zbMATH — the first resource for mathematics

On viscosity iterative methods for variational inequalities. (English) Zbl 1115.49024
Summary: Let \(\widetilde J\) be a commutative family of nonexpansive mappings of a closed convex subset \(C\) of a reflexive Banach space \(X\) such that the set of common fixed-point is nonempty. In this paper, we suggest and analyze a new viscosity iterative method for a commutative family of nonexpansive mappings. We also prove that the approximate solution obtained by the proposed method converges to a solution of a variational inequality. Our method of proof is simple and different from the other methods. Results proved in this paper may be viewed as an improvement and refinement of the previously known results.

49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49M30 Other numerical methods in calculus of variations (MSC2010)
Full Text: DOI
[1] Noor, M. Aslam, Some development in general variational inequalities, Appl. math. comput., 152, 199-277, (2004) · Zbl 1134.49304
[2] Browder, F.E., Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Arch. ration. mech. anal., 24, 82-90, (1967) · Zbl 0148.13601
[3] Halpern, B., Fixed points of nonexpansive maps, Bull. amer. math. soc., 73, 957-961, (1967) · Zbl 0177.19101
[4] Lions, P.L., Approximation de points fixes de contractions, C. R. acad. sci. Paris ser. A-B, 284, 1357-1359, (1977) · Zbl 0349.47046
[5] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. math. anal. appl., 75, 287-292, (1980) · Zbl 0437.47047
[6] Suzuki, T., Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. math. anal. appl., 305, 227-239, (2005) · Zbl 1068.47085
[7] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math., 59, 486-491, (1992) · Zbl 0797.47036
[8] Shioji, N.; Takahashi, W., Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. amer. math. soc., 125, 12, 3461-3465, (1997)
[9] Cho, Y.J.; Kang, S.M.; Zhou, H.Y., Some control conditions on iterative methods, Comm. appl. nonlinear anal., 12, 2, 27-34, (2005) · Zbl 1088.47053
[10] Xu, H.K., Iterative algorithms for nonlinear operators, J. London math. soc., 66, 240-256, (2002) · Zbl 1013.47032
[11] Benavides, T.D.; Acedo, G.L.; Xu, H.K., Construction of sunny nonexpansive retractions in Banach spaces, Bull. austral. math. soc., 66, 9-16, (2002) · Zbl 1017.47037
[12] Jung, J.S.; Morales, C., The Mann process for perturbed m-accretive operators in Banach spaces, Nonlinear anal., 46, 231-243, (2001) · Zbl 0997.47042
[13] Goebel, K.; Kirk, W.A., Topics in metric fixed point theory, Cambridge stud. adv. math., vol. 28, (1990), Cambridge Univ. Press Cambridge, UK · Zbl 0708.47031
[14] Liu, L.S., Iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces, J. math. anal. appl., 194, 114-125, (1995) · Zbl 0872.47031
[15] Takahashi, W.; Ueda, Y., On Reich’s strong convergence theorems for resolvents of accretive operators, J. math. anal. appl., 104, 546-553, (1984) · Zbl 0599.47084
[16] Lim, T.C., A fixed point theorem for families on nonexpansive mappings, Pacific J. math., 53, 487-493, (1974) · Zbl 0291.47032
[17] Moudafi, A., Viscosity approximation methods for fixed point problems, J. math. anal. appl., 241, 46-55, (2000) · Zbl 0957.47039
[18] Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060
[19] A. Aleyner, Y. Censor, Best approximation to common fixed points of a semigroup of nonexpansive operators, J. Convex Anal., in press · Zbl 1071.41031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.