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International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 753965, 8 pages
http://dx.doi.org/10.1155/2013/753965
Research Article

Some Nonunique Common Fixed Point Theorems in Symmetric Spaces through Property

1Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey
2Department of Mathematics and Humanities, S. V. National Institute of Technology, Surat, Gujarat 395 007, India
3Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India

Received 3 October 2012; Revised 23 December 2012; Accepted 24 December 2012

Academic Editor: N. Hussain

Copyright © 2013 E. Karapınar 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.

Abstract

We introduce a new class of mappings satisfying the “common limit range property” in symmetric spaces and utilize the same to establish common fixed point theorems for such mappings in symmetric spaces. Our results generalize and improve some recent results contained in the literature of metric fixed point theory. Some illustrative examples to highlight the realized improvements are also furnished.

1. Introduction

In 1986, Jungck [1] generalized the idea of weakly commuting pair of mappings due to Sessa [2] by introducing the notion of compatible pair of mappings and also showed that compatible pair of mappings commute on the set of coincidence points of the involved mappings. Recall that a point is called a coincidence point of the pair of self-mappings defined on if while the point is then called a point of coincidence for the pair . In the recent past and even now, the concept of compatible mappings is frequently used to prove results on the existence of common fixed points. The study of common fixed points of noncompatible pairs is also equally natural and fascinating. Pant [3] initiated the study of noncompatible pairs employing the idea of pointwise -weakly commuting pairs. Pant [4] proved an interesting fixed point theorem for maps satisfying Lipschitz type conditions. In recent years, the result of Pant [4] was generalized and improved by Sastry and Murthy [5] (see also [6]) by introducing the idea of tangential maps (or the property (E.A)) and -continuity. In continuation of this, Imdad and Soliman [7] and Soliman et al. [8] extended the results of Sastry and Murthy [5] as well as Pant [4] to symmetric space utilizing the idea of weakly compatible pair together with common property (E.A) (a notion due to Liu et al. [9]). For more references on the recent development of common fixed point theory in symmetric spaces, we refer readers to [1014]. Most recently, Gopal et al. [15] improved these results by utilizing the idea of absorbing pair which is essentially due to Gopal et al. [16].

In this paper, we introduce a new notion called the common limit range property and show that this new notion buys a typically required condition up to a pair of mappings along with the notion of absorbing property in proving common fixed point theorems for Lipschitz type mappings in symmetric spaces. Consequently, the relevant recent fixed point theorems due to Soliman et al. [8] and Gopal et al. [15] are generalized and improved.

2. Preliminaries

A symmetric on a nonempty set is a function which satisfies and (for all ). If is a symmetric on a set , then for and , we write . A topology on is given by the sets (along with empty set) in which for each , one can find some such that . A set is a neighbourhood of if and only if there is a containing such that . A symmetric is said to be a semimetric if for each and for each is a neighbourhood of in the topology . Thus a symmetric (resp. a semimetric) space is a topological space whose topology on is induced by a symmetric (resp. a semimetric) . Notice that if and only if in the topology . The distinction between a symmetric and a semimetric is apparent as one can easily construct a semimetric such that need not be a neighbourhood of in .

Since symmetric spaces are not necessarily Hausdorff and the symmetric is not generally continuous, in the course of proving fixed point theorems, some additional axioms are required. The following axioms are relevant to this note which are available in the papers of Aliouche [17], Galvin and Shore [18], Hicks and Rhoades [19], and Wilson [20].() [20] Given and in with and imply .() [20] Given and an in with and imply .() [17] Given and an in with and imply .() [18] A symmetric is said to be -continuous if implies . () [18] A symmetric is said to be continuous if and imply where are sequences in and .

Clearly, implies but not conversely. Also implies and implies but converse implications are not true. All other possible implications amongst , and are not true in general. A nice illustration via demonstrative examples is given by Cho et al. [21]. However, implies all the remaining four conditions namely: , and .

Recall that a sequence in a semimetric space is said to be -Cauchy if it satisfies the usual metric condition. Here, one needs to notice that in a semimetric space, Cauchy convergence criterion is not a necessary condition for the convergence of a sequence but this criterion becomes a necessary condition if semimetric is suitably restricted (see Wilson [20]). In [22], Burke furnished an illustrative example to show that a convergent sequence in semimetric spaces need not admit a Cauchy subsequence. He was able to formulate an equivalent condition under which every convergent sequence in semimetric space admits a Cauchy subsequence. There are several concept of completeness in semimetric space for example, -completeness, -Cauchy completeness, strong and weak completeness (see Wilson [20]). We omit the details of these notions which are not relevant to this paper.

