Viscosity approximation process for a sequence of quasinonexpansive mappings
© Aoyama and Kohsaka; licensee Springer. 2014
Received: 27 September 2013
Accepted: 18 December 2013
Published: 22 January 2014
We study the viscosity approximation method due to Moudafi for a fixed point problem of quasinonexpansive mappings in a Hilbert space. First, we establish a strong convergence theorem for a sequence of quasinonexpansive mappings. Then we employ our result to approximate a solution of the variational inequality problem over the common fixed point set of the sequence of quasinonexpansive mappings.
MSC:47H09, 47H10, 41A65.
for , where is a real number in , is a contraction-like mapping on C, and is a quasinonexpansive mapping on C. This iterative method (1.1) is called the viscosity approximation method . In Section 3, we establish that, under some appropriate assumptions, the sequence converges strongly to a certain common fixed point of by using the technique developed in . Then, in Section 4, we apply our result to approximate a solution of a variational inequality problem over the common fixed point set of .
The viscosity approximation method (1.1) is based on the study of Moudafi , who considered a fixed point problem of a single nonexpansive mapping and proved strong convergence of sequences generated by the method. After that, Xu  extended Moudafi’s results  in the framework of Hilbert spaces and Banach spaces; Suzuki  gave simple proofs of Xu’s results ; Aoyama and Kimura  investigated a relationship between viscosity approximation methods and Halpern  type iterative methods for a sequence of nonexpansive mappings.
On the other hand, Maingé  adopted the viscosity approximation method for a fixed point problem of a single quasinonexpansive mapping; Wongchan and Saejung  extended Maingé’s result . Our main result (Theorem 3.1) is a generalization of Wongchan and Saejung’s result  and is closely related to the study in . Moreover, it is also applicable to an approximation method, which is called the hybrid steepest descent method [9, 10], for a variational inequality problem over the common fixed point set of a sequence of quasinonexpansive mappings.
Throughout the present paper, H denotes a real Hilbert space, the inner product of H, the norm of H, C a nonempty closed convex subset of H, I the identity mapping on H, ℝ the set of real numbers, and ℕ the set of positive integers. Strong convergence of a sequence in H to is denoted by and weak convergence by .
Let be a mapping. The set of fixed points of T is denoted by . A mapping T is said to be quasinonexpansive if and for all and ; T is said to be nonexpansive if for all ; T is said to be strongly quasinonexpansive if T is quasinonexpansive and whenever is a bounded sequence in C and for some point ; T is demiclosed at 0 if whenever is a sequence in C such that and . We know that if is quasinonexpansive, then is closed and convex; see [, Theorem 1].
Such a point is denoted by and is called the metric projection of H onto C. It is known that the metric projection is nonexpansive; see .
Let be a mapping, F a nonempty subset of C, and θ a real number in . A mapping f is said to be a θ-contraction with respect to F if for all and ; f is said to be a θ-contraction if f is a θ-contraction with respect to C. By definition, it is easy to check the following results.
Lemma 2.1 Let F be a nonempty subset of C and a θ-contraction with respect to F, where . If F is closed and convex, then is a θ-contraction on F, where is the metric projection of H onto F.
Lemma 2.2 Let be a θ-contraction, where and a quasinonexpansive mapping. Then is a θ-contraction with respect to .
Let D be a nonempty subset of C. A sequence of mappings of C into H is said to be stable on D  if is a singleton for every . It is clear that if is stable on D, then for all and .
A function is said to be eventually increasing  if and for all . By definition, we easily obtain the following.
Lemma 2.3 Let be an eventually increasing function and a sequence of real numbers such that . Then .
The following is a direct consequence of [, Lemma 3.1].
Lemma 2.4 ([, Lemma 3.4])
Let be a sequence of nonnegative real numbers which is not convergent. Then there exist and an eventually increasing function such that for all and for all .
Under the assumptions of Lemma 2.4, we cannot choose a strictly increasing function τ; see [, Example 3.3].
Let be a sequence of mappings of C into H such that is nonempty. Then
is said to be strongly quasinonexpansive type if each is quasinonexpansive and whenever is a bounded sequence in C and
for some point ;
Remark 2.5 Since if and only if for all bounded sequences and in , is strongly quasinonexpansive type if and only if it is a strongly relatively nonexpansive sequence in the sense of [2, 17]. See also [18, 19].
