Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations

We establish new theorems for the composition of pseudo almost periodic and pseudo almost automorphic functions in Banach spaces. Our results extend the recent ones [H. Li, F. Huang and J. Li, Composition of pseudo almost-periodic functions and semilinear differential equations, J. Math. Anal. Appl. 255 (2001), pp. 436–446; J. Liang, J. Zhang, T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2001), pp. 1493–1499]. We also study some sufficient conditions for the continuity of the superposition operator. As an application to the abstract results, we give some existence theorems of pseudo almost periodic/automorphic solutions for some semilinear evolution equations and examples with the heat equation.     

Weighted pseudo almost periodic solutions for some partial functional differential equations

In this work, we give sufficient conditions for the existence and uniqueness of a weighted pseudo almost periodic solution for some partial functional differential equations. To illustrate our main result, we study the existence of a weighted pseudo almost periodic solution for some diffusion equation with delay.     

Positive pseudo almost periodic solutions for a class of differential iterative equations with biological background

The article considers a class of differential iterative equations with biological background. We establish some sufficient conditions for the existence of positive pseudo almost periodic solutions by applying exponential dichotomy and contraction mapping principle. The obtained results extend some known ones.     

Weighted pseudo‐almost periodic functions on time scales with applications to cellular neural networks with discrete delays

In this paper, we first propose a concept of weighted pseudo‐almost periodic functions on time scales and study some basic properties of weighted pseudo‐almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo‐almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo‐almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new. Copyright © 2016 John Wiley & Sons, Ltd.     

Pseudo almost periodic solutions for a model of hematopoiesis with a feedback control

In this paper, a generalized model of hematopoiesis is considered with the introduction of a feedback control and continuously distributed delays. By using Lyapunov functional method and differential inequality techniques, we obtain some sufficient conditions for the existence and global exponential stability of positive pseudo almost periodic solutions of this model. We also provide numerical simulations to support the theoretical results.     

Existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations

This paper is concerned with the study of weighted pseudo almost periodic solutions to some classes of hyperbolic differential equations. Upon making some suitable assumptions, the existence and uniqueness of a weighted pseudo almost periodic solution is obtained. To illustrate our main result, we study the existence of a weighted pseudo almost periodic solution to a system of partial differential equations arising in control systems described by abstract retarded functional-differential equations with feedback control.     

Pseudo almost periodic solutions for a class of first order differential iterative equations

This paper is concerned with a class of first order differential iterative equations. Under proper conditions, we employ a novel argument to establish a criterion on the existence of pseudo almost periodic solutions. The obtained result complements with some existing ones.     

DYNAMICS OF NEW CLASS OF HOPFIELD NEURAL NETWORKS WITH TIME-VARYING AND DISTRIBUTED DELAYS

In this paper,we investigate the dynamics and the global exponential stability of a new class of Hopfield neural network with time-varying and distributed delays.In fact,the properties of norms and the contraction principle are adjusted to ensure the existence as well as the uniqueness of the pseudo almost periodic solution,which is also its derivative pseudo almost periodic.This results are without resorting to the theory of exponential dichotomy.Furthermore,by employing the suitable Lyapunov function,some delay-independent sufficient conditions are derived for exponential convergence.The main originality lies in the fact that spaces considered in this paper generalize the notion of periodicity and almost periodicity.Lastly,two examples are given to demonstrate the validity of the proposed theoretical results.     

Existence of positive pseudo almost periodic solutions to a class of neutral integral equations

In this work, a new fixed point theorem in partially ordered Banach spaces is established, and then used to prove the existence of positive pseudo almost periodic solutions to a class of neutral integral equations. Our existence theorem extends some recent results due to [E. Ait Dads, P. Cieutat, L. Lhachimi, Positive pseudo almost periodic solutions for some nonlinear infinite delay integral equations, Mathematical and Computer Modelling 49 (2009) 721–739] to a more general class of integral equations.     

On a new almost periodic type solution of a class of singularly perturbed differential equations with piecewise constant argument

A new class of ergodic sequences, pseudo almost periodic sequence, is introduced, and the existence of pseudo almost periodic sequence to difference equation is studied. Based on these, we investigate the existence of pseudo almost periodic solutions for a nonautonomous, singularly perturbed differential equations with piecewise constant argument.     

