Anti-periodic Boundary Value Problems of <i>φ</i>-Laplacian Impulsive Differential Equations

Abstract

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.

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

“…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].…”

“…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 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).…”