A Weak Convergence Theorem for Common Fixed Points of Two Nonlinear Mappings in Hilbert Spaces

Ibaraki, Takanori; Kajiba, Shunsuke; Takeuchi, Yukio

doi:https://doi.org/10.1155/2022/9568060

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2022 | Article ID 9568060 | https://doi.org/10.1155/2022/9568060

A Weak Convergence Theorem for Common Fixed Points of Two Nonlinear Mappings in Hilbert Spaces

Takanori Ibaraki,¹Shunsuke Kajiba,^2,3and Yukio Takeuchi⁴

Academic Editor: Gaston Mandata N gu r kata

Received30 Jun 2022

Accepted05 Aug 2022

Published26 Aug 2022

Abstract

In this paper, by using properties of attractive points, we study an iteration scheme combining simplified Baillon type and Mann type to find a common fixed point of commutative two nonlinear mappings in Hilbert spaces. Then, we apply the obtained results to prove a new weak convergence theorem.

1. Introduction

In 1963, DeMarr [1] proved a common fixed point theorem for a family of commuting nonexpansive mappings in a Banach space. After DeMarr, many researchers studied this subject (see [2–6] and others).

On the other hand, in 1975, Baillon [7] proved a mean convergence theorem known as the first nonlinear ergodic theorem in a Hilbert space. After Baillon, many researchers have studied topics related to his mean convergence theorem. In 1997, Shimizu and Takahashi [8] introduced the iteration scheme that combines Baillon type and Halpern type [9]. Then, they proved a strong convergence theorem to a common fixed point of a finite family of commutative nonexpansive mappings in Hilbert spaces. In 1998, Atsushiba and Takahashi [10] introduced the iteration scheme that combines Baillon type and Mann type [11] and proved a weak convergence theorem to a common fixed point of commutative two nonexpansive mappings in uniformly convex Banach spaces. In 2002, Suzuki [12] studied for common fixed points of commutative two nonexpansive mappings in general Banach spaces. Then, he proved a strong convergence theorem using Atsushiba and Takahashi’s iteration scheme. Stimulated by Suzuki [12], Takeuchi [13] introduced a new iteration scheme combining simplified Baillon type and Mann type and proved the following strong convergence theorem in general Banach spaces.

Theorem 1 (see [13]). Let be a Banach space and let be a compact convex subset of . Let and be nonexpansive self-mappings on with . Let be a sequence in satisfying . Let and define a sequence in by for each . Then, converges strongly to some common fixed point of and .

Also, some researchers studied topics related to common fixed points of various nonlinear mappings or semigroups of nonlinear mappings, and some convergence theorems were proved; for example, see [14–16] and others. In addition, Baillon’s theorem [7] evolved as follows. In 2011, Takahashi and Takeuchi [17] proposed the notion of an attractive point of a mapping . They denote by the set of all attractive points of and by the metric projection from onto the closed convex set . An attractive point is an important notion related to fixed points (see [18–21] and therein). Then, Takahashi and Takeuchi [17] proved the following Baillon type mean convergence theorem finding an attractive point for a wide class of nonlinear mappings called generalized hybrid [22].

Theorem 2 (see [17]). Let be a Hilbert space and let be a nonempty subset of . Let be a generalized hybrid mapping from into itself. Let and be sequences defined by for each . Suppose is bounded. Then, the following hold: (1) is nonempty, closed, and convex(2) converges weakly to such that

Motivated by the works as above, considering properties of attractive points, we study the iteration scheme proposed by Takeuchi [13]. Then, using the obtained results, we prove a new weak convergence theorem for common fixed points of commutative two nonlinear mappings in Hilbert spaces.

