Research Article
A-Statistical Cluster Points in Finite Dimensional Spaces and
Application to Turnpike Theorem
Pratulananda Das,
1
Sudipta Dutta,
1
S. A. Mohiuddine,
2
and Abdullah Alotaibi
2
1
Department of Mathematics, Jadavpur University, Kolkata West Bengal, 700032, India
2
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Correspondence should be addressed to S. A. Mohiuddine; mohiuddine@gmail.com
Received 6 August 2013; Accepted 19 December 2013; Published 28 January 2014
Academic Editor: Andrew Pickering
Copyright © 2014 Pratulananda Das et al. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
In the frst part of the paper, following the works of Pehlivan et al. (2004), we study the set of all A-statistical cluster points of
sequences in m-dimensional spaces and make certain investigations on the set of all A-statistical cluster points of sequences in
m-dimensional spaces. In the second part of the paper, we apply this notion to study an asymptotic behaviour of optimal paths
and optimal controls in the problem of optimal control in discrete time and prove a general version of turnpike theorem in line of
the work of Mamedov and Pehlivan (2000). However, all results of this section are presented in terms of a more general notion of
I-cluster points.
1. Introduction and Background
Troughout this paper, let be a nonnegative regular matrix
and N will denote the set of all positive integers. Let and
be two sequence spaces and = (
) be an infnite matrix.
If for each ∈ the series
() = ∑
∞
=1
converges
for each and the sequence = {
()} ∈ , we say that
A maps into . By (,) we denote the set of all matrices
which maps into . In addition if the limit is preserved,
then we denote the class of such matrices by (,)
reg
.A
matrix is called regular if ∈ (, ) and lim
→∞
() =
lim
→∞
for all = {
}
∈N
∈ when , as usual, stands for
the set of all convergent sequences. It is well known that the
necessary and sufcient condition for to be regular are
(a) ‖‖ = sup
∑
|
|<∞;
(b) lim
=0, for each ;
(c) lim
∑
=1.
Te idea of -statistical convergence was introduced
by Kolk [1] using a nonnegative regular matrix . For a
nonnegative regular matrix = (
), a set ⊂ N will be
said to have -density if
() := lim
∑
∈
exists. Te
real number sequence = {
}
∈N
is said to be -statistically
convergent to provided that for every >0 the set () :=
{ ∈ N : |
− | ≥ } has -density zero. Note that the
idea of -statistical convergence is an extension of the idea of
statistical convergence introduced by Fast [2] using the idea of
asymptotic density and later studied by Fridy [3, 4], Connor
[5], and
ˇ
Sal´ at [6] (see also [1, 7–11] for more references). Let
= {() : (1) < (2) < (3) < ⋅ ⋅ ⋅ } ⊂ N and {}
= {
()
}
be a subsequence of . If the set has -density zero then
the subsequence {}
of the sequence is called an -thin
subsequence. If the set does not have -density zero then
the subsequence {}
is called an -nonthin subsequence of
. Te statement
() ̸ =0 means that either
() > 0 or
() does not exist.
A family I ⊂2
of subsets of a nonempty set is said
to be an ideal in if (i) , ∈ I imply ∪∈ I;
(ii) ∈ I, ⊂ implies ∈ I, while an admissible ideal
I of further satisfes {} ∈ I for each ∈. If I is a
proper ideal in (i.e, ∉ I, ̸ =0), then the family of sets
(I) = { ⊂ : there exists ∈ I : = \ } is a
flter in . It is called the flter associated with the ideal I.
Troughout I will stand for a proper nontrivial admissible
ideal of N. Let = {() : (1) < (2) < (3) < ⋅ ⋅ ⋅ } ⊂ N and
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 354846, 7 pages
http://dx.doi.org/10.1155/2014/354846