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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 102980, 11 pages
http://dx.doi.org/10.1155/2012/102980
Research Article

Common Fixed Points, Invariant Approximation and Generalized Weak Contractions

Department of Mathematics, Khalsa College of Engineering & Technology (Punjab Technical University), Ranjit Avenue, Amritsar-143001, India

Received 23 March 2012; Revised 23 August 2012; Accepted 31 August 2012

Academic Editor: Harvinder S. Sidhu

Copyright © 2012 Sumit Chandok. 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

Sufficient conditions for the existence of a common fixed point of generalized -weakly contractive noncommuting mappings are derived. As applications, some results on the set of best approximation for this class of mappings are obtained. The proved results generalize and extend various known results in the literature.

1. Introduction and Preliminaries

It is well known that Banach’s fixed point theorem for contraction mappings is one of the pivotal result of analysis. Let be a metric space. A mapping is said to be contraction if there exists such that for all , If the metric space is complete, then the mapping satisfying (1.1) has a unique fixed point.

A natural question is that whether we can find contractive conditions which will imply existence of fixed point in a complete metric space but will not imply continuity. Kannan [1, 2] proved the following result, giving an affirmative answer to the above question.

Theorem 1.1. If , where is a complete metric space, satisfies where and , then has a unique fixed point.

The mappings satisfying (1.2) are called Kannan type mappings. A similar type of contractive condition has been studied by Chatterjea [3] and he proved the following result.

Theorem 1.2. If , where is a complete metric space, satisfies where and , then has a unique fixed point.

In Theorems 1.1 and 1.2 there is no requirement of continuity of .

A map is called a weakly contractive (see [46]) if for each , where is continuous and nondecreasing, if and only if and .

If we take , , then a weakly contractive mapping is called contraction.

A map is called -weakly contractive (see [7]) if for each , where is a self-mapping, is continuous and nondecreasing, if and only if and .

If we take , , then a -weakly contractive mapping is called -contraction. Further, if identity mapping and , , then a -weakly contractive mapping is called contraction.

A map is called a generalized weakly contractive (see [5]) if for each , where is continuous such that if and only if .

If we take , , then inequality (1.6) reduces to (1.3). Choudhury [5] shows that generalized weakly contractive mappings are generalizations of contractive mappings given by Chatterjea (1.3), and it constitutes a strictly larger class of mappings than given by Chatterjea.

A map is called a generalized -weakly contractive [8] if for each , where is a self-mapping, is continuous such that if and only if .

If identity mapping, then generalized -weakly contractive mapping is generalized weakly contractive.

For a nonempty subset of a metric space and , an element is said to be a best approximant to or a best -approximant to if . The set of all such is denoted by .

A subset of a normed linear space is said to be a convex set if for all and . A set is said to be -starshaped, where , provided for all and , that is, if the segment joining to is contained in for all . is said to be starshaped if it is -starshaped for some .

Clearly, each convex set is starshaped but converse is not true.

Suppose that is a subset of normed linear space . A mapping from to is said to be demiclosed if for every sequence such that converges weakly to and converges strongly to imply . is said to be demiclosed at 0, if for every sequence in converging weakly to and converging strongly to , then .

For a convex subset of a normed linear space , a mapping is said to be affine if for all , for all .

The ordered pair of two self-maps of a metric space is called a Banach operator pair [9], if the set of fixed points of is -invariant, that is, . A point is a coincidence point (common fixed point) of and if . The set of fixed points (resp., coincidence points) of and is denoted by (resp., ). The pair is called commuting if for all . Obviously, commuting pair, is a Banach operator pair but not conversely (see [9]). If is a Banach operator pair then need not be a Banach operator pair (see [9]). If the self-maps and of satisfy , for all and for some , whenever , that is, , then is a Banach operator pair. This class of noncommuting mappings is different from the known classes of noncommuting mappings namely, -weakly commuting, -subweakly commuting, compatible, weakly compatible, -commuting, and so forth, existing in the literature.

