Kostrykin and Oleynik Fixed Point Theory and Applications 2012, 2012:211
http://www.fixedpointtheoryandapplications.com/content/2012/1/211
RESEARCH Open Access
An intermediate value theorem for monotone
operators in ordered Banach spaces
Vadim Kostrykin1* and Anna Oleynik1,2
*Correspondence:
kostrykin@mathematik.uni-mainz.de Abstract
1FB 08 - Institut für Mathematik,
Johannes Gutenberg-Universität We consider a monotone increasing operator in an ordered Banach space having u–
Mainz, Staudinger Weg 9, Mainz, and u+ as a strong super- and subsolution, respectively. In contrast with the
D-55099, Germany well-studied case u+ < u–, we suppose that u– < u+. Under the assumption that the
Full list of author information is
available at the end of the article order cone is normal and minihedral, we prove the existence of a fixed point located
in the order interval [u–,u+].
MSC: 47H05; 47H10; 46B40
Keywords: fixed point theorems in ordered Banach spaces
It is an elementary consequence of the intermediate value theorem for continuous real-
valued functions f : [a,a]→R that if either
f (a) > a and f (a) < a ()
or
f (a) < a and f (a) > a, ()
then f has a fixed point in [a,a]. It is a natural question whether this result can be ex-
tended to the case of ordered Banach spaces. A number of fixed point theorems with as-
sumptions of type () are well known; see, e.g., [, Section .]. However, to the best of our
knowledge, fixed point theorems with assumptions of type () have not been known so
far. In the present note, we prove the following fixed point theorem of this type.
Theorem  Let X be a real Banach space with an order cone K satisfying
(a) K has a nonempty interior,
(b) K is normal and minihedral.
Assume that there are two points in X, u–  u+, and a monotone increasing compact con-
tinuous operator T : [u–,u+]→ X. If u– is a strong supersolution of T and u+ is a strong
subsolution, that is,
Tu–  u– and Tu+  u+,
then T has a fixed point u∗ ∈ [u–,u+].
© 2012 Kostrykin andOleynik; licensee Springer. This is anOpen Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Kostrykin and Oleynik Fixed Point Theory and Applications 2012, 2012:211 Page 2 of 4
http://www.fixedpointtheoryandapplications.com/content/2012/1/211
Here [u–,u+] denotes the order interval {u ∈ X : u– ≤ u≤ u+}.
Theorem  generalizes an idea developed by the present authors in [], where the exis-
tence of solutions to a certain nonlinear integral equation of Hammerstein type has been
shown.
Before we present the proof, we rec◦all some n◦otions. We write u≥ v if u– v ∈ K , u > v if
u≥ v and u 	= v, and u v if u – v ∈K , where K is the interior of the cone K .
A coneK is calledminihedral if for any pair {x, y}, x, y ∈ X, bounded above in order there
exists the least upper bound sup{x, y}, that is, an element z ∈ X such that
() x≤ z and y≤ z,
() x≤ z′ and y≤ z′ implies that z≤ z′.
Obviously, a cone K is minihedral if and only if for any pair {x, y}, x, y ∈ X, bounded below
in order there exists the greatest lower bound inf{x, y}. If aminihedral cone has a nonempty
interior, then any pair x, y ∈ X is bounded above in order. Hence, sup{x, y} and inf{x, y} exist
for all x, y ∈ X.
A coneK is called normal if there exists a constantN >  such that x≤ y, x, y ∈ K implies
‖x‖X ≤N‖y‖X .
By the Kakutani-Krein brothers theorem [, Theorem .] a real Banach space X with
an order cone K satisfying assumptions (a) and (b) of Theorem  is isomorphic to the
Banach space C(Q) of continuous functions on a compact Hausdorff space Q. The image
of K under this isomorphism is the cone of nonnegative continuous functions on Q.
An operatorT acting in theBanach spaceX is calledmonotone increasing ifu≤ v implies
Tu≤ Tv.
Consider the operator T̂ : [u–,u+]→ X defined by
{ }
T̂u := sup inf{Tu,u+},u– . ()
Since inf{Tu+,u+} = u+ and sup{u+,u–} = u+, u+ is a fixed point of the operator T̂ . Similarly,
one shows that u– is also a fixed point.
Lemma  The operator T̂ is continuous,monotone increasing, compact and maps the or-
der interval [u–,u+] into itself.
Proof For any v ∈ K , the maps u 
→ sup{u, v} and u 
→ inf{u, v} are continuous; see, e.g.,
Corollary .. in []. Due to the continuity of T , it follows immediately that T̂ is con-
tinuous as well. The operator T̂ is monotone increasing since inf and sup are monotone
increasing with respect to each argument. Therefore, for any u ∈ [u–,u+], we have
u– = T̂u– ≤ T̂u≤ T̂u+ = u+.
