A Note on a Beam Equation with Nonlinear Boundary Conditions
© Paolamaria Pietramala. 2011
Received: 14 May 2010
Accepted: 31 July 2010
Published: 12 August 2010
We present new results on the existence of multiple positive solutions of a fourth-order differential equation subject to nonlocal and nonlinear boundary conditions that models a particular stationary state of an elastic beam with nonlinear controllers. Our results are based on classical fixed point index theory. We improve and complement previous results in the literature. This is illustrated in some examples.
that models a bar with the left end being simply supported (hinged) and the right end being sliding clamped. This problem, and its generalizations, has been studied previously by Davies and coauthors , Graef and Henderson  and Yao .
model a bar where the displacement and the bending moment at are zero, and there are relations, not necessarily linear, between the shearing force and the angular attitude at and the displacement in two other points of the beam.
involving Riemann-Stieltjes integrals.
The conditions (1.5)-(1.6) cover a variety of cases and include, as special cases when , multipoint and integral boundary conditions. These are widely studied objects in the case of fourth-order BVPs; see, for example, [5–14]. BCs of nonlinear type also have been studied before in the case of fourth-order equations; see, for example, [15–20] and references therein.
The study of positive solutions of BVPs that involve Stieltjes integrals has been done, in the case of positive measures, in [21–24]. Signed measures were used in [12, 25]; here, as in [21, 22], due to some inequalities involved in our theory, the functionals are assumed to be given by positive measures.
and gives a simple general method to avoid long technical calculations.
In our paper the approach of  is applied to BVP (1.1)–(1.6): we rewrite this BVP as a perturbed Hammerstein integral equation, and we prove the existence of multiple positive solutions under a suitable oscillatory behavior of the nonlinearity . We observe that our results are new even for the local BCs, when . We illustrate our theory with some examples. We also point out that this approach may be utilized for other sets of nonlinear BCs that have a physical interpretation, this is done in the last section.
2. The Boundary Value Problem
and we show that we can study this problem by means of a perturbation of the Hammerstein integral equation (2.2).
Our assumptions are the following:
We observe, as in , that leaves invariant and is compact. We give the proof in the Carathéodory case for completeness.
Hence we have . Moreover, the map is compact since it is a sum of two compact maps: the compactness of is well known, and since , and are continuous, the perturbation maps bounded sets into bounded subsets of a 1-dimensional space.
The proofs of the following results can be immediately deduced from the analogous results in , where the proofs involve a careful analysis of fixed point index and utilize order-preserving matrices. The only difference here is that we allow nonlinearity to be Carathéodory instead of continuous. The lines of proof are not effected and therefore the proofs are omitted.
The two lemmas above lead to the following result on existence of one or two positive solutions for the integral equation (2.20). Note that, if the nonlinearity has a suitable oscillatory behavior, it is possible to state, with the same arguments as in , a theorem on the existence of more than two positive solutions.
3. Optimal Constants and Examples
with the BCs (1.4)–(1.6).
We seek the "optimal" for which is a minimum. This type of problems has been tackled in the past, for example, in the second-order case for heat-flow problems in  and for beam equations (under different BCs) in [9, 12, 13].
These results are new and improve and complement the previous ones. Gupta  and Yao  studied the problem with more general nonlinearity but established existence results only. Davies and co-authors  and Graef and Henderson  obtain the existence of multiple positive solutions for a -order differential equation subject to our boundary conditions in the case . In  the choice gives the values and which replace our constants and in the growth conditions of . The growth conditions of the nonlinearity in Theorem in  cannot be compared with ours, but we do not require the restriction .
and conditions (2.34) and (2.33) read and . Since holds, from Theorem 2.4 it follows that this BVP has a nontrivial solution in . A nonlinearity that easily verifies , for example, is the function for every and every .
We now give an example with continuously distributed positive measures.
4. Other Nonlinear BCs
As in , where a different set of BCs were investigated, we point out that these nonlocal boundary conditions can be interpreted as feedback controls: for example, the BCs (4.1) can be seen as a control on the displacement at the left end and a device handling the shear force at .
The author would like to thank professor Salvatore Lopez of the Faculty of Engineering, University of Calabria, for shedding some light on the physical interpretation of this problem. The author wishes to thank the anonymous referees for their constructive comments.
