Science Journal of Applied Mathematics and Statistics
Volume 4, Issue 6, December 2016, Pages: 298-302

Existence of Coupled Solutions of BVP for ϕ-Laplacian Impulsive Differential Equations

Xiufeng Guo

College of Sciences, Hezhou University, Hezhou, China

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To cite this article:

Xiufeng Guo. Existence of Coupled Solutions of BVP for ϕ-Laplacian Impulsive Differential Equations. Science Journal of Applied Mathematics and Statistics. Vol. 4, No. 6, 2016, pp. 298-302. doi: 10.11648/j.sjams.20160406.18

Received: November 4, 2016; Accepted: November 25, 2016; Published: December 14, 2016


Abstract: In this paper, we study the existence of coupled solutions of anti-periodic boundary value problems for impulsive differential equations with ϕ-Laplacian operator. Based on a pair of coupled lower and upper solutions and appropriate Nagumo condition, we prove the existence of coupled solutions for anti-periodic impulsive differential equations boundary value problems with ϕ-Laplacian operator.

Keywords: Boundary Value Problems, Coupled Solutions, Impulsive Differential Equations, ϕ-Laplacian Operator


1. Introduction

In recent years, the study boundary value problems (BVPs for short) with -Laplacian operator has been emerging as an important area and obtained a considerable attention. Since -Laplacian operator appears in the study of flow through porous media (), nonlinear elasticity (), glaciology () and so on, there are many works about existence of solutions for differential equations with -Laplacian operator [24, 25]. Usually, -Laplacian operator is replaced by abstract and more general version -Laplacian operator, which lead to clearer expositions and a better understanding of the methods which ware employed to derive the existence results [12, 22, 23].

Moreover, impulsive differential equations have become an important aspect in some mathematical models of real processes and phenomena in science. There has a significant development in impulsive differential equations and impulse theory(see [2, 3, 14]). Moreover, -Laplacian operator arises in turbulent filtration in porous media, non-Newtonian fluid flows and in many other application areas [10, 12].

Furthermore, the study of anti-periodic problem for nonlinear evolution equations is closely related to the study of periodic problem which was initiated by Okochi [17]. Anti-periodic problem which is a very important area of research has been extensively studied during the past decades, such as anti-periodic trigonometric polynomials [11] and anti-periodic wavelets [4]. Moreover, anti-periodic boundary conditions also appear in physics in a variety of situations (see [1, 13]) and difference and differential equations (see [6, 8, 19, 20]). The anti-periodic problem is a very important area of research.

In addition, we known that every -anti-periodic solution gives rise to a -periodic solution if the nonlinearity  satisfy some symmetry condition. Indeed, the periodic and anti-periodic boundary value problems have attracted many researchers great interest (see [6, 8, 9, 15, 16, 19, 20, 21] and references therein). Recently, Guo and Gu [22] study a class of nonlinear impulsive differential equation with anti-periodic boundary condition:

(1)

(2)

(3)

where is an increasing homeomorphism from  to ,  is a Carathéodory function. , ,  are impulsive functions. will be given later. In [22], the authors obtained the existence of solution for anti-periodic boundary value problems (1)-(3) for impulsive differential equations with -Laplacian operator. In this paper, we will continuous to consider the existence of coupled solutions for boundary value problems (1)-(3).

This paper is organized as follows: In section 2, we will state some preliminaries that will be used throughout the paper. In section 3, we will obtain the existence of coupled solutions for anti-periodic-Laplacian impulsive differential equations boundary value problems (1)-(3).

2. Preliminaries

In this section, we will introduce some definitions and preliminaries which are used throughout this paper.

For a given Banachspace , let  be the set of all continuous functions . Let  be the set of functions  which are  times continuously differentiable on  with finite norm

For , Let  be the set of Lebesgue measurable functions  on  such that  is finite. denotes the set of absolutely continuous functions  on  satisfy . denotes the set of functions  and  with finite norm

It is easy to see that  and are Banach spaces and  is a usual Sobolev space.

Let . A finite subset  of the interval  defined by

Let  and  for all . For and , we denote

It is easy to verify that the spaces  and are Banach spaces with the norms

We say that ) satisfies the restricted Carathéodory conditions on  if

i.       for each  the function  is measurable on ;

ii.     the function  is continuous on a.e. ;

iii.    for every compact set , there exists a nonnegative function  such that

In this paper, we use Car() to denote the set of functions satisfying the restricted Carathéodory conditionson . In what follows,  and  denote the Dini derivatives.

