Modified Halpern-type iterative methods for relatively nonexpansive mappings and maximal monotone operators in Banach spaces
© Wu and Cheng; licensee Springer. 2014
Received: 1 July 2014
Accepted: 18 November 2014
Published: 4 December 2014
We obtain the modified Halpern-type iterative method for finding a common element of the fixed point set of a relatively nonexpansive mapping and the zero set of a maximal monotone operator in a uniformly convex and uniformly smooth Banach space. Our results extend and improve the recent results of Chuang, Lin and Takahashi (J. Glob. Optim. 56:1591-1601, 2013) and Nilsrakoo and Saejung (Appl. Math. Comput. 217:6577-6586, 2011).
MSC:47H05, 47H09, 47J25.
for all , ;
If T satisfies (i) and (ii), then T is called relatively quasi-nonexpansive (see ). In a Hilbert space, relatively quasi-nonexpansive mappings coincide with quasi-nonexpansive mappings. Quasi-nonexpansive mappings are investigated by Chuang et al. , Yamada and Ogura , Kim , etc.
Iterative methods for finding the fixed points of relatively nonexpansive mappings have been studied by many researchers. Matsushita and Takahashi  established the Mann-type iteration for relatively nonexpansive mappings. Nilsrakoo and Saejung  constructed the Halpern-Mann iterative methods for relatively nonexpansive mappings and proved the strong convergence theorem. Matsushita and Takahashi  presented the hybrid methods for relatively nonexpansive mappings.
Let A be a maximal monotone operator from E to . Several problems in nonlinear analysis and optimization can be formulated to find a point such that . We denote by the set of all with . There has been tremendous interest in developing the method for solving zero point problems of maximal monotone operators and related topics (see [13–23]). Zeng et al. [20–22] proposed hybrid proximal-type and hybrid shrinking projection algorithms for maximal monotone operators, relatively nonexpansive mappings and equilibrium problems. Klin-Eam et al.  introduced the Halpern iterative method for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping in a Banach space by using hybrid methods. It is helpful to point out that the methods in [20–23] involve the generalized projections. However, even in Hilbert spaces, sometimes it is hard to compute the generalized projection.
Motivated by Chuang et al.  and Nilsrakoo and Saejung , we present the modified Halpern-type iterative method for finding a common element of the fixed point set of a relatively nonexpansive mapping and the zero set of a maximal monotone operator. This iterative method is practicable since it does not involve the generalized projections. Our results extend and improve the recent results of some authors.
The paper is organized as follows. Section 2 contains some important concepts and facts. Section 3 is devoted to introducing an iterative scheme and proving a strong convergence theorem. Section 4 provides some examples and numerical results.
for every . By the Hahn-Banach theorem, Jx is nonempty for all . In a Hilbert space, the normalized duality mapping J is the identity (see  for more details).
(p1) A Banach space E is uniformly smooth if and only if is uniformly convex.
(p2) If E is strictly convex, then J is one-to-one.
(p3) If E is smooth, then J is single-valued.
(p4) If E is reflexive, then J is onto.
(p5) If E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.
for and .
Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. From Alber  and Kamimura and Takahashi , the generalized projection from E onto C is defined by for all . In a Hilbert space H, the generalized projection coincides with the nearest metric projection from H onto C.
Let A be a set-valued mapping from E to with graph and domain . It is said to be monotone if for all . A monotone operator is maximal if its graph is not properly contained in the graph of any other monotone operator. For a maximal monotone operator A and , the resolvent of A is defined by for . It is easy to see that . The Yosida approximation of A is defined by for . Note that .
The following lemmas are useful in the sequel.
Lemma 2.1 
Lemma 2.2 
for all , and .
Lemma 2.3 
Let E be a uniformly convex and smooth Banach space. Suppose that and are two sequences of E such that or is bounded. If , then .
Lemma 2.4 
If if and only if for all ;
for all .
Lemma 2.5 
Lemma 2.6 
Lemma 2.7 
and for all .
3 Strong convergence theorems
In this section, we present the modified Halpern-type iterative method for a relatively nonexpansive mapping and a maximal monotone operator in a uniformly convex and uniformly smooth Banach space.
where and are sequences in and the sequence is contained in E. Suppose that the following conditions are satisfied:
(c1) and ;
(c3) for some ;
Then the sequence converges strongly to .
Proof It follows from [, Proposition 2.4] that the set is closed and convex. The set is closed and convex since is closed and convex. For simplicity, we write .
which yields that is bounded. So are and .
We divide the rest of the proof into two cases.
Thus, inequality (3.7) holds.
Using (3.2), (3.7) and Lemma 2.6, we see that the sequence converges strongly to .
This together with (3.17) implies that . It follows from (3.18) that . Then we have according to the fact that . The proof is completed. □
Remark 1 Letting in our result, we obtain the algorithm for minimal-norm solutions of the corresponding problem.
Remark 2 When (that is, the subdifferential of the indicator function of C) and , Theorem 3.1 improves and extends the result of Nilsrakoo and Saejung [, Theorem 3.4] in which the variable is reduced to the constant u.
The set of solutions of (3.19) is denoted by . Numerous problems in physics, optimization and economics can be reduced to finding a solution of the equilibrium problem (for instance, see ). The equilibrium problem has been studied extensively (see [7, 8, 28–33]).
For solving the equilibrium problem, we assume that the bifunction f satisfies the following conditions:
(a1) for all ;
(a2) f is monotone, i.e., for all ;
(a3) For every , ;
(a4) is convex and lower semicontinuous for all .
Takahashi and Zembayashi  obtained the following result.
Proposition 3.2 
for all . Then the following hold:
(r1) is single-valued;
(r2) is a firmly nonexpansive-type mapping, i.e., for , ;
(r4) is closed and convex.
We call the resolvent of f for . The following result is a specialized case of the result of Aoyama et al. [, Theorem 3.5].
Then is a maximal monotone operator with and . Furthermore, for , the resolvent of f coincides with the resolvent of .
Using Theorem 3.1 and Proposition 3.3, we get the following result.
where and are sequences in and the sequence is contained in E. If conditions (c1)-(c4) are satisfied, then the sequence converges strongly to .
Remark 3 Corollary 3.4 improves and extends Theorem 3.4 of Chuang et al. .
4 Numerical experiments
In this section, we give some examples and numerical results to illustrate our result in the preceding section.
Remark 4 Figures 1 and 2 show that when an iteration step n is greater than 100 and 60 in Examples 4.1 and 4.2 respectively, the term is close to the desired element. Therefore, our iterative method is effective.
The authors would like to thank reviewers and editors for their valuable comments and suggestions. This work was supported by graduate funds of Beijing University of Technology (no. ykj-2013-9422).
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