Journal of Function Spaces

Volume 2016, Article ID 4093524, 7 pages

http://dx.doi.org/10.1155/2016/4093524

## Strong Convergence Theorems for an Implicit Iterative Algorithm for the Split Common Fixed Point Problem

^{1}School of Mathematics and Statistics, Xidian University, Xi’an 710071, China^{2}Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Received 13 September 2016; Revised 11 November 2016; Accepted 21 November 2016

Academic Editor: Ismat Beg

Copyright © 2016 Huimin He et al. 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.

#### Abstract

The aim of this paper is to construct a novel implicit iterative algorithm for the split common fixed point problem for the demicontractive operators , , and , , where , and we obtain the sequence which strongly converges to a solution of this problem, and the solution satisfies the variational inequality. , , where denotes the set of all solutions of the split common fixed point problem.

#### 1. Introduction

The split feasibility problem (SFP) is to find a pointwhere is a nonempty closed convex subset of a Hilbert space , is a nonempty closed convex subset of a Hilbert space , and is a bounded linear operator.

This problem was proposed by Censor and Elfving [1] in 1994.

Since the SFP can extensively be applied in fields such as intensity-modulated radiation therapy, signal processing, and image reconstruction, then the SFP has received so much attention by so many scholars; see [2–23].

In 1994, Censor and Elfving [1] proposed the original algorithm in ,where and are nonempty closed convex subsets of , in the finite-dimensional is a matrix, and is the projection operator from onto .

As we know, the computation of the inverse is not easy if the inverse of existed. So, the algorithm (2) does not become popular.

In 2002 and 2004, Byrne [2, 3] gave the so-called algorithm as follows:where with taken as the largest eigenvalue of the operator and and denote the projection operators from and onto the sets , , respectively.

For the stepsize of algorithm (3) is fixed and closely related to spectral radius of , then the projection operators and are not easily calculated usually.

The split common fixed point problem (SCFP) is to find a pointwhere and , and and denote the fixed point sets of and .

This problem was proposed by Censor and Segal [12] in 2009. Note that the SCFP is closely related to SFP and it is a particular case of SFP.

In 2009, Censor and Segal [12] introduced the original algorithm for directed operators as follows:where the step size satisfies , and they obtained that weakly converges to a solution of the SCFP (4) if the solution of SCFP exists. But it is obvious that the choice of the step size depends on the norm of operator, , which is the disadvantage of this algorithm.

The next two years, some extension results on the operators are obtained, such as Moudafi (2010) [24], Moudafi (2011) [25], and Wang and Xu (2011) [14].

In order to overcome this disadvantage, Cui and Wang [26] proposed the following algorithm in 2014:where and the step size is chosen by the following way:and they proved that the sequence converges weakly to a solution of the SCFP (4). Note that the advantage of this algorithm is that the step size searches automatically and does not depend on the norm of operator .

Recently, Byrne et al. [27] introduced the split common null point problem (SCNPP) for set-valued maximal monotone mappings in Hilbert spaces. Given set-valued mappings , , and , , respectively, and the bounded linear operators , , the SCNPP is formulated as follows:As we know, the SCNPP generalizes the split common fixed point problem and the split variational inequality problem [28, 29].

Motivated by the viscosity idea of [30], in this paper, we construct a novel algorithm for demicontractive operators to approximate the solution of the SCFP (4), that is, the following implicit iterative algorithm:where and the step size is also chosen as (7).

The research highlight of this paper is that the strong convergence of the SCFP (4) is constructed; that is to say the sequence generated by (9) converges strongly to a solution of the SCFP.

#### 2. Preliminaries

Throughout this paper, we denote the set of all solutions of the SCFP (4) by . We use to indicate that converges weakly to . Similarly, symbolizes the sequence which converges strongly to .

Let , , and be Hilbert spaces endowed with the inner product and norm , and and are nonempty closed convex subsets of and , respectively.

Some concepts and lemmas are given in the following and they are useful in proving our main results.

*Definition 1. *A operator is said to be

(i) nonexpansive if(ii) quasi-nonexpansive if(iii) directed if(iv) -demicontractive with if

Note that (12) is equivalent to

*Definition 2. *Let be an operator, then is said to be demiclosed at zero, if for any in , the following implication holds

As we know, the nonexpansive mappings are demiclosed at zero [31].

*Definition 3. *Let be a nonempty closed convex subset of a Hilbert space , the metric (nearest point) projection from to is defined as follows: Given , is the only point in with the property

Lemma 4 (see [32]). *Let be a nonempty closed convex subset of a Hilbert space , is a nonexpansive mapping from onto and is characterized as: Given , there holds the inequality *

Lemma 5 (see [32]). *Let be a Hilbert space, then the following inequality holds,*

Lemma 6 (Cui and Wang [26]). *Let be a bounded linear operator and a -demicontractive operator with . If , then*(a)*, .*(b)*In addition, for * *where , and *

*Lemma 7 (Cui and Wang [26]). Let be a -demicontractive operator with . Denote for . Then for any and ,*

*3. Main Results*

*Proposition 8. Based on the definitions in preliminaries, the classes of -demicontractive operators, directed operators, quasi-nonexpansive operators, and nonexpansive operators have close relations. We can visually use the following Venn diagram (Figure 1) to denote their relations.*