On Strong Convergence of Halpern’s Method Using Averaged Type Mappings

Abstract

Under suitable hypotheses on control coefficients, we study Halpern’s method to approximate strongly common fixed points of a nonexpansive mapping and of a nonspreading mapping or a fixed point of one of them. A crucial tool in our results is the regularization with the averaged type mappings.

1. Introduction

Let be a real Hilbert space with the inner product , which induces the norm .

Let be a nonempty, closed, and convex subset of . Let be a nonlinear mapping of into itself; we denote by the set of fixed points of , that is, .

We recall that a mapping is said to be nonexpansive if

The problem of finding fixed points of nonexpansive mappings has been widely investigated by many authors.

Halpern [1] was the first to consider the following explicit method: where and is fixed.

Moreover, Halpern proved in [1] the following theorem on the convergence of (2) for a particular choice of .

Theorem 1. Let be a bounded, closed, and convex subset of Hilbert space and let be a nonexpansive mapping. For any initialization and anchor , define a sequence in by where . Then converges strongly to the element of nearest to .

He also showed that the control conditions,  , ,

are necessary for the convergence of (2) to a fixed point of .

Subsequently, several authors carefully studied the following problem: are the control conditions and sufficient for the convergence of (2)?

In this direction, C. E. Chidume and C. O. Chidume [2] and Suzuki [3], independently, proved that the conditions and are sufficient to assure the strong convergence to a fixed point of of the following iterative sequence: Recently, in the setting of Banach spaces, Song and Chai [4], under the same conditions and but under stronger hypotheses on the mapping, obtained strong convergence of Halpern iterations (2). In particular, they assumed that is a real reflexive Banach space with a uniformly Gateâux differentiable norm and with the fixed point property for nonexpansive self-mappings, and considered an important subclass of nonexpansive mappings which is the firmly type nonexpansive mappings.

Let be a mapping with domain . is said to be firmly type nonexpansive [4] if for all , there exists such that A more general class of firmly type nonexpansive mappings is the class of the strongly nonexpansive mappings. Recall that a mapping is said to be strongly nonexpansive if (1) is nonexpansive;(2), whenever and are sequences in such that is bounded and .

Saejung [5] proved the strong convergence of Halpern’s iterations (2) for strongly nonexpansive mappings in a Banach space such that one of the following conditions is satisfied:(i) is uniformly smooth;(ii) is reflexive, strictly convex with a uniformly Gateâux differentiable norm.

In the setting of Hilbert spaces, Kohsaka and Takahashi [6] defined a nonspreading mapping if The following Lemma is a useful characterization of nonspreading mapping.

Lemma 2 (see [7]). Let be a nonempty closed subset of Hilbert space . Then, a mapping is nonspreading if and only if

Observe that if is a nonspreading mapping from into itself and , then is quasi-nonexpansive; that is,

Further, the set of fixed points of a quasi-nonexpansive mapping is closed and convex [8].

Osilike and Isiogugu [9] studied Halpern’s type for -strictly pseudononspreading mappings , which are a more general class of the nonspreading mappings.

To obtain the strong convergence of (2), they replaced the mapping with the averaged type mapping , that is, with the mapping Iemoto and Takahashi [7] approximated common fixed points of a nonexpansive mapping and of a nonspreading mapping in a Hilbert space using Moudafi’s iterative scheme [10]. They obtained the following Theorem that states the weak convergence of their iterative method.

Theorem 3. Let be a Hilbert space and let be a nonempty, closed, and convex subset of . Assume that . Define a sequence as follows: for all , where , . Then, the following hold. (i)If and , then converges weakly to .(ii)If and , then converges weakly to .(iii)If and , then converges weakly to .

In this paper, inspired by Iemoto and Takahashi [7], we introduce an iterative method of Halpern’s type involving the averaged type mappings and , where is a nonexpansive mapping and is a nonspreading mapping. The averaged type mappings and have a regularizing role in order to prove the strong convergence of our iterative scheme. In particular, we prove that the method strongly converges to the unique solution of the variational inequality where is an anchor and, depending on the hypotheses on control coefficients, is the set of fixed points of , the set of fixed points of , or the set of common fixed points of and .

Suitable tools in our proofs are Maingé’s Lemma [11] and some techniques used by Maingé in [12] to study the strong convergence of the viscosity approximation method. However, Wongchan and Saejung [13] found a small mistake in Maingé’s proof.

