Abstract
This paper presents some theorems of the fixed point for decreasing operators in Banach spaces with lattice structure. The results are applied to nonlinear secondorder elliptic equations.
MSC: 47H10, 34B15.
Keywords:
decreasing operators; lattice structure; nonlinear; elliptic equations1 Introduction and preliminaries
The fixed point theory for monotone operators in ordered Banach spaces has been investigated extensively in the past 30 years [18]. Many new fixed point theorems have been proved under the nonlinear contractive condition by using the theorem of cone and monotone iterative technique. These results have been applied to study the ordinary differential equations, partial differential equations, and integral equations.
In this paper, we investigate decreasing operators in ordered Banach spaces with lattice structure. The theoretical results of fixed points are extended by using the famous Schauder fixed point theorem for the operators. We weaken the conditions of the Schauder fixed point theorem. The results of this paper have no need for the closed bounded and convex property of domains for the operators. To demonstrate the applicability of our results, we apply them to study a problem of nonlinear secondorder elliptic equations in the final section of the paper, and the existence of solution is obtained.
Let E be a Banach space and P be a cone of E. We define a partial ordering ≤ with respect to P by
Let E be a partially ordered set. We call E a lattice in the partial ordering ≤. For arbitrary
Let
Lemma 1.1[9]
LetEbe a real Banach space,
Lemma 1.2[10]
LetEbe a real Banach space,
Lemma 1.3[11]
LetEbe a real Banach space,
Remark 1 Lemma 1.1 is the famous Sadovskii fixed point theorem; Lemma 1.2 is the famous Schauder fixed point theorem; Lemma 1.3 is the famous Darbo fixed point theorem.
2 Main results
Theorem 2.1LetEbe an ordered Banach space with lattice structure,
Proof For any
Since E is a Banach space with lattice structure and
That is,
Since A is a decreasing operator, we have
(2.1) and (2.2) show that
Similar to the proof of (2.3), there exists
That is,
Since A is a decreasing operator, we have
(2.4) and (2.5) show that
(2.3) and (2.6) together with
For any
By (2.7), we have
It is easy to know that
Theorem 2.2LetEbe an ordered Banach space with lattice structure,
Proof For any
Since E is a Banach space with lattice structure, there exists
That is,
Since A is a decreasing operator, we have
(2.8) and (2.9) show that
Similar to the proof of (2.10), there exist
That is,
Since A is a decreasing operator, we have
(2.11) and (2.12) show that
(2.10) and (2.13) together with
For any
By (2.14), we have
It is easy to know that
3 Corollaries and relative results
Similar to the proof of Theorem 2.1, by Lemma 1.2 and Lemma 1.3, we can get the following corollaries and relative results.
Corollary 3.1LetEbe an ordered Banach space with lattice structure,
Corollary 3.2LetEbe an ordered Banach space with lattice structure,
Corollary 3.3LetEbe an ordered Banach space with lattice structure,
Corollary 3.4LetEbe an ordered Banach space with lattice structure,
4 Applications
In this section, we use Theorem 2.1 to show the existence of a solution for the uniformly
elliptic differential problem. Let Ω be a bounded convex domain in
i.e., there exists a positive constant
Considering the Dirichlet problem
we have the following conclusions.
Theorem 4.1Suppose that
Proof It is easy to know that
where
Hence, the linear integral operator
is a completely continuous operator from E into E. Clearly, the superposition operator
Moreover, the mapping A is decreasing in u. In fact, by hypotheses, for
implies that
so A is decreasing.
So, the condition of Theorem 2.1 holds, Theorem 4.1 is proved. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
XL carried out the the main theorem and the main conclusion. ZW carried out the application of the main theorem. All authors read and approved the final manuscript.
Acknowledgements
The first author was supported financially by the NSFC (71240007, 11001151), NSFSP (ZR2010AM005).
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