Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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}.

17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom