On Strong Convergence by the Hybrid Method for Equilibrium and Fixed Point Problems for an Inifnite Family of Asymptotically Nonexpansive Mappings
© G. Cai and C. S. Hu. 2009
Received: 17 April 2009
Accepted: 9 July 2009
Published: 4 August 2009
We introduce two modifications of the Mann iteration, by using the hybrid methods, for equilibrium and fixed point problems for an infinite family of asymptotically nonexpansive mappings in a Hilbert space. Then, we prove that such two sequences converge strongly to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of an infinite family of asymptotically nonexpansive mappings. Our results improve and extend the results announced by many others.
Let be a nonempty closed convex subset of a Hilbert space . A mapping is said to be nonexpansive if for all we have . It is said to be asymptotically nonexpansive  if there exists a sequence with and such that for all integers and for all . The set of fixed points of is denoted by .
The set of solutions of (1.1) is denoted by . In 2005, Combettes and Hirstoaga  introduced an iterative scheme of finding the best approximation to the initial data when is nonempty, and they also proved a strong convergence theorem.
In the case of , . In the case of , is denoted by . The problem (1.2) is very general in the sense that it includes, as special cases, some optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others (see, e.g., [3, 4]).
for all . It is obvious that any inverse strongly monotone mapping is monotone and Lipschitz continuous.
Construction of fixed points of nonexpansive mappings and asymptotically nonexpansive mappings is an important subject in nonlinear operator theory and its applications, in particular, in image recovery and signal processing (see, e.g., [1, 8–10]). Fixed point iteration processes for nonexpansive mappings and asymptotically nonexpansive mappings in Hilbert spaces and Banach spaces including Mann  and Ishikawa  iteration processes have been studied extensively by many authors to solve nonlinear operator equations as well as variational inequalities; see, for example, [11–13]. However, Mann and Ishikawa iteration processes have only weak convergence even in Hilbert spaces (see, e.g., [11, 12]).
where denotes the metric projection from onto a closed convex subset of . They proved that if the sequence bounded above from one, then defined by (1.6) converges strongly to .
where , as . They proved that if for all and for some , then the sequence generated by (1.7) converges strongly to .
where , as and and for all . They proved that the sequence generated by (1.8) converges strongly to a common fixed point of two asymptotically nonexpansive mappings and .
for every , where for some and satisfies . Further, they proved that and converge strongly to , where .
Inspired and motivated by the above facts, it is the purpose of this paper to introduce the Mann iteration process for finding a common element of the set of common fixed points of an infinite family of asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem. Then we prove some strong convergence theorems which extend and improve the corresponding results of Tada and Takahashi , Inchan and Plubtieng , Zegeye and Shahazad , and many others.
We will use the following notations:
(1)" " for weak convergence and " " for strong convergence;
(2) denotes the weak -limit set of .
for all .
We need some facts and tools in a real Hilbert space which are listed as follows.
Lemma 2.1 ().
Let be an asymptotically nonexpansive mapping defined on a nonempty bounded closed convex subset of a Hilbert space . If is a sequence in such that and , then .
Lemma 2.2 ().
Let be a nonempty closed convex subset of and also give a real number . The set is convex and closed.
Lemma 2.3 ().
For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions (see ):
(A1) for all ;
(A2) is monotone, that is, for any ;
(A3) is upper-hemicontinuous, that is, for each
(A4) is convex and weakly lower semicontinuous for each .
The following lemma appears implicity in .
Lemma 2.4 ().
The following lemma was also given in .
Lemma 2.5 ().
for all . Then, the following holds
(1) is single-valued;
(2) is firmly nonexpansive, that is, for any , .
This implies that , that is, is a nonexpansive mapping:
(4) is a closed and convex set.
Definition 2.6 (see ).
Such a mapping is called the modified -mapping generated by and .
Lemma 2.7 ([10, Lemma 4.1]).
for all , and ;
(iii) if , and is closed convex.
Lemma 2.8 ([10, Lemma 4.4]).
