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An intermediate value theorem for monotone operators in ordered Banach spaces

Vadim Kostrykin1* and Anna Oleynik12

Author Affiliations

1 FB 08 - Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 9, Mainz, D-55099, Germany

2 Department of Mathematics, University of Uppsala, P.O. Box 480, Uppsala, S-75106, Sweden

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Fixed Point Theory and Applications 2012, 2012:211  doi:10.1186/1687-1812-2012-211

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/211


Received:5 June 2012
Accepted:5 November 2012
Published:22 November 2012

© 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), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a monotone increasing operator in an ordered Banach space having <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M1">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M2">View MathML</a> as a strong super- and subsolution, respectively. In contrast with the well-studied case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M3">View MathML</a>, we suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M4">View MathML</a>. Under the assumption that the order cone is normal and minihedral, we prove the existence of a fixed point located in the order interval <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M5">View MathML</a>.

MSC: 47H05, 47H10, 46B40.

Keywords:
fixed point theorems in ordered Banach spaces

Research

It is an elementary consequence of the intermediate value theorem for continuous real-valued functions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M6">View MathML</a> that if either

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M7">View MathML</a>

(1)

or

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M8">View MathML</a>

(2)

then f has a fixed point in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M9">View MathML</a>. It is a natural question whether this result can be extended to the case of ordered Banach spaces. A number of fixed point theorems with assumptions of type (1) are well known; see, e.g., [[1], Section 2.1]. However, to the best of our knowledge, fixed point theorems with assumptions of type (2) have not been known so far. In the present note, we prove the following fixed point theorem of this type.

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, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M10">View MathML</a>, and a monotone increasing compact continuous operator<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M11">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M1">View MathML</a>is a strong supersolution ofTand<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M2">View MathML</a>is a strong subsolution, that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M14">View MathML</a>

thenThas a fixed point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M15">View MathML</a>.

Here <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M5">View MathML</a> denotes the order interval <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M17">View MathML</a>.

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 onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M18">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M19">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M20">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M18">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M22">View MathML</a>, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M23">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M24">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M25">View MathML</a> is the interior of the cone K.

A cone K is called minihedral if for any pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M26">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M27">View MathML</a>, bounded above in order there exists the least upper bound <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M28">View MathML</a>, that is, an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M29">View MathML</a> such that

(1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M30">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M31">View MathML</a>,

(2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M32">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M33">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M34">View MathML</a>.

Obviously, a cone K is minihedral if and only if for any pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M26">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M27">View MathML</a>, bounded below in order there exists the greatest lower bound <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M37">View MathML</a>. If a minihedral cone has a nonempty interior, then any pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M27">View MathML</a> is bounded above in order. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M28">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M37">View MathML</a> exist for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M27">View MathML</a>.

A cone K is called normal if there exists a constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M42">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M43">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M44">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M45">View MathML</a>.

By the Kakutani-Krein 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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M46">View MathML</a> 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 operator T acting in the Banach space X is called monotone increasing if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M47">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M48">View MathML</a>.

Consider the operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M49">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M50">View MathML</a>

(3)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M51">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M52">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M2">View MathML</a> is a fixed point of the operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54">View MathML</a>. Similarly, one shows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M1">View MathML</a> is also a fixed point.

Lemma 2The operator<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54">View MathML</a>is continuous, monotone increasing, compact and maps the order interval<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M5">View MathML</a>into itself.

Proof For any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M58">View MathML</a>, the maps <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M59">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M60">View MathML</a> are continuous; see, e.g., Corollary 3.1.1 in [4]. Due to the continuity of T, it follows immediately that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54">View MathML</a> is continuous as well. The operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54">View MathML</a> is monotone increasing since inf and sup are monotone increasing with respect to each argument. Therefore, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M63">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M64">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M65">View MathML</a> be an arbitrary sequence in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M5">View MathML</a>. Since T is compact, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M67">View MathML</a> has a subsequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M68">View MathML</a> converging to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M69">View MathML</a>. From the continuity of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54">View MathML</a>, it follows that the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M71">View MathML</a> converges to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M72">View MathML</a>, thus, proving that the range of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54">View MathML</a> is relatively compact. □

Lemma 3There exist<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M74">View MathML</a>with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M75">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M76">View MathML</a>

Proof Due to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M77">View MathML</a>, there is a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M78">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M79">View MathML</a>. The preimage of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M80">View MathML</a> under the continuous mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M81">View MathML</a> contains a ball <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M82">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M83">View MathML</a> holds for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M84">View MathML</a>. By the same argument, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M85">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M86">View MathML</a>. Choosing <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M87">View MathML</a> sufficiently small, we can achieve that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M88">View MathML</a>.

Set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M89">View MathML</a>. We choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M90">View MathML</a> so small that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M91">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M92">View MathML</a> so close to 1 that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M93">View MathML</a>. Then we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M94">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M95">View MathML</a>

Due to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M96">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M97">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M98">View MathML</a>. Further, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M99">View MathML</a>

From <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M97">View MathML</a> it follows that there is an element <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M101">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M102">View MathML</a>. Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M103">View MathML</a>. Then we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M104">View MathML</a>. However, in view of the Kakutani-Krein brothers theorem, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M105">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M106">View MathML</a>. Thus, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M107">View MathML</a> and, therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M108">View MathML</a>. Similarly one shows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M109">View MathML</a>. □

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,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M110">View MathML</a>

and a monotone increasing image compact operator<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M111">View MathML</a>such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M112">View MathML</a>

Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54">View MathML</a>has a third fixed pointpsatisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M114">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M115">View MathML</a>, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M116">View MathML</a>.

Recall that the operator is called image compact if it is continuous and its image is a relatively compact set.

We choose <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M117">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M118">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M119">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M120">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M121">View MathML</a> is as in Lemma 3. Since the cone K is normal, by Theorem 1.1.1 in [1], <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M5">View MathML</a> is norm bounded. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54">View MathML</a> is image compact.

Theorem 4 yields the existence of a fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M124">View MathML</a> of the operator <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M54">View MathML</a> satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M126">View MathML</a>. Obviously, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/211/mathml/M124">View MathML</a> is a fixed point of the operator T as well. This observation completes the proof of Theorem 1.

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 2936/4-1.

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