Existence of periodic solutions in shifts delta(+/-) for neutral nonlinear dynamic systems

Abstract

This paper focuses on the existence of a periodic solution of the delay neutral nonlinear dynamic systems x(Delta)(t) = A(t)x(t) + Q(Delta)(t, x(delta (-) (s, t))) + G(t, x(t), x(delta (-) (s, t))). In our analysis, we utilize a new periodicity concept in terms of shifts operators, which allows us to extend the concept of periodicity to time scales where the additivity requirement t +/- T is an element of T for all t is an element of T and for a fixed T > 0, may not hold. More importantly, the new concept will easily handle time scales that are not periodic in the conventional way such as; (q(z)) over bar and boolean OR(infinity)(k-1) [3(+/- k), 2.3(+/- k)] boolean OR {0}. Hence, we will develop the tool that enables us to investigate the existence of periodic solutions of q-difference systems. Since we are dealing with systems, in order to convert our equation to an integral systems, we resort to the transition matrix of the homogeneous Floquet system y(Delta)(t) = A(t)y(t) and then make use of Krasnoselskii's fixed point theorem to obtain a fixed point. (C) 2014 Elsevier Inc. All rights reserved.