Pseudo-almost Automorphic Solutions Research Articles

Weighted Pseudo Almost-Automorphic Solutions of Quaternion-Valued RNNs With Mixed Delays

Weighted Pseudo Almost-Automorphic Solutions of Quaternion-Valued RNNs With Mixed Delays

Second-order nonlinear differential equations: existence, uniqueness and global exponential stability of doubly measure pseudo-almost automorphic solutions

The main aim of this work is to study a second-order nonlinear differential equation with mixed delay and time-varying coefficients. By employing the fixed-point theorem and some properties of the doubly measure pseudo almost automorphic functions, we prove the existence, uniqueness and global exponential stability of doubly measure pseudo-almost automorphic solution to the second-order delay differential equation. Our approach generalizes the classical results on weighted pseudo almost automorphic functions. Finally, a numerical example is provided to illustrate the effectiveness and feasibility of the obtained results.

Study of genetic regulatory networks with Stepanov-like pseudo-weighted almost automorphic coefficients

In this article, we investigated a class of genetic regulatory networks (GRNs) with time-varying delays and Stepanov-like weighted pseudo-almost automorphic coefficients. By Banach fixed point theorem, we firstly established the existence results of the weighted pseudo-almost automorphic solutions for the proposed GRNs. Secondly, by using suitable Lyapunov functional and mathematical analysis skills, new criteria are derived for ensuring the global exponential stability of the solutions. Finally, a numerical example and remarks are provided to substantiate the efficiency of obtained results. The derived results of this article are new and complement many earlier works and the ideas of this work can be applied to investigate other similar systems.

An analysis for a special class of solution of a Duffing system with variable delays

<abstract> <p>In this study, we are concerned the existence of pseudo almost automorphic (PAA) solutions and globally exponential stability of a Duffing equation system with variable delays. Some differential inequalities and the well-known Banach fixed point theorem are used for the existence and uniqueness of PAA solutions. Also, with the help of Lyapunov functions, sufficient conditions are obtained for globally exponential stability of PAA solutions. Since the PAA is more general than the almost and pseudo almost periodicity, this work is new and complementary compared to previous studies. In addition, an example is given to show the correctness of our results.</p> </abstract>

Generalization of Weighted Stepanov-Like Pseudo-Almost Automorphic Space

In this paper we introduce and study a new class of functions called weighted Stepanov-like pseudo-almost automorphic functions with variable exponents, which generalizes the class of weighted Stepanov-like pseudo-almost automorphic functions. Basic properties of these new spaces are established. The existence of weighted pseudo-almost automorphic solutions to some first-order differential equations with Sp,q(x)-pseudo-almost automorphic coefficients will also be studied.

$$(\mu ,\nu )$$(μ,ν)-Pseudo-almost automorphic solutions for high-order Hopfield bidirectional associative memory neural networks

This article is concerned with a high-order Hopfield bidirectional associative memory neural networks with time-varying coefficients and mixed delays. Sufficient conditions are derived for the existence, the uniqueness and the exponential stability of $$(\mu ,\nu )$$-pseudo-almost automorphic solutions of the considered model. Banach fixed-point theorem is applied for the existence and the uniqueness results. Global exponential stability is derived via differential inequalities. Finally, two examples are provided to support the feasibility of the theoretical results.

Besicovitch almost automorphic solutions of nonautonomous differential equations of first order

The main purpose of this paper is to analyze the existence and uniqueness of Besicovitch almost automorphic solutions and weighted Besicovitch pseudo-almost automorphic solutions of nonautonomous differential equations of first order. We provide an interesting application of our abstract theoretical results.

Weighted pseudo-almost automorphic solutions of high-order Hopfield neural networks with neutral distributed delays

In this paper, a class of high-order Hopfield neural networks with neutral distributed delays is considered. Some sufficient conditions are obtained for the existence, uniqueness and global exponential stability of weighted pseudo-almost automorphic solutions for this class of networks by employing the Banach fixed-point theorem and differential inequality techniques. The results of this paper are completely new. An example is given to show the effectiveness of the proposed method and results.

