抽象半线性发展方程正周期解的存在唯一性

讨论有序Banach空间E中半线性发展方程 u′(t)+Au(t)=f(t,u(t)),t∈R, ω-周期解的存在性,其中A为E中正C0-半群的生成元,f:R×E→E连续,关于t 以ω为周期．我们对相应的线性发展方程建立了周期解的存在唯一性,并对周期解算子的谱半径作了精确估计．借助于这个估计,我们用单调迭代方法获得了半线性发展方程正ω-周期解的存在唯一性．

EXISTENCE AND UNIQUENESS OF POSITIVE PERIODIC SOLUTIONS FOR ABSTRACT SEMILINEAR EVOLUTION EQUATIONS

This paper discusses the existence of $\omega$-periodic solutions for semilinear evolution equation $$u'(t)+A\,u(t) =f(t,\,u(t))\,,\quad t \in\R\,,$$ in an ordered Banach space $E$, where $A$ is the infinitesimal generator of a positive $C_0$ -semigroup, and $f:\R\times E\to E$ is a continuous mapping which is $\omega$-periodic in $t$. The existence and uniqueness results of periodic solutions for the associated linear evolution equation is established, and the spectral radius of periodic resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness results of positive periodic solutions are obtained by using monotone iterative technique.