On the Strong Convergence of Viscosity Approximation Process for Quasinonexpansive Mappings in Hilbert Spaces
We improve the viscosity approximation process for approximation of a fixed point of a quasi-nonexpansive mapping in a Hilbert space proposed by Maingé (2010). An example beyond the scope of the previously known result is given.
Let be a real Hilbert space with inner product and the induced norm . In this paper, we denote the strong and weak convergence by and , respectively. For a subset of , a mapping is said to be nonexpansive if for all ; and it is quasinonexpansive if its fixed-point set is nonempty and for all and . It is clear that every nonexpansive mapping with a nonempty fixed-point set is quasinonexpansive, but the converse is not true. The process for approximation of a fixed point of a nonexpansive or quasinonexpansive mapping is one of interesting problems in mathematics and it has been investigated by many researchers. One of the effective processes for this problem is given by Moudafi . Let be a closed convex subset of , and is a nonexpansive mapping with a nonempty fixed-point set . Moudafi proposed the following scheme which is known as Moudafi's viscosity approximation process: where is a contraction; that is, there exists an such that for all and is a sequence in satisfying (M1),(M2),(M3).
It was also proved that converges to an element satisfying the following inequality: for all .
In the literature, Moudafi's scheme has been widely studied and extended (see [2–5] and references therein). For example, Xu  improved this result to a Banach space. The interesting improvement of this result given by Maingé  is our starting point. His result is given below.
Theorem 1.1. Let be a closed convex subset of a Hilbert space , and is a quasinonexpansive mapping such that is demiclosed at zero, that is, whenever is a sequence in such that and . Suppose that is a contraction. Let be a sequence in defined by where , is an identity mapping, and is a sequence in satisfying (C1), (C2). Then the sequence converges to an element and the following inequality holds for all .
It should be noted that Maingé's result is more widely applicable than Moudafi's. However, after a careful reading, we find that there is a small mistake in Maingé's proof. The following fact (see [7, Remark 2.1(i3)]) is used: if is quasinonexpansive and where , then for all and . Note that the inequality above is equivalent to But this fails; for example, let us consider the nonexpansive mapping defined by for all . It is clear that and .
Recall the following identities in a Hilbert space : for , (i); (ii).
The correction of Maingé's result is as follows.
Proposition 1.2. Let be a subset of a Hilbert space and be a mapping with a nonempty fixed-point set . Suppose that where . Then is quasinonexpansive if and only if for all and .
Proof. Notice that and On the other hand, Hence
Remark 1.3. Unfortunately, this effects the main result (see [7, Theorem 3.1]) in Maingé's paper. More precisely, inequality (32) of its proof (page 78, line 22) should read rather than Therefore, Theorem 1.1 above is valid for only under the same technique.
The purpose of this paper is to simultaneously present a correction of the proof of Theorem 1.1 which is valid for all , and extend his scheme to a wider class of mappings including average mappings, that is, mappings of the form . Our result is more general than Maingé's theorem. An example of a quasinonexpansive mapping which is not applied by Maingé's theorem but applied by our result is given.
First, let us recall some lemmas which are needed for proving the main result.
Lemma 2.1 (see [8, Lemma 2.3]). Let be a sequence of nonnegative real numbers, a sequence of with , a sequence of nonnegative real numbers with , and a sequence of real numbers with . Suppose that for all . Then .
The following nice result was proved by Maingé (see [7, Lemma 2.1]).
Lemma 2.2. Let be a sequence of nonnegative real numbers. If there exists a subsequence of such that for all , then there exists a subsequence of such that for all .
For a closed convex subset of a Hilbert space , the metric projection is defined for each as the unique element such that It is well known that (see, e.g., ) for and For , the following inequality is known as the subdifferential inequality:
A mapping is said to be strongly quasinonexpansive  if it is quasinonexpansive and whenever is a bounded sequence in such that for some . It is known that every metric projection is strongly quasinonexpansive.
We are now ready to present our main result.
Theorem 2.3. Let be a closed convex subset of a Hilbert space and is a strongly quasinonexpansive mapping such that is demiclosed at zero. Suppose that is a contraction. Let be a sequence in defined by where is a sequence in satisfying (C1), (C2). Then the sequence converges to an element and the following inequality holds for all .
Before we give the proof, we note that is closed and convex (see  for more general setting). Hence the mapping is a contraction. Then it follows from the well-known Banach's contraction principle that there exists a unique element such that . In particular, and for all .
Let us assume that for all where is a real number in .
Lemma 2.4. The sequence is bounded.
Proof. We consider the following inequality: By induction, we conclude that the sequence is bounded and hence so is the sequence .
Lemma 2.5. The following inequality holds for all :
Proof. It follows from the subdifferential inequality that
Lemma 2.6. If there exists a subsequence of such that , then .
Proof. First, we note that and let us consider the following inequality: This implies that . Since is a strongly quasinonexpansive mapping, . In particular, . Because is bounded, so there exists a subsequence of such that and It follows from the demiclosedness of at zero that . Then Hence , as desired.
Proof of Theorem 2.3. Let us consider the following two cases.Case 1. There exists an such that for all . It follows then that exists and hence . This implies that . By Lemma 2.5, for all , Notice that and By Lemma 2.1, we have .Case 2. There exists a subsequence of such that for all . In this case, it follows from Lemma 2.2 that there exists a subsequence of such that for all . It follows from that . Moreover, by Lemma 2.5, we have In particular, it follows that This implies that Hence Then . This completes the proof.
Remark 2.7. If is a convex subset of a Hilbert space and is a quasinonexpansive mapping, then the mapping is strongly quasinonexpansive whenever (see ). This means that Maingé's result is included in ours as a special case.
Remark 2.8. There is a strongly quasinonexpansive mapping such that is not of the form where and is a quasinonexpansive mapping. This means that there is an example which is beyond the scope of Maingé's result (see Remark 1.3, Theorem 1.1 with his old proof is valid for only ).
Example 2.9. Let . It is clear that is a closed and convex subset of . Notice that is a strongly quasinonexpansive mapping and . Suppose that where and is a quasinonexpansive mapping. Then, by Proposition 1.2, we have It is easy to see that . In particular, That is , a contradiction.
We propose a viscosity approximation process for approximation of a fixed point of a quasinonexpansive mapping. This not only corrects Maingé's result but also essentially improves his result to a more general relaxation.
The first author is supported by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Commission on Higher Education. The second author is supported by Thailand Research Fund, the Commission on Higher Education and Khon Kaen University under Grant RMU5380039.
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