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Asymptotically Pseudocontractions, Banach Operator Pairs and Best Simultaneous Approximations
Fixed Point Theory and Applications volumeÂ 2011, ArticleÂ number:Â 812813 (2011)
Abstract
The existence of common fixed points is established for the mappings where is asymptotically pseudocontraction on a nonempty subset of a Banach space. As applications, the invariant best simultaneous approximation and strong convergence results are proved. Presented results are generalizations of very recent fixed point and approximation theorems of Khan and Akbar (2009), Chen and Li (2007), Pathak and Hussain (2008), and several others.
1. Introduction and Preliminaries
We first review needed definitions. Let be a subset of a normed space . The set is called the set of best approximants to out of , where . Suppose that and are bounded subsets of . Then, we write
The number is called the Chebyshev radius of w.r.t. , and an element is called a best simultaneous approximation of w.r.t. . If , then and is the set of all best approximations, , of from . We also refer the reader to Milman [1], and Vijayraju [2] for further details. We denote by and (), the set of positive integers and the closure (weak closure) of a set in , respectively. Let be mappings. The set of fixed points of is denoted by . A point is a coincidence point (common fixed point) of and if . The pair is called
(1)commuting [3] if for all ,
(2)compatible (see [3, 4]) if whenever is a sequence such that for some in ,
(3)weakly compatible if they commute at their coincidence points; that is, if whenever ,
(4)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 [5, 6]. If is a Banach operator pair, then need not be a Banach operator pair (see, e.g., [5, 7, 8]). The set is called starshaped with , if the segment joining to is contained in for all . The map defined on a starshaped set is called affine if
Suppose that is starshaped with and is both  and invariant. Then, and are called,
(5)subweakly commuting on (see [9]) if for all , there exists a real number such that ,
(6)uniformlysubweakly commuting on (see [10]) if there exists a real number such that , for all and . The map is said to be demiclosed at 0 if, for every sequence in converging weakly to and converges to , then .
The classical Banach contraction principle has numerous generalizations, extensions and applications. While considering Lipschitzian mappings, a natural question arises whether it is possible to weaken contraction assumption a little bit in Banach contraction principle and still obtain the existence of a fixed point. In this direction the work of Edelstein [11], Jungck [3], Park [12â€“18] and Suzuki [19] is worth to mention.
Schu [20] introduced the concept of asymptotically pseudocontraction and proved the existence and convergence of fixed points for this class of maps (see also [21]). Recently, Chen and Li [5] introduced the class of Banach operator pairs, as a new class of noncommuting maps and it has been further studied by Hussain [6], Ä†iriÄ‡ et al. [7], Khan and Akbar [22, 23] and Pathak and Hussain [8]. More recently, Zhou [24] established a demiclosedness principle for a uniformly Lipschitzian asymptotically pseudocontraction map and as an application obtained a fixed point result for asymptotically pseudocontraction in the setup of a Hilbert space. In this paper, we are able to join the concepts of uniformly Lipschitzian, asymptotically pseudocontraction and Banach operator pair to get the result of Zhou [24] in the setting of a Banach space. As a consequence, the common fixed point and approximation results of AlThagafi [25], Beg et al. [10], Chidume et al. [26], Chen and Li [5], Cho et al. [27], Khan and Akbar [22, 23], Pathak and Hussain [8], Schu [28] and Vijayraju [2] are extended to the class of asymptotically pseudocontraction maps.
2. Main Results
Let be a real Banach space and be a subset of . Let be mappings. Then is called
(a)an contraction if there exists such that for any ; if , then is called nonexpansive,
(b)asymptoticallynonexpansive [2] if there exists a sequence of real numbers with and such that
for all and for each ; if , then is called asymptotically nonexpansive map,
(c)pseudocontraction if and only if for each , there exists such that
where is the normalized duality mapping defined by
(d)strongly pseudocontraction if and only if for each , there exists and such that
(e)asymptoticallypseudocontractive if and only if for each and , there exists and a constant with such that
If in (2.5), then is called asymptoticallypseudocontractive [20, 24, 27],
(f)uniformlyLipschitzian if there exists some such that
for all and for each ; if , then is called uniformlyLipschitzian [20, 24, 29].
The map is called uniformly asymptotically regular [2, 10] on , if for each , there exists such that for all and all .
The class of asymptotically pseudocontraction contains properly the class of asymptotically nonexpansive mappings and every asymptotically nonexpansive mapping is a uniformly Lipschitzian [2, 24]. For further details, we refer to [21, 24, 27, 29, 30].
In 1974, Deimling [30] proved the following fixed point theorem.
Theorem D.
Let be selfmap of a closed convex subset of a real Banach space . Assume that is continuous strongly pseudocontractive mapping. Then, has a unique fixed point.