Let be a pair of self-mappings defined on a nonempty set equipped with a symmetric (semimetric) . Then for the pair , we recall some relevant concepts as follows.

A pair of self-mappings is said to be(i)compatible (cf. [1]) if whenever is a sequence such that for some in ,(ii)noncompatible (cf. [4, 23]) if there exists at least one sequence such that = = for some in but is either nonzero or nonexistent, (iii)tangential (or satisfying the property (E.A)) (cf. [5, 24]) if there exists a sequence in such that for some .Let be an arbitrary set and be a nonempty set equipped with symmetric (semimetric) . Then the pairs and of mappings from into are said to have (iv)(cf. [9]) the common property (E.A) if there exist two sequences and in such that while the pair is said to have (v)the common limit range property with respect to the map (denoted by (CLR) (cf. [2529]) if there exists a sequence in such that for some ,(vi)let be an arbitrary set and be a nonempty set equipped with symmetric (semimetric) . Then is said to be -continuous (cf. [5]) if whenever is a sequence in and . (vii)a pair of self-mappings defined on a set is said to be weakly compatible (or partially commuting or coincidentally commuting (cf. [5, 30])) if the pair commutes on the set of coincidence points that is, (for ) implies that , (viii)let and be two self-mappings defined on a symmetric (or semimetric) space , then is called -absorbing if there exists some real number such that for all in . Analogously, will be called -absorbing (cf. [16]) if there exists some real number such that for all in . The pair of self maps will be called absorbing if it is both -absorbing as well as -absorbing,(ix)let and be two self-mappings defined on a symmetric (or semimetric) space , then is called pointwise -absorbing if for given in , there exists some such that .

On similar lines, we can define pointwise -absorbing map. If we take , the identity map on , then is trivially -absorbing. Similarly, is -absorbing in respect of any . It has been shown in [16] that a pair of compatible or -weakly commuting pair need not be -absorbing or -absorbing. Also absorbing pairs are neither a subclass of compatible pairs nor a subclass of noncompatible pairs as the absorbing pairs need not commute at their coincidence points. For other properties and related results for absorbing pair of maps, one can consult [16].

For the sake of completeness, we state below some theorems contained in Soliman et al. [8] and Gopal et al. [15].

Theorem 1 (see cf. [8]). Let be an arbitrary nonempty set while be another nonempty set equipped with a symmetric (semimetric) which enjoys (Hausdorffness of ) and (). Let be four mappings which satisfy the following conditions: (i) is -continuous and is -continuous, (ii)the pairs and share the common property (E.A), (iii) and are -closed (-closed) subset of (resp., and ).
Then there exist such that .
Moreover, if   along with (iv)the pairs and are weakly compatible and (v),  , whenever the right hand side is nonzero. Then, , and have a common fixed point in .

Theorem 2 (see cf. [15]). Let be an arbitrary nonempty set while be another nonempty set equipped with a symmetric (semimetric) which enjoys   (Hausdorffness of ) and (). Let be four mappings which satisfy the following conditions: (i) is -continuous and is -continuous, (ii)the pairs and share the common property (E.A), (iii) is a -closed (-closed) subset of and (resp., is a -closed (-closed) subset of and ). Then, there exist such that .
Moreover, if , then and have a common fixed point provided the pairs and are pointwise absorbing.

Theorem 3 (see cf. [15]). Let be an arbitrary set while be a symmetric (semimetric) space equipped with a symmetric (semimetric) which enjoys   (Hausdorffness of )  and (HE). Let be four mappings which satisfy the following conditions: (i)the pair satisfies the property (E.A) (resp., satisfies the property (E.A)), (ii) is a -closed (-closed) subset of and (resp., is a -closed (-closed) subset of and ) and (iii) for any where and , , . Then, there exist such that .
Moreover, if , then and have a common fixed point provided the pairs and are pointwise absorbing.

