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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 459085, 9 pages
Common Fixed-Point Problem for a Family Multivalued Mapping in Banach Space
Department of Mathematics and Computer Science, Chongqing Three Gorges University, Wanzhou 404000, China
Received 11 December 2010; Revised 15 March 2011; Accepted 17 March 2011
Academic Editor: Vittorio Colao
Copyright © 2011 Zhanfei Zuo. 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.
It is our purpose in this paper to prove two convergents of viscosity approximation scheme to a common fixed point of a family of multivalued nonexpansive mappings in Banach spaces. Moreover, it is the unique solution in to a certain variational inequality, where stands for the common fixed-point set of the family of multivalued nonexpansive mapping .
Let be a Banach space with dual , and let be a nonempty subset of . A gauge function is a continuous strictly increasing function such that and . The duality mapping associated with a gauge function is defined by , , where denotes the generalized duality pairing. In the particular case , the duality map is called the normalized duality map. We note that . It is known that if is smooth, then is single valued and norm to weak* continuous (see ). When is a sequence in , then will denote strong (weak, weak*) convergence of the sequence to . s
Following Browder , we say that a Banach space has the weakly continuous duality mapping if there exists a gauge function for which the duality map is single valued and weak to weak* sequentially continuous, that is, if is a sequence in weakly convergent to a point , then the sequence converges weak* to . It is known that spaces have a weakly continuous duality mapping with a gauge . Setting it is easy to see that is a convex function and , for , where denotes the subdifferential in the sense of convex analysis. We will denote by the family of all subsets of , by the family of all nonempty closed bounded subsets of , and by the family of all nonempty compact subsets of . A multivalued mapping is said to be nonexpansive (resp., contractive) if where denotes the Hausdorff metric on defined by Since Banach's contraction mapping principle was extended nicely to multivalued mappings by Nadler in 1969 (see ), many authors have studied the fixed-point theory for multivalued mappings.
In this paper, we construct two viscosity approximation sequences for a family of multivalued nonexpansive mappings in Banach spaces. Let be a nonempty closed convex subset of Banach space and let , be a family of multivalued nonexpansive mapping, is a contraction mapping with constant . Let , . For any given , let such that From Nadler Theorem (see ), we can choose such that Inductively, we can get the sequence as follows: where, for each , such that Similarly, we also have the following multivalued version of the modified Mann iteration: and such that . Then, is said to satisfy Condition () if for any subsequence and implies that , where is the common fixed-point set of the family of multivalued mapping . We give an example of a family of multivalued nonexpansive mappings with Condition () as follows.
Example 1.1. Take and (for all ), where is defined by Let and , , then and the iteration (1.6), reduced to where , and it satisfies Condition (). In fact, if , then for all and Condition () is automatically satisfied. If , then there exists an integer , such that Then, ; hence, for all , from which we deduce that Condition () is satisfied.
Let be a closed convex and a mapping of onto , then is said to be sunny if for all and . A mapping of into is said to be a retraction if . A subset of is said to be a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction of onto , and it is said to be a nonexpansive retract of . If , the metric projection is a sunny nonexpansive retraction from to any closed convex subset of . The following Lemmas will be useful in this paper.
Lemma 2.1 (see ). Let be a nonempty convex subset of a smooth Banach space , let be the (normalized) duality mapping of , and let be a retraction, then the following are equivalent: (1) for all and ,(2) is both sunny and nonexpansive. We note that Lemma 2.1 still holds if the normalized duality map is replaced with the general duality map , where is a gauge function.
Lemma 2.2 (see ). Let be a sequence of nonnegative real numbers satisfying the property where and is a real number sequence such that (i),(ii)either or , then converges to zero, as .
Lemma 2.3 (see ). Let be a real Banach space, then for all , one gets that
Lemma 2.4 (see ). Let and be bounded sequences in a Banach space such that where is a sequence in such that Assume that , then .
3. Main Results
Theorem 3.1. Let be a reflexive Banach space with weakly sequentially continuous duality mapping for some gauge , let be a nonempty closed convex subset of , and let , , be a family of multivalued nonexpansive mappings with which is sunny nonexpansive retract of with a nonexpansive retraction. Furthermore, for any fixed-point , is defined by (1.6), and satisfies the following conditions: (1) as , (2),(3) satisfies Condition . Then, converges strongly to a common fixed-point of a family , , as . Moreover, is the unique solution in to the variational inequality
Proof. First, we show the uniqueness of the solution to the variational inequality (3.1) in . In fact, let be another solution of (3.1) in , then we have From (3.2), we have that We must have and the uniqueness is proved. Let , then, from iteration (1.6), we obtain that Using an induction, we obtain , for all integers , thus, is bounded and so are and . This implies that Next, we will show that Since is reflexive and is bounded, we may assume that such that From (3.5) and satisfying Condition (), we obtain that . On the other hand, we notice that the assumption that the duality mapping is weakly continuous implies that is smooth; from Lemma 2.1, we have Finally, we will show that as . From iteration (1.6) and Lemma 2.3, we get that Lemma 2.2 gives that as . Moreover, satisfying the variational inequality follows from the property of .
Let in iteration (1.6) be a constant mapping, then . In fact, we have the following corollary.
Corollary 3.2. Let and be as in Theorem 3.1, , then converges strongly to a common fixed-point of a family , , as . Moreover, is the unique solution in to the variational inequality If , then the condition that is a sunny nonexpansive retract of in Theorem 3.1 is not necessary, and one has the following Corollary.
Corollary 3.3. Let be a Hilbert space with weakly sequentially continuous duality mapping for some gauge , and let and be as in Theorem 3.1, then converges strongly to a common fixed-point of a family of , , where is the metric projection from onto .
Proof. It is well known that is reflexive; by Propositions 2.3 and 2.6(ii) of , we get that is closed and convex, and hence the projection mapping is sunny nonexpansive retraction mapping, and the result follows from Theorem 3.1.
Corollary 3.4. Let be a real smooth Banach space, let be a nonempty compact subset of , and let and be as in Theorem 3.1, then converges strongly to a common fixed-point of a family of , , as . Moreover, is the unique solution in to the variational inequality
Proof. Following the method of the proof of Theorem 3.1, we get that Next, we will show that Since is compact and is bounded, we can assume that there exists a subsequence of such that , From (3.12) and satisfying Condition (), we obtain that . On the other hand, from the fact that is smooth, the duality being norm to weak* continuous, and the standard characterization of retraction on , we obtain that Now, following the method of the proof of Theorem 3.1, we get the required result.
Theorem 3.5. Let be a reflexive Banach space with weakly sequentially continuous duality mapping for some gauge , let be a nonempty closed convex subset of , and let , , be a family of multivalued nonexpansive mappings with which is sunny nonexpansive retract of with a nonexpansive retraction. for arbitrary . Furthermore, for any fixed-point . is defined by (1.8) and , satisfy the following conditions: (i) as , (ii), (iii). If satisfies Condition , then converges strongly to a common fixed-point of a family of , , as . Moreover, is the unique solution in to the variational inequality
Proof. We first show that the sequence defined by (1.8) is bounded. In fact, take , noting that , we have It follows from induction that so are and . Thus, we have that Next, we show that Let and , then Therefore, we have for some appropriate constant that the following inequality: holds. Thus, . By Lemma 2.4, we obtain Therefore, we have Using (3.20) and satisfying Condition (), we can use the same argumentation as Theorem 3.1 proves that and Finally, we show that as . In fact, from iteration (1.8) and Lemma 2.3, we have From (ii) and (3.25), it then follows that Apply Lemma 2.2 to conclude that .
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