Abstract
We give new criteria for the existence of nontrivial fixed points on cones assuming some monotonicity of the operator on a suitable conical shell. Moreover, we give an application to the existence of multiple solutions for a nonlocal boundary value problem that models the displacement of a beam subject to some feedback controllers.
MSC:47H10, 34B10, 34B18.
Similar content being viewed by others
1 Introduction and preliminaries
The classical cone compression-expansion fixed point theorem of Krasnosel’skiĭ (see Theorem 1.2 below) and the monotone iterative technique (see Theorem 1.1 below) are among the most popular and fruitful tools to deal with the existence of solutions for nonlinear problems. Following earlier ideas of Persson [1] valid in the finite-dimensional setting, both methods were combined in [2] to obtain the existence of a fixed point, assuming the operator T to be monotone non-decreasing with some conditions on the set of supersolutions. This result was improved in [3] by relaxing the monotonicity condition. More recently, in [4] the authors were able to present a refinement of the results of [2, 3], by allowing a comparison between a point and a boundary, instead of on two boundaries as in Krasnosel’skiĭ’s theorem. This approach, which required a monotonicity assumption on the operator on a conical shell, has proved to be well suited to establish multiplicity results. Our aim in this paper is to pursue this line of research by obtaining new fixed point theorems, valid not only for non-decreasing (Section 3) but also for non-increasing operators (Section 2). We point out that this type of theorems can be combined in the applications to obtain the existence of multiple non-trivial solutions. This fact is illustrated in Section 4, where the existence of multiple positive solutions for a nonlocal boundary value problem modeling the displacement of a beam is discussed.
We now recall some definitions that will be useful in the sequel. A subset K of a real Banach space N is a cone if and only if it is closed, , for all and . A cone K defines the partial ordering in N given by if and only if . The notation means and . The cone K is called normal with a normal constant if and only if for all with . Whenever , the symbol means and the cone is said to be solid. ∂K denotes the boundary of K and is the distance of x to the boundary of K.
If satisfies the conditions
and
then it is called a cone compression when and a cone expansion when .
The closed ball of center and radius is denoted by
and for , with , we define the interval
Now we recall the two classical fixed point results mentioned above. The first one is known as the monotone iterative method (see, for example, [[5], Theorem 7.A] or [6]).
Theorem 1.1 Let N be a real Banach space with a normal order cone K. Suppose that there exist such that is a completely continuous monotone nondecreasing operator with and . Then T has a fixed point and the iterative sequence , with , converges to the greatest fixed point of T in , and the sequence , with , converges to the smallest fixed point of T in .
The second one, which is widely used for the search of positive fixed points, is due to Krasnosel’skiĭ (see [[5], Theorem 13.D]).
Theorem 1.2 Let N be a real Banach space with order cone K. Suppose that the operator is completely continuous and either a cone compression or expansion. Then T has a fixed point x on K and
2 Non-increasing operators
Firstly, we present a result under the assumption of the existence of a lower solution in a solid cone.
Theorem 2.1 Let N be a real Banach space, K be a solid cone and be a completely continuous operator. Assume that
-
(1)
there exist , with , and such that ;
-
(2)
the map T is monotone non-increasing in the set
-
(3)
there exists , with , such that for all with .
Then the map T has at least one non-zero fixed point such that
Proof First, note that . Since , it is clear that if with , then . Since T is non-increasing in and , we have that
If , we have a fixed point such that , on the contrary, we deduce that for all , which together with (3) implies, by Theorem 1.2, that there exists a non-zero fixed point with the desired localization property. □
Remark 2.2 Note that if , in the proof of the previous result, it is showed that the map T has at least one non-zero fixed point such that
As a direct consequence of Theorem 2.1, we obtain the following ‘dual result’ for a non-increasing operator in a solid cone with an upper solution.
