Convergence of Some Iterative Algorithms for System of Generalized Set-Valued Variational Inequalities

Akram, M.; Khan, Aysha; Dilshad, M.

doi:https://doi.org/10.1155/2021/6674349

Journal of Function Spaces

On this page

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

Special Issue

Sequence Spaces, Function Spaces and Approximation Theory

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 6674349 | https://doi.org/10.1155/2021/6674349

Convergence of Some Iterative Algorithms for System of Generalized Set-Valued Variational Inequalities

M. Akram,¹Aysha Khan,²and M. Dilshad³

Academic Editor: Mohammad Mursaleen

Received07 Oct 2020

Revised24 Nov 2020

Accepted27 Nov 2020

Published05 Jan 2021

Abstract

In this article, we consider and study a system of generalized set-valued variational inequalities involving relaxed cocoercive mappings in Hilbert spaces. Using the projection method and Banach contraction principle, we prove the existence of a solution for the considered problem. Further, we propose an iterative algorithm and discuss its convergence. Moreover, we establish equivalence between the system of variational inequalities and altering points problem. Some parallel iterative algorithms are proposed, and the strong convergence of the sequences generated by these iterative algorithms is discussed. Finally, a numerical example is constructed to illustrate the convergence analysis of the proposed parallel iterative algorithms.

1. Introduction

The theory of variational inequality was planted in the early 1960’s by Stampacchia [1] in the framework of obstacle constraint minimization problems. The first evolutionary variational inequality was solved in the seminal paper of Lions and Stampacchia [2]. Since its inception, it has enjoyed a vigorous development for the last few decades. This subject has developed in multiple directions using innovative techniques to solve fundamental problems to be insurmountable previously. This field is influential and experiencing an explosive growth in theory as well as applications. Consequently, some of these developments in this area enriched other mutual areas of mathematical and engineering sciences such as economics, transportation, nonlinear programming, and operations research. It has been shown that this theory provides the most natural, direct, simple, unified, and efficient framework for a general treatment of a wide class of unrelated linear and nonlinear problems; see, for example, [3–9] and the references cited therein.

Recently, fixed-point methods have been extensively investigated for solving monotone variational inequalities. Among the fixed point algorithms, Mann-like iterative algorithms are useful for solving several nonlinear problems. The proximal point algorithm is a widely used tool for solving a variety of (single objective) convex optimization problems such as finding zeros of maximal monotone operators and fixed points of nonexpansive mappings, as well as minimizing convex functions. In 2007, Agarwal et al. [10] introduced the following iteration process. Let be a nonempty convex subset of a normed linear space and let be an operator. Then, for arbitrary , estimate by the following -iterative scheme:where and are real sequences in (0,1) satisfying some suitable conditions. Sahu [11] proved that the -iteration process is more applicable than the Picard [12], Mann [13], and Ishikawa [14] iteration algorithms because it converges faster than these iteration processes for contraction mappings and also works for nonexpansive mappings. Recently, Cholamjiak et al. [15] applied the -iteration process for finding a minimizer of a convex function and fixed points of nonexpansive mappings in CAT(0) spaces. In [16], Amir et al. studied the convex constraint multiobjective optimization problem as the constrained set of fixed points of the nonexpansive mapping.

On the other hand, a number of numerical methods such as auxiliary principle, projection method, Wiener-Hopf equations, dynamical systems, and decomposition have been developed for solving the variational inequalities and related optimization problems. Among these methods, the projection method and its variant forms have been proved innovative and important tools for finding the approximate solutions of variational inequalities. In this technique, the concept of projection is used for the fixed-point formulation of variational inequality. This alternative formulation has played a significant role in developing various projection-type methods for solving variational inequalities. It is well known that the convergence of the projection methods requires that the operator must be strongly monotone and Lipschitz continuous. It is also known that the relaxed cocoercive mappings are more general than strongly monotone mappings, and under some mild conditions, relaxed cocoercive mappings can be reduced to strongly monotone mappings. For some related works, see, [17–24].

