H. Can Koyuncuoglu | Searcharxiv

arXiv2015

Almost automorphic solutions of discrete delayed neutral system

We study almost automorphic solutions of the discrete delayed neutral dynamic system% \[ x(t+1)=A(t)x(t)+ΔQ(t,x(t-g(t)))+G(t,x(t),x(t-g(t))) \] by means of a fixed point theorem due to Krasnoselskii. Using discrete variant of exponential dichotomy and proving uniqueness of projector of discrete exponential dichotomy we invert the equation and obtain some limit results leading to sufficient conditions for the existence of almost automorphic solutions of the neutral system. Unlike the existing literature we prove our existence results without assuming boundedness of inverse matrix $A\left( t\right) ^{-1}$. Hence, we significantly improve the results in the existing literature. We provide two examples to illustrate effectiveness of our results. Finally, we also provide an existence result for almost periodic solutions of the system.

math.FA↗

arXiv2014

Existence of periodic solutions in shifts $δ_{\pm}$ for neutral nonlinear dynamic systems

In this study, we focus on the existence of a periodic solution for the neutral nonlinear dynamic systems with delay% \[ x^Δ(t)=A(t)x(t)+Q^Δ\left(t,x\left(δ_{-}(s,t)\right) \right) +G\left(t,x(t),x\left(δ_{-}(s,t)\right) \right) . \] We utilize the new periodicity concept in terms of shifts operators, which allows us to extend the concept of periodicity to time scales where the additivity requirement $t\pm T\in\mathbb{T}$ for all $t\in\mathbb{T}$ 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; $\overline{q^{\mathbb{Z}}}$ and $\cup_{k=1}^{\infty}\left[ 3^{\pm k},2.3^{\pm k}\right] \cup\left\{0\right\} .$ Hence, we develop a tool that enables the investigation 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 $y^Δ(t)=A(t)y(t)$ and then make use of Krasnoselskii's fixed point theorem to obtain a fixed point.

math.CA↗