Measure pseudo almost periodic solution for a class of nonlinear delayed stochastic evolution equations driven by Brownian motion

Abstract

In this work, we present a new concept of measure-ergodic process to define the space of measure pseudo almost periodic process in the p-th mean sense. We show some results regarding the completeness, the composition theorems and the invariance of the space consisting in measure pseudo almost periodic process. Motivated by above mentioned results, the Banach fixed point theorem and the stochastic analysis techniques, we prove the existence, uniqueness and the global exponential stability of doubly measure pseudo almost periodic mild solution for a class of nonlinear delayed stochastic evolution equations driven by Brownian motion in a separable real Hilbert space. We provide an example to illustrate the effectiveness of our results.

Existence and exponential stability of the piecewise pseudo almost periodic mild solution for some partial impulsive stochastic neutral evolution equations

“…In this section, we shall give some preliminary results which will be used in the sequel. The involvement of these results is stated with reference to previous studies [2,26,[30][31][32]. Throughout this paper, we shall introduce the following notations:…”

“…In this section, we shall give some preliminary results which will be used in the sequel. The involvement of these results is stated with reference to previous studies [2, 26, 30–32]. Throughout this paper, we shall introduce the following notations: $$$ \left(\mathcal{H},{\left\Vert \cdotp \right\Vert}_{\mathcal{H}}\right) $$$ and $$$ \left(\mathcal{K},{\left\Vert \cdotp \right\Vert}_{\mathcal{K}}\right) $$$ are real separable Hilbert spaces. $$$ \left(\Omega, \mathcal{F},P\right) $$$ is a complete probability space. Let $$$ {L}^r\left(P,\mathcal{H}\right) $$$ a Banach space defined by $$$$ {L}^r\left(P,\mathcal{H}\right):= \left\{Y:\mathcal{H}-\mathrm{valued}\ \mathrm{random}\ \mathrm{variable}\right\}\cdotp $$$$ With the norm $$$ \mathcal{L}\left(\mathcal{K},\mathcal{H}\right)=\left\{Y:\mathcal{K}\to \mathcal{H}\kern0.1em \mathrm{linear}\ \mathrm{bounded}\ \mathrm{operators}\right\} $$$.…”

“…Zuomao and Fangxia [25] studied the existence of $$$ r $$$‐mean piecewise pseudo almost periodic to the following partial evolution equations: $$$$ {\displaystyle \begin{array}{cc}\hfill dx(t)=& \kern0.3em \left[A(t)x(t)+\varphi \left(t,x(t)\right)\right] dt+\psi \left(t,x(t)\right) dB(t),t\in \mathrm{\mathbb{R}},t\ne {t}_k,k\in \mathrm{\mathbb{Z}},\hfill \\ {}\hfill \Delta x\left({t}_k\right)=& \kern0.3em x\left({t}_k^{+}\right)-x\left({t}_k^{-}\right)={K}_k\left(x\left({t}_k\right)\right),k\in \mathrm{\mathbb{Z}},\hfill \end{array}} $$$$ where $$$ A(t) $$$ is a family of densely defined closed linear operators and $$$ \varphi, \psi $$$ are some appropriate functions. Motivated by the above description, and unlike works [26, 27], where we showed the uniqueness of the solution by Banach's fixed bridge theorem, assuming that t...…”

“…where A(t) is a family of densely defined closed linear operators and 𝜑, 𝜓 are some appropriate functions. Motivated by the above description, and unlike works [26,27], where we showed the uniqueness of the solution by Banach's fixed bridge theorem, assuming that the coefficient functions of the differential equation are Lipschitzian, in this paper, we used another theorem of fixed point, and subsequently, we lost the uniqueness, and in this paper, we investigate the existence and stability of double measure r-mean piecewise pseudo almost periodic solution of the following stochastic differential equation:…”

Stochastic Nicholson’s blowflies model with delays

“…where (A(t)) t∈R is a family of densely defined closed linear operators satisfying Acquistapace-Terreni conditions; f, θ are two stochastic processes and ψ a function deterministic. The notion of pseudo almost periodicity with measure (see [2,6,8,10,[14][15][16][17]) is a generalization of the almost periodicity and pseudoalmost periodicity introduced by Zhang [20]; it is also a generalization of weighted pseudo almost periodicity firstly introduced by Diagana [9]. The concept of measure almost periodicity is of great importance in probability for investigating stochastic processes.…”