## 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.

Zbl 1089.47058
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### References:

  Das, G.; Debata, J.P., Fixed points of quasi-nonexpansive mappings, Indian J. pure appl. math., 17, 1263-1269, (1986) · Zbl 0605.47054  Ishikawa, S., Fixed points by a new iteration method, Proc. amer. math. soc., 44, 147-150, (1974) · Zbl 0286.47036  Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 44, 506-510, (1953) · Zbl 0050.11603  Takahashi, W.; Kim, G.E., Approximating fixed points of nonexpansive mappings in Banach spaces, Math. japon., 48, 1-9, (1998) · Zbl 0913.47056  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  Zhou, H.Y., Nonexpansive mappings and iterative methods in uniformly convex Banach spaces, Acta math. sinica, 20, 829-836, (2004) · Zbl 1083.47060  Takahashi, W.; Tamura, T., Convergence theorems for a pair of nonexpansive mappings, J. convex anal., 5, 45-58, (1995)  Shahzad, N., Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear anal., 61, 1031-1039, (2005) · Zbl 1089.47058  Goebel, K.; Kirk, W.A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. amer. math. soc., 35, 171-174, (1972) · Zbl 0256.47045  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  Schu, J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. austral. math. soc., 43, 153-159, (1991) · Zbl 0709.47051  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  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  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  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  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  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  Khan, S.H.; Takahashi, W., Approximating common fixed points of two asymptotically nonexpansive mappings, Sci. math. japon., 53, 143-148, (2001) · Zbl 0985.47042  Takahashi, W.; Shimoji, K., Convergence theorems for nonexpansive mappings and feasibility problems, Math. comput. modelling, 32, 1463-1471, (2000) · Zbl 0971.47040  Xu, H.K.; Ori, R.G., An implicit iteration process for nonexpansive mappings, Numer. funct. anal. optim., 22, 767-773, (2001) · Zbl 0999.47043  Kaczor, W., Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups, J. math. anal. appl., 272, 565-574, (2002) · Zbl 1058.47049  Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. math. soc., 73, 591-597, (1967) · Zbl 0179.19902  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  Senter, H.F.; Dotson, W.G., Approximating fixed points of nonexpansive mappings, Proc. amer. math. soc., 44, 375-380, (1974) · Zbl 0299.47032
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