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. Alber and Guerre-Delabriere [4] introduced the concept of weakly contractive mappings and proved the existence of fixed points for single-valued weakly contractive mappings in Hilbert spaces. Thereafter, Rhoades [6] proved a fixed point theorem which is one of the generalizations of Banach's Fixed Point Theorem in 1922, because the weakly contractions contain many contractions as a special case, and he also showed that some results of [4] are true for any Banach spaces. In fact, weakly contractive mappings are closely related to the mappings introduced by Boyd and Wong [10] and Reich [11]. Many other nonlinear contractive type mappings like Chatterjea, Ciric, Kannan, Reich type, and so forth and their generalizations have been investigated by many authors. Fixed point and common fixed point theorems for different types of nonlinear contractive mappings have been investigated extensively by various researchers (see [121] and references cited therein). In this paper, sufficient conditions for the existence of a common fixed point of generalized -weakly contractive noncommuting mappings are obtained. As applications, we also establish some results on the set of best approximation for this class of mappings. The proved results generalize and extend the corresponding results of [5, 8, 9, 1214, 1820] and of few others.

2. Main Results

The following result is a consequence of the main theorem of Choudhury [5].

Lemma 2.1. Let be a subset of a metric space and is a self-mapping of such that . If is complete and satisfies where is a continuous mapping such that if and only if , for all , then has a unique fixed point in .

Corollary 2.2 (see [5]). Let be a self-mapping of , where is a complete metric space. If satisfies where is a continuous mapping such that if and only if , for all , then has a unique fixed point.

If , , we have Theorem 1.2.

Theorem 2.3. Let be a subset of a metric space , and and are self-mappings of such that . If is complete, is nonempty, and and satisfy where is a continuous mapping such that if and only if , for all , then is a singleton.

Proof. being a subset of is complete and and . So for all , we have Thus, by Lemma 2.1, has a unique fixed point in and consequently, is a singleton.

Corollary 2.4. Let be a subset of a metric space , and and are self-mappings of . If is complete, is a Banach operator pair, is nonempty and closed, and and satisfy where is a continuous mapping such that if and only if , for all , then is a singleton.

Example 2.5. Let and is a metric defined on . Let and be self-mappings of such that , , , , , and . Then is a generalized -weakly contraction, and is the coincidence point of and .
If identity mapping, this example given in [5].

If , , we have the following result.

Corollary 2.6. Let be a subset of a metric space , and and be self-mappings of such that . If is complete, is nonempty, and and satisfy where , for all , then is a singleton.

Corollary 2.7. Let be a subset of a metric space and and be self-mappings of . If is complete, is a Banach operator pair, is nonempty and closed, and and satisfy where , for all , then is a singleton.

Theorem 2.8. Let be a nonempty subset of a normed (resp., Banach) space , and be self-mappings of . Suppose that is -starshaped, (resp., ), is compact (resp., is weakly compact, and is demiclosed at ), and satisfies where is a continuous mapping such that if and only if , for all . Then .

Proof. For each , define by , where is a sequence in such that . Since is -starshaped and (resp., ), we have for all and so (resp., ) for each . Consider for all . As is compact, is compact for each and hence complete. Now by Corollary 2.6, there exists such that is a common fixed point of and for each . The compactness of implies that there exists a subsequence of such that . Since is a sequence in , . Now, as , we have and . Further, we have on taking limit, we get and so .
Next, the weak compactness of implies that is weakly compact and hence complete. Hence, by Corollary 2.6, for each , there exists such that . The weak compactness of implies that there is subsequence of such that . Since is a sequence in , . Also, we have .
If is demiclosed at , then and so .

Let be a nonempty subset of a metric space . Suppose that , where .

Corollary 2.9. Let be a normed (resp., Banach) space, and , are self-mappings of . If , , is -starshaped, (resp., ), is compact (resp., is weakly compact and is demiclosed at ), and satisfies the inequality (2.8) for all , then is nonempty.

Corollary 2.10. Let be a normed (resp., Banach) space and are self-mappings of . If , , is -starshaped, (resp., ), is compact (resp., is weakly compact and is demiclosed at ), and satisfies the inequality (2.8) for all , then is nonempty.

Remark 2.11. Theorem 2.8 extends and generalizes the corresponding results of [8, 9, 13, 14, 18, 19].
Let denote the class of closed convex subsets of a normed space containing . For and , let . Then (see [12, 20]).

