A STRONG CONVERGENCE HALPERN-TYPE INERTIAL ALGORITHM FOR SOLVING SYSTEM OF SPLIT VARIATIONAL INEQUALITIES AND FIXED POINT PROBLEMS

A. A. Mebawondu; L. O. Jolaoso; H. A. Abass; O. K Oyewole; K. O. Aremu

doi:10.11948/20200411

2021 Volume 11 Issue 6

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A. A. Mebawondu, L. O. Jolaoso, H. A. Abass, O. K Oyewole, K. O. Aremu. A STRONG CONVERGENCE HALPERN-TYPE INERTIAL ALGORITHM FOR SOLVING SYSTEM OF SPLIT VARIATIONAL INEQUALITIES AND FIXED POINT PROBLEMS[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2762-2791. doi: 10.11948/20200411

Citation:

A. A. Mebawondu, L. O. Jolaoso, H. A. Abass, O. K Oyewole, K. O. Aremu. A STRONG CONVERGENCE HALPERN-TYPE INERTIAL ALGORITHM FOR SOLVING SYSTEM OF SPLIT VARIATIONAL INEQUALITIES AND FIXED POINT PROBLEMS[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2762-2791. doi: 10.11948/20200411

A STRONG CONVERGENCE HALPERN-TYPE INERTIAL ALGORITHM FOR SOLVING SYSTEM OF SPLIT VARIATIONAL INEQUALITIES AND FIXED POINT PROBLEMS

1.
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, 4001, South Africa
2.
DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoEMaSS)
3.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 94 Medunsa 0204, Pretoria, South Africa
4.
Department of Mathematics, Usmanu Danfodiyo University Sokoto, Sokoto state, Nigeria
5.
Department of Mathematics, Mountain Top University, Prayer City, Ogun statee, Nigeria

Corresponding author: Email: jollatanu@yahoo.co.uk, lateef.jolaoso@smu.ac.za(L. O. Jolaoso)
Fund Project: A.A.Mebawondu and H.A.Abass acknowledge with thanks the bursary and financial support from Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS) Post-Doctoral Bursary

Abstract

Abstract

In this paper, we propose a new Halpern-type inertial extrapolation method for approximating common solutions of the system of split variational inequalities for two inverse-strongly monotone operators, the variational inequality problem for monotone operator, and the fixed point of composition of two nonlinear mappings in real Hilbert spaces. We establish that the proposed method converges strongly to an element in the solution set of the aforementioned problems under certain mild conditions. In addition, we present some numerical experiments to show the efficiency and applicability of our method in comparison with some related methods in the literature. This result improves and generalizes many recent results in this direction in the literature.
- Generalized system of variational inequality problem /
- split feasibility problem /
- inertial iterative scheme /
- firmly nonexpansive /
- fixed point problem
MSC: 47H06, 47H09, 47J05, 47J25

