 1993.

    MATH  Google Scholar 

  2. Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.

    MATH  Google Scholar 

  3. Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.

    Google Scholar 

  4. Sabatier J, Agrawal OP, Machado JAT (Eds): Advances in Fractional Calculus: Theoretical Developments and Applicationsin Physics and Engineering. Springer, Dordrecht; 2007.

    Google Scholar 

  5. Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.

    MATH  Google Scholar 

  6. Baleanu D, Golmankhaneh AK, Golmankhaneh AK: Fractional Nambu mechanics. Int. J. Theor. Phys. 2009, 48: 1044-1052. 10.1007/s10773-008-9877-9

    Article  MATH  MathSciNet  Google Scholar 

  7. Ahmad B, Ntouyas SK: Nonlinear fractional differential equations and inclusions of arbitrary orderand multi-strip boundary conditions. Electron. J. Differ. Equ. 2012., 2012: Article ID 98

    Google Scholar 

  8. Ahmad B, Nieto JJ: Sequential fractional differential equations with three-point boundaryconditions. Comput. Math. Appl. 2012, 64: 3046-3052. 10.1016/j.camwa.2012.02.036

    Article  MATH  MathSciNet  Google Scholar 

  9. Ahmad B, Otero-Espinar V: Existence of solutions for fractional differential inclusions withanti-periodic boundary conditions. Bound. Value Probl. 2009., 2009: Article ID 625347

    Google Scholar 

  10. Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions withboundary conditions. Math. Comput. Model. 2009, 49: 605-609. 10.1016/j.mcm.2008.03.014

    Article  MATH  MathSciNet  Google Scholar 

  11. Cernea A: On the existence of solutions for nonconvex fractional hyperbolicdifferential inclusions. Commun. Math. Anal. 2010, 9(1):109-120.

    MATH  MathSciNet  Google Scholar 

  12. Ahmad B: Existence results for fractional differential inclusions with separatedboundary conditions. Bull. Korean Math. Soc. 2010, 47: 805-813.

    Article  MATH  MathSciNet  Google Scholar 

  13. Ahmad B, Ntouyas SK: Some existence results for boundary value problems fractional differentialinclusions with non-separated boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2010., 2010: Article ID 71

    Google Scholar 

  14. Ahmad B, Ntouyas SK: A four-point nonlocal integral boundary value problem for fractionaldifferential equations of arbitrary order. Electron. J. Qual. Theory Differ. Equ. 2011., 2011: Article ID 22

    Google Scholar 

  15. Ahmad B, Ntouyas SK: Boundary value problems for q -difference inclusions. Abstr. Appl. Anal. 2011., 2011: Article ID 292860

    Google Scholar 

  16. Ahmad B, Nieto JJ: Solvability of nonlinear Langevin equation involving two fractional orderswith Dirichlet boundary conditions. Int. J. Differ. Equ. 2010., 2010: Article ID 649486

    Google Scholar 

  17. Agarwal RP, Ahmad B: Existence theory for anti-periodic boundary value problems of fractionaldifferential equations and inclusions. Comput. Math. Appl. 2011, 62: 1200-1214. 10.1016/j.camwa.2011.03.001

    Article  MATH  MathSciNet  Google Scholar 

  18. Sudsutad W, Tariboon J: Existence results of fractional integro-differential equations with m -point multi-term fractional order integral boundaryconditions. Bound. Value Probl. 2012., 2012: Article ID 94

    Google Scholar 

  19. Coffey WT, Kalmykov YP, Waldron JT: The Langevin Equation. 2nd edition. World Scientific, Singapore; 2004.

    MATH  Google Scholar 

  20. Lim SC, Li M, Teo LP: Langevin equation with two fractional orders. Phys. Lett. A 2008, 372: 6309-6320. 10.1016/j.physleta.2008.08.045

    Article  MATH  MathSciNet  Google Scholar 

  21. Lim SC, Teo LP: The fractional oscillator process with two indices. J. Phys. A, Math. Theor. 2009., 42: Article ID 065208

    Google Scholar 

  22. Uranagase M, Munakata T: Generalized Langevin equation revisited: mechanical random force andself-consistent structure. J. Phys. A, Math. Theor. 2010., 43: Article ID 455003

    Google Scholar 

  23. Denisov SI, Kantz H, Hänggi P: Langevin equation with super-heavy-tailed noise. J. Phys. A, Math. Theor. 2010., 43: Article ID 285004

    Google Scholar 

  24. Lozinski A, Owens RG, Phillips TN: The Langevin and Fokker-Planck equations in polymer rheology. 16. Handbook of Numerical Analysis 2011, 211-303.

