An Implicit Algorithm for Finding a Fixed Point of a <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.838901pt" id="M1" height="14.2383pt" version="1.1" viewBox="-0.0657574 -11.3994 12.5675 14.2383" width="12.5675pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-82" d="M699 368C699 549 574 666 407 666C186 666 23 488 23 277C23 113 129 -3 288 -13L307 -26C431 -111 501 -139 533 -147C559 -154 613 -163 658 -164L666 -141C597 -111 507 -66 430 -11L416 -1C580 42 699 190 699 368ZM601 371C601 227 518 54 381 22L354 40L278 24C175 47 120 145 120 269C120 451 235 631 398 631C540 631 601 521 601 371Z"/></g></svg>-</span>Nonexpansive Mapping in Locally Convex Spaces

Soori, E.; Omidi, M.R.; Farajzadeh, A.P.; Wang, Yuanheng

doi:https://doi.org/10.1155/2021/9913909

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Volume 2021 | Article ID 9913909 | https://doi.org/10.1155/2021/9913909

An Implicit Algorithm for Finding a Fixed Point of a -Nonexpansive Mapping in Locally Convex Spaces

E. Soori,¹M.R. Omidi,²A.P. Farajzadeh,³and Yuanheng Wang⁴

Academic Editor: Zoran Mitrovic

Received26 Mar 2021

Accepted29 Jul 2021

Published13 Aug 2021

Abstract

In this paper, the notion of the -duality mappings in locally convex spaces is introduced. An implicit method for finding a fixed point of a -nonexpansive mapping is provided. Finally, the convergence of the proposed implicit scheme is investigated. Some examples in order to illustrate of the main results are presented.

1. Introduction

Let be a nonempty closed and convex subset of a Banach space and be the dual space of . Let denote the pairing between and . The normalized duality mapping is defined byfor all . It is well known that if is a single-valued mapping, then it is norm- continuous (see, for more details, Theorem 2.6.10 in [1]). In this paper, the duality mapping of locally convex space is studied and denoted by for the seminorm .

Suppose that is a family of seminorms on a locally convex space which determines the topology of that will be denoted by . Let be a nonempty closed and convex subset of . A mapping is called -nonexpansive if , for all and . Also, the function is -contraction if there exists such that , for all and .

The methods of iterations, which find a number of real applications, are popular for problems in sciences and engineering; see, e.g., [2–6]. Recently, various iterative algorithms have been investigated for fixed points of the nonlinear mapping, especially, for the mappings of nonexpansive type; see, e.g., [7–11] and the references therein.

Also, the general implicit algorithm for finding a fixed point of a -nonexpansive mapping is given bywhere is a -contraction and is a -nonexpansive mapping. On the contrary, it is proven that there exists a sunny -nonexpansive retraction of onto and such that the the sequence converges to with respect to the topology .

Moreover, some notions and concepts in locally convex spaces are recalled. The Hahn–Banach theorem and Banach contraction principle will be generalized to locally convex spaces. The results of this article can be viewed as implicit and locally convex space version of some results given in [1, 12–17].

2. Methods and Preliminaries

Suppose that is a family of seminorms on a locally convex space which determines the topology of will be denoted by .

Let and be a retraction of onto , that is, , for each . Then, is said to be sunny if, for each and with , we have

A subset of is said to be sunny -nonexpansive retract of if there exists a sunny -nonexpansive retraction of onto . We know that if is a smooth Banach space and a retraction of onto , then is sunny nonexpansive if and only iffor each and . For more details, see [18]. In the sequel, we will prove inequality (4) for a real locally convex space.

Now, we recall the following definitions:(1)The locally convex topology is separated if and only if the family of seminorms possesses the following property: for each , there exists such that (see [19]) or equivalently(2)Let be a locally convex topological vector space over or . If , the polar of is denoted bysee [20].(3)Suppose is a Hausdorff topological group, and is a net in defined on a directed set . is left Cauchy when the following happens: for every neighborhood of the identity of , there exists an such thatsee [20].

Suppose that is a family of seminorms on a locally convex space which determines the topology of and . Let be a subset of and

Also,for every linear functional on . Notice that if, for each , , and , thenfor more details, see [12].

We will make use of the following theorems.