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References

[1]	H. A. Abass, K. O. Aremu, L. O. Jolaoso and O. T. Mewomo, An inertial forward-backward splitting method for approximating solutions of certain optimization problem, J. Nonlinear Funct. Anal., 2020, Article ID 6. Google Scholar
[2]	F. Alvarez and H. Attouch, An inertial proximal method formaximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 2001, 9, 3-11. doi: 10.1023/A:1011253113155 CrossRef Google Scholar
[3]	H. Attouch, X. Goudon and P. Redont, The heavy ball with friction. I. The continuous dynamical system, Commun Contemp Math., 2000, 21(2), 1-34. Google Scholar
[4]	H. Attouch, and M. O. Czarnecki, Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria, J Diff Eq., 2002, 179, 278-310. doi: 10.1006/jdeq.2001.4034 CrossRef Google Scholar
[5]	A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2009, 2(1), 183-202. doi: 10.1137/080716542 CrossRef Google Scholar
[6]	C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Prob., 2002, 18, 441-453. doi: 10.1088/0266-5611/18/2/310 CrossRef Google Scholar
[7]	L. Ceng, C. Wang and J. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res., 2008, 67, 375-390. doi: 10.1007/s00186-007-0207-4 CrossRef Google Scholar
[8]	Y. Censor, T. Elfving, N. Kopt and T. Bortfeld, The multiple-sets split feasibility problem and its applications, Inverse Prob., 2005, 21, 2071-2084. doi: 10.1088/0266-5611/21/6/017 CrossRef Google Scholar
[9]	Y. Censor and Y. Segal, The split common fixed point for directed operators, J. Convex. Anal., 2009, 16, 587-600. Google Scholar
[10]	Y. Censor and T. Elfving. A multi projection algorithm using Bregman projections in a product space, Numer. Algor., 1994, 8(2), 221-239. doi: 10.1007/BF02142692 CrossRef Google Scholar
[11]	Y. Censor, A. Gibali, and S. Reich, The split variational inequality problem. The Technion-Israel Institue of Technology, Haifa, 2010. Google Scholar
[12]	L. Dong, Y. J. Cho, L. Zhong and M. Th. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Glob. Optim., 2018, 70, 687-704. doi: 10.1007/s10898-017-0506-0 CrossRef Google Scholar
[13]	G. Ficher, Sul pproblem elastostatico di signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 1963, 34, 138-142. Google Scholar
[14]	G. Ficher, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincci. Cl. Sci. Fis. Mat. Nat., Sez., 1964, 7, 91-140. Google Scholar
[15]	A. Gibali, D. V. Thong and P. A. Tuan. Two simple projection-type methods for solving variational inequalities in Euclidean spaces, J. Nonlinear Anal. Optim., 2015, 6, 41-51. Google Scholar
[16]	A. Gibali, S. Reich and R. Zalas, Outer approximation methods for solving variational inequalities in Hilbert space, Optimization, 2017, 66, 417-437. doi: 10.1080/02331934.2016.1271800 CrossRef Google Scholar
[17]	R. Glowinski, J. L. Lions and R. Trémoliéres, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981. Google Scholar
[18]	K. Goebel and S. Reich, convexity, hyperbolic geometry, and nonexpansive mappings. New York, Marcel Dekker, 1984. Google Scholar
[19]	A. A. Goldstein, Convex programming in Hilbert space, Bull. Am. Math. Soc., 1964, 70, 709-710. doi: 10.1090/S0002-9904-1964-11178-2 CrossRef Google Scholar
[20]	Y. He. A new double projection algorithm for variational inequalities, J. Comput. Appl. Math., 2006, 185(1), 166-173. doi: 10.1016/j.cam.2005.01.031 CrossRef Google Scholar
[21]	L. O. Jolaoso and Y. Shehu, Single Bregman projection method for solving variational inequalities in reflexive Banach spaces, Appl. Analysis., 2021, 1-22. https://doi.org/10.1080/00036811.2020.1869947. doi: 10.1080/00036811.2020.1869947 CrossRef Google Scholar
[22]	L. O. Jolaoso, Y. Shehu and J. C. Cho, Convergence Analysis for Variational Inequalities and Fixed Point Problems in Reflexive Banach Spaces, J. Inequal. and Appl., 2021, 44. https://doi.org/10.1186/s13660-021-02570-6 doi: 10.1186/s13660-021-02570-6 CrossRef Google Scholar
[23]	G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Èkon. Mat. Metody., 1976, 12, 747-756. Google Scholar