Definition 1. The functions  such that  are said to be a pair of coupled lower and upper solutions of problem (1)-(3) if  satisfy the following conditions:

(i)       for all . Moreover, if  such that , then there exists  such that

,

and

(ii)     for all . Moreover, if  such that , then there exists  such that

,

and

(iii)   For all ,  are injective and there exist , , ,  such that

and there exist , , ,  such that

Definition 2. Given a function  is called a solution of the problem (1)-(3) if  and  satisfies (1) and fulfills conditions (2) and (3).

Definition 3. Assume that Car() and  satisfying for . We say that  satisfies a Nagumo condition with respect to  and  if, for , there exist  and , such that  on ,

Moreover, there exists a constant  with , such that

(4)

where and  Any constant such  will be called a Nagumo constant.

Throughout this paper, we impose the following hypotheses:

(H)The function  is a continuous and strictly increasing.

(H)The BVP (1)-(3) has a pair of coupled lower and upper solutions and .

(H)Car() and satisfies a Nagumo condition with respect to and.

(H)The functions  are non-decreasing in the first variable for , and the functions  are non-increasing in the third variable and non-decreasing in the fourth and fifth variables.

3. Existence Results of Coupled Solutions

This section is devoted to proving the existence of coupled solutions for anti-periodic impulsive differential equations boundary value problems with -Laplacian operator. Firstly, we state the following existence and uniqueness result.

Lemma 1.(Lemma 7 of [23])Assume that  and  for each . Suppose that  is a strictly increasing function satisfies . Then the non homogeneous impulsive Dirichlet problem

has a unique solution , which can be written in the form

where is the unique solution of the equation

Next, let us consider the following functions

where  is the constant introduced in definition 2.3,

coupled with functionals given by

Moreover, for each  we consider a function  defined by

The function  is well defined according to the following result (by redefining function  as zero when it does not exist). It can be proved in a similar way to Lemma 2 in [24].

Lemma 2. For given  such that in , then

(i)     exists for a.e. ;

(ii)   for a.e. .

Now, we can define a strictly increasing homeomorphism  by:

In the following, we are in a position to prove the existence theorem for our considering problems.

Lemma 3. (Theorem 3.3 of [22])Assume that (H1)-(H4) hold. Then there exists at least one solution  of the problem (1)-(3) such that

and

where is the constant introduced in Definition 2.3.

Next, we are devoted to the existence of coupled solutions. We first introduce the following definition.

Definition 4. The functions  are called coupled solutions of problems (1)-(3) if  and satisfy (1)-(2) and

(5)

(6)

(7)

(8)

Remark If the coupled solutions  and  of problem (1)-(3) satisfy , the  is a solution of problem (1)-(3).

Next, we give the existence of coupled solutions for problems (1)-(3).

Theorem 5. Assume hypotheses (H)-(H) hold. Then there exists at least a pair of coupled solutions  of the impulsive differential equations boundary value problem (1)-(3) such that

(9)

and

where is the constant introduced in Definition 2.3.

Proof. Let us define  for each  in the same way as above, and construct a modified problem  similar to the proof of Lemma 3, that is

where

From the proof of the Lemma 3, there exists a couple of solutions  such that

and

Furthermore,  satisfy the condition (2). Now, to prove that (5)-(8) is verified, it suffices to prove that

(10)

(11)

(12)

(13)

Firstly, we will prove (10), by contradiction, if , then by , we have

which contradict to . Moreover,  can be proved similarly.

As the same way, we can obtain that the inequality (10) is holds. Thus we have

(14)

Assume that the first inequality if (11) isn’t holds, as a consequence, we have

and

.

From (14) and , we have

From these facts and the relation , we have

,

thus

It is a contradiction. Moreover, the inequality in (13) be obtain in a similar way. Hence inequalities (11)-(12) are hold, that is to say  satisfy (5)-(8).

Therefore, the functions  is a coupled solutions of the problem (1)-(3), which completes the proof.

4. Conclusion

In this paper, we mainly discuss the existence of coupled solutions of anti-periodic boundary value problems for impulsive differential equations with -Laplacian operator. To give the existence results of coupled solutions for the problem (1)-(3), we first introduce a pair of coupled lower and upper solutions (see Definition 1), Then, we provide and prove the existence results of coupled solutions for anti-periodic -Laplacian impulsive differential equations boundary value problems based on a pair of coupled lower and upper solutions and appropriate Nagumo condition (Theorem 5).

Acknowledgments

The work was partially supported by NNSF of China Grants No.11461021, NNSF of Guangxi Grant No. 2014GXNSFAA118028, the Scientific Research Foundation of Guangxi Education Department No. KY2015YB306, the Scientific Research Project of Hezhou University Nos. 2015ZZZK16, 2016HZXYSX07, and Guangxi Colleges and Universities Key Laboratory of Symbolic Computation and Engineering Data Processing.


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