Let (un) be an arbitrary sequence in [u–,u+]. Since T is compact, (Tun) has a subsequence
(Tunk ) converging to some v ∈ X. From the continuity of T̂ , it follows that the sequence
(T̂unk ) converges to sup{inf{v,u+},u–}, thus, proving that the range of T̂ is relatively com-
pact. 
Lemma  There exist p± ∈ X with
u–  p–  p+  u+
Kostrykin and Oleynik Fixed Point Theory and Applications 2012, 2012:211 Page 3 of 4
http://www.fixedpointtheoryandapplications.com/content/2012/1/211
and
T̂p– < p–, T̂p+ > p+.
◦
Proof Due to Tu–  u–, there is a δ >  such that Bδ(u– – Tu–) ⊂ K . The preimage of
Bδ(u– – Tu–) under the continuous mapping u →
 u – Tu contains a ball B(u–). Hence,
u – Tu  holds for all u ∈ B(u–). By the same argument, u – Tu  for all u ∈ B(u+).
Choosing  >  sufficiently small, we can achieve that B(u–)∩ B(u+) =∅.
Set p(t) := {( – t)u– + tu+|t ∈ [, ]}. We choose t– ∈ (, ) so small that p– := p(t–) ∈
B(u–) and t+ ∈ (, ) so close to  that p+ := p(t+) ∈ B(u+). Then we have u–  p– 
p+  u+ and
Tp–  p–, Tp+  p+.
Due to p–  u+ and Tp–  p–, we have inf{Tp–,u+} = Tp–. Further, we obtain
sup{Tp–,u–} ≤ sup{p–,u–} = p–.
From Tp–  p– it follows that there is an element z  such that Tp– = p– + z. Assume
that sup{Tp–,u–} = p–. Thenwe have sup{z,u– –p–} = .However, in view of the Kakutani-
Krein brothers theorem, u– – p–   implies sup{z,u– – p–}  . Thus, it follows that
sup{Tp–,u–} 	= p– and, therefore, T̂p– < p–. Similarly one shows that T̂p+ > p+. 
Themain tool for the proof of Theorem  is Amann’s theorem on three fixed points (see,
e.g., [, Theorem .F and Corollary .]):
Theorem  Let X be a real Banach space with an order cone having a nonempty interior.
Assume there are four points in X,
p  p < p  p,
and a monotone increasing image compact operator T̂ : [p,p]→ X such that
T̂p = p, T̂p < p, T̂p > p, T̂p = p.
Then T̂ has a third fixed point p satisfying p < p < p, p ∈/ [p,p], and p ∈/ [p,p].
Recall that the operator is called image compact if it is continuous and its image is a
relatively compact set.
We choose p = u–, p = p–, p = p+, p = u+, where p± is as in Lemma . Since the cone
K is normal, by Theorem .. in [], [u–,u+] is norm bounded. Thus, T̂ is image compact.
Theorem  yields the existence of a fixed point u∗ of the operator T̂ satisfying
u– < u∗ < u+. Obviously, u∗ is a fixed point of the operator T as well. This observation
completes the proof of Theorem .
Competing interests
The authors declare that they have no competing interests.
Kostrykin and Oleynik Fixed Point Theory and Applications 2012, 2012:211 Page 4 of 4
http://www.fixedpointtheoryandapplications.com/content/2012/1/211
Authors’ contributions
All authors contributed equally. All authors read and approved the final manuscript.
Author details
1FB 08 - Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 9, Mainz, D-55099, Germany.
2Current address: Department of Mathematics, University of Uppsala, P.O. Box 480, Uppsala, S-75106, Sweden.
Acknowledgements
The authors thank H.-P. Heinz for useful comments. This work has been supported in part by the Deutsche
Forschungsgemeinschaft, Grant KO 2936/4-1.
Received: 5 June 2012 Accepted: 5 November 2012 Published: 22 November 2012
References
1. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, Boston (1988)
2. Kostrykin, V, Oleynik, A: On the existence of unstable bumps in neural networks. Preprint. arXiv:1112.2941 [math.DS]
(2011)
3. Krasnosel’skij, MA, Lifshits, JA, Sobolev, AV: Positive Linear Systems. The Method of Positive Operators, Sigma Series in
Applied Mathematics, vol. 5. Heldermann, Berlin (1989)
4. Chueshov, I: Monotone Random Systems Theory and Applications. Lecture Notes in Mathematics, vol. 1779. Springer,
Berlin (2002)
5. Zeidler, E: Nonlinear Functional Analysis and Its Applications: I: Fixed-Point Theorems. Springer, New York (1986)
doi:10.1186/1687-1812-2012-211
Cite this article as: Kostrykin and Oleynik: An intermediate value theorem for monotone operators in ordered
Banach spaces. Fixed Point Theory and Applications 2012 2012:211.