Let be a nonempty closed convex subset of . Let be a family of asymptotically nonexpansive mappings of into itself with Lipschitz constants , that is, ( ) such that . Let for every , where for every and with for every and for every and let for every . Then, the following holds:
- (ii)if is bounded and , for every sequence in C,(2.9)
if , and is closed convex.
3. Main Results
In this section, we will introduce two iterative schemes by using hybrid approximation method for finding a common element of the set of common fixed points for a family of infinitely asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert space. Then we show that the sequences converge strongly to a common element of the two sets.
where and and and . Then and converge strongly to .
We show first that the sequences and are well defined.
We observe that is closed and convex by Lemma 2.2. Next we show that for all . we prove first that is nonexpansive. Let . Since is -inverse strongly monotone and , we have
Thus is nonexpansive.
By Lemma 2.5, we have , .
Let , it follows the definition of that
Again by Lemma 2.5, we have , .
Since and are nonexpansive, one has
So for all and hence for all . This implies that is well defined. From Lemma 2.4, we know that is also well defined.
Next, we prove that , , , , as .
It follows from that
Since is bounded, then and are bounded.
From and , we have
Hence, is nodecreasing, and so exists.
Next, we can show that . Indeed, From (2.1) and (3.13), we obtain
Next, we claim that . Let , it follows from (3.8) that
Next, we prove that there exists a subsequence of which converges weakly to , where .
Since is bounded and is closed, there exists a subsequence of which converges weakly to , where . From (3.28), we have . Noticing (3.29) and (3.32), it follows from Lemma 2.7 that . Next we prove that . Since , for any we have
This implies that . Therefore .
Finally we show that , , where .
Putting and consider the sequence . Then we have and by the weak lower semicontinuity of the norm and by the fact that for all which is implied by the fact that , we obtain
It follows that , and hence . Since is an arbitrary (weakly convergent) subsequence of , we conclude that . From (3.28), we know that also. This completes the proof of Theorem 3.1.
where and and . Then and converge strongly to .
We divide the proof of Theorem 3.2 into four steps.
(i)We show first that the sequences and are well defined.
it follows that is convex. So, is a closed convex subset of for any .
Next, we show that . Indeed, let and let be a sequence of mappings defined as in Lemma 2.5. Similar to the proof of Theorem 3.1, we have
Therefore, for all .
Next, we prove that , . For , we have . Assume that . Since is the projection of onto , by Lemma 2.3, we have
This implies that is well defined. From Lemma 2.4, we know that is also well defined.
(ii)We prove that , , , , as .
Since is a nonempty closed convex subset of , there exists a unique such that .
From , we have
This shows that the sequence is nondecreasing. So, exists.
It follows from (2.1) and (3.53) that
We prove that there exists a subsequence of which converges weakly to , where .
Finally we show that , , where .
Since is an arbitrary subsequence of , we conclude that converges strongly to . By (3.58), we have also. This completes the proof of Theorem 3.2.
where and and and such that . Then and converge strongly to .
Putting , the conclusion of Corollary 3.3 can be obtained as in the proof of Theorem 3.1.
Corollary 3.3 extends the Theorem of Tada and Takahashi  in the following senses:
from computation point of view, the algorithm in Corollary 3.3 is also simpler and, more convenient to compute than the one given in .
where and . Then converges strongly to .
Putting , , and , for all in Theorem 3.1, we have , therefore . The conclusion of Corollary 3.5 can be obtained from Theorem 3.1 immediately.
Corollary 3.5 extends Theorem 3.1 of Inchan and Plubtieng  from two asymptotically nonexpansive mappings to an infinite family of asymptotically nonexpansive mappings.
where and . Then converges strongly to .
Putting , , and , for all in Theorem 3.2, we have , therefore . The conclusion of Corollary 3.7 can be obtained from Theorem 3.2.
Corollary 3.7 extends Theorem 3.1 of Zegeye and Shahzad  from a finite family of asymptotically nonexpansive mappings to an infinite family of asymptotically nonexpansive mappings.
This research is supported by the National Science Foundation of China under Grant (10771175), and by the key project of chinese ministry of education(209078) and the Natural Science Foundational Committee of Hubei Province (D200722002).
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