Following the facts and discussion mentioned above, in this paper, we consider a system of generalized set-valued variational inequalities defined over closed and convex subsets of a real Hilbert space. We establish an equivalence between the system of generalized set-valued variational inequalities and nonlinear projection equations using the projection technique. Further, by virtue of the projection method and Banach contraction principle, we prove an existence result. Furthermore, we propose an iterative algorithm and show that the approximate solution generated by the proposed algorithm converges strongly to the unique solution of the system of generalized set-valued variational inequalities involving relaxed cocoercive and relaxed monotone mappings in Hilbert space. Moreover, we establish equivalence between the system of variational inequalities and altering points problem. Parallel Mann and parallel -iterative algorithms [11] have been proposed for solving the considered system of variational inequalities. Finally, the convergence analysis of the proposed parallel iterative algorithms is discussed. A numerical example is constructed to illustrate the convergence analysis of the proposed parallel iterative algorithms. The results presented in this paper can be viewed as generalizations and refinements of several results existing in the literature and include general variational inequality and some other classes of variational inequalities as special cases.

Now, we enumerate some basic notions, definitions, and results that are worthwhile tools in succeeding analysis and will be utilized in the rest of this paper.

Throughout the paper, unless otherwise specified, let be a real Hilbert space with inner product and induced norm .

Definition 1 [25]. A mapping is said to be(i)-strongly monotone, if there exists a constant such that(ii)-cocoercive, if there exists a constant such that(iii)Relaxed -cocoercive, if there exists a constant such that(iv)Relaxed -cocoercive, if there exist constants such that(v)-Lipschitz continuous, if there exists a constant such that(vi)-contraction, if there exists a constant such that is called nonexpansive, if .

Remark 2. We remark that every -strongly monotone mapping is relaxed -cocoercive mapping and every -cocoercive mapping is -Lipschitz continuous.
Define the norm on byNote that is a Banach space.

Definition 3 [26]. Let be a single-valued mapping. A set-valued mapping is said to be a relaxed monotone with respect to if and only if there exists a constant such thatLet be a nonempty closed and convex subset of a real Hilbert space . Then, for any , there exists a unique nearest point of such thatThe mapping is called the metric projection [27] from onto . Note that the metric projection mapping is nonexpansive from onto (see, Agarwal et al. [28]), i.e.,

Lemma 4 [27]. Let be a closed and convex subset in . Then, for any , the projection of onto satisfiesif and only if

Lemma 5. [29]. Let be a nonnegative real sequence satisfying the following conditionwhere , , and . Then, .

2. System of Generalized Set-Valued Variational Inequalities

Let be a real Hilbert space and and be the nonempty closed and convex subsets of . Let be an index set; for each , let be nonlinear single-valued mappings and be a set-valued mapping. We consider the problem of finding with and such thatwhere and are positive constants. We call problem (15) a system of generalized set-valued variational inequalities. Some special cases of problem (15) are listed below.(i)If are the single-valued mappings and are the identity mappings, then problem (15) reduces to the equivalent problem of finding such that

Problem (16) is called the system of variational inequalities.(ii)If , then problem (16) coincides with the following problem of finding such that

Problem (17) was studied by Sahu et al. [11].(iii)If and , then problem (17) reduces to the following system of nonlinear variational inequalities of finding such that

Problem (18) was studied by Verma [30]. He extended the concept of variational inequalities to the system of nonlinear variational inequalities.(iv)If and are the single-valued mappings, then problem (15) coincides to the following system of extended general variational inequalities of finding such that

An equivalent form of problem (19) was studied by Noor et al. [31].(v)If and , then problem (19) reduces to the following system of variational inequalities of finding , such that

The problem of type (20) is called a system of extended general variational inequalities with four nonlinear operators.

Next, we establish the following lemma which plays a crucial role to prove the existence result for the unique solution of the system of generalized set-valued variational inequalities (15) and to propose an iterative algorithm for studying convergence analysis.

