Strong Convergence for the Split Common Fixed-Point Problem for Total Quasi-Asymptotically Nonexpansive Mappings in Hilbert Space

Abstract

In this paper, we study and modify the algorithm of Kraikaew and Saejung for the class of total quasi-asymptotically nonexpansive case so that the strong convergence is guaranteed for the solution of split common fixed-point problems in Hilbert space. Moreover, we justify our result through an example. The results presented in this paper not only extend the result of Kraikaew and Saejung but also extend, improve, and generalize some existing results in the literature.

1. Introduction

Let be an inner product space, the corresponding norm, a Banach space, , two Hilbert spaces, a bounded linear operator, and an adjoint of . Let and be a nonempty, closed, convex subsets of and , respectively.

A Banach space is said to satisfy Opial’s condition (see [1]) if, for any sequence in , as implies that And also, a Banach space is said to have Kadec-Klee property (see [1]), if, for any sequence in , and as implies that

Remark 1. It is well known that each Hilbert space satisfied Opial and Kadec-Klee property.

The mapping is said to be demiclosed at zero, if any sequence in there holds the following implication: is said to be -strongly quasi-nonexpansive if there exists with the property ,   and ; this is equivalent to is said to be quasi-nonexpansive, if such that ,   and , and -quasi-asymptotically nonexpansive mapping, if and there exists a sequence with such that, for each , ,   and , and it is said to be -total quasi-asymptotically nonexpansive mapping if ; and there exist nonnegative real sequences , in with and and a strictly continuous function with such that, for each ,

Remark 2. It is known that, the class of quasi-nonexpansive mapping contained in the class of -quasi-asymptotically nonexpansive mapping and the class of -quasi-asymptotically nonexpansive mapping is contained in the class of -total quasi-asymptotically nonexpansive mapping, see [2].

The mapping is said to be uniformly -Lipschitzian if a constant such that, for each , ,  , , and it is said to be semicompact, if, for any bounded sequence with , there exists subsequence such that converges strongly to some point .

The convex feasibility problems (CFP) are finding a vector satisfying The problem of solving 6 has been intensively studied by numerous authors due to its various application in several physical problems such as approximation theorem, image recovery, signal processing, control theory, biomedical engineering, communication and geophysics (see [3–5]) and reference therein.

In 2005, Censor et al. (see [6]) introduced and studied the problem of multiple set split feasibility problems (MSSFP) which is formulated as finding a vector with the property If, in 7, we take , we get Equation 8 is known as the split feasibility problems (SFP) (see [7]), where and are nonempty, closed, and convex subsets of and , respectively. Since every closed convex subset of Hilbert space is the fixed-point set of its associating projection, then 6 and 7 become Equations 9 and 10 are called the common fixed-point problems (CFPP) and split common fixed-point problems (SCFPP), respectively, where   () and   () are some nonlinear operators.

If we take , problem 10 is reduced to find a point with property Equation 11 is known as the two-set SCFPP.

In 2009, Censor and Segal [8] introduced the concept of SCFPP 10 in finite dimensional Hilbert space, who invented an algorithm for solving 11 which generate a sequence according to the following iterative procedure: where the initial guess is choosing arbitrarily and .

In 2011, Moudafi [9] studied the convergence properties of relaxed algorithm for solving 10 for the class of quasi-nonexpansive operators such that is demiclosed at zero and he obtained the weak convergence results. Note that, in finite dimensional Hilbert space, weak and strong convergence are equivalent. Differently, in infinite dimensional cases, they are not the same. Moudafi’s results guarantee only weak convergence results. In 2013, Mohammed [10, 11] utilized the strongly quasi-nonexpansive operators and quasi-nonexpansive operators to solve Moudafi’s algorithm and he obtained weak and strong convergence results, respectively.

In 2014, Kraikaew and Saejung [12] also modified Moudafi’s algorithm [9] and they obtained the strong convergent results as shown below.

Theorem 3 (see [12]). let be a strongly quasi-nonexpansive operator and let be a quasi-nonexpansive operator such that both and are demiclosed at zero. Let be a bounded linear operator with . Suppose that . Let be a sequence generated by where the parameters and satisfy the following conditions: (a); (b) and . Then .

Motivated by these results, in this paper, we studied and modified the algorithm of Kraikaew and Saejung [12] for the class of total quasi-asymptotically nonexpansive mappings to solve the split common fixed-point problems 10 in the frame work of infinite dimensional Hilbert space. The results presented in this paper not only improve and extend some recent results of Kraikaew and Saejung [12], but also improve and extend some recent results of Censor and Segal [8], Moudafi [9], and Mohammed [10, 11] and many existing results.

Throughout this paper, we adopt the following notations.(i) is the identity operator.(ii) is the fixed-point set of ; that is, .(iii)“” and “” denote the strong and weak convergence, respectively.(iv) denote the set of the cluster point of in the weak topology, that is, .(v) is the solution set of split common fixed-point problems 10; that is,

2. Preliminaries

In the sequel, we will make use of the following lemmas in proving our main results.

Lemma 4 (see [2]). Let be a -total quasi-asymptotically nonexpansive mapping. Then for each and , the following inequalities are equivalent: for each

Lemma 5 (see [2]). Let , , be a sequences of nonnegative real numbers satisfying If and , then exists.

