Abstract
Some problems that appear in the classical bending theory of elastic beams can be modelled by boundary value problems for fourth-order nonlinear differential equations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
Agarwal, Ravi. Boundary Value Problems for Higher Order Differential Equations, World Scientific, Singapore, 1986.
Ambrosetti A, Rabinowitz P. Dual variational methods in critical point theory and applications. J. Funct. Anal., 1973;14:349–381.
Ambrosetti A. Critical Points and Nonlinear Variational Problems. Scuola Normale Superiore, Pisa, Preprint N 118, Nov. 1991.
Ambrosetti A, Badiale M. The dual Variational Principle and Elliptic Problems with Discontinuous Nonlinearities. J. Math. Anal. Appl., 1989;140:363–373.
Caklovic L., Li S. and Willem M. A note on Palais-Smale condition and coercivity, Difr. Int. Eqs., 1990;3:799–800.
Clarke F. A classical variational principle for Hamiltonian trajectories, Proc. Amer. Math. Soc., 1979;76:186–188.
Clarke F, Ekeland I. Hamiltonian Trajectories Having Prescribed Minimal Period. Comm. Pure Appl. Math., 1980;XXXIII:103–116.
Feireisl E. Non-zero time periodic solutions to an equation of Petrovsky type with nonlinear boundary conditions: slow oscillations of beams on elastic bearings, Ann. Sc. Norm. Sup. Pisa, 1993;20:133–146.
De Figueiredo DG. Lectures on Ekeland variational principle with applications and detours, TATA Inst. Fund. Res., Springer Verlag, Heidelberg, 1989.
Grossinho MR, Ma TF. Symmetric equilibria for a beam with a nonlinear elastic foundation, Portugal. Math. 1994; 51:375–393.
Grossinho MR, Ma TF. Existence results for a fourth order ODE with nonlinear boundary conditions, Proc. of CDE’V, Univ. Rousse, Bulgaria, ed. S. Bilchev and S. Tersian,1995:34–41.
Grossinho MR, Ma TM. Nontrivial solutions for a fourth order O.D.E with singular boundary conditions, “Proceedings of the Seventh International Colloquium on Differential Equations” ( Bainov, ed.), Bulgaria, VSP, Netherlands, 1996:123–130.
Grossinho MR, Tersian S. The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation, Nonlinear Anal., 2000;41:417–431.
Gupta C. Existence and uniqueness result for the bending of an elastic beam equation at resonance, J. Math. Anal. Appl. 1988;135:208–225.
Monteiro Marques, Manuel D P. Differential Inclusions in Nonsmooth Mechanical Problems. Basel: Birkhäuser, 1993.
Mawhin, Jean and Willem, Michael. Critical Point Theory and Hamiltonian Systems. N.Y.: Springer-Verlag, 1988.
Rabinowitz P. Minimax Methods in Critical Point Theory and Applications to Differential Equations. CBMS Reg. Conf. 65, AMS, Providence, R.I., 1986.
O’Regan D. Second and higher order systems of boundary value problems, J. Math. Anal. Appl., 1991;156:0–149.
Rockafellar RT. Convex Analysis, Princeton, N.J.: Princeton Univ. Press, 1970.
Stampacchia G. Le problème de Dirichlet pour les équations elliptiques du second ordre a coefficients discontinus, Ann. Inst. Fourier (Grenoble) 1965;15:189–258.
Sanchez L. Boundary value problems for some fourth order ordinary differential equations, Applicable Analysis, 1990;38:161–177.
Weaver W Jr, Timoshenko S and Young DH. Vibrations Problems in Engineering, 5th ed, John Wiley and Sons, New York, 1990.
Willem, Michael. Analyse Convexe et Optimisation. Louvain laNeuve: CIACO, 1987.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
do Rosário Grossinho, M., Tersian, S.A. (2001). Dual Variational Method and Applications to Boundary Value Problems. In: An Introduction to Minimax Theorems and Their Applications to Differential Equations. Nonconvex Optimization and Its Applications, vol 52. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3308-2_5
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3308-2_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4849-6
Online ISBN: 978-1-4757-3308-2
eBook Packages: Springer Book Archive