Pseudo-almost Periodic and Pseudo-almost Automorphic Solutions to Evolution Equations in Hilbert Spaces

In this work we prove some existence and uniqueness results for pseudo-almost periodic and pseudo-almost automorphic solutions to a class of semi-linear differential equations in Hilbert spaces using theoretical measure theory. The main technique is based upon some appropriate composition theorems combined with the Banach contraction mapping principle and the method of the invariant subspaces for unbounded linear operators. A few illustrative examples will be discussed at the end of the paper.

Weighted Stepanov-like pseudo-almost automorphic mild solutions for semilinear fractional differential equations

This work is concerned with the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions for a class of semilinear fractional differential equations, $D_{t}^{\alpha}x(t)=Ax(t)+D_{t}^{\alpha-1}F(t,x(t))$ , $t\in \mathbb{R}$ , where $1<\alpha<2$ , A is a linear densely defined operator of sectorial type of $\omega<0$ on a complex Banach space X and F is an appropriate function defined on phase space. The fractional derivative is understood in the Riemann-Liouville sense. The results obtained are utilized to study the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions for a fractional relaxation-oscillation equation.

Existence of pseudo-almost automorphic solutions for nonlinear differential equations

In this paper, we discuss the existence of pseudo-almost automorphic solutions to linear differential equation which has an exponential trichotomy, and the results also hold for some nonlinear equations with the form x′(t) = f(t, x(t)) + λ g (t, x(t)), where f, g are pseudo-almost automorphic functions. We prove our main result by the application of Leray-Schauder fixed point theorem.

Weighted pseudo almost automorphic classical solutions and optimal mild solutions for fractional differential equations and application in fractional reaction–diffusion equations

In this paper, we are concerned with a class of fractional differential equations given by $$\begin{aligned} \hbox {D}_{t}^{\alpha }x(t)=Ax(t)+f(t,x(t)). \end{aligned}$$ Our main results concern the existence, uniqueness of weighted pseudo-almost automorphic classical solutions and optimal mild solutions. Moreover, as example and applications, we study the weighted pseudo-almost automorphic classical solutions and optimal mild solutions for a fractional reaction–diffusion equation to illustrate the practical usefulness of the analytical results that we establish in the paper.

Existence results for some damped second-order Volterra integro-differential equations

In this paper we make a subtle use of operator theory techniques and the well-known Schauder fixed-point principle to establish the existence of pseudo-almost automorphic solutions to some second-order damped integro-differential equations with pseudo-almost automorphic coefficients. In order to illustrate our main results, we will study the existence of pseudo-almost automorphic solutions to a structurally damped plate-like boundary value problem.

Some properties of weighted Stepanov-like pseudo almost auto-morphic functions and applications to Volterra integral equations

This paper is concerned with weighted Stepanov-like pseudo-almost automorphic functions. We rst study the relationship between a weighted Stepanov-like pseudo-almost automorphic function and its Stepanov-like almost automorphic component. With the help of the relationship, previous composition theorems of these functions are improved. Secondly, we study the convolution operators generated by absolutely integrable functions in weighted Stepanov-like pseudo-almost automorphic function spaces. Finally, by using contraction mapping principle, we study the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic solutions for two classes of Volterra integral equations. These results are new and improve some known ones.

Pseudo-Almost Periodic and Pseudo-Almost Automorphic Solutions to Some Evolution Equations Involving Theoretical Measure Theory

