## Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations.(English)Zbl 0958.47030

Let $$K$$ be a closed convex nonempty subset of a Hilbert space $$H$$ and let $$T: K\to K$$ be a Lipschitz pseudocontraction mapping with a nonempty set of fixed points. Weak and strong convergence theorems for iterative approximations of fixed points are proved. Some applications to monotone operators in Hilbert spaces are presented.

### MSC:

 47H10 Fixed-point theorems 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H05 Monotone operators and generalizations
Full Text:

### References:

 [1] 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 [2] Hicks, T.L.; Kubicek, J.D., On the Mann iteration process in a Hilbert space, J. math. anal. appl., 59, 498-504, (1977) · Zbl 0361.65057 [3] Naimpally, S.A.; Singh, K.L., Extensions of some fixed point theorems of rhoades, J. math. anal. appl., 96, 437-446, (1983) · Zbl 0524.47033 [4] Qihou, L., On naimpally and Singh’s open questions, J. math. anal. appl., 124, 157-164, (1987) · Zbl 0625.47044 [5] 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 [6] Deimling, K., Nonlinear functional analysis, (1980), Springer-Verlag [7] Browder, F.E., Nonlinear operators and nonlinear equations of evolution in Banach spaces, () · Zbl 0176.45301 [8] Ishikawa, I.S., Fixed points by a new iteration method, (), 147-150 · Zbl 0286.47036 [9] Chidume, C.E.; Moore, C., Fixed point iteration for pseudocontractive maps, (), 1163-1170 · Zbl 0913.47052 [10] Xu, Y., Ishikawa and Mann iteration processes with errors for nonlinear strongly accretive operator equations, J. math. anal. appl., 224, 91-101, (1998) · Zbl 0936.47041 [11] Reinermann, J., Über fixpunkte kontrahievuder abbidungen und schwach konvergente tooplite-verfahren, Arch. math., 20, 59-64, (1969) · Zbl 0174.19401 [12] 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 [13] Liu, L., Ishikawa and Mann iteration processes with errors for nonlinear strongly accretive mappings in Banach spaces, J. math. anal. appl., 194, 1, 114-125, (1995) · Zbl 0872.47031 [14] Maruster, T., The solution by iteration of nonlinear equations in Hilbert spaces, (), 69-73 · Zbl 0355.47037 [15] Opial, Z., Weak convergence of the sequences of approximations for nonexpansive mappings, Bull. amer. math. soc., 73, 591-597, (1967) · Zbl 0179.19902 [16] Verma, R.U., A fixed-point theorem involving Lipschitzian generalized pseudo-contractions, (), 83 · Zbl 0907.47052 [17] Osilike, M.O., Ishikawa and Mann iteration methods with errors for nonlinear equations of the accretive type, J. math. anal. appl., 213, 1, 91-105, (1997) · Zbl 0904.47056 [18] Osilike, M.O., Stable iteration procedures for nonlinear pseudocontractive and accretive operators in arbitrary Banach spaces, Indian J. pure appl. math., 28, 8, 1017-1029, (1997) · Zbl 0898.47047 [19] Chidume, C.E.; Osilike, M.O., Nonlinear accretive and pseudocontractive operator equations in Banach spaces, Nonlinear analysis, 31, 7, 779-789, (1998) · Zbl 0901.47037
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.