Positivity 6: 47–57, 2002.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
Stability of Positive Fixed Points of
Department of Mathematics, Computer Science, and Engineering University of Pittsburgh at
Bradford, Bradford PA 16701, USA. E-mail: email@example.com
(Received 23 November 1999; accepted 9 May 1999)
Abstract. This paper considers the stability of positive ﬁxed points of nonlinear operators under
purturbations. Through the use of the Thompson’s metric, we obtain the results in ordered Banach
spaces by proving the corresponding results in metric spaces ﬁrst. An application to the existence of
periodic solutions for a parametrized system of ordinary differential equations is given.
Mathematics Subject Classitication (2000): Primary 47H07, 47H09; Secondary 47H10
Key words: cone, contraction type mapping, ﬁxed point, metric convexity, monotone operator,
ordered Banach space, Thompson’s metric
Let f be a nonlinear operator deﬁned in the interior of the positive closed convex
cone of an ordered Banach space, and satisfy
f(tx) φ(t)f (t)
for t ∈ (0, 1],whereφ : (0, 1]→(0, 1]. The existence of the attractive ﬁxed
point of f was discussed under various assumptions on φ by many authors. For
recent works, see [2, 5, 6] and the references therein. This paper discusses the
stability of the ﬁxed point of f under perturbation. In Section 2, we prove that for
a parametrized family of contraction type mapping in a complete metric space, a
ﬁxed point is stable under small changes of the parameter and the perturbed ﬁxed
points are globally attracting. These results are applied to the study of the para-
metrized family of nonlinear operators in ordered Banach spaces by appealing to
the Thompson’s metric. The nonlinear operators studied in that section are related
to the ascending operators deﬁned by Krause [5, 6]. In Section 4, we discuss the
existence of periodic solutions for a parametrized system of ordinary differential
equations to exemplify the applications of our results.