2. Preliminaries

To begin, we collected some lemmas which we will use in our proofs in the next section.

In the sequel, we denote by a real Hilbert space and by a nonempty closed convex subset of .

Lemma 4. The following known results hold:  (1), for all and for all , (2), for all .

We recall that for every point , there exists a unique nearest point in , denoted by , such that Such is called the metric projection of onto .

Lemma [14] characterizes the projection .

Lemma 5. Let be a closed and convex subset of a real Hilbert space and let be the metric projection from onto . Given that and , then if and only if there holds the inequality

By Lemma 5, if is fixed, is the unique solution of the variational inequality (13).

To prove our main theorem, we need some fundamental properties of the involved mappings in the variational inequality.

The following result summarizes some significant properties of if is a nonexpansive mapping ([15, 16]).

Lemma 6. Let be a nonempty closed convex subset of and let be nonexpansive. Then,  (1) is -inverse strongly monotone, that is,  for all ; (2)moreover, if , is demiclosed at ; that is, for every sequence weakly convergent to such that as , it follows .

Iemoto and Takahashi showed the demiclosedness of at and a suitable property of . These results are summarized in the following two Lemmas.

Lemma 7 (see [7]). Let be a nonempty, closed, and convex subset of . Let be a nonspreading mapping such that . Then, is demiclosed at .

Lemma 8 (see [7]). Let be a nonempty, closed, and convex subset of . Let be a nonspreading mapping. Then, for all .

If is a nonexpansive mapping of into itself, Byrne [15] defined the averaged mapping as follows: where .

Moreover, Byrne [15] and successively Moudafi [17] proved some properties of the averaged mappings; in particular, they showed that is a nonexpansive mapping. In this paper, inspired by [15, 17], we define the averaged type mapping as in (16) for a nonlinear mapping ; we notice that . It is easy to verify that if is a nonspreading mapping of into itself and , the averaged type mapping is quasi-nonexpansive and consequently the set of fixed points of is closed and convex.

Actually, it follows from [9] that is quasi-firmly type nonexpansive mapping; that is, it is a firmly type nonexpansive mapping (5) on fixed points of . For completeness we include the easy proof.

Proposition 9. Let be a nonempty closed and convex subset of and let be a nonspreading mapping such that is nonempty. Then the averaged type mapping is quasi-firmly type nonexpansive mapping with coefficient .

Proof. We obtain Hence, we have In particular, choosing in (19) , where we obtain

The following lemma is useful in the proof of our main result.

Lemma 10. Let be a nonempty closed and convex subspace of , fixed, a nonexpansive mapping from into itself, and a nonspreading mapping from into itself such that . Consider a bounded sequence . Then,  (1)if , as , then  where is the unique point in that satisfies the variational inequality  (2)If , as , then  where is the unique point in that satisfies the variational inequality  (3)If and , as , then  where is the unique point in that satisfies the variational inequality

Proof. Let satisfy (22). Let be a subsequence of for which Select a subsequence of such that (this is possible by boundedness of ). By the hypothesis , as , and by demiclosedness of at we have and
so the claim follows by (22).
The proof is the same of since also is demiclosed in .
Select a subsequence of such that where satisfies (26). Now select a subsequence of such that . Then by demiclosedness of and at and by the hypotheses and , as , we obtain that , that is, . So, so the claim follows by (26).

A pertinent tool for us is the well-known lemma of Xu [18].

Lemma 11. Let be a sequence of nonnegative real numbers satisfying the following relation: where,(i), ;(ii);(iii), .Then,

Finally, a crucial tool for our results is the following lemma proved by Maingé.

Lemma 12 (see [11]). Let be a sequence of real numbers such that there exists a subsequence of such that , for all . Consider the sequence of integers defined by Then, is a nondecreasing sequence for all , satisfying (i);(ii), ;(iii), .

3. Main Result

In all sections we denote by a nonexpansive mapping, a nonspreading mapping, and the respectively averaged type mappings. Moreover, denotes a real sequence and denotes the convex combination of and , that is, Further we assume that (i);(ii) a real sequence such that and ;(iii) is any bounded real sequence.We start with the following lemma:

Lemma 13. Let be an anchor and let be the sequence defined by where Then(1) is quasi-nonexpansive for all ;(2), , , , , are bounded sequences.

