Nodal Solutions for a Class of Fourth-Order Two-Point Boundary Value Problems
© J. Xu and X. Han. 2010
Received: 18 February 2010
Accepted: 27 April 2010
Published: 31 May 2010
We consider the fourth-order two-point boundary value problem , , , where is a parameter, is given constant, with on any subinterval of , satisfies for all , and , , for some . By using disconjugate operator theory and bifurcation techniques, we establish existence and multiplicity results of nodal solutions for the above problem.
where is continuous, is a parameter. Since the problem (1.1) cannot transform into a system of second-order equation, the treatment method of second-order system does not apply to the problem (1.1). Thus, existing literature on the problem (1.1) is limited. In 1984, Agarwal and chow  firstly investigated the existence of the solutions of the problem (1.1) by contraction mapping and iterative methods, subsequently, Ma and Wu  and Yao [3, 4] studied the existence of positive solutions of this problem by the Krasnosel'skii fixed point theorem on cones and Leray-Schauder fixed point theorem. Especially, when , Korman  investigated the uniqueness of positive solutions of the problem (1.1) by techniques of bifurcation theory. However, the existence of sign-changing solution for this problem have not been discussed.
under the assumptions:
(H1) is a parameter, is given constant,
(H2) with on any subinterval of ,
for some .
This can be achieved under (H1) by using disconjugacy theory in .
is disconjugacy on , moreover, we establish some preliminary properties on the eigenvalues and eigenfunctions of the generalized eigenvalue problem (1.4). Finally in Section 3, we state and prove our main result.
For other results on the existence and multiplicity of positive solutions and nodal solutions for boundary value problems of ordinary differential equations based on bifurcation techniques, see Ma [8–12], An and Ma , Yang  and their references.
2. Preliminary Results
be th-order linear differential equation whose coefficients are continuous on an interval .
Definition 2.1 (see [7, Definition , page 2]).
Equation (2.1) is said to be disconjugate on an interval if no nontrivial solution has zeros on , multiple zeros being counted according to their multiplicity.
Lemma 2.2 (see [7, Theorem , page 3]).
Lemma 2.3 (see [7, Theorem , page 9]).
Now using Lemmas 2.2 and 2.3, we will prove some preliminary results.
Let (H1) hold. Then
We divide the proof into three cases.
. The case is obvious.
Using (2.14), we conclude that is equivalent to (2.8).
From and , we have , so on
Using (2.18), we conclude that is equivalent to (2.8).
This completes the proof of the theorem.
Let (H1) hold and satisfy (H2). Then
(iii) To each eigenvalue there corresponding an essential unique eigenfunction which has exactly simple zeros in and is positive near 0.
(iv) Given an arbitrary subinterval of , then an eigenfunction which belongs to a sufficiently large eigenvalue change its sign in that subinterval.
(v) For each , the algebraic multiplicity of is 1.
(i)–(iv) are immediate consequences of Elias [6, Theorems ] and Theorem 2.4. we only prove (v).
which is a contradiction!
Theorem 2.6 (Maximum principle).
Then on .
where is Green's function.
Using (2.28), when on with in , then on .
3. Statement of the Results
Then there are at least nontrivial solutions of the problem (1.2). In fact, there exist solutions such that for , has exactly simple zeros on the open interval and and there exist solutions such that for , has exactly simple zeros on the open interval and .
with the norm Then is completely continuous, here is given as in (2.20).
as a bifurcation problem from the trivial solution .
Further we note that for near 0 in .
In what follows, we use the terminology of Rabinowitz .
Let under the product topology. Let denote the set of function in which have exactly interior nodal (i.e., nondegenerate) zeros in and are positive near , set , and . They are disjoint and open in . Finally, let and .
The results of Rabinowitz  for (3.8) can be stated as follows: for each integer , , there exists a continuum of solutions of (3.8), joining to infinity in . Moreover, .
