一类半线性脉冲发展方程$\left(\omega ,c\right)$-周期解的存在性

Existence of $\left(\omega ,c\right)$-Periodic Solutions for a Class of Semilinear Impulsive Evolution Equations

摘要: 文章用算子半群理论和Schauder不动点定理证明了Banach空间中一类半线性脉冲发展方程{x′(t)=Ax(t)+f(t,x(t)),t∈R+,t≠τi,i∈Ν:={1,2,⋯},Δx|t=τi=x(τi+)−x(τi−)=Bx(τi−)+ci,(ω,c)-周期mild解的存在性。其中，A是稠定闭线性算子，生成X中的C0半群T(t)(t≥0)，B是有界线性算子，f∈C(R+×X,X)，且f满足f(t+ω,cx)=cf(t,x)。x(τi−)和x(τi+)分别表示x(t)在t=τi处的左右极限。

Abstract: The existence of(ω,c)-periodic mild solutions for a class of semilinear impulsive evolution equations in Banach space is proved by operator semigroup theory and Schauder fixed point theorem in this paper.{x′(t)=Ax(t)+f(t,x(t)),t∈R+,t≠τi,i∈Ν:={1,2,⋯},Δx|t=τi=x(τi+)−x(τi−)=Bx(τi−)+ci,Where A is a coherently closed linear operator that generates aC0semigroupT(t)(t≥0)in X, B is the bounded operator,f∈C(R+×X,X), and f satisfiesf(t+ω,cx)=cf(t,x),x(τi−)andx(τi+)represent the left and right limits ofx(t)att=τi.