Corollary 2.3 Let N be a real Banach space, K be a solid cone and be a completely continuous operator. Assume that condition (3) is fulfilled together with
-
(4)
there exist , with , and such that ;
-
(5)
the map T is monotone non-increasing in the set
Then the map T has at least one non-zero fixed point in , and
Proof As in the proof of Theorem 2.1, we deduce that .
If , then , which implies that , i.e., is a fixed point of T. Obviously, if this is the case, and the result holds.
On the other hand, if , we know that and, since T is non-increasing in and belongs to the interior of K, we deduce that , i.e., is a non-zero lower solution of the operator T.
As consequence, all the conditions of Theorem 2.1 are fulfilled and the result holds. □
Now, we give a fixed point result for a non-increasing operator under the assumption of the existence of an upper solution in a cone not necessarily solid, but it verifies an extra condition.
Theorem 2.4 Let N be a real Banach space, K be a cone that satisfies the following condition:
and is a completely continuous operator. Assume that
-
(6)
there exists , , such that ;
-
(7)
the map T is monotone non-increasing in the set
-
(8)
there exists , with , such that for all with .
Then the map T has at least one non-zero fixed point such that
Proof By the property (2.1), we have that if with , then . The definition of says that ; so, since T is non-increasing on , we have
If , we have a fixed point such that ; on the contrary, for all such that which, together with (8), implies by Theorem 1.2 the existence of a fixed point with the desired localization property. □
Remark 2.5 (i) Note that if , in the proof of the two previous results, it is showed that the map T has at least one non-zero fixed point such that
The same conclusion holds if in condition (2.1) we assume that the constant . To verify this, it is enough to take into account that if there is a fixed point with , then and, as a consequence, .
(ii) An example of a cone satisfying condition (2.1) is, for instance, the one used in [7]
Lemma 2.6 Condition (2.1) is equivalent to the following one:
Proof Obviously, if condition (2.2) is fulfilled, then (2.1) also holds.
Suppose now that (2.1) is satisfied, and let be such that ; in consequence, there is such that . Condition (2.1) shows that , i.e., condition (2.2) holds. This proves the result. □
In an analogous way to Corollary 2.3, we arrive at the following ‘dual result’.
Corollary 2.7 Let N be a real Banach space, K be a cone that satisfies the condition (2.1) and be a completely continuous operator. Assume that
-
(9)
there exists , such that ;
-
(10)
the map T is monotone non-increasing in the set
-
(11)
there exists , with , such that for all with .
Then the map T has at least one non-zero fixed point such that
Proof By the property (2.1), we have that if with , then .
Suppose now that . From Lemma 2.6, we have that , which implies that is a fixed point with and , so the result holds.
When , by the definition of , it is obvious that ; so, since T is non-increasing on , we have
and the results holds from Theorem 2.4. □
Remark 2.8 Note that if or , in the proof of the previous result, it is showed that the map T has at least one non-zero fixed point such that
3 Non-decreasing operators
For non-decreasing operators, in the case of an upper solution, it was proved in [4] the following result which is an improvement of those in [2, 3].
Theorem 3.1 [4]
Let N be a real Banach space, K be a normal solid cone with a normal constant and be a completely continuous operator. Assume that
-
(12)
there exist , with , and such that ;
-
(13)
the map T is monotone non-decreasing in the set
-
(14)
there exists , with , such that for all with .
Then the map T has at least one non-zero fixed point in K that either belongs to or is such that
In the sequel, we prove a fixed point result in this direction for a not necessarily normal cone that satisfies (2.1).
Theorem 3.2 Let N be a real Banach space, K be a solid cone that satisfies condition (2.1), and be a completely continuous operator. Assume that conditions (12) and (14) hold and
(13′) the map T is monotone non-decreasing in the set
Then the map T has at least one non-zero fixed point in K that either belongs to or is such that
Proof Since , it is clear that if with , then .
Suppose first that there is with and . Notice that in this case .
If , then , which implies that , i.e., is a fixed point of T and the result is fulfilled.
