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:
(1)commuting if for all ;(2)-weakly commuting if for all and for all , there exists such that If , then the maps are called weakly commuting [4];(3)compatible [5] if for all , whenever is a sequence such that for some in ;(4)weakly compatible if they commute at their coincidence points, that is, whenever .Suppose that is -starshaped with and is both - and -invariant. Then and are called:
(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];(6)-subweakly commuting on (see [7]) if for all and for all , there exists a real number such that , where and ;(7)-commuting [8, 9] if for all , where and .If then we define the set, , of best -approximations to as , for all . A mapping is called demiclosed at if converges weakly to and converges to , then we have . A locally convex space satisfies Opial's condition if for every net in weakly convergent to the inequality
holds for all and .
In 1963, Meinardus [10] employed the Schauder fixed point theorem to prove a result regarding invariant approximation. Singh [11], Sahab et al. [12], and Jungck and Sessa [13] proved similar results in best approximation theory. Recently, Hussain and Khan [6] have proved more general invariant approximation results for 1-subcommuting maps which extend the work of Jungck and Sessa [13] and Al-Thagafi [14] to locally convex spaces. More recently, with the introduction of noncommuting maps to this area, Pant [15], Pathak et al. [16], Hussain and Jungck [7], and Jungck and Hussain [9] 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 [22] 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 [5], which will be needed in the sequel.
Theorem 1.1. Let be a complete metric space, and let be selfmaps of . Suppose that and are continuous, the pairs and are compatible such that . If there exists such that for all
then there is a unique point in such that .
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 [14], Chen and Li [23], Hussain [24], Hussain and Berinde [25], Hussain and Jungck [7], Hussain and Khan [6], Hussain and Rhoades [8], Jungck and Sessa [13], Khan and Akbar [19, 20], Pathak and Hussain [21], Sahab et al. [12], Sahney et al. [26], Singh [11, 27], Tarafdar [3], and Taylor [28].
2. Subcompatible Maps in Locally Convex Spaces
Recently, Khan et al. [29] introduced the class of subcompatible mappings as follows:
Definition 2.1. Let be a -starshaped subset of a normed space . For the selfmaps and of with we define where and . Now and are subcompatible if for all sequences .
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 [29]). Let with usual norm and Let and for all . Let Then is -starshaped with . Note that and are compatible. For any sequence in with , we have, . However, . Thus and are not subcompatible maps.
Note that -subweakly commuting and -subcommuting maps are subcompatible. The following simple example reveals that the converse is not true, in general.
Example 2.3 (see [29]). 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 [30]). 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].
Lemma 2.5. Let and be compatible selfmaps of a -bounded subset of a Hausdorff locally convex space . Then and are compatible on with respect to
Proof. By hypothesis, for each whenever for some . Taking supremum on both sides, we get
whenever
This implies that
whenever
Hence whenever as desired.
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.
Theorem 2.6. Let be a nonempty -bounded, -complete subset of a Hausdorff locally convex space and let and be selfmaps of Suppose that and are nonexpansive, the pairs and are compatible such that . If there exists such that for all and for all
then there is a unique point in such that .
Proof. Since the norm topology on has a base of neighbourhoods of consisting of -closed sets and is -sequentially complete, therefore is - sequentially complete in see [3, the proof of Theorem 1.2]. By Lemma 2.5, the pairs and are compatible maps of . From (2.5) we obtain for any ,
Thus
As and are nonexpansive on -bounded set , and are also nonexpansive with respect to and hence continuous (cf. [6]). 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.
Theorem 2.7. 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, for all and for all
then provided that one of the following conditions holds: (i) is -sequentially compact, and is continuous ( stands for closure);(ii) is -sequentially compact, and is continuous;(iii) is weakly compact in and is demiclosed at .
Proof. Define by
for all and a fixed sequence of real numbers ) converging to . Then, each is a selfmap of and for each , since and are affine and As is affine and the pair is subcompatible, so for any with , we have
Thus the pair is compatible on for each . Similarly, the pair is compatible for each .
