Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T01:39:21.887Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Browder, F. E., ‘The solvability of nonlinear functional equations,’ Duke Math. J. 30 (1963), 557–566.CrossRefGoogle Scholar
[2]Browder, F. E., ‘Nonlinear mappings of nonexpansive and accretive type in Banach spaces,’ Bull. Amer. Math. Soc. 73 (1967), 875–882.CrossRefGoogle Scholar
[3]Browder, F. E., ‘Nonlinear monotone and accretive operators in Banach spaces,’ Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 388–393.CrossRefGoogle ScholarPubMed
[4]Browder, F. E. and Petryshyn, W. V., ‘Construction of fixed points of nonlinear mappings in Hubert space,’ J. Math. Anal. Appl. 20 (1967), 197–228.CrossRefGoogle Scholar
[5]Deimling, K., ‘Zeros of accretive operators,’ Manuscripta Math. 13 (1974), 365–374.CrossRefGoogle Scholar
[6]Dotson, W. G., ‘An iterative process for nonlinear monotonic nonexpansive operators in Hilbert space,’ Math. Comp. 32 (141) (1978), 223–225.CrossRefGoogle Scholar
[7]Kato, T., ‘Nonlinear semigroups and evolution equations,’ J. Math. Soc. Japan 19 (1967), 508–520.CrossRefGoogle Scholar
[8]Kirk, W. A., ‘A fixed point theorem for mappings which do not increase distance,’ A mer. Math. Monthly 72 (1965), 1004–1006.CrossRefGoogle Scholar
[9]Kirk, W. A., ‘On zeros of accretive operators in uniformly convex spaces,’ Boll. Un. Mat. Ital., to appear.Google Scholar
[10]Kirk, W. A. and Schoneberg, R., ‘Some results on pseudo-contractive mappings,’ Pacific J. Math. 71 (1977), 89–100.CrossRefGoogle Scholar
[11]Mann, W. R., ‘Mean value methods in iteration,’ Proc. Amer. Math. Soc. 4 (1953), 506–510.CrossRefGoogle Scholar
[12]Martin, R. H. Jr, ‘A global existence theorem for autonomous differential equations in Banach space,’ Proc. Amer. Math. Soc. 26 (1970), 307–314.CrossRefGoogle Scholar
[13]Minty, G. J., ‘Monotone (nonlinear) operators in Hubert space’, Duke Math. J. 29 (1962), 541–546.CrossRefGoogle Scholar
[14]Zarantonello, E. H., Solving functional equations by contractive averaging, (Technical Report No. 160, U. S. Army Math. Res. Centre, Madison, Wisconsin, 1960).Google Scholar