凸幂凝聚算子的不动点定理及其对抽象半线性发展方程的应用

The Fixed Point Theorem of Convex-Power Condensing Operator and Applications to Abstract Semilinear Evolution Equations

从应用问题的需要出发,给出了一类新的算子-凸幂凝聚算子的定义,推广了凝聚算子的概念,并证明了这类新算子的不动点定理,从而推广了著名的Schauder不动点定理和Sadovskii不动点定理.作为应用,获得了Banach空间中一类具有非紧半群的半线性发展方程初值问题整体mild解和正mild解的存在性.

We give the definition of a kind of new operator-convex-power condensing operator for the need of applied questions and generalize the definition of condensing operator. The fixed point theorem of this kind of new operator is proved. This generalizes the famous Schauder fixed point theorem and Sadovskii fixed point theorem. As applications, the existence of global mild solutions and positive mild solutions of initial value problem for a class of semilinear evolution equations with noncompact semigroup in Banach spaces is obtained.