Periodic boundary value problem for the first order functional differential equations with impulses

Abstract

This paper is concerned with the existence and approximation of solutions for a class of first order impulsive functional differential equations with periodic boundary value conditions.A new comparison result is presented and the previous results are extended. §1 IntroductionThe theory of impulsive differential equations has become an important aspect of differential equations (see [1][2][3][4]). Periodic boundary value problems (PBVP) for impulsive differential equations have drawn much attention, see [5][6][7][8][9][10][11][12][13] and the references therein. Recently, Ding Wei and Mi Junrong [14] have discussed the following impulsive functional differential equation under a new concept of lower and upper solutions:, k = 1, 2, · · · , p. In this paper, via a new comparison result and monotone iterative method, we study PBVP(1.1), and extend the previous results.The paper is organized as follows: In §2, we establish a new comparison principle. In §3, using Banach's fixed point theorem, we discuss the existence and uniqueness of the solutions for impulsive functional differential equations. In §4, we first introduce the new concept of lower and upper solutions, then by using the method of upper and lower solutions and the monotone iterative technique we obtain the existence of extreme solutions for PBVP (1.1).

Nonlinear boundary value problems for discontinuous delayed differential equations

“…Later, the case of Carathéodory system(ODEs) with discontinuous upper and lower solutions were studied(see [12,13,15] for details). Most recently, functional boundary value problems for ODEs and periodic boundary value problems for FDEs have been considered and more general existence results were obtained by S. Heikkiä, A. Cabada, V. Lakshmikantham, R. L. Pouso, W. Ding, J. J. Nieto, R. X. Liang, and others(see [2][3][4][5][7][8][9]11,14,16,[19][20][21]). We also want to point out that the method of upper and lower solutions has also succeeded in solving periodic(or multi-point) boundary value problems for impulsive FDEs(see [3][4][5]7,11,14,19]).…”