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;Binmathscr 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}.
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