STRONG CONVERGENCE OF COMPOSITE ITERATIVE METHODS FOR NONEXPANSIVE MAPPINGS

Title & Authors
STRONG CONVERGENCE OF COMPOSITE ITERATIVE METHODS FOR NONEXPANSIVE MAPPINGS
Jung, Jong-Soo;

Abstract
Let E be a reflexive Banach space with a weakly sequentially continuous duality mapping, C be a nonempty closed convex subset of E, f : C $\small{\rightarrow}$C a contractive mapping (or a weakly contractive mapping), and T : C $\small{\rightarrow}$ C a nonexpansive mapping with the fixed point set F(T) $\small{{\neq}{\emptyset}}$. Let {$\small{x_n}$} be generated by a new composite iterative scheme: $\small{y_n={\lambda}_nf(x_n)+(1-{\lambda}_n)Tx_n}$, $\small{x_{n+1}=(1-{\beta}_n)y_n+{\beta}_nTy_n}$, ($\small{n{\geq}0}$). It is proved that {$\small{x_n}$} converges strongly to a point in F(T), which is a solution of certain variational inequality provided the sequence {$\small{\lambda_n}$} $\small{\subset}$ (0, 1) satisfies $\small{lim_{n{\rightarrow}{\infty}}{\lambda}_n}$ = 0 and $\small{\sum_{n=0}^{\infty}{\lambda}_n={\infty}}$, {$\small{\beta_n}$} $\small{\subset}$ [0, a) for some 0 < a < 1 and the sequence {$\small{x_n}$} is asymptotically regular.
Keywords
viscosity approximation method;nonexpansive mapping;composite iterative scheme;contractive mapping;weakly contractive mapping;weakly sequentially continuous duality mapping;variational inequality;
Language
English
Cited by
1.
Convergence of a General Composite Iterative Method for a Countable Family of Nonexpansive Mappings, ISRN Applied Mathematics, 2012, 2012, 1
2.
Strong convergence of a new composite iterative method for equilibrium problems and fixed point problems, Applied Mathematics and Computation, 2010, 215, 11, 3891
3.
Strong convergence of composite iterative methods for equilibrium problems and fixed point problems, Applied Mathematics and Computation, 2009, 213, 2, 498
References
1.
Ya. I. Alber and A. N. Iusem, Extension of subgradient techniques for nonsmooth optimization in Banach spaces, Set-Valued Anal. 9 (2001), no. 4, 315-335.

2.
Ya. I. Alber, S. Reich, and J. C. Yao, Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces, Abstr. Appl. Anal. 2003 (2003), no. 4, 193-216.

3.
Y. J. Cho, S. M. Kang, and H. Y. Zhou, Some control conditions on iterative methods, Comm. Appl. Nonlinear Anal. 12 (2005), no. 2, 27-34.

4.
I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, 1990.

5.
K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.

6.
B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957-961.

7.
J. S. Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005), no. 2, 509-520.

8.
J. S. Jung, Convergence theorems of iterative algorithms for a family of finite nonexpansive mappings, Taiwanese J. Math. 11 (2007), no. 3, 883-902.

9.
J. S. Jung and C. Morales, The Mann process for perturbed m-accretive operators in Banach spaces, Nonlinear Anal. 46 (2001), no. 2, Ser. A: Theory Methods, 231-243.

10.
J. S. Jung and D. R. Sahu, Convergence of approximating paths to solutions of variational inequalities involving non-Lipschitzian mappings, J. Korean Math. Soc. 45 (2009), no. 2, 377-392.

11.
T. H. Kim and H. K Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005), no. 1-2, 51-60.

12.
P. L. Lions, Approximation de points fixes de contractions, C. R. Acad. Sci. Paris Ser. A-B 284 (1977), no. 21, A1357-A1359.

13.
L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), no. 1, 114-125.

14.
A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000), no. 1, 46-55.

15.
J. G. O'Hara, P. Pillay, and H. K. Xu, Iterative approaches to convex feasibility problems in Banach spaces, Nonlinear Anal. 64 (2006), no. 9, 2022-2042.

16.
B. E. Rhodes, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (2001), no. 4, 2683-2693.

17.
S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), no. 1, 287-292.

18.
S. Reich, Approximating fixed points of nonexpansive mappings, Panamer. Math. J. 4 (1994), no. 2, 23-28.

19.
N. Shioji and W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3641-3645.

20.
R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel) 58 (1992), no. 5, 486-491.

21.
H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2) 66 (2002), no. 1, 240-256.

22.
H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), no. 1, 279-291.

23.
H. K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006), no. 2, 631-643.