Extragradient Method for Fixed Points in CAT(0) Spaces

Lv, Yu-Pei; Shabbir, Khurram; Shahzeen, Sundus; Ali, Farman; Kafle, Jeevan

doi:https://doi.org/10.1155/2021/7808255

Journal of Function Spaces

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Special Issue

Unique and Non-Unique Fixed Points and their Applications

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 7808255 | https://doi.org/10.1155/2021/7808255

Extragradient Method for Fixed Points in CAT(0) Spaces

Yu-Pei Lv,¹Khurram Shabbir,²Sundus Shahzeen,³Farman Ali,⁴and Jeevan Kafle⁵

Academic Editor: Liliana Guran

Received11 Jun 2021

Revised05 Aug 2021

Accepted12 Aug 2021

Published26 Aug 2021

Abstract

This paper is dedicated to construct a viscosity extragradient algorithm for finding fixed points in a CAT(0) space. The mappings we consider are nonexpansive. Strong convergence of the algorithm is obtained. The results established in this work extend and improve some recent discovers in the literature.

1. Introduction

Let be a CAT(0) space and be any closed convex subset of . Let be a bifunction with for all . The main equilibrium problem is to get satisfying

Finding solutions for equilibrium problems aids in solving problems in other areas of science like physics, optimization, and economics. The set of all solutions of the equilibrium problem is denoted by , i.e.,

Numerous iterative algorithms for monotome equilibrium have been investigated previously; for finding the solutions, see [1, 2]. Here, we will find the iterative algorithm for pseudomonotone bifunction. The bifunction is said to be pseudomonotone if

An extragradient method, to solve a pseudomonotone equilibrium problem in , was introduced in [3]. The method of extragradient is as follows: given , find successively and by where and is such that the Lipschitz (type) condition holds. In [4], the following algorithm was introduced by Anh for finding a fixed point of a nonexpansive mapping which is also the solution of the equilibrium problem for pseudomonotone bifunction in a Hilbert space: where and is such that the Lipschitz (type) condition holds. The convergence (strong) of with and under certain considerations on .

In 2012, an algorithm for hybrid projection was considered by Vuong et al. [5] for where and is such that the Lipschitz (type) condition holds. The convergence proved was strong.

The Armijo-type method for pseudomonotone equilibrium problems was formulated in [6] in the setting of Hilbert spaces. After that, the authors in [7] presented the convergence of weak and strong types for the algorithms in order to solve the equilibrium problem. The admirable outcomes are due to Dinh and Kim in which there is no restriction on monotonicity of the bifunction. The current results of equilibrium problems are given for pseudomonotone type. For further references, see [8, 9]. To the current knowledge, the authors modified the “hybrid projection algorithm” in order to get convergence of strong type for iterative algorithms of equilibrium problems of pseudomonotone type [10, 11].

The aim of this paper is to construct an extragradient algorithm of viscosity type for finding the same element for the solution set of a pseudomonotone equilibrium problem and fixed point set of a nonexpansive mapping in the framework of a CAT(0) space and derive its strong convergence.

2. Definitions and Known Results

In this section, we present basic definitions and known results. The notions that are not defined in this paper can be seen in [12–14].

Throughout this paper, denotes the geodesic metric space with a geodesic triangle in , where , , and represent the vertices of in . We will represent as henceforth. A comparison triangle in is a triangle in the Euclidean plane such that for

A geodesic space is called a CAT(0) space, if all the geodesic triangles satisfy the following comparison axiom

Let , and by Lemma 2.1(iv) of [15] for each , there exists a unique point such that for all with the corresponding points of in .

Lemma 1 (see [16]). Let be a CAT(0) space. Then the following assertions are true: (i)For any and ,(ii)For any and ,

Let be a uniquely geodesic metric space; that is, for each , there exists a unique isometry such that and , and in this case, we write . For each , we write for the element such that and .

