Abstract
In this paper, we investigate the pth mean almost periodic solution to a neutral stochastic evolution equation with infinite delay and Poisson jumps. We give a sufficient condition for the existence and uniqueness of pth mean almost periodic solution and the condition depends on the continuity of coefficients and the power of a fractional operator. We give an example to illustrate the abstract results.
Highlights
The theory of almost periodicity was introduced in the 1920s by Bochner and Bohr, from on, it has been well developed in dynamical systems and differential equations
Almost periodic solutions of stochastic differential equations driven by white Gaussian noise have been studied extensively for the reason that random fluctuations come ubiquitously with all kinds of natural and unnatural systems in the real world
The concept of pth mean almost periodicity has been widely studied in stochastic differential equations since it was firstly considered by Bezandry [4] in 2007
Summary
The theory of almost periodicity was introduced in the 1920s by Bochner and Bohr, from on, it has been well developed in dynamical systems and differential equations. Definition 2.4 A function F : R × Lp(Ω; K) × Z → Lp(Ω; K) is said to be Poisson pth mean almost periodic in t ∈ R uniformly in compact subsets of Lp(Ω, K) if F is continuous in the following sense:. (H3) Let the function g : R × Lp(Ω, K) × B → L(Ω, Lp(Ω, K)) be pth mean almost periodic in t uniformly in compact subsets of L(Ω, Lp(Ω, K)), and there exists a constant Lg > 0 such that g(t, x1, y1) – g(t, x2, y2) ≤ Lg x1 – x2 + y – y2 B for any (x1, y1), (x2, y2) ∈ Lp(Ω, K) × B. The problem (1.1) has a unique pth mean almost periodic mild solution whenever
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