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An approximation method for monotone Lipschitzian operators in Hilbert spaces

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  09 April 2009

C. E. Chidume
C. E. Chidume
Affiliation:
Department of Mathematics, University of Nigeria, Nsukka (Anambra State), Nigeria
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Abstract

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Suppose H is a complex Hilbert space and K is a nonempty closed convex subset of H. Suppose T: K → H is a monotomc Lipschitzian mapping with constant L ≧ 1 such that, for x in K and h in H, the equation x + Tx Tx = h has a solution q in K. Given x0 in K, let {Cn}∞n=0 be a real sequence satisfying: (i) C0 = 1, (ii) 0 ≦ Cn < L-2 for all n ≧ 1, (iii) ΣnCn(1 − Cn) diverges. Then the sequence {Pn}∞n=0 in H defined by pn = (1 − Cn)xn + CnSxn, n ≧ 0, where {xn}∞n=0 in K is such that, for each n ≧ 1, ∥ xn – Pn−1 ∥ = infx ∈ k ∥ Pn−1 − x ∥, converges strongly to a solution q of x + Tx = h. Explicit error estimates are given. A similar result is also proved for the case when the operator T is locally Lipschitzian and monotone.

MSC classification

Secondary: 47H10: Fixed-point theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 1 , August 1986 , pp. 59 - 63
Copyright
Copyright © Australian Mathematical Society 1986

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