Hadamard spaces are the complete CAT(0) spaces; for details, see [16]. If are points of a CAT(0) space and is the midpoint of the segment , which we will denote by , then the CAT(0) inequality gives

This inequality is called the (CN) inequality of Bruhat and Tits [16]. In fact, a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality (cf. [16], page 163). Berg and Nikolaev [17] introduced the idea of the quasilinearization as follows: Let us denote the pair by and call it a vector. Then, quasilinearization is defined as a map defined as

It is easy to see that , , and for all . We say that satisfies the Cauchy-Schwarz inequality if for all . It is well known [17] that a geodesically connected metric space is a CAT(0) space if and only if it satisfies the Cauchy-Schwarz inequality.

In 1976, Lim gave the definition of -convergence in a general metric space case (see [18]). He named this type of convergence as strong -convergence, proving that CAT(0) spaces provide a natural framework for the Lim concept and, in the setting of -convergence, provide many properties of the usual notion of weak convergence in Banach spaces. Let us recall this type of convergence for the case of CAT(0) spaces, as follows.

Definition 2. Let be a CAT(0) space. A sequence in is said to -converge to if and only if is the unique asymptotic center of all subsequences of . In this case, we write , and is called the -limit of .

Note that the asymptotic center of is the set and the asymptotic radius of is given by where for is a bounded sequence in .

Ahmadi Kakavandi and Amini introduced in [19] another variant of weak convergence in complete CAT(0) spaces, taking into account the concept of quasilinearization.

Definition 3 (see [19]). Let be a complete CAT(0) space. A sequence in is said to -converge to an element if for each , .

Obviously, the convergence in the metric implies -convergence, and the -convergence implies -convergence (see Proposition 2.5 in [19]). But the converse is not true (see [20]). The following result proves an explicit connection between -convergence and -convergence.

Theorem 4 (see [20]). Let be a complete CAT(0) space. Then a sequence in -converges to if and only if, for every , .

Further, let us recall some definitions for the case of a CAT(0) space.

Definition 5. Let be a CAT(0) space and be a mapping. Then is called nonexpansive if

Definition 6. Let be a CAT(0) space and be a mapping. Then is called a contraction if

Throughout this paper, we denote by the set of fixed points of .

Concerning the convexity in CAT(0) spaces, we remark that, in this type of spaces, angles exist in a strong sense, the distance function is convex, and one has both uniform convexity and orthogonal projection onto convex subsets. Moreover, CAT(0) spaces turn up to represent a real framework for convexity theory.

Remark 7. Considering the CAT(0) space case, a subset is said to be convex if includes every geodesic segment joining any two of its points, i.e., , for every and .

Definition 8. Let be a CAT(0) space. A function is said to be convex if for all , and .

Let be a CAT(0) space and a convex and closed subset. Further, let a bifunction ; then is called (1)-strong monotone on if , we have(2)monotone on if for each , one has(3)pseudomonotone on if for each , one has

The above bifunction is Lipschitz-type continuous on , if there exist two constants and such that

In [19], Kakavandi and Amini define the notion of a subdifferential of a function as follows.

Definition 9 (see [19]). Let be a proper function with efficient domain , then the subdifferential of is the multifunction defined by when and .

We mention that denote the notion of dual space of the metric space . For more details, see [19].

In [21], Georgiou and Papadopoulos gave a strong discussion concerning the convergence types and topologies on function spaces. Then, let us consider as three topological spaces. A mapping into is called weakly continuous at if for every open neighbourhood of there exists an open neighbourhood of such that . denotes the closure of . The mapping is weakly continuous on , if it is weakly continuous at each point of . In the following, denotes the set of all weakly continuous maps of into . If is a topology on the set , then the corresponding topological space is denoted by .

In this conditions, we can recall the notion of jointly weakly continuous function.

Definition 10. A topology on is called weakly jointly continuous if for every , the weak continuity of a map implies the weak continuity of the map .

For more details and results concerning the topology and the convergence types in function spaces, see [21–23].

Further, taking into account the previous notions, let us consider the following properties of : (1) for all , and is taken pseudomonotone on the subset (2) is taken to be Lipschitz-type continuous on the subset (3)For all , is subdifferentiable and convex(4) is taken jointly weakly continuous on

With conditions (6), (7), (8), and (10), the set is convex and closed.