The following result extends and improves Theorem 3.4 of Beg et al. [10], Theorem 2.10 in [22], Theorems 2.2 of [25] and Theorem 4 in [31].
Theorem 2.1.
Let be selfmaps of a subset of a real Banach space . Assume that is closed (resp., weakly closed) and convex, is uniformly Lipschitzian and asymptotically pseudocontractive which is also uniformly asymptotically regular on . If is compact (resp., is weakly compact and is demiclosed at 0) and , then .
Proof.
For each , define a selfmap on by
where and is a sequence of numbers in such that and . Since and is convex with , it follows that maps into . As is convex and (resp. ), so ) (resp. ) for each . Since is a strongly pseudocontractive on , by Theorem D, for each , there exists such that . As is bounded, so as . Now,
Since for each , and , therefore . Thus . Also is uniformly asymptotically regular, we have from (2.8)
as . Thus as . As is compact, so there exists a subsequence of such that as . Since is a sequence in and , therefore . Moreover,
Taking the limit as , we get . Thus, proves the first case.
Since a weakly closed set is closed, by Theorem D, for each , there exists such that . The weak compactness of implies that there is a subsequence of converging weakly to as . Since is a sequence in and , so . Moreover, we have, as . If is demiclosed at 0, then . Thus, .
Remark 2.2.
By comparing Theorem 3.4 of Beg et al. [10] with the first case of Theorem 2.1, their assumptions " is closed and starshaped, , , are continuous, is linear, , is compact, is asymptotically nonexpansive and and are uniformly subweakly commuting on " are replaced with " is nonempty set, is closed, convex, , is compact, is uniformly Lipschitzian and asymptotically pseudocontractive".
If is weakly closed and is weakly continuous, then is weakly closed and hence closed, thus we obtain the following.
Corollary 2.3.
Let be selfmaps of a weakly closed subset of a Banach space . Assume that is weakly continuous, is nonempty and convex, is uniformly Lipschitzian and asymptotically pseudocontractive which is also uniformly asymptotically regular on . If is compact (resp. is weakly compact and is demiclosed at 0) and is a Banach operator pair, then .
A mapping on is called pointwise asymptotically nonexpansive [32, 33] if there exists a sequence of functions such that
for all and for each where pointwise on .
An asymptotically nonexpansive mapping is pointwise asymptotically nonexpansive. A pointwise asymptotically nonexpansive map defined on a closed bounded convex subset of a uniformly convex Banach space has a fixed point and is closed and convex [32, 33]. Thus we obtain the following.
Corollary 2.4.
Let be a pointwise asymptotically nonexpansive selfmap of a closed bounded convex subset of a uniformly convex Banach space . Assume that is a selfmap of which is uniformly Lipschitzian, asymptotically pseudocontractive and uniformly asymptotically regular. If is compact (resp. is weakly compact and is demiclosed at 0) and , then .
Corollary 2.5 (see [24, Theorem 3.3]).
Let be selfmap of a closed bounded and convex subset of a real Hilbert space . Assume that is uniformly Lipschitzian and asymptotically pseudocontractive which is also uniformly asymptotically regular on . Then, .
Corollary 2.6.
Let be a Banach space and and be selfmaps of . If , , is closed (resp. weakly closed) and convex, is compact (resp. is weakly compact and is demiclosed at 0), is uniformly Lipschitzian and asymptotically pseudocontractive which is also uniformly asymptotically regular on , and , then .
Remark 2.7.
Corollary 2.6 extends Theorems 4.1 and 4.2 of Chen and Li [5] to a more general class of asymptotically pseudocontractions.
Theorem 2.1 can be extended to uniformly Lipschitzian and asymptotically pseudocontractive map which extends Theorem 2.10 of [22] to asymptotically  pseudocontractions.
Theorem 2.8.
Let be selfmaps of a subset of a Banach space . Assume that is closed (resp. weakly closed) and convex, is uniformly Lipschitzian and asymptotically pseudocontractive which is also uniformly asymptotically regular on . If is compact (resp. is weakly compact and is demiclosed at 0) and , then .
Proof.
For each , define a selfmap on by
where and is a sequence of numbers in such that and . Since and is convex with , it follows that maps into . As is convex and (resp. ), so ) (resp. ) for each . Further, since is a strongly pseudocontractive on , by Theorem D, for each , there exists such that . Rest of the proof is similar to that of Theorem 2.1.
Corollary 2.9.
Let be selfmaps of a subset of a Banach space . Assume that is closed (resp. weakly closed) and convex, is uniformly Lipschitzian and asymptotically pseudocontractive which is also uniformly asymptotically regular on . If is compact (resp. is weakly compact and is demiclosed at 0) and and are Banach operator pairs, then .
Corollary 2.10.
Let be a Banach space and , , and be selfmaps of . If , , where is the set of best simultaneous approximations of w.r.t . Assume that is closed (resp. weakly closed) and convex, is compact (resp. is weakly compact and is demiclosed at 0), is uniformly Lipschitzian and asymptotically pseudocontractive which is also uniformly asymptotically regular on , and , then .
Remark 2.11.

