Existence of Coupled Solutions of BVP for ϕ-Laplacian Impulsive Differential Equations
College of Sciences, Hezhou University, Hezhou, China
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
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 . 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  and anti-periodic wavelets . 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  study a class of nonlinear impulsive differential equation with anti-periodic boundary condition:
where is an increasing homeomorphism from to , is a Carathéodory function. , , are impulsive functions. will be given later. In , 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).
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
(ii) for all . Moreover, if such that , then there exists such that
(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
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 )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 .
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 )Assume that (H1)-(H4) hold. Then there exists at least one solution of the problem (1)-(3) such that
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
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
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
From the proof of the Lemma 3, there exists a couple of solutions such that
Furthermore, satisfy the condition (2). Now, to prove that (5)-(8) is verified, it suffices to prove that
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
Assume that the first inequality if (11) isn’t holds, as a consequence, we have
From (14) and , we have
From these facts and the relation , we have
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.
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).
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.