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DEPENDENCE OF EIGENVALUES OF SIXTH-ORDER BOUNDARY VALUE PROBLEMS ON THE BOUNDARY

Part of: Special classes of linear operators Boundary value problems

Published online by Cambridge University Press:  09 September 2014

SUQIN GE ,
WANYI WANG  and
QIUXIA YANG
SUQIN GE*
Affiliation:
School of Mathematics, Physics and Biological Engineering, Inner Mongolia University of Science and Technology, BaoTou 014010, PR China email 15647280518@163.com School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, PR China
WANYI WANG
Affiliation:
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, PR China email wwy@imu.edu.cn
QIUXIA YANG
Affiliation:
College of Information and Management, Dezhou University, Dezhou 253023, PR China email yawenxuan@21cn.com
*
✉15647280518@163.com
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Abstract

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In this paper, we consider the dependence of eigenvalues of sixth-order boundary value problems on the boundary. We show that the eigenvalues depend not only continuously but also smoothly on boundary points, and that the derivative of the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$th eigenvalue as a function of an endpoint satisfies a first-order differential equation. In addition, we prove that as the length of the interval shrinks to zero all higher eigenvalues of such boundary value problems march off to plus infinity. This is also true for the first (that is, lowest) eigenvalue.

Keywords

sixth-order boundary value problemsboundary conditioneigenvalues

MSC classification

Secondary: 34B20: Weyl theory and its generalizations 34B24: Sturm-Liouville theory 47B25: Symmetric and selfadjoint operators (unbounded)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 3 , December 2014 , pp. 457 - 468
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Bailey, P. B., Everitt, W. N., Weidmann, J. and Zettl, A., ‘Regular approximations of singular Sturm–Liouville problems’, Results Math. 23 (1993), 3–22.Google Scholar
Baldwin, P., ‘Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global phase-integral methods’, Phil. Trans. R. Soc. Lond. A 322 (1987), 281–305.Google Scholar
Baldwin, P., ‘A localized instability in a Bénard layer’, Appl. Anal. 24 (1987), 117–156.CrossRefGoogle Scholar
Boutayeb, A. and Twizell, E. H., ‘Numerical methods for the solution of special sixth-order boundary value problems’, Int. J. Comput. Math. 45 (1992), 207–233.Google Scholar
Cao, X., Kong, Q., Wu, H. and Zettl, A., ‘Sturm–Liouville problems whose leading coefficient function changes sign’, Canad. J. Math. 55 (2003), 724–749.CrossRefGoogle Scholar
Chandrasekhar, S., Hydrodynamics and Hydromagnetic Stability (Dover, New York, 1981).Google Scholar
Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. 1 (Wiley-Interscience, New York, 1989).Google Scholar
Dauge, M. and Helffer, B., ‘Eigenvalues variation, I. Neumann problem for Sturm–Liouville operators’, J. Differential Equations 104 (1993), 243–262.Google Scholar
Dauge, M. and Helffer, B., ‘Eigenvalues variation, II. Multidimensional problems’, J. Differential Equations 104 (1993), 263–297.Google Scholar
Ge, S. Q., Wang, W. Y. and Suo, J. Q., ‘Dependence of eigenvalues of a class of fourth-order Sturm–Liouville problems on the boundary’, Appl. Math. Comput. 220 (2013), 268–276.Google Scholar
Glatzmaier, G. A., ‘Numerical simulations of stellar convection dynamics at the base of the convection zone’, Geophys. Fluid Dyn. 31 (1985), 137–150.Google Scholar
Kong, Q. and Zettl, A., ‘Eigenvalues of regular Sturm–Liouville problems’, J. Differential Equations 131 (1996), 1–19.CrossRefGoogle Scholar
Kong, Q. and Zettl, A., ‘Dependence of eigenvalues of Sturm–Liouville problems on the boundary’, J. Differential Equations 126 (1996), 389–407.CrossRefGoogle Scholar
Kong, Q., Wu, H. and Zettl, A., ‘Dependence of the nth Sturm–Liouville eigenvalues on the problem’, J. Differential Equations 156 (1999), 328–354.Google Scholar
Kong, Q., Wu, H. and Zettl, A., ‘Geometric aspects of Sturm–Liouville problems I. Structures on spaces of boundary conditions’, Proc. Roy. Soc. Edinburgh 130 (2000), 561–589.CrossRefGoogle Scholar
Kong, Q., Wu, H. and Zettl, A., ‘Limits of Sturm–Liouville eigenvalues when the interval shrinks to an end point’, Proc. Roy. Soc. Edinburgh 138 (2008), 323–338.Google Scholar
Scott, M. R., Shampine, L. F. and Wing, G. M., ‘Invariant imbedding and the calculation of eigenvalues for Sturm–Louville Systems’, Computing 4 (1969), 10–23.CrossRefGoogle Scholar
Scott, M. R., ‘An initial value method for the eigenvalue problem for systems of ordinary differential equations’, J. Comput. Phys. 12 (1973), 334–347.Google Scholar
Siddiqi, S. S. and Twizell, E. H., ‘Spline solutions of linear sixth-order boundary value problems’, Int. J. Comput. Math. 60 (1996), 295–304.CrossRefGoogle Scholar
Toomore, J., Zahn, J. R., Latour, J. and Spiegel, E. A., ‘Stellar convection theory II: single-mode study of the second convection zone in A-type stars’, Astrophys. J. 207 (1976), 545–563.CrossRefGoogle Scholar
Twizell, E. H. and Boutayeb, A., ‘Numerical methods for the solution of special and general sixth-order boundary-value problems, with applications to Bénard layer eigenvalue problems’, Proc. R. Soc. Lond. A 431 (1990), 433–450.Google Scholar
Wang, A. P., Sun, J. and Zettle, A., ‘The classification of self-adjoint boundary conditions: separated, coupled and mixed’, J. Funct. Anal. 255 (2008), 1554–1573.Google Scholar
Weidmann, J., Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, 1258 (Springer, Berlin, 1987).Google Scholar