Suppose now that . If we know, by condition (2.1), that and, as a consequence, () is a fixed point of T.
So, if we have that and, arguing as before, we have that either or are fixed points of T, or .
By recurrence we verify that the set .
Since T is monotone non-decreasing on , we have that the same holds in P. Moreover, . Now, the completely continuous character of the operator T implies that is a compact set of N. We can ensure the existence of a fixed point on from [[8], Proposition 1.1.7].
Now suppose that for all with . By (14) there exists such that for all with . Therefore by Theorem 1.2 there exists a non-zero fixed point. □
Remark 3.3 An example of a solid cone satisfying condition (2.1) is the following one:
Now, we give a result under the assumption of the existence of a lower solution.
Theorem 3.4 Let N be a real Banach space, K be a normal cone (not necessarily solid) with a normal constant that satisfies condition (2.1), and be a completely continuous operator. Assume that there is a lower solution as in (9), and
-
(15)
the map T is monotone non-decreasing in the set
-
(16)
there exists , with , such that for all with .
Then the map T has at least one non-zero fixed point in K that either belongs to or is such that
Proof By the property (2.1), we have that if with , then .
Suppose first that we can choose with and . Since and due to the normality of the cone K, we have that , which implies that T is nondecreasing on . Then we can apply Theorem 1.1 to ensure the existence of the extremal fixed points of T on .
Now suppose that for all with . By (16) there exists such that for all with . Therefore by Theorem 1.2, there exists a non-zero fixed point in the required set. □
Remark 3.5 We stress that the above theorems can be combined to prove the existence of multiple fixed points. The idea is to use a nesting argument similar to those utilized, for example, in [9, 10], where the authors used the classical fixed point index, and in [11, 12], where Theorem 1.2 was used. In the next section we do this in the case of the existence of two non-trivial fixed points, and we refer to Theorem 3.4 of [4] to give an idea of the type of results that may be stated in the case of n fixed points.
4 Applications to a nonlocal BVP
We now discuss the existence of positive solutions of the nonlocal boundary value problem (BVP)
where, for , and , , a.e. and is continuous. BVP (4.1)-(4.3) that has been studied in [13] models the displacement of a beam with feedback controllers; in particular, the boundary conditions mean that the shear force and the angular attitude vanish at , and in they are related to the displacement and to the bending moment registered in other points of the beam.
This BVP can be rewritten as a Hammerstein integral equation of the form
where the Green’s function and its properties are given in the following result.
Lemma 4.1 [13]
Let and . The Green’s function for the linear fourth-order boundary value problem
is given by
Moreover, for , we have
where
Note that implies that and so . Now, with the above conditions, it is routine to prove that leaves invariant the cone
where in we are considering the supremum norm . It is also known that K is a normal solid cone with a normal constant .
We will make use of the numbers
note that and , in the notation of [13].
Now we present the main result of this section.
Theorem 4.2 Assume that the hypotheses in Lemma 4.1 hold. Moreover, let be such that
Assume that g satisfies that and, moreover,
-
(i)
f is non-decreasing on ;
-
(ii)
f is non-increasing on ;
-
(iii)
and .
Then BVP (4.1)-(4.3) has at least two positive solutions for any satisfying
Proof The main idea in the proof is to apply Theorems 2.1 and 3.1 in two disjoint conical shells in order to get two different non-trivial fixed points. Firstly, we are going to check that the conditions of Theorem 2.1 are satisfied.
-
(1.a)
and there exists such that .
We have because of the inequality . On the other hand, the inequality implies that . Indeed, let , that is,
Since , we have and then for all . Moreover, it is easy to check that
which means that .
-
(1.b)
The map T is monotone non-increasing in the set
This fact is a consequence of the assumption (ii).
-
(1.c)
There exists , with , such that for all with .
By the second part of the assumption (iii), we have this result for large enough.
