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

Abstract

The purpose of this paper is devoted to studying representation of measures of non generalized compactness, in particular, measures of noncompactness, of non-weak compactness, and of non-super weak compactness, etc, 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 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, regular measure, homogeneous measure, sublinear measure of non generalized compactness µ on X, there is a convex function ̥ on the cone V = T(B) which is Lipschitzian on each bounded set of V such thatAs 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 Goebel, Rzymowski, and Banaś.