Banach空间中积分-微分方程的周期边值问题

On periodic initial value problems of nonlinear integro- differential equations in banach spaces

考虑Banach空间中非线性积分-微分方程的周期边值问题u ′=f(t,u,Tu),u(0)=u(2π) ∈E。其中Tu=∫0t h(t,s)u(s)ds, h > 0,f∈C[J×E×E,E].利用抽象锥、推广了的比较定理和定义域与值域不同的非线性算子的不动点定理,构造出两个单调迭代序列,证明了Banach空间中非线性积分-徽分方程具有周期边值的最小值、最大解存在定理。

By considering the periodic initial problems of nonlinear integro-differential equations in Banach spaces u ′=f(t,u,Tu) u(0)=u(2π) ∈E where Tu=∫0t h(t,s)u(s)ds, h≥0,f∈C[J×E×E,E],and applying abstract cone and the theorem of fixed points of nonlinear operators which are different from definition region and value region and extended compared theorem,this paper constructs two monotone iterative serials and proves the existence of minimal solutions and maximal solutions with periodic initial problems in Banach spaces.