So, Theorem 2.1 implies the existence of a solution such that
Now, we are going to check that the conditions of Theorem 3.1 are satisfied.
-
(2.a)
and there exists such that .
We have because of the inequality . Again, implies that by reasoning as in the proof of claim (1.a).
-
(2.b)
The map T is monotone non-decreasing in the set
This fact is a consequence of the assumption (i).
-
(2.c)
There exists , with , such that for all with .
By the first part of the assumption (iii), we have this result for small enough.
Then, applying Theorem 3.1, we get the existence of a solution such that . Since , we have that and the theorem is proven. □
The following example illustrates our previous theorem.
Example 4.3 We consider the BVP
with
In Figures 1, 2 and 3 you can see the behavior of the function f on different intervals.
In this example , and by [13] we know that and . Moreover, since and , we have .
Now, it is easy to check that all the assumptions of Theorem 4.2 are satisfied by taking , , and . So, by Theorem 4.2, BVP (4.8)-(4.9) has at least two positive solutions provided that
References
Persson H: A fixed point theorem for monotone functions. Appl. Math. Lett. 2006, 19: 1207–1209. 10.1016/j.aml.2006.01.008
Cabada A, Cid JA: Existence of a non-zero fixed point for nondecreasing operators via Krasnoselskii’s fixed point theorem. Nonlinear Anal. 2009, 71: 2114–2118. 10.1016/j.na.2009.01.045
Cid JA, Franco D, Minhós F: Positive fixed points and fourth-order equations. Bull. Lond. Math. Soc. 2009, 41: 72–78. 10.1112/blms/bdn105
Franco D, Infante G, Perán J: A new criterion for the existence of multiple solutions in cones. Proc. R. Soc. Edinb. A 2012, 142: 1043–1050. 10.1017/S0308210511001016
Zeidler E: Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems. Springer, New York; 1986.
Amann H: On the number of solutions of nonlinear equations in ordered Banach spaces. J. Funct. Anal. 1972, 11: 346–384. 10.1016/0022-1236(72)90074-2
Cabada A, Cid JA: A note on fixed point theorems for T -monotone operators. Comput. Math. Appl. 2004, 47: 853–857. 10.1016/S0898-1221(04)90069-7
Heikkilä S, Lakshmikantham V Monographs and Textbooks in Pure and Applied Mathematics 181. In Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations. Dekker, New York; 1994.
Infante G, Webb JRL: Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations. Proc. Edinb. Math. Soc. 2006, 49: 637–656. 10.1017/S0013091505000532
Lan KQ: Multiple positive solutions of semilinear differential equations with singularities. J. Lond. Math. Soc. 2001, 63: 690–704. 10.1112/S002461070100206X
Graef JR, Qian C, Yang B: Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations. Proc. Am. Math. Soc. 2003, 131: 577–585. 10.1090/S0002-9939-02-06579-6
Karakostas GL, Tsamatos PCh: Existence of multiple positive solutions for a nonlocal boundary value problem. Topol. Methods Nonlinear Anal. 2002, 19: 109–121.
Infante, G, Pietramala, P: The displacement of a sliding bar subject to nonlinear controllers. In: Proceedings of the International Conference on Differential & Difference Equations and Applications. Springer, Berlin (in press)
Acknowledgements
Alberto Cabada and José Ángel Cid were partially supported by Ministerio de Educación y Ciencia, Spain, and FEDER, Project MTM2010-15314. This paper was partially written during the visit of Gennaro Infante to the Departamento de Análise Matemática of the Universidade de Santiago de Compostela. Gennaro Infante is grateful to the people of the aforementioned Departamento for their kind and warm hospitality.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors contributed equally and significantly in writing this article. All the authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Cabada, A., Cid, J.Á. & Infante, G. New criteria for the existence of non-trivial fixed points in cones. Fixed Point Theory Appl 2013, 125 (2013). https://doi.org/10.1186/1687-1812-2013-125
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-125