非紧性测度和不动点定理及其应用(英)

MEASURES OF NONCOMPACTNESS AND FIXED POINT THEOREM AND THEIR APPLICATION

在Ｂａｎａｃｈ空间中研究微分方程的解的存在性问题时，经常用到Ｋｕｒａｔｏｗｓｋｉ引进的非紧性测度α（·）．本文在ｋ阶导数连续的函数空间定义了一类新的集函数Ω（·），我们称其为Ω－非紧性测度．其性质与非紧性测度α（·）的非常相似．然后又给出了一个不动点定理．利用Ω－非紧性测度和不动点定理，我们给出了两个例子，证明了Ｂａｎａｃｈ空间微分方程的解的存在性定理．其方法较以往要简练得多．特别是例１的结果有了很大的改近．

While discussing the existence of solution of differential equations in Banach spaces we often use the measureα(·) of noncompactness by name of Kuratowsiki. In this paper we define a kind of new set functions Ω(·) that we name it Ω-noncompact measure in the function spaces which k-order derivative functions are continuous. The properties of Ω(·) are very similar to that of α (· ). We prove a theorem of fixed point. Using the Ω-noncompact measure and the fixed point theorem we give two examples to prove existence theorems of differential equations in Banach spaces. The methods are more short-cat than before.