Lemma 11 (see [24]). Let be a CAT(0) space. Assume that and . Further, let be solutions of strongly convex problems, where , then

Lemma 12 (the demiclosedness principle). Let be a nonempty closed convex subset of the CAT(0) space and such that Then, . (Here, (respectively, ) denotes strong (respectively, weak) convergence.) The notion of weak convergence is the same as defined in [25].

Lemma 13 (see [26]). Let be a CAT(0) space and be any closed convex subset of . For each point , there exists a unique nearest point of , denoted by , such that for all . Such a is metric projection onto from . Then, (1)for all and , iff(2)for all and , it holds

For more information, see Section 3 of [27].

Lemma 14. Let be a complete CAT(0) space. For all and , the following hold: (1)(2) for all

Lemma 15 (see [28]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence with (1)(2) or Then, .

Another interesting result useful in the proof of our main results is the following.

Lemma 16 (see [29]). Assume that is a sequence of nonnegative real numbers such that there exists subsequences of such that for all . Then there exists a nondecreasing sequence such that as , and the following properties are satisfied all (sufficiently large) number : In fact, .

Example 17. Let and . Consider a bifunction defined as for each . Then we have the inequality Then, is Lipschitz-type continuous on with .

3. Strong Convergence of the Proposed Algorithm

Suppose that a nonexpansive mapping and a bifunction satisfy the conditions (6)–(10) and . Further, let be a -contraction. Because is a contraction mapping on , we have , such that .

Before presenting the first result of this section, let us recall a crucial theorem given by Lim in [18], which is used in the proof of our main results.

Theorem 18 ([18], Theorem 5.2). Every bounded sequence in a Hadamard space has a -convergent subsequence.

The following algorithm is useful in finding a common element of a solution set of pseudomonotone equilibrium problem on and fixed point set of .

Consider with , and . Consider .

Step 1.

Step 2. If , then , stop the process; otherwise, go to Step 3.

Step 3. Generate Set and go to Step 1. Obviously, if for , by using (26), it gives for all and . It gives : from . Further, for convergence of the algorithm, let us consider that Step 2 is not true for .

Lemma 20. Consider to be bounded sequence. If , , as , then

Proof. We consider the following bounded sequence . Using Theorem 18, there exists a subsequence of such that weakly converges to with Since , we have .
By (26) with and , we have Taking on , (6), and (10), we achieve This gives . Also .
By Lemma 12 with , we obtain . Then . Using Lemma 13 with (36), the conclusion follows.☐

Theorem 21. Consider the sequences and satisfying the conditions: then the sequence strongly converges to .

Proof. Consider . Lemma 11 follows Then Hence, is bounded; then, , , and are bounded too. On the other hand, by (41), we have Let us consider Combining (43) and (44), we get By Lemma 14 and (40), we have Therefore, where . Taking for all . As , then . By the conditions on , we get Then, we have The rest of the proof will be divided into two parts.☐

Case 1. Suppose that there exists such that is nonincreasing. In this situation, is convergent. This together with the hypothesis on , , , and (45) gives On the other hand, by Lemmas 11 and 14, we have Hence, Then, converges. Also, , and conditions on and gives Combining (49) and (52), we get Lemma 20 and (49) and (53) give The conclusion follows from Lemma 15 and (48), (49), and (54).

Case 2. Let be a subsequence of with Then, by Lemma 16, there exists a subsequence such that : The above expression with (44) concludes By the hypothesis on , and , it follows that Using (51), we have By the hypothesis on and , it follows that In a similar way as Case 1, we arrive at Note that Since we have As , we have From (61), it follows that as . By the above arguments, using relations (61) and (62) and conditions of , we obtain with .
As for all , we get when

Corollary 22. Consider a CAT(0) space and nonempty, convex, and closed subset of . Further, consider a bifunction that fulfils conditions (6)–(10) and a nonexpensive mapping with . Then, define a sequence as follows: for any , consider Initially, we choose with , and take . Set and the sequences , as in Theorem 21, then converges strongly to .

