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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 579819, 10 pages
http://dx.doi.org/10.1155/2011/579819
Research Article

Invariant Points and -Simultaneous Approximation

Department of Mathematics, Guru Nanak Dev University, Amritsar-143005, India

Received 10 December 2010; Revised 18 March 2011; Accepted 19 April 2011

Academic Editor: S. M. Gusein-Zade

Copyright © 2011 Sumit Chandok and T. D. Narang. 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.

Abstract

We generalize and extend Brosowski-Meinardus type results on invariant points from the set of best approximation to the set of -simultaneous approximation. As a consequence some results on -approximation and best approximation are also deduced. The results proved in this paper generalize and extend some of the known results on the subject.

1. Introduction and Preliminaries

Fixed point theory has gained impetus, due to its wide range of applicability, to resolve diverse problems emanating from the theory of nonlinear differential equations, theory of nonlinear integral equations, game theory, mathematical economics, control theory, and so forth. For example, in theoretical economics, such as general equilibrium theory, a situation arises where one needs to know whether the solution to a system of equations necessarily exists; or, more specifically, under what conditions will a solution necessarily exist. The mathematical analysis of this question usually relies on fixed point theorems. Hence finding necessary and sufficient conditions for the existence of fixed points is an interesting aspect.

Fixed point theorems have been used in many instances in best approximation theory. It is pertinent to say that in best approximation theory, it is viable, meaningful, and potentially productive to know whether some useful properties of the function being approximated is inherited by the approximating function. The idea of applying fixed point theorems to approximation theory was initiated by Meinardus [1]. Meinardus introduced the notion of invariant approximation in normed linear spaces. Brosowski [2] proved the following theorem on invariant approximation using fixed point theory by generalizing the result of Meinardus [1].

Theorem 1.1. Let be a linear and nonexpansive operator on a normed linear space . Let be a -invariant subset of and a -invariant point. If the set of best -approximants to is nonempty, compact, and convex, then it contains a -invariant point.

Subsequently, various generalizations of Brosowski's results appeared in the literature. Singh [3] observed that the linearity of the operator and convexity of the set in Theorem 1.1 can be relaxed and proved the following.

Theorem 1.2. Let be a nonexpansive self-mapping on a normed linear space . Let be a -invariant subset of and a -invariant point. If the set is nonempty, compact, and star shaped, then it contains a -invariant point.

Singh [4] further showed that Theorem 1.2 remains valid if is assumed to be nonexpansive only on . Since then, many results have been obtained in this direction (see Chandok and Narang [5, 6], Mukherjee and Som [7], Mukherjee and Verma [8], Narang and Chandok [911], Rao and Mariadoss [12], and references cited therein).

In this paper we prove some similar types of results on -invariant points for the set of -simultaneous approximation in a metric space . Some results on -invariant points for the set of -approximation and best approximation are also deduced. The results proved in the paper generalize and extend some of the results of [6, 813] and of few others.

Let be a nonempty subset of a metric space and let be a nonempty bounded subset of . For , let , and . An element is said to be a best simultaneous approximation of with respect to .

For , we define . An element is said to be a -simultaneous approximation of with respect to .

It can be easily seen that for , the set is always a nonempty bounded set and is closed if is closed.

In case , , then elements of are called best approximations to in and of are called -approximation to in .

A sequence in is called a -minimizing sequence for , if . The set is said to be -simultaneous approximatively compact with respect to if for every , each -minimizing sequence in has a subsequence converging to an element of .

Let be a metric space. A continuous mapping is said to be a convex structure on if for all and , holds for all . The metric space together with a convex structure is called a convex metric space [14].

A convex metric space is said to satisfy Property (I) [15] if for all and ,

A normed linear space and each of its convex subset are simple examples of convex metric spaces with given by for and . There are many convex metric spaces which are not normed linear spaces (see [14]). Property (I) is always satisfied in a normed linear space.

A subset of a convex metric space is said to be (i)a convex set [14] if for all and ; (ii)-star shaped [16] where , provided for all and ; (iii)star shaped if it is -star shaped for some .

Clearly, each convex set is star shaped but not conversely.

A self-map on a metric space is said to be (i)contraction if there exists , such that for all ;(ii)nonexpansive if for all ;(iii)quasi-nonexpansive if the set of fixed points of is nonempty and for all and .

A nonexpansive mapping on with is quasi-nonexpansive, but not conversely. A linear quasi-nonexpansive mapping on a Banach space is nonexpansive. But there exist continuous and discontinuous nonlinear quasi-nonexpansive mappings that are not nonexpansive.

2. Main Results

To start with, we prove the following proposition on -simultaneous approximation which will be used in the sequel.

