Iterative Scheme for Split Variational Inclusion and a Fixed-Point Problem of a Finite Collection of Nonexpansive Mappings

Dilshad, M.; Aljohani, A. F.; Akram, M.

doi:https://doi.org/10.1155/2020/3567648

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Volume 2020 | Article ID 3567648 | https://doi.org/10.1155/2020/3567648

Iterative Scheme for Split Variational Inclusion and a Fixed-Point Problem of a Finite Collection of Nonexpansive Mappings

M. Dilshad,¹A. F. Aljohani,¹and M. Akram²

Academic Editor: Mohammad Mursaleen

Received31 Jul 2020

Revised22 Aug 2020

Accepted31 Aug 2020

Published09 Oct 2020

Abstract

This article is aimed at introducing an iterative scheme to approximate the common solution of split variational inclusion and a fixed-point problem of a finite collection of nonexpansive mappings. It is proven that under some suitable assumptions, the sequences achieved by the proposed iterative scheme converge strongly to a common element of the solution sets of these problems. Some consequences of the main theorem are also given. Finally, the convergence analysis of the sequences achieved from the iterative scheme is illustrated with the help of a numerical example.

1. Introduction

Let and be two real Hilbert spaces endowed with inner product and induced norm . A mapping is called contraction, if such that , , . If , then becomes nonexpansive. A mapping is said to have a fixed point, if such that . Further, if , is a finite collection of nonexpansive mappings. Then, the fixed-point problem (FPP) is defined as find such that

It is easy to show that if , then is closed and convex. Many iterative methods have been adopted to examine the solution of a fixed-point problem for nonexpansive mappings and its variant forms, see [1–5] and references therein.

We know that most of the techniques for solving the fixed-point problems can be acquired from Mann’s iterative technique [3], namely, for arbitrary , compute where is a nonexpansive mapping from a nonempty closed convex subset of Hilbert space to itself and is a control sequence, which force to converge (weak) to a fixed point of . To obtain the strong convergence result, Moudafi [4] proposed the viscosity approximation method by combining the nonexpansive mapping with a contraction of given mapping over . For an arbitrary , compute the sequence generated by where goes slowly to zero. The sequence achieved from this iterative method converges strongly to a fixed point of .

On the other hand, let us recall some work about split variational inequality/inclusion problems. A multivalued mapping is called maximal monotone, if its graph is not properly contained by the graph of any other monotone mapping. A monotone mapping is maximal monotone if and only if for , for every implies that . If is maximal monotone, then operator is well defined, nonexpansive, and known as the resolvent of with parameter , which is defined at every point of the domain.

The idea of split variational inequality problem (SVIP) given by Censor et al. [6], which amounts to saying find a solution of variational inequality whose image, under a given bounded linear operator, solves another variational inequality. Find such that and such that where and are closed, convex subsets of Hilbert spaces and , respectively; is a bounded linear operator, and and are two operators. They studied the weak convergent result to solve SVIP.

Moudafi [7] generalized SVIP and introduced split monotone variational inclusion problem (S_pMVIP): find such that and such that where and are multivalued monotone mappings, is a bounded linear operator, and are two single-valued operators. The author also composed an iterative algorithm to solve (S_pMVIP) and showed that the sequence achieved by the proposed algorithm converges weakly to the solution of (S_pMVIP). Numerous iterative methods have been investigated for split variational inequality/inclusion problems, split common fixed-point problems, split feasibility problems, and split zero problems and their generalizations, see [6, 8–16] and references therein.

If in S_pMVIP, then we obtain the split variational inclusion problem (S_pVIP) considered in [8], stated as find such that such that

Byrne et al. [8] proposed the following iterative scheme for S_pVIP and studied the strong and weak convergence. For arbitrary , compute the iterative sequence achieved by the following scheme: for

Recently, Kazmi and Rizvi [17] suggested and examined an iterative algorithm to estimate the common solution for S_pVIP and a fixed-point problem of a nonexpansive mapping in Hilbert spaces. Puangpee and Sauntai [18] studied the split variational inclusion problem and fixed-point problem in Banach spaces. Haitao and Li [19] investigated the split variational inclusion problem and fixed-point problem of nonexpansive semigroup without prior calculation of operator norm. Later, many authors studied the common solution of split variational inequality/inclusion problem and fixed-point problem of nonexpansive mappings in the framework of Hilbert/Banach spaces, see for example [18–24] and references therein.

