Bifurcation from interval and positive solutions for a class of fourth-order two-point boundary value problem
© Shen and He; licensee Springer. 2013
Received: 14 November 2012
Accepted: 8 July 2013
Published: 22 July 2013
We consider the fourth-order two-point boundary value problem , , , which is not necessarily linearizable. We give conditions on the parameters k, l and that guarantee the existence of positive solutions. The proof of our main result is based upon topological degree theory and global bifurcation techniques.
Keywordstopological degree fourth-order ordinary differential equation bifurcation positive solution eigenvalue
where is continuous, is a parameter and l is a given constant. Since problem (1.1) cannot transform into a system of second-order equations, the treatment method of the second-order system does not apply to it. Thus, the existing literature on problem (1.1) is limited. When , the existence of positive solutions of problem (1.1) has been studied by several authors, see [1–5]. Especially, when , Xu and Han  studied the existence of nodal solutions of problem (1.1) by applying disconjugate operator theory and bifurcation techniques.
- (i)k, l satisfying are given constants with(1.3)
- (ii)k, l satisfying are given constants with(1.4)
In this paper, we consider bifurcation from interval and positive solutions for problem (1.2). In order to prove our main result, condition (A1) and the following weaker conditions are satisfied throughout this paper:
uniformly for .
(H2) for and .
It is the purpose of this paper to study the existence of positive solutions of (1.2) under conditions (A1), (H1), (H2) and (H3). The main tool we use is the following global bifurcation theorem for the problem which is not necessarily linearizable.
Theorem A (Rabinowitz )
is unbounded in , or
Remark 1.1 For other results on the existence and multiplicity of positive solutions and nodal solutions for boundary value problems of fourth-order ordinary differential equations based on bifurcation techniques, see [11–20].
2 Hypotheses and lemmas
Theorem 2.1 (see [, Theorem 2.4])
- (i)is disconjugate on , and has a factorization(2.2)
- (ii)if and only if(2.3)
Theorem 2.2 (see [, Theorem 2.7])
- (i)the problem(2.5)
to each eigenvalue , there corresponds an essential unique eigenfunction which has exactly simple zeros in and is positive near 0;
given an arbitrary subinterval of , an eigenfunction that belongs to a sufficiently large eigenvalue changes its sign in that subinterval;
for each , the algebraic multiplicity of is 1.
Theorem 2.3 (see [, Theorem 2.8]) (Maximum principle)
then on .
Let with the norm . Let with its usual norm . By a positive solution of (1.2), we mean x is a solution of (1.2) with (i.e., in and ).
Then is a closed operator and is completely continuous.
Since for , we have for . Thus x is a nonnegative solution of (2.11), and the closure of the set of nontrivial solutions of (2.13) in is exactly Σ.
For , let , and let denote the degree of on with respect to 0.
Proof Suppose to the contrary that there exist sequences and in , in E, such that for all , then in .
Thus, . This contradicts . □
Corollary 2.6 For and , .
which ends the proof. □
where is the nonnegative eigenfunction corresponding to .
where and . Since on and , we have from (2.32) that .
This contradicts (2.33). □
Corollary 2.8 For and , .
Now, using Theorem A, we may prove the following.
is unbounded, or
By Lemma 2.5, the case (ii) cannot occur. Thus is unbounded bifurcated from in . Furthermore, we have from Lemma 2.5 that for any closed interval , if , then in E is impossible. So, must be bifurcated from in . □
3 Main results
then problem (1.2) has at least one positive solution.
We note that for all since is the only solution of (2.15) for and .
Case 1. .
We divide the proof into two steps.
Step 1. We show that is bounded.
Let denote the nonnegative eigenfunction corresponding to .
Step 2. We show that C joins to .
So, C joins to .
Case 2. .
Again C joins to and the result follows. □
This work is supported by the NSF of Gansu Province (No. 1114-04).
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