Strong convergence theorems for relatively quasi-nonexpansive mappings, variational inequality problems and systems of generalized mixed equilibrium problems

• Autorzy: Shehu Y. , Oguguo O.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we construct a new iterative scheme by hybrid methods to approximate a common element in the fixed points set of an infinite family of relatively quasi-nonexpansive mappings, the solutions set of a variational inequality problem and the solutions set of a system of generalized mixed equilibrium problems in a 2-uniformly convex real Banach space which is also uniformly smooth. Then, we prove strong convergence of the scheme to a common element of the three sets. We give several applications of our results in a Banach space. Our results extend many known recent results in the literature.

Słowa kluczowe

EN

relatively quasi-nonexpansive mappings; generalized mixed equilibrium problems; variational inequality problem; hybrid method; Banach spaces

PL

odwzorowanie nieekspansywne; problem równowagi; nierówności wariacyjne; metoda hybrydowa; przestrzenie Banacha

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2012
• Tom: Vol. 18, nr 1
• Strony: 69--95
• Opis fizyczny: Bibliogr. 42 poz.

Twórcy

autor

Shehu Y.

autor

Oguguo O.

• Department of Mathematics, University of Nigeria, Nsukka, Nigeria,

Bibliografia

• [1] Y. I. Alber, Metric and generalized projection operator in Banach spaces: properties and applications, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics 178, Dekker, New York, (1996), 15-50.
• [2] Y. I. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Pan. Amer. Math. J. 4 (1994), 39-54.
• [3] K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2006), 2350-2360.
• [4] J. B. Baillon and G. Haddad, Quelques proprietes des operateurs angle-bornes et n-cycliquement monotones, Isreal J. Math. 26 (1977), 137-150.
• [5] B. Beauzamy, Introduction to Banach Spaces and Their Geometry, 2nd ed., North-Holland, 1985.
• [6] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems. Math. Student 63 (1994), 123-145.
• [7] D. Butnariu, S. Reich and A. J. Zaslavski, Asymptotic behaviour of relatively non-expansive operators in Banach spaces, J. Appl. Anal. 7 (2001), 151-174.
• [8] D. Butnariu, S. Reich and A. J. Zaslavski, Weak convergence of orbits of nonlinear operator in reflexive Banach spaces, Numer. Fund Anal. Optim. 24 (2003), 489-508.
• [9] L. C. Ceng and J. C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008), 186-201.
• [10] Y. Censor and S. Reich, Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 37 (1996), 323-339.
• [11] C. E. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Lecture Notes in Math. 1965, Springer, 2009.
• [12] P. Cholamjiak, A hybrid iterative scheme for equilibrium problems, variatonal inequality problems and fixed points problems in Banach spaces. Fixed Point Theory Appl. (2009), article 719360.
• [13] W. Chuayjan and S. Thianwan, Strong convergence of a hybrid projection algorithm for equilibrium problems, variational inequality problems and fixed point problems in a Banach space, Abstr. Appl. Anal. (2009), article 613524.
• [14] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic, Dordrecht, 1990.
• [15] P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), 117-136.
• [16] S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (2002), 938-945.
• [17] D. Kinderlehrer and G. Stampaccia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1984.
• [18] M. Liu, S. Chang and P. Zuo, On a hybrid method for generalized mixed equilibrium problem and fixed point problem of a family of quasi-0-asymptotically nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. (2010), article 157278.
• [19] Y. Liu, A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 71 (2009), 4852-4861.
• [20] S. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory 134 (2005), 257-266.
• [21] A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems, J. Nonlinear Convex Anal. 9 (2008), 37-43.
• [22] W. Nilsrakoo and S. Saejung, Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings. Fixed Points Theory Appl. (2008), article 312454.
• [23] J. W. Peng and J. C. Yao, Strong convergence theorems of an iterative scheme based on extragradient method for mixed equilibrium problems and fixed points problems, Math. Com. Model. 49 (2009), 1816-1828.
• [24] N. Petrot, K. Wattanawitoon and P. Kumam, A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces. Nonlinear Anal. Hybrid Syst. 4 (2010), 631-643.
• [25] S. Plubtieng and R. Punpaeng A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput. 197 (2008), 548-558.
• [26] S. Plubtieng and K. Sombut, Weak convergence theorems for a system of mixed equilibrium problems and nonspreading mappings in a Hilbert space, J. Ineq. Appl. (2010), article 246237.
• [27] X. Qin, Y. J. Cho and S. M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math. 225 (2009), 20-30.
• [28] X. Qin, Y. J. Cho and S. M. Kang, Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 72(2010),99-112.
• [29] X. Qin, M. Shang and Y. Su, Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Com. Model. 48 (2008), 1033-1046.
• [30] Y. Shehu, Fixed point solutions of generalized equilibrium problems for nonexpansive mappings, J. Comput. Appl. Math. 234 (2010), 892-898.
• [31] Y. Su, M. Shang and X. Qin, An iterative method of solution for equilibrium and optimization problems. Nonlinear Anal. 69 (2008), 2709-2719.
• [32] S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69 (2008), 1025-1033.
• [33] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), 506-518.
• [34] W. Takahashi, Nonlinear Functional Analysis - Fixed Point Theory and Applications (in Japanese), Yokohama Publishers, Yokohama, 2000.
• [35] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.
• [36] W. Takahashi and K. Zembayashi, Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, Fixed Point Theory Appl. (2008), article 528476.
• [37] W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009), 45-57.
• [38] R. Wangkeeree, An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings. Fixed Point Theory and Appl. (2008), article 134148.
• [39] K. Wattanawitoon and P. Kumam, Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi nonexpansive mappings, Nonlinear Anal. Hybrid Syst. 3 (2009), 11-20.
• [40] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), 1127-1138.
• [41] Y. Yao, Y. C. Liou and J. C. Yao, A new hybrid iterative algorithm for fixed point problems, variational inequality problems and mixed equilibrium problems. Fixed Points Theory Appl. (2008), article 312454.
• [42] S. Zhang, Generalized mixed equilibrium problems in Banach spaces, Appl. Math. Mech. (English Ed.) 30 (2009), 1105-1112.