Periodic Mild Solutions Research Articles

Existence and exponential stability of pseudo almost periodic solutions for impulsive nonautonomous partial stochastic evolution equations

In this paper, the concept of p-mean piecewise pseudo almost periodic for stochastic processes is first introduced. Using the exponential dichotomy techniques and a fixed point strategy with stochastic analysis theory, we establish the existence of p-mean piecewise pseudo almost periodic mild solutions for a class of impulsive nonautonomous partial stochastic evolution equations in Hilbert spaces. Moreover, the exponential stability of p-mean piecewise pseudo almost periodic mild solutions is investigated. Finally, an example is provided to illustrate the obtained theory.

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

AbstractIn 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.

Pseudo [formula omitted]-asymptotically periodic solutions of second-order abstract Cauchy problems

In this work we discuss the existence of pseudo S-asymptotically periodic mild solutions for a second order abstract Cauchy problem. In particular, we show that pseudo S-asymptotically ω-periodic cosine functions of operators are ω-periodic. The paper is completed with an application to the nonautonomous wave equation.

Asymptotic Periodicity for Flexible Structural Systems and Applications

We study the existence of pseudo S$S$-asymptotically ?$\omega$-periodic mild solutions for an abstract version of the damped wave equation u?+?u?=βΔu+?Δu?+f$u''+\alpha u'''=\beta\Delta u +\gamma\Delta u'+f$ which model the vibrations of flexible structures possessing internal material damping and external force f$f$. We show concrete applications of our methods to flexible structures, heat conduction, fractional equations and viscoelasticity.

Asymptotic Almost Periodicity to Some Evolution Equations

In this paper, we introduce a new notion of semi-Lipschitz continuity for the class of asymptotically almost periodic functions and establish new existence theorems for asymptotically almost periodic mild solutions to some semi linear abstract evolution equations upon making some suitable assumptions. As one would expect, the results presented here would generalize and improve some recent results in this area.

Periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations in Banach spaces

In this paper, we are concerned with the periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations with periodic inhomogenous terms in Banach spaces, where the operators in the linear part (possibly unbounded) depend on the time [Formula: see text] and generate an evolution family of linear operators. We first establish two new Gronwall–Bellman-type inequalities, and then prove a new and general existence theorem for periodic mild solutions to the nonautonomous impulsive delay evolution equations, which extends essentially some existing results even for the autonomous case as well as for the case when impulsive perturbations or delays are absent.

Global Solutions for Abstract Differential Equations with Non-Instantaneous Impulses

In this note we study the existence of global solutions for a class of impulsive abstract differential equations with non-instantaneous impulses. Specifically, we establish the existence of mild solutions on $${[0, \infty)}$$ and the existence of $${\mathcal{S}}$$ -asymptotically $${\omega}$$ -periodic mild solutions. Our results are based on the Hausdorff measure of non-compactness. Some applications involving partial differential equations are considered.

Weighted pseudo asymptotically periodic mild solutions of evolution equations

In this paper, we propose a new class of functions called weighted pseudo S-asymptotically periodic function in the Stepanov sense and explore its properties in Banach space including composition theorems. Furthermore, the existence, uniqueness of the weighted pseudo S-asymptotically periodic mild solutions to partial evolution equations and nonautonomous semilinear evolution equations are investigated. Some interesting examples are presented to illustrate the main findings.

Pseudo almost periodic and automorphic mild solutions to nonautonomous neutral partial evolution equations

AbstractIn this work, we present a new concept of Stepanov weighted pseudo almost periodic and automorphic functions which is more generale than the classical one, and we obtain a new existence result of μ-pseudo almost periodic and μ-pseudo almost automorphic mild solutions for some nonautonomous evolution equations with Stepanov μ-pseudo almost periodic terms. An example is shown to illustrate our results.

Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional Brownian motion

In this work, we establish a new concept of pseudo almost periodic processes in p-th mean sense using the measure theory. We use the μ-ergodic process to define the spaces of μ-pseudo almost periodic process in the p-th mean sense. We establish many interesting results on the functional space of such processes like completeness and composition theorems. The main objective of this paper is to use those results and some stochastic analysis approaches to study the existence, the uniqueness and the global attractiveness for a μ-pseudo almost periodic mild solution to a class of abstract stochastic evolution equations driven by fractional Brownian motion. We provide an example to illustrate our results.

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.

Periodicity and ergodicity for abstract evolution equations with critical nonlinearities

We study the existence of pseudo almost periodic mild solutions for the abstract evolution equation \(u'(t)=Au(t)+f(t,u(t))\), \(t\in\mathbb {R,}\) when the nonlinearity f satisfies certain critical conditions. We apply our abstract results to the heat equation.

Existence of Stepanov-like square-mean pseudo almost periodic solutions to partial stochastic neutral differential equations

In this paper, we introduce the concept of Stepanov-like square-mean pseudo almost periodic functions and establish the existence and uniqueness of square-mean almost periodic mild solutions for several neutral partial stochastic differential equations with Stepanov-like almost periodic coefficients in a real separable Hilbert space. Moreover, two examples are given to illustrate the general theorems.

On almost periodic mild solutions for neutral stochastic evolution equations with infinite delay

In this paper, a class of neutral stochastic functional evolution equations with infinite delay is investigated. Under some suitable assumptions, the existence and uniqueness of quadratic mean almost periodic mild solutions for these equations are discussed by means of semigroups of operators and the fixed point method. Moreover, two examples are given to illustrate our results.

Periodic Motions of Stokes and Navier–Stokes Flows Around a Rotating Obstacle

We prove the existence and uniqueness of periodic motions to Stokes and Navier–Stokes flows around a rotating obstacle \({D \subset \mathbb{R}^3}\) with the complement \({\Omega = \mathbb{R}^3 \backslash D}\) being an exterior domain. In our strategy, we show the Cb-regularity in time for the mild solutions to linearized equations in the Lorentz space \({L^{3,\infty}(\Omega)}\) (known as weak-L3 spaces) and prove a Massera-typed Theorem on the existence and uniqueness of periodic mild solutions to the linearized equations in weak-L3 spaces. We then use the obtained results for such equations and the fixed point argument to prove such results for Navier–Stokes equations around a rotating obstacle. We also show the stability of such periodic solutions.

Asymptotically Periodic Solutions to Nonlocal Cauchy Problems Governed by Compact Evolution Families

We study a class of abstract Cauchy problems for semilinear nonautonomous evolution equations involving nonlocal initial conditions. Combining the theory of evolution families with the Krasnosel’skii fixed-point theorem and a decomposition technique, we prove the existence of asymptotically periodic mild solutions to these problems. Our results generalize and improve some previous results, since the condition of (local) Lipschitz continuity of the nonlinearity is not required. A partial differential equation is also presented as an application.

Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations

This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equationsdxt=Atxt+Ft,xt,xtdt+H(t,xt,xt)∘dW(t). A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup{Tt}t≥0is essentially removed, which is generated by the linear densely defined operatorA∶D(A)⊂L2(ℙ,ℍ)→L2(ℙ,ℍ), only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain. An example is also given to illustrate our results.

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.

ON PSEUDO -ASYMPTOTICALLY PERIODIC FUNCTIONS

AbstractWe introduce the concept of pseudo$ \mathcal{S} $-asymptotically periodic functions and study some of the qualitative properties of functions of this type. In addition, we discuss the existence of pseudo$ \mathcal{S} $-asymptotically periodic mild solutions for abstract neutral functional differential equations. Some applications involving ordinary and partial differential equations with delay are presented.

Asymptotically periodic solutions of neutral partial differential equations with infinite delay

In this paper we discuss the existence and uniqueness of asymptotically almost automorphic and $S$-asymptotically $\omega$-periodic mild solutions to some abstract nonlinear integro-differential equation of neutral type with infinite delay. We apply our results to neutral partial differential equations with infinite delay.