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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
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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
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