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The shrinking projection method for solving variational inequality problems and fixed point problems in Banach spaces. (English) Zbl 1184.49019
Summary: We consider a hybrid projection algorithm based on the shrinking projection method for two families of quasi-$$\varphi$$-nonexpansive mappings. We establish strong convergence theorems for approximating the common element of the set of the common fixed points of such two families and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. As applications, at the end of the paper we first apply our results to consider the problem of finding a zero point of an inverse-strongly monotone operator and we finally utilize our results to study the problem of finding a solution of the complementarity problem. Our results improve and extend the corresponding results announced by recent results.

MSC:
 49J40 Variational inequalities 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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References:
 [1] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, vol. 88 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1980. · Zbl 0457.35001 [2] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. · Zbl 0712.47043 [3] W. Takahashi, Convex Analysis and Approximation Fixed points, vol. 2 of Mathematical Analysis Series, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 1089.49500 [4] M. M. Vainberg, Variational Methods and Method of Monotone Operators, John Wiley & Sons, New York, NY, USA, 1973. · Zbl 0279.47022 [5] Ya. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A.G. Kartsatos, Ed., vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15-50, Marcel Dekker, New York, NY, USA, 1996. · Zbl 0883.47083 [6] Ya. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in Banach spaces,” PanAmerican Mathematical Journal, vol. 4, no. 2, pp. 39-54, 1994. · Zbl 0851.47043 [7] S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938-945, 2002. · Zbl 1101.90083 [8] S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 313-318, Marcel Dekker, New York, NY, USA, 1996. · Zbl 0943.47040 [9] D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operators in Banach spaces,” Journal of Applied Analysis, vol. 7, no. 2, pp. 151-174, 2001. · Zbl 1010.47032 [10] D. Butnariu, S. Reich, and A. J. Zaslavski, “Weak convergence of orbits of nonlinear operators in reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 24, no. 5-6, pp. 489-508, 2003. · Zbl 1071.47052 [11] Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,” Optimization, vol. 37, no. 4, pp. 323-339, 1996. · Zbl 0883.47063 [12] D. Butnariu and A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, vol. 40 of Applied Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. · Zbl 0960.90092 [13] S. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257-266, 2005. · Zbl 1071.47063 [14] S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 149, no. 2, pp. 103-115, 2007. · Zbl 1137.47056 [15] X. Qin and Y. Su, “Strong convergence theorems for relatively nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 6, pp. 1958-1965, 2007. · Zbl 1124.47046 [16] X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20-30, 2009. · Zbl 1165.65027 [17] H. Iiduka and W. Takahashi, “Weak convergence of a projection algorithm for variational inequalities in a Banach space,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 668-679, 2008. · Zbl 1129.49012 [18] F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, no. 2, pp. 197-228, 1967. · Zbl 0153.45701 [19] H. Iiduka, W. Takahashi, and M. Toyoda, “Approximation of solutions of variational inequalities for monotone mappings,” PanAmerican Mathematical Journal, vol. 14, no. 2, pp. 49-61, 2004. · Zbl 1060.49006 [20] F. Liu and M. Z. Nashed, “Regularization of nonlinear ill-posed variational inequalities and convergence rates,” Set-Valued Analysis, vol. 6, no. 4, pp. 313-344, 1998. · Zbl 0924.49009 [21] F. E. Browder, “Fixed-point theorems for noncompact mappings in Hilbert space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 53, pp. 1272-1276, 1965. · Zbl 0125.35801 [22] B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957-961, 1967. · Zbl 0177.19101 [23] C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400-2411, 2006. · Zbl 1105.47060 [24] K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372-379, 2003. · Zbl 1035.47048 [25] X. L. Qin, Y. J. Cho, S. M. Kang, and H. Y. Zhou, “Convergence of a hybrid projection algorithm in Banach spaces,” Acta Applicandae Mathematicae, vol. 108, no. 2, pp. 299-313, 2009. · Zbl 1178.47048 [26] W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276-286, 2008. · Zbl 1134.47052 [27] S. Reich, “Book Review: Geometry of Banach spaces, duality mappings and nonlinear problems by loana Cioranescu,” Bulletin of the American Mathematical Society, vol. 26, no. 2, pp. 367-370, 1992. [28] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. · Zbl 0712.47043 [29] W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Application, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 0997.47002 [30] K. Ball, E. A. Carlen, and E. H. Lieb, “Sharp uniform convexity and smoothness inequalities for trace norms,” Inventiones Mathematicae, vol. 115, no. 3, pp. 463-482, 1994. · Zbl 0803.47037 [31] Y. Takahashi, K. Hashimoto, and M. Kato, “On sharp uniform convexity, smoothness, and strong type, cotype inequalities,” Journal of Nonlinear and Convex Analysis, vol. 3, no. 2, pp. 267-281, 2002. · Zbl 1030.46012 [32] B. Beauzamy, Introduction to Banach Spaces and Their Geometry, vol. 68 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 2nd edition, 1985. · Zbl 0585.46009 [33] R. E. Bruck and S. Reich, “Nonexpansive projections and resolvents of accretive operators in Banach spaces,” Houston Journal of Mathematics, vol. 3, no. 4, pp. 459-470, 1977. · Zbl 0383.47035 [34] Y. J. Cho, H. Y. Zhou, and G. Guo, “Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings,” Computers & Mathematics with Applications, vol. 47, no. 4-5, pp. 707-717, 2004. · Zbl 1081.47063 [35] R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75-88, 1970. · Zbl 0222.47017
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