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.
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 )
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).
- Agarwal RP, Chow YM: Iterative methods for a fourth-order boundary value problem. J. Comput. Appl. Math. 1984, 10(2):203–217. 10.1016/0377-0427(84)90058-XMathSciNetView ArticleMATHGoogle Scholar
- Ma R, Wu HP: Positive solutions of a fourth-order two-point boundary value problem. Acta Math. Sci., Ser. A 2002, 22(2):244–249.MathSciNetMATHGoogle Scholar
- Yao Q: Positive solutions for eigenvalue problems of fourth-order elastic beam equations. Appl. Math. Lett. 2004, 17(2):237–243. 10.1016/S0893-9659(04)90037-7MathSciNetView ArticleMATHGoogle Scholar
- Yao Q: Solvability of an elastic beam equation with Caratheodory function. Math. Appl. 2004, 17(3):389–392.MathSciNetMATHGoogle Scholar
- Korman P: Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems. Proc. R. Soc. Edinb. A 2004, 134(1):179–190. 10.1017/S0308210500003140MathSciNetView ArticleMATHGoogle Scholar
- Xu J, Han XL: Nodal solutions for a fourth-order two-point boundary value problem. Bound. Value Probl. 2010., 2010: Article ID 570932Google Scholar
- Shen WG: Existence of nodal solutions of a nonlinear fourth-order two-point boundary value problem. Bound. Value Probl. 2012. 10.1186/1687-2770-2012-31Google Scholar
- Elias U:Eigenvalue problems for the equations . J. Differ. Equ. 1978, 29(1):28–57. 10.1016/0022-0396(78)90039-6MathSciNetView ArticleGoogle Scholar
- Elias U Mathematics and Its Applications 396. In Oscillation Theory of Two-Term Differential Equations. Kluwer Academic, Dordrecht; 1997.View ArticleGoogle Scholar
- Rabinowitz PH: Some aspects of nonlinear eigenvalue problems. Rocky Mt. J. Math. 1973, 3: 161–202. 10.1216/RMJ-1973-3-2-161MathSciNetView ArticleMATHGoogle Scholar
- Ma R: Existence of positive solutions of a fourth-order boundary value problem. Appl. Math. Comput. 2005, 168(2):1219–1231. 10.1016/j.amc.2004.10.014MathSciNetView ArticleMATHGoogle Scholar
- Ma R: Nodal solutions for a fourth-order two-point boundary value problem. J. Math. Anal. Appl. 2006, 314(1):254–265. 10.1016/j.jmaa.2005.03.078MathSciNetView ArticleMATHGoogle Scholar
- Ma R: Nodal solutions of boundary value problem of fourth-order ordinary differential equations. J. Math. Anal. Appl. 2006, 319(2):424–434. 10.1016/j.jmaa.2005.06.045MathSciNetView ArticleMATHGoogle Scholar
- Ma R, Thompson B: Nodal solutions for a nonlinear fourth-order eigenvalue problem. Acta Math. Sin. Engl. Ser. 2008, 24(1):27–34. 10.1007/s10114-007-1009-6MathSciNetView ArticleMATHGoogle Scholar
- Ma R, Xu J: Bifurcation from interval and positive solutions of a fourth-order boundary value problem. Nonlinear Anal., Theory Methods Appl. 2010, 72(1):113–122. 10.1016/j.na.2009.06.061MathSciNetView ArticleMATHGoogle Scholar
- Bai Z, Wang H: On positive solutions of some nonlinear fourth-order beam equations. J. Math. Anal. Appl. 2006, 270(1):357–368. 10.1016/S0022-247X(02)00071-9MathSciNetMATHGoogle Scholar
- Ma R, Gao CH, Han XL: On linear and nonlinear fourth-order eigenvalue problems with indefinite weight. Nonlinear Anal., Theory Methods Appl. 2011, 74(18):6965–6969. 10.1016/j.na.2011.07.017MathSciNetView ArticleMATHGoogle Scholar
- Ma R, Gao CH: Nodal solutions of a nonlinear eigenvalue problem of the Euler-Bernoulli equation. J. Math. Anal. Appl. 2012, 387(2):1160–1166. 10.1016/j.jmaa.2011.10.019MathSciNetView ArticleMATHGoogle Scholar
- Ma R, Chen TL: Existence of positive solutions of fourth-order problems with integral boundary conditions. Bound. Value Probl. 2011., 2011: Article ID 297578Google Scholar
- Ma R, Xu L: Existence of positive solutions of a nonlinear fourth-order boundary value problem. Appl. Math. Lett. 2010, 23(5):537–543. 10.1016/j.aml.2010.01.007MathSciNetView ArticleMATHGoogle Scholar
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