Positive pseudo almost periodic solutions for a delayed differential neoclassical growth model

This paper is concerned with a non-autonomous delayed differential neoclassical growth model with an oscillating death rate. Under proper conditions, we employ a novel argument to establish a criterion on the existence and global exponential stability of positive pseudo almost periodic solution, which improves and extends some known relevant results. Finally, we present an example along with its numerical simulations to demonstrate the validity of the proposed result.     

Pseudo Almost Periodic Mild Solution of Nonautonomous Impulsive Integro-Differential Equations

In this paper, we investigate the existence, uniqueness and stability of pseudo almost periodic mild solution to nonautonomous impulsive integro-differential equations in Banach space. The working tools are based on the fixed point theorems and Gronwall–Bellman inequality. To illustrate our main results, we study pseudo almost periodic solution of the heat equations with Dirichlet conditions.     

Pseudo almost periodic solutions for a Lasota–Wazewska model with an oscillating death rate

This paper is concerned with a Lasota–Wazewska model with an oscillating death rate. Under proper conditions, we employ a novel argument to establish a criterion on the existence and stability of positive pseudo almost periodic solutions. The obtained result complements with some existing ones.     

Stepanov ergodic perturbations for some neutral partial functional differential equations

In this work, we give sufficient conditions for the existence and uniqueness of μ?pseudo almost periodic integral solutions for some neutral partial functional differential equations with Stepanov μ?pseudo almost periodic forcing functions. Our working tools are based on the variation of constant formula and the spectral decomposition of the phase space. To illustrate our main results, we give applications to a neutral model arising in physical systems, as well as an application to heat equations with discrete and continuous delay. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of μ−pseudo almost periodic solutions to some evolution equations

In this article, a new approach for pseudo almost periodic solution under the measure theory, under Acquistpace‐Terreni conditions. We make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of μ‐pseudo almost periodic solutions to some classes of nonautonomous partial evolution equations. For illustration, we propose some application to a nonautonomous heat equation. Copyright © 2017 John Wiley & Sons, Ltd.     

Weighted pseudo almost periodic functions and applications

This Note introduces a new class of functions called weighted pseudo almost periodic functions, which generalize in a natural fashion the classical pseudo almost periodic functions due to C. Zhang. Properties of those weighted pseudo almost periodic functions are discussed including a composition result of weighted pseudo almost periodic functions, which plays a crucial role for the solvability of some weighted pseudo almost periodic semilinear differential and partial differential equations. To cite this article: T. Diagana, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Weighted Pseudo Almost Periodic Solutions for a Class of Hematopoiesis Model with Time-Varying Delay

In this paper, firstly, a notion of a class of generalized weighted pseudo al-most periodic function is introduced, then we investigate some basic and essential properties of the space that consists of these functions. Finally, we study the exis-tence of weighted pseudo almost periodic solutions to hematopoiesis model with time-varying delay.     

Pseudo almost periodic solutions to some non-autonomous neutral functional differential equations with unbounded delay

In this paper we give some sufficient conditions, which ensure the existence of pseudo almost periodic solutions to some classes of non-autonomous first-order partial neutral functional-differential with unbounded delay.     

Exponential stability of pseudo almost periodic delayed Nicholson‐type system with patch structure

In this paper, we consider the existence and global exponential stability of positive pseudo almost periodic solutions for a delayed Nicholson‐type system with patch structure. With the aid of ergodic theory and differential inequality technique, we prove the existence and global exponential stability of positive pseudo almost periodic solutions for the addressed system, which generalize and improve some known results. Specially, an example and its numerical simulations are provided to manifest the effectiveness of theoretical results.     

Pseudo almost periodic solution of fractional-order Clifford-valued high-order Hopfield neural networks

In this work, based on the principle of contraction mapping, we deduce sufficient conditions ensuring the existence of pseudo almost periodic solutions of fractional-order Clifford-valued high-order Hopfield neural networks (FCHHNNs). In addition, we employ a kind of Gronwall inequality to study the finite-time stability of pseudo almost periodic solutions of FCHHNNs. The results and methods of our paper are new. Finally, we give a numerical example to illustrate the effectiveness of the results obtained.