# On two classes of third order boundary value problems with finite spectrum

Document Type: Research Paper

Author

School of Mathematical Sciences‎, ‎Inner Mongolia University‎, ‎Hohhot 010021‎, ‎China‎ and College of Sciences‎, ‎Inner Mongolia University of Technology‎, ‎Hohhot 010051‎, ‎China.

Abstract

‎The spectral analysis of two classes of third order boundary value problems is investigated‎. ‎For every positive integer $m$ we construct two classes of regular third order boundary value problems with at most $2m+1$‎ ‎eigenvalues‎, ‎counting multiplicity‎. ‎These kinds of finite spectrum results are previously known only for even order boundary value problems‎.

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Main Subjects

### References

D. Anderson and J.M. Davis, Multiple solutions and eigenvalues for third-order right focal boundary value problems, J. Math. Anal. Appl. 267 (2002), no. 1, 135--157.

J.J. Ao, F.Z. Bo and J. Sun, Fourth order boundary value problems with finite spectrum, Appl. Math. Comput. 244 (2014) 952--958.

J.J. Ao, J. Sun and A. Zettl, Equivalence of fourth order boundary value problems and matrix eigenvalue problems, Result. Math. 63 (2013), no. 1-2, 581--595.

J.J. Ao, J. Sun and A. Zettl, Finite spectrum of 2nth order boundary value problems, Appl. Math. Lett. 42 (2015) 1--8.

F.V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York/London, 1964.

B. Chanane, Accurate solutions of fourth order Sturm-Liouville problems, J. Comput. Appl. Math. 234 (2010), no. 10, 3064--3071.

W.N. Everitt and A. Poulkou, Kramer analytic kernels and first-order boundary value problems, J. Comput. Appl. Math. 148 (2002), no. 1, 29--47.

W.N. Everitt and D. Race, On necessary and sufficient conditions for the existence of Caratheodory solutions of ordinary differential equations, Quaest. Math. 2 (1977/78), no. 4, 507--512.

L. Greenberg and M. Marletta, Numerical methods for higher order Sturm-Liouville problems, J. Comput. Appl. Math. 125 (2000), no. 1-2, 367--383.

M. Greguš, Third Order Linear Differential Equations, Reidel, Dordrecht, 1987.

Q. Kong, H. Wu and A. Zettl, Sturm-Liouville problems with finite spectrum, J. Math. Anal. Appl. 263 (2001), no. 2, 748--762.

Y.Y. Wu and Z.Q. Zhao, Positive solutions for third-order boundary value problems with change of signs, Appl. Math. Comput. 218 (2011), no. 6, 2744--2749.

A. Zettl, Sturm-Liouville Theory, Math. Surveys and Monogr. 121, Amer. Math. Soc. Providence, RI, 2005.

### History

• Receive Date: 08 November 2015
• Revise Date: 02 May 2016
• Accept Date: 02 May 2016