Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Continuation method for $\alpha$-sublinear mappings

Author: Yong-Zhuo Chen
Journal: Proc. Amer. Math. Soc. 129 (2001), 203-210
MSC (1991): Primary 47H07, 47H09; Secondary 47H10
Published electronically: August 29, 2000
MathSciNet review: 1694453
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $B$ be a real Banach space partially ordered by a closed convex cone $P$ 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 [Enhancements On Off] (What's this?)

  • [1] Y.-Z. Chen, Thompson's metric and mixed monotone operators, J. Math. Anal. Appl. 117 (1993), 31-37. MR 94d:47055
  • [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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47H07, 47H09, 47H10

Retrieve articles in all journals with MSC (1991): 47H07, 47H09, 47H10

Additional Information

Yong-Zhuo Chen
Affiliation: Division of Natural Sciences, University of Pittsburgh at Bradford, Bradford, Pennsylvania 16701

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
Published electronically: August 29, 2000
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society