In this paper, we provide a unified approach to certain theorems in symmetric (semimetric) spaces using a blend of common limit range property along with absorbing pair property and obtain generalizations of various results due to Gopal et al. [15], Soliman et al. [8], Pant [31], Sastry and Murthy [5], Imdad et al. [7], Cho et al. [21], and some others.

3. Main Results

We start to section with the following definition.

Definition 4. Let , and be four self-mappings defined on a symmetric space . Then the pairs and are said to have the common limit range property (with respect to and ), often denoted by , if there exist two sequences and in such that with , for some .

If and , then the above definition implies property due to Sintunavarat and Kumam [28]. Also notice that the preceeding definition implies the common property (E.A) but the converse implication is not true in general. The following example substantiates this fact.

Example 5. Consider equipped with the symmetric defined by for all which satisfies and (). Define self mappings and on as

For sequences and , we have which shows that the pairs and share the common property (E.A). However, there does not exist points and in for which .

In view of the preceeding example, the following proposition is predictable.

Proposition 6. If the pairs and share the common property (E.A) and as well as are closed subsets of , then the pairs also share the   property.

We now prove our first result employing -continuity of and -continuity of instead of utilizing some Lipschitz or contractive type condition.

Theorem 7. Let be an arbitrary nonempty set while be a nonempty set equipped with a symmetric (semimetric) which enjoys (Hausdorffness of and (). If are four mappings which satisfy the following conditions: (i) is -continuous and is -continuous, (ii)the pairs and satisfy the property, then, and have a coincidence point. Moreover, if , then , and have a common fixed point provided the pairs and are pointwise absorbing.

Proof. Since the pairs and satisfy the property, therefore there exist two sequences and in such that with , for some .
On using -continuity of along with the condition , we get which shows that is a coincidence point of the mappings and . Similarly, using the -continuity of along with the condition , we obtain which shows that is a coincidence point of and . Owing to property, we have .
As the pairs and are pointwise absorbing, we can write which show that is a common fixed point of and . This concludes the proof.

With a view to demonstrate the utility of Theorem 7 over Theorem 1 and Theorem 2, we adopt the following example.

Example 8. Consider equipped with the symmetric defined by for all which satisfies and (). Define self mappings , and on as

Consider sequences and in . Clearly, with , and which show that the pairs and share the property while the map is -continuous and the map is -continuous. Further   =               and   =               and evidently none of the involved subspaces are closed. Also, by a routine calculation, one can easily verify that the pairs and are pointwise absorbing. Thus, the involved pairs of maps and satisfy all the conditions of Theorem 7 and have two common fixed points namely: and .

Notice that at , the involved maps do not satisfy the condition whenever the right hand side is nonzero. Moreover, it can also be verified that at points and , the involved maps do not satisfy the Lipschitz type condition employed in [4]. Thus, this example substantiates the fact that Theorem 7 is genuine extension of Theorems 1 and 2.

By restricting , , , and suitably, one can derive corollaries involving two as well as three mappings. Here, it may be pointed out that any result involving three maps is itself a new result. For the sake of brevity, we opt to mention just one such corollary by restricting Theorem 7 to three mappings , and which is still new and presents yet another sharpened form of a relevant theorem contained in [15] besides admitting a nonself setting upto coincidence points.

Corollary 9. Let be an arbitrary set while be a symmetric (semimetric) space equipped with a symmetric (semimetric) which enjoys (Hausdorffness of ) and (). If are three mappings which satisfy the following conditions: (i) is -continuous and is -continuous, (ii)the pairs and satisfy the property, then, there exist such that . Moreover, if , then and have a common fixed point provided the pairs and are pointwise absorbing.

The following example illustrates the preceding corollary involving a pair of two self-mappings.

Example 10. Consider equipped with the symmetric defined by , for all which satisfies and (). Define self mappings as

By routine calculations, one can easily verify that the maps in the pair satisfies all the conditions of Corollary 9 and have two common fixed points, namely: 2 and 11. Also, the present example does not satisfy the Lipschitz type condition utilized in [4]. To view this claim, consider and , then we have , which is a contradiction. Also, observe that at , the involved maps do not satisfy the condition: whenever the right hand side is nonzero. Here, it is worth noting that none of the earlier relevant theorems for example, Imdad and Soliman [7], Soliman et al. [8] and Gopal et al. [15] can be used in the context of this example as Corollary 9 does not require conditions on containment and closedness amongst the ranges of the involved mappings.

