In this work we study the existence of periodic and asymptotically periodic solutions of a system of nonlinear Volterra difference equations with infinite delay. By means of fixed point theory, we furnish conditions that guarantee the existence of such periodic solutions.

We study the existence of almost automorphic solutions of the delayed neutral dynamic system on hybrid domains that are additively periodic. We use exponential dichotomy and prove the uniqueness of projector of exponential dichotomy to obtain some limit results leading to sufficient conditions for existence of almost automorphic solutions to neutral system. Unlike the existing literature we prove our existence results without assuming boundedness of the coefficient matrices in the system. Hence, we significantly improve the results in the existing literature. Finally, we also provide an existence result for almost periodic solutions of the system.

a r t i c l e i n f o MSC: 34K13 34C25 39A13 34N05 Keywords: Floquet Hybrid system Lyapunov Periodicity Shift operators Stability a b s t r a c tUsing the new periodicity concept based on shifts, we construct a unified Floquet theory for homogeneous and nonhomogeneous hybrid periodic systems on domains having continuous, discrete or hybrid structure. New periodicity concept based on shifts enables the construction of Floquet theory on hybrid domains that are not necessarily additive periodic. This makes periodicity and stability analysis of hybrid periodic systems possible on non-additive domains. In particular, this new approach can be useful to know more about Floquet theory for linear q-difference systems defined on q Z := {q n : n ∈ Z} ∪ {0} where q > 1. By constructing the solution of matrix exponential equation we establish a canonical Floquet decomposition theorem. Determining the relation between Floquet multipliers and Floquet exponents, we give a spectral mapping theorem on closed subsets of reals that are periodic in shifts. Finally, we show how the constructed theory can be utilized for the stability analysis of dynamic systems on periodic time scales in shifts.

We study almost automorphic solutions of the discrete delayed neutral dynamic systemby 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 (t) −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.

a b s t r a c tThis paper focuses on the existence of a periodic solution of the delay neutral nonlinear dynamic systemsx D ðtÞ ¼ AðtÞxðtÞ þ Q D ðt; xðd À ðs; tÞÞÞ þ Gðt; xðtÞ; xðd À ðs; tÞÞÞ: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 AE T 2 T for all t 2 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 and [ 1 k¼1 3 AEk ; 2:3 AEk h i [ f0g. 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 D ðtÞ ¼ AðtÞyðtÞ and then make use of Krasnoselskii's fixed point theorem to obtain a fixed point.

Abstract:We study the existence of an almost periodic solution of discrete Volterra systems by means of xed point theory. Using discrete variant of exponential dichotomy, we provide su cient conditions for the existence of an almost periodic solution. Hence, we provide an alternative solution for the open problem proposed in the literature.

Let T be a nonempty, closed, and arbitrary set of real numbers, namely a time scale, and consider the following delay dynamical equationwhere ϑ stands for the abstract delay function. The main goal of this study is three-fold: Obtaining the existence of an equi-bounded solution, proving the asymptotic stability of the zero solution, and showing the existence of a periodic solution based on new periodicity concept on time scales for the given delayed equation under certain conditions. In our analysis, we propose an alternative variation of parameters formulation by using an auxiliary function to invert a mapping for the utilization of fixed point theory.

Inspired by Cooke and York's work, we concentrate on the delay dynamic equation on a nonempty, arbitrary, closed set of real numbers, so‐called a time scale. The utilization of an abstract delay function δ in the above‐given equation relaxes the condition that each individual has a fixed constant life span when it fits a growth process.The main result is obtained by establishing a linkage between the delay dynamic equation and an integral equation. By constructing a phase space for a given initial function, we show the unique solution converges to a predetermined constant.