- Open Access
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.
- boundary value problem
- fixed point index
- positive solution
- nonlocal boundary condition
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 .
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. □
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.
- Cabada A, Cid JA: Existence of a non-zero fixed point for non-decreasing operators via Krasnoselskii’s fixed point theorem. Nonlinear Anal. 2009, 71: 2114-2118. 10.1016/j.na.2009.01.045MathSciNetView ArticleMATHGoogle Scholar
- Cabada A, Cid JA, Infante G: New criteria for the existence of non-trivial fixed points in cones. Fixed Point Theory Appl. 2013., 2013: 10.1186/1687-1812-2013-125Google Scholar
- Cid JA, Franco D, Minhós F: Positive fixed points and fourth-order equations. Bull. Lond. Math. Soc. 2009, 41: 72-78. 10.1112/blms/bdn105MathSciNetView ArticleMATHGoogle Scholar
- Franco D, Infante G, Perán J: A new criterion for the existence of multiple solutions in cones. Proc. R. Soc. Edinb., Sect. A 2012, 142: 1043-1050. 10.1017/S0308210511001016View ArticleMathSciNetMATHGoogle Scholar
- Persson H: A fixed point theorem for monotone functions. Appl. Math. Lett. 2006, 19: 1207-1209. 10.1016/j.aml.2006.01.008MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, O’Regan D, Wong PJY: Constant-Sign Solutions of Systems of Integral Equations. Springer, Cham; 2013.View ArticleMATHGoogle Scholar
- Cheng X, Zhang Z: Existence of positive solutions to systems of nonlinear integral or differential equations. Topol. Methods Nonlinear Anal. 2009, 34: 267-277.MathSciNetMATHGoogle Scholar
- Cheng X, Zhong C: Existence of positive solutions for a second-order ordinary differential system. J. Math. Anal. Appl. 2005, 312: 14-23. 10.1016/j.jmaa.2005.03.016MathSciNetView ArticleMATHGoogle Scholar
- Dunninger DR, Wang H: Existence and multiplicity of positive solutions for elliptic systems. Nonlinear Anal. 1997, 29: 1051-1060. 10.1016/S0362-546X(96)00092-2MathSciNetView ArticleMATHGoogle Scholar
- Dunninger DR, Wang H: Multiplicity of positive radial solutions for an elliptic system on an annulus. Nonlinear Anal. 2000, 42: 803-811. 10.1016/S0362-546X(99)00125-XMathSciNetView ArticleMATHGoogle Scholar
- Franco D, Infante G, O’Regan D: Nontrivial solutions in abstract cones for Hammerstein integral systems. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2007, 14: 837-850.MathSciNetMATHGoogle Scholar
- Goodrich CS: Nonlocal systems of BVPs with asymptotically superlinear boundary conditions. Comment. Math. Univ. Carol. 2012, 53: 79-97.MathSciNetMATHGoogle Scholar
- Goodrich CS: Nonlocal systems of BVPs with asymptotically sublinear boundary conditions. Appl. Anal. Discrete Math. 2012, 6: 174-193. 10.2298/AADM120329010GMathSciNetView ArticleMATHGoogle Scholar
- Henderson J, Luca R: Existence and multiplicity for positive solutions of a system of higher-order multi-point boundary value problems. NoDEA Nonlinear Differ. Equ. Appl. 2013, 20: 1035-1054. 10.1007/s00030-012-0195-9MathSciNetView ArticleMATHGoogle Scholar
- Henderson J, Luca R: Positive solutions for systems of second-order integral boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-70Google Scholar
- Infante G, Pietramala P: Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations. Nonlinear Anal. 2009, 71: 1301-1310. 10.1016/j.na.2008.11.095MathSciNetView ArticleMATHGoogle Scholar
- Karakostas GL: Existence of solutions for an n -dimensional operator equation and applications to BVPs. Electron. J. Differ. Equ. 2014., 2014: 10.1186/1687-1847-2014-71Google Scholar
- Lan KQ: Multiple positive solutions of semilinear differential equations with singularities. J. Lond. Math. Soc. 2001, 63: 690-704. 10.1112/S002461070100206XView ArticleMathSciNetMATHGoogle Scholar
- Lan KQ, Lin W: Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations. J. Lond. Math. Soc. 2011, 83: 449-469. 10.1112/jlms/jdq090MathSciNetView ArticleMATHGoogle Scholar
- Lan KQ, Lin W: Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli. Nonlinear Anal. 2011, 74: 7184-7197. 10.1016/j.na.2011.07.038MathSciNetView ArticleMATHGoogle Scholar
- Yang Z: Positive solutions for a system of nonlinear Hammerstein integral equations and applications. Appl. Math. Comput. 2012, 218: 11138-11150. 10.1016/j.amc.2012.05.006MathSciNetView ArticleMATHGoogle Scholar
- Yang Z, Zhang Z: Positive solutions for a system of nonlinear singular Hammerstein integral equations via nonnegative matrices and applications. Positivity 2012, 16: 783-800. 10.1007/s11117-011-0146-4MathSciNetView ArticleMATHGoogle Scholar
- Goodrich CS: On nonlinear boundary conditions involving decomposable linear functionals. Proc. Edinb. Math. Soc. 2014.Google Scholar
- Zeidler E: Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems. Springer, New York; 1986.View ArticleMATHGoogle Scholar
- 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)90074-2MathSciNetView ArticleMATHGoogle Scholar
- Amann H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 1976, 18: 620-709. 10.1137/1018114MathSciNetView ArticleMATHGoogle Scholar
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, Boston; 1988.MATHGoogle Scholar
- Webb JRL: A class of positive linear operators and applications to nonlinear boundary value problems. Topol. Methods Nonlinear Anal. 2012, 39: 221-242.MathSciNetMATHGoogle Scholar
- Webb JRL: Remarks on positive solutions of three point boundary value problems. Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations 2003, 905-915. (Wilmington, NC, 2002)Google Scholar
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