Also by (2.8),
for each , and . By Theorem 2.6, for each , there exists such that .
(i) The compactness of implies that there exists a subsequence of and a such that as . Since , also converges to Since , and are continuous, we have Thus
(ii) Proof follows from (i).
(iii) 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 .
Theorem 2.7 extends and improves [14, Theorem 2.2], [7, Theorems 2.2-2.3, and Corollaries 2.4–2.7], [13, Theorem 6], and the main results of Tarafdar [3] and Taylor [28](see also [6, Remarks 2.4]).
Theorem 2.8. 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 (i) is -sequentially compact;(ii) is -sequentially compact;(iii) is weakly compact in , is demiclosed at .(iv) is weakly compact in an Opial space .
Proof. (i)–(iii) follow from Theorem 2.7.
(iv) As in (iii) we have and as If , then by the Opial's condition of and -nonexpansiveness of we get,
which is a contradiction. Thus and hence .
As -subcommuting maps are subcompatible, so by Theorem 2.8, we obtain the following recent result of Hussain and Khan [6] 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: (i) is -sequentially compact;(ii) is -sequentially compact;(iii) is weakly compact in , is demiclosed at .(iv) is weakly compact in an Opial space .
The following theorem improves and extends the corresponding approximation results in [6–8, 11–14, 25, 27].
Theorem 2.10. Let be a nonempty subset of a Hausdorff locally convex space and let be mappings such that for some and . Suppose that and are affine and nonexpansive on with is -bounded, -sequentially complete, -starshaped and . If the pairs and are subcompatible and, for all and ,
then , provided that one of the following conditions holds
(i) is -sequentially compact, and is continuous;(ii) is -sequentially compact, and is continuous;(iii) is weakly compact, and is demiclosed at .
Proof. Let . Then for each , . Note that for any ,
It follows that the line segment and the set are disjoint. Thus is not in the interior of and so . Since , must be in . Also since , and satisfy (2.13), we have for each ,
Thus Consequently, . Now Theorem 2.7 guarantees that .
Remark 2.11. 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 [7] in the setting of a Hausdorff locally convex space.
We define and denote by the class of closed convex subsets of containing . For , we define for each . It is clear that .
The following result extends [14, Theorem 4.1] and [7, Theorem 2.14].
Theorem 2.12. Let be selfmaps of a Hausdorff locally convex space with and such that . Suppose that and for all and for each where is compact. Then (i) is nonempty, closed, and convex,(ii)(iii) provided and are subcompatible, affine, and nonexpansive on , and, for some and for all is continuous, the pairs and are subcompatible on and satisfy for all
for all and for each .
Proof. (i) We follow the arguments used in [7] and [8]. Let for each .
Then there is a minimizing sequence in such that As is compact so has a convergent subsequence with (say) in Now by using
we get for each ,
Hence Thus is nonempty closed and convex.(i)Follows from [7, Theorem 2.14].(ii)By Theorem 2.7(i), so it follows that there exists such that Hence (iii) follows from Theorem 2.7(i).
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.
Theorem 3.1. Let be a -bounded subset of a Hausdorff locally convex space , and let and let be weakly compatible self-maps of . Assume that , is -sequentially complete, and and satisfy, for all , and for some
Then is a singleton.
As an application of Theorem 3.1, the analogue of all the results due to Hussain and Berinde [25], and Hussain and Rhoades [8] 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 [23] introduced the class of Banach operator pairs, as a new class of noncommuting maps and it has been further studied by Hussain [24], Ciric et al. [32], Khan and Akbar [19, 20], and Pathak and Hussain [21]. 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 [23] 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 [23] in the setup of a Hausdorff locally convex space.
Lemma 3.3. Let be a nonempty -bounded subset of Hausdorff locally convex space , and let and be self-maps of If is nonempty, , is -sequentially complete, and , and satisfy for all and for some
then is singleton.
Proof. Note that being a subset of is -sequentially complete. Further, for all , we have
Hence is a generalized contraction on and . By Theorem 3.1 (with = identity map),