Proposition 2.1. Let be a nonempty bounded subset of a metric space , and let be a non-empty subset of . If is -simultaneous approximatively compact with respect to , then the set is a nonempty compact subset of .

Proof. Since , is nonempty. We now show that is compact. Let be a sequence in . Then , that is, is an -minimizing sequence for . Since is -simultaneous approximatively compact with respect to , there is a subsequence such that . Consider This implies that . Thus we get a subsequence of converging to an element . Hence is compact.

For , we have the following result on the set of -approximation.

Corollary 2.2 (see [9]). If is an -approximatively compact set in a metric space then is a nonempty compact set.

For and , we have the following result on the set of best approximation.

Corollary 2.3 (see [10]). Let be a nonempty approximatively compact subset of a metric space , , and be the metric projection of onto defined by . Then is a nonempty compact subset of .

We will be using the following result of Hardy and Rogers [17] in proving our first theorem.

Lemma 2.4. Let be a mapping from a complete metric space into itself satisfying for any , where , and are nonnegative numbers such that . Then has a unique fixed point in . In fact for any , the sequence converges to .

Theorem 2.5. Let be a continuous self-map on a complete convex metric space with Property (I) and satisfying inequality (2.2), let be a -invariant subset of , and let be a nonempty bounded subset of such that for all . If is compact, and star shaped, then it contains a -invariant point.

Proof. Let be arbitrary. Then by (2.2), we have for all This gives since . Therefore, using definition of , we get Hence . Therefore is a self-map on .
Let be the star-center of . Define as , where is a sequence in such that . Consider where . Therefore by Lemma 2.4, each has a unique fixed point in . Since is compact, there is a subsequence of such that . We claim that . Consider , as is continuous. Thus and consequently, , that is, is a -invariant point.

Since for an -simultaneous approximatively compact subset of a metric space the set of -simultaneous -approximant is nonempty and compact (Proposition 2.1), we have the following result.

Corollary 2.6. Let be a continuous self-map on a complete convex metric space with Property (I) and satisfying inequality (2.2), let be a nonempty bounded subset of such that for all , and let be a -invariant subset of . If is -simultaneous approximatively compact with respect to and is star shaped, then it contains a -invariant point.

Corollary 2.7 (see [8]). Let be a continuous self-map on a Banach space satisfying (2.2), let be an approximatively compact and -invariant subset of . Let for some not in . If the set of best simultaneous -approximants to is star shaped, then it contains a -invariant point.

Corollary 2.8 (see [11]). Let be a mapping on a metric space , let be a -invariant subset of and a -invariant point. If is a nonempty, compact set for which there exists a contractive jointly continuous family of functions and is nonexpansive on then contains a -invariant point.

Corollary 2.9. Let be a mapping on a convex metric space with Property (I),let be an approximatively compact, -star shaped, -invariant subset of and let be a -invariant point. If is nonexpansive on , then contains a -invariant point.

Corollary 2.10 (see [10, Theorem  4]). Let be a quasi-nonexpansive mapping on a convex metric space with Property (I), let be a -invariant subset of , and let be a -invariant point. If is nonempty, compact, and star shaped, and is nonexpansive on , then contains a -invariant point.

Corollary 2.11 (see [10, Theorem 5]). Let be a quasi-nonexpansive mapping on a convex metric space with Property (I), let be an approximatively compact, -invariant subset of , and let be a -invariant point. If is star shaped and is nonexpansive on , then contains a -invariant point.

Remark 2.12. Theorem 2.5 improves and generalizes Theorem  1 of Narang and Chandok [9] and of Rao and Mariadoss [12].

Definition 2.13. A subset of a metric space is said to be contractive if there exists a sequence of contraction mappings of into itself such that for each .

Theorem 2.14. Let be a nonexpansive self-mapping on a metric space , let be a -invariant subset of , and let be a nonempty bounded subset of such that for all . If the set is compact and contractive, then the set contains a -invariant point.

Proof. Proceeding as in Theorem 2.5, we can prove that is a self-map of . Since is contractive, there exists a sequence of contraction mapping of into itself such that for every .
Clearly, is a contraction on the compact set for each and so by Banach contraction principle, each has a unique fixed point, say in . Now the compactness of implies that the sequence has a subsequence . We claim that is a fixed point of . Let be given. Since and , there exist a positive integer such that for all Again, Hence that is, for all and so . But and therefore .

Using Proposition 2.1 we have the following result.

Corollary 2.15. Let be a nonexpansive self-mapping on a metric space , let be a -invariant subset of , and let be a nonempty bounded subset of such that for all . If is -simultaneous approximatively compact with respect to and the set is contractive, and -invariant, then contains a -invariant point.

