Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Continuation method for $\alpha$ -sublinear mappings

Author(s): Yong-Zhuo Chen
Journal: Proc. Amer. Math. Soc. 129 (2001), 203-210.
MSC (1991): Primary 47H07, 47H09; Secondary 47H10
Posted: August 29, 2000
MathSciNet review: 1694453
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Abstract: Let be a real Banach space partially ordered by a closed convex cone with nonempty interior $\overset{\circ}P$ . We study the continuation method for the monotone operator $A:\, \overset{\circ}P \rightarrow \overset{\circ}P$ which satisfies

$\begin{eqnarraystar}A(tx) \geq t^{\alpha (a,b)}\, A(x), \end{eqnarraystar}$

for all $x \in \overset{\circ}P$ , $t \in [a,\, b] \subset (0,\, 1)$ , where $\alpha (a,b) \in (0,\, 1)$ . Thompson's metric is among the main tools we are using.

References:

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[2]: J. Dugundji and A. Granas, Fixed Point Theory, Vol. I, Monografie Mat., Warszawa, 1982. MR 83j:54038
[3]: A. Granas, Continuation method for contractive maps, Topological Methods in Nonlinear Analysis, 3(1994), 375-379. MR 95d:54034
[4]: D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988. MR 89k:47084
[5]: M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin, 1984. MR 85b:47057
[6]: U. Krause, A nonlinear extension of the Birkhoff-Jentzsch theorem, J. Math. Anal. Appl. 114 (1986), 552-568. MR 87d:47069
[7]: A. C. Thompson, On certain contraction mappings in a partially ordered vector space, Proc. Amer. Math. Soc. 14 (1963), 438-443. MR 26:6727

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Additional Information:

Yong-Zhuo Chen
Affiliation: Division of Natural Sciences, University of Pittsburgh at Bradford, Bradford, Pennsylvania 16701
Email: yong@imap.pitt.edu

DOI: 10.1090/S0002-9939-00-05514-3
PII: S 0002-9939(00)05514-3
Keywords: $\alpha$-sublinear, cone, fixed point, generalized contraction, monotone operator, ordered Banach space, Thompson's metric
Received by editor(s): September 15, 1997
Received by editor(s) in revised form: April 5, 1999
Posted: August 29, 2000
Communicated by: Dale Alspach
Copyright of article: Copyright 2000, American Mathematical Society

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