Strong Convergence Theorem for Finding a Common Solution of Convex Minimization and Fixed Point Problems in CAT(0) Spaces

Ndlovu, P. V.; Jolaoso, L. O.; Aphane, Maggie

doi:https://doi.org/10.1155/2022/2960135

Abstract and Applied Analysis

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Abstract Introduction Preliminaries Numerical Example Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 2960135 | https://doi.org/10.1155/2022/2960135

Strong Convergence Theorem for Finding a Common Solution of Convex Minimization and Fixed Point Problems in CAT(0) Spaces

P. V. Ndlovu,¹L. O. Jolaoso,^1,2and Maggie Aphane¹

Academic Editor: Christopher Goodrich

Received14 Jun 2021

Accepted22 Jan 2022

Published15 Feb 2022

Abstract

In this paper, we introduce a proximal point algorithm for approximating a common solution of finite family of convex minimization problems and fixed point problems for -demicontractive mappings in complete CAT(0) spaces. We prove a strong convergence result and obtain other consequence results which generalize and extend some recent results in the literature. We further provide a numerical example to illustrate the convergence behaviour of the sequence generated by our algorithm.

1. Introduction

Let be a geodesic metric space and be a proper, convex, and lower semicontinuous function. One of the problems arising in optimization theory is to find a minimizer of , that is, find such that

We denote the set of minimizers of by . A strong tool used for solving such problems is known as the proximal point algorithm (PPA) which was first initiated by Martinet [1]. Rockafellar [2] generalized the study to Hilbert spaces as follows: for where for all . In 2013, Bačák [3] extended the PPA to CAT(0) spaces as follows: where for all . It was shown that if has a minimizer and then the sequence -converges to its minimizer. Bačák [4] further employed a split version of the PPA for sum of convex operators in CAT(0) spaces. Furthermore, Cholamjiak [5] modified the PPA in CAT(0) space using the Halpern method as follows: where , , and . He showed that the sequence strongly converges to its minimizer.

On the other hand, one of the most fundamental findings in fixed point theory is the Banach contraction principle [6]. It has been used to prove the existence of the solutions of different nonlinear functional equations. Recently, efforts have been done to obtain a fixed point in partially ordered sets. Ran and Reurings (see [7]) generalized the Banach contraction to ordered metric space. Nieto and Rodriguez-Lopez [8] extended the results of fixed point theory in partially ordered metric space and used them to study the existence of the solution of differential equations. Fixed point theory in CAT(0) spaces was introduced by Kirk [9]; since then, many others have studied fixed point theory for different types of mappings, such as nonexpansive, Lipschitz, quasinonexpansive, -strictly pseudocontractive, and -demicontractive together with nonlinear optimization problems (see [10–15]).

Definition 1 (see [16]). Let be a nonempty subset of a metric space and be a nonlinear mapping. A point is called a fixed point of if The set of fixed points of is denoted by i.e.,

Definition 2 (see [17]). Let be a Hadamard space and be a nonempty closed and convex subset of . A mapping is said to be (i)a contraction, if there exists , such thatwhen , then is said to be nonexpansive. (ii)quasinonexpansive, if and(iii)-strictly pseudocontractive, if there exists such that(iv)-demicontractive, if and there exists such that

We note that the class of -demicontractive mappings contains the class of nonexpansive, quasinonexpansive, and -strictly pseudocontractive mappings (see [18, 19]).

In 2014, Cholamjiak and Abdou [20] modified (4) in CAT(0) spaces for approximating the minimizers of and common fixed point of nonexpansive mappings and as follows: given

Also, they established the -convergence for sequence generated by (9). Motivated by the results of Liu and Chang [17], the authors in [21] introduced a modified PPA for finding a common solution of convex minimization problems and fixed point of nonexpansive mappings in CAT(0) spaces. In particular, the authors proved the -convergence for the following results.

Theorem 3. Let be a nonempty closed convex subset of a complete CAT(0) space. Let be a finite family of proper, convex, and lower semicontinuous functions of into and let be a finite family of nonexpansive mappings of into itself. Suppose that is nonempty. For , let be a sequence in defined by where , , and are sequences in such that , , and for all and for all . Then, the sequence generated by (1.3) -converges to an element

Recently, Chidume et al. [22] proved the following theorem for approximating the common fixed points of finite family of -demicontractive mappings in CAT(0) spaces.

Theorem 4 (see [22]). Let be a nonempty closed convex subset of a complete CAT(0) space. Let , be a family of demicontractive mappings with constants , , such that . Suppose that is -demiclosed at 0 for all and for all for arbitrary , define a sequence by where , such that and . Then, -converges to a point .