(1)
Theorem 2.2 and 2.7 of Khan and Akbar [23] are particular cases of Corollary 2.10.

(2)
By comparing Theorem 2.2 of Khan and Akbar [23] with the first case of Corollary 2.10, their assumptions " is nonempty, compact, starshaped with respect to an element , is invariant under , and , and are Banach operator pairs on , and are starshaped with , and are continuous and is asymptotically nonexpansive on ," are replaced with ", is closed and convex, , is compact and is uniformly Lipschitzian and asymptotically pseudocontractive on ."

(3)
By comparing Theorem 2.7 of Khan and Akbar [23] with the second case of Corollary 2.10, their assumptions " is nonempty, weakly compact, starshaped with respect to an element , is invariant under , and , and are Banach operator pairs on , and are starshaped with , and are continuous under weak and strong topologies, is demiclosed at 0 and is asymptotically nonexpansive on ," are replaced with ", is weakly closed and convex, , is weakly compact and is demiclosed at 0 and is uniformly Lipschitzian and asymptotically pseudocontractive on ."
We denote by the class of closed convex subsets of containing 0. For , we define = . It is clear that (see [9, 25]).
Theorem 2.12.
Let be selfmaps of a Banach space . If and such that , is compact (resp. is weakly compact) and for all , then is nonempty, closed and convex with . If, in addition, , is closed (resp. weakly closed) and convex, is compact (resp. is weakly compact and is demiclosed at 0), is uniformly Lipschitzian and asymptotically pseudocontractive which is also uniformly asymptotically regular on , and , then .
Proof.
We may assume that . If , then . Note that
Thus, . If is compact, then by the continuity of norm, we get for some .
If we assume that is weakly compact, using Lemma 5.5 in [34, page 192], we can show the existence of a such that .
Thus, in both cases, we have
for all . Hence and so is nonempty, closed and convex with . The compactness of (resp. weak compactness of ) implies that is compact (resp. is weakly compact). The result now follows from Theorem 2.8.
Remark 2.13.
Theorem 2.12 extends Theorems 4.1 and 4.2 in [25], Theorem 8 in [31], and Theorem 2.15 in [22].
Definition 2.14.
Let be a nonempty closed subset of a Banach space , be mappings and . Then and are said to satisfy property [10, 27] if the following holds: for any bounded sequence in , implies .
The normal structure coefficient of a Banach space is defined [10, 26] by , where is the Chebyshev radius of relative to itself and is diameter of . The space is said to have the uniform normal structure if . A Banach limit is a bounded linear functional on such that and for all bounded sequences in . Let be bounded sequence in . Then we can define the realvalued continuous convex function on by for all .
The following lemmas are well known.
Let be a Banach space with uniformly GÃ¢teaux differentiable norm and . Let be bounded sequence in . Then if and only if for all , where is the normalized duality mapping and denotes the generalized duality pairing.
Let be a convex subset of a smooth Banach space , be a nonempty subset of and be a retraction from onto . Then is sunny and nonexpansive if and only if for all and .
