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Approximation of fixed points of strongly pseudocontractive mappings in uniformly smooth Banach spaces. (English) Zbl 1124.47050

The author proves the following result. Let \(E\) be a real uniformly smooth Banach space, \(K\) be a nonempty closed convex subset of \(E\), and the map \(T: K \to K\) be continuous and strongly pseudocontractive. Then the Ishikawa iteration scheme converges to the unique fixed point of \(T\). The author buttresses his result with an example of a strongly pseudocontractive map \(T:=T_{1} + T_{2}\) which is neither Lipschitzian nor has a bounded range.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H06 Nonlinear accretive operators, dissipative operators, etc.
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References:

[1] Bogin, J., On strict pseudo-contractions and a fixed point theorem
[2] Zhou, H.; Jia, Y., Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption, Proceedings of the American Mathematical Society, 125, 6, 1705-1709 (1997) · Zbl 0871.47045 · doi:10.1090/S0002-9939-97-03850-1
[3] Chidume, C. E.; Osilike, M. O., Ishikawa iteration process for nonlinear Lipschitz strongly accretive mappings, Journal of Mathematical Analysis and Applications, 192, 3, 727-741 (1995) · Zbl 0862.47045 · doi:10.1006/jmaa.1995.1200
[4] Deimling, K., Zeros of accretive operators, Manuscripta Mathematica, 13, 4, 365-374 (1974) · Zbl 0288.47047 · doi:10.1007/BF01171148
[5] Zhou, H.; Jia, Y., Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption, Proceedings of the American Mathematical Society, 125, 6, 1705-1709 (1997) · Zbl 0871.47045 · doi:10.1090/S0002-9939-97-03850-1
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