Representation of measures of noncompactness and its applications related to an initial value problem in Banach spaces  

作　　者：Xiaoling Chen Lixin Cheng 

机构地区：[1]School of Science,Jimei University,Xiamen 361021,China [2]School of Mathematical Sciences,Xiamen University,Xiamen 361005,China

出　　处：《Science China Mathematics》2023年第4期745-776,共32页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11731010)。

摘　　要：This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spaces and its applications.With the aid of a three-time order-preserving embedding theorem,we show that for every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K and an order-preserving affine mapping T from the super space B of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)^(+) of C(K),such that for every convex measure,in particular,the regular measure,the homogeneous measure and the sublinear measure of non-generalized compactnessμon X,there is a convex function F on the cone V=T(B)which is Lipschitzian on each bounded set of V such that F(T(B))=μ(B),■B∈B.As its applications,we show a class of basic integral inequalities related to an initial value problem in Banach spaces,and prove a solvability result of the initial value problem,which is an extension of some classical results due to Bana′s and Goebel(1980),Goebel and Rzymowski(1970)and Rzymowski(1971).

关 键 词：representation of measures of noncompactness convex analysis Lebesgue-Bochner measurability integral inequality initial value problem in Banach spaces 

分 类 号：O177.2[理学—数学]