Common Fixed Point and Approximation Results for Noncommuting Maps on Locally Convex Spaces
© F. Akbar and A. R. Khan. 2009
Received: 21 February 2009
Accepted: 14 April 2009
Published: 11 May 2009
Common fixed point results for some new classes of nonlinear noncommuting maps on a locally convex space are proved. As applications, related invariant approximation results are obtained. Our work includes improvements and extension of several recent developments of the existing literature on common fixed points. We also provide illustrative examples to demonstrate the generality of our results over the known ones.
1. Introduction and Preliminaries
In the sequel, will be a Hausdorff locally convex topological vector space. A family of seminorms defined on is said to be an associated family of seminorms for if the family where and , forms a base of neighborhoods of zero for . A family of seminorms defined on is called an augmented associated family for if is an associated family with property that the seminorm for any . The associated and augmented associated families of seminorms will be denoted by and , respectively. It is well known that given a locally convex space there always exists a family of seminorms defined on such that (see [1, page 203]).
The following construction will be crucial. Suppose that is a -bounded subset of . For this set we can select a number for each such that where Clearly, is -bounded, -closed, absolutely convex and contains . The linear span of in is The Minkowski functional of is a norm on . Thus is a normed space with as its closed unit ball and for each (for details see [1–3]).
Let be a subset of a locally convex space . Let be mappings. A mapping is called -Lipschitz if there exists such that for any and for all . If (resp., ), then is called an -contraction (resp., -nonexpansive). A point is a common fixed (coincidence) point of and if ( ). The set of coincidence points of and is denoted by and the set of fixed points of is denoted by The pair is called:
(2) -weakly commuting if for all and for all , there exists such that If , then the maps are called weakly commuting ;
(3)compatible  if for all , whenever is a sequence such that for some in ;
(5) -subcommuting on if for all and for all , there exists a real number such that for each . If , then the maps are called -subcommuting ;
(6) -subweakly commuting on (see ) if for all and for all , there exists a real number such that , where and ;
In 1963, Meinardus  employed the Schauder fixed point theorem to prove a result regarding invariant approximation. Singh , Sahab et al. , and Jungck and Sessa  proved similar results in best approximation theory. Recently, Hussain and Khan  have proved more general invariant approximation results for 1-subcommuting maps which extend the work of Jungck and Sessa  and Al-Thagafi  to locally convex spaces. More recently, with the introduction of noncommuting maps to this area, Pant , Pathak et al. , Hussain and Jungck , and Jungck and Hussain  further extended and improved the above-mentioned results; details on the subject may be found in [17, 18]. For applications of fixed point results of nonlinear mappings in simultaneous best approximation theory and variational inequalities, we refer the reader to [19–21]. Fixed point theory of nonexpansive and noncommuting mappings is very rich in Banach spaces and metric spaces [13–17]. However, some partial results have been obtained for these mappings in the setup of locally convex spaces (see  and its references). It is remarked that the generalization of a known result in Banach space setting to the case of locally convex spaces is neither trivial nor easy (see, e.g., [2, 22]).
The following general common fixed point result is a consequence of Theorem 3.1 of Jungck , which will be needed in the sequel.
The aim of this paper is to extend the above well-known result of Jungck to locally convex spaces and establish general common fixed point theorems for generalized -nonexpansive subcompatible maps in the setting of a locally convex space. We apply our theorems to derive some results on the existence of common fixed points from the set of best approximations. We also establish common fixed point and approximation results for the newly defined class of Banach operator pairs. Our results extend and unify the work of Al-Thagafi , Chen and Li , Hussain , Hussain and Berinde , Hussain and Jungck , Hussain and Khan , Hussain and Rhoades , Jungck and Sessa , Khan and Akbar [19, 20], Pathak and Hussain , Sahab et al. , Sahney et al. , Singh [11, 27], Tarafdar , and Taylor .
2. Subcompatible Maps in Locally Convex Spaces
Recently, Khan et al.  introduced the class of subcompatible mappings as follows:
We can extend this definition to a locally convex space by replacing the norm with a family of seminorms.
Clearly, subcompatible maps are compatible but the converse does not hold, in general, as the following example shows.
Example 2.2 (see ).
Example 2.3 (see ).
Let with usual norm and Let if and if , and if and if . Then is -starshaped with and . Note that and are subcompatible but not -weakly commuting for all . Thus and are neither -subweakly commuting nor -subcommuting maps.
We observe in the following example that the weak commutativity of a pair of selfmaps on a metric space depends on the choice of the metric; this is also true for compatibility, -weak commutativity, and other variants of commutativity of maps.
Example 2.4 (see ).
Let with usual metric and Let and . Then and . Thus the pair is not weakly commuting on with respect to usual metric. But if is endowed with the discrete metric , then for . Thus the pair is weakly commuting on with respect to discrete metric.
Next we establish a positive result in this direction in the context of linear topologies utilizing Minkowski functional; it extends [6, Lemma 2.1].
There are plenty of spaces which are not normable (see [31, page 113]). So it is natural and essential to consider fixed point and approximation results in the context of a locally convex space. An application of Lemma 2.5 provides the following general common fixed point result.
