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

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.


Introduction
In recent years, the study boundary value problems (BVPs for short) with p -Laplacian operator has been emerging as an important area and obtained a considerable attention. Since p -Laplacian operator appears in the study of flow through porous media ( 3 2 p = / ), nonlinear elasticity ( 2 p ≥ ), glaciology ( 1 4 3 p ≤ ≤ / ) and so on, there are many works about existence of solutions for differential equations with p -Laplacian operator [24,25]. Usually, p -Laplacian operator is replaced by abstract and more general version φ -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, p -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 T -anti-periodic solution gives rise to a 2T -periodic solution if the nonlinearity f 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: where φ is an increasing homeomorphism from R to R , are impulsive functions. 1 P C 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 φ -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 φ -Laplacian impulsive differential equations boundary value problems (1)-(3).

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

⋯ ⋯
We say that iii. for every compact set K S ⊂ , there exists a nonnegative function 1 , satisfy the following conditions: and there exist (0) Moreover, there exists a constant ( ) will be called a Nagumo constant.
Throughout this paper, we impose the following hypotheses: (

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 φ -Laplacian operator. Firstly, we state the following existence and uniqueness result.  [0 ] t T P ∈ , , ; for a.e.
[0 ] t T P ∈ , , . Now, we can define a strictly increasing homeomorphism R R ϕ : → by: In the following, we are in a position to prove the existence theorem for our considering problems.
is the constant introduced in Definition 2.3.
Next, we are devoted to the existence of coupled solutions. We first introduce the following definition. x y C , ∈ and satisfy (1)-(2) and (0) ( ) y x T ′ ′ = − . x y C , ∈ of the impulsive differential equations boundary value problem and ( ) is the constant introduced in Definition 2.3.
Proof. Let us define for each 1 k p = , , ⋯ in the same way as above, and construct a modified problem ( ) P * similar to the proof of Lemma 3, that is Furthermore, x y , satisfy the condition (2). Now, to prove that (5)-(8) is verified, it suffices to prove that Firstly, we will prove (10) can be proved similarly. As the same way, we can obtain that the inequality (10) Assume that the first inequality if (11) isn't holds, as a consequence, 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 x y , satisfy (5)- (8).
Therefore, the functions x y , is a coupled solutions of the problem (1)-(3), which completes the proof.

Conclusion
In this paper, we mainly discuss the existence of coupled solutions of anti-periodic boundary value problems for impulsive differential equations with φ -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 φ -Laplacian impulsive differential equations boundary value problems based on a pair of coupled lower and upper solutions and appropriate Nagumo condition (Theorem 5).