Notice that we have used the fact that if is a nontrivial solution of (3.7), then all zeros of on are simply under (H1), (H2), and (H3).
clearly satisfies (H2). So Theorem 2.5(iii) yields that all zeros of on are simple.
Proof of Theorem 3.1.
Furthermore, since is completely continuous, we may assume that there exist with such that as .
By Theorem 2.6, we know that in . This means is the first eigenvalue of and is the corresponding eigenfunction. Hence . Since is open and , we have that for large. But this contradict the assumption that and , so (3.12) is wrong, which completes the proof.
This work is supported by the NSFC (no. 10671158), the Spring-sun program (no. Z2004-1-62033), SRFDP (no. 20060736001), the SRF for ROCS, SEM (2006[ ]), NWNU-KJCXGC-SK0303-23, and NWNU-KJCXGC-03-69.
- Agarwal RP, Chow YM: Iterative methods for a fourth order boundary value problem. Journal of Computational and Applied Mathematics 1984,10(2):203-217. 10.1016/0377-0427(84)90058-XMATHMathSciNetView ArticleGoogle Scholar
- Ma R, Wu HP: Positive solutions of a fourth-order two-point boundary value problem. Acta Mathematica Scientia A 2002,22(2):244-249.MATHMathSciNetGoogle Scholar
- Yao Q: Positive solutions for eigenvalue problems of fourth-order elastic beam equations. Applied Mathematics Letters 2004,17(2):237-243. 10.1016/S0893-9659(04)90037-7MATHMathSciNetView ArticleGoogle Scholar
- Yao Q: Solvability of an elastic beam equation with Caratheodory function. Mathematica Applicata 2004,17(3):389-392.MATHMathSciNetGoogle Scholar
- Korman P: Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems. Proceedings of the Royal Society of Edinburgh A 2004,134(1):179-190. 10.1017/S0308210500003140MATHMathSciNetView ArticleGoogle Scholar
- Elias U:Eigenvalue problems for the equations . Journal of Differential Equations 1978,29(1):28-57. 10.1016/0022-0396(78)90039-6MathSciNetView ArticleGoogle Scholar
- Elias U: Oscillation Theory of Two-Term Differential Equations, Mathematics and Its Applications. Volume 396. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:viii+217.View ArticleGoogle Scholar
- Ma R: Existence of positive solutions of a fourth-order boundary value problem. Applied Mathematics and Computation 2005,168(2):1219-1231. 10.1016/j.amc.2004.10.014MATHMathSciNetView ArticleGoogle Scholar
- Ma R: Nodal solutions for a fourth-order two-point boundary value problem. Journal of Mathematical Analysis and Applications 2006,314(1):254-265.MATHMathSciNetView ArticleGoogle Scholar
- Ma R: Nodal solutions of boundary value problems of fourth-order ordinary differential equations. Journal of Mathematical Analysis and Applications 2006,319(2):424-434. 10.1016/j.jmaa.2005.06.045MATHMathSciNetView ArticleGoogle Scholar
- Ma R, Xu J: Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2010,72(1):113-122. 10.1016/j.na.2009.06.061MATHMathSciNetView ArticleGoogle Scholar
- Ma R, Wang H: On the existence of positive solutions of fourth-order ordinary differential equations. Applicable Analysis 1995,59(1–4):225-231.MATHMathSciNetGoogle Scholar
- An Y, Ma R:Global behavior of the components for the second order -point boundary value problems. Boundary Value Problems 2008, 2008:-10.Google Scholar
- Yang Z: Existence and uniqueness of positive solutions for higher order boundary value problem. Computers & Mathematics with Applications 2007,54(2):220-228. 10.1016/j.camwa.2007.01.018MATHMathSciNetView ArticleGoogle Scholar
- Rabinowitz PH: Some global results for nonlinear eigenvalue problems. Journal of Functional Analysis 1971,7(3):487-513. 10.1016/0022-1236(71)90030-9MATHMathSciNetView ArticleGoogle Scholar
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