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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 354846, 7 pages
A-Statistical Cluster Points in Finite Dimensional Spaces and Application to Turnpike Theorem
1Department of Mathematics, Jadavpur University, Kolkata West Bengal, 700032, India
2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 6 August 2013; Accepted 19 December 2013; Published 28 January 2014
Academic Editor: Andrew Pickering
Copyright © 2014 Pratulananda Das et al. This 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 first 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 -cluster points.
1. Introduction and Background
Throughout this paper, let be a nonnegative regular matrix and will denote the set of all positive integers. Let and be two sequence spaces and be an infinite matrix. If for each the series 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 . A matrix is called regular if and for all when , as usual, stands for the set of all convergent sequences. It is well known that the necessary and sufficient condition for to be regular are(a);(b), ;(c).
The idea of -statistical convergence was introduced by Kolk  using a nonnegative regular matrix . For a nonnegative regular matrix , a set will be said to have -density if exists. The real number sequence is said to be -statistically convergent to provided that for every the set has -density zero. Note that the idea of -statistical convergence is an extension of the idea of statistical convergence introduced by Fast  using the idea of asymptotic density and later studied by Fridy [3, 4], Connor , and Šalát  (see also [1, 7–11] for more references). Let 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 . The statement means that either or does not exist.
A family of subsets of a nonempty set is said to be an ideal in if (i) imply ; (ii), implies , while an admissible ideal of further satisfies for each . If is a proper ideal in (i.e, ), then the family of sets there exists is a filter in . It is called the filter associated with the ideal . Throughout will stand for a proper nontrivial admissible ideal of . Let and be a subsequence of . If the set belongs to , then the subsequence of the sequence is called a -thin subsequence and if the set does not belong to , then the subsequence is called a -nonthin subsequence of .
In this context, it should be mentioned that ideals were first used to generalize the idea of statistical convergence by Kostyrko et al. . For more recent applications of ideals one can see [1, 4, 13–21] where many more references can be found.
The concept of statistical cluster points was introduced by Fridy . A real number is a statistical cluster point of if for every ; the set does not have asymptotic density zero. It was shown that the set of all statistical cluster points is nonempty and compact. Later the notion of statistical cluster points was extended to -cluster points in  (see also ) and to -statistical cluster point by Demirci  (see also ) and very recently to -statistical cluster point by Pehlivan et al. . A number is said to be an -statistical cluster point of the number sequence provided that for every , , where . By , we denote the set of all -statistical cluster points of . Many interesting results concerning statistical and then -cluster points were proved in  and then in .
In this paper, following the line of , we first investigate certain properties of the set of -statistical cluster points in and the concept of -statistical convergence and examine some of its consequences. Though some of our results were proved in  for arbitrary ideals of , we present the results for a specific ideal consisting of -density zero sets with alternative methods of proofs. Finally, we present an application by establishing a general version of turnpike theorem in line of the results proved by Pehlivan and Mamedov [27, 28], which appears to be valid for a special class of ideals.
2. Characterization of
In this section, we investigate some properties of the set of all -statistical cluster points in with usual norm . Consider a sequence and a point . Following , we consider the following definitions.
Definition 1. A sequence is -statistically convergent to if for every
Definition 2. A point is called an -statistical cluster point of the sequence if for every
We denote the set of all -statistical cluster points of the sequence by . Now from Definition 2 it readily follows that where That is, Hence,
Definition 3. A sequence is said to be -statistically bounded if there exists a compact set such that .
The above definitions can be generalized via ideals as follows.
We denote the set of all -cluster points by .
Definition 6. A sequence is said to be -bounded if there exists a compact set such that .
Let stand for the distance of a point from a closed set and it is defined as . Let be the open -neighborhood of .
Lemma 7. Let be a compact subset of such that . Then .
