Strong convergence of implicit and explicit iterations for a class of variational inequalities in Banach spaces

Volume 10, Issue 7, pp 3502--3518 http://dx.doi.org/10.22436/jnsa.010.07.13

Publication Date: July 22, 2017 Submission Date: April 29, 2017

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Authors

Lu-Chuan Ceng - Department of Mathematics, Shanghai Normal University, Shanghai 200234, China. Ching-Feng Wen - Center for Fundamental Science, and Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 80702, Taiwan. - Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 80702, Taiwan.

Abstract

In this paper, we introduce and analyze implicit and explicit iteration methods for solving a variational inequality problem over the set of common fixed points of an infinite family of nonexpansive mappings on a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Strong convergence results are given. Our results improve and extend the corresponding results in the literature.

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Keywords

• Nonexpansive mapping
• fixed point
• variational inequality
• global convergence.

MSC

•  49J30
•  47H09
•  47J20
•  49M05

References

• [1] K. Aoyama, H. Iiduka, W. Takahashi, Weak convergence of an iterative sequence for accretive operators in Banach spaces, Fixed Point Theory Appl., 2006 (2006), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228.
  • View Article
  • MathSciNet
  • Google Scholar


• [3] N. Buong, N. T. H. Phuong, Strong convergence to solutions for a class of variational inequalities in Banach spaces by implicit iteration methods, J. Optim. Theory Appl., 159 (2013), 399–411.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] N. Buong, N. T. Quynh Anh, An implicit iteration method for variational inequalities over the set of common fixed points for a finite family of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2011 (2011), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] L.-C. Ceng, Q. H. Ansari, J.-C. Yao, Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces, Numer. Funct. Anal. Optim., 29 (2008), 987–1033.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] L.-C. Ceng, S.-M. Guu, J.-C. Yao, Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems, Fixed Point Theory Appl., 2012 (2012), 19 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] L.-C. Ceng, A. Petruşel, J.-C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of Lipschitz pseudocontractive mappings, J. Math. Inequal., 1 (2007), 243–258.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] L.-C. Ceng, J.-C. Yao, Relaxed viscosity approximation methods for fixed point problems and variational inequality problems, Nonlinear Anal., 69 (2008), 3299–3309.
  • View Article
  • MathSciNet
  • Google Scholar


• [9] Y. Censor, A. Gibali, S. Reich, Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2012), 1119–1132.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] S.-S. Chang, Some problems and results in the study of nonlinear analysis, Proceedings of the Second World Congress of Nonlinear Analysts, Part 7, Athens, (1996), Nonlinear Anal., 33 (1997), 4197–4208.
  • View Article
  • MathSciNet
  • Google Scholar


• [11] I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1990)
  • View Article
  • MathSciNet
  • MATH


• [12] F. Facchinei, J.-S. Pang, Finite-dimensional variational inequalities and complementarity problems, Vol. I, Springer Series in Operations Research, Springer-Verlag, New York (2003)
  • View Article
  • MathSciNet
  • MATH


• [13] R. Glowinski, J.-L. Lions, R. Trémolières, Numerical analysis of variational inequalities, Translated from the French, Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York (1981)


• [14] A. N. Iusem, B. F. Svaiter, A variant of Korpelevich’s method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309–321.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] J. S. Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., (2005), 509–520.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] M. Kikkawa, W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Hilbert spaces, RIMS Kokyuroku, 1484 (2006), 105–113.
  • Google Scholar


• [17] M. Kikkawa, W. Takahashi, Strong convergence theorems by the viscosity approximation method for a countable family of nonexpansive mappings, Taiwanese J. Math., 12 (2008), 583–598.
  • View Article
  • MathSciNet
  • Google Scholar


• [18] N. Kikuchi, J. T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1988)
  • View Article
  • MathSciNet
  • Google Scholar


• [19] D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1980)
  • MathSciNet
  • Google Scholar


• [20] I. V. Konnov, Combined relaxation methods for variational inequalities, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin (2001)
  • Google Scholar


• [21] I. V. Konnov, Equilibrium models and variational inequalities, Mathematics in Science and Engineering, Elsevier B. V., Amsterdam (2007)
  • MathSciNet
  • Google Scholar


• [22] G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, (Russian) Èkonom. i Mat. Metody, 12 (1976), 747–756.
  • MathSciNet
  • Google Scholar
  • MATH


• [23] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46–55.
  • View Article
  • Google Scholar


• [24] J. G. O’Hara, P. Pillay, H.-K. Xu, Iterative approaches to convex feasibility problems in Banach spaces, Nonlinear Anal., 64 (2006), 2022–2042.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] K. Shimoji, W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math., 5 (2001), 387–404.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc., 125 (1997), 3641–3645.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] M. V. Solodov, B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765–776.
  • View Article
  • Google Scholar


• [28] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227–239.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] W. Takahashi, Weak and strong convergence theorems for families of nonexpansive mappings and their applications, Proceedings of Workshop on Fixed Point Theory, Kazimierz Dolny, (1997), Ann. Univ. Mariae Curie-Skodowska Sect. A, 51 (1997), 277–292.
  • MathSciNet
  • Google Scholar
  • MATH


• [30] S.-H. Wang, L.-X. Yu, B.-H. Guo, An implicit iterative scheme for an infinite countable family of asymptotically nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. , 2008 (2008), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240–256.
  • View Article
  • Google Scholar


• [32] I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Inherently parallel algorithms in feasibility and optimization and their applications, Haifa, (2000), Stud. Comput. Math., North-Holland, Amsterdam, 8 (2001), 473–504.
  • Google Scholar


• [33] Y.-H. Yao, R.-D. Chen, H.-K. Xu, Schemes for finding minimum-norm solutions of variational inequalities, Nonlinear Anal., 72 (2010), 3447–3456.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] Y.-H. Yao, Y.-C. Liou, S. M. Kang, Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method, Comput. Math. Appl., 59 (2010), 3472–3480.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [35] Y.-H. Yao, M. A. Noor, Y.-C. Liou, Strong convergence of a modified extragradient method to the minimum-norm solution of variational inequalities, Abstr. Appl. Anal., 2012 (2012), 9 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [36] Y.-H. Yao, M. A. Noor, Y.-C. Liou, S. M. Kang, Iterative algorithms for general multivalued variational inequalities, Abstr. Appl. Anal., 2012 (2012), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [37] Y.-H. Yao, M. Postolache, Y.-C. Liou, Z.-S. Yao, Construction algorithms for a class of monotone variational inequalities, Optim. Lett., 10 (2016), 1519–1528.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [38] H. Zegeye, N. Shahzad, Y.-H. Yao, Minimum-norm solution of variational inequality and fixed point problem in Banach spaces, Optimization, 64 (2015), 453–471.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [39] E. Zeidler, Nonlinear functional analysis and its applications, III, Variational methods and optimization, Translated from the German by Leo F. Boron, Springer-Verlag, New York (1985)
  • View Article
  • MathSciNet
  • MATH


• [40] L.-C. Zeng, J.-C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear Anal., 64 (2006), 2507–2515.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [41] H.-Y. Zhou, L. Wei, Y. J. Cho, Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces , Appl. Math. Comput., 173 (2006), 196–212.
  • View Article
  • MathSciNet
  • Google Scholar