In this paper, we
introduce some new classes of the totally quasi-G-asymptotically nonexpansive
mappings and the totally quasi-G-asymptotically nonexpansive semigroups. Then,
with the generalized f-projection operator, we prove some strong convergence
theorems of a new modified Halpern type hybrid iterative algorithm for the
totally quasi-G-asymptotically nonexpansive semigroups in Banach space. The
results presented in this paper extend and improve some corresponding ones by
many others.
Ya. I. Alber, C. E. Chidume and J. L. Li, “Stochastic Approximation Method for Fixed Point Problems,” Applied Mathematics, Vol. 2012, No. 3, 2012, pp. 2123-2132.
L. J. Chen and J. H. Huang, “Strong Convergence of an Iterative Method for Generalized Mixed Equilibrium Problems and Fixed Point Problems,” Applied Mathematics, Vol. 2011, No. 2, 2011, pp. 1213-1220.
S. S. Chang, L. H. W. Joseph, C. K. Chan and W. B. Zhang, “A Modified Halpern-Type Iteration Algorithm for Totally Quasi-ø-Asymptotically Nonexpansive Mappings with Applications,” Applied Mathematics and Computation, Vol. 218, No. 11, 2012, pp. 6489-6497. http://dx.doi.org/10.1016/j.amc.2011.12.019
S. S. Chang, L. H. W. Joseph, C. K. Chan and L. Yang, “ Approximation Theorems for Total Quasi-ø-Asymptotically Nonexpansive Mappings with Application,” Applied Mathematics and Computation, Vol. 218, No. 6, 2011, pp. 2921-2931. http://dx.doi.org/10.1016/j.amc.2011.08.036
S. S. Zhang, L. Wang and Y. H. Zhao, “Multi-Valued Totally Quai-Phi-Asymptotically Nonexpansive Semigrops and Strong Convergence Theorems in Banach Spaces,” Acta Mathematica Scientia, Vol. 33B, No. 2, 2013, pp. 589-599. http://dx.doi.org/10.1016/S0252-9602(13)60022-3
Y. Li, “Fixed Point of a Countable Family of Uniformly Totally Quasi-Phi-Asymptotically Nonexpansive Multi-Valued Mappings in Reflexive Banach Spaces with Applications,” Applied Mathematics, Vol. 2013, No. 4, 2013, pp. 6-12.
X. R. Wang, S. S. Chang, L. Wang, Y. K. Tang and Y. G. Xu, “Strong Convergence Theorem for Nonlinear Operator Equations with Total Quasi-ø-Asymptotically Nonexpansive Mappings and Applications,” Fixed Point Theory and Applications, Vol. 2012, 2012, p. 34. http://dx.doi.org/10.1186/1687-1812-2012-34
J. Quan, S. S. Chang and X. R. Wang, “Strong Convergence for Total Quasi-ø-asymptotically Nonexpansive Semigroup in Banach Spaces,” Fixed Point Theory and Applications, Vol. 2012, 2012, p. 142.
K. Wu and N. J. Huang, “The Generalized f-Projection Operator and an Application,” Bulletin of the Australian Mathematical Society, Vol. 73, No. 2, 2006, pp. 307-317. http://dx.doi.org/10.1017/S0004972700038892
X. Li, N. J. Huang and D. R. Regan, “Strong Convergence Theorems for Relatively Nonexpansive Mappings in Banach Spaces with Applications,” Computers & Mathematics with Applications, Vol. 60, No. 5, 2010, pp. 1322-1331. http://dx.doi.org/10.1016/j.camwa.2010.06.013
S. Saewan, P. Kanjanasamranwong, P. Kumam and Y. J. Cho, “The Modified Mann Type Iterative Algorithm for a Countable Family of Totally Quasi-ø-Asymptotically Nonexpansive Mappings by the Hybrid Generalized f-Projection Method,” Fixed Point Theory and Applications, Vol. 2013, 2013, p. 63. http://dx.doi.org/10.1186/1687-1812-2013-63
Y. H. Wang, “Strong Convergence Theorems for Asymptotically Weak G-Pseudo-φ-Contractive Nonself Mappings with the Generalized Projection in Banach Spaces,” Abstract and Applied Analysis, Vol. 2012, 2012, Article ID: 651304.
Y. H. Wang and Y. H. Xia, “Strong Convergence for Asymptotically Qseudo-Contractions with the Demiclosedness Principle in Banach Spaces,” Fixed Point Theory and Applications, Vol. 2012, 2012, p. 45.
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, 2008, pp. 276-286. http://dx.doi.org/10.1016/j.jmaa.2007.09.062
S. Saewan, P. Kumam and K. Wattanawitoon, “Convergence Theorem Based on a New Hybrid Projection Method for Finding a Common Solution of Generalized Equilibrium and Variational Inequality Problems in Banach Spaces,” Abstract and Applied Analysis, Vol. 2010, 2010, Article ID: 734126. http://dx.doi.org/10.1155/2010/734126
X. L. Qin, Y. J. Cho, S. M. Kang and H. Y. Zhou, “Convergence of a Modified Halpern-Type Iterative Algorithm for Quasi-ø-Nonexpansive Mappings,” Applied Mathematics Letters, Vol. 22, No. 7, 2009, pp. 1051-1055. http://dx.doi.org/10.1016/j.aml.2009.01.015
Y. F. Su, H. K. Xu and X. Zhang, “Strong Convergence Theorems for Two Countable Families of Weak Relatively Nonexpansive Mappings and Applications,” Nonlinear Analysis, Vol. 73, No. 12, 2010, pp. 3890-3906. http://dx.doi.org/10.1016/j.na.2010.08.021
Z. M. Wang, Y. F. Su, D. X. Wang and Y. C. Dong, “A Modified Halpern-Type Iteration Algorithm for a Family of HemiRelative Nonexpansive Mappings and Systems of Equilibrium Problems in Banach Spaces,” Journal of Computational and Applied Mathematics, Vol. 235, No. 8, 2011, pp. 2364-2371. http://dx.doi.org/10.1016/j.cam.2010.10.036
