×

zbMATH — the first resource for mathematics

A cyclic method for solutions of a class of split variational inequality problem in Banach space. (English) Zbl 1474.49018
Summary: In this paper, a cyclic algorithm for approximating a class of split variational inequality problem is introduced and studied in some Banach spaces. A strong convergence theorem is proved. Some applications of the theorem are presented. The results presented here improve, unify, and generalize certain recent results in the literature.
MSC:
49J40 Variational inequalities
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J20 Variational and other types of inequalities involving nonlinear operators (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Zeng, L.-C.; Yao, J.-C., Strong convergence theorems by an extragradient method for fixed point problems and variational inequality problems, Taiwanese Journal of Mathematics, 10, 5, 1293-1303 (2006) · Zbl 1110.49013
[2] Ganciaruso, F.; Marino, G.; Muglia, L.; Yao, Y. H., On a two-step algorithm for hierarchical fixed point problems and variational inequalities, Journal of Inequalities and Applications, 2009, 1 (2009) · Zbl 1180.47040
[3] Luxin, A. N., An iterative algorithm for the variational inequality problem, Applied Mathematics and Computation, 13, 103-114 (1994) · Zbl 0811.65049
[4] Qin, X.; Cho, S. Y., Convergence analysis of a monotone projection algorithm in reflexive banach spaces, Acta Mathematica Scientia, 37, 2, 488-502 (2017) · Zbl 1389.47163
[5] Yao, Y.; Chen, R.; Xu, H.-K., Schemes for finding minimum-norm solutions of variational inequalities, Nonlinear Analysis: Theory, Methods & Applications, 72, 7-8, 3447-3456 (2010) · Zbl 1183.49012
[6] Lions, J. L.; Stampacchia, G., Variational inequalities, Communications on Pure and Applied Mathematics, 20, 3, 493-519 (1967) · Zbl 0152.34601
[7] Antipin, A., Methods for solving variationalinequalities with relaxed constraints, Computational Mathematics and Mathematical Physics, 40, 1239-1254 (2000) · Zbl 0999.65055
[8] Berinde, V., Iterative approximation of fixed points, Lecture Notes in Mathematics (2007), London, UK: Springer, London, UK · Zbl 1165.47047
[9] Buong, N., Strong convergence theorem of an iterative method for variational inequalities and fixed point problems in Hilbert spaces, Applied Mathematics and Computation, 217, 1, 322-329 (2010) · Zbl 1251.65095
[10] Censor, Y.; Gibali, A.; Reich, S.; Sabach, S., Common solutions to variational inequalities, Set-Valued and Variational Analysis, 20, 2, 229-247 (2012) · Zbl 1296.47060
[11] Gibali, A.; Reich, S.; Zalas, R., Iterative methods for solving variational inequalities in Euclidean space, Journal of Fixed Point Theory and Applications, 17, 4, 775-811 (2015) · Zbl 1332.47044
[12] Tian, M.; Jiang, B., Weak convergence theorems for a class of split variational inequality problem and applications in a Hilbert space, Journal of Inequalities and Application, 2017, 1, 123 (2017) · Zbl 1367.58007
[13] Chidume, C. E.; Adamu, A.; Okereke, L., A Krasnoselskii-type algorithm for approximating solutions of variational inequality problems and convex feasibility problems, Journal of Nonlinear and Variational Analysis, 2, 2, 203-218 (2018) · Zbl 07015095
[14] Korpelevch, G. M., An extragradient method for solving saddle points and for other problems, Ekon. Mat. Metody, 12, 747-756 (1976) · Zbl 0342.90044
[15] Censor, Y.; Elfving, T., A multiprojection algorithm using Bregman projections in a product space, Numerical Algorithms, 8, 2, 221-239 (1994) · Zbl 0828.65065
[16] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20, 1, 103-120 (2004) · Zbl 1051.65067
[17] Censor, Y.; Gibali, A.; Reich, S., The Split Variational Inequality Problem (2010), Haifa, Israel: The Technion-Israel Institute of Technology, Haifa, Israel
[18] Nilsrakoo, W.; Saejung, S., Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in banach spaces, Applied Mathematics and Computation, 217, 14, 6577-6586 (2011) · Zbl 1215.65104
[19] Alber, Y.; Kartsatos, A. G., Metric and generalized projection operators in Banach spaces, properties and applications, Theory and Application of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure Applied Mathematics, 178, 15-50 (1996), New York, USA: Marcel Dekker, New York, USA · Zbl 0883.47083
[20] Chidume, C. E., Geometric properties of banach spaces and nonlinear iterations, Vol. 1905 of Lecture Notes in Mathematics (2009), London, UK: Springer, London, UK · Zbl 1167.47002
[21] Xu, H.-K., Inequalities in banach spaces with applications, Nonlinear Analysis: Theory, Methods & Applications, 16, 12, 1127-1138 (1991) · Zbl 0757.46033
[22] Rockafellar, R. T., On the maximality of sums of nonlinear monotone operators, Transactions of the American Mathematical Society, 149, 1, 75 (1970) · Zbl 0222.47017
[23] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type Algorithm in a banach space, SIAM Journal on Optimization, 13, 3, 938-945 (2002) · Zbl 1101.90083
[24] Qin, X.; Cho, Y. J.; Kang, S. M., Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, Journal of Computational and Applied Mathematics, 225, 1, 20-30 (2009) · Zbl 1165.65027
[25] Kohsaka, F.; Takahashi, W., Existence and approximation of fixed points of firmly nonexpansive type mappings in banach spaces, SIAM Journal on Optimization, 19, 2, 824-835 (2009) · Zbl 1168.47047
[26] Kamimura, S.; Takahashi, W.; Tanaka, T., The proximal point algorithm in a Banach space, Proceedings of the Third International Conference on Nonlinear Analysis and Convex Analysis, Yokohama Publishers · Zbl 1086.47513
[27] Matsushita, S.-Y.; Takahashi, W., Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications, 2004, 1, 829453 (2004) · Zbl 1088.47054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.