Strong Convergence Algorithms of the Split Common Fixed Point Problem for Total Quasi-Asymptotically Pseudocontractive Operators

Abstract

We present a new algorithm for solving the two-set split common fixed point problem with total quasi-asymptotically pseudocontractive operators and consider the case of quasi-pseudocontractive operators. Under some appropriate conditions, we prove that the proposed algorithms have strong convergence. The results presented in this paper improve and extend the previous algorithms and results of Censor and Segal (2009), Moudafi (2011 and 2010), Mohammed (2013), Yang et al. (2011), Chang et al. (2012), and others.

1. Introduction

Let and be nonempty closed convex subsets of real Hilbert spaces and , respectively. Let be a bounded linear operator. To allow for constraints both in the domain and range of , Censor and Elfving [1] originally formulated the split feasibility problem (SFP), which is to find a member of set A recent generalization, due to Censor and Segal in [2], is called the split common fixed point problem (SCFPP), which is to find a point satisfying where () and ( are some nonlinear operators and is also a bounded linear operator. Denote the solution set of SCFPP by

In particular, if , problem (2) is reduced to the two-set SCFPP, where and , and the SFP can be retrieved by picking as operators and orthogonal projections.

Censor and Segal [2] invented the following CQ-algorithm with directed operators to solve the two-set SCFPP: where and ; is the largest eigenvalue of the matrix .

Inspired by the work of Censor and Segal, for , Moudafi presented the following iteration with the demicontractive mappings and quasi-nonexpansive operators in papers [3] and [4], respectively: Moudafi's results are weak convergence. In [5, 6], Mohammed utilized the strongly quasi-nonexpansive operators and quasi-nonexpansive operators to solve recursion (5) and obtain weak and strong convergence, respectively. Strong convergence of (5) with pseudo-demicontractive and firmly pseudo-demicontractive mappings can be found in [7, 8]. Furthermore, for several different strong convergence recursions with nonexpansive operators for solving the SCFPP see [9, 10]. For the purpose of generalization, papers [11–13] discussed the total asymptotically strictly pseudocontractive mappings and asymptotically strict pseudocontractive mappings for solving (2) and multiple-set fixed point problem (MSSFP) by the following iteration: which is of weak convergence; when is semicompact, strong convergence of (6) can be deduced. Obviously, (5) is the particular case of (6). On the other hand, papers [14, 15] presented cyclic algorithms of the SCFPP for directed operators and demicontractive mappings, and the results converge weakly.

However, we found that the strong convergence of (6) needs the condition of to be semicompact. In order to obtain strong algorithm for the two-set SCFPP without more constraints on or and continue to generalize the operators, in this paper, we propose a different iteration, which can ensure the strong convergence with more general case when the operators are total quasi-asymptotically pseudocontractive, demiclosed at the origin. We can choose an initial data arbitrarily and define the sequence by the recursion: where is a -contraction with ,   and are total quasi-asymptotically pseudocontractive mappings, and , , and are three real sequences satisfying appropriate conditions. Under some mild conditions, we prove that the sequence generated by (7) converges strongly to the solution of the two-set SCFPP.

2. Preliminaries

In order to reach the main results, we first recall the following facts.

Let be a nonempty closed and convex subset of a real Hilbert space with the inner product and norm . Denote by the set of fixed points of a mapping ; that is, .

Definition 1 (see [2, 3, 16, 17]). (i) Recalled that is said to be a directed or firmly quasi-nonexpansive operator; if , then
(ii) Let be a closed convex nonempty set of ; is nonexpansive; we say that is attracting with respect to , if, for every , ,
(iii) A mapping is said to be paracontracting or quasi-nonexpansive; if , then
(iv) A mapping is said to be demicontractive or strictly quasi-pseudocontractive; for , there exists a constant such that

Definition 2 (see [11, 18]). (i) Let be a total quasi-asymptotically strictly pseudocontractive if , and there exist a constant , sequences , and with and as such that where is a continuous and strictly increasing function with .
(ii) A mapping is said to be total quasi-asymptotically pseudocontractive if , and there exist sequences and with and as such that where is a continuous and strictly increasing function with .
(iii) A mapping is said to be quasi-pseudocontractive if , such that
(iv) A mapping is said to be uniformly -Lipschitzian if there is a constant , such that

Remark 3. Note that the classes of directed operators and attracting operators belong to the class of paracontracting operators. The class of paracontracting operators belongs to the class of demicontractive operators, while the class of quasi-pseudocontractive operators includes the class of demicontractive operators. Further, the class of total quasi-asymptotically pseudocontractive operators, with quasi-pseudocontractive operators as a special case, includes the class of total quasi-asymptotically strictly pseudocontractive operators.