2. Preliminaries

We present some of fundamental concepts and some symbols used throughout this paper. We denote by the set of all real numbers, by the set of all positive integers, and by the set of all nonnegative integers. Also, we denote by the set for each and by the set for each with . Obviously, .

always denotes a real Hilbert space with inner product and induced norm . Let be a sequence in . Sometimes the strong convergence and weak convergence of to a point are denoted by and , respectively. So, implies , and implies for each . Then, we know the following basic facts: (i)A nonempty closed convex subset of is weakly closed(ii)A bounded sequence of has a weakly convergent subsequence(iii) if every weak cluster point of and are the same

A Hilbert space has the Opial property; that is, if a sequence in converges weakly to a point , then for all with .

Let be a nonempty subset of and let be a mapping from into . Sometimes, we denote by the identity mapping on . Then, we denote by the set of all fixed points of and by the set of all attractive points of , that is,

is called nonexpansive if for all . We say that is demiclosed at 0 if holds whenever is a sequence in such that for some and . Then, is demiclosed at 0 if is nonexpansive. is called quasi-nonexpansive if . It is easy to see that a nonexpansive mapping with is quasi-nonexpansive. Aoyama et al. [23] proposed -hybrid mappings for . is called –hybrid if for all . We can easily verify that a mapping of several important classes of nonlinear mappings is -hybrid for some . For example, a nonspreading mapping [24] is 0-hybrid; a hybrid mapping [25] is -hybrid and a nonexpansive mapping is 1-hybrid. satisfies if is -hybrid. So a -hybrid mapping is quasi-nonexpansive if . According to Falset et al. [26], is said to satisfy the condition () if there is such that for all . If satisfies the condition (), then . So is quasi-nonexpansive if satisfies the condition () and .

Let be a subset of and let and be self-mappings on . We denote by the common fixed point set and by the common attractive point set . We place importance on the condition . Therefore, we present some facts relevant to this condition. It is easy to see the following: (a) implies neither nor (b) does not imply without the assumption (c)In the case when is closed and convex, implies . However, does not imply (d)If and are quasi-nonexpansive mappings, then (e)Even if , neither nor need be quasi-nonexpansive

To clearly understand such situations, a specific example from [20] is given below.

Example 1 (see [20]). Let . Then, is compact and convex. Let and be self-mappings on defined by for each .

In this example, we easily see

From this, we can easily confirm the following: (i) holds(ii)Neither nor holds(iii)Neither nor is quasi-nonexpansive

Refer to [17–21] for more details of attractive points.

3. Lemmas

We begin this section with preparing the required symbols for the iteration scheme we are dealing with and then present some lemmas which are needed to prove our main result.

Let be a nonempty subset of a Hilbert space . Let and be self-mappings on . For each , define mappings , , and from into , respectively, by

for each . Then, for each , holds if , and holds if . In the case of , the following holds: for all , .

Here are some lemmas which are needed to get our main result. First, since a Hilbert space has the Opial property, the following lemma is easily obtained; for example, see Atsushiba et al. [18].

Lemma 3. Let be a Hilbert space and let be a nonempty subset of . Let be a sequence in such that converges for each . Suppose and are subsequences of which converge weakly to , respectively. Then, .

The following lemma is due to Ibaraki and Takeuchi [20].

Lemma 4 (see [20]). Let be a Hilbert space and let be a nonempty subset of . Let be a mapping from into . Let , , and . Suppose . Then, the following holds: Suppose further that is bounded. Let . Then,

We need the following trivial lemma to prove Lemma 6.

Lemma 5. Let be a sequence in . Then, for each , there exists satisfying , and therefore, the following holds:

Proof. Fix any . It is trivial that there exists such that In the case when is odd, and satisfies In the case when is even, satisfies Thus, we see that satisfies (13).

Lemma 6. Let be a Hilbert space and let be a nonempty subset of . Let and be self-mappings on satisfying . Let satisfy and let be a sequence in . For each , let be as in (9). Let be a bounded sequence in and define by for each . Suppose . Then, there is a sequence such that for each and

Proof. For each , let and be as in (9). Note that for all . Fix any . Since is a bounded sequence in , there is an satisfying . For each , by (12) in Lemma 4, we see that Then, we have In the same way, we also have By (20) and (21), the following holds: For each , we regard as in Lemma 5. From Lemma 5, for each , there exists satisfying (13). So, we have the sequence which consists of such . Then, for each , it follows from (13) that From this, by (22), we see Then, by , we immediately see Thus, by and , we obtain In the same way, we also have the following:

The following lemmas for a -hybrid mapping and a mapping satisfying the condition () were known; for example, see Ibaraki and Takeuchi [20]. Of course, these are extensions of the demiclosed principle in the Hilbert space setting.