We know several examples of strongly quasinonexpansive type sequences satisfying the condition (Z); see  and Example 4.5 in Section 4.
The following lemma follows from [, Lemma 3.5] and Remark 2.5.
Lemma 2.6 Let be a sequence of mappings of C into H such that is nonempty, an eventually increasing function, and a bounded sequence in C such that for some . If is strongly quasinonexpansive type, then .
In order to prove our main result in Section 3, we need the following lemmas.
Lemma 2.7 ([, Lemma 3.6])
Let be a sequence of mappings of C into H such that is nonempty, an eventually increasing function, and a bounded sequence in C such that . Suppose that satisfies the condition (Z). Then every weak cluster point of belongs to F.
Lemma 2.8 ([, Lemma 3.7])
Lemma 2.9 Let be a sequence of nonnegative real numbers, a sequence of real numbers, and a sequence in . Suppose that for every , , and . Then .
3 Strong convergence of a viscosity approximation process
In this section, we prove the following strong convergence theorem.
for . Suppose that is strongly quasinonexpansive type and satisfies the condition (Z). Then converges strongly to , where w is the unique fixed point of a contraction .
Note that Lemma 2.1 implies that is a θ-contraction on F and hence it has a unique fixed point on F.
hold for every .
Therefore, it turns out that and are bounded, and moreover, is also bounded.
Equation (3.2) follows from (3.4).
for every . Therefore, (3.3) holds. □
Lemma 3.3 The following hold:
for every ;
It follows from that .
it follows that .
Finally, we prove . Suppose that . Then it is clear that for every . Thus, . Next, we suppose that . Then it is clear that for every . Thus, . This completes the proof. □
Lemma 3.4 is convergent.
for every .
as . Since is bounded and is strongly quasinonexpansive type, Lemma 2.6 implies that .
for every .
and hence , which is a contradiction. □
for every , so that by Lemma 3.4, , and Lemma 3.2. Since is strongly quasinonexpansive type and is bounded, we conclude that .
for every . Therefore, noting that and using Lemma 2.9, we conclude that . □
A direct consequence of Theorem 3.1 is the following corollary, which is a slight generalization of [, Theorem 2.3].
for . Suppose that is demiclosed at 0. Then converges strongly to , where w is the unique fixed point of a contraction .
Proof Set and for . Then it is clear that , is strongly quasinonexpansive type, satisfies the condition (Z), and is stable on C. Thus Theorem 3.1 implies the conclusion. □
4 Application to a variational inequality problem
In this section, applying Theorem 3.1, we study an approximation method for the following variational inequality problem.
The solution set of Problem 4.1 is denoted by . Under the assumptions of Problem 4.1, it is known that the following hold; see, for example, .
, and is a θ-contraction, where ;
Problem 4.1 has a unique solution and .
Remark 4.2 The assumption that in Problem 4.1 is not restrictive. Indeed, let F be a nonempty closed convex subset of H and a -strongly monotone and -Lipschitz continuous mapping, where and . Set , , and , where μ is a positive constant such that . Then it is easy to verify that A is κ-strongly monotone and η-Lipschitz continuous, , and moreover, .
Using Theorem 3.1, we obtain the following convergence theorem for Problem 4.1.
for . Suppose that is strongly quasinonexpansive type and satisfies the condition (Z). Then converges strongly to the unique solution of Problem 4.1.
for . Thus Theorem 3.1 implies that converges strongly to , which is the unique solution of Problem 4.1. □
We finally construct an example of in Theorem 4.3 by using the notion of a subgradient projection.
(S2) for all and ;
(S3) if is bounded for each bounded subset V of H, then is demiclosed at 0.
and is not nonexpansive.
Using (S1), (S2), and (S3), we show the following.
for all ;
is strongly quasinonexpansive type;
satisfies the condition (Z).
Proof Since for all , the part (i) obviously follows from (S1).
for all and . Since , we know that each is quasinonexpansive.
and hence by . Thus is strongly quasinonexpansive type.
for all . Since , we obtain . Consequently, by (S1) and (S3), we know that satisfies the condition (Z). □
This paper is dedicated to Professor Wataru Takahashi on the occasion of his 70th birthday.
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