Following the works in [4, 7, 8, 17] and by the current research in this flow, we propose an iterative scheme to approximate a common solution of FPP and S_pVIP. We prove that the sequences achieved by the proposed iterative scheme strongly converge to the common solution of FPP and S_pVIP. The iterative scheme and results discussed in this article are new and can be viewed as generalization and refinement of the previously published work in this area.

2. Prelude and Auxiliary Results

In this section, we assembled some underlying definitions and supporting results.

Definition 1. Let , the metric projection onto the set is defined as and .
is also characterised by the facts that , and

Remark 2 (see [25, 26]). For a nonexpansive mapping and projection onto , the following results hold in Hilbert spaces: (i)(ii)For all , Thus, for all , we get (iii)For all ,

Definition 3. A mapping is said to be (i)monotone, if (ii)-strongly monotone, if there exists a constant such that (iii)-inverse strongly monotone, if there exists a constant such that (iv)firmly nonexpansive, if Some important characteristics of an averaged operator are mentioned below; for more details, we refer to [7, 27, 28].

Definition 4. A mapping is called an averaged if and only if is the average of identity mapping and a nonexpansive mapping, that is, , where and is nonexpansive.

Thus, firmly nonexpansive mappings are averaged. It can also be seen that averaged mappings are nonexpansive.

Proposition 5. (i)Let be an averaged and be a nonexpansive mapping, then is averaged for (ii)If the composite is averaged and have a nonempty common fixed point, then (iii)If is -inverse strongly monotone, then for , is -inverse strongly monotone(iv) is averaged if its compliment is -inverse strongly monotone for some

Lemma 6 (see [29]). Assume that is nonexpansive self-mapping of a closed convex subset of a Hilbert space . If has a fixed point, then is demiclosed, i.e., whenever is a sequence in converging weakly to some and the sequence converges strongly to some , then , where is the identity mapping on .

Lemma 7 (see [5]). If is a sequence of nonnegative real numbers such that where is a sequence in (0,1) and is a sequence in such that (i)(ii)Then, .
We denote the solution set of S_pVIP by and of FPP by .

3. Iterative Scheme and Its Convergence

In this section, we present the iterative scheme and show that the sequences obtained from the proposed iterative scheme converge strongly to the common solution of FPP and S_pVIP.

For integer , we define the mapping with the mod function, which is taking values from the set , that is, if for some integer and , then if and if .

Iterative Scheme 8. Step 0. Take . Choose arbitrary and let .

Step 1. Given , compute as update and go to Step 1.

Condition C. We assume that , , is a finite number of nonexpansive mappings such that and

Lemma 9. and are solutions of S_pVIP, if and only if and , for some .

Proof. The proof of the lemma follows immediately from the definitions of resolvent operators.

Remark 10. If is the resolvent of maximal monotone mapping , is the adjoint operator of and is the spectral radius of . Then, using the properties of averaged mapping, one can easily show that the operator is averaged with , .

Now, we prove the following lemma which guarantees the contractivity of , which is needed to prove our main result.

Lemma 11. Let and be two real Hilbert spaces and be a bounded linear operator. Suppose that and are maximal monotone operators and be a nonexpansive mapping. Let be a -contraction mapping with constant . For any , we define a mapping on by where , is the spectral radius of the operator , and is the adjoint operator of . Then, the mapping is a contraction with constant ; hence, has a unique fixed point.

Proof. The operators and are averaged being firmly nonexpansive. For , the operators and are averaged and hence nonexpansive. Thus, for all , , we have Since implies that is a contraction, hence has a unique fixed point.

Theorem 12. Let and be two real Hilbert spaces and be a bounded linear operator. Assume that and are two maximal monotone operators and is a contraction with constant . Let , , be a finite collection of nonexpansive mappings satisfying the Condition such that . Let be a spectral radius of , where is the adjoint of such that and be a sequence in (0,1) such that , , and . Then, the iterative sequences and generated by Iterative Scheme 8 converge strongly to , where .

