Anti-periodic Boundary Value Problems of φ-Laplacian Impulsive Differential Equations

This paper concerned with the existence of solutions of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator. Firstly, the definition a pair of coupled lower and upper solutions of the problem is introduced. Then, under the approach of coupled upper and lower solutions together with Nagumo condition, we prove that there exists at least one solution of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator.


Introduction
In this paper, we will study a class of nonlinear impulsive where ϕ is an increasing homeomorphism from R to R , and 2 [0 ] f T R R : , × → is a Carathéodory function. C will be given later. 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 impulse theory and impulsive differential equations (see [1,2,3]). Moreover, p-Laplacian operator arises in non-Newtonian fluid flows, turbulent filtration in porous media and in many other application areas (see [5,7] and references therein). 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 (see [12,13]). Recently, Cabada and Tomecek [4] focussed on the ϕ-Laplacian differential equations (1) subject to impulsive functions (2) with non-local boundary conditions Later, the paper [6] generalize the problem of [4] and considered a more general ϕ-Laplacian differential equations coupled with the compatible impulses and boundary conditions = , ⋯ . After make a study of the monotonicity of the boundary condition functions in [4,6,8], we found that their boundary problems didn't include the anti-periodic boundary condition problems.
Motivated by above the mentioned papers, we try to find some appropriate conditions to make sure the existence solutions for anti-periodic problem (1)-(3). As far as we known, although the papers [4,6,8] study such a general ϕ-Laplacian problems with nonlinear boundary conditions, but its didn't contain the anti-periodic problems. Besides, there are few dependent references for studying the ϕ-Laplacian impulsive functional differential equations with anti-periodic boundary condition. Furthermore, the anti-periodic problem is a very important area of research. The study of anti-periodic problem for nonlinear evolution equations is closely related to the study of periodic problem which was initiated by Okochi [14]. Anti-periodic problem has been extensively studied during the past decades, such as anti-periodic trigonometric polynomials ( [15]). Moreover, anti-periodic boundary conditions also appear in difference and differential equations (see [16,17] and references therein).
This paper is organized as follows: In section 2, we state some preliminaries that will be used throughout the paper. In section 3, we obtain existence solutions for problem (1)-(3) by the approach of coupled upper and lower solutions together with Nagumo condition. Finally, we give the conclusion of our main work in section 4.

Preliminaries
It is easy to verify that the spaces m P C and m q P W , are Banach spaces with the norms We say that (iii) for every compact set K S ⊂ , there exists a nonnegative function for a e t T and all x K µ | , |≤ . . ∈ , ∈ .
Throughout this paper, we impose the following hypotheses: (

Existence Results
In this section, we will deduce that there exists at least one solution of the problem (1)
[0 ] t T P ∈ , ， . Now, define a strictly increasing homeomorphism R R ϕ : → by: In what follows, we are in a position to prove the existence theorem.
Theorem 3. Assume that the hypotheses (H 1 )-(H 4 ) hold. Then there exists at least one solution u of the problem Proof. First, we consider the following auxiliary problem From the definitions of functions ρ and K δ , Lemma 3.2 and hypothesis (H 3 ), for Define an operator

∫ ∫ ɶ
The proof of the existence of a fixed point u of operator F follows in a similar way to the one given in Theorem 9 of [4]. In view of Lemma 3.1, we know that the fixed piont u is a solution of the problem ( ) P * . Next, we prove that u  Similarly, we can prove that u β − is non-negative on [0 ] T , .