Corollary 23. Consider a CAT(0) space and nonempty, convex, and closed subset of . Further consider a bifunction that fulfils conditions (6)–(10) such that , the sequence with and where with If the sequences and are same as in Theorem 21, then the converges strongly to .

Example 24. Let and . Let for all, . Then is Lipschitz-type continuous with the constants .
It is easy to see that satisfies conditions (6)–(10). Let and for all . It follows that is a nonexpansive mapping and is a contraction. Take the initial point and put the sequences , , and for all . It is easy to see that We give some and by Table 1: from Table 1, we see that after 13 iterations .

Corollary 25. Consider a CAT(0) space and nonempty, convex, and closed subset of . Further consider a bifunction that fulfils conditions (6)–(10) and a nonexpensive mapping with . Then define a sequence as follows.
For any , consider Initially choose with and take . Set and the sequences , as in Theorem 21.
Then converges strongly to .

Corollary 26. Consider a CAT(0) space and nonempty, convex, and closed subset of . Further consider a bifunction that fulfils conditions (6)–(10) such that , the sequence with and where with If the sequences and are the same as in Theorem 21, then the sequence converges strongly to .

Remark 27. Our Theorem 21 is an analog of Theorem 3.1 of Wang et al. [30] for the Hilbert space case.

Remark 28. Our results can be extended to any space with , since any space is a space, for any (see [17]).

4. Conclusions

It is well known that finding solutions for equilibrium problems play an important role in solving problems in other areas of science like physics, optimization, and economics. In this paper, we construct a new viscosity extragradient algorithm in order to find fixed points in a CAT(0) space for the case of nonexpansive mappings. Also, we prove a strong convergence of the proposed algorithm, and we give some examples to support our results.

Data Availability

All data required for this research is included within the paper.

Conflicts of Interest

There are no competing interests among the authors.

Authors’ Contributions

Yu-Pei Lv and Farman Ali contributed equally to this work.

