Fixed point theorems for a class of nonlinear operators in Hilbert spaces with lattice structure and application
© Cui and Sun; licensee Springer. 2013
Received: 14 June 2013
Accepted: 1 December 2013
Published: 17 December 2013
We discuss the existence of a fixed point for a class of nonlinear operators in Hilbert spaces with lattice structure by a combination of variational and partial ordered methods. An application to second-order ordinary differential equations is included.
and are called the positive part and the negative part of x, respectively, and obviously . Take , then . One can refer to  for the definition and properties of the lattice.
In recent years, many mathematicians have studied a fixed point theorem of nonlinear operators in an ordered Banach space by using topological methods and partial ordered methods (see [4–14] and references therein). However, to the best of our knowledge, few authors have studied the fixed point theorem in Hilbert spaces with lattice structure by applying variational and partial ordered methods. As a result, the goal of this paper is to fill the gap in this area.
Motivated by [7, 8], we obtain some new theorems for nonlinear operators which are not cone mappings by means of variational and partial ordered methods. This paper is organized as follows. Section 2 is devoted to our main results. Section 3 gives examples to indicate the application of our main results.
At the end of this section, we give the following basic concept and lemma from literature which will be used in Section 2.
We consider two real ordered Hilbert spaces in this paper, H with the inner product, norm and cone , , P, and X with the inner product, norm and cone , and we assume that and , X is dense in H, the injection being continuous.
where and are defined by (1.1). Note that if , (1.2) becomes .
Let be a bounded linear operator. B is said to be positive if . In this case, B is an increase operator, namely for , implies . We have the following conclusion.
2 Fixed point theorems
Let H be an ordered real Hilbert space with an ordering given by a closed cone P and suppose that the gradient of a given functional has the expression . Obviously, the critical points of the functional Φ are the fixed points of the operator A, and vice versa.
- (PS)Every sequence , satisfying the conditions has a subsequence which converges strongly in M.
is a critical value of Φ and there is at least one critical point in M corresponding to this value, where .
- (i)Φ satisfies the PS condition on H and its gradient admits the decomposition such that , where is quasi-additive on lattice, is a positive bounded linear operator satisfying:
for , ;
- (2)There exists with such that
that is, φ is the normalized first eigenfunction.
- (ii)There exist and such that(2.1)
- (iii)There exist and such that(2.2)
There exist such that , where denotes the norm of .
- (v)There exist and a positive number r such that
Then A has a nontrivial fixed point.
Remark 2.1 From Theorem 3.1 below, we point out that conditions (i) and (iv) of Theorem 2.1 appear naturally in the applications for nonlinear differential equations and integral equations.
It is easy to see that is a closed convex subset of H.
Therefore , such that .
So we have as . Therefore there exists satisfying , set , is as required. Hence Theorem 2.1 holds by Lemma 2.1. □
Here f is a continuous function from ℝ into ℝ.
and , then H is a lattice under the partial ordering induced by P.
and hence and so condition (iv) of Theorem 2.1 holds with .
and so condition (1) of Theorem 2.1 holds.
The algebraic multiplicities of every eigenvalue are simple, and the spectral radius and .
Then (3.1) has at least one nontrivial solution.
One has that , which implies that in H, where B is given by (3.2) and . Furthermore, from the properties of H, the regularity properties of Φ, it is easy to assert that the solutions to (3.1) are precisely the critical points of Φ.
and so conditions (ii), (iii) and (iv) of Theorem 2.1 hold.
which is a contradiction. Thus, is bounded in H and it has a convergent subsequence, as a standard consequence. □
The authors would like to thank the referees for carefully reading this article and making valuable comments and suggestions. This work is supported by the Foundation items: NSFC (10971179, 11371221, 11071141, 61201431), the Postdoctoral Science Foundation of Shandong Province, NSF (BS2010SF023, BS2012SF022) of Shandong Province.
- Deimling K: Nonlinear Functional Analysis. Springer, New York; 1985.MATHView ArticleGoogle Scholar
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, San Diego; 1988.MATHGoogle Scholar
- Luxemburg WAJ, Zaanen AC 1. In Riesz Space. North-Holland, London; 1971.Google Scholar
- Cui Y, Sun J: Fixed point theorems for a class of nonlinear operators in Hilbert spaces and applications. Positivity 2011, 15: 455–464. 10.1007/s11117-010-0095-3MATHMathSciNetView ArticleGoogle Scholar
- Krasnoselskii MA: Positive Solutions of Operator Equations. Noordhoff, Groningen; 1964.Google Scholar
- Liu X, Sun J: Computation of topological degree of unilaterally asymptotically linear operators and its applications. Nonlinear Anal. 2009, 71: 96–106. 10.1016/j.na.2008.10.032MATHMathSciNetView ArticleGoogle Scholar
- Sun J, Liu X: Computation of topological degree for nonlinear operators and applications. Nonlinear Anal. 2008, 69: 4121–4130. 10.1016/j.na.2007.10.042MATHMathSciNetView ArticleGoogle Scholar
- Sun J, Liu X: Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations. J. Math. Anal. Appl. 2008, 348: 927–937. 10.1016/j.jmaa.2008.05.023MATHMathSciNetView ArticleGoogle Scholar
- Sun J, Liu X: Computation for topological degree and its applications. J. Math. Anal. Appl. 1996, 202: 785–796. 10.1006/jmaa.1996.0347MATHMathSciNetView ArticleGoogle Scholar
- Nieto JJ, Pouso RL, Rodríguez-López R: Fixed point theorems in ordered abstract spaces. Proc. Am. Math. Soc. 2007, 135: 2505–2517. 10.1090/S0002-9939-07-08729-1MATHView ArticleMathSciNetGoogle Scholar
- O’Regan D, Petruşel A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 2008, 341: 1241–1252. 10.1016/j.jmaa.2007.11.026MATHMathSciNetView ArticleGoogle Scholar
- Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. 2007, 23: 2205–2212. 10.1007/s10114-005-0769-0MATHView ArticleMathSciNetGoogle Scholar
- Chifu C, Petruşel G: Fixed-point results for generalized contractions on ordered gauge spaces with applications. Fixed Point Theory Appl. 2011., 2011: Article ID 979586Google Scholar
- Siegfried C, Seppo H: Fixed Point Theory in Ordered Sets and Applications. From Differential and Integral Equations to Game Theory. Springer, New York; 2011.MATHGoogle Scholar
- Chong KG: Infinite Dimensional Theory and Multiple Solution Problems. Birkhäuser, Boston; 1993.View ArticleGoogle Scholar
- Rabinowitz PH CBMS Reg. Conf. Ser. Math. 65. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Conference Board of the Mathematical Sciences, Washington; 1986.View ArticleGoogle Scholar
- Liu Z, Sun J: Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations. J. Differ. Equ. 2001, 172: 257–299. 10.1006/jdeq.2000.3867MATHView ArticleMathSciNetGoogle Scholar
- Guo D, Sun J, Qi G: Some extensions of the mountain pass lemma. Differ. Integral Equ. 1988, 1: 351–358.MATHMathSciNetGoogle Scholar
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