Lemma 7. Let be a Hilbert space and let be a subset of . Let be a -hybrid mapping from into . Suppose is a sequence in which converges weakly to some and satisfies . Then, .

Lemma 8. Let be a Hilbert space and let be a subset of . Let be a mapping from into which satisfies the condition . Suppose is a sequence in which converges weakly to some and satisfies . Then, .

4. Main Result and Applications

We present a weak convergence theorem which is our main result.

Theorem 9. Let be a Hilbert space and let be a nonempty closed convex subset of . Let and be self-mappings on satisfying . Let satisfy and let be a sequence in . For each , let be as in (9). Let and define a sequence in by for each . Set and . Suppose and is demiclosed at for each . Then, the following hold: (1)A weakly convergent subsequence of exists, and every weakly convergent subsequence of converges weakly to a point of (2)In the case of , converges weakly to some

Proof. Fix any . It is trivial that for all . Then, by (11) in Lemma 4, we see for each . From this, we see that for all ; that is, converges. So, since for all , we have Furthermore, we know that for each , By (30) and (31), holds. For each , we may regard as in Lemma 6. Thus, since is bounded, by Lemma 6, there is a sequence such that for each and We show that (1) holds. Since is bounded, has a weakly convergent subsequence. Let be a subsequence of which converges weakly to some . Since is weakly closed and is a sequence in , we see .
We show . By and (32), we see From the latter, we see . Since converges weakly to , by , also converges weakly to . So, since is demiclosed at 0, we see .
We show . Similarly to the discussion above, we have the following: So, converges weakly to and Since is demiclosed at 0, we see . Thus, .
By the argument so far, we see that a weakly convergent subsequence of exists and any weakly convergent subsequence of has to converge weakly to a point of . So, we confirmed that (1) holds.
We show that (2) holds. We already know that converges for each . Suppose . Then, by (1), , and Lemma 3, all weak cluster points of coincide. That is, itself converges weakly to a point .

Next, we present some results derived from Theorem 9.

Theorem 10. Let be a Hilbert space and let be a nonempty closed convex subset of . Let and be quasi-nonexpansive self-mappings on satisfying . Assume that and is demiclosed at for each . Let satisfy and let be a sequence in . For each , let be as in (9). Let and define a sequence in by for each . Then, converges weakly to some .

Proof. Since and are quasi-nonexpansive, we know that holds. Thus, by Theorem 9, we have the result.

Theorem 11. Let be a Hilbert space and let be a nonempty closed convex subset of . Let and be nonexpansive self-mappings on satisfying . Assume . Let satisfy and let be a sequence in . For each , let be as in (9). Let and define a sequence in by for each . Then, converges weakly to some .

Proof. If a mapping is nonexpansive and , then is quasi-nonexpansive. We also know that is demiclosed at if is nonexpansive. Thus, by Theorem 10, we have the result.

By considering Lemmas 7 and 8, we have the following theorems.

Theorem 12. Let be a Hilbert space and let be a nonempty closed convex subset of . Let and be self-mappings on satisfying . Assume that is -hybrid, is -hybrid, and . Let satisfy and let be a sequence in . For each , let be as in (9). Let and define a sequence in by for each . Then, converges weakly to some .

Proof. Since is -hybrid and is -hybrid, by , we know that and are quasi-nonexpansive. By Lemma 7, we also know that and are demiclosed at 0. Thus, by Theorem 10, we have the result.

Theorem 13. Let be a Hilbert space and let be a nonempty closed convex subset of . Let and be self-mappings on satisfying . Assume that and satisfy the condition and . Let satisfy and let be a sequence in . For each , let be as in (9). Let and define a sequence in by for each . Then, converges weakly to some .

