We consider a monotone increasing operator in an ordered Banach space having and as a strong super- and subsolution, respectively. In contrast with the well-studied case , we suppose that . Under the assumption that the order cone is normal and minihedral, we prove the existence of a fixed point located in the order interval .
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 that if either
then f has a fixed point in . 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., [, 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, , and a monotone increasing compact continuous operator . If is a strong supersolution ofTand is a strong subsolution, that is,
thenThas a fixed point .
Here denotes the order interval .
Theorem 1 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 if , if and , and if , where is the interior of the cone K.
A cone K is called minihedral if for any pair , , bounded above in order there exists the least upper bound , that is, an element such that
(1) and ,
(2) and implies that .
Obviously, a cone K is minihedral if and only if for any pair , , bounded below in order there exists the greatest lower bound . If a minihedral cone has a nonempty interior, then any pair is bounded above in order. Hence, and exist for all .
A cone K is called normal if there exists a constant such that , implies .
By the Kakutani-Krein brothers theorem [, 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 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 implies .
Consider the operator defined by
Since and , is a fixed point of the operator . Similarly, one shows that is also a fixed point.
Lemma 2The operator is continuous, monotone increasing, compact and maps the order interval into itself.
Proof For any , the maps and are continuous; see, e.g., Corollary 3.1.1 in . Due to the continuity of T, it follows immediately that is continuous as well. The operator is monotone increasing since inf and sup are monotone increasing with respect to each argument. Therefore, for any , we have
Let be an arbitrary sequence in . Since T is compact, has a subsequence converging to some . From the continuity of , it follows that the sequence converges to , thus, proving that the range of is relatively compact. □
Lemma 3There exist with
Proof Due to , there is a such that . The preimage of under the continuous mapping contains a ball . Hence, holds for all . By the same argument, for all . Choosing sufficiently small, we can achieve that .
Set . We choose so small that and so close to 1 that . Then we have and
Due to and , we have . Further, we obtain
From it follows that there is an element such that . Assume that . Then we have . However, in view of the Kakutani-Krein brothers theorem, implies . Thus, it follows that and, therefore, . Similarly one shows that . □
The main tool for the proof of Theorem 1 is Amann’s theorem on three fixed points (see, e.g., [, 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 such that
Then has a third fixed pointpsatisfying , , and .
Recall that the operator is called image compact if it is continuous and its image is a relatively compact set.
We choose , , , , where is as in Lemma 3. Since the cone K is normal, by Theorem 1.1.1 in , is norm bounded. Thus, is image compact.
Theorem 4 yields the existence of a fixed point of the operator satisfying . Obviously, is a fixed point of the operator T as well. This observation completes the proof of Theorem 1.
The authors declare that they have no competing interests.
All authors contributed equally. All authors read and approved the final manuscript.
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.