In , it was shown that for the ideal of density zero sets, the above result is not true for open or unbounded subsets of ; that is, the assumption of compactness is essential. However, in , no such example was presented to show the essentiality of compactness. Below we present two examples which show that the assumption of compactness is essential even if we take the ideal of subsets of with -density zero.
Next we show that the compactness of is essential for Lemma 7. The result may fail if one of closedness or boundedness hypotheses is relaxed.
Remark 8. Let be the unbounded closed set . Consider the following nonnegative regular matrix: Now for the sequence , as . In this case, we have but .
Remark 9. Let be the bounded open set . Consider the nonnegative regular matrix: Clearly for the sequence , we have . In this case also, we have , but we have .
Lemma 10. If a sequence has a bounded -nonthin subsequence then the set is nonempty and closed.
Lemma 11. If is -statistically bounded, then the set is nonempty and compact.
Letavaj  proved the ideal version of the above lemma. Next we prove the following interesting theorem.
Theorem 12. If is an -statistically bounded sequence then for every ,
Proof. Since is an -statistically bounded sequence, there exists a compact set such that . Now is nonempty and by Lemma 11. Suppose that . Then there exists a number , such that , where . Define and let . Consequently, is compact and contains an -nonthin subsequence of . Hence, from Lemma 7 it follows that . Therefore contains an -statistical cluster point which is a contradiction. This proves the result.
3. -Statistical Convergence to a Set
Definition 14. Let be a closed set satisfying the property Then set is said to be an -minimal closed set if for every closed set , there exists a number such that
Definition 15. A sequence is said to be -statistically convergent to the set if is a nonempty -minimal closed set.
Generalizing the above concepts using ideals we can have the following defintion.
Definition 16. Let be a closed set satisfying the property Then set is said to be an -minimal closed set if for every closed set , there exists a number such that A sequence is called -convergent to the set if is a nonempty -minimal closed set.
Following the line of Theorem 4 in , we now give the following theorem.
Theorem 17. If is -statistically bounded then it is -statistically convergent to the set .
Proof. From Lemma 11 and Theorem 12, is a nonempty compact set and , for every . We need to show that is an -minimal set.
On the contrary, suppose that is not -minimal. Then there exists a closed set with . Hence, there is a point such that and so there exists a number such that . Now since is an -statistical cluster point so , where . Since , Therefore , where which is a contradiction. This completes the proof.
Remark 18. Define the sequence by that is, is the number of factors of in the prime factorization of . If we consider the matrix given in Remark 8, then we can show that the sequence is not -statistically bounded. Here and the sequence is -statistically convergent to .
Next we study the uniqueness of the -limit set. We prove the following theorem in line of Theorem 5 in .
Theorem 19. If is -statistically convergent to the set , then .
Proof. We show that . On the contrary, let there be a point such that . Now closedness of implies that there exists a number for which which consequently implies that
Now since , so we get . This implies that , a contradiction. Hence, .
Now we show that . Suppose , but . Then there is a number such that for every . Now point may be an isolated point or a limit point of . So we consider the following cases.
Case 1. Let be an isolated point . Then there is a number such that . This implies that . Hence, Therefore where , and . It now readily follows that and so . This shows that is not an -minimal set, a contradiction.
Case 2. If is a limit point of the set then there is a sequence in such that converges to and for . Let be given. Choose such that with .
We claim that . Let and be such that . If then and so . Again if then . But and so . Hence and so which implies that Hence, and so is not the -minimal set. This completes the proof of the theorem.
4. An Application to Turnpike Theorem
We consider the following system: where is the assigned initial point, function is continuous, , and is a compact set. The sequences and are called, a path and a control respectively, if (21) is satisfied for every . The pair is called a process. We assume that there exists a bounded closed set such that for every path that is -bounded. The point is called a stationary point if there exists such that . We denote the set of all stationary points by . It is clear that is a closed set. Let be a given continuous function by which we shall define a functional.