Proof. Any convex combination of quasi-nonexpansive mappings is quasi-nonexpansive too. So is every , since and are quasi-nonexpansive.
The boundedness of follows the fact that is quasi-nonexpansive. In fact, let . Then Since and by induction we assume that and then Thus is bounded. The boundedness of the other sequences follows by boundedness of and by the quasi-nonexpansivity of involved mappings.

Now, we prove our strong convergence theorem.

Theorem 14. Let be a Hilbert space and let be a nonempty closed and convex subset of . Let be a nonexpansive mapping and let be a nonspreading mapping such that . Let and be the averaged type mappings. Suppose that is a real sequence in satisfying the conditions (1),(2).If is a sequence in , we define a sequence as follows: Then, the following hold. (i)If , then strongly converges to which is the unique solution in of the variational inequality , for all .(ii)If , then strongly converges to which is the unique solution in of the variational inequality , for all .(iii)If , then strongly converges to which is the unique solution in of the variational inequality , for all .

Proof. (i) We rewrite the sequence as where is bounded, that is, .
We begin to prove that .
Let the unique solution in of the variational inequality We have and hence We turn our attention to the monotony of the sequence .
We consider the following two cases.
Case  A. is definitively nonincreasing.
Case  B. There exists a subsequence such that
Case  A. Since is definitively nonincreasing, exists. From (45), , and , we have so, we can conclude that
By Lemma 10, it follows that
Finally, we prove that converges strongly to .
We compute that
If we put and , we have
Hence, from assumption and , from (49) and we can apply Xu’s Lemma 11.
Case  B. There exists a subsequence such that
Then by Maingé Lemma 12 there exists a sequence of integers that satisfies (a) is nondecreasing;(b);(c);(d).
Consequently, so By (45), we have and from (55), and we get By Lemma 10 and (57) we have Finally, we show that converges strongly to .
As in Case , we can obtain then, from property of Maingé Lemma 12 and (55) we can conclude

Proof. (ii) Now, we rewrite the sequence as where is bounded, that is, .
We begin to prove that .
Let the unique solution in of the variational inequality , for all . We have and hence Again, we turn our attention to the monotony of the sequence . We consider the following two cases.
Case  A. is definitively nonincreasing.
Case  B. There exists a subsequence such that
Case  A. Since is definitively nonincreasing, exists. From (45), , and , we have and hence By Lemma 10, it follows that Finally, we prove that converges strongly to .
We compute that
If we put and , we have So, from assumption and , from (67) and we can apply Xu’s Lemma 11.
Case  B. There exists a subsequence such that
Then by Maingé Lemma there exists a sequence of integers that satisfies (a) is nondecreasing;(b);(c);(d).
Consequently, so By (63), we obtain and from (73), and we get By Lemma 10 and (75) we get Finally, we show that converges strongly to .
As in Case  , we obtain and then from property of Maingé Lemma and (73) we can conclude that

Proof. (iii) We recall that the sequence is defined as where .
We first show that .
Let be the unique solution of the variational inequality , for all . We compute that So, we get We have From (82), we derive Now, also we consider two cases.
Case  A. is definitively nonincreasing.
Case  B. There exists a subsequence such that
Case  A. Since is definitively nonincreasing, exists. From (83) and and since we conclude that Furthermore, from (84) we have since by (86) and (87) we obtain
Then, since by (86) and (89) we get By Lemma 10, we have Finally, converges strongly to .
We compute that If we set , we have From , (91), and we conclude that strongly converges to .
Case  B. is not definitively nonincreasing. This means that there exists a subsequence such that Then by Maingé Lemma 12 there exists a sequence of integers that satisfies some properties defined previously.
Consequently, and hence By (83) we get and by(84) we have As in Case  , we get(a),(b),(c).
By Lemma 10, (a) and (b) we have Finally, we prove that converges strongly to .
As in Case  , using , the assumption , and (100) we can apply Xu’s Lemma 11 and conclude that and then from property of Maingé Lemma and (97) we can derive that

Remark 15. We remark that in order to prove that we do not follow the line of the proof of Song and Chai in [4] because their techniques seem questionable.

Remark 16. The main result of this paper gives a positive answer to the question of Kurokawa and Takahashi; see Remark page   in [19].

Conflict of Interests

The authors declare that they have no competing interests.