×

zbMATH — the first resource for mathematics

Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces. (English) Zbl 1134.47049
Three-step iterative schemes with errors for two and three nonexpansive maps are introduced in the paper. Finding common fixed points of maps acting on a Banach space is a problem that often arises in applied mathematics. In fact, many algorithms have been introduced for different classes of maps with nonempty set of common fixed points [see, e.g., N. Shahzad, Nonlinear Anal. 61, No. 6 (A), 1031–1039 (2005; Zbl 1089.47058)]. Let \(C\) be a nonempty convex subset of a real Banach space \(E\) and let \(T_i: C \to C\) be nonexpansive maps \((i=1,2,3)\). The following three-step iterative scheme with errors is considered: \( x_1 \in C\), \(z_n=\alpha^{(3)}_n x_n + \beta^{(3)}_n T_3 x_n + \gamma^{(3)}_n u^{(3)}_n\), \(y_n=\alpha^{(2)}_n x_n + \beta^{(2)}_n T_2 z_n + \gamma^{(2)}_n u^{(2)}_n\), \(x_{n+1}=\alpha^{(1)}_n x_n + \beta^{(1)}_n T_1 y_n + \gamma^{(1)}_n u^{(1)}_n\), for all \(n \geq 1\), where \(\{ u^{(j)}_n \}\) is a bounded sequence in \(C\) and \(\{ \alpha^{(j)}_n \}\), \(\{ \beta^{(j)}_n \}\), \(\{ \gamma^{(j)}_n \}\) are sequences in \([0,1]\) satisfying \(\alpha^{(j)}_n + \beta^{(j)}_n + \gamma^{(j)}_n =1\), \(n \geq 1\), \(j=1,2,3\).
Under suitable conditions, the weak and strong convergence of the above scheme to a common fixed point of nonexpansive maps in a uniformly convex Banach space is proved. By modifying the iteration schemes, the corresponding results can be proved for asymptotically nonexpansive mappings with suitable changes. The convergence theorems improve and generalize some important results in the current literature.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Das, G.; Debata, J.P., Fixed points of quasi-nonexpansive mappings, Indian J. pure appl. math., 17, 1263-1269, (1986) · Zbl 0605.47054
[2] Ishikawa, S., Fixed points by a new iteration method, Proc. amer. math. soc., 44, 147-150, (1974) · Zbl 0286.47036
[3] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 44, 506-510, (1953) · Zbl 0050.11603
[4] Takahashi, W.; Kim, G.E., Approximating fixed points of nonexpansive mappings in Banach spaces, Math. japon., 48, 1-9, (1998) · Zbl 0913.47056
[5] Tan, K.K.; Xu, H.K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. math. anal. appl., 178, 301-308, (1993) · Zbl 0895.47048
[6] Zhou, H.Y., Nonexpansive mappings and iterative methods in uniformly convex Banach spaces, Acta math. sinica, 20, 829-836, (2004) · Zbl 1083.47060
[7] Takahashi, W.; Tamura, T., Convergence theorems for a pair of nonexpansive mappings, J. convex anal., 5, 45-58, (1995)
[8] Shahzad, N., Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear anal., 61, 1031-1039, (2005) · Zbl 1089.47058
[9] Goebel, K.; Kirk, W.A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. amer. math. soc., 35, 171-174, (1972) · Zbl 0256.47045
[10] Bose, S.C., Weak convergence to the fixed point of an asymptotically nonexpansive map, Proc. amer. math. soc., 68, 305-308, (1978) · Zbl 0377.47037
[11] Schu, J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. austral. math. soc., 43, 153-159, (1991) · Zbl 0709.47051
[12] Tan, K.K.; Xu, H.K., Fixed point iteration processes for asymptotically nonexpansive mapping, Proc. amer. math. soc., 122, 733-739, (1994) · Zbl 0820.47071
[13] Xu, B.; Noor, M.A., Fixed-points iteration for asymptotically nonexpansive mappings in Banach spaces, J. math. anal. appl., 267, 444-453, (2002) · Zbl 1011.47039
[14] Cho, Y.J.; Zhou, H.Y.; Guo, G., Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. math. appl., 47, 707-717, (2004) · Zbl 1081.47063
[15] 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
[16] Fukhar-ud-din, H.; Khan, S.H., Convergence of two-step iterative scheme with errors for two asymptotically nonexpansive mappings, Internat. J. math. math. sci., 37, 1965-1971, (2004) · Zbl 1086.47048
[17] Khan, S.H.; Fukhar-ud-din, H., Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear anal., 8, 1295-1301, (2005) · Zbl 1086.47050
[18] Khan, S.H.; Takahashi, W., Approximating common fixed points of two asymptotically nonexpansive mappings, Sci. math. japon., 53, 143-148, (2001) · Zbl 0985.47042
[19] Takahashi, W.; Shimoji, K., Convergence theorems for nonexpansive mappings and feasibility problems, Math. comput. modelling, 32, 1463-1471, (2000) · Zbl 0971.47040
[20] Xu, H.K.; Ori, R.G., An implicit iteration process for nonexpansive mappings, Numer. funct. anal. optim., 22, 767-773, (2001) · Zbl 0999.47043
[21] Kaczor, W., Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups, J. math. anal. appl., 272, 565-574, (2002) · Zbl 1058.47049
[22] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. math. soc., 73, 591-597, (1967) · Zbl 0179.19902
[23] Bruck, R.E., A simple proof of the Mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. math., 32, 107-116, (1979) · Zbl 0423.47024
[24] Senter, H.F.; Dotson, W.G., Approximating fixed points of nonexpansive mappings, Proc. amer. math. soc., 44, 375-380, (1974) · Zbl 0299.47032
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.