Our next theorem is essentially inspired by Theorem 3 due to Gopal et al. [15].

Theorem 11. Let be an arbitrary set while be a symmetric (semimetric) space equipped with a symmetric (semimetric) which enjoys (or Hausdorffness of ) and . If are four mappings which satisfy the following conditions: (i)the pairs and satisfy the property, (ii), for any , where and , , , then, there exist such that . Moreover, if , then , and have a common fixed point provided the pairs and are pointwise absorbing.

Proof. Since the pairs and share the property, therefore there exist two sequences and in such that with , for some .
On using condition (ii), we have which on letting , gives rise . Now appealing to , we get so that .
Next, we show that . To accomplish this, using (ii), we have so that and hence in all which shows that both the pairs have a point of coincidence.
On using pointwise absorbing property of the pairs and , we have which show that is a common fixed point of , and .

The following example demonstrates Theorem 11.

Example 12. Consider equipped with the symmetric for all which satisfies and (). Set and . Define as follows:

Then, by a routine calculation, it can be easily verified that and satisfy condition (ii) (of Theorem 11) for . Also, the mappings and satisfies the property with the sequence . The verification of the pointwise absorbing property of the pair is straight forward. Thus and satisfy all the conditions of Theorem 11 and have two common fixed points, namely: and .

Observe that      and none of and is closed. Further, it is also worth noting that for all with and with and , the involved pair does not satisfy the condition whenever the right hand side is nonzero. Thus, this example also establishes the utility of Theorem 11 over corresponding results proved in Soliman et al. [8] and Gopal et al. [15].

Remark 13. Choosing in Theorem 11, we can derive a slightly sharpened form of a theorem due to Cho et al. [21] as conditions on the ranges of involved mappings are completely relaxed.

By restricting , , , and suitably, one can derive corollaries for two as well as three mappings. For the sake of brevity, we derive just one corollary by restricting Theorem 11 to three mappings which is yet another sharpened and unified form of a theorem due to Gopal et al. [15] in symmetric spaces and also remains relevant to some results in Pant [4] and Pant [31].

Corollary 14. Suppose that (in the setting of Theorem 11) satisfies and . If are three mappings which satisfy the following conditions: (i)the pairs and satisfy the property, (ii), for any , where and , , ,then, there exist such that . Moreover, if , then ,, and have a common fixed point provided the pair is pointwise -absorbing while the pair is pointwise -absorbing.

Corollary 15. Let be symmetric (semimetric) space wherein satisfies (Hausdoffness of and (). If are four self mappings of which satisfy the following conditions: (i)the pairs and satisfy the property, (ii), where , , then there exist such that .
Moreover, if , then , and have a unique common fixed point provided the pair is pointwise -absorbing whereas the pair is pointwise -absorbing.

Proof. Proof follows from Theorem 11 by setting .

Our next theorem is essentially inspired by a Lipschitzian condition utilized by Cho et al. [21] as well as Gopal et al. [15].

Theorem 16. Theorem 11 remains true if is replaced by while condition (ii) (of Theorem 11) is replaced by the following condition (ii′) besides retaining rest of the hypotheses: (ii′), for any , where together with , and wherein .

Proof. The proof can be completed on the lines of proof of Theorem 11, hence details are not included.

By restricting , and suitably, one can derive corollaries for two as well as three mappings. For the sake of brevity, we derive just one corollary by restricting Theorem 16 to three mappings which is yet another sharpened form of a theorem contained in [15] which also remains relevant to some results in Pant [4] and Pant [31].

Corollary 17. Suppose that (in the setting of Theorem 16) satisfies and . If are three mappings which satisfy the following conditions: (i)the pairs and satisfy the property, (ii), for any , where together with , and = then, there exist such that . Moreover, if , then , and have a common fixed point provided the pair is pointwise -absorbing while the pair is pointwise -absorbing.

Corollary 18. Let be symmetric (semimetric) space wherein satisfies and . If , and are four self mappings of which satisfy the following conditions: (i)the mappings satisfy the property, (ii), where with , then, there exist such that . Moreover, if  , then , and have a unique common fixed point provided the pair is pointwise -absorbing while the pair is pointwise -absorbing.

Proof. The proof can be completed on the lines of proof of Theorem 11.

Acknowledgments

The authors are grateful to two anonymous referees for their helpful comments and suggestions.

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