Lemma 6. Let be a Hilbert space, a nonempty closed convex subset of , and a metric projection of onto satisfying , and then ,  .

Proof. Let ; then

Lemma 7 (see [5]). If a sequence is Fejer monotone with respect to nonempty closed convex subset , then the following hold:(i) if and only if ;(ii)the sequence converges strongly to some point in ;(iii)if , then

3. Main Results

Theorem 8. Let , be two Hilbert spaces and , be , -total quasi-asymptotically nonexpansive mappings and uniformly -Lipschitzian continuous mappings such that and are both demiclosed at zero. Let be a bounded linear operator and let be an adjoint of with . Let and be positive constants such that ,  . Assume that the solution set of SCFPP 14 is nonempty, and let be a metric projection of onto satisfying . Define the sequence by where the parameter , , , , , and satisfy the following conditions: (a), , where ;(b), , and .Then, the sequence defined by 18 converges strongly to .

Proof. To show that as , it suffices to show and as .
The proof is divided into five steps as follows.
Step 1. In this step, we show that, for each , the following limit exists: Let ; this implies that and . From 18 and Lemma 4, we have On the other hand, by Lemma 4, it follows that By substituting 22 and 23 into 21, we obtained Substituting 24 into 20 and then simplifying, we have and from 25, we deduce that Therefore, from 26, we have where Clearly, and . Moreover, and as .
By Lemma 5, we conclude that exists.
We now prove that, for each , exists.
From 25, we deduce that From 29, we deduce that From 24, 30, and the fact that exists, then exists. Moreover, from 20, we deduce that

Step 2. In this step, we show that

Proof. It follows from 18 that and in view of 30, we obtain that Similarly, it follows from 30 and 34 that

Step 3. In this step, we show that

Proof. From the fact that and and is uniformly -Lipschitzian continuous, it follows that Similarly, from the fact that , , and is uniformly -Lipschitzian continuous, it follows that .

Step 4. In this step, we show that

Proof. Since is bounded, then there exists a subsequence such that From 39 and 36, we have From 39 and 40 and the fact that is demiclosed at zero, we get that .
Moreover, from 18, 39, and the fact , as , we have By the definition of , we get In view of 36, we get From 42 and 43 and the fact that is demiclosed at zero, we have , and this implies that .
Now, we show that is unique.
Suppose to the contrary that there exists another subsequence such that with by virtue of 19 and opial property of Hilbert space; we have which is contradiction. Therefore . By using 18 and 30, we have

Step 5. In this step, we show that To show 46, it suffices to show that as .

Proof. From Lemmas 6, 7, 27, and the fact that and , we have From 38 and 46, we conclude that , as .

Corollary 9. Let , , and , be as in Theorem 8. Let and be quasi-asymptotically nonexpansive and uniformly -Lipschitzians continuous mappings such that and are both demiclosed at zero. Let , and let and be constants such that ,  . Assume that the solution set of SCFPP 14 is nonempty, and let be a metric projection of onto satisfying . Let the sequence be defined as in Theorem 8 where the parameters , , , and satisfy the following conditions: (a), where ;(b). Then the sequence defined as in Theorem 8 converges strongly to .

Proof. By Remark 2   and are -total quasi-asymptotically nonexpansive mappings with , , and , . Therefore, all the conditions in Theorem 8 are satisfied. The conclusions of this corollary follow directly from Theorem 8.

Corollary 10. Let , , and be as in Theorem 8. Let and be two quasi-nonexpansive and uniformly -Lipschitzian continuous mappings such that and are both demiclosed at zero. Let , and let and be positive constants such that ,  . Assume that the solution set of SCFPP 14 is nonempty, and let be a metric projection of H onto satisfying . Let the sequence be defined as in Theorem 8 where the parameters , , and , satisfy the following conditions:(a), , where .Then, the sequence defined as in 18 converges strongly to .

Proof. By Remark 2   and are -quasi-asymptotically nonexpansive mappings. Therefore, all the conditions in Corollary 9 are satisfied. The conclusions of this corollary follow directly from Corollary 9.

Now we give an example of our theorem.

Example 11. Let be a unit ball in a real Hilbert space , and let be a mapping define by where is a sequence in such that .
It is proved in Goebel and Kirk [13] that(a), (b)   and .Let such that , for ; then Let ,  , let ,  , let and be a nonnegative real sequence such that as . From (a), (b) and  , and , we have Again, since and , this implies that . From the above equation, we have This show that is total quasi-asymptotically nonexpansive mapping.

Example 12. Let be a real Hilbert spaces, two unit balls in , and , two mappings defined by such that and are both demiclosed at zero, where and are sequences in such that . Let , , , , and be as in Theorem 8. And assume that conditions (a)–(b) in Theorem 8 are satisfied. Then, the sequence defined in Theorem 8 converges strongly to .

Proof. By Example 11, it follows that and are both -total quasi-asymptotically nonexpansive mappings; moreover from Example 11 (b) we have that and are both uniformly -Lipschitzian with and ; also by our hypothesis and are both demiclosed at zero. Therefore, all the conditions in Theorem 8 are satisfied. Hence, the conclusions of this corollary follow directly from Theorem 8.

Conflict of Interests

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

Acknowledgment

The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under GP-IBT Grant Scheme having project number GP-IBT/2013/9420100.