An intermediate value theorem for monotone operators in ordered Banach spaces

Abstract

We consider a monotone increasing operator in an ordered Banach space having [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.] as a strong super- and subsolution, respectively. In contrast with the well-studied case [InlineEquation not available: see fulltext.], we suppose that [InlineEquation not available: see fulltext.]. Under the assumption that the order cone is normal and minihedral, we prove the existence of a fixed point located in the order interval [InlineEquation not available: see fulltext.].

MSC: 47H05, 47H10, 46B40.[PUBLICATION ABSTRACT]

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R E S E A R C H Open Access

An intermediate value theorem for monotone operators in ordered Banach spaces

Vadim Kostrykin1* and Anna Oleynik1,2

*Correspondence: mailto:[email protected]

Web End [email protected]

1FB 08 - Institut fr Mathematik, Johannes Gutenberg-Universitt Mainz, Staudinger Weg 9, Mainz, D-55099, GermanyFull list of author information is available at the end of the article

Abstract

We consider a monotone increasing operator in an ordered Banach space having u and u+ as a strong super- and subsolution, respectively. In contrast with the well-studied case u+ < u, we suppose that u < u+. Under the assumption that the order cone is normal and minihedral, we prove the existence of a xed point located in the order interval [u, u+].

MSC: 47H05; 47H10; 46B40

Keywords: xed 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 xed point in [a, a]. It is a natural question whether this result can be extended to the case of ordered Banach spaces. A number of xed point theorems with assumptions of type () are well known; see, e.g., [, Section .]. However, to the best of our knowledge, xed point theorems with assumptions of type () have not been known so far. In the present note, we prove the following xed 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 continuous 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 xed point u [u, u+].

2012 Kostrykin and Oleynik; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Here [u, u+] denotes the order interval {u X : u u u+}.

Theorem generalizes an idea developed by the present authors in [], 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 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 cone K is called minihedral 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 a minihedral 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 cone K is called normal if there exists a constant N > 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 Hausdor space Q. The image of K under this isomorphism is the cone of nonnegative continuous functions on Q.

An operator T acting in the Banach space X is called monotone increasing if u v implies Tu Tv.

Consider the operator

[hatwide]

one shows that u is also a xed point.

Lemma The operator

[hatwide]

T is continuous, monotone increasing, compact and maps the order 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

[hatwide]

T is con-

[hatwide]

T is monotone increasing since inf and sup are monotone increasing with respect to each argument. Therefore, for any u [u, u+], we have

u =

tinuous as well. The operator

[hatwide]

Tu

[hatwide]

T is relatively compact.

Lemma There exist p X with

u p p+ u+

[hatwide]

Tunk) converges to sup{inf{v, u+}, u}, thus, proving that the range of

[hatwide]

T : [u, u+] X dened by

[bracerightbig]. ()

Since inf{Tu+, u+} = u+ and sup{u+, u} = u+, u+ is a xed point of the operator

[hatwide]

T. Similarly,

Tu := sup

[braceleftbig]inf{Tu, u+}, u

[hatwide]

Tu+ = 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

[hatwide]

T, it follows that the sequence

(

[hatwide]

Tu

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and

[hatwide]

Tp+ > p+.

Proof Due to Tu u, there is a > such that B(u Tu)

K. The preimage of B(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 > suciently 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. Then we 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,

[hatwide]

Tp < p. Similarly one shows that

[hatwide]

T : [p, p] X such that

[hatwide]

Tp = p.

[hatwide]

T has a third xed point p satisfying p < p < p, p /

[hatwide]

T is image compact.

Theorem yields the existence of a xed point u of the operator

[hatwide]

T satisfying

u < u < u+. Obviously, u is a xed 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.

[hatwide]

Tp < p,

[hatwide]

Tp+ > p+.

The main tool for the proof of Theorem is Amanns theorem on three xed 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

[hatwide]

Tp = p,

[hatwide]

Tp < p,

[hatwide]

Tp > p,

Then

[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,

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Authors contributions

All authors contributed equally. All authors read and approved the nal manuscript.

Author details

1FB 08 - Institut fr Mathematik, Johannes Gutenberg-Universitt 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

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3. Krasnoselskij, MA, Lifshits, JA, Sobolev, AV: Positive Linear Systems. The Method of Positive Operators, Sigma Series in Applied Mathematics, vol. 5. Heldermann, Berlin (1989)

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5. Zeidler, E: Nonlinear Functional Analysis and Its Applications: I: Fixed-Point Theorems. Springer, New York (1986)

doi:10.1186/1687-1812-2012-211Cite 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.