Proof. Since and satisfy the condition (), by , we know that and are quasi-nonexpansive. By Lemma 8, we also know that and are demiclosed at 0. Thus, by Theorem 10, we have the result.

5. Supplement

In this section, we present some examples that complement the argument so far. In advance, recall that weak and strong topologies on a Euclidean space coincide.

In the previous section, Theorem 9, neither nor need to be quasi-nonexpansive. However, in Theorems 10–13, we deal only with quasi-nonexpansive mappings and . We therefore give an example where all conditions of Theorem 9 are satisfied and is not quasi-nonexpansive.

Example 2. Let . Let and be continuous self-mappings on defined, respectively, by for each .

In this example, noting and , we see

From these, we can easily verify the following: (i) is a nonempty compact convex subset of (ii), that is, is not quasi-nonexpansive(iii) is nonexpansive and ; that is, is quasi-nonexpansive(iv) and are demiclosed at , since and are continuous(v) and

So, all conditions of Theorem 9 are satisfied and is not quasi-nonexpansive. This implies that the sequence generated by the procedure in Theorem 9 converges strongly to .

Recall that the condition is unnecessary to prove (1) of Theorem 9. That is, without the condition , we proved that the sequence in Theorem 9 has a subsequence which converges weakly to a point . So, to gain a better understanding the contents of this paper, we are interested in what happens when the condition is missing from Theorem 9. Then, from this point of view, we give the following example.

Example 3. Let . Let and be continuous self-mappings on defined, respectively, by

For this example, we easily see

From these, we see the following: (i) is a nonempty compact convex subset of (ii)Neither nor is quasi–nonexpansive(iii) and are demiclosed at , since and are continuous(iv) and

So, without , all conditions of Theorem 9 are satisfied.

In this example, for ease of verification, we have chosen and which are special in the following sense: (i)We can regard and as self-mappings on (ii)We can regard and as self-mappings on

Accordingly, the following should be noted: (i) and if and are considered like the former(ii) and if and are considered like the latter

We know that the sequence generated by the procedure in Theorem 9 has a subsequence which converges strongly to a point of . Especially, in this example, we can confirm that the sequence itself converges strongly to . That is, by the argument so far, we can easily verify the following: (i) converges strongly to if (ii) converges strongly to if (iii) converges strongly to if

From this, we simultaneously see the following: Example 3 is illustrative example such that may not hold when the condition is missing.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Japan Society for the Promotion of Science KAKENHI (Grant Numbers JP19K03632 and JP19H01479).

