(&omega,c)- PSEUDO ALMOST PERIODIC FUNCTIONS, (&omega,c)- PSEUDO ALMOST AUTOMORPHIC FUNCTIONS AND APPLICATIONS

In this paper, we introduce the classes of $(\omega, c)$-pseudo almost periodicfunctions and $(\omega, c)$-pseudo almost automorphicfunctions. These collections include $(\omega, c)$-pseudo periodicfunctions, pseudo almost periodic functions and their automorphic analogues.We present an application to the abstract semilinear first-order Cauchy inclusions in Banach spaces.

Influence of time delay on weighted pseudo-almost periodic dynamics in SICNNs

Purpose The purpose of this paper is to investigate the weighted pseudo-almost periodic solutions of shunting inhibitory cellular neural networks (SICNNs) with time-varying delays and distributed delays. Design/methodology/approach The principle of weighted pseudo-almost periodic functions and some new mathematical analysis skills are applied. Findings A set of sufficient criteria which guarantee the existence and exponential stability of the weighted pseudo-almost periodic solutions of the considered SICNNs are established. Originality/value The derived results of this paper are new and complement some earlier works. The innovation of this paper concludes two points: a new sufficient criteria guaranteeing the existence and exponential stability of the weighted pseudo-almost periodic solutions of SICNNs are established; and the ideas of this paper can be applied to investigate some other similar neural networks.

New results for a Lasota–Wazewska model

In nature there is no phenomenon that is purely periodic, and this gives the idea to consider the measure pseudo almost periodic oscillation. In this paper, by employing a suitable fixed point theorem, the properties of the measure pseudo almost periodic functions and differential inequality, we investigate the existence and uniqueness of the measure pseudo almost periodic solutions for some models of Lasota–Wazewska equation with measure pseudo almost periodic coefficients and mixed delays. We suppose that the linear part has almost periodic and the nonlinear part is assumed to be measure pseudo almost periodic. Moreover, the global attractivity and the exponential stability of the measure pseudo almost periodic solutions are also considered for the system. As application, an illustrative numerical example is given to demonstrate the effectiveness of the obtained results.

Pseudo Almost Periodicity and Its Applications to Impulsive Nonautonomous Partial Functional Stochastic Evolution Equations

In this paper, we establish a new composition theorem for pseudo almost periodic functions under non-Lipschitz conditions. We apply this new composition theorem together with a fixed-point theorem for condensing maps to investigate the existence of$p$-mean piecewise pseudo almost periodic mild solutions for a class of impulsive nonautonomous partial functional stochastic evolution equations in Hilbert spaces, and then, the exponential stability of$p$-mean piecewise pseudo almost periodic mild solutions is studied. Finally, an example is given to illustrate our results.

THE CONNECTION BETWEEN PSEUDO ALMOST PERIODIC FUNCTIONS DEFINED ON TIME SCALES AND ON THE REAL LINE

A necessary and sufficient condition for a continuous function $g$ to be almost periodic on time scales is the existence of an almost periodic function $f$ on $\mathbb{R}$ such that $f$ is an extension of $g$. Our aim is to study this question for pseudo almost periodic functions. We prove the necessity of the condition for pseudo almost periodic functions. An example is given to show that the sufficiency of the condition does not hold for pseudo almost periodic functions. Nevertheless, the sufficiency is valid for uniformly continuous pseudo almost periodic functions. As applications, we give some results on the connection between the pseudo almost periodic (or almost periodic) solutions of dynamic equations on time scales and of the corresponding differential equations.

Piecewise weighted pseudo almost periodic functions and applications to impulsive differential equations

In this paper, we introduce a concept of piecewise weighted pseudo almost periodic functions on a Banach space and present the essential properties of this kind of functions, including the composition theorems and the uniqueness of decomposition of the functions. As an application, we give some results on the existence and stability of piecewise weighted pseudo almost periodic mild solutions to an abstract impulsive differential equation. Moreover, a concrete example is given to illustrate our abstract results.