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Fixed point theorems for a class of nonlinear operators in Hilbert spaces with lattice structure and application
Fixed Point Theory and Applications volume 2013, Article number: 345 (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.
We call E a lattice under the partial ordering ‘≤’ if and exist for arbitrary . For , let
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.
Definition 1.1 Let and be a nonlinear operator. F is said to be quasi-additive on lattice, if there exists such that
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.
Suppose that is a positive bounded linear operator. If the spectral radius , then exists and is a positive bounded linear operator. Furthermore,
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.
We will show that, under additional assumptions on the operator A, Φ satisfies the Palais-Smale compactness condition on a closed convex set , which ensures the existence of a critical point of Φ, see [15, 16].
Every sequence , satisfying the conditions has a subsequence which converges strongly in M.
Lemma 2.1 Assume that H is a Hilbert space, M is a closed convex subset of H, Φ is a functional defined on H, can be expressed in the form , and . Assume also that Φ satisfies the PS condition on M, Ω is an open subset of M, and there are two points , such that . Then
is a critical value of Φ and there is at least one critical point in M corresponding to this value, where .
Theorem 2.1 Let X, H be two ordered real Hilbert spaces. Suppose that satisfies the following hypotheses.
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 , ;
There exists with such that
that is, φ is the normalized first eigenfunction.
There exist and such that(2.1)
There exist and such that(2.2)
There exist such that , where denotes the norm of .
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.
Proof of Theorem 2.1 It follows from condition (v) that
Thus, we have
By virtue of (2.1) and (2.2), we have
where and . Since , Lemma 1.1 yields that is a positive linear operator. Let and , then and . This shows . For , by we infer . On account of (2.3) and (2.4), we arrive at
It is easy to see that is a closed convex subset of H.
By the definition of gradient operator and , we have
Noticing , we claim that there is a mountain surrounding θ and , and with . Indeed, from (v), we have such that
Replacing x by in the above inequality, we have . Thus,
Similarly, we have
Thus by (i), for , we have
Let . Then by (iv) we get
Since , Lemma 1.1 yields that exists and
It follows from that . So we know that
Combining with (2.5), we have
Therefore , such that .
Next, we take , where is the normalized first eigenfunction of B, and is to be determined. Set
From (2.3), we have
So we have as . Therefore there exists satisfying , set , is as required. Hence Theorem 2.1 holds by Lemma 2.1. □
Consider the two-point boundary value problem
Here f is a continuous function from ℝ into ℝ.
Let with the inner product and norm
and , then H is a lattice under the partial ordering induced by P.
Let with the cone
For any , it is evident that
and hence and so condition (iv) of Theorem 2.1 holds with .
It is easy to see that is a positive bounded linear operator satisfying . For any , , we have
and so condition (1) of Theorem 2.1 holds.
As is well known, B has an unbounded sequence of eigenvalues:
The algebraic multiplicities of every eigenvalue are simple, and the spectral radius and .
Theorem 3.1 Let be continuous. Suppose that there exists such that
Then (3.1) has at least one nontrivial solution.
Proof We define the functional as
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.
By means of (3.3), (3.4) and (3.5), we know that there exist such that
and therefore there exists a constant such that
and so conditions (ii), (iii) and (iv) of Theorem 2.1 hold.
From the above discussion, in order to apply Theorem 2.1, we only need to verify that Φ satisfies the PS condition. Suppose that satisfies as and . Taking the inner product of and , we have
Then we have
which gives a bound for . Then both and are bounded. In order to find a bound for , we use a contradiction argument and assume that as . Defining and selecting a subsequence if necessary, we have weakly in H and strongly in X as for some , . Since for and is bounded, it follows from (3.6) that
which implies that . The embedding theorem and the boundedness of in H guarantee that there exists such that for all n. So is bounded for . Taking the inner product of and , we see that
Letting , we obtain
which is a contradiction. Thus, is bounded in H and it has a convergent subsequence, as a standard consequence. □
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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.
The authors declare that they have no competing interests.
All authors contributed equally and significantly to this research work. All authors read and approved the final manuscript.
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Cui, Y., Sun, J. Fixed point theorems for a class of nonlinear operators in Hilbert spaces with lattice structure and application. Fixed Point Theory Appl 2013, 345 (2013). https://doi.org/10.1186/1687-1812-2013-345
- Hilbert Space
- Lattice Structure
- Fixed Point Theorem
- Bounded Linear Operator
- Nonlinear Operator