Approximating a common fixed point of finite family of asymptotically quasi-nonexpansive mappings in Banach spaces
Tóm tắt
The purpose of this paper is to prove convergence of a one-step iterative algorithm to approximate a common fixed point of finite family of asymptotically quasi-nonexpansive mappings in a uniformly convex Banach space by assuming some control conditions on the parameters. Our results extend and improve the corresponding results in the literature.
Tài liệu tham khảo
Chidume, C.E.: Geometric properties of Banach spaces and nonlinear iterations. In: Springer Verlag Series: Lecture Notes in Mathematics, vol. 1965 (2009) (ISBN 978-1-84882-189-7)
Goebel, K., Kirk, W.A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171–174 (1972)
Abbas, M., Khan, S.H., Kim, J.K.: A new one-step iterative process for common fixed points in Banach spaces. J. Inequal. Appl. 2008(548627). doi:10.1155/2008/548627 (10 pages)
Bose, S.C.: Weak convergence to the fixed point of an asymptotically nonexpansive mapping. Proc. Am. Math. Soc. 68, 305–308 (1978)
Khan, S.H.: Convergence of a one-step iteration scheme for quasi-asymptotically nonexpansive mappings. World Acad. Sci. Eng. Technol. 63, 504–506 (2012)
Lim, T.C., Xu, H.K.: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Anal. 22, 1345–1355 (1994)
Lin, P.K., Tan, K.K., Xu, H.K.: Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings. Nonlinear Anal. Theory Methods Appl. 24(6), 929–936 (1995)
Passty, G.B.: Construction of fixed points for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 84, 213–216 (1982)
Schu, J.: Approximations of fixed points of asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 112, 143–151 (1991)
Schu, J.: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 158, 407–412 (1991)
Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bul. Aust. Math. Soc. 43, 153–159 (1991)
Tan, K.K., Xu, H.K.: Nonlinear ergodic theorem for asymptotically nonexpansive mappings. Bul. Aust. Math. Soc. 45, 25–26 (1992)
Tan, K.K., Xu, H.K.: Fixed point iteration processes for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 122, 733–739 (1994)
Xu, H.K.: Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. Nonlinear Anal. 16, 1139–1146 (1991)
Chidume, C.E., Ofoedu, E.U., Zegeye, H.: Strong and weak convergence theorems for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 280(2), 364–374 (2003)
Zegeye, H., Shahzad, N.: Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62, 4007–4014 (2011)
Singh, A.: Stability of common fixed points. Int. J. Funct. Anal. Oper. Theory Appl. 6(2), 73–84 (2014)
Osilike, M.O., Aniagbosor, S.C.: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. In: Mathematical and Computer Modelling, vol. 32, pp. 1181-1191 (2000)
Ćirić, L., Rafiq, A., Cakić, N., Ume, J.S.: On fixed points of two asymptotically quasi-nonexpansive mappings. Int. J. Appl. Math. Mech. 6(4), 68–81 (2010)
Senter, H.F., Dotson, W.G.: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 44(2), 375–380 (1974)
Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)
Yao, Y., Noor, M.A.: Convergence of three-step iterations for asymptotically nonexpansive mappings. Appl. Math. Comput. 187, 883–892 (2007)
Chidume, C.E., Ali, B.: Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in banach spaces. J. Math. Anal. Appl. 330, 377–387 (2007)
Yao, Y., Chen, Y.: Weak and strong convergence of a modified Mann iteration for asymptotically nonexpansive mappings. Nonlinear Funct. Anal. Appl. 12, 307–315 (2007)
Khan, S.H., Abbas, M., Khan, A.R.: Common fixed points of two nonexpansive mappings by a new one step iterative scheme. Iran. J. Sci. Technol. Trans. A Sci. 33(A3), 249–257 (2009)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)
Zhao, J., He, S., Sue, Y.: Weak and strong convergence theorems for nonexpansive mappings in Banach spaces. In: Fixed Point Theory and Applications (2008)
Takahashi, W.: Nonlinear Functional Analysis. Kindikagaku, Tokyo (1988)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems. Springer, New York (1986)
Sangago, M.G.: Convergence of iterative schemes for nonexpansive mappings. Asian–Eur. J. Math. 4, 671–682 (2011)
Purtas, Y., Kiziltunc, H.: Weak and strong convergence of an explicit iteration process for an asymptotically quasi-i-nonexpansive mapping in banach spaces. J. Nonlinear Sci. Appl. 5, 403-411 (2012) (research article)
Chang, S.S., Cho, Y.J., Zhou, H.: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. J. Korean Math. Soc. 38(6), 1245–1260 (2001)