Existence of periodic solutions in shifts δ±δ± for neutral nonlinear dynamic systems

This paper focuses on the existence of a periodic solution of the delay neutral nonlinear dynamic systemsxΔ(t)=A(t)x(t)+QΔ(t,x(δ-(s,t)))+G(t,x(t),x(δ-(s,t))).xΔ(t)=A(t)x(t)+QΔ(t,x(δ-(s,t)))+G(t,x(t),x(δ-(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∈Tt±T∈T for all t∈Tt∈T and for a fixed T>0T>0, may not hold. More importantly, the new concept will easily handle time scales that are not periodic in the conventional way such as; qZ‾ and ∪k=1∞3±k,2.3±k∪{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 yΔ(t)=A(t)y(t)yΔ(t)=A(t)y(t) and then make use of Krasnoselskii’s fixed point theorem to obtain a fixed point.