具有非紧半群的发展方程非局部问题mild解的存在性

Existence of Mild Solutions for Nonlocal Problem of Evolution Equations with Noncompact Semigroups

本文研究抽象空间中一类具有非紧半群的半线性发展方程非局部问题.在非线性项满足适当增长条件的情形下,运用算子半群理论、Sadovskii不动点定理及凝聚映射的拓扑度不动点定理获得了所研究问题mild解的存在性.特别地,我们发现本文所得结论对抽象空间中的常微分方程非局部问题同样成立.最后,我们给出一个具体的抛物型偏微分方程非局部问题的例子来说明本文所得抽象结果的可行性.

In this paper,we study the nonlocal problem of a class of semilinear evolution equations with noncompact semigroup in abstract spaces.By using the theory of operator semigroup,Sadovskii＇s fixed point theorem and topological degree fixed point theorem with respect to condensing operator,the existence of mild solutions is obtained under some appropriate growth conditions of the nonlinear term.Particularly,we find that the results obtained are also valid for the nonlocal problem of ordinary differential equations in abstract spaces.At last,an example of concrete parabolic partial differential equations with nonlocal conditions is also given to illustrate the feasibility of our abstract results.