Strong convergence of the Modified Halpern-type iterative algorithms in Banach spaces

The purpose of this paper is to introduce a modified Halpern-type iteration algorithm and prove strong convergence of the algorithm for quasi-�-asymptotically non-expansive mappings. Our results improve and extend the corresponding results announced by many others.

[1]  X. Qin,et al.  Strong convergence of modified Ishikawa iterations for nonlinear mappings , 2007, 0707.1955.

[2]  Ravi P. Agarwal,et al.  Generalized Projection Algorithms for Nonlinear Operators , 2007 .

[3]  Haiyun Zhou,et al.  Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces ✩ , 2008 .

[4]  Hong-Kun Xu,et al.  Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups , 2006 .

[5]  S. Reich,et al.  Asymptotic Behavior of Relatively Nonexpansive Operators in Banach Spaces , 2001 .

[6]  Kasamsuk Ungchittrakool,et al.  Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space , 2007, J. Approx. Theory.

[7]  Meijuan Shang,et al.  Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings , 2007 .

[8]  Meijuan Shang,et al.  Strong Convergence Theorems for Asymptotically Nonexpansive Mappings by Hybrid Methods , 2008 .

[9]  Yongfu Su,et al.  Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups , 2006 .

[10]  Dan Butnariu,et al.  Weak Convergence of Orbits of Nonlinear Operators in Reflexive Banach Spaces , 2003 .

[11]  B. Halpern Fixed points of nonexpanding maps , 1967 .

[12]  Yongfu Su,et al.  Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators , 2008 .

[13]  Y. Alber Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications , 1993, funct-an/9311001.

[14]  I. Ciorǎnescu Geometry of banach spaces, duality mappings, and nonlinear problems , 1990 .

[15]  Arif Rafiq STRONG CONVERGENCE THEOREMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS , 2008 .

[16]  Wataru Takahashi,et al.  Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces , 2008 .

[17]  W. Takahashi Nonlinear Functional Analysis , 2000 .

[18]  Rudong Chen,et al.  Strong convergence and control condition of modified Halpern iterations in Banach spaces , 2006, Int. J. Math. Math. Sci..

[19]  Wataru Takahashi,et al.  A strong convergence theorem for relatively nonexpansive mappings in a Banach space , 2005, J. Approx. Theory.

[20]  William A. Kirk,et al.  A FIXED POINT THEOREM FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS , 1972 .

[21]  Hong-Kun Xu,et al.  Strong convergence of the CQ method for fixed point iteration processes , 2006 .

[22]  Yeol Je Cho,et al.  Convergence of a modified Halpern-type iteration algorithm for quasi-phi-nonexpansive mappings , 2009, Appl. Math. Lett..

[23]  Yongfu Su,et al.  Strong convergence theorems for relatively nonexpansive mappings in a Banach space , 2007 .

[24]  W. Takahashi,et al.  Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups , 2003 .

[25]  Hong-Kun Xu,et al.  Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions , 2008 .

[26]  Yeol Je Cho,et al.  Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces , 2009 .

[27]  Wataru Takahashi,et al.  Strong Convergence of a Proximal-Type Algorithm in a Banach Space , 2002, SIAM J. Optim..