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Bifurcation and positive solutions of a nonlinear fourth-order dynamic boundary value problem on time scales
Advances in Difference Equations volume 2013, Article number: 64 (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.
In 2006, Luo and Ma  considered the second-order dynamic boundary value problem on time scales
where . They proved that (1.1) has at least one positive solution if either
and is the first eigenvalue of the linear problem
Notice that conditions (1.2) and (1.3) are optimal! Inspired by this paper, we have a natural question if we could establish some optimal results for the fourth-order dynamic boundary value problem
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.
For the continuous cases, Ma and Xu  discussed
where is continuous. And for the discrete cases, there exists no work corresponding exactly to the continuous cases, but we can refer the reader to Xu, Gao and Ma  and Ma and Gao  and Gao and Xu  for the similar discussions. Moreover, nothing is known about the time scales extension analog (1.4) except for  for second-order dynamic equations. The likely reason is that few properties of the associated linear eigenvalue problem
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
For a compact interval , is a Banach space with the norm
For , let us consider the linear problem
Let for . Then
Now, (2.1) is equivalent to
And we can easily check that
To sum up, we have proved the following.
Lemma 2.1 For each , problem (2.1) has a unique solution
Let : , . Then solves the problem
and solves the problem
from [, pp.92-93].
It is clear that
and so there exist such that
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 .
and define the norm of by
where . It is easy to check that is a Banach space. Let
Then the cone P is normal and has nonempty interiors intP.
Lemma 2.2 For ,
Proof Since , there exist : , and : , , we have
Let us make the assumption
(H0) with or on .
Define a linear operator by
Then (1.5) is equivalent to
Definition 2.1 We say λ is an eigenvalue of the linear problem
if (2.16) has nontrivial solutions.
Theorem 2.1 Let (H0) hold and T be not an ldrs point of the time scale . Then
, 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.
For , we have
for some positive δ and γ. From (2.15), the condition (H0), (2.11) and (2.12), we have
By (2.15), the condition (H0) and (2.12), we have
Similarly, we get
Thus . Furthermore, since , we have that is compact.
Next, we show that is strongly positive.
For , if on , then on for some constant , and subsequently,
it follows that there exists such that on . Thus
If on , then on for some constant , and subsequently,
Then there exists such that on . So,
Similarly, for any , if on , then there exists such that for ,
If on , then there exists such that for ,
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 .
(H2) is continuous and there exist functions with and on such that
uniformly for , and
uniformly for . Here
(H3) for and .
(H4) There exist functions and such that
Theorem 3.1 Let (H1)-(H4) hold. Assume that either
Then problem (1.4) has at least one positive solution.
Denote by setting
From [, Lemma 3.7] and standard properties of compact linear operators, we can get that is compact.
Let be continuous and satisfy
Obviously, (H2) implies that
Then is nondecreasing and
Next we consider
as a bifurcation problem from the trivial solution . It is easy to check that (3.11) can be converted to the equivalent equation
From Theorem 2.1, we have that for each fixed , the operator ,
is compact and strongly positive. Define by
Then we have from (3.7) and Lemma 2.2 that
such that .
Proof of Theorem 3.1 It is clear that any solution of (3.11) of the form yields a solution u of (1.4). We will show that crosses the hyperplane in . To do this, it is enough to show that joins to . Let satisfy
We note that for all since is the only solution of (3.11) for and .
Case 1. .
In this case, we show that
We divide the proof into two steps.
Step 1. We show that if there exists a constant number such that
then C joins to .
If (3.17) holds, we have that . We divide the equation
by and set . Since is bounded in X, choosing a subsequence and relabeling if necessary, we see that for some with . Moreover, from Lemma 2.2, we have
and from (3.10),
where , again choosing a subsequence and relabeling if necessary. Therefore,
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.
Since , we have
Therefore from (H4), we get
where . Set , we have , , and
From , . Now we can obtain , ∀n for some positive constant M according to [, Lemma 2.2].
Case 2. .
According to Step 2 of Case 1, we have
for some . Then if is such that
applying a similar argument to that used in Step 1 of Case 1, after taking a subsequence and relabeling if necessary, it follows that
Again joins to and the result follows. □
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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).
The author declares that they have no competing interests.
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Luo, H. Bifurcation and positive solutions of a nonlinear fourth-order dynamic boundary value problem on time scales. Adv Differ Equ 2013, 64 (2013). https://doi.org/10.1186/1687-1847-2013-64
- fourth-order dynamic equations
- positive solutions
- Krein-Rutman theorem
- time scales