References

S.-S. Chang, H. W. Joseph Lee, and K. K. Chan, “A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization,” Nonlinear Analysis: Theory Methods & Applications, vol. 70, no. 9, pp. 3307–3319, 2009.
View at: Publisher Site | Google Scholar
P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1996.
View at: Publisher Site | Google Scholar
D. Q. Tran, M. L. Dung, and V. H. Nguyen, “Extragradient algorithms extended to equilibrium problem,” Optimization, vol. 57, pp. 749–776, 2008.
View at: Google Scholar
P. N. Anh, “A hybrid extragradient method extended to fixed point problems and equilibrium problems,” Optimization, vol. 62, no. 2, pp. 271–283, 2013.
View at: Publisher Site | Google Scholar
P. T. Vuong, J. J. Strodiot, and V. H. Nguyen, “Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems,” Journal of Optimization Theory and Applications, vol. 155, no. 2, pp. 605–627, 2012.
View at: Publisher Site | Google Scholar
P. N. Anh and H. A. Le Thi, “An Armijo-type method for pseudomonotone equilibrium problems and its applications,” Journal of Global Optimization, vol. 57, no. 3, pp. 803–820, 2013.
View at: Publisher Site | Google Scholar
B. V. Dinh and D. S. Kim, “Projection algorithms for solving nonmonotone equilibrium problems in Hilbert space,” Journal of Computational and Applied Mathematics, vol. 302, pp. 106–117, 2016.
View at: Publisher Site | Google Scholar
P. N. Anh, “Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities,” Journal of Optimization Theory and Applications, vol. 154, no. 1, pp. 303–320, 2012.
View at: Publisher Site | Google Scholar
P. N. Anh and L. T. H. An, “The subgradient extragradient method extended to equilibrium problems,” Optimization, vol. 64, no. 2, pp. 225–248, 2015.
View at: Publisher Site | Google Scholar
P. N. Anh, “A hybrid extragradient method for pseudomonotone equilibrium problems and fixed point problems,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 36, pp. 107–116, 2013.
View at: Google Scholar
D. V. Hieu, L. D. Muu, and P. K. Anh, “Parallel hybrid extragradient methods for pseudomotone equilibrium problems and nonexpansive mappings,” Numerical Algorithms, vol. 73, pp. 197–217, 2016.
View at: Google Scholar
M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, vol. 319, Springer Science & Business Media, 2013.
R. Espínola and A. Fernández-León, “CAT-spaces, weak convergence and fixed points,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 410–427, 2009.
View at: Publisher Site | Google Scholar
K. J. Horadam, Hadamard Matrices and Their Applications, Princeton University Press, 2012.
S. Dhompongsa and B. Panyanak, “On △-convergence theorems in CAT(0) spaces,” Computers and Mathematics with Applications, vol. 56, no. 10, pp. 2572–2579, 2008.
View at: Publisher Site | Google Scholar
F. Bruhat and J. Tits, Groupes réductifs sur un corps local. I.Donnes radicielles values, vol. 41, Institut des Hautes Études Scientifiques. Publications Mathématiques, 1972.
I. D. Berg and I. G. Nikolaev, “Quasilinearization and curvature of Aleksandrov spaces,” Geometriae Dedicata, vol. 133, no. 1, pp. 195–218, 2008.
View at: Publisher Site | Google Scholar
T. C. Lim, “Remarks on some fixed point theorems,” Proceedings of the American Mathematical Society, vol. 60, no. 1, pp. 179–182, 1976.
View at: Publisher Site | Google Scholar
B. Ahmadi Kakavandi and M. Amini, “Duality and subdifferential for convex functions on complete CAT(0) metric spaces,” Nonlinear Analysis, vol. 73, no. 10, pp. 3450–3455, 2010.
View at: Publisher Site | Google Scholar
B. A. Kakavandi, “Weak topologies in complete CAT(0) metric spaces,” Proceedings of the American Mathematical Society, vol. 141, no. 3, pp. 1029–1039, 2013.
View at: Google Scholar
D. N. Georgiou and B. K. Papadopoulos, “Weakly continuous, weakly -?continuous, super-continuous and topologies on function spaces,” Scientiae Mathematicae Japonicae Online, vol. 4, pp. 315–328, 2001.
View at: Google Scholar
A. di Concilio, “Exponential law and θ-continuous functions,” Quaestiones Mathematicae, vol. 8, no. 2, pp. 131–142, 1985.
View at: Publisher Site | Google Scholar
R. Arens and J. Dugundji, “Topologies for function spaces,” Pacific Journal of Mathematics, vol. 1, no. 1, pp. 5–31, 1951.
View at: Publisher Site | Google Scholar
K. O. Aremu, L. O. Jolaoso, C. Izuchukwu, and O. T. Mewomo, “Approximation of common solution of finite family of monotone inclusion and fixed point problems for demicontractive multivalued mappings in CAT(0) spaces,” Ricerche di Matematica, vol. 69, no. 1, pp. 13–34, 2020.
View at: Publisher Site | Google Scholar
S. Dhompongsa, W. A. Kirk, and B. Panyanak, “Nonexpansive set-valued mappings in metric and Banach spaces,” Journal of nonlinear and convex analysis, vol. 8, pp. 35–45, 2007.
View at: Google Scholar
H. Dehghan and J. Rooin, “Metric projection and convergence theorems for nonexpansive mapping in Hadamard spaces,” 2014, https://arxiv.org/abs/1410.1137.
View at: Google Scholar
K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984.
H. K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, article 240256, pp. 240–256, 2002.
View at: Publisher Site | Google Scholar
P. E. Maingé, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008.
View at: Publisher Site | Google Scholar
S. Wang, M. Zhao, P. Kumam, and Y. J. Cho, “A viscosity extragradient method for an equilibrium problem and fixed point problem in Hilbert space,” Journal of Fixed Point Theory and Applications, vol. 20, no. 1, p. 19, 2018.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Yu-Pei Lv 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.

PDF Download Citation

Download other formats

Order printed copies

Views

385

Downloads

462

Citations