×

zbMATH — the first resource for mathematics

Asymptotically strict pseudocontractive mappings in the intermediate sense. (English) Zbl 1221.47122
Summary: It is proved that the modified Mann iteration process \(x_{n+1}=(1-\alpha _n)x_n+\alpha _nT^nx_n\), \(n\in \mathbb N\), where \(\{\alpha _n\}\) is a sequence in (0,1) with \(\delta \leq \alpha _n\leq 1 - \kappa - \delta \) for some \(\delta \in (0,1)\), converges weakly to a fixed point of an asymptotically \(\kappa \)-strict pseudocontractive mapping \(T\) in the intermediate sense which is not necessarily Lipschitzian. We also develop a CQ method for this modified Mann iteration process which generates a strongly convergent sequence.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
46B20 Geometry and structure of normed linear spaces
47H10 Fixed-point theorems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agarwal, R.P.; O’Regan, Donal; Sahu, D.R., Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. nonlinear convex anal., 8, 1, 61-79, (2007) · Zbl 1134.47047
[2] Bruck, R.E.; Kuczumow, T.; Reich, S., Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloq. math., 65, 169-179, (1993) · Zbl 0849.47030
[3] Chang, S.S.; Cho, Y.J.; Zhou, H., Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings, J. Korean math. soc., 38, 1245-1260, (2001) · Zbl 1020.47059
[4] Chidume, C.E.; Ali, B., Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces, J. math. anal. appl., 326, 2, 960-973, (2007) · Zbl 1112.47053
[5] Chidume, C.E.; Ofoedu, E.U.; Zegeye, H., Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. math. anal. appl., 280, 364-374, (2003) · Zbl 1057.47071
[6] Chidume, C.E.; Shahzad, N.; Zegeye, H., Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense, Numer. funct. anal. optim., 25, 239-257, (2004) · Zbl 1061.47048
[7] Goebel, K.; Kirk, W.A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. amer. math. soc., 35, 1, 171-174, (1972) · Zbl 0256.47045
[8] Gornicki, J., Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces, Comment math. univ. carolin., 30, 2, 249-252, (1989) · Zbl 0686.47045
[9] Huang, N.J.; Lan, H.Y., A new iterative approximation of fixed points for asymptotically contractive type mappings in Banach spaces, Indian J. pure appl. math., 35, 4, 441-453, (2004) · Zbl 1081.47065
[10] Khan, S.H.; Fukharuddin, H., Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear anal., 61, 1295-1301, (2005) · Zbl 1086.47050
[11] Kim, G.E.; Kim, T.H., Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces, Comp. math. appl., 42, 1565-1570, (2001) · Zbl 1001.47048
[12] Kim, T.H.; Xu, H.K., Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear anal., 64, 5, 1140-1152, (2006) · Zbl 1090.47059
[13] Kim, T.H.; Xu, H.K., Convergence of the modified mann’s iteration method for asymptotically strict pseudocontractions, Nonlinear anal., 68, 2828-2836, (2008) · Zbl 1220.47100
[14] Liu, Z.Q.; Kang, S.M., Weak and strong convergence for fixed points of asymptotically nonexpansive mappings, Acta math. sinica, 20, 1009-1018, (2004) · Zbl 1098.47059
[15] Marino, G.; Xu, H.K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. math. anal. appl., 329, 336-346, (2007) · Zbl 1116.47053
[16] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. math. anal. appl., 279, 372-379, (2003) · Zbl 1035.47048
[17] Nammanee, K.; Noor, M.A.; Suantai, S., Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings, J. math. anal. appl., 314, 320-334, (2006) · Zbl 1087.47054
[18] Osilike, M.O.; Aniagbosor, S.C., Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. comput. model., 32, 1181-1191, (2000) · Zbl 0971.47038
[19] Rhoades, B.E., Fixed point iterations for certain nonlinear mappings, J. math. anal. appl., 183, 118-120, (1994) · Zbl 0807.47045
[20] Schu, J., Iterative construction of fixed points of asymptotically nonexpansive mapping, J. math. anal. appl., 159, 407-413, (1991) · Zbl 0734.47036
[21] Schu, J., Weak and strong convergence of fixed points of asymptotically nonexpansive maps, Bull. austral. math. soc., 43, 153-159, (1991) · Zbl 0709.47051
[22] Tan, K.K.; Xu, H.K., Nonlinear ergodic theorems for asymptotically nonexpansive mappings in Banach spaces, Proc. amer. math. soc., 114, 399-404, (1992) · Zbl 0781.47045
[23] Tan, K.K.; Xu, H.K., Fixed point iteration process for asymptotically nonexpansive mappings, Proc. amer. math. soc., 122, 733-739, (1994) · Zbl 0820.47071
[24] Wang, L., Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. math. anal. appl., 323, 1, 550-557, (2006) · Zbl 1126.47055
[25] Martinez-Yanes, C.; Xu, H.K., Strong convergence of the CQ method for fixed point iteration processes, Nonlinear anal., 64, 11, 2400-2411, (2006) · Zbl 1105.47060
[26] Xu, H.K., Existence and convergence for fixed points for mappings of asymptotically nonexpansive type, Nonlinear anal., 16, 1139-1146, (1991) · Zbl 0747.47041
[27] Xu, B.L.; Noor, M.A., Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. math. anal. appl., 267, 444-453, (2002) · Zbl 1011.47039
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.