Periodic Boundary Value Problem for First Order Differential Equations with Impulses at Variable Times

Abstract

In this paper we show the validity of the method of upper and lower solutions to obtain an existence result for a periodic boundary value problem of first order impulsive differential equations at variable times. ᮊ

Nonconvex-valued impulsive functional differential inclusions with variable times

“…The theory of impulsive differential equations with variable times is relatively less developed due to the difficulties created by state-dependent impulses. Recently, some interesting extensions to impulsive differential equations with variable times have been done by Bajo and Liz [4], Frigon and O'Regan [5][6][7], Kaul et al [8], Kaul and Liu [9,10], Lakshmikantham et al [11,12], Liu and Ballinger [13], and Vatsala and Vasundara Devi [14,15]. Very recently, by using the Schaefer theorem and the concept of upper and lower solutions, Benchohra et al [16][17][18] have considered different classes of impulsive functional differential equations.…”

Impulsive Boundary Value Problems for First-order Ordinary Differential Inclusions

“…That is why the perturbations are considered to take place in the form of impulses. The theory of impulsive differential equations has undergone considerable development in recent years; see the monographs of Lakshmikantham, et al [10,11] , Bajo I. and E. Liz [1] , Pierson-Gorez C. [15] and Erbe and Krawcewicz [8] and detailed bibliographies therein. Recently, different tools such as fixed point theorem, the LeraySchauder alternative, Granas's topological transversality method and the lower and upper solutions method have been applied to various initial and boundary value problems for impulsive differential inclusions.…”

“…When the right hand side is a single-valued function, the first-order impulsive ordinary differential equations or inclusions were considered by J. J. Nieto [13,14] , Benchohra et al [3] , Bajo I. and E. Liz [1] and Pierson Gorez C. [15] . In this paper, we establish the existence results for Problem (1)-(3) by using the Schaefer's fixed point theorem.…”

Sufficient conditions for pulse phenomena of nonlinear systems with state-dependent impulses

“…Refs. [4,13] dealt with impulsive differential equations with no pulse phenomena and Refs. [1,3,6,9] dealt with pulse phenomena.…”