- Gupta CP: Existence and uniqueness theorems for the bending of an elastic beam equation. Applicable Analysis 1988, 26(4):289-304. 10.1080/00036818808839715View ArticleMathSciNetGoogle Scholar
- Davis JM, Erbe LH, Henderson J: Multiplicity of positive solutions for higher order Sturm-Liouville problems. The Rocky Mountain Journal of Mathematics 2001, 31(1):169-184. 10.1216/rmjm/1008959675View ArticleMathSciNetGoogle Scholar
- Graef JR, Henderson J:Double solutions of boundary value problems for th-order differential equations and difference equations. Computers & Mathematics with Applications 2003, 45(6–9):873-885.View ArticleMathSciNetGoogle Scholar
- Yao Q: An existence theorem for a nonlinear elastic beam equations with all order derivatives. Journal of Mathematical Study 2005, 38(1):24-28.MathSciNetGoogle Scholar
- Eggesperger M, Kosmatov N: Positive solutions of a fourth-order multi-point boundary value problem. Communications in Mathematical Analysis 2009, 6(1):22-30.MathSciNetGoogle Scholar
- Graef JR, Qian C, Yang B: A three point boundary value problem for nonlinear fourth order differential equations. Journal of Mathematical Analysis and Applications 2003, 287(1):217-233. 10.1016/S0022-247X(03)00545-6View ArticleMathSciNetGoogle Scholar
- Graef JR, Yang B: Positive solutions to a multi-point higher order boundary value problem. Journal of Mathematical Analysis and Applications 2006, 316(2):409-421. 10.1016/j.jmaa.2005.04.049View ArticleMathSciNetGoogle Scholar
- Henderson J, Ma D: Uniqueness of solutions for fourth-order nonlocal boundary value problems. Boundary Value Problems 2006, 2006:-12.Google Scholar
- Infante G, Pietramala P: A cantilever equation with nonlinear boundary conditions. Electronic Journal of Qualitative Theory of Differential Equations 2009, (15):1-14.View ArticleMathSciNetGoogle Scholar
- Karna BK, Kaufmann ER, Nobles J: Comparison of eigenvalues for a fourth-order four-point boundary value problem. Electronic Journal of Qualitative Theory of Differential Equations 2005, (15):-9.MathSciNetGoogle Scholar
- Kelevedjiev PS, Palamides PK, Popivanov NI: Another understanding of fourth-order four-point boundary-value problems. Electronic Journal of Differential Equations 2008, (47):-15.MathSciNetGoogle Scholar
- Webb JRL, Infante G: Non-local boundary value problems of arbitrary order. Journal of the London Mathematical Society 2009, 79(1):238-258.View ArticleMathSciNetGoogle Scholar
- Webb JRL, Infante G, Franco D: Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions. Proceedings of the Royal Society of Edinburgh. Section A 2008, 138(2):427-446.View ArticleMathSciNetGoogle Scholar
- Yude J, Guo Y: The existence of countably many positive solutions for nonlinear nth-order three-point boundary value problems. Boundary Value Problems 2005, 2005:-18.Google Scholar
- Alves E, Ma TF, Pelicer ML: Monotone positive solutions for a fourth order equation with nonlinear boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(9):3834-3841. 10.1016/j.na.2009.02.051View ArticleMathSciNetGoogle Scholar
- Amster P, Cárdenas Alzate PP: A shooting method for a nonlinear beam equation. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(7):2072-2078. 10.1016/j.na.2007.01.032View ArticleMathSciNetGoogle Scholar
- Cabada A, Minhós FM: Fully nonlinear fourth-order equations with functional boundary conditions. Journal of Mathematical Analysis and Applications 2008, 340(1):239-251. 10.1016/j.jmaa.2007.08.026View ArticleMathSciNetGoogle Scholar
- Cabada A, Pouso RL, Minhós FM: Extremal solutions to fourth-order functional boundary value problems including multipoint conditions. Nonlinear Analysis: Real World Applications 2009, 10(4):2157-2170. 10.1016/j.nonrwa.2008.03.026View ArticleMathSciNetGoogle Scholar
- Franco D, O'Regan D, Perán J: Fourth-order problems with nonlinear boundary conditions. Journal of Computational and Applied Mathematics 2005, 174(2):315-327. 10.1016/j.cam.2004.04.013View ArticleMathSciNetGoogle Scholar
- Minhós F: Location results: an under used tool in higher order boundary value problems. In Proceedings of the International Conference on Boundary Value Problems, 2009, Mathematical Models in Engineering, Biology, and Medicine Edited by: Cabada A, Liz E, Nieto JJ. 1124: 244-253.Google Scholar
- Infante G: Nonlocal boundary value problems with two nonlinear boundary conditions. Communications in Applied Analysis 2008, 12(3):279-288.MathSciNetGoogle Scholar
- Infante G, Pietramala P: Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(3-4):1301-1310. 10.1016/j.na.2008.11.095View ArticleMathSciNetGoogle Scholar
- Infante G, Webb JRL: Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. Proceedings of the Edinburgh Mathematical Society. Series II 2006, 49(3):637-656. 10.1017/S0013091505000532View ArticleMathSciNetGoogle Scholar
- Karakostas GL, Tsamatos PCh: Existence of multiple positive solutions for a nonlocal boundary value problem. Topological Methods in Nonlinear Analysis 2002, 19(1):109-121.MathSciNetGoogle Scholar
- Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. Journal of the London Mathematical Society. Second Series 2006, 74(3):673-693. 10.1112/S0024610706023179View ArticleMathSciNetGoogle Scholar
- Infante G: Positive solutions of differential equations with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems. Series A 2003, (supplement):432-438.Google Scholar
- Infante G: Nonzero solutions of second order problems subject to nonlinear BCs. In Dynamic Systems and Applications, Vol. 5. Dynamic, Atlanta, Ga, USA; 2008:222-226.Google Scholar
- Lan K, Webb JRL: Positive solutions of semilinear differential equations with singularities. Journal of Differential Equations 1998, 148(2):407-421. 10.1006/jdeq.1998.3475View ArticleMathSciNetGoogle Scholar
- Amann H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review 1976, 18(4):620-709. 10.1137/1018114View ArticleMathSciNetGoogle Scholar
- Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.Google Scholar
- Lan KQ: Multiple positive solutions of semilinear differential equations with singularities. Journal of the London Mathematical Society. Second Series 2001, 63(3):690-704. 10.1112/S002461070100206XView ArticleMathSciNetGoogle Scholar
- Webb JRL: Optimal constants in a nonlocal boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2005, 63(5–7):672-685.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.