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An Iterative Shrinking Metric -Projection Method for Finding a Common Fixed Point of a Closed and Quasi-Strict -Pseudocontraction and a Countable Family of Firmly Nonexpansive Mappings and Applications in Hilbert Spaces

DOI: 10.1155/2013/589282

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We create some new ideas of mappings called quasi-strict -pseudocontractions. Moreover, we also find the significant inequality related to such mappings and firmly nonexpansive mappings within the framework of Hilbert spaces. By using the ideas of metric -projection, we propose an iterative shrinking metric -projection method for finding a common fixed point of a quasi-strict -pseudocontraction and a countable family of firmly nonexpansive mappings. In addition, we provide some applications of the main theorem to find a common solution of fixed point problems and generalized mixed equilibrium problems as well as other related results. 1. Introduction It is well known that the metric projection operators in Hilbert spaces and Banach spaces play an important role in various fields of mathematics such as functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, and complementarity problem (see, e.g., [1, 2]). In 1994, Alber [3] introduced and studied the generalized projections from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [1] presented some applications of the generalized projections to approximately solve variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li [2] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solve the variational inequality in Banach spaces. Later, Wu and Huang [4] introduced a new generalized -projection operator in Banach spaces. They extended the definition of the generalized projection operators introduced by [3] and proved some properties of the generalized -projection operator. Fan et al. [5] presented some basic results for the generalized -projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces. Let be a real Hilbert space; a mapping with domain and range in is called firmly nonexpansive if nonexpansive if Throughout this paper, stands for an identity mapping. The mapping is said to be a strict pseudocontraction if there exists a constant such that In this case, may be called a -strict pseudocontraction. We use to denote the set of fixed points of (i.e. . is said to be a quasi-strict pseudocontraction if the set of fixed point is nonempty and if there exists a constant such that Construction of fixed

References

[1]  Y. 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., pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.
[2]  J. Li, “The generalized projection operator on reflexive Banach spaces and its applications,” Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 55–71, 2005.
[3]  Ya. Alber, “Generalized projection operators in Banach spaces: properties and applications,” in Proceedings of the Israel Seminar, Ariel, Israel, vol. 1 of Functional Differential Equation, pp. 1–21, 1994.
[4]  K.-Q. Wu and N.-J. Huang, “The generalised -projection operator with an application,” Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 307–317, 2006.
[5]  J. Fan, X. Liu, and J. Li, “Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 70, no. 11, pp. 3997–4007, 2009.
[6]  W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.
[7]  C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, vol. 20, no. 1, pp. 103–120, 2004.
[8]  S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979.
[9]  K.-K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993.
[10]  R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol. 58, no. 5, pp. 486–491, 1992.
[11]  A. Genel and J. Lindenstrauss, “An example concerning fixed points,” Israel Journal of Mathematics, vol. 22, no. 1, pp. 81–86, 1975.
[12]  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.
[13]  A. Jarernsuk and K. Ungchittrakool, “Strong convergence by a hybrid algorithm for solving equilibrium problem and fixed point problem of a Lipschitz pseudo-contraction in Hilbert spaces,” Thai Journal of Mathematics, vol. 10, no. 1, pp. 181–194, 2012.
[14]  S.-y. 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.
[15]  K. Ungchittrakool, “Strong convergence by a hybrid algorithm for finding a common fixed point of Lipschitz pseudocontraction and strict pseudocontraction in Hilbert spaces,” Abstract and Applied Analysis, vol. 2011, Article ID 530683, 14 pages, 2011.
[16]  K. Ungchittrakool and A. Jarernsuk, “Strong convergence by a hybrid algorithm for solving generalized mixed equilibrium problems and fixed point problems of a Lipschitz pseudo-contraction in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2021, article 147, 14 pages, 2012.
[17]  K. Ungchittrakool, “An iterative shrinking projection method for solving fixed point problems of closed and ?-quasi-strict pseudocontractions along with generalized mixed equilibrium problems in Banach spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 536283, 20 pages, 2012.
[18]  K. Ungchittrakool, “A strong convergence theorem for a common fixed points of two sequences of strictly pseudocontractive mappings in Hilbert spaces and applications,” Abstract and Applied Analysis, vol. 2010, Article ID 876819, 17 pages, 2010.
[19]  S. Plubtieng and K. Ungchittrakool, “Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 72, no. 6, pp. 2896–2908, 2010.
[20]  S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 3, pp. 431–450, 2007.
[21]  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.
[22]  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.
[23]  C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 64, no. 11, pp. 2400–2411, 2006.
[24]  G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007.
[25]  S. Plubtieng and K. Ungchittrakool, “Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 67, no. 7, pp. 2306–2315, 2007.
[26]  T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 64, no. 5, pp. 1140–1152, 2006.
[27]  E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1, pp. 123–145, 1994.
[28]  S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007.
[29]  A. Tada and W. Takahashi, “Strong convergence theorem for an equilibrium problem and a nonexpansive mapping,” in Nonlinear Analysis and Convex Analysis, W. Takahashi and T. Tanaka, Eds., pp. 609–617, Yokohama Publishers, Yokohama, Japan, 2007.
[30]  A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007.
[31]  W. Takahashi and K. Zembayashi, “Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 528476, 11 pages, 2008.
[32]  W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 70, no. 1, pp. 45–57, 2009.
[33]  S. Saewan and P. Kumam, “A modified Mann iterative scheme by generalized -projection for a countable family of relatively quasi-nonexpansive mappings and a system of generalized mixed equilibrium problems,” Fixed Point Theory and Applications, vol. 2011, article 104, 21 pages, 2011.
[34]  S. Saewan and P. Kumam, “A generalized -projection method for countable families of weak relatively nonexpansive mappings and the system of generalized Ky Fan inequalities,” Journal of Global Optimization, vol. 56, no. 2, pp. 623–645, 2013.
[35]  X. Li, N.-j. Huang, and D. O'Regan, “Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1322–1331, 2010.
[36]  W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, Japan, 2009.
[37]  H. H. Bauschke and P. L. Combettes, “A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces,” Mathematics of Operations Research, vol. 26, no. 2, pp. 248–264, 2001.
[38]  U. Kamraksa and R. Wangkeeree, “Existence and iterative approximation for generalized equilibrium problems for a countable family of nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2011, article 11, 12 pages, 2011.
[39]  K. Nakajo, K. Shimoji, and W. Takahashi, “Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces,” Taiwanese Journal of Mathematics, vol. 10, no. 2, pp. 339–360, 2006.
[40]  K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp. 11–34, 2007.
[41]  K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
[42]  S.-S. Zhang, “Generalized mixed equilibrium problem in Banach spaces,” Applied Mathematics and Mechanics. English Edition, vol. 30, no. 9, pp. 1105–1112, 2009.

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