%0 Journal Article
%T Periodic Boundary Value Problems for a Class of Impulsive Functional Differential Equations of Hybrid Type
%A Guoping Chen
%J ISRN Applied Mathematics
%D 2014
%R 10.1155/2014/657629
%X This paper is concerned with the existence of extreme solutions of periodic boundary value problems for a class of first-order impulsive functional differential equations of hybrid type. We obtain the sufficient conditions for existence of extreme solutions by using upper and lower solutions method coupled with monotone iterative technique. 1. Introduction The theory of impulsive differential equations is now being recognized to be not only richer than the corresponding theory of differential equations without impulses but also represents a more natural framework for mathematical modeling of many real-world phenomena [1¨C3]. Significant progress has been made in the theory of systems of impulsive differential equations in recent years (see [4¨C18] and the references cited therein). It is well known that the monotone iterative technique offers an approach for obtaining approximate solutions of nonlinear differential equations; for details, see [19] and the references therein. There also exist several works devoted to the applications of this technique to periodic boundary value problems of impulsive differential equations; see [20¨C26]. In [27, 28], the authors introduce a new concept of upper and lower solutions for periodic boundary value problems of a class of first-order functional differential equations. In paper [23], the authors applied this new concept to study the periodic boundary value problems for first-order impulsive functional differential equations. Motivated by [23, 27, 28], we will study periodic boundary value problem for the first-order impulsive functional differential equation of hybrid type where , , , , denotes the jump of at , , and represent the right and left limits of at , respectively. Denote . The integral part in (1) is defined by where , , , , , , and . Let , is continuous for , , , and exist, and , . is continuously differentiable for , . and are Banach spaces with the norms By a solution of (1), we mean a for which problem (1) is satisfied. Note that (1) has a very general form; as special instances resulting from (1), one can have impulsive differential equations with deviating arguments and impulsive differential equations with the Volterra or Fredholm operators. For example, if does not include and , then (1) reduces to periodic boundary problem for impulsive differential equations with deviating arguments, which is discussed in [22, 23]; when , (1) is the following periodic boundary problem for impulsive integrodifferential equations of mixed type: similar problems are also discussed in [24¨C26]. 2. Preliminaries To
%U http://www.hindawi.com/journals/isrn.applied.mathematics/2014/657629/