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

Abstract

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, 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 $$\mathbb{T}$$ from the super space $${\cal B}$$ 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, homogeneous measure and sublinear measure of non-generalized compactness μ on X, there is a convex function F on the cone $$V = \mathbb{T}\left( {\cal B} \right)$$ which is Lipschitzian on each bounded set of V such that $$\digamma\left(\mathbb{T}{\left( B \right)} \right) = \mu \left( B \right),\,\,\,\,\,\,\forall \,B \in {\cal B}$$ 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 Banaś and Goebel (1980), Goebel and Rzymowski (1970) and Rzymowski (1971).