Abstract
We consider a monotone increasing operator in an ordered Banach space having
MSC: 47H05, 47H10, 46B40.
Keywords:
fixed point theorems in ordered Banach spacesResearch
It is an elementary consequence of the intermediate value theorem for continuous realvalued
functions
or
then f has a fixed point in
Theorem 1LetXbe a real Banach space with an order coneKsatisfying
(a) Khas a nonempty interior,
(b) Kis normal and minihedral.
Assume that there are two points inX,
thenThas a fixed point
Here
Theorem 1 generalizes an idea developed by the present authors in [2], where the existence of solutions to a certain nonlinear integral equation of Hammerstein type has been shown.
Before we present the proof, we recall some notions. We write
A cone K is called minihedral if for any pair
(1)
(2)
Obviously, a cone K is minihedral if and only if for any pair
A cone K is called normal if there exists a constant
By the KakutaniKrein brothers theorem [[3], Theorem 6.6] a real Banach space X with an order cone K satisfying assumptions (a) and (b) of Theorem 1 is isomorphic to the Banach space
An operator T acting in the Banach space X is called monotone increasing if
Consider the operator
Since
Lemma 2The operator
Proof For any
Let
Lemma 3There exist
and
Proof Due to
Set
Due to
From
The main tool for the proof of Theorem 1 is Amann’s theorem on three fixed points (see, e.g., [[5], Theorem 7.F and Corollary 7.40]):
Theorem 4LetXbe a real Banach space with an order cone having a nonempty interior. Assume there are four points inX,
and a monotone increasing image compact operator
Then
Recall that the operator is called image compact if it is continuous and its image is a relatively compact set.
We choose
Theorem 4 yields the existence of a fixed point
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally. All authors read and approved the final manuscript.
Acknowledgements
The authors thank H.P. Heinz for useful comments. This work has been supported in part by the Deutsche Forschungsgemeinschaft, Grant KO 2936/41.
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