    Google Scholar 

  25. Lizana L, Ambjörnsson T, Taloni A, Barkai E, Lomholt MA: Foundation of fractional Langevin equation: harmonization of a many-bodyproblem. Phys. Rev. E 2010., 81: Article ID 051118

    Google Scholar 

  26. Ahmad B, Eloe PW: A nonlocal boundary value problem for a nonlinear fractional differentialequation with two indices. Commun. Appl. Nonlinear Anal. 2010, 17: 69-80.

    MATH  MathSciNet  Google Scholar 

  27. Ahmad B, Nieto JJ, Alsaedi A, El-Shahed M: A study of nonlinear Langevin equation involving two fractional orders indifferent intervals. Nonlinear Anal., Real World Appl. 2012, 13: 599-606. 10.1016/j.nonrwa.2011.07.052

    Article  MATH  MathSciNet  Google Scholar 

  28. Kac V, Cheung P: Quantum Calculus. Springer, New York; 2002.

    Book  MATH  Google Scholar 

  29. Bangerezako G: Variational q -calculus. J. Math. Anal. Appl. 2004, 289: 650-665. 10.1016/j.jmaa.2003.09.004

    Article  MATH  MathSciNet  Google Scholar 

  30. Dobrogowska A, Odzijewicz A: Second order q -difference equations solvable by factorizationmethod. J. Comput. Appl. Math. 2006, 193: 319-346. 10.1016/j.cam.2005.06.009

    Article  MATH  MathSciNet  Google Scholar 

  31. Gasper G, Rahman M: Some systems of multivariable orthogonal q -Racah polynomials. Ramanujan J. 2007, 13: 389-405. 10.1007/s11139-006-0259-8

    Article  MATH  MathSciNet  Google Scholar 

  32. Ismail MEH, Simeonov P: q -Difference operators for orthogonal polynomials. J. Comput. Appl. Math. 2009, 233: 749-761. 10.1016/j.cam.2009.02.044

    Article  MATH  MathSciNet  Google Scholar 

  33. Bohner M, Guseinov GS: The h -Laplace and q -Laplace transforms. J. Math. Anal. Appl. 2010, 365: 75-92. 10.1016/j.jmaa.2009.09.061

    Article  MATH  MathSciNet  Google Scholar 

  34. El-Shahed M, Hassan HA: Positive solutions of q -difference equation. Proc. Am. Math. Soc. 2010, 138: 1733-1738.

    Article  MATH  MathSciNet  Google Scholar 

  35. Ahmad B: Boundary-value problems for nonlinear third-order q -differenceequations. Electron. J. Differ. Equ. 2011., 2011: Article ID 94

    Google Scholar 

  36. Ahmad B, Alsaedi A, Ntouyas SK: A study of second-order q -difference equations with boundaryconditions. Adv. Differ. Equ. 2012., 2012: Article ID 35

    Google Scholar 

  37. Ahmad B, Ntouyas SK, Purnaras IK: Existence results for nonlinear q -difference equations with nonlocalboundary conditions. Commun. Appl. Nonlinear Anal. 2012, 19: 59-72.

    MATH  MathSciNet  Google Scholar 

  38. Ahmad B, Nieto JJ: On nonlocal boundary value problems of nonlinear q -differenceequations. Adv. Differ. Equ. 2012., 2012: Article ID 81

    Google Scholar 

  39. Ahmad B, Ntouyas SK: Boundary value problems for q -difference inclusions. Abstr. Appl. Anal. 2011., 2011: Article ID 292860

    Google Scholar 

  40. Zhou W, Liu H: Existence solutions for boundary value problem of nonlinear fractional q -difference equations. Adv. Differ. Equ. 2013., 2013: Article ID 113

    Google Scholar 

  41. Yu C, Wang J: Existence of solutions for nonlinear second-order q -differenceequations with first-order q -derivatives. Adv. Differ. Equ. 2013., 2013: Article ID 124

    Google Scholar 

  42. Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.

    Book  MATH  Google Scholar 

  43. Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.

    MATH  Google Scholar 

  44. Benchohra M, Henderson J, Ntouyas SK 2. In Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York; 2006.

    Chapter  Google Scholar 

  45. Tariboon J, Ntouyas SK: Quantum calculus on finite intervals and applications to impulsive differenceequations. Adv. Differ. Equ. 2013., 2013: Article ID 282

    Google Scholar 

Download references

Acknowledgements

We would like to thank the reviewers for their valuable comments and suggestionson the manuscript. The research of J Tariboon is supported by KingMongkut’s University of Technology North Bangkok, Thailand.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jessada Tariboon.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally in this article. They read and approved the finalmanuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tariboon, J., Ntouyas, S.K. Nonlinear second-order impulsive q-difference Langevin equation with boundary conditions. Bound Value Probl 2014, 85 (2014). https://doi.org/10.1186/1687-2770-2014-85

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2014-85

Keywords