A New Iterative Algorithm for Approximating Common Fixed Points for Asymptotically Nonexpansive Mappings

Suppose that is a nonempty closed convex subset of a real uniformly convex and smooth Banach space with as a sunny nonexpansive retraction. Let be two weakly inward and asymptotically nonexpansive mappings with respect to with sequences , , respectively. Suppose that is a sequence in generated iteratively by , , for all , where , , and are three real sequences in for some which satisfy condition . Then, we have the following. (1) If one of and is completely continuous or demicompact and , then the strong convergence of to some is established. (2) If is a real uniformly convex Banach space satisfying Opial's condition or whose norm is Fréchet differentiable, then the weak convergence of to some is proved.

[1]  B. Rhoades,et al.  Fixed Point Iterations for Certain Nonlinear Mappings , 1994 .

[2]  Yisheng Song,et al.  Viscosity approximation methods for nonexpansive nonself-mappings☆ , 2006 .

[3]  Naseer Shahzad,et al.  Approximating fixed points of non-self nonexpansive mappings in Banach spaces , 2005 .

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

[5]  Gregory B. Passty Construction of fixed points for asymptotically nonexpansive mappings , 1982 .

[6]  W. Takahashi,et al.  Strong convergence theorems for nonexpansive nonself-mappings in Banach spaces , 1998 .

[7]  Jarosław Górnicki,et al.  Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces , 1989 .

[8]  Safeer Hussain Khan,et al.  Weak and strong convergence of a scheme with errors for two nonexpansive mappings , 2005 .

[9]  高橋 渉 Nonlinear functional analysis : fixed point theory and its applications , 2000 .

[10]  Jürgen Schu,et al.  Iterative construction of fixed points of asymptotically nonexpansive mappings , 1991 .

[11]  J. Schu,et al.  Weak and strong convergence to fixed points of asymptotically nonexpansive mappings , 1991, Bulletin of the Australian Mathematical Society.

[12]  M. O. Osilike,et al.  Weak and strong convergence theorems for fixed points of asymptotically nonexpensive mappings , 2000 .

[13]  Charles E. Chidume,et al.  Strong and weak convergence theorems for asymptotically nonexpansive mappings , 2003 .

[14]  M. Birkner,et al.  Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach , 2002 .

[15]  Hong-Kun Xu,et al.  Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process , 1993 .

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

[17]  Hong-Kun Xu,et al.  Fixed point iteration processes for asymptotically nonexpansive mappings , 1994 .

[18]  Hong-Kun Xu,et al.  The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces , 1992 .

[19]  S. C. Bose,et al.  Weak convergence to the fixed point of an asymptotically nonexpansive map , 1978 .

[20]  Yeol Je Cho,et al.  DEMI-CLOSED PRINCIPLE AND WEAK CONVERGENCE PROBLEMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS , 2001 .

[21]  S. Reich Weak convergence theorems for nonexpansive mappings in Banach spaces , 1979 .

[22]  Yeol Je Cho,et al.  Iterative methods for nonlinear operator equations in banach spaces , 2002 .

[23]  S. Ishikawa Fixed points and iteration of a nonexpansive mapping in a Banach space , 1976 .

[24]  Daishi Kuroiwa,et al.  Strong convergence of averaging iterations of nonexpansive nonself-mappings , 2004 .