Motivated by the above results, in this paper, we introduce a modified Halpern method together with the PPA method for approximating a common solution of finite family of convex minimization problems and common fixed point of -demicontractive mappings in complete CAT(0) spaces. We prove a strong convergence result and provide some consequence results which generalize some other related results in the literature. We also give a numerical example to show the convergence behaviour of the sequence generated by our algorithm.

2. Preliminaries

Definition 5 (see [23]). Let be a metric space and . A geodesic path from to is a map such that , , and for all . Its image is called a geodesic segment. If every pair of points are joined by a geodesic, we call the space a geodesic space, and a space is said to be a uniquely geodesic if every pair of points of are joined by exactly one geodesic segment. A geodesic triangle in a geodesic metric space contains three points (vertices of ), and the geodesic segment between each pair of vertices (edge of ) of a comparison triangle for the geodesic triangle is a triangle in the Euclidean plane such that for all . A geodesic space is said to be a CAT(0) space if for each geodesic triangle in X and its comparison , the CAT(0) space inequality is satisfied for all and comparison points . If is the midpoint and are points in a CAT(0) space, then the CAT(0) inequality implies

Equation (13) is called the CN-inequality of Bruhat and Tits [23]. Moreover, a geodesic space is a CAT(0) space if and only if CN-inequality is satisfied. Examples of CAT(0) spaces are complete, simply connected Riemannian manifold having nonpositive sectional curvature, -Trees, Euclidean buildings, pre-Hilbert spaces, the complex Hilbert ball, with hyperbolic spaces. A complete CAT(0) space is called the Hadamard space. The normed linear spaces satisfy the CN-inequality if and only if it satisfies the parallelogram identity (see [24]). It is not so unexpected to have an inner product-like notion in Hadamard spaces.

Definition 6 (see [25]). The concept of quasilinearization was introduced by Berg and Nikolaev [25] as follows: let us first denote a pair by and call it a vector. Then, quasilinearization is defined as a map defined by It can be seen that , , and for all Also the Cauchy-Schwartz inequality is satisfied if and only if

Lemma 7. Let be CAT(0) space, , and Then, (i) [26](ii) [10](iii) [23]

Definition 8 (see [10]; demiclosedness principle). Let be a nonempty closed and convex subset of a Hadarmard space and be a mapping. is said to be -demiclosed at zero if there exists a bounded sequence such that -converges to and , then
The notion of asymptotic properties was introduced in a general framework of Hadamard space as follows.

Definition 9 (see [10]). Let be a bounded sequence in Hadamard space . For any we put (1)The asymptotic radius of is given by .(2)The asymptotic centre of is the set

In Hadamard space, consist of exactly one point.

Lemma 10 (see [10]). If is a bounded sequence in Hadamard space with is the subsequence of with and the sequence converges, then
Let be the proper, convex, and lower semicontinuous function. For all , we define the Moreau-Yosida resolvent of in Hadamard space as follows: It was proven in [24] that the set of a fixed point of the resolvent associated with coincides with the set , i.e., the set of minimizers of . Also, for , the resolvent of is nonexpansive (see [27]).

Lemma 11 (see [28]). Let be a Hadamard space and be a proper, convex, and lower semicontinuous function. Then, for all and , we have

Lemma 12 (see [29]). Let be a CAT(0) space and . Let and be the real numbers in such that . Then, the following inequality holds:

Lemma 13 (see [30]). Let be a sequence of nonnegative real numbers, be a sequence of real numbers in with a sequence of nonnegative real numbers with and be a sequence of real numbers with . Suppose that for all . Then,

Lemma 14 (see [31]). Let be a complete CAT(0) space, a sequence in , and , then -convergence to if and only if for all

3. Iterative Algorithm

In this section, we present our main result which deals with approximation of the common solution of the finite family of convex minimization problems and common fixed point of -demicontractive mappings in complete CAT(0) space. First, we give a coincise statement for our algorithm as follows.

Let be a complete CAT(0) space and be a nonempty closed convex subset of . For , let be a finite family of proper, convex, and lower semicontinuous functions, and for , let be a finite family of -demicontractive mappings such that is demiclosed at zero. Suppose the solution set

Let and be sequences in such that

(C1) and

(C2) and where

(C3)

Choose arbitrarily and let be a sequence generated by the following algorithm.

Algorithm 15.

We begin the proof of the convergence of Algorithm 15 with the following lemmas.

Lemma 16. The sequence generated by Algorithm 15 is bounded.

Proof. Let then for all and for all Thus, we have Hence, for Since and is nonexpansive, then we have Similarly, for all and Thus, we obtain Moreover, then we obtain Therefore, from (24), (25), and (26), we have Furthermore, it follows from Lemma 7 (ii) and Lemma 12 that Thus, from (C2), we obtain Therefore, from Lemma 7 (v), we get This implies that is bounded. Therefore, are bounded.

