Abstract
In this paper we investigate the existence of the periodic solutions of a nonlinear impulsive differential system with piecewise alternately advanced and retarded arguments, in short IDEPCAG, 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 system are obtained. In the process we use the Green's function for impulsive periodic solutions and convert the given the IDEPCAG into an equivalent integral equation system. 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 impulsive differential systems. We also use the contraction mapping principle to show the existence of a unique impulsive periodic solution. Appropriate examples are given to show the feasibility of our results.
Highlights
Among the functional di¤erential equations, Myshkis [27] proposed to study di¤erential equations with piecewise constant arguments: DEPCA
In this paper we investigate the existence of the periodic solutions of a nonlinear impulsive di¤erential system with piecewise alternately advanced and retarded arguments, in short IDEPCAG, that is, the argument is a general step function
In the process we use the Green’s function for impulsive periodic solutions and convert the given the IDEPCAG into an equivalent integral equation system
Summary
Among the functional di¤erential equations, Myshkis [27] proposed to study di¤erential equations with piecewise constant arguments: DEPCA. Impulsive di¤erential equation, piecewise constant arguments of generalized type, Green’s function, periodic solutions, ...xed point theorems. Chiu [19], using Green’s function and ...xed point theory, obtained some su¢ cient conditions for the existence and uniqueness of periodic (or harmonic) and subharmonic solutions of quasilinear DEPCA systems of generalized type. The main purpose of this paper is to establish some simple criteria for the existence of periodic solutions of a nonlinear system of impulsive di¤erential systems with alternately of advanced and retarded arguments of generalized type (in short IDEPCAG): z0(t) = A(t)z(t) + f t; z(t); z( (t)) + g t; z(t); z( (t)) ; t 6= tk; (2a) zjtk = Jk(z(tk )); k 2 Z;. Appropriate examples are provided in the last section to show the feasibility of our results
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