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Bifurcation and positive solutions of a nonlinear fourth-order dynamic boundary value problem on time scales
© Luo; licensee Springer 2013
Received: 29 September 2012
Accepted: 2 March 2013
Published: 20 March 2013
This paper discusses the spectrum properties of a linear fourth-order dynamic boundary value problem on time scales and obtains the existence result of positive solutions to a nonlinear fourth-order dynamic boundary value problem. The key condition which makes nonlinear problem have at least one positive solution is related to the first eigenvalue of the associated linear problem. The proof of the main result is based upon the Krein-Rutman theorem and the global bifurcation techniques on time scales.
where is a time scale and , , is continuous. To the best of our knowledge, few papers can be found to study such a problem in the literature. Wang and Sun  and Luo and Gao  applied the Schauder fixed point theorem to show the existence of positive solutions of a fourth-order dynamic boundary value problem under different boundary value conditions. However, the key conditions in these two papers are not directly related to the first eigenvalue of the associated linear eigenvalue problems.
In this paper, we establish spectrum properties of (1.5) in Section 2 and use the global bifurcation techniques on time scales (see Davidson and Rynne [, Theorem 7.1] or Luo and Ma ) to discuss the existence of positive solutions for problem (1.4) in Section 3. Our existence result is related to the first eigenvalue of the associated linear eigenvalue problem (1.5) and thus should be optimal.
2 Generalized eigenvalues
To sum up, we have proved the following.
from [, pp.92-93].
Remark 2.2 (2.11) holds for any time scale with , but (2.12) does not hold for the case of , i.e., T is an left-dense and right-scattered point (abbreviated to ldrs point) of the time scale .
Then the cone P is normal and has nonempty interiors intP.
Let us make the assumption
(H0) with or on .
if (2.16) has nontrivial solutions.
, and is strongly positive;
Problem (1.5) has an algebraically simple eigenvalue, , with a nonnegative eigenfunction ;
There is no other eigenvalue with a nonnegative eigenfunction.
Proof We prove firstly.
Thus . Furthermore, since , we have that is compact.
Next, we show that is strongly positive.
It follows from (2.21)-(2.24) that .
Now, by the Krein-Rutman theorem ([, Theorem 7.C] or [, Theorem 19.3]), G has an algebraically simple eigenvalue with an eigenfunction . Moreover, there is no other eigenvalue with a nonnegative eigenfunction. □
3 The main result
In this section, we will make the following assumptions:
(H1) T is not an ldrs point of the time scale .
(H3) for and .
Then problem (1.4) has at least one positive solution.
From [, Lemma 3.7] and standard properties of compact linear operators, we can get that is compact.
such that .
We note that for all since is the only solution of (3.11) for and .
Case 1. .
We divide the proof into two steps.
then C joins to .
This together with Theorem 2.1 yields . Thus joins to .
Step 2. We show that there exists a constant M such that for all n.
From , . Now we can obtain , ∀n for some positive constant M according to [, Lemma 2.2].
Case 2. .
Again joins to and the result follows. □
We would like to thank the referees for carefully reading this paper and suggesting many valuable comments. This work was supported by China Postdoctoral Science Foundation Funded Project (Nos. 201104602, 20100481239), General Project for Scientific Research of Liaoning Educational Committee (Nos. L2011200, L2012409), Teaching and Research Project of DUFE (No. YY12012) and the NSFC (Nos. 71171035, 71201019).
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