Theorem 1 (see [12]). Suppose that is a family of seminorms on a real locally convex space which determines the topology of and is a continuous seminorm and is a vector subspace of such that

Let be a real linear functional on such that . Then, there exists a continuous linear functional on that extends such that

Theorem 2 (see [12]). Suppose that is a family of seminorms on a real locally convex space which determines the topology of and a nonzero continuous seminorm. Let be a point in . Then, there exists a continuous linear functional on such that and .

Next, we bring the following known results for easy reference.

Theorem 3 (see Proposition 2.5.2 in [1]). Let be a topological space and a function. Then, the following statements are equivalent: (a) is lower semicontinuous; (b) for each , the level set is closed.

Theorem 4 (see Theorem 3.26 (Banach–Alaoglu) in [20]). Suppose that is a locally convex space and is a neighborhood of zero in . Then, , the polar set of , is compact.

For general topological spaces, we know the useful property as follows.

Theorem 5 (see Theorem 3.3.18 in [21]). For a topological space , the following are equivalent:(i) is compact(ii)Each net in has a convergent subnet

3. Main Result

First, we define our notation of the -duality mapping as follows.

Suppose that is a family of seminorms on a real locally convex space which determines the topology of , is a continuous seminorm, and is the dual space of . A multivalued mapping , defined byis called -duality mapping. Obviously, and . Indeed, let ; if , then , and if , by applying Theorem 2, there exists a linear functional such that and . Putting , thenand then, .

Now, we present the following result that extends Proposition 4.1.1 in [1].

Lemma 1. Let be a family of seminorms on a Hausdorff and complete locally convex space which determines the topology of , a bounded below and lower semicontinuous function, for each , and a sequence in such that , for all and . Then, converges to a point , and for each ,for all .

Proof. Since , for each and , then is a decreasing sequence for each . Moreover,for . Letting , we haveand then, , for each . This implies that is a left Cauchy sequence in . Since is Hausdorff and complete, there exists a unique such that . Let with . Then,for each . Since is lower semicontinuous and is continuous (Theorem 1.4 in [19]), letting , we conclude thatfor all and .

Now, we state an extension of Banach contraction principle to locally convex spaces, and we call it Banach -contraction principle.

Theorem 6 (Banach -contraction principle). Let be a family of seminorms on a separated and complete locally convex space which determines the topology of and a -contraction mapping with Lipschitz constant . Then,(a)T has a unique fixed point .(b)For arbitrary , the Picard iteration process defined by converges to .(c), for all and .

Proof. (a)For each , let be a function defined by for each . Since is continuous, is also a continuous function. Also, is a -contraction mapping; then, for each and , which conclude that Hence, for each . Let be an arbitrary and fixed element in . We define the sequence in by (). By using (24), we have for each and . Since is Hausdorff, by applying Lemma 1, there exists an element such that for each and . Notice that is continuous (page 3 in [19]) and ; therefore, . Let be another fixed point for and for some . Then, which is a contradiction. Therefore, , for each . Since is separated, we have . Hence, has unique fixed point .(b)This assertion follows from part (a).(c)By applying (22), we have , for each . This implies from (26) that , for each .

Next, we prove a generalization of Proposition 2.4.7 in [1] as follows.

Lemma 2. Let be a locally convex space. Then, for and for each with , the following statements are equivalent:(a), for all with (b)There exists such that

Proof. (a) (b). Let and , and we define . It is clear that and ; then, Putting , it is clear that from Theorem 4, is compact; then, by applying Theorem 5, the net has a limit point such that . From the above equations , let ; by limiting, we have . Also, since , by using (28) and limiting, we have . Therefore, which conclude that and . Set ; then, and Hence, and .(b) (a). Suppose, for and with , there exists such that . We know thatHence, for with , we havewhich implies that . This completes the proof.

The next theorem is significant in the sequel, and it extends Proposition 2.10.21 in [1].

Theorem 7. Let be a separated locally convex space, a nonempty convex subset of , and a nonempty subset of . Also, let be a single valued, for every and a retraction of onto . Then, for each , the following statements are equivalent:(a)(b) is sunny -nonexpansive

Proof. (a) (b). First, we show that is sunny. For and , set . Since is convex, , for each . Hence, from (a), we have Because and , we have Combining (34) and (35), we have Then, , for each . Thus, from the fact that is separated, . This means that is sunny. Now, we show that is -nonexpansive. For and , we have from (a) that and . Hence, (notice that is single valued). On the contrary, Therefore, be applying (37) and (38), we have it follows that is -nonexpansive.(b) (a). Conversely, suppose that is the sunny -nonexpansive retraction from onto , and . Then, and . Since is convex, , for . Also, is sunny; then,and therefore,Hence, from Lemma 2, there exists such thatSince is single valued, we can say that

Next, as a direct consequence of Theorem 6.5.3 in [18], we have the following result. To apply Theorem 6.5.3 in [18] in Theorems 8, 9, 11, and 12 and Corollary 1, it is considered that , for each , when is in (28) and (34), , , and , respectively.