Let be a -bounded number sequence and let be the set of -cluster points of this sequence. We denote by the minimal element in . If is a path to (21) then is a -bounded number sequence. So on the paths to (21), we can consider the problem in line of :
First we prove the following lemma.
Lemma 20. Let be a continuous function. Then the functional may be expressed in the form
Proof. Since the path is -bounded, so is nonempty and compact. Again since the function is continuous, so there exists a point such that .
Let denote the set of -cluster points of the sequence . Now we will show that . Since , for every , . Let be given. Then there exists such that for every . Therefore which implies that and so that is, . Hence .
Now assume that and . Let . Since is continuous, there exists such that . Put . Obviously . Then , that is, and so . Again implies . Therefore . This shows that can not be satisfied for . Hence which implies that . Hence . Therefore .
Now we will establish a general version of the Turnpike Theorem (see [27, 29–33] for more details and history) which turns out to be valid for a special class of ideals, namely those ideals which are invariant under translation.
A map is a submeasure on if(i),(ii). It is lower semicontinuous if for all , we have , where . For every lower semicontinuous submeasure on , let be the submeasure defined by Let . It is clear that is an ideal for an arbitrary submeasure . An ideal on is an analytic -ideal if for some lower semicontinuous submeasure on . Following , we say that an ideal is invariant under translation if for each , and where . If is invariant under translations, then the ideal is invariant under translations.
Let be such that and . Then is called an Erdös-Ulam ideal . The ideal consisting of all sets with statistical density zero is an Erdös-Ulam ideal (generated by any constant positive function ). Every Erdös-Ulam ideal is an analytic -ideal of the form , where and . Observe that if for every then is invariant under translations. So Erdös-Ulam ideals are examples of ideals which are invariant under translation under certain conditions.
Condition 1 (in short C1). The maximum of the function on is reached at the unique stationary point such that .
Condition 2 (in short C). There exists a process such that as .
Condition 3 (in short C). There exists a vector such that for every and where .
Proof. By C for the process we have . Hence from Lemma 20. Therefore the maximal quantity of the functional (22) is not less than . As is an optimal process, so . Thus .
Define the function . It is clear that is a continuous function. From C it follows that for every , the inequality is satisfied and at the point we have .
Case 1. Suppose that there is a point such that and . We have . Since is continuous, so there exists a number such that where .
Let . Then from (25), or for every . In particular for the point we have . As and , so we obtain . Thus . On the other hand for every point , . Then . Thus if then .
Let , and consider the set . As is a -cluster point so . Since is assumed to be invariant under translation so we have . But and so , a contradiction. Thus is a unique -cluster point for which , that is; for every .
Case 2. Let now be such that and . Let . Now C implies that . As is continuous, for the number we can find a number such that for every .
Let . Then or for every . In particular for the point , . But . Therefore . On the other hand for every point we can write . Thus if .
Consider the set . Obviously and therefore we obtain that Again if , then it is clear that is compact, and . By C, we have for every . As is continuous there exists a number such that for every .
Let . Then and in particular for the point , . It is clear that . Therefore . On the other hand for every , . Then . Thus we obtain that
Now let . It is clear that and therefore . Then from (26) and (27) it follows if , then which implies that if , then .
Let , and . As is a -cluster point so which consequently implies . Again and so , a contradiction. Thus that is, the path is -convergent to .
Remark 22. We consider the system where is a given initial point and is a multivalued mapping having compact images and is continuous in the Hausdorff metric.
Now, following the line of Mamedov and Pehlivan , we can show the following.
Theorem 23 (Turnpike Theorem). Let conditions C1, C2, and C3 hold and let be an asymptotical optimal path satisfying the conditions of the systems (28) and (22). Then has , where the ideal is invariant under translation.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia. The first author is thankful to CSIR for granting Project no. 25(0186)/10/EMR-II during the tenure of which this work was done. The second author is thankful to CSIR, India, for giving JRF during the tenure of which this work was done.
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