Periodic Solutions for Nonlinear Integro-Differential Systems with Piecewise Constant Argument

Kuo-Shou Chiu

doi:10.1155/2014/514854

Abstract

We investigate the existence of the periodic solutions of a nonlinear integro-differential system with piecewise alternately advanced and retarded argument of generalized type, in short DEPCAG; that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use Green's function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.

Highlights

Among the functional differential equations, Myshkis [1] proposed to study differential equations with piecewise constant arguments: DEPCA
Great attention has been paid to the study of the existence of periodic solutions of several different types of differential equations
In this paper, comparing the three DEPCAG inequalities of Gronwall type and remarked new Gronwall lemma requests a weaker condition than the other Gronwall lemmas and has a better estimate. It is well-known that there are many subjects in physics and technology using mathematical methods that depend on the linear and nonlinear integro-differential equations, and it became clear that the existence of the periodic solutions and its algorithm structure from more important problems in the present time

Summary

Introduction

Among the functional differential equations, Myshkis [1] proposed to study differential equations with piecewise constant arguments: DEPCA. Chiu and Pinto [23], using Poincareoperator, a new Gronwall type lemma and fixed point theory, obtained some sufficient conditions for the existence and uniqueness of periodic (or harmonic) and subharmonic solutions of quasilinear differential equation with a general piecewise constant argument of the form y󸀠 (t) = A (t) y (t) + f (t, y (t) , y (γ (t))) ,. In this paper, comparing the three DEPCAG inequalities of Gronwall type and remarked new Gronwall lemma requests a weaker condition than the other Gronwall lemmas and has a better estimate It is well-known that there are many subjects in physics and technology using mathematical methods that depend on the linear and nonlinear integro-differential equations, and it became clear that the existence of the periodic solutions and its algorithm structure from more important problems in the present time.

Green’s Function and Periodicity

Existence of Periodic Solutions

Applications and Illustrative Examples