Theorem 8. Let be a family of seminorms on a locally convex space which determines the topology of and be a convex subset of , and . Then, for each , the following are equivalent:(1).(2)There exists an such that , for each , , and(3)There exists an such that , for each , , and

The next theorem is similar to Corollary 6.5.5 in [18].

Theorem 9. Let be a family of seminorms on a real locally convex space which determines the topology of and be a convex subset of , and . Then, for each , the following statements are equivalent:(1)(2)There exists an such that , for every

Proof. (1). Let and . Then, from Theorem 8, there exists an such that , for each , , and Set . It is easy to see that ; therefore, Then, . Hence, from (46) and (47), we have . Since , thenAlso, , for every ; therefore,for every ; hence, we havefor all . This means that

Corollary 1. Let be a family of seminorms on a locally convex space which determines the topology of , a nonempty closed convex subset of , , and be a sunny -nonexpansive retraction of onto . Let be single-valued duality mapping for each ; then,for each and .

Proof. Let and . Set , for each ; then, andfor each . Now, from Theorem 9, there exists such that . Since is single valued, we can say thatand since , then

We now extend the result of Theorem 2.6.10 in [1] as follows.

Theorem 10. Let be a locally convex space and be a single-valued duality mapping. Then, is continuous from to topology, for each .

Proof. Let be fixed and arbitrary, , and such that in , and we prove that in the topology. First, assume that ; then, (from (13)). It is clear that , also is continuous Theorem 1.4 in [19]; therefore,and hence, , for every that (from (9)). On the contrary, if , then ; therefore, . So, we conclude that , for every . Hence, in the topology. Second, assume that . Sinceand is single-valued set . Then,Since is bounded with respect to , then there exists such that , for each . Now, it is shown that is bounded in .
One can see that is a locally convex space and the family of seminorms defines the topology on it, where , for each (Chapter 5, Definition 1.1 in [22]). Hence,when . Also, if ,for each . By applying (60) and (61), there exists related to each and , such that , for each . This means that is bounded in the topology of . Putting , we have , for each , where is the polar of and it is compact (Theorem 4). Then, from Theorem 5, there exists a subnet of such that in the topology, equivalentlyWe know that the function on is lower semicontinuous in the topology. Indeed, if such that in the topology, thenfor each ; therefore, and , i.e.,So, by applying Theorem 3, is lower semicontinuous in the topology on . Hence, we haveSince in and in weak topology, thenTherefore, using (62) and (66) implies thatOn the contrary, . Thus,Since , using (65), we haveand therefore,This means that . Finally, since , from (62), we conclude that . This completes the proof.

In the next theorem, we prove an existence theorem of a sunny -nonexpansive retract.

Theorem 11. Let be a family of seminorms on a real separated and complete locally convex space which determines the topology of , a nonempty closed convex and bounded subset of such that every sequence in has a convergent subsequence, and be a -nonexpansive mapping such that , the fixed points set of . Let be single valued for every . Then, is a sunny -nonexpansive retract of and the sunny -nonexpansive retraction of onto is unique.

Proof. Let be fixed; then, there exists a sequence in such thatFor this means, we define the function , withfor each and . It is clear that is -nonexpansive becausefor every . Then, by applying Theorem 6, there exists a unique point in such that , for each ; therefore,Since is bounded in , there exists such that , for each and . Also,and then,for each . Next, we show that the sequence converges to an element of . In the other words, we show that the limit set of (denoted by ) is a subset of . First, for each and , we havebecause andAlso, since is single valued, from definition (13), we haveTherefore,From our assumption, has a subsequence converges to a point in . Let and be subsequences of such that and converge to and , respectively. Then, (76) implies thatfor each . Since is separated, ; hence, and similarly .
Now, and , and we claim thatIndeed, from the fact that is single valued and continuous from to topology, we haveand hence, by using (77),Similarly, since ; then,that is, , for each , and since is separated, . Thus, converges to an element of . Therefore, a mapping can be defined by . Then, since , by using (84) and for each ,It follows from Theorem 7 that is a sunny -nonexpansive retraction of onto . Let be another sunny -nonexpansive retraction of onto . Then, from Corollary 1,for each and . Putting in (86) and in (88), we have and ; then,for each . Since is separated, . This completes the proof.