Proof. Let , then we have , , and , , then using Iterative Scheme 8, we evaluate Denoting and using (16), we have using (28), (27) becomes Since , we obtain Now, we show that is bounded. Hence, is bounded, which implies that the sequences , , and are also bounded. It follows from nonexpansiveness of , , and Lipschitz continuity of with constant that that is, where . Since, , the operator is average and hence nonexpansive, then we have From (34), (33) becomes let , , by using Lemma 7, we conclude that Now, we show that as . From (29), it follows that Therefore, Since , , and , we get Since and using (27) and (29), we obtain Thus, we get By (37) and (41), we have that is, Since , , as , we get We recognized that the following relation holds: By Iterative Scheme 8, we can easily see that as . From (44) and nonexpansiveness of , it follows that By using (36) and (45), we conclude that Now, using (47) and (44), we write as , that is, Boundedness of implies that there exists a subsequence of , converging weakly to . Because the collection of mappings is finite, we can say for some integer Thus, from (49), we have Therefore, using Lemma 6, we conclude that Thus, by the assumptions of Condition , we have . On the other hand, We know that the graph of a maximal monotone operator is weakly strongly closed; hence, by taking and using (37) and (44), we get Since , have the same asymptotical behaviour, converges weakly to . Therefore, by (39), the nonexpansive property of and Lemma 6, we get . Thus, .
Now, we need to show that , where .
We have since .
Finally, we show that which implies that From Lemma 7 and (55), we conclude that and from , , and as , we achieve that . This completes the proof.

4. Consequences

Suppose and are closed convex subsets of Hilbert spaces and , respectively. Then, find such that is called the split feasibility problem (SFP), where is a bounded linear operator. Byrne [9] introduced the algorithm to approximate the solution of (58): where and are orthogonal projections onto and , respectively.

The split common fixed-point problem (SCFPP) is an extension of Problem (58), which has been widely investigated in the present scenario. The SCFPP is the inverse problem design to search a vector in a fixed-point set so that its image under a bounded linear operator corresponds to other fixed-point set, that is, find such that where and are nonexpansive mappings. By putting and , in (59), we can have an iterative scheme, which converges to the solution of SCFPP.

We denote the solution set of SFP (58) and SCFPP (60) by , and , respectively. The following corollaries are given as consequences of Theorem 12.

Corollary 13. Let and be two closed convex subsets of Hilbert spaces and , respectively. Let be a bounded linear operator and be a contraction mapping with constant . Let , , be a finite collection of nonexpansive mappings satisfying the condition such that . Let be the spectral radius of , where is the adjoint of such that and be a sequence in (0,1) with , , and . Then, the iterative sequences and generated by Iterative Scheme 8 with and converge to , where .

Corollary 14. Let and be two real Hilbert spaces and be a bounded linear operator. Assume that and are maximal monotone operators and is a -contraction mapping with constant . Let , , be a finite collection of nonexpansive mappings satisfying the condition such that . Let be spectral radius of , where is the adjoint of such that and be a sequence in (0,1) with , , and . Then, the iterative sequences and obtained from Iterative Scheme 8 with and converge to , where .

Remark 15. If we take , a nonexpansive mapping, then we can obtain the iterative scheme and its convergence theorem for the common solution of S_pVIP and a nonexpansive mapping , studied in [17].
At last, we illustrate the convergence analysis of the proposed iterative scheme with the help of the following numerical example.

5. Numerical Example

Let and defined as and defined as . For , we compute the resolvents of and as

It can be easily seen that, here, . Further, let , and are three nonexpansive mappings, defined by such that

Let be a contraction mapping defined as and be a bounded linear operator defined as with adjoint operator such that .

Since and , so we choose and ; then, the sequences and generated by the iterative scheme are evaluated as or for some positive integer , and , we can write

From Table 1, we conclude that for two arbitrary different initial points and , the sequences and converge approximately to a point .

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

No potential conflict of interest is reported by the authors.

Authors’ Contributions

All authors read and approved the final manuscript.

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Copyright © 2020 M. Dilshad 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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