Corollary 2.16 (see [9]). Let be a self-mapping on a metric space , let be a -invariant subset of , and let be a -invariant point. If the set of -approximant to is compact, contractive and is nonexpansive on , then contains a -invariant point.

Remark 2.17. Theorem 2.14 also improves and generalizes the corresponding results of Brosowski [2], Mukherjee and Verma [8, 13], Chandok and Narang [9], Rao and Mariadoss [12], and of Singh [3].

Definition 2.18. For each bounded subset of a metric space , the Kuratowski's measure of noncompactness of , is defined as A mapping is called condensing if for all bounded sets , .

We will be using the following result of [18] on fixed points of nonexpansive condensing maps.

Lemma 2.19. Let be a complete contractive metric space with contractions . Let be a closed bounded subsets of and is nonexpansive and condensing, then has a fixed point in .

Using the above lemma and Theorem 2.5, we now prove the following result.

Theorem 2.20. Let be a complete, contractive metric space with contractions . Let be a closed and bounded subset of and let be a nonempty bounded subset of . If is a nonexpansive and condensing self-map on such that for all , then has a -invariant point.

Proof. As is closed and bounded, is nonempty, closed and bounded. Using Theorem 2.5, we can prove that is a self-map of . Now a direct application of Lemma 2.19, gives a -invariant point in .

Corollary 2.21 (see [8, Theorem 3.1]). Let be a complete, contractive metric space with contractions . Let be a closed and bounded subset of . If is a nonexpansive and condensing self-map on such that and for some , and is nonempty, then it has a -invariant point.

Corollary 2.22 (see [12, Theorem 4]). Let be a complete, contractive metric space with contractions . Let be a closed and bounded subset of . If is a nonexpansive and condensing self-map on such that for some , and is nonempty, then it has a -invariant point.

Definition 2.23. A mapping on a metric space is called a Kannan mapping [19] if there exists such that for all .

Kannan [19] proved that if is complete, then every Kannan mapping has a unique fixed point.

For -simultaneous approximation, we have the following result.

Theorem 2.24. Let be a nonempty subset of a complete metric space and let be a nonempty bounded subset of . Let be a self-map on with for all and satisfies, for some positive integer , all and some fixed . If is compact, then it has a unique fixed point of .

Proof. As , for all positive integers . Let . Then, for , which implies that Further, we have Therefore, , . Since satisfies the conditions of Kannan map, has a unique fixed point in . Now, , implies that is a fixed point of . But the fixed point of is unique and equals . Therefore and hence is a unique fixed point of in .

Corollary 2.25. Let be a nonempty bounded subset of a complete metric space and a subset of . Let be -simultaneous approximatively compact with respect to , and a self map on with for all , and satisfies for some positive integer , all and some fixed then has a unique fixed point of .

Remarks 2.26. Theorem 2.24 extends and generalizes Theorem 3.2 of Mukherjee and Verma [8] and Theorem 5 of Rao and Mariadoss [12] from the set of best simultaneous approximation and best approximation, respectively, to -simultaneous approximation.

For , we define . An element is said to be a -simultaneous coapproximation of with respect to .

A mapping satisfies condition (A) (see [13]) if for all .

We now prove a result for -invariant points from the set of -simultaneous coapproximations.

Theorem 2.27. Let be a self-map satisfying condition (A) and inequality (2.2) on a convex metric space satisfying Property (I), let be a subset of , and let be a nonempty bounded subset of such that is compact and star shaped. Then contains a -invariant point.

Proof. Let . Consider and so , that is, . Since is star shaped, there exists such that for all , . Let , be a sequence of real numbers such that as . Define as , . Since is a self-map on and is star shaped, each is a well defined and maps into . Moreover, where . So by Lemma 2.4 each has a unique fixed point , that is, for each . Since is compact, has a subsequence . Now, we claim that . As . On letting , we have . Therefore , that is, is -invariant, hence the result.

For , , we have the following result on the set of -coapproximation.

Corollary 2.28 . (see [9, Theorem 4]). Let be a self-map satisfying condition (A) on a convex metric space satisfying Property (I), let be a subset of such that is nonempty compact, star shaped, and let be nonexpansive on . Then there exists a such that .

Remarks 2.29. (i) Theorem 2.27 also improves and generalizes Theorem  4.1 of Mukherjee and Verma [13] from the set of best approximation to -simultaneous approximation.
(ii) By taking , , the set (respectively, ) is the set of -simultaneous approximation (respectively, -simultaneous coapproximation) to the pair of points and so the results of this paper generalize and extend the corresponding results proved in [6].

Acknowledgments

The authors are thankful to the learned referee for careful reading and very valuable suggestions leading to an improvement of the paper. The research work of the second author has been supported by the UGC India under the Emeritus Fellowship.

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