Abstract
We consider the following boundary value problem (-1)ny(2n)=F(t,y), n≥ 1, t ∈ (0,1), y(2i)(0)=y(2i)(1)=0, 0≧i≧n-1. Criteria are developed for the existence of two and three positive solutions of the boundary value problem. In addition, for special cases we establish upper and lower bounds for these positive solutions. Several examples are also included to dwell upon the importance of the results obtained.
Similar content being viewed by others
References
R. P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific (Singapore, 1986).
R. P. Agarwal, Focal Boundary Value Problems for Differential and Difference Equations, Kluwer Academic Publishers (Dordrecht, 1998).
R. P. Agarwal and G. Akrivis, Boundary value problems occurring in plate deflection theory, J. Comp. Appl. Math., 8 (1982), 145–154.
R. P. Agarwal, D. O'Regan and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers (Dordrecht, 1999).
R. P. Agarwal and P. J. Y. Wong, On Lidstone polynomials and boundary value problems, Computers Math. Applic., 17 (1989), 1397–1421.
R. P. Agarwal and P. J. Y. Wong, Quasilinearization and approximate quasilinearization for Lidstone boundary value problems, Intern. J. Computer Math., 42 (1992), 99–116.
R. P. Agarwal and P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and their Applications, Kluwer Academic Publishers (Dordrecht, 1993).
P. Baldwin, Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global phase-integral method, Phil. Trans. R. Soc. London, A322 (1987), 281–305.
P. Baldwin, A localized instability in a Bénard layer, Applicable Analysis, 24 (1987), 117–156.
A. Boutayeb and E. H. Twizell, Finite-difference methods for twelfth-order boundary value problems, J. Comp. Appl. Math., 35 (1991), 133–138.
A. Boutayeb and E. H. Twizell, Numerical methods for the solution of special sixth-order boundary-value problems, Intern. J. Computer Math., 45 (1992), 207–233.
A. Boutayeb and E. H. Twizell, Finite difference methods for the solution of eighth-order boundary-value problems, Intern. J. Computer Math., 48 (1993), 63–75.
M. M. Chawla and C. P. Katti, Finite difference methods for two-point boundary value problems involving higher order differential equations, BIT, 19 (1979), 27–33.
J. Davis and J. Henderson, Uniqueness implies existence for fourth-order Lidstone boundary value problems, PanAmerican Math. J., 8 (1998), in press.
P. Forster, Existenzaussagen und Fehlerabschätzungen bei gewissen nichtlinearen Randwertaufgaben mit gewöhnlichen Differentialgleichungen, Numerische Mathematik, 10 (1967), 410–422.
D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press (San Diego, 1988).
M. A. Krasnosel'skii, Positive Solutions of Operator Equations, Noordhoff (Groningen, 1964).
T. H. Lamer, Analysis of a 2n-th Order Differential Equation with Lidstone Boundary Conditions, Ph.D. Thesis, Auburn University, 1997.
T. H. Lamar, Existence of positive solutions in a cone for a class of 2nth-order nonlinear boundary value problems with Lidstone boundary conditions, Applicable Analysis, to appear.
R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana University Math. J., 28 (1979), 673–688.
Mubammad Aslam Noor and S. I. Tirmizi, Numerical methods for unilateral problems, J. Comp. Appl. Math., 16 (1986), 387–395.
J. Toomre, J. R. Jahn, J. Latour and E. A. Spiegel, Stellar convection theory II: single-mode study of the second convection zone in an A-type star, Astrophys. J., 207 (1976), 545–563.
E. H. Twizell and S. I. A. Tirmizi, A sixth order multiderivative method for two beam problems, Intern. J. Numer. Methods Engg., 23 (1986), 2089–2102.
E. H. Twizell, Numerical methods for sixth-order boundary value problems, International Series of Numerical Mathematics, 86 (1988), 495–506.
E. H. Twizell and S. I. A. Tirmizi, Multiderivative methods for nonlinear beam problems, Comm. Appl. Numer. Methods. 4 (1988), 43–50.
E. H. Twizell and A. Boutayeb, Numerical methods for the solution of special and general sixth-order boundary value problems, with applications to Bénard layer eigenvalue problems, Proc. R. Soc. London, A431 (1990), 433–450.
E. H. Twizell, A. Boutayeb and K. Djidjli, Numerical methods for eighth-, tenth-, and twelfth-order eigenvalue problems arising in thermal instability, Advances in Computational Mathematics, 2 (1994), 407–436.
R. A. Usmani, Solving boundary value problems in plate deflection theory, Simulation, December (1981), 195–206.
P. J. Y. Wong and R. P. Agarwal, Eigenvalue characterization for (n, p) boundary value problems, J. Austral. Math. Soc. Ser. B, 39 (1998), 386–407.
P. J. Y. Wong and R. P. Agarwal, Eigenvalues of Lidstone boundary value problems, Applied Math. Comput., 104 (1999), 15–31.
P. J. Y. Wong and R. P. Agarwal, On eigenvalues and twin positive solutions of (n, p) boundary value problems, Functional Differential Equations, 4 (1997), 443–476.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wong, P.J.Y., Agarwal, R.P. Results and Estimates on Multiple Solutions of Lidstone Boundary Value Problems. Acta Mathematica Hungarica 86, 137–168 (2000). https://doi.org/10.1023/A:1006751703693
Issue Date:
DOI: https://doi.org/10.1023/A:1006751703693