As and are nonexpansive on -bounded set , and are also nonexpansive with respect to and hence continuous (cf. ). A comparison of our hypothesis with that of Theorem 1.1 tells that we can apply Theorem 1.1 to as a subset of to conclude that there exists a unique in such that .
We now prove the main result of this section.
Proof follows from (i).
Since is weakly compact, there is a subsequence of converging weakly to some . But, and being affine and continuous are weakly continuous, and the weak topology is Hausdorff, so we have . The set is bounded, so as Now the demiclosedness of at guarantees that and hence .
Let be a nonempty -bounded, -sequentially complete, -starshaped subset of a Hausdorff locally convex space and let and be selfmaps of Suppose that and are affine and nonexpansive with , and . If the pairs and are subcompatible and is -nonexpansive, then provided that one of the following conditions holds
As -subcommuting maps are subcompatible, so by Theorem 2.8, we obtain the following recent result of Hussain and Khan  without the surjectivity of . Note that a continuous and affine map is weakly continuous, so the weak continuity of is not required as well.
Corollary 2.9 ([6, Theorem 2.2]).
Let be a nonempty -bounded, -sequentially complete, -starshaped subset of a Hausdorff locally convex space and let be selfmaps of Suppose that is affine and nonexpansive with , and . If the pair is -subcommuting and is -nonexpansive, then provided that one of the following conditions holds:
One can now easily prove on the lines of the proof of the above theorem that the approximation results are similar to those of Theorems 2.11-2.12 due to Hussain and Jungck  in the setting of a Hausdorff locally convex space.
(i)Follows from [7, Theorem 2.14].
3. Banach Operator Pair in Locally Convex Spaces
Utilizing similar arguments as above, the following result can be proved which extends recent common fixed point results due to Hussain and Rhoades [8, Theorem 2.1] and Jungck and Hussain [9, Theorem 2.1] to the setup of a Hausdorff locally convex space which is not necessarily metrizable.
As an application of Theorem 3.1, the analogue of all the results due to Hussain and Berinde , and Hussain and Rhoades  can be established for -commuting maps and defined on a -bounded subset of a Hausdorff locally convex space. We leave details to the reader.
Recently, Chen and Li  introduced the class of Banach operator pairs, as a new class of noncommuting maps and it has been further studied by Hussain , Ciric et al. , Khan and Akbar [19, 20], and Pathak and Hussain . The pair is called a Banach operator pair, if the set is -invariant, namely, . Obviously, commuting pair is a Banach operator pair but converse is not true, in general; see [21, 23]. If is a Banach operator pair, then need not be a Banach operator pair (cf. [23, Example 1]).
Chen and Li  proved the following.
Theorem 3.2 ([23, Theorems 3.2-3.3]).
Let be a -starshaped subset of a normed space and let , be self-mappings of Suppose that is -starshaped and is continuous on . If is compact (resp., is weakly continuous, is complete, is weakly compact, and either is demiclosed at or satisfies Opial's condition), is a Banach operator pair, and is -nonexpansive on , then .
In this section, we extend and improve the above-mentioned common fixed point results of Chen and Li  in the setup of a Hausdorff locally convex space.
If is compact, for each , is -compact and hence -sequentially complete. By Lemma 3.3, for each there exists such that The compactness of implies that there exists a subsequence of such that as . Since is a sequence in and , therefore . Further, . By the continuity of , we obtain . Thus, proves the first case.
The weak compactness of implies that is weakly compact and hence -sequentially complete due to completeness of . From Lemma 3.3, for each there exists such that Moreover, we have as . The weak compactness of implies that there is a subsequence of converging weakly to as . Since is a sequence in , therefore . Also we have, as . If is demiclosed at , then . Thus
Let be a nonempty -bounded subset of Hausdorff locally convex (resp., complete) space and let and be self-maps of Suppose that is -starshaped, and -closed (resp., -weakly closed), is compact (resp., is weakly compact), is continuous on (resp., is demiclosed at ), and are Banach operator pairs and satisfy (3.4) for all then .
Let be a Hausdorff locally convex (resp., complete) space and and be self-maps of If , , is -starshaped, (resp., ], is compact (resp., is weakly compact), is continuous on (resp., is demiclosed at ), and (3.4) holds for all then .
Let be a Hausdorff locally convex (resp., complete) space and and be self-maps of If , , is -starshaped, (resp., ), is compact (resp., is weakly compact), is continuous on (resp., is demiclosed at ), and (3.4) holds for all then .
Let be self-maps of a Hausdorff locally convex space . If and such that , is compact and for all , then is nonempty, closed, and convex with . If, in addition, , is -starshaped, , is continuous on and (3.4) holds for all then .
We utilize Corollary 3.5 instead of Theorem 2.7 in the proof of Theorem 2.12.
The author A. R. Khan gratefully acknowledges the support provided by the King Fahd University of Petroleum & Minerals during this research. The authors would like to thank the referees for their valuable suggestions to improve the presentation of the paper.
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