Hybrid Proximal-Type Algorithms for Generalized Equilibrium Problems, Maximal Monotone Operators, and Relatively Nonexpansive Mappings
© Lu-Chuan Zeng et al. 2011
Received: 24 September 2010
Accepted: 18 October 2010
Published: 2 November 2010
The purpose of this paper is to introduce and consider new hybrid proximal-type algorithms for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a relatively nonexpansive mapping , and the set of zeros of a maximal monotone operator in a uniformly smooth and uniformly convex Banach space. Strong convergence theorems for these hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequences generated by these various algorithms converge strongly to the same point in . These new results represent the improvement, generalization, and development of the previously known ones in the literature.
This algorithm was first introduced by Martinet  and generally studied by Rockafellar  in the framework of a Hilbert space. Later many authors studied its convergence in a Hilbert space or a Banach space. See, for instance, [16–21] and the references therein.
Let be a reflexive, strictly convex, and smooth Banach space with the dual and be a nonempty closed convex subset of . Let be a maximal monotone operator with domain and be a relatively nonexpansive mapping. Let be an α-inverse-strongly monotone mapping and be a bifunction satisfying (A1)–(A4): (A1) , ; (A2) is monotone, that is, , ; (A3) , ; (A4) the function is convex and lower semicontinuous. The purpose of this paper is to introduce and investigate two new hybrid proximal-type Algorithms 1.1 and 1.2 for finding an element of .
In this paper, strong convergence results on these two hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequence generated by Algorithm 1.1 and the sequence generated by Algorithm 1.2, converge strongly to the same point . These new results represent the improvement, generalization and development of the previously known ones in the literature including Solodov and Svaiter , Kamimura and Takahashi , Qin and Su , and Ceng et al. .
Throughout this paper the symbol ⇀ stands for weak convergence and → stands for strong convergence.
It is also said to be uniformly smooth if the limit is attained uniformly for . Recall also that if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subsets of . A Banach space is said to have the Kadec-Klee property if for any sequence , whenever and , we have . It is known that if is uniformly convex, then has the Kadec-Klee property; see [26, 27] for more details.
Let be a nonempty closed convex subset of a real Hilbert space and be the metric projection of onto C, then is nonexpansive. This fact actually characterizes Hilbert spaces and hence, it is not available in more general Banach spaces. Nevertheless, Alber  recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.
Let be a nonempty closed convex subset of , and let be a mapping from into itself. A point is called an asymptotically fixed point of  if contains a sequence which converges weakly to such that . The set of asymptotical fixed points of will be denoted by . A mapping from into itself is called relatively nonexpansive [32–34] if and for all and .
We remark that if is a reflexive, strictly convex and smooth Banach space, then for any if and only if . It is sufficient to show that if then . From (2.5), we have . This implies that . From the definition of , we have . Therefore, we have ; see [26, 27] for more details.
We need the following lemmas for the proof of our main results.
Lemma 2.1 (see ).
Lemma 2.4 (see ).
The following result is according to Blum and Oettli .
Lemma 2.5 (see ).
Lemma 2.6 (see ).
Using Lemma 2.6, one has the following result.
Lemma 2.7 (see ).
Utilizing Lemmas 2.5, 2.6 and 2.7 as above, Chang  derived the following result.
Proposition 2.8 (see [39, Lemma 2.5]).
Let be a smooth, strictly convex and reflexive Banach space and be a nonempty closed convex subset of . Let be an α-inverse-strongly monotone mapping, let be a bifunction from to satisfying (A1)–(A4), and let , then there hold the following:
3. Main Results
Throughout this section, unless otherwise stated, we assume that is a maximal monotone operator with domain , is a relatively nonexpansive mapping, is an α-inverse-strongly monotone mapping and is a bifunction satisfying (A1)–(A4), where is a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space . In this section, we study the following algorithm.
First we investigate the condition under which the Algorithm 3.1 is well defined. Rockafellar  proved the following result.
Lemma 3.2 (Rockafellar ).
Utilizing this result, we can show the following lemma.
We are now in a position to prove the main theorem.
So we have . Utilizing the Kadec-Klee property of , we conclude that converges strongly to . Since is an arbitrary weakly convergent subsequence of , we know that converges strongly to . This completes the proof.
Let be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space . Let be a maximal monotone operator with domain , be a relatively nonexpansive mapping, be an α-inverse-strongly monotone mapping and be a bifunction satisfying (A1)–(A4). Assume that is a sequence in satisfying and that is a sequences in satisfying .
Finally, we prove that . Indeed, for arbitrarily fixed, there exists a subsequence of such that , then . Now let us show that . Since , we have that . Moreover, since is uniformly norm-to-norm continuous on bounded subsets of , and , we obtain that . It follows from and the monotonicity of that for all and . This implies that for all and . Thus from the maximality of , we infer that . Further, let us show that . Since and , from we obtain that and .
It follows from the definition of that and hence . So we have . Utilizing the Kadec-Klee property of , we know that . Since is an arbitrary weakly convergent subsequence of , we know that . This completes the proof.
This research was partially supported by the Leading Academic Discipline Project of Shanghai Normal University (no. DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (no. 09ZZ133), National Science Foundation of China (no. 11071169), Ph.D. Program Foundation of Ministry of Education of China (no. 20070270004), Science and Technology Commission of Shanghai Municipality Grant (no. 075105118), and Shanghai Leading Academic Discipline Project (no. S30405). Al-Homidan is grateful to KFUPM for providing research facilities.
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