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

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.

Existence and Global Exponential Stability of Equilibrium for Impulsive Cellular Neural Network Models with Piecewise Alternately Advanced and Retarded Argument

“…Applications of DEPCAs are discussed in [30]. Theory and practice of DEPCA of general type, have been discussed extensively in [31][32][33][34][35][36][37]. Piecewise constant systems exist in widely expanded areas such as biomedicine, chemistry, mechanical engineering, and physics.…”

“…, , ∈ N. It is clear that the stability of the zero solution of (36) is equivalent to that of the equilibrium * of the ICNNs with IDEPCA system (4a)-(4b). Therefore, we restrict our discussion to the stability of the zero solution of (36).…”

Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments

“…Indeed, corresponds to , and corresponds to . The theory of DEPCAG has been developed by few researchers; see, for instance, .…”

“…The study of the existence of oscillatory and periodic solutions is one of the most interesting topics for applications. The oscillatory and periodic solutions with deviating arguments has been the subject of many investigations; see, for instance, [1,2,6,7,10,12,14,16,18,21,22,[30][31][32]34] and the references cited therein. Of particular importance, however, has been the study of oscillatory and periodic solutions which are caused by the argument deviations and which do not appear in the corresponding ordinary differential equations; see [23].…”

“…Indeed, 1 ( ) = [ ] corresponds to 1 | ∈ = = ∈ ℤ, and 2 ( ) = [ + 1] corresponds to 2 | ∈ = +1 = + 1 ∈ ℤ. The theory of DEPCAG has been developed by few researchers; see, for instance, [6][7][8][9][10][11][12][13]25,26].…”