Now, we are ready to prove strong convergence to nearest common fixed points of asymptotically pseudocontraction mappings.
Theorem 2.17.
Let be a subset of a reflexive real Banach space with uniformly GÃ¢teaux differentiable norm. Let and be selfmaps on such that is closed and convex, is continuous, uniformly asymptotically regular, uniformly Lipschitzian and asymptotically pseudocontractive with a sequence . Let be sequence of real numbers in such that and . If , then we have the following:
(A)For each , there is exactly one in such that
(B)If is bounded and and satisfy property , then converges strongly to , where is the sunny nonexpansive retraction from onto .
Proof.
Part (A) follows from the proof of Theorem 2.1.

(B)
As in Theorem 2.1, we get . Since is bounded, we can define a function by for all . Since is continuous and convex, as and is reflexive, for some . Clearly, the set is nonempty. Since is bounded and and satisfy property , it follows that . Suppose that , then by Lemma 2.15, we have
(2.16)
In particular, we have
From (2.8), we have
Now, for any , we have
for some . It follows from (2.18) that
Hence we have
This together with (2.17) implies that .
Thus there is a subsequence of which converges strongly to . Suppose that there is another subsequence of which converges strongly to (say). Since is continuous and , is a fixed point of . It follows from (2.21) that
Adding these two inequalities, we get
Consequently, converges strongly to . We can define now a mapping from onto by . From (2.21), we have for all and . Thus by Lemma 2.16, is the sunny nonexpansive retraction on . Notice that and , so by the same argument as in the proof of Theorem 2.1 we obtain, .
Remark 2.18.
Theorem 2.17 extends Theorem 1 in [27]. Notice that the conditions of the continuity and linearity of are not needed in Theorem 3.6 of Beg et al. [10]; moreover, we have obtained the conclusion for more general class of uniformly Lipschitzian and asymptotically pseudocontractive map without any type of commutativity of and .
Corollary 2.19 (see [26, Theorem 3.1]).
Let be a closed convex bounded subset of a real Banach space with uniformly GÃ¢teaux differentiable norm possessing uniform normal structure. Let be an asymptotically nonexpansive mapping with a sequence . Let be fixed, be sequence of real numbers in such that and . Then,
(A)for each , there is unique in such that
(B)if , then converges strongly to a fixed point of .
Remark 2.20.

(1)
Theorem 2.17 improves and extends the results of Beg et al. [10], Cho et al. [27], and Schu [20, 28] to more general class of Banach operators.

(2)
It would be interesting to prove similar results in Modular Function Spaces (cf. [29]).

(3)
Let with the usual norm and . A mapping is defined by , for and , for and on . Clearly, is not nonexpansive [21]â€‰â€‰(e.g., and ). But, is a pseudocontractive mapping.
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Hussain, N. Asymptotically Pseudocontractions, Banach Operator Pairs and Best Simultaneous Approximations. Fixed Point Theory Appl 2011, 812813 (2011). https://doi.org/10.1155/2011/812813
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DOI: https://doi.org/10.1155/2011/812813