Toka DiaganaHoward University,2441 6th Street N.W.,Washington, D.C. 20059,USA.tdiagana@howard.eduKhalil EzzinbiUniversit´e Cadi Ayyad,Faculte´ des Sciences Semlalia,D´epartement deMath´ematiques,BP 2390, Marrakesh, MarocMohsen MiraouiInstitut sup´erieur des Etudestechnologiques de Kairouan,Rakkada-3191 Kairouan,Tunisie.ABSTRACTMotivated by the recent works by the first and the second named authors, in this paperwe introduce the notion of doubly-weighted pseudo-almost periodicity (respectively,doubly-weighted pseudo-almost automorphy) using theoretical measure theory. Basicproperties of these new spaces are studied. To illustrate our work,westudy,underAcquistapace–Terreni conditions and exponential dichotomy, the existence of (µ,ν)-pseudo almost periodic (respectively, (µ,ν)-pseudo almost automorphic) solutions tosome nonautonomous partial evolution equations in Banach spaces. A few illustrativeexamples will be discussed at the end of the paper.RESUMENMotivado por los trabajos recientes del primer y segundo autor, en este art´iculo intro-ducimos la noci´on de seudo-casi periodicidad con doble peso (seudo-casi automorf´ia condoble peso respectivamente) usando Teor´ia de la Medida. Se estudian las propiedadesbasicas de estos espacios nuevos. Para ilustrar nuestro trabajo, bajo las condiciones deAcquistapace-Terreni y dicotom´ia exponencial estudiamos la existencia de soluciones(respectivamente, (µ,ν) seudo-casi peri´odicas (µ,ν) seudo-casi automorficas) para al-gunas ecuaciones parciales de evoluci´on autonomas en espacios de Banach. Algunosejemplos ilustrativos se discutiran al final del art´iculo.Keywords and Phrases: Evolution family; exponential dichotomy; Acquistapace–Terreni con-ditions; pseudo-almost periodic; pseudo-almost automorphic; evolution equation; nonautonomousequation; doubly-weighted pseudo-almost periodic; doubly-weighted pseudo-almost automorphy;(µ,ν)-pseudo-almost periodicity; (µ,ν)-pseudo-almost automorphy; neutral systems; positive mea-sure.2010 AMS Mathematics Subject Classification: 34C27; 34K14; 34K30; 35B15; 43A60;47D06; 28Axx; 58D25; 65J08.

Existence of Weighted Pseudo-almost Automorphic Mild Solutions to Non-autonomous Abstract Neutral Differential Equations with Deviating Arguments

In this article we obtain sufficient conditions for the existence and uniqueness of a weighted pseudo-almost automorphic mild solution to a non-autonomous neutral differential equation with a deviating argument in an arbitrary Banach space. We use the exponentially stable evolution semigroup and the Banach fixed point theorem for establishing the main results. Extension of the main results to some integro-differential equation has also been considered.

Pseudo-almost automorphic mild solutions to semilinear integral equations in a Banach space

In this paper, we consider the existence of pseudo-almost automorphic solutions of the semilinear integral equation x ( t ) = ∫ − ∞ t a ( t − s ) [ A x ( s ) + f ( s , x ( s ) ) ] d s , t ∈ R in a Banach space X , where a ∈ L 1 ( R + ) , A is the generator of an integral resolvent family of linear bounded operators defined on the Banach space X , and f : R × X → X is a pseudo-almost automorphic function. The main results are proved by using integral resolvent families combined with the theory of fixed points.

Weighted pseudo-almost automorphic solutions for some partial functional differential equations

In this paper, we study the existence and uniqueness of a weighted pseudo-almost automorphic solution for some nonhomogeneous partial functional differential equations. We use the variation of constants formula developed in Ezzinbi and N’Guérékata (2007) [11] and the spectral decomposition of the phase space to show the main result of this work. To illustrate our main result, we study the existence and uniqueness of a weighted pseudo-almost automorphic solution for some diffusion equations with delay.

A composition theorem for weighted pseudo-almost automorphic functions and applications

In this paper, we establish a composition theorem for weighted pseudo-almost automorphic functions under a weaker Lipschitz condition. Our composition theorem generalizes some known results. Moreover, the existence and uniqueness of pseudo-almost automorphic solutions for abstract semilinear evolution equations are studied.

Almost automorphic and pseudo-almost automorphic mild solutions to an abstract differential equation in Banach spaces

In this paper we prove some existence results for almost automorphic and pseudo-almost automorphic mild solutions to a class of abstract differential equations in Banach spaces. The main technique is based on some composition theorems combined with the contraction mapping theorem. Finally, we present an application to a semilinear partial differential equation with Dirichlet conditions.