Theorem 2.12. Let be a normed (resp., Banach) space, and , are self-mappings of . If and such that , is compact (resp., is weakly compact), and for all , then is nonempty, closed, and convex with . If, in addition, is a subset of , is -starshaped, (resp., and is demiclosed at ), and satisfies inequality (2.8) for all , then is nonempty.

Proof. If , then the results are obvious. So assume that . If , then and so . Thus . Since is compact, and by the continuity of norm, there exists such that .
On the other hand, if is weakly compact, then using Lemma 5.5 of Singh et al. [22, page 192], we can show that there exists such that .
Hence, in both cases, we have for all . Therefore, , that is, , that is, and so is nonempty. The closedness and convexity of follow from that of . Now to prove , let . Then for . Consider and so as , that is, .
The compactness of (resp., weakly compactness of ) implies that is compact (resp., is weakly compact). Hence, the result follows from Corollary 2.10.

Corollary 2.13. Let be a normed (resp., Banach) space, and , are self-mappings of . If and such that , is compact (resp., is weakly compact), and for all , then is nonempty, closed, and convex with . If, in addition, is a subset of , is -starshaped and closed (resp., weakly closed and is demiclosed at ), is a Banach operator pair on , and satisfies inequality (2.8) for all , then is nonempty.

Remark 2.14. Theorem 2.12 extends and generalizes the corresponding results of Al-Thagafi [12], Al-Thagafi and Shahzad [13], Habiniak [18], Narang and Chandok [20], and Shahzad [21].

The following result will be used in the sequel.

Lemma 2.15. Let be a nonempty subset of a metric space , , self-maps of , . Suppose that is complete, and , , satisfy for all and , If is nonempty and , then there is a common fixed point of and .

Proof. , being a closed subset of the complete set , is complete. Further for all , we have Hence, is a generalized contraction on and . So by Lemma 3.1 of [13], has a unique fixed point in and consequently is a singleton.

Remark 2.16. If , then Theorem 3.2 of Al-Thagafi and Shahzad [13] is a particular case of Lemma 2.15.
The following result extends and improves the corresponding results of [9, 1214, 18, 20].

Theorem 2.17. Let be self-mappings of a Banach space . If and such that , is compact, and , , for all , then(i) is nonempty, closed, and convex,(ii),(iii), provided is continuous, is -starshaped, , and the pair satisfies the inequality (2.8) for all , is -starshaped with , and satisfy for all and then there is a common fixed point of , , , and .

Proof. Proceeding as in Theorem 2.12, we can prove (i) and (ii).
By (ii), the compactness of implies that and are compact. Hence, Theorem 2.8 implies that .
For each , define by , for each , where is a sequence in such that . Then each is a self-mapping of . Since , is -starshaped with , so for each . Consider for all . As is compact, is compact for each and hence complete. Now by Lemma 2.15, there exists such that is a common fixed point of and for each . The compactness of implies that there exists a subsequence of such that . Since is a sequence in , and , then . Now, as , we have is continuous, we have and hence .

Remark 2.18. (i) Let . Let be defined by . We set and for all . Define by Then for , we have From (2.21) and (2.22), without loss of generality assume that . Hence, we have two cases:
Case  1. If , from (2.22), we have
Case  2. If , from (2.22), we have Thus inequality (2.3) is satisfied and by Theorem 2.3; is a common fixed point of and .
(ii) It may be noted that the assumption of linearity or affinity for is necessary in almost all known results about common fixed points of maps , such that is -nonexpansive under the conditions of commuting, weakly commuting, -subweakly commuting, or compatibility (see [9, 12, 14, 20, 21] and the literature cited therein), but our results in this paper are independent of the linearity or affinity.
(iii) Consider with usual metric , , . Define and on as and . Obviously, is -nonexpansive, -asymptotically nonexpansive, but is not linear or affine. Moreover, , and . Thus, , which is not a compatible pair (see [9]), is convex, starshaped for any , and is a common fixed point of and .

Acknowledgment

The author is thankful to the learned referees for the valuable suggestions.

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