Proposition 1. Let be a family of seminorms on a separated locally convex space which determines the topology of . Then,for all and such that .

Proof. Let , . Then,

Theorem 12. Let be a family of seminorms on a real separated and complete locally convex space which determines the topology of , a nonempty closed convex and bounded subset of such that every sequence in has a convergent subsequence, and a -nonexpansive mapping such that . Let be single valued for each and be a -contraction on ; also, is a sequence in such that . Then, there exists a unique and sunny -nonexpansive retraction of onto such that the following sequence , generated byconverges to .

Proof. Since is a -contraction, there exists such thatfor each and . We divide the proof into five steps. Step 1: the existence of which satisfies (92). This follows immediately from the fact that, for every , the mapping , given by is a -contraction. To see this, put , . Then, we have Therefore, by Theorem 6, there exists a unique point such that , that is, Step 2: , for each . Since is bounded in , there exists such that , for each and . Then, for each . Step 3: , where denotes the set of -limit points of subsequences of . Let and be a subsequence of such that . By using (97), we have for each . Hence, , for each , and since is separated, we conclude that . Step 4: there exists a unique sunny -nonexpansive retraction of onto and such that We know from Theorem 10 that is continuous from to topology; then, by Theorem 11, there exists a unique sunny -nonexpansive retraction of onto . Theorem 6 guarantees that has a unique fixed point . Since by the definition of , there exists subsequence of such that On the contrary, every sequence in has a convergent subsequence; then, there exists subsequence of such that and . By Step 3, we have . From Theorem 7, we have and since is continuous, then Therefore, by applying (101)–(103), we have . Step 5: converges to in .Since , thus . Now, from Proposition 1 and our assumption, we havefor each . Hence,for each . Since , from (99) and (105), we conclude thatfor each . That is, in .

4. Numerical Example

The following examples illustrate Theorem 12. The first example is in the setting of a locally convex space that is not normable and the rest in finite dimensional spaces.

Example 1. Let be an arbitrary positive number, , the set of all continuous complex valued functions, and . For each , the seminorm on is defined by , for each . Note that is a locally convex space with the topology induced by . It can be proved by Urysohn’s Lemma that it is not normable.
In Theorem 12, let be convex neighborhood of 0 with and . Suppose that is the identity mapping on the complex numbers, and let be defined by . The following examples illustrate Theorem 12. The first example is in the setting of a locally convex space that is not normable and the rest in finite dimensional spaces.

Example 2. Let be an arbitrary positive number and be the set of all continuous complex valued functions and . For each , we define the seminorm on by , for each . Note that is a locally convex space with the topology induced by . It can be proved by Urysohn’s Lemma which is not normable.
In Theorem 12, let be convex neighborhood of 0 with and . Suppose that is the identity mapping on the complex numbers, and let be defined by and , respectively. It is obvious that is -nonexpansive and is -contraction mapping on . It can be verified that is a sunny -nonexpansive retraction from onto ; then, obviously, we have the sequence generated by (92) which converges to with respect to the topology induced by .

Example 3. In Theorem 12, let , , and be the only seminorm on , i.e., . Let and be a -nonexpansive mapping and be a -contraction mapping on . It is obvious is a sunny -nonexpansive retraction from onto and the sequence generated by (92) converges to and , respectively. It is obvious that is -nonexpansive and is -contraction mapping on . It can be verified that is a sunny -nonexpansive retraction from onto ; then, obviously, we have the sequence generated by (92) converges to with respect to the topology induced by .

Example 4. In Theorem 12, let , , and be the only seminorm on , i.e., . Let and be a -nonexpansive mapping and be a -contraction mapping on . It is obvious is a sunny -nonexpansive retraction from onto and the sequence , generated by (92), converges to .

Example 5. In Theorem 12, assume and , and and are two seminorms on , i.e., . Let and be a -nonexpansive mapping and be a -contraction mapping on in Theorem 11. It is easy to check that is a sunny -nonexpansive retraction from onto , and the sequence generated by (92) converges to .

Data Availability

The data used to support the findings of the study are available within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11671365).

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Copyright © 2021 E. Soori 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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