The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces.(English)Zbl 1142.47326

Summary: A recent trend in the iterative methods for constructing fixed points of nonlinear mappings is to use the viscosity approximation technique. The advantage of this technique is that one can find a particular solution to the associated problems, and in most cases this particular solution solves some variational inequality. In this paper, we try to extend this technique to find a particular common fixed point of a finite family of asymptotically nonexpansive mappings in a Banach space which is reflexive and has a weakly continuous duality map. Both implicit and explicit viscosity approximation schemes are proposed and their strong convergence to a solution to a variational inequality is proved.

MSC:

 47H06 Nonlinear accretive operators, dissipative operators, etc. 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47J05 Equations involving nonlinear operators (general) 47J25 Iterative procedures involving nonlinear operators
Full Text:

References:

 [1] Browder, F.E., Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z., 100, 201-225, (1967) · Zbl 0149.36301 [2] Browder, F.E.; Petryshyn, W.V., Construction of fixed points of nonlinear mappings in Hilbert spaces, J. math. anal. appl., 20, 197-228, (1967) · Zbl 0153.45701 [3] Chang, S.S.; Tan, K.K.; Lee, H.W.J.; Chan, C.K., On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings, J. math. anal. appl., 313, 273-283, (2006) · Zbl 1086.47044 [4] Chidume, C.E.; Li, J.; Udomene, A., Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings, Proc. amer. math. soc., 133, 2, 473-480, (2004) · Zbl 1073.47059 [5] Dominguez Benavides, T.; Lorenzo Ramirez, P., Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings, Proc. amer. math. soc., 129, 12, 3549-3557, (2001) · Zbl 0985.47041 [6] Goebel, K.; Kirk, W.A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. amer. math. soc., 35, 171-174, (1972) · Zbl 0256.47045 [7] Lim, T.C.; Xu, H.K., Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear anal., 22, 1345-1355, (1994) · Zbl 0812.47058 [8] Lions, P.L., Approximation de points fixes de contractions, C. R. acad. sci. Sèr. A-B Paris, 284, 1357-1359, (1977) · Zbl 0349.47046 [9] Moudafi, A., Viscosity approximation methods for fixed-point problems, J. math. anal. appl., 241, 46-55, (2000) · Zbl 0957.47039 [10] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. math. soc., 73, 595-597, (1967) · Zbl 0179.19902 [11] Shahzad, N.; Udomene, A., Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces, Nonlinear anal., 64, 558-567, (2006) · Zbl 1102.47056 [12] Wu, X.; Yao, J.C.; Zeng, L.C., Uniform normal structure and strong convergence theorems for asymptotically pseudocontractive mappings, J. nonlinear convex anal., 6, 3, 453-463, (2005) · Zbl 1100.46008 [13] Xu, H.K.; Ori, R.G., An implicit iteration process for nonexpansive mappings, Numer. funct. anal. optim., 22, 767-773, (2001) · Zbl 0999.47043 [14] Xu, H.K., Iterative algorithms for nonlinear operators, J. London math. soc., 66, 240-256, (2002) · Zbl 1013.47032 [15] Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060 [16] Xu, Z.B.; Roach, G.F., Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces, J. math. anal. appl., 157, 189-210, (1991) · Zbl 0757.46034 [17] Yao, Y.; Liou, Y.C., Strong convergence to common fixed points of a finite family of asymptotically nonexpansive mappings, Taiwanese J. math., 11, 849-866, (2007) · Zbl 1219.47135 [18] Zeng, L.C.; Yao, J.C., Implicit iteration scheme with perturbed mappings for common fixed points of a finite family of nonexpansive mappings, Nonlinear anal., 64, 2507-2515, (2006) · Zbl 1105.47061
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.