[24]	P. E. Mainge, Regularized and inertial algorithms for common fixed points of nonlinear operators, J Math Anal Appl., 2008, 34, 876-887. Google Scholar
[25]	A. Moudafi and M. Oliny, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 2013, 155, 447-454. Google Scholar
[26]	Y. Nesterov, A method of solving a convex programming problem with convergence rate O(1/k²), Soviet Math. Doklady., 1983, 27, 372-376. Google Scholar
[27]	F. U. Ogbuisi and O. T. Mewomo, Convergence analysis of an inertial accelerated iterative algorithm for solving split variational inequality problem, Adv. Pure Appl. Math., 2019, 10(4), 1-15. Google Scholar
[28]	M. O. Osilike and D. I. Igbokwe, Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations, Comput. Math. Appl., 2000, 40, 559-567. doi: 10.1016/S0898-1221(00)00179-6 CrossRef Google Scholar
[29]	B. T. Polyak, Some methods of speeding up the convergence of iterates methods, U. S. S. R Comput. Math. Phys., 1964, 4(5), 1-17. doi: 10.1016/0041-5553(64)90137-5 CrossRef Google Scholar
[30]	S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 2012, 75, 742-750. doi: 10.1016/j.na.2011.09.005 CrossRef Google Scholar
[31]	D. R. Sahu, S. M. Kang and A. Kumar, Convergence analysis of parallel S-iteration process for system of generalized variational inequalities, J. Funct. Spaces., Article ID 5847096, 2017, 10, 1-10. Google Scholar
[32]	D. R. Sahu, Altering points and applications, Nonlinear Stud., 2014, 21(2), 349-365. Google Scholar
[33]	D. R. Sahu and A. K. Singh, Inertial iterative algorithms for common solution of variational inequality and system of variational inequalities problems, Journal of Applied Mathematics and Computing, 2020, 1-28. doi: 10.1007/s12190-020-01395-8 CrossRef Google Scholar
[34]	Y. Shehu and P. Cholamjiak, Iterative method with inertial for variational inequalities in Hilbert spaces, Calcolo, 2019, 56. https://doi.org/10.1007/s10092-018-0300-5. doi: 10.1007/s10092-018-0300-5 CrossRef Google Scholar
[35]	Y. Shehu and O. S. Iyiola, Projection methods with alternating inertial steps for variational inequalities: Weak and linear convergence, Appl. Numer. Math., 2020, 157, 315-337. doi: 10.1016/j.apnum.2020.06.009 CrossRef Google Scholar
[36]	G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Math. Acad. Sci., Paris, 1964, 258, 4413-4416. Google Scholar
[37]	W. Takahashi, Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, 2000. Google Scholar
[38]	S. Suantai, N. Pholosa and P. Cholamjiak, The Modified Inertial Relaxed CQ Algorithm for Solving the Split Feasibility Problems, J. Industrial and Management Optimization, 2017. DoI:10.3934/jimo.2018023. CrossRef Google Scholar
[39]	W. Takahashi, H. Xu, and J. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Anal., 2015, 23(2), 205-221. doi: 10.1007/s11228-014-0285-4 CrossRef Google Scholar
[40]	M. Tian, and B. Jiang, Weak convergence theorem for a class of split variational inequality problems and applications in a Hilbert space, J. Ineq and Appl., 2017, 1, 1-17. Google Scholar
[41]	M. Tian, and B. Jiang, Viscosity Approximation Methods for a Class of Generalized Split Feasibility Problems with Variational Inequalities in Hilbert Space, Numer. Funct. Anal. Optim., 2019, 40(8), 902-923. doi: 10.1080/01630563.2018.1564763 CrossRef Google Scholar
[42]	D. V. Thong and N. T. Vinh, Inertial methods for fixed point problems and zero point problems of the sum of two monotone mappings, Optimization, 2019, 68(5), 1037-1072. doi: 10.1080/02331934.2019.1573240 CrossRef Google Scholar
[43]	D. V. Thong and D. V. Hieu, Weak and strong convergence theorems for variational inequality problems, Numer. Algor., 2017. DOI10.1007/s11075-017-0412-z. Google Scholar
[44]	R. U. Verma, On a new system of nonlinear variational inequalities and associated iterative algorithms, Mathematical Sciences Research, 1999, 3(8), 65-68. Google Scholar
[45]	F. Wang and H. Xu, Approximating curve and strong convergence of the CQ algorithm for split feasibility problem, J. Inequal. Appl., 2010, Article ID 102085. Google Scholar
[46]	H. Xu, Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl., 2011, 150, 360-378. doi: 10.1007/s10957-011-9837-z CrossRef Google Scholar