An inertia-based algorithm for pseudomonotone variational inequality and fixed point problems in real Hilbert space

Ezeora, J. N.¹ ; Ogbonna, R. C. ² ; Bazuaye, F. E. ¹

¹ Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria
² Department of Computer Science and Mathematics, Evangel University, Aka-eze, Aka-eze

Journal of nonlinear sciences and its applications, Tome 15 (2022) no. 3, p. 209-224.

Résumé

The aim of this work is to study a pseudomonotone variational inequality and a fixed point problem involving pseudocontractive mappings in real Hilbert spaces. We introduce an inertia-based iterative algorithm for finding a common solution to this problem. The strong convergence of the proposed algorithm is proved. Finally, numerical examples are provided and also meaningful comparisons of these results with those in [Y. Yao, M. Postolache, J. C. Yao, Mathematics, $\textbf{7}$ (2019), 14 pages], proving that at our proposed numerical schemes are more efficient.

Détail

DOI : 10.22436/jnsa.015.03.04

Classification : 47H09, 47H10, 49M05, 54H25
Keywords: Pseudomonotone variational inequality, pseudocontractive mapping, fixed point problem, Hilbert space

Bibliographie

[1] Acedo, G. L.; Xu, H.-K. Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., Volume 67 (2007), pp. 2258-2271

[2] Ansari, Q. H.; Yao, J. C. Iterative schemes for solving mixed variational-like inequalities, J. Optim. Theory Appl., Volume 108 (2001), pp. 527-541

[3] Cruz, J. Y. Bello; Iusem, A. N. A strongly convergent direct method for monotone variational inequalities in Hilbert space, Numer. Funct. Anal. Optim., Volume 30 (2009), pp. 23-36

[4] Bot, R. I.; Csetnek, E. R.; Vuong, P. T. The forward-backward-forward method from discrete and continuous perspective for pseudomonotone variational inequalities in Hilbert spaces, arXiv, Volume 2018 (2018), pp. 1-12

[5] Ceng, L. C.; ; Petrus¸el, A.; Yao, J. C.; Yao, Y. Hybrid viscosity extragradient method for systems of variational inequalities, fixed points of nonexpansive mappings, zero points of accretive operators in Banach spaces, Fixed Point Theory, Volume 19 (2018), pp. 487-501

[6] Censor, Y.; Gibali, A.; Reich, S. Extensions of Korpelevich´s extragradient method for the variational inequality problem in Euclidean space, Optimization, Volume 61 (2012), pp. 1119-1132

[7] Chidume, C. Geometric Properties of Banach Spaces and Nonlinear Iterations, Springer-Verlag, London, 2009

[8] Chidume, C; Mutangadura, S An example on the Mann iteration method for Lipschitz pseudocontractioons, Proc. Amer. Math. Soc., Volume 129 (2001), pp. 2359-2363

[9] Cholamjiak, P.; Thong, D. V.; Cho, Y. J. A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems, Acta Appl. Math., Volume 169 (2020), pp. 217-245

[10] Dadashi, V.; Postolache, M. Forward-backward splitting algorithm for fixed point problems and zeros of the sum of monotone operators, Arab. J. Math. (Springer), Volume 9 (2020), pp. 89-99

[11] Dong, Q. L.; Cho, Y. J.; Zhong, L. L.; Rassias, T. M. Inertial projection and contraction algorithms for variational inequalities, J. Global Optim., Volume 70 (2018), pp. 687-704

[12] Ezeora, J. N. Approximation of solution of a system of variational inequality problems in real Banach spaces, JP J. Fixed Point Theory Appl., Volume 11 (2017), pp. 207-220

[13] He, B.-S.; Yang, Z.-H.; Yuan, X.-M. An approximate proximal-extragradient type method for monotone variational inequalities, J. Math. Anal. Appl., Volume 300 (2004), pp. 362-374

[14] Lorenz, D. A.; Pock, T. An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vision, Volume 51 (2015), pp. 311-325

[15] Mainge, P.-E. Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math., Volume 219 (2008), pp. 223-236

[16] Polyak, B. T. Some methods of speeding up the convergence of iteration methods, Comput. Math. Phys., Volume 4 (1964), pp. 1-17

[17] Shehu, Y.; Iyiola, O. S. Projection methods with alternating inertial steps for variational inequalities: weak and linear convergence, Appl. Numer. Math., Volume 157 (2020), pp. 315-337

[18] Sintunavarat, W.; Pitea, A. On a new iteration scheme for numerical reckoning fixed points of Berinde mappings with convergence analysis, J. Nonlinear Sci. Appl., Volume 9 (2016), pp. 2553-2562

[19] Tian, M.; Jiang, B.-N. Weak convergence theorem for a class of split variational inequality problems and applications in a Hilbert space, J. Inequal. Appl., Volume 2017 (2017), pp. 1-17

[20] Hieu, D. Van; Anh, P. K.; Muu, L. D. Modified extragradient-like algorithms with new stepsizes for variational inequalities, Comput. Optim. Appl., Volume 73 (2019), pp. 913-932

[21] Hieu, D. Van; Muu, L. D.; Quy, P. K.; Vy, L. Van Explicit extragradient-like method with regularization for variational inequalities, Results Math., Volume 74 (2019), pp. 1-20

[22] Yang, J.; Liu, H. W. Strong convergence result for solving monotone variational inequalities in Hilbert space, Numer. Algorithms, Volume 80 (2019), pp. 741-752

[23] Yao, Y.; Liou, Y. C.; Yao, J. C. Split common fixed point problem for two quasi-pseudocontractive operators and its algorithm construction, Fixed Point Theory Appl., Volume 2015 (2015), pp. 1-19

[24] Yao, Y.; Postolache, M.; Yao, J. C. Iterative algorithms for pseudomonotone variational inequalities and fixed point problems of pseudocontractive operators, Mathematics, Volume 7 (2019), pp. 1-14

[25] Zhou, H. Y. Strong convergence of an explicit iterative algorithm for continuous pseudocontractions in Banach, Nonlinear Anal., Volume 70 (2009), pp. 4039-4046