Lemma 6. Let and be the nonempty closed convex subsets of a real Hilbert space . Let be an index set; for each , let and be the nonlinear single-valued mappings. Let be a set-valued mapping. Then, the system of generalized set-valued variational inequalities (15) has a solution , if and only if , and satisfieswhere and are positive constants.

Proof. Let and is a solution of the system of generalized set-valued variational inequalities (15). ThenThen, from Lemma 4, we haveConversely, suppose that and satisfiesAgain, it follows from Lemma 4 thatThus, , and is solution of the system of generalized set-valued variational inequalities (15).

Now, by virtue of the Banach contraction principle, we shall show the existence of the unique solution for the system of generalized set-valued variational inequalities (15).

Theorem 7. Let be a real Hilbert space and be an index set, for each ; let be a closed and convex subset of . Let and be nonlinear single-valued mappings such that is relaxed -cocoercive and -Lipschitz type continuous, and is relaxed -cocoercive and -Lipschitz type continuous. Let be a set-valued mapping such that is a relaxed monotone with respect to with constant , -Lipschitz type continuous with constant , and is -Lipschitz type continuous in the first argument and -Lipschitz type continuous in the second argument. In addition, satisfieswhere andThen, the system of generalized set-valued variational inequalities (15) admits unique solution.

Proof. For any given , we define a mapping bywhere are the single-valued mappings defined bywhere and are positive constants. Utilizing the fact that the projection mapping is nonexpansive, it follows from (31) thatSince the mapping is relaxed -cocoercive and -Lipschitz type continuous, then we havewhich implies thatAgain, using the fact that the mapping is relaxed -cocoercive and -Lipschitz type continuous, then we havewhich implies thatSince is relaxed monotone with respect to with constant in the first argument, -Lipschitz type continuous with constant and is -Lipschitz type continuous in the first argument, then we havewhich implies thatSince is -Lipschitz type continuous in the second argument and is -Lipschitz type continuous with constant , then we haveThus, from (34), (36), (38), and (39), (32) becomesAgain, utilizing the fact that the projection mapping is nonexpansive, it follows from (30) thatSince the mapping is relaxed -cocoercive and -Lipschitz type continuous, then we havewhich implies thatAgain, using the fact that the mapping is relaxed -cocoercive and -Lipschitz type continuous, then we havewhich implies thatSince is relaxed monotone with respect to with constant in the first argument, -Lipschitz type continuous with constant and is -Lipschitz type continuous in the first argument, then we havewhich implies thatSince is -Lipschitz type continuous in the second argument and is -Lipschitz type continuous with constant , then we haveThus, from (43), (45), (47), and (48), (41) becomesNow, it follows from (40) and (49) thatwhere andIt follows from (8), (29), and (50) thatIt follows from conditions (27) and (52) that is a contraction mapping. Therefore, there exists unique such that . Thus, we haveTherefore, by Lemma 6, one can conclude that , and is the unique solution of the system of generalized set-valued variational inequalities (15).

3. Iterative Algorithm and Convergence Result

In this section, we suggest an iterative algorithm to analyse the convergence of the system of generalized set-valued variational inequalities (15).

By utilizing (21) and (22) of Lemma 6, we can offer the following iterative forms:where and are positive constants and the real sequences . Now, we propose the following iterative algorithm.

Algorithm 8. For any given , we choose and . For and and from (54) and (55), letSince and , by Nadler’s theorem [32], there exist and such thatwhere is the Hausdorff metric. LetAgain, it follows from Nadler’s theorem [32] that there exist and such thatContinuing in the same manner, we can figure out the sequences , and by the following iterative process:and for , choose and such thatNow, we are accessible to study the convergence of the proposed iterative algorithm for the system of generalized set-valued variational inequalities (15).

