Dependence of eigenvalues of regular Sturm-Liouville operators on the boundary condition

Abstract

In this paper, we study the continuous dependence of eigenvalue of Sturm-Liouville differential operators on the boundary condition by using of implicit function theorem. The work not only provides a new and elementary proof of the above results, but also explicitly presents the expressions for derivatives of the n-th eigenvalue with respect to given parameters. Furthermore, we obtain the new results of the position and number of the generated double eigenvalues under the real coupled boundary condition.

Graphical abstract

Keywords

Regular Sturm-Liouville operator / eigenvalue / implicit function theorem

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References

[1] Bailey P B, Everitt W N, Zettl A. Regular and singular sturm-Liouville problems with coupled boundary conditions. Proc Roy Soc Edinburgh Sect A 1996; 126(3): 505–514

[2] Bailey P B, Gordon M K, Shampine L F. Automatic solution of the Sturm-Liouville problem. ACM Trans Math Software 1978; 4(3): 193–208

[3] CoddingtonE ALevinsonN. Theory of Ordinary Differential Equations. New York: McGraw-Hill, 1955

[4] EasthamM S P. The Spectral Theory of Periodic Differential Equations. Edinburgh: Scottish Academic Press, 1973

[5] Eastham M S P, Kong Q, Wu H, Zettl A. Inequalities among eigenvalues of Sturm-Liouville problems. Journal of Inequalities and Applications 1999; 3(1): 25–43

[6] EverittW NMöllerMZettlA. Discontinuous dependence of the n-th Sturm-Liouville eigenvalue. In: General Inequalities, 7 (Oberwolfach, 1995), Internat Ser Numer Math, Vol 123. Basel: Birkhäuser, 1997: 145–150

[7] Kong Q, Wu H, Zettl A. Dependence of eigenvalues on the problem. Math Nachr 1997; 188: 173–201

[8] Kong Q, Wu H, Zettl A. Geometric aspects of Sturm-Liouville problems, I. Structures on spaces of boundary conditions. Proc Roy Soc Edinburgh Sect A 2000; 130(3): 561–589

[9] Kong Q, Zettl A. Dependence of eigenvalues of Sturm-Liouville problems on the boundary. J Differential Equations 1996; 126(2): 389–407

[10] Kong Q, Zettl A. Eigenvalues of regular Sturm-Liouville problems. J Differential Equations 1996; 131(1): 1–19

[11] PöeschelJTrubowitzE. Inverse Spectral Theory. Pure and Applied Mathematics, Vol 130. Boston, MA: Academic Press, 1987

[12] RudinW. Principles of Mathematical Analysis, Third Edition. Beijing: China Machine Press, 2004

[13] Wei G, Xu Z. Inequalities among eigenvalues between indefinite and definite Sturm-Liouville problems. Acta Math Sinica (Chin Ser) 2005; 48(4): 773–780

[14] WeidmannJ. Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics, Vol 1258. Berlin: Springer-Verlag, 1987

[15] ZettlA. Sturm-Liouville Theory. Mathematical Surveys and Monographs, Vol 121. Providence, RI: AMS, 2005

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant No. 11571212). The author would like to thank Professor Guangsheng Wei for discussions and patient instructions to the paper.

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