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Algorithms with variant anchors for pseudocontractive mappings
Journal of Inequalities and Applications volume 2014, Article number: 386 (2014)
Abstract
The purpose of this paper is to find the fixed points of pseudocontractive mappings by using the iterative technique. Two algorithms with variant anchors have been introduced. Strong convergence results are given. Especially, we can find the minimum-norm fixed point of pseudocontractive mappings.
MSC:47H05, 47H10, 47H17.
1 Introduction
In this paper, we assume that H is a real Hilbert space and is a nonempty closed convex subset. Recall that a mapping is said to be Lipschitzian if
where is a constant, which is in general called the Lipschitz constant. If , T is called nonexpansive.
A mapping is said to be pseudocontractive if
We use to denote the set of fixed points of T.
In the literature, there are a large number references associated with the fixed point algorithms for the pseudocontractive mappings. See, for instance, [1–24]. (The interest of pseudocontractions lies in their connection with monotone operators; namely, T is a pseudocontraction if and only if the complement is a monotone operator.)
Now there exists an example which shows that Mann iteration does not converge for the pseudocontractive mappings [2]. At present, it is still an interesting topic to construct algorithms for finding the fixed points of the pseudocontractive mappings.
On the other hand, there are perturbations always occurring in the iterative processes because the manipulations are inaccurate. Recently, in order to find the fixed points of the nonexpansive mappings, Yao and Shahzad [25] introduced the following algorithms with perturbations and obtained the strong convergence results.
Algorithm 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a nonexpansive mapping. For given , define a sequence in the following manner:
where is a sequence in and the sequence is a small perturbation for the m-step iteration satisfying as .
Theorem 1.2 Suppose . Then, as , the sequence generated by the implicit method (1.1) converges to , which is the minimum-norm fixed point of T.
Algorithm 1.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a nonexpansive mapping. For given , define a sequence in the following manner:
where and are two sequences in and the sequence is a perturbation for the n-step iteration.
Theorem 1.4 Suppose that . Assume the following conditions are satisfied:
-
(i)
and ;
-
(ii)
;
-
(iii)
.
Then the sequence generated by the explicit iterative method (1.2) converges to , which is the minimum-norm fixed point of T.
Note that the idea of the iterative algorithms with perturbations has been extended to the other topics, see, for example, [26].
Motivated by the above ideas and the results in the literature, in the present paper, we present two algorithms with variant anchors for finding the fixed points of the pseudocontractive mappings in Hilbert spaces. Strong convergence results are given. As special cases, we can find the minimum-norm fixed point of the pseudocontractive mappings.
2 Preliminaries
Recall that the metric projection is defined by
It is obvious that satisfies
and is characterized by
The following two lemmas will be useful for our main results.
Lemma 2.1 ([24])
Let C be a closed convex subset of a Hilbert space H. Let be a Lipschitzian pseudocontractive mapping. Then is a closed convex subset of C and the mapping is demiclosed at 0, i.e., whenever is such that and , then .
Lemma 2.2 ([27])
Assume is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence in R such that
-
(i)
;
-
(ii)
or .
Then .
3 Main results
In the sequel, we assume that C is a nonempty closed convex subset of a real Hilbert space H and is a κ-Lipschitzian pseudocontractive mapping with nonempty fixed points set .
The first result is on the convergence of the path for the pseudocontractive mappings. Now, we define our path as follows.
For fixed and , we define a mapping by
where is the metric projection from H on C.
Next, we show that the mapping is strongly pseudocontractive. Indeed, for , we have
Since , . Hence, is a strongly pseudocontractive mapping. By [2], has a unique fixed point . That is, satisfies
Remark 3.1 can be seen as a perturbation.
Next, we prove the convergence of the path (3.1).
Theorem 3.2 If , then the path defined by (3.1) converges strongly to .
Proof Let . We get from (3.1) that
It follows that
Since , there exists a constant such that . So,
Thus, is bounded.
By (3.1), we have
Therefore,
Let be a sequence satisfying as . Put . By (3.2), we get
By (3.1), we obtain
Hence,
It follows that
Here is a constant such that . In particular, we obtain
Since is bounded, there exists a subsequence of satisfying weakly. By (3.3), we get
Applying Lemma 2.1 to (3.6) to deduce .
By (3.5), we derive
Since and , we deduce that by (3.7). By (3.5), we have
Assume that there exists another subsequence of satisfying weakly. Similarly, we can prove that , which satisfies
In (3.8), we pick up to get
In (3.9), we pick up to get
Adding (3.10) and (3.11), we deduce
Thus, . This indicates that the weak limit set of is singleton and the path converges strongly to by (3.8). This completes the proof. □
Corollary 3.3 The path defined by
converges strongly to , which is the minimum-norm fixed point of T.
Now, we introduce another algorithm, which is an explicit manner.
Algorithm 3.4 Let and be two real number sequences in . Let be a sequence. For arbitrarily, let the sequence be generated by
Theorem 3.5 Assume the following conditions are satisfied:
-
(C1)
;
-
(C2)
.
Then we have
-
(1)
the sequence is bounded;
-
(2)
the sequence is asymptotically regular, that is, .
Further, if and , then the sequence converges strongly to .
Proof By the condition (C1), we can find a sufficiently large positive integer m such that
Let . For fixed m, we pick up a constant such that
Next, we show that . Set for all . Thus, we have for all .
Since is monotone, we have
By (3.12), we obtain
Note that
Then we have
Hence,
By (3.12), we have
From condition (C1), we deduce and as . Therefore, we get
That is, the sequence is asymptotically regular.
By (3.15) and (3.16), we have
This together with (3.13) and (3.14) imply that
By induction, we get
So is bounded.
By (3.12), we have
It follows that
By the condition and the assumption , we deduce
Let the net be defined by . By Theorem 3.2, we know that converges strongly to . Next, we prove
By the definition of , we have
It follows that
Since , by the characteristic inequality of metric projection, we have
Then
which implies that
By (3.17), we deduce
Note the fact that the two limits and are interchangeable. This together with , and (3.18) implies that
Note that and . We derive
Finally, we prove that . Note that
and
Then
By (3.12), (3.16), and (3.20), we get
where and
It is clear that and . We can therefore apply Lemma 2.2 to (3.21) and conclude that as . This completes the proof. □
Corollary 3.6 Let and be two real number sequences in . For arbitrarily, let the sequence be generated by
Assume . Then we have
-
(1)
the sequence is bounded;
-
(2)
the sequence is asymptotically regular, that is, .
Further, if and , then the sequence converges strongly to , which is the minimum-norm fixed point of T.
Proof Letting in (3.12), we obtain (3.22). Consequently, by Theorem 3.5, we find that the sequence generated by (3.22) converges strongly to , which is the minimum-norm fixed point of T. □
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Guo, L., Kang, S.M. & Jung, C.Y. Algorithms with variant anchors for pseudocontractive mappings. J Inequal Appl 2014, 386 (2014). https://doi.org/10.1186/1029-242X-2014-386
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DOI: https://doi.org/10.1186/1029-242X-2014-386