Theorem 9. Let be a real Hilbert space and be an index set, for each ; let be a closed convex subset of . Let and be nonlinear single-valued mappings such that is relaxed -cocoercive, -Lipschitz type continuous; is relaxed -cocoercive, -Lipschitz type continuous; and is -Lipschitz type continuous in the first argument and -Lipschitz type continuous in the second argument. Let be a set-valued mapping such that is a relaxed monotone with respect to with constant and -Lipschitz type continuous with constant . In addition, for each , satisfy the following conditions:(i),(ii)(iii) such thatwhereThen, the approximate solution generated by Algorithm 8 converges strongly to the unique solution of the system of generalized set-valued variational inequalities (15).

Proof. Let and be a solution of (15); then, from (55) and (61) of Algorithm 8 and utilizing the fact that the projection mapping is nonexpansive, we haveSince the mapping is relaxed -cocoercive and -Lipschitz type continuous, then we havewhich implies thatAgain, using the fact that mapping is relaxed -cocoercive and -Lipschitz type continuous, then we havewhich implies thatSince is relaxed monotone with respect to with constant in the first argument, -Lipschitz type continuous with constant and is -Lipschitz type continuous in the first argument, then we havewhich implies thatSince is -Lipschitz type continuous in the second argument and is -Lipschitz type continuous with constant , then, we haveThus, from (68), (70), (72), and (73), (66) becomesAgain, from (54) and (60) of Algorithm 8 and utilizing the fact that the projection mapping is nonexpansive, we haveSince the mapping is relaxed -cocoercive and -Lipschitz type continuous, then we havewhich implies thatAgain, using the fact that mapping is relaxed -cocoercive and -Lipschitz type continuous, then we havewhich implies thatSince is relaxed monotone with respect to with constant in the first argument, -Lipschitz type continuous with constant and is -Lipschitz type continuous in the first argument, then we havewhich implies thatSince is -Lipschitz type continuous in the second argument and is -Lipschitz type continuous with constant , then, we haveThus, from (77), (79), (81), and (82), (75) becomesFor the sake of simplicity, we putThen, it follows from (74) and (83) thatwhich implies thatwhere
and .
Thus, from (86), Lemma 5, and conditions, we havewhich implies thatHence, and . It follows from (62) and (63) that and are Cauchy sequences; we can assume that and , strongly. Next, we show that and . Since , then we haveHence, , so as . Similarly, it is easy to show that . Thus, by Lemma 6, one can deduce that and is a solution to the system of generalized set-valued variational inequalities (15).

4. Altering Points Problem

In this section, the concept of altering points problem is used to find the solution of the considered system of variational inequalities (16). We propose parallel Mann and parallel -iterative algorithms, and the strong convergence of the sequences generated by these parallel iterative algorithms is discussed.

Definition 10. [11]. Let and be nonempty subsets of a metric space . Then, the points and are the altering points of mappings and , ifAlt is called the set of altering points of the mappings and .

Lemma 11 [33]. Let and be nonempty closed subsets of a complete metric space . Let and be Lipschitz continuous mappings with constants and , respectively, such that . Then, the following conditions hold:(i)There exists unique point such that and are altering points of the mappings and .(ii)For arbitrary , the sequence generated byconverges to .

Lemma 12. Let and be nonempty closed convex subsets of a real Hilbert space . Let be an index set, for each ; let and be nonlinear single-valued mappings. Then, is the solution to the system of variational inequalities (16), if and only if satisfieswhere and are positive constants.
Define the mappings and as follows:where are constants in . By virtue of Lemma 4, we can formulate the system of variational inequalities (16) into the following equivalent altering points problem.
Find such thatNow, we propose the following parallel Mann iterative algorithm to solve the system of variational inequalities (16) as follows.

Algorithm 13. Let be nonempty closed convex subsets of a real Hilbert space . For any , let be an iterative sequence in defined bywhere and are the mappings defined by (93) and (94), respectively.

Also, we propose the following parallel -iterative algorithm to solve the system of variational inequalities (16). Note that the parallel -iterative algorithm is more general than the parallel Mann iterative algorithm.

Algorithm 14. Let and be nonempty closed convex subsets of a real Hilbert space . For any , let be an iterative sequence in defined bywhere and are the mappings defined by (93) and (94), respectively.

Theorem 15 [33]. Let and be nonempty closed convex subsets of a Banach space . Let and be Lipschitz continuous mappings with constants and , respectively. Then, the sequence generated by the parallel -iterative algorithm (98)–(99) converges strongly to the unique point such that and are altering points of the mappings and .

Next, we prove the following proposition, which plays a crucial role to prove the convergence of parallel iterative algorithms.

Proposition 16. Let and be nonempty closed convex subsets of a real Hilbert space . For each , let be the single valued mappings such that is -strongly monotone and -Lipschitz type continuous. Let and be the single-valued mappings such that is -strongly monotone and -Lipschitz type continuous with respect to the second argument. Then, the mappings and defined by (93) and (94) are Lipschitz type continuous with constant and , respectively, where and .

Proof. For any given , (93), and using the nonexpansiveness of the projection mapping , we haveSince is -strongly monotone and -Lipschitz type continuous, then, we getwhich implies thatAgain, using the fact that is -strongly monotone and -Lipschitz type continuous with respect to the second argument, we havewhich implies thatUtilizing (102) and (104), (100) becomesi.e., is -Lipschitz continuous, where and . Similarly, one can prove that is -Lipschitz continuous.

Now, we prove the convergence of the parallel Mann iterative scheme (96)–(97).

Theorem 17. Let and be nonempty closed convex subsets of a real Hilbert space . For each , let be the single-valued mapping such that is -strongly monotone and -Lipschitz type continuous. Let and be the single-valued mappings such that is -strongly monotone and -Lipschiz type continuous with respect to the second argument. For any , let be an iterative sequence in generated by Algorithm 13, where satisfying and are the mappings defined by (93) and (94), respectively. In addition, the following condition holds:where and , for . Then(i)There exists unique point , which is the solution of the system of variational inequalities (16).(ii)The sequence generated by parallel Mann iterative Algorithm 13 converges strongly to the point .

Proof. (i)Conclusion follows from Lemma 11 and (95).(ii)It follows from (94), (95), (96), and Lipschitz continuity of mapping thatAgain, utilizing (93), (95), (97), and Lipschitz continuity of mapping , we getLetting ; then, from (107) and (108), we haveThus, from (8) and (109), we haveSince and from condition (106), we have . Thus, from Lemma 5, we can conclude that . Therefore, . Hence, the sequences and generated by parallel Mann iterative algorithm converge strongly to and , respectively.

Now, we prove the convergence of the parallel -iterative algorithm (98)–(99).

Theorem 18. Let and be nonempty closed and convex subsets of a real Hilbert space . For each , let be the single-valued mapping such that is -strongly monotone and -Lipschitz type continuous. Let and be the single-valued mappings such that is -strongly monotone and -Lipschitz type continuous with respect to the second argument. For any , let be an iterative sequence in generated by Algorithm 14, where , , and are the mappings defined by (93) and (94), respectively. In addition, the following condition holds:where and , for . Then(i)There exists unique point , which is the solution of the system of variational inequalities (16).(ii)The sequence generated by parallel -iterative Algorithm 14 converges strongly to the point .

Proof. (i)The conclusion follows from Lemma 11 and (95).(ii)It follows from (94), (95), (98), and Lipschitz continuity of , we getUsing the same facts, we haveLetting , then from (112) and (113), we haveThus, from (8) and (114), we haveNotice that and from condition (111), we have . Thus, we can conclude that . Therefore, . Hence, the sequences and generated by parallel -iterative algorithm converge strongly to and , respectively.

Finally, we discuss an example which illustrates the convergence analysis of parallel Mann and parallel -iterative algorithms.

