Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive
  Mappings   Let $(A, B)$ be a nonempty bounded closed convex proximal parallel pair in a
nearly uniformly convex Banach space and $T: A\cup B \rightarrow A\cup B$ be a
continuous and asymptotically relatively nonexpansive map. We prove that there
exists $x \in A\cup B$ such that $\|x - Tx\| = \emph{dist}(A, B)$ whenever
$T(A) \subseteq B$, $T(B) \subseteq A$. Also, we establish that if $T(A)
\subseteq A$ and $T(B) \subseteq B$, then there exist $x \in A$ and $y\in B$
such that $Tx = x$, $Ty = y$ and $\|x - y\| = \emph{dist}(A, B)$. We prove the
aforesaid results when the pair $(A, B)$ has the rectangle property and
property $UC$. In case of $A = B$, we obtain, as a particular case of our
results, the basic fixed point theorem for asymptotically nonexpansive maps by
Goebel and Kirk.

4
The categories of this paper are: cond-mat
18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
17
The categories of this paper are: q-fin
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph