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.
Keywords:topological degree; fourth-order ordinary differential equation; bifurcation; positive solution; eigenvalue
The deformations of an elastic beam in an equilibrium state with fixed both endpoints can be described by the fourth-order boundary value problem
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.
Recently, motivated by , when k, l satisfy (A1), Shen  studied the existence of nodal solutions of a general fourth-order boundary value problem by applying disconjugate operator theory [8,9] and Rabinowitz’s global bifurcation theorem
(A1) one of following conditions holds:
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:
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 )
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])
Let (A1) hold. Then
Theorem 2.2 (see [, Theorem 2.7])
(i) the problem
has an infinite sequence of positive eigenvalues
Theorem 2.3 (see [, Theorem 2.8]) (Maximum principle)
Integrating by parts, we obtain
From Lemma 2.4, we have
Similarly, we deduce from the second inequality in (2.19) that
which ends the proof. □
Applying Lemma 2.4 and (2.37), it follows that
This contradicts (2.33). □
Now, using Theorem A, we may prove the following.
Proof For fixed with , let us take that , and . It is easy to check that for , all of the conditions of Theorem A are satisfied. So, there exists a connected component of solutions of (2.15) containing , and either
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
Theorem 3.1Let (A1), (H1), (H2), (H3) hold. Assume that either
then problem (1.2) has at least one positive solution.
Proof of Theorem 3.1 It is clear that any solution of (2.15) of the form yields a solution x of (1.2). We will show that C crosses the hyperplane in . To do this, it is enough to show that C joins to . Let satisfy
In this case, we show that
We divide the proof into two steps.
From (3.4), we have
By Lemma 2.4, we have
which implies that
relabeling if necessary. Thus, (3.9) yields that
which implies that
From Lemma 2.4, integrating by parts, we obtain that
Similarly, we deduce from the second inequality in (3.12) that
The authors declare that they have no competing interests.
WS conceived of the study, and participated in its design and coordination and helped to draft the manuscript. TH drafted the manuscript. All authors read and approved the final manuscript.
This work is supported by the NSF of Gansu Province (No. 1114-04).
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