Stepanov-like pseudo-almost periodic mild solutions to perturbed nonautonomous evolution equations with infinite delay

In this paper, we prove the invariance of Stepanov-like pseudo-almost periodic functions under bounded linear operators. Furthermore, we obtain existence and uniqueness theorems of pseudo-almost periodic mild solutions to evolution equations u^′(t)=A(t)u(t)+h(t) and on , assuming that A(t) satisfy “Acquistapace–Terreni” conditions, that the evolution family generated by A(t) has exponential dichotomy, that R(λ₀,A()) is almost periodic, that B,C(t,s)_t≥s are bounded linear operators, that f is Lipschitz with respect to the second argument uniformly in the first argument and that h, f, F are Stepanov-like pseudo-almost periodic for p>1 and continuous. To illustrate our abstract result, a concrete example is given.     

The existence of mild solutions for impulsive fractional partial differential equations

This paper is concerned with the existence of mild solutions for a class of impulsive fractional partial semilinear differential equations. Some errors in Mophou (2010) [2] are corrected, and some previous results are generalized.     

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 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.     

The existence of pseudo-almost periodic solutions of third-order neutral differential equations with piecewise constant argument

In this paper, we present some existence theorems for pseudo-almost periodic solutions of differential equations with piecewise constant argument by means of pseudo-almost periodic solutions of relevant difference equations.     

Weighted pseudo-almost periodic solutions of a class of abstract differential equations

For abstract linear functional differential equations with a weighted pseudo-almost periodic forcing term, we prove that the existence of a bounded solution on R⁺ implies the existence of a weighted pseudo-almost periodic solution. Our results extend the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations. To illustrate the results, we consider the Lotka-Volterra model with diffusion.     

Almost periodic solutions for forced perturbed impulsive differential equations

Sufficirnt condition for the existence of almost periodic solutions of forced perturbed systems of impulsive differential equations with impulsive effect at fixed Moments are considered.     

Almost periodic solutions of discrete Volterra equations

For discrete Volterra equations with or without delay, we obtain several results concerning almost periodic solutions and asymptotically almost periodic solutions under certain conditions. We also investigate the relations among solutions of equations discussed and give an example to illustrate our results.     

Existence of almost periodic solutions for strong stable impulsive differential equations

Conditions for strong stability and the existence of almostperiodic solutions of systems of impulsive differential equationswith impulsive effect at fixed moments are obtained. The investigationsare carried out by means of piecewise continuous functions whichare analogues of Lyapunov functions.     

Variational solutions for partial differential equations driven by a fractional noise

In this article we develop an existence and uniqueness theory of variational solutions for a class of nonautonomous stochastic partial differential equations of parabolic type defined on a bounded open subset D⊂R^d and driven by an infinite-dimensional multiplicative fractional noise. We introduce two notions of such solutions for them and prove their existence and their indistinguishability by assuming that the noise is derived from an L²(D)-valued fractional Wiener process W^H with Hurst parameter , whose covariance operator satisfies appropriate integrability conditions, and where γ∈(0,1] denotes the Hölder exponent of the derivative of the nonlinearity in the stochastic term of the equations. We also prove the uniqueness of solutions when the stochastic term is an affine function of the unknown random field. Our existence and uniqueness proofs rest upon the construction and the convergence of a suitable sequence of Faedo-Galerkin approximations, while our proof of indistinguishability is based on certain density arguments as well as on new continuity properties of the stochastic integral we define with respect to W^H.     

Almost periodic linear differential equations with non-separated solutions

A celebrated result by Favard states that, for certain almost periodic linear differential systems, the existence of a bounded solution implies the existence of an almost periodic solution. A key assumption in this result is the separation among bounded solutions. Here we prove a theorem of anti-Favard type: if there are bounded solutions which are non-separated (in a strong sense) sometimes almost periodic solutions do not exist. Strongly non-separated solutions appear when the associated homogeneous system has homoclinic solutions. This point of view unifies two fascinating examples by Zhikov-Levitan and Johnson for the scalar case. Our construction uses the ideas of Zhikov-Levitan together with the theory of characters in topological groups.     

Existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations

Fractional stochastic differential equations have gained considerable importance due to their application in various fields of science and engineering. This paper is concerned with the square-mean pseudo almost automorphic solutions for a class of fractional stochastic differential equations in a Hilbert space. The main objective of this paper is to establish the existence and uniqueness of square-mean pseudo almost automorphic mild solutions to a linear and semilinear case of these equations. A new set of sufficient conditions is obtained to achieve the required result by using the stochastic analysis theory and fixed point strategy. Finally, an example is provided to illustrate the obtained theory.     

Existence of almost periodic solutions to some stochastic differential equations

The article introduces and studies the concept of p-mean almost periodicity for stochastic processes. Our abstract results are, subsequently, applied to studying the existence of square-mean almost periodic solutions to some semilinear stochastic equations.     

Almost periodic solutions for nonlinear duffing equations

The main purpose of this paper is to investigate the existence of almost periodic solutions for the Duffing differential equation. By combining the theory of exponential dichotomies with Liapunov functions, we obtain an intersting result on the existence of almost periodic solutions. This work is supported by NSF of China, No.19401013     

Asymptotically almost periodic solutions of stochastic functional differential equations

In this paper, we investigate a class of stochastic functional differential equations of the form

dx(t)=(Ax(t)+F(t,x(t),x_t))dt+G(t,x(t),x_t)°dW(t).     

On almost periodic mild solutions for stochastic functional differential equations

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