Open Access

An intermediate value theorem for monotone operators in ordered Banach spaces

Fixed Point Theory and Applications20122012:211

DOI: 10.1186/1687-1812-2012-211

Received: 5 June 2012

Accepted: 5 November 2012

Published: 22 November 2012

Abstract

We consider a monotone increasing operator in an ordered Banach space having u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq1_HTML.gif and u + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq2_HTML.gif as a strong super- and subsolution, respectively. In contrast with the well-studied case u + < u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq3_HTML.gif, we suppose that u < u + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq4_HTML.gif. Under the assumption that the order cone is normal and minihedral, we prove the existence of a fixed point located in the order interval [ u , u + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq5_HTML.gif.

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 1 , a 2 ] R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq6_HTML.gif that if either
f ( a 1 ) > a 1 and f ( a 2 ) < a 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_Equ1_HTML.gif
(1)
or
f ( a 1 ) < a 1 and f ( a 2 ) > a 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_Equ2_HTML.gif
(2)

then f has a fixed point in [ a 1 , a 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq7_HTML.gif. 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 1 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 + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq8_HTML.gif, and a monotone increasing compact continuous operator T : [ u , u + ] X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq9_HTML.gif. If u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq1_HTML.gif is a strong supersolution of T and u + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq2_HTML.gif is a strong subsolution, that is,
T u u and T u + u + , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_Equa_HTML.gif

then T has a fixed point u [ u , u + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq10_HTML.gif.

Here [ u , u + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq5_HTML.gif denotes the order interval { u X : u u u + } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq11_HTML.gif.

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 u v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq12_HTML.gif if u v K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq13_HTML.gif, u > v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq14_HTML.gif if u v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq12_HTML.gif and u v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq15_HTML.gif, and u v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq16_HTML.gif if u v K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq17_HTML.gif, where K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq18_HTML.gif is the interior of the cone K.

A cone K is called minihedral if for any pair { x , y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq19_HTML.gif, x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq20_HTML.gif, bounded above in order there exists the least upper bound sup { x , y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq21_HTML.gif, that is, an element z X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq22_HTML.gif such that
  1. (1)

    x z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq23_HTML.gif and y z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq24_HTML.gif,

     
  2. (2)

    x z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq25_HTML.gif and y z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq26_HTML.gif implies that z z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq27_HTML.gif.

     

Obviously, a cone K is minihedral if and only if for any pair { x , y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq19_HTML.gif, x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq20_HTML.gif, bounded below in order there exists the greatest lower bound inf { x , y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq28_HTML.gif. If a minihedral cone has a nonempty interior, then any pair x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq20_HTML.gif is bounded above in order. Hence, sup { x , y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq21_HTML.gif and inf { x , y } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq28_HTML.gif exist for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq20_HTML.gif.

A cone K is called normal if there exists a constant N > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq29_HTML.gif such that x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq30_HTML.gif, x , y K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq31_HTML.gif implies x X N y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq32_HTML.gif.

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 C ( Q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq33_HTML.gif 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 u v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq34_HTML.gif implies T u T v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq35_HTML.gif.

Consider the operator T ˆ : [ u , u + ] X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq36_HTML.gif defined by
T ˆ u : = sup { inf { T u , u + } , u } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_Equ3_HTML.gif
(3)

Since inf { T u + , u + } = u + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq37_HTML.gif and sup { u + , u } = u + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq38_HTML.gif, u + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq2_HTML.gif is a fixed point of the operator T ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq39_HTML.gif. Similarly, one shows that u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq1_HTML.gif is also a fixed point.

Lemma 2 The operator T ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq39_HTML.gif is continuous, monotone increasing, compact and maps the order interval [ u , u + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq5_HTML.gif into itself.

Proof For any v K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq40_HTML.gif, the maps u sup { u , v } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq41_HTML.gif and u inf { u , v } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq42_HTML.gif are continuous; see, e.g., Corollary 3.1.1 in [4]. Due to the continuity of T, it follows immediately that T ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq39_HTML.gif is continuous as well. The operator T ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq39_HTML.gif is monotone increasing since inf and sup are monotone increasing with respect to each argument. Therefore, for any u [ u , u + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq43_HTML.gif, we have
u = T ˆ u T ˆ u T ˆ u + = u + . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_Equb_HTML.gif

