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 automorphic solutions for dissipative differential equations in Banach spaces

This work aims to study the existence and uniqueness of pseudo compact almost automorphic solution for some dissipative ordinary and functional differential equations. We prove the existence and uniqueness of pseudo compact almost automorphic solution for dissipative differential equations in Banach spaces and then we apply this result to show the existence of pseudo compact almost automorphic solutions for some functional differential equations.     

Pseudo almost periodic in distribution solutions to impulsive partial stochastic functional differential equations

In this paper, we study the piecewise pseudo almost periodicity in distribution for a stochastic process. Using the analytic semigroup theory and fixed point strategy with stochastic analysis theory, we obtain the existence and the exponential stability of piecewise pseudo almost periodic in distribution mild solutions for impulsive partial neutral stochastic functional differential equations under non-Lipschitz conditions. Moreover, an example is given to illustrate the general theorems.     

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.     

Pseudo almost periodic solutions of differential equations with piecewise constant arguments

In this paper, we give a definition of pseudo almost periodic sequence and study the existence of pseudo almost periodic sequence to difference equation. Based on these, we investigate the existence of pseudo almost periodic solutions to differential equations with piecewise constant arguments which was considered by K. L. Cooke & J. Wiener and found applications in certain biomedical problems.     

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.     

The almost periodic type difference equations

In this paper, we study the existence of almost periodic type solutions of difference equations by means of exponential dichotomy and trichotomy of linear difference equations.     

Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces

In this paper, we prove that the space is complete. This not only gives an affirmative answer to a basic problem in this field, but also enables us to obtain an existence and uniqueness theorem of pseudo almost automorphic mild solutions to semilinear differential equations in Banach spaces. An example is given to illustrate our theorem. The work was supported partly by the National Natural Science Foundation of China, the NCET-04-0572 and Research Fund for the Key Program of the Chinese Academy of Sciences.     

Asymptotically almost periodic solutions for abstract neutral integro-differential equations

We study the existence of asymptotically almost periodic classical solutions for a class of abstract neutral integro-differential equation with unbounded delay. A concrete application to partial neutral integro-differential equations which arise in the study of heat conduction in fading memory material is considered.     

Generalized almost periodic and ergodic solutions of linear differential equations on the half-line in Banach spaces

The Bohl-Bohr-Amerio-Kadets theorem states that the indefinite integral y=Pφ of an almost periodic (ap) is again ap if y is bounded and the Banach space X does not contain a subspace isomorphic to c₀. This is here generalized in several directions: Instead of it holds also for φ defined only on a half-line , instead of ap functions abstract classes with suitable properties are admissible, can be weakened to φ in some “mean” class , then ; here contains all f∈L¹_loc with in for all h>0 (usually strictly); furthermore, instead of boundedness of y mean boundedness, y in some , or in , ergodic functions, suffices. The Loomis-Doss result on the almost periodicity of a bounded Ψ for which all differences Ψ(t+h)−Ψ(t) are ap for h>0 is extended analogously, also to higher order differences. Studying “difference spaces” in this connection, we obtain decompositions of the form: Any bounded measurable function is the sum of a bounded ergodic function and the indefinite integral of a bounded ergodic function. The Bohr-Neugebauer result on the almost periodicity of bounded solutions y of linear differential equations P(D)y=φ of degree m with ap φ is extended similarly for ; then provided, for example, y is in some with U=L^∞ or is totally ergodic and, for the half-line, Reλ?0 for all eigenvalues P(λ)=0. Analogous results hold for systems of linear differential equations. Special case: φ bounded and Pφ ergodic implies Pφ bounded. If all Reλ>0, there exists a unique solution y growing not too fast; this y is in if , for quite general .     

Existence of anti-periodic mild solutions for semilinear evolution equations

In this paper we study the existence of anti-periodic mild solutions for a class of semilinear evolution equations in Banach spaces and extend some related results in this direction. An example is given to illustrate our results.     

Pseudo almost periodic solutions for equation with piecewise constant argument

By using the roughness theory of exponential dichotomies and the contraction mapping, some sufficient conditions are obtained for the existence and uniqueness of pseudo almost periodic solution of the above differential equation with piecewise constant argument

     

Almost periodic and almost automorphic solutions to semilinear parabolic boundary differential equations

In this paper, we study the existence and uniqueness of almost periodic and almost automorphic solutions to the semilinear parabolic boundary differential equations

where AA_m|kerL generates a hyperbolic analytic semigroup on a Banach space X. The functions h and are defined on some intermediate subspaces X_β,0<β<1, and take values in X and in a boundary space ∂X respectively.     

Pseudo almost periodic solutions to parabolic boundary value inverse problems

We first define the pseudo almost periodic functions in a more general setting.Then we show the existence,uniqueness and stability of pseudo almost periodic solutions of parabolic inverse problems for a type of boundary value problems.     

Existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional-differential equations and applications

The paper considers the existence and uniqueness of pseudo almost periodic (mild) solutions to some classes of first-order partial neutral functional-differential equations. Upon making some suitable assumptions, existence and uniqueness results are obtained. Applications include both a partial integro-differential equation arising in control systems and the scalar reaction-diffusion equation with delay.     

Pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument

In this paper, we investigate the existence and uniqueness of new almost periodic type solutions, so-called pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument by means of introducing the notion of pseudo almost periodic vector sequences.     

On almost periodic mild solutions for stochastic functional differential equations

In this paper, a class of stochastic functional differential equations given by

     

Square-mean almost periodic solutions to some stochastic evolution equations

This paper concerns the square-mean almost periodic mild solutions to a class of abstract nonautonomous functional integro-differential stochastic evolution equations in a real separable Hilbert space. By using the so-called "Acquistapace–Terreni" conditions and the Banach fixed point theorem, we establish the existence, uniqueness and the asymptotical stability of square-mean almost periodic solutions to such nonautonomous stochastic differential equations. As an application, almost periodic solution to a concrete nonautonomous stochastic integro-differential equation is considered to illustrate the applicability of our abstract results.     

Pseudo almost automorphy of two-term fractional functional differential equations

In this paper, by measure theory, we introduce and investigate the concepts of (Stepanov-like) $(\mu,\nu)$-pseudo almost automorphic of class $r$ and class infinity, respectively. As applications, we establish some sufficient criteria for the existence, uniqueness of pseudo almost automorphic mild solutions to two-term fractional functional differential equations with finite or infinite delay. The working tools are based on the generalization of semigroup theory, Banach contraction mapping principle and Leray-Schauder alternative theorem. Finally, we explore the same topic for a fractional partial functional differential equation with delay.