## On Mann iteration in Hilbert spaces.(English)Zbl 1136.47047

The author proves the strong convergence of certain Mann iterates of a hemicontractive map in a Hilbert space. Not all results, however, seem to be correct, as was pointed out by Y. Qing [ibid. 68, No. 2 (A), 460 (2008; Zbl 1136.47048), see the following review].

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.

### Keywords:

Hilbert space; Mann iteration; pseudocontractive maps

Zbl 1136.47048
Full Text:

### References:

 [1] Browder, F.E., Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. sympos. pure math., XVIII, 2, (1976) · Zbl 0176.45301 [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] Chidume, C.E., Iterative approximation of Lipschitz strictly pseudocontractive mappings, Proc. amer. math. soc., 99, 2, 283-288, (1987) · Zbl 0646.47037 [4] Chidume, C.E., Approximation of fixed points of strongly pseudocontractive mappings, Proc. amer. math. soc., 120, 2, 545-551, (1994) · Zbl 0802.47058 [5] Chidume, C.E., Global iteration schemes for strongly pseudocontractive maps, Proc. amer. math. soc., 126, 9, 2641-2649, (1998) · Zbl 0901.47046 [6] Chidume, C.E., Iterative solution of nonlinear equations of strongly accretive type, Math. nachr., 189, 49-60, (1998) · Zbl 0911.47063 [7] Chidume, C.E.; Moore, C., Fixed point iteration for pseudocontractive maps, Proc. amer. math. soc., 127, 4, 1163-1170, (1999) · Zbl 0913.47052 [8] Chidume, C.E.; Osilike, M.O., Ishikawa iteration process for nonlinear Lipschitz strongly accretive mappings, J. math. anal. appl., 192, 727-741, (1995) · Zbl 0862.47045 [9] Chidume, C.E.; Osilike, M.O., Nonlinear accretive and pseudocontractive operator equations in Banach spaces, Nonlinear anal., 31, 7, 779-789, (1998) · Zbl 0901.47037 [10] Chidume, C.E.; Mutangadura, S.A., An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. amer. math. soc., 129, 8, 2359-2363, (2001) · Zbl 0972.47062 [11] Crandall, M.G.; Pazy, A., On the range of accretive operators, Israel J. math., 27, 235-246, (1977) · Zbl 0355.47039 [12] Deng, L., On chidume’s open problems, J. math. anal. appl., 174, 2, 441-449, (1993) · Zbl 0784.47051 [13] Deng, L., Iteration process for nonlinear Lipschitzian strongly accretive mappings in $$L_p$$ spaces, J. math. anal. appl., 188, 128-140, (1994) · Zbl 0828.47042 [14] Deng, L.; Ding, X.P., Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth Banach spaces, Nonlinear anal., 24, 7, 981-987, (1995) · Zbl 0827.47041 [15] Hicks, T.L.; Kubicek, J.R., On the Mann iteration process in Hilbert space, J. math. anal. appl., 59, 498-504, (1977) · Zbl 0361.65057 [16] Ishikawa, S., Fixed point by a new iteration method, Proc. amer. math. soc., 4, 1, 147-150, (1974) · Zbl 0286.47036 [17] Liu, L.S., Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces, J. math. anal. appl., 194, 114-125, (1995) · Zbl 0872.47031 [18] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 4, 506-610, (1953) · Zbl 0050.11603 [19] Qihou, L., On naimpally and singh’s open questions, J. math. anal. appl., 124, 157-164, (1987) · Zbl 0625.47044 [20] Qihou, L., The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings, J. math. anal. appl., 148, 55-62, (1990) · Zbl 0729.47052 [21] Qihou, L., Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear anal., 26, 11, 1835-1842, (1996) · Zbl 0861.47047 [22] Reich, S., An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear anal., 2, 85-92, (1978) · Zbl 0375.47032 [23] Reich, S., Constructive techniques for accretive and monotone operators, (), 335-345 [24] Reich, S., Constructing zeros of accretive operators, I II, Appl. anal., 9, 159-163, (1979) · Zbl 0424.47034 [25] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. math. anal. appl., 67, 274-276, (1979) · Zbl 0423.47026 [26] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. math. anal. appl., 75, 287-292, (1980) · Zbl 0437.47047 [27] Rhoades, B.E., Comments on two fixed point iteration procedures, J. math. anal. appl., 56, 741-750, (1976) · Zbl 0353.47029 [28] Schu, J., On a theorem of C.E. Chidume concerning the iterative approximation of fixed points, Math. nachr., 153, 313-319, (1991) · Zbl 0796.47047 [29] Schu, J., Iterative construction of fixed points of strictly pseudocontractive mappings, Appl. anal., 40, 67-72, (1991) · Zbl 0697.47061 [30] Schu, J., Iterative construction of fixed points of asymptotically nonexpansive mappings, J. math. anal. appl., 158, 407-413, (1991) · Zbl 0734.47036 [31] 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 [32] Weng, X.L., Fixed point iteration for local strictly pseudocontractive mappings, Proc. amer. math. soc., 113, 727-731, (1991) · Zbl 0734.47042 [33] Xu, Y., Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. math. anal. appl., 224, 91-101, (1998) · Zbl 0936.47041 [34] Xu, Z.B.; Roach, G.F., A necessary and sufficient condition for convergence of steepest descent approximation to accretive operator equations, J. math. anal. appl., 167, 340-354, (1992) · Zbl 0818.47061
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