Let ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq44_HTML.gif be an arbitrary sequence in [ u , u + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq5_HTML.gif. Since T is compact, ( T u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq45_HTML.gif has a subsequence ( T u n k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq46_HTML.gif converging to some v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq47_HTML.gif. From the continuity of T ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq39_HTML.gif, it follows that the sequence ( T ˆ u n k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq48_HTML.gif converges to sup { inf { v , u + } , u } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq49_HTML.gif, thus, proving that the range of T ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq39_HTML.gif is relatively compact. □

Lemma 3 There exist p ± X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq50_HTML.gif with
u p p + u + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_Equc_HTML.gif
and
T ˆ p < p , T ˆ p + > p + . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_Equd_HTML.gif

Proof Due to T u u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq51_HTML.gif, there is a δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq52_HTML.gif such that B δ ( u T u ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq53_HTML.gif. The preimage of B δ ( u T u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq54_HTML.gif under the continuous mapping u u T u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq55_HTML.gif contains a ball B ϵ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq56_HTML.gif. Hence, u T u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq57_HTML.gif holds for all u B ϵ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq58_HTML.gif. By the same argument, u T u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq59_HTML.gif for all u B ϵ ( u + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq60_HTML.gif. Choosing ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq61_HTML.gif sufficiently small, we can achieve that B ϵ ( u ) B ϵ ( u + ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq62_HTML.gif.

Set p ( t ) : = { ( 1 t ) u + t u + | t [ 0 , 1 ] } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq63_HTML.gif. We choose t ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq64_HTML.gif so small that p : = p ( t ) B ϵ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq65_HTML.gif and t + ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq66_HTML.gif so close to 1 that p + : = p ( t + ) B ϵ ( u + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq67_HTML.gif. Then we have u p p + u + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq68_HTML.gif and
T p p , T p + p + . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_Eque_HTML.gif
Due to p u + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq69_HTML.gif and T p p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq70_HTML.gif, we have inf { T p , u + } = T p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq71_HTML.gif. Further, we obtain
sup { T p , u } sup { p , u } = p . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_Equf_HTML.gif

From T p p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq70_HTML.gif it follows that there is an element z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq72_HTML.gif such that T p = p + z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq73_HTML.gif. Assume that sup { T p , u } = p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq74_HTML.gif. Then we have sup { z , u p } = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq75_HTML.gif. However, in view of the Kakutani-Krein brothers theorem, u p 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq76_HTML.gif implies sup { z , u p } 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq77_HTML.gif. Thus, it follows that sup { T p , u } p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq78_HTML.gif and, therefore, T ˆ p < p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq79_HTML.gif. Similarly one shows that T ˆ p + > p + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq80_HTML.gif. □

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 4 Let X be a real Banach space with an order cone having a nonempty interior. Assume there are four points in X,
p 1 p 2 < p 3 p 4 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_Equg_HTML.gif
and a monotone increasing image compact operator T ˆ : [ p 1 , p 4 ] X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq81_HTML.gif such that
T ˆ p 1 = p 1 , T ˆ p 2 < p 2 , T ˆ p 3 > p 3 , T ˆ p 4 = p 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_Equh_HTML.gif

Then T ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq39_HTML.gif has a third fixed point p satisfying p 1 < p < p 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq82_HTML.gif, p [ p 1 , p 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq83_HTML.gif, and p [ p 3 , p 4 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq84_HTML.gif.

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

We choose p 1 = u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq85_HTML.gif, p 2 = p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq86_HTML.gif, p 3 = p + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq87_HTML.gif, p 4 = u + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq88_HTML.gif, where p ± https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq89_HTML.gif is as in Lemma 3. Since the cone K is normal, by Theorem 1.1.1 in [1], [ u , u + ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq5_HTML.gif is norm bounded. Thus, T ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq39_HTML.gif is image compact.

Theorem 4 yields the existence of a fixed point u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq90_HTML.gif of the operator T ˆ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq39_HTML.gif satisfying u < u < u + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq91_HTML.gif. Obviously, u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-211/MediaObjects/13663_2012_Article_277_IEq90_HTML.gif is a fixed point of the operator T as well. This observation completes the proof of Theorem 1.

Declarations

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.

Authors’ Affiliations

(1)
FB 08 - Institut für Mathematik, Johannes Gutenberg-Universität Mainz
(2)
Department of Mathematics, University of Uppsala

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Copyright

© Kostrykin and Oleynik; licensee Springer 2012

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