Example 1. Let and Let be the single-valued mappings defined bySuppose that the mappings are defined by

Then, it is easy to check that for each , is -strongly monotone and -Lipschitz continuous with and and is -strongly monotone and -Lipschitz continuous in the second argument with and . Let the mappings and be defined as follows:

Then, for ,

It is easy to verify that is -Lipschitz continuous and is -Lipschitz continuous. Also,

Thus, all the conditions of Theorem 17 and Theorem 18 are satisfied. Hence, by using Proposition 16, Algorithm 13, and Algorithm 14, the conclusions of Theorem 17 and Theorem 18 follow.

5. Concluding Remarks

In this paper, a system of generalized set-valued variational inequalities involving relaxed cocoercive mappings in Hilbert spaces is considered. Using the projection method and Banach contraction principle, an existence result is proved. Also, we proposed an iterative algorithm, and its convergence is discussed. Moreover, we established an equivalence between the system of variational inequalities and altering points problem. Some parallel iterative algorithms are proposed, and the strong convergence of the sequences generated by these iterative algorithms is discussed. Finally, a numerical example is constructed to illustrate the convergence analysis of the proposed parallel iterative algorithms.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this article.

Authors’ Contributions

All authors contributed equally to the writing of this manuscript. All authors read and approve the final version.

Acknowledgments

The second author would like to thank the Deanship of Scientific Research, Prince Sattam bin Abdulaziz University for supporting this work.

