Positivity 12 (2008), 643–652
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040643-10, published online May 1, 2008
The existence of a ﬁxed point for the sum
of two monotone operators
Abstract. Let A and H be two operators deﬁned in an ordered Banach space
H(tx) ≥ tHx for all t ∈ (0, 1),
A(tx) ≥ t
Ax for all t ∈ (0, 1),
where α ∈ (0, 1). This paper discusses the conditions which will guarantee the
existence of an asymptotically attractive ﬁxed point for T = A + H.
Mathematics Subject Classiﬁcation (2000). 47H07, 47H09.
Keywords. Cone, convex metric space, ﬁxed point, monotone operator, ordered
Banach space, Thompson’s metric.
Let B be a Banach space partially ordered by a closed convex cone P and A :
P → P be a monotone operator, i.e., Ax ≥ Ay for x ≥ y. A monotone operator is
called α-concave if there exists α ∈ (0, 1) such that
A(tx) ≥ t
Ax for all t ∈ (0, 1). (1)
It is well-known that the monotone discrete dynamical system deﬁned by A has
an asymptotically attractive ﬁxed point in
(e.g., see [6, Theorem 2.2.6., p. 72]).
However, in application, we often have to deal with the operators which satisfy
equation (1) but with α = 1, i.e.,
A(tx) ≥ tAx for all t ∈ (0, 1). (2)
We call these monotone operators which satisfy (2) subhomogeneous operators.
Generally speaking, a subhomogeneous operator does not guarantee a ﬁxed point.
In , Nussbaum discussed the operator which is the sum of a subhomogeneous
operator and an α-concave operator. He proved the convergence property under