Remark 4. Let be a total quasi-asymptotically pseudocontractive, if , for each and ; from (13) we can easily obtain the following equivalent inequalities:

Lemma 5 (see [19]). Consider
(i), for all ;
(ii), for all and .

Lemma 6 (see [18]). Let be a bounded and closed convex subset of a real Hilbert space . Let be a uniformly -Lipschitz and total quasi-asymptotically pseudocontractive mapping with . Suppose there exist positive constants and , for the function in (13), for all such that Then is a closed convex subset of .

Lemma 7 (see [20]). A mapping is said to be demiclosed at zero, if for any sequence , such that and as ; then .

Lemma 8 (see [21]). Let , , and be sequences of nonnegative real numbers satisfying If and , then the limit exists.

Lemma 9 (see [22]). Let a sequence satisfy and . Let be a sequence of nonnegative real numbers that satisfies any of the following conditions.(i)For all , there exists an integer such that, for all , (ii),  , where satisfies ;(iii), where .Then .

3. Main Results

In this section, we will prove the strong convergence of (7) to solve the two-set SCFPP.

Theorem 10. Let and be nonempty closed convex subsets of real Hilbert spaces and , respectively. Let be a uniformly -Lipschitz and -total quasi-asymptotically pseudocontractive mapping, a uniformly -Lipschitz, and -total quasi-asymptotically pseudocontractive mappings satisfying the following conditions: , ;, , , and , ; and .
Let be a -contraction with . Let be a bounded linear operator. For , sequence can be generated by the iteration (7), where the sequence satisfies (i) and (ii) , with , and with being the largest eigenvalue of the matrix . Assume that and are demiclosed at zero. If , then generated by (7) converges strongly to a solution of the two-set SCFPP.

Proof. (1) First of all, we show that, for ,   generated by (7) is bounded.
From (7), (16), and Lemma 6, we have
Since substituting (25) into (24), we have where ; by condition (), we know
Next, from (7), (13), and Lemma 5, we can get we also can see that then substituting (29) into (28) and from (26), we have where , and we also know that
From (7) and Lemma 5, we also have
Substituting (30) into (32) and simplifying it we have set (33) can be rewritten as by condition (), (27), and (31), we know that and . Thus it follows from Lemma 8 that the following limit exists: Therefore, we obtain that is bounded, so is . Set . Then is also bounded.
(2) Next we prove , =0.
For each , , assume there exists such that for . Then for all , and by virtue of (16), we have
which implies that Now we take in (38); multiplying and on the two side of (38), respectively, and then adding up, we can obtain Letting in (39), we have
From (7), we know that Letting in (41) and by condition (i) in Theorem 10, we know Similarly, from (40) the limit of exists and
Therefore, when we take limit on both sides of (22), we can deduce that
Then, In view of (40) and (45) we have that Similarly, it follows from (7), (45), and (47) that
(3) Next we prove that , as .
From (40) and (48), we have
By the same way, from (45) and (47) we can also prove that
Therefore, from (44) and (49), we know Since is bounded, there exists a subsequence of which converges weakly to a point . Without loss of generality, we may assume that converges weakly to . Therefore, from (49)–(51) and Lemma 7, we have .
(4) Finally, we prove that in norm. To do this, we calculate Therefore, we have
Substituting (23) into (28), we have Since and substituting (53) into (51), we get Let Equation (55) can be rewritten as Evidently, from (40), (45), and Lemma 9 (ii), we can conclude that .
This completes the proof.

The following theorem can be concluded from Theorem 10 immediately.

Theorem 11. Let and be nonempty closed convex subsets of real Hilbert spaces and , respectively. Let be a uniformly -Lipschitz and quasi-pseudocontractive mapping with . Let be a uniformly -Lipschitz and quasi-pseudocontractive mapping with . Let be a -contraction with . Let be a bounded linear operator. For , sequence can be generated by the iteration: where the sequence satisfies (i) and (ii) , with , and with being the largest eigenvalue of the matrix . Assume that and are demiclosed at zero. If , then generated by (58) converges strongly to a solution of the two-set SCFPP.

Proof. For each , if we take , , and , and follow the proof of Theorem 10, we can also prove that converges strongly to by the same way.

Remark 12. Algorithm (7) and Theorems 10 and 11 improve and extend the corresponding results of Censor and Segal [2], Moudafi [3, 4], Mohammed [5, 6], Chang et al. [11, 13], Yang et al. [12], and others.

4. Concluding Remarks

In this work, we develop the split common fixed point problem with more general classes of total quasi-asymptotically pseudocontractive and quasi-pseudocontractive operators; corresponding algorithms are improved based on the viscosity iteration; thus we can obtain strong convergence without more constraints on operators.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the associate editor and the referees for their comments and suggestions. This research was supported by the National Natural Science Foundation of China (11071053).