Lemma 17. Let be a subsequence of the sequence generated by Algorithm 15 such that for all Thus, the following conclusions hold: (i) exists for all (ii) for (iii) for

Proof. From (30), we have Hence, Therefore, exists.
(ii) Also, from Lemma 7, we have Also from Lemma 11, we obtain Thus, from (32), we get Therefore, Now, we show that From the fact that and we get Hence, Continuing in this manner, we get Hence, by induction, In addition, using the triangle inequality with (35), (37), and (39), we get the following: Therefore, (iii) Now, we show that From (28), we have
(30)
Thus, From (32), we have Hence, This implies that

Now, we prove the strong convergence of the iterative scheme.

Theorem 18. The sequence generated by Algorithm 15 converges strongly to an element , where

Proof. Let From Lemma 7 (i), we obtain Since so from (47), we obtain Moreover, since we have Hence, from (50), we have where Obviously, , and from condition (C1), we have Also, since we obtain , and from Lemma 14, we have Thus, Clearly, Therefore, from Lemma 13, we obtain that This implies that Therefore, converges strongly to

The following can be obtained as consequence results from our main theorem.

(i) If , then we have the following result for approximating a common solution of minimization and fixed point problem for a demicontractive mapping .

Corollary 19. Let be a complete CAT(0) space and be a nonempty closed convex subset of . Let be a proper, convex, and lower semicontinuous function, and let be -demicontractive mappings such that is demiclosed at zero. Suppose Let and be sequences in such that
(C1) and
(C2)
(C3)

Choose arbitrarily and let be a sequence generated by the following algorithm.

Algorithm 20.

Then, the sequence converges strongly to a solution

Also, changing from demicontractive into quasinonexpansive mapping, we have the following result.

Corollary 21. Let be a complete CAT(0) space and be a nonempty closed convex subset of . For , let be a finite family of proper, convex, and lower semicontinuous functions, and for , let be a finite family of quasinonexpansive mappings such that is demiclosed at zero. Suppose Let and be sequences in such that
(C1) and
(C2) and
(C3)

Choose arbitrarily and let be a sequence generated by the following algorithm.

Algorithm 22.

Then, the sequence converges strongly to a solution

4. Numerical Example

In this section, we present a computational example to illustrate the convergence behaviour of the proposed iterative scheme. We perform the numerical experiment using MATLAB programming.

Example 23. Let (Euclidean -dimensional space) with the Euclidean norm defined by and inner product

It is clear that is a complete CAT(0) space. Let

Also, for , let be defined by where is a matrix and is a vector for all . It is easy to see that is proper, convex, and lower semicontinuous.

Also, for , let be defined by

We need to show that is a -demicontractive mapping, for all .

Recall that is -demicontactive if

Clearly,

Also,

Then,

Now, since

Therefore, where . Hence, is -demicontractive, where .

Since and then . Hence,

We choose ; clearly,

Since , then for . Also, , Hence, conditions (1)–(3) are satisfied. In this case, our algorithm becomes

In our computation, the initial values are generated randomly in where the values of , and are as follows: (i)Case I: (ii)Case II: (iii)Case III: (iv)Case IV:

We compare the performance of Algorithm 15 with Algorithm (3) (which refers to Iterative scheme 10 in Theorem 3), choosing , and We also use as the tolerance level for our computation. The numerical results are shown in Table 1 and Figure 1.

(a)

(b)

(c)

(d)

5. Conclusions

In this paper, we introduced a modified Halpern-type algorithm for approximating the common solution of the finite family of convex minimization problems and fixed point problems for -demicontractive mappings in CAT(0) spaces. A strong result is proven under a mild condition, and a numerical example is provided to illustrate the convergence behaviour of the sequence generated by our iterative scheme. This result improves and generalizes many other recent results such as [32] from Hilbert to CAT(0) spaces. We are hopeful that this result can be extend to the split minimization, split inclusion, and the split fixed problems (see, e.g., [32–34]) in CAT(0) spaces. We wish to consider this in our future research.

Data Availability

No data is need for this research.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors acknowledge the Department of Mathematics and Applied Mathematics at the Sefako Makgatho Health Sciences University for making their facilities available for the research. This research was completed in part while the second author was visiting the Federal University of Agriculture Abeokuta, Nigeria, from August to October 2021. The second author thanks the institution for making their facilities available for the research. L.O. Jolaoso is supported by the Postdoctoral Research Funding from the Research Office, Sefako Makgatho Health Sciences University, South Africa.

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Copyright © 2022 P. V. Ndlovu 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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