Abstract
In this paper, we study a class of second-order neutral impulsive stochasticevolution equations with infinite delay (SNISEEIs in short), in which theinitial value belongs to the abstract space . Sufficient conditions for the existence of themild solutions for SNISEEIs are derived by means of the Krasnoselskii-Schaeferfixed point theorem. Two examples are given to illustrate the obtainedresults.
Highlights
In this paper, we consider the second-order neutral impulsive stochastic evolution equations with infinite delay (SNISEEIs in short) of the following form:d x (t) – g(t, xt) = Ax(t) + f (t, xt) dt + σ (t, xt) dw(t), t ∈ J := [, T], ( ) x = φ ∈ Bh,x ( ) = ψ ∈ H, x(tk) = Ik(xtk ), x = Ik(xtk ), k =, . . . , m.Here, the state x(·) takes values in a separable real Hilbert space H with inner product (·, ·) and norm ·, where A : D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous cosine family C(t) on H
We present the abstract phase space Bh
In Section, we prove the existence of the mild solutions for SNISEEIs by means of the Krasnoselskii-Schaefer fixed point theorem
Summary
1 Introduction In this paper, we consider the second-order neutral impulsive stochastic evolution equations with infinite delay (SNISEEIs in short) of the following form: d x (t) – g(t, xt) = Ax(t) + f (t, xt) dt + σ (t, xt) dw(t), t ∈ J := [ , T], ( ) LQ(K, H) denotes the space of all Q-Hilbert-Schmidt operators from K into H, which will be defined .
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