A positive fixed point theorem with applications to systems of Hammerstein integral equations
© Cabada et al.; licensee Springer. 2014
Received: 21 October 2014
Accepted: 24 November 2014
Published: 11 December 2014
We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of positive solutions for systems of nonlinear Hammerstein integral equations. An example is also presented to show the applicability of our results.
MSC: 47H10, 34B10, 34B18, 45G15, 47H30.
Keywordscone boundary value problem fixed point index positive solution nonlocal boundary condition system
In this manuscript we pursue the line of research developed in the recent papers – in order to deal with fixed point theorems on cones that mix monotonicity assumptions and conditions in one boundary, instead of imposing conditions on two boundaries as in the celebrated cone compression/expansion fixed point theorem of Krasnosel’skiĭ. In order to do this we employ the well-known monotone iterative method, combined with the classical fixed point index. In Section 2 we prove two results concerning non-decreasing and non-increasing operators in a shell, in presence of an upper or of a lower solution; in Remark 2.4 we present a comparison with previous results in this direction.
(with the obvious meaning when), then the BVP (1.1) has at least a positive solution.
providing, under suitable assumptions on the kernels and the nonlinearities, the existence of a positive solution.
computing all the constants that occur in our theory. We also prove that the system (1.4) has a solution for every . A similar result has been proven recently, in the context of one equation subject to nonlinear boundary conditions, by Goodrich .
2 Two fixed point theorems in cones
The cone K is normal if there exists such that for all with then .
Let N be a real Banach space with normal order cone K. Suppose that there existsuch thatis a completely continuous monotone non-decreasing operator withand. 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.
In the next proposition we recall the main properties of the fixed point index of a completely continuous operator relative to a cone, for more details see , . In the sequel the closure and the boundary of subsets of K are understood to be relative to K.
If there exists such that for all and all , then .
If for , then .
If for all and all , then .
Let be open in X such that . If and , then T has a fixed point in . The same holds if and .
We state our first result on the existence of non-trivial fixed points.
Let X be a real Banach space, K a normal cone with normal constantand nonempty interior (i.e. solid) anda completely continuous operator.
(1):there exist, with, andsuch that,
(3):there exists a (relatively) open bounded setsuch thatand eitheror.
Since , if with , then .
Suppose first that we can choose with and . Since and due to the normality of the cone K we have , which implies that T is non-decreasing on . Then we can apply the Theorem 2.1 to ensure the existence of a fixed point of T on , which, in particular, is a non-trivial fixed point.
Now suppose that such α does not exist. Thus for all with , which by Proposition 2.2(iii) implies that . Since, by assumption, we get the existence of a non-trivial fixed point belonging to the set (when ) or to the (when ). □
We note that we can use either Proposition 2.2(i), or Proposition 2.2(ii), in order to check the assumption (3) in Theorem 2.3. We also stress that is contained in the set . Therefore Theorem 2.3 is a genuine generalization of the previous fixed point theorems obtained in –. Moreover, we show in the applications that in many cases it is useful to apply Theorem 2.3 with a set V different from .
We observe that, following some ideas introduced in , Theorem 2.1], it is possible to modify the assumptions of Theorem 2.3 in order to deal with non-increasing operators. The next result describes precisely this situation.
Let X be a real Banach space, K a cone with nonempty interior (i.e. solid) anda completely continuous operator.
(1′):there exist, with, andsuch that,
(3′):there exists a (relatively) open bounded setsuch thatand eitheror.
Now, if for some is the case that then we are done. If not, for all which by Proposition 2.2 implies that . This result together with (3′) gives the existence of a non-zero fixed point with the desired localization property. □
3 An application to a system of Hammerstein integral equations
where we assume the following assumptions:
(H1):, for .
(H2): is continuous, for .
(H3): is continuous, for all , for .
(H4): is continuous, for .
which is a normal cone with .
leaves K invariant and is completely continuous.
Now we present our main result concerning the existence of positive solutions for the system (3.1).
Assume that the assumptions (H1)-(H5) hold and moreover:
(H6):There exist constantssuch that for every, is non-decreasing on (that is, ifwithfor, thenfor).
On the other hand, for let as in (H7) and fix .
Claim 1. and .
As a consequence, we have , and the claim is proven.
Claim 2. T is non-decreasing on the set.
therefore , , so , and since , T is also non-decreasing on .
Firstly, note that since then we have .
If not, there exist and such that .
Thus, we obtain , a contradiction.
Therefore by Proposition 2.2 we have and the proof is finished. □
The following condition, similar to the one given in , implies (H7) and it is easier to check.
: For every , , uniformly w.r.t. , .
In order to deal with negative kernels we can require conditions (H2), (H3), and (H5) on the absolute value of the kernel such that and conditions (H4), (H6), and (H7) on .
The authors would like to thank the anonymous referee for his/her valuable comments, which have improved the correctness and the presentation of the manuscript. A Cabada was partially supported by Ministerio de Educación y Ciencia, Spain, and FEDER, Projects MTM2010-15314 and MTM2013-43014-P, JA Cid was partially supported by Ministerio de Educación y Ciencia, Spain, and FEDER, Project MTM2013-43404-P and G Infante was partially supported by G.N.A.M.P.A. - INdAM (Italy). This paper was partially written during a visit of G Infante to the Departamento de Análise Matemática of the Universidade de Santiago de Compostela. G Infante is grateful to the people of the aforementioned Departamento for their kind and warm hospitality.
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