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

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 $\mathbb T$ from the super space $\mathscr B$ 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 $\mu$ on $X$, there is a convex function $\digamma$ on the cone $V=\mathbb T(\mathscr B)$ which is Lipschitzian on each bounded set of $V$ such that $$\digamma(\mathbb T(B))=\mu(B),\;\;\forall\;B\in\mathscr B.$$ 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 Goebel, Rzymowski, and Bana\'{s}.