References

G. Stampacchia, “Formes bilineaires coercitives sur les ensembles convexes,” Académie des Sciences de Paris, vol. 258, pp. 4413–4416, 1964.
View at: Google Scholar
J.-L. Lions and G. Stampacchia, “Variational inequalities,” Communications on Pure and Applied Mathematics, vol. 20, pp. 493–519, 1967.
View at: Google Scholar
M. Akram and M. Dilshad, “New generalized system of variational inclusions involving H(.,.)-co-accretive mapping,” Afrika Matematika, vol. 29, pp. 1173–1187, 2018.
View at: Publisher Site | Google Scholar
M. Akram, “Existence and iterative approximation of solution for generalized Yosida inclusion problem,” Iranian Journal of Mathematical Sciences and Informatics, vol. 15, no. 2, pp. 147–161, 2020.
View at: Google Scholar
P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, pp. 117–136, 2005.
View at: Google Scholar
K. S. Kim, “System of extended general variational inequalities for relaxed cocoercive mappings in Hilbert space,” Mathematics, vol. 6, no. 10, 2018.
View at: Publisher Site | Google Scholar
C. E. Lemke, “Bimatrix equilibrium points and mathematical programming,” Management science, vol. 11, no. 7, pp. 681–689, 1965.
View at: Publisher Site | Google Scholar
J.-F. Rodrigues, “Obstacle problems in mathematical physics,” in North-Holland Mathematics Studies, vol. 134, Elsevier, 1987.
View at: Google Scholar
M. J. Smith, “The existence, uniqueness and stability of traffic equilibria,” Transportation Research Part B: Methodological, vol. 13, no. 4, pp. 295–304, 1979.
View at: Publisher Site | Google Scholar
R. P. Agarwal, D. ÓRegan, and D. R. Sahu, “Iterative construction of fixed points of nearly asymptotically nonexpansive mappings,” Journal of Nonlinear and convex Analysis, vol. 8, no. 1, pp. 61–79, 2007.
View at: Google Scholar
D. R. Sahu, S. M. Kang, and A. Kumar, “Convergence analysis of parallel -iteration process for system of generalized variational inequalities,” Journal of Function Spaces, vol. 2017, Article ID 5847096, 10 pages, 2017.
View at: Publisher Site | Google Scholar
E. Picard, “Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives,” Journal de Mathématiques pures et appliquées, vol. 16, pp. 145–210, 1890.
View at: 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
S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, no. 1, pp. 147–150, 1974.
View at: Publisher Site | Google Scholar
P. Cholamjiak, A. A. Abdou, and Y. J. Cho, “Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces,” Fixed Point Theory and Applications, vol. 2015, Article ID 227, 2015.
View at: Publisher Site | Google Scholar
F. Amir, A. Farajzadeh, and N. Petrot, “Hybrid proximal point algorithm for solution of convex multiobjective optimization problem over fixed point constraint,” Thai Journal of Mathematics, vol. 18, no. 3, pp. 841–850, 2020.
View at: Google Scholar
M. Alansari, M. Akram, and M. Dilshad, “Iterative algorithms for a generalized system of mixed variational-like inclusion problems and altering points problem,” Statistics, Optimization & Information Computing, vol. 8, no. 2, pp. 549–564, 2020.
View at: Publisher Site | Google Scholar
T. T. Anh, L. B. Long, and T. V. Anh, “A projection method for bilevel variational inequalities,” Journal of Inequalities and Applications, vol. 2014, Article ID 205, 2014.
View at: Publisher Site | Google Scholar
A. Farajzadeh, M. Mursaleen, and A. Shafie, “On mixed vector equilibrium problems,” Azerbaijan Journal of Mathematics, vol. 6, no. 2, pp. 87–102, 2016.
View at: Google Scholar
P. Lohawech, A. Kaewcharoen, and A. Farajzadeh, “Algorithms for the common solution of the split variational inequality problems and fixed point problems with applications,” Journal of Inequalities and Applications, vol. 2015, Article ID 358, 2018.
View at: Publisher Site | Google Scholar
N. Nimana, A. Farajzadeh, and N. Petrot, “Adaptive subgradient method for the split quasi-convex feasibility problems,” Optimization, vol. 65, no. 10, pp. 1885–1898, 2016.
View at: Publisher Site | Google Scholar
M. A. Noor, K. I. Noor, and A. Bnouhachem, “Some new iterative methods for solving variational inequalities,” Canadian Journal of Applied Mathematics, vol. 2, no. 2, pp. 1–17, 2020.
View at: Google Scholar
Y. Shehu, O. S. Iyiola, X. Li, and Q. Dong, “Convergence analysis of projection method for variational inequalities,” Computational and Applied Mathematics, vol. 38, 2019.
View at: Publisher Site | Google Scholar
Y. Yao, X. Qin, and J.-C. Yao, “Projection methods for firmly type nonexpansive operators,” Journal of Nonlinear and Convex Analysis, vol. 19, pp. 407–415, 2018.
View at: Google Scholar
R. U. Verma, “Generalized system for relaxed cocoercive variational inequalities and Projection methods,” Journal of Optimization Theory and Applications, vol. 121, pp. 203–210, 2004.
View at: Publisher Site | Google Scholar
R. Ahmad, M. Akram, and J. C. Yao, “Generalized monotone mapping with an application for solving a variational inclusion problem,” Journal of Optimization Theory and Applications, vol. 157, no. 2, pp. 324–346, 2013.
View at: Publisher Site | Google Scholar
D. Kinderlehrar and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
R. P. Agarwal, D. ÓRegan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer, New York, 2009.
X. Weng, “Fixed point iteration for local strictly pseudo-contractive mappings,” Proceedings of the American Mathematical Society, vol. 113, pp. 727–731, 1991.
View at: Publisher Site | Google Scholar
R. U. Verma, “Projection methods, algorithms and a new system of nonlinear variational inequalities,” Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 1025–1031, 2001.
View at: Publisher Site | Google Scholar
M. A. Noor, K. I. Noor, A. G. Khan, and R. Latif, “Some new algorithms for solving a system of general variational inequalities,” Applied Mathematics & Information Sciences, vol. 10, no. 2, pp. 557–564, 2016.
View at: Publisher Site | Google Scholar
S. B. Nadlar, “Multivalued contraction mapping,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969.
View at: Google Scholar
D. R. Sahu, “Altering points and applications,” Nonlinear Studies, vol. 21, no. 2, pp. 349–365, 2014.
View at: Google Scholar

Copyright

Copyright © 2021 M. Akram 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

517

Downloads

578

Citations