References

R. DeMarr, “Common fixed points for commuting contraction mappings,” Pacific Journal of Mathematics, vol. 13, no. 4, pp. 1139–1141, 1963.
View at: Publisher Site | Google Scholar
L. P. Belluce and W. A. Kirk, “Fixed-point theorems for families of contraction mappings,” Pacific Journal of Mathematics, vol. 18, no. 2, pp. 213–217, 1966.
View at: Publisher Site | Google Scholar
F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, no. 4, pp. 1041–1044, 1965.
View at: Publisher Site | Google Scholar
R. E. Bruck, “A common fixed point theorem for a commuting family of nonexpansive mappings,” Pacific Journal of Mathematics, vol. 53, no. 1, pp. 59–71, 1974.
View at: Publisher Site | Google Scholar
S. Ishikawa, “Common fixed points and iteration of commuting nonexpansive mappings,” Pacific Journal of Mathematics, vol. 80, no. 2, pp. 493–501, 1979.
View at: Publisher Site | Google Scholar
T. Suzuki, “Convergence theorems to common fixed points for infinite families of n onexpansive mappings in strictly convex Banach spaces,” Nihonkai Mathematical Journal, vol. 14, pp. 43–54, 2003.
View at: Google Scholar
J. B. Baillon, “Un theoreme de type ergodique pour les contractions non lineaires dans unespace de Hilbert, C,” Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Paris, Série A-B, vol. 280, pp. 1511–1514, 1975.
View at: Google Scholar
T. Shimizu and W. Takahashi, “Strong convergence to common fixed points of families of nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71–83, 1997.
View at: Publisher Site | Google Scholar
B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 959–961, 1976.
View at: Google Scholar
S. Atsushiba and W. Takahashi, “Approximating common fixed points of two nonexpansive mappings in Banach spaces,” Bulletin of the Australian Mathematical Society, vol. 57, no. 1, pp. 117–127, 1998.
View at: Publisher Site | Google Scholar
W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, no. 3, pp. 506–510, 1953.
View at: Publisher Site | Google Scholar
T. Suzuki, “Strong convergence theorem to common fixed points of two nonexpansive mappings in general Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 3, pp. 381–391, 2002.
View at: Google Scholar
Y. Takeuchi, “An iteration scheme finding a common fixed point of commuting two nonexpansive mappings in general Banach spaces,” Linear and Nonlinear Analysis, vol. 2, pp. 317–327, 2016.
View at: Google Scholar
K. Eshita and W. Takahashi, “Strong convergence theorems for commutative semigroups ofcontinuous linear operators on Banach spaces,” Taiwanese Journal of Mathematics, vol. 9, pp. 531–550, 2005.
View at: Google Scholar
F. Kohsaka, “Existence and approximation of common fixed points of two hybrid mappings in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 16, pp. 2193–2205, 2015.
View at: Google Scholar
W. Takahashi, “Iterative methods for approximation of fixed points and their applications,” Journal of the Operations Research Society of Japan, vol. 43, no. 1, pp. 87–108, 2000.
View at: Publisher Site | Google Scholar
W. Takahashi and Y. Takeuchi, “Nonlinear ergodic theorem without convexity for generalized hybrid mappings in a Hilbert space,” Journal of Nonlinear and Convex Analysis, vol. 12, pp. 399–406, 2011.
View at: Google Scholar
S. Atsushiba, S. Iemoto, R. Kubota, and Y. Takeuchi, “Convergence theorems for some classes of nonlinear mappings in Hilbert space, linear,” Nonlinear Analysis, vol. 2, pp. 125–153, 2016.
View at: Google Scholar
S. Atsushiba and W. Takahashi, “Nonlinear ergodic theorems without convexity for nonex- pansive semigroups in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 14, pp. 209–219, 2013.
View at: Google Scholar
T. Ibaraki and Y. Takeuchi, “New convergence theorems for common fixed points of a wide range of nonlinear mappings,” Journal of Nonlinear Analysis and Optimization: Theory & Applications, vol. 9, pp. 95–114, 2018.
View at: Google Scholar
T. Ibaraki and Y. Takeuchi, “A mean convergence theorem finding a common attractive point of two nonlinear mappings,” Yokohama Mathematical Journal, vol. 66, pp. 61–77, 2020.
View at: Google Scholar
P. Kocourek, W. Takahashi, and J.-C. Yao, “Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces,” Taiwanese Journal of Mathematics, vol. 14, no. 6, pp. 2497–2511, 2010.
View at: Publisher Site | Google Scholar
K. Aoyama, S. Iemoto, F. Kohsaka, and W. Takahashi, “Fixed point and ergodic theorems for -hybrid mappings in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 11, pp. 335–343, 2010.
View at: Google Scholar
F. Kohsaka and W. Takahashi, “Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces,” Archiv der Mathematik, vol. 91, no. 2, pp. 166–177, 2008.
View at: Publisher Site | Google Scholar
W. Takahashi, “Fixed point theorems for new nonlinear mappings in a Hilbert space,” Journal of Nonlinear and Convex Analysis, vol. 11, pp. 79–88, 2010.
View at: Google Scholar
J. G. Falset, E. L. Fuster, and T. Suzuki, “Fixed point theory for a class of generalized nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 375, no. 1, pp. 185–195, 2011.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Takanori Ibaraki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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