Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces. (English) Zbl 1102.47056

Shahzad, Naseer; Udomene, Aniefiok

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 64, No. 3, 558-567 (2006); erratum ibid. 66, No. 12, 2980-2981 (2007).

Let \(E\) be a real Banach space with uniformly Gâteaux differentiable norm, \(K\) a nonempty bounded closed convex subset of \(E\), and \(T: K \to K\) an asymptotically nonexpansive mapping, i.e., \(\| T^n x- T^n y \| \leq k_n \| x-y \|\) for all \(x, y \in K\), where \(k_n \in [1, \infty)\), \(\lim_{n \to \infty} k_n=1\). Let \(f: K \to K\) be a contraction. The authors study approximating properties of the sequence \(\{ x_n\}\) defined by \(x_n=(1-k_n^{-1} t_n) f(x_n)+k_n^{-1} t_n T^n x_n\) (\(t_n>0\), \(\lim_{n \to \infty} t_n=1\)) with respect to the set of fixed points of the mapping \(T\).

Reviewer: Mikhail Yu. Kokurin (Yoshkar-Ola)

MSC:

47J25	Iterative procedures involving nonlinear operators
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H06	Nonlinear accretive operators, dissipative operators, etc.
47J20	Variational and other types of inequalities involving nonlinear operators (general)

Keywords:

viscosity approximation; asymptotically nonexpansive mapping; variational inequality; normal structure

References:

[1]	Aksoy, A.G.; Khamsi, M.A., Nonstandard methods in fixed point theory, (1990), Springer New York · Zbl 0713.47050
[2]	Bynum, W.L., Normal structure coefficients for Banach spaces, Pacific J. math., 86, 427-436, (2001) · Zbl 0442.46018
[3]	Chidume, C.E.; Jinlu Liu; Udomene, A., Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings, Proc. amer. math. soc., 113, 473-480, (2005) · Zbl 1073.47059
[4]	Cioranescu, I., Geometry of Banach spaces, duality mappings and nonlinear problems, (1990), Kluwer Academic Publishers Amsterdam · Zbl 0712.47043
[5]	Goebel, K.; Kirk, W.A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. amer. math. soc., 35, 171-174, (1972) · Zbl 0256.47045
[6]	Goebel, K.; Kirk, W.A., Topics of metric fixed point theory, (1990), Cambridge University Press Cambridge · Zbl 0708.47031
[7]	Kim, T.H.; Xu, H.K., Remarks on asymptotically nonexpansive mappings, Nonlinear anal., 41, 405-415, (2000) · Zbl 0957.47037
[8]	Lim, T.C.; Xu, H.K., Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear anal., 22, 1345-1355, (1994) · Zbl 0812.47058
[9]	Moudafi, A., Viscosity approximation methods for fixed-point problems, J. math. anal. appl., 241, 46-55, (2000) · Zbl 0957.47039
[10]	Xu, H.K., Iterative algorithms for nonlinear operators, J. London math. soc., 66, 240-256, (2002) · Zbl 1013.47032
[11]	Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060