On the Finite Spectrum of Three-Point Boundary Value Problems

The article is devoted to the solution of one of John Locker's problems, namely, to clarify whether boundary-value problem can have a finite spectrum. It is shown that if a differential equation does not have multiple roots of the characteristic equation, then the spectrum of the corresponding three-point boundary value problem cannot be finite. The proof of the theorem is based on the fact that the corresponding characteristic determinant is an entire function of class K and the results of V.B. Lidsky and V.A. Sadovnichy, from which it follows that the number of its roots (if any) of an entire function of class K is infinite and, if the characteristic determinant is not identical to zero, has asymptotic representations written out in the works of V.B. Lidsky and V.A. Gardener. If the roots of the characteristic equation are multiple, then the spectrum can be finite. Moreover, there are boundary value problems with a predetermined finite spectrum. The problems of determining the finite spectrum from a predetermined finite spectrum can be viewed as the inverse problems of determining the coefficients of a differential equation from a given spectrum.

Keywords

three-point boundary value problem; John Locker problem; finite spectrum; infinite or empty spectrum.

References

1. Locker J. Eigenvalues and Completeness for Regular and Simply Irregular Two-Point Differential Operators. Providence, American Mathematical Society, 2008.
2. Naimark M.A. Lineinye Differentsialnye Operatory [Linear Differential Operators]. Moscow, Nauka, 1969. (in Russian)
3. Shiryaev E.A., Shkalikov A.A. Regular and Completely Regular Differential Operators. Mathematical Notes, 2007, vol. 81, no. 4, pp. 566-570.
4. Sadovnichii V.A., Sultanaev Ya.T., Akhtyamov A.M. General Inverse Sturm-Liouville Problem with Symmetric Potential. Azerbaijan Journal of Mathematics, 2015, vol. 5, no. 2, pp. 96-108.
5. Sadovnichii V.A., Kanguzhin B.E. [On the Relation Between the Spectrum of a Differential Operator with Symmetric Coefficients and Boundary Conditions]. Doklady Akademii Nauk SSSR, 1982, vol. 267, no. 2, pp. 310–313. (in Russian)
6. Akhtyamov A.M. On the Spectrum of an Odd-Order Differential Operator. Mathematical Notes, 2017, vol. 101, no. 5, pp. 755-758.
7. Lang P. Spectral theory of Two-Point Differential Operators Determined by D^2. I. Spectral properties. Journal of Mathematical Analysis and Applications, 1989, vol. 146, no. 1, pp. 538-558.
8. Kal'menov T.Sh., Suragan D. Determination of the Structure of the Spectrum of Regular Boundary Value Problems for Differential Equations by V.A. Il'in's Method of Anti-A'Priori Estimates. Doklady Mathematics. 2008, vol. 78, no. 3, pp. 913-915.
9. Akhtyamov A.M. Finiteness of the Spectrum of Boundary Value Problems. Differential Equations, 2019, vol. 55, no. 1, pp. 142-144.
10. Zhongmin Sun. Positive Solutions of Singular Third-Order Three-Point Boundary Value Problems. Journal of Mathematical Analysis and Applications, 2005, vol. 306, pp. 589-603.
11. Wong P.J.Y. Eigenvalues of a General Class of Boundary Value Problem with Derivative-Dependent Nonlinearity. Applied Mathematics and Computation, 2015, vol. 259, pp. 908-930.
12. Zhongmin Sun, Suoquan Ren, Zhenlin Wu, Huabin Zhang. Multiple Positive Solutions to Third-Order Three-Point Singular Semipositone Boundary Value Problem. International Journal of Engineering and Manufacturing, 2004, vol. 114, pp. 409-422.
13. Anderson D., Davis J.M. Multiple Solutions and Eigenvalues for Third-Order Right Focal Boundary Value Problems. Journal of Mathematical Analysis and Applications, 2002, vol. 267, pp. 135-157.
14. Hopkins W. Some Convergent Developments Associated with Irregular Boundary Conditions. Transactions of the American Mathematical Society, 1919, vol. 20, pp. 249-259.
15. Minghe Pei, Sung Kag Chang. Solvability of $n$-th Order Lipschitz Equations with Nonlinear Three-Point Boundary Conditions. Boundary Value Problems, 2014, vol. 1, pp. 183-197.
16. Sansone G. Sopra una famiglia di cubiche con infiniti punti razionali. Rendiconti Istituto Lombardo, 1929, vol. 62, no. 2, pp. 354-360. (in Italian)
17. Lidskii V.B., Sadovnichii V.A. Regularized Sums of Zeros of a Class of Entire Functions. Konstruktivnaya Teoriya Funktsii i Funktsional'nyi Analiz, 1967, vol. 1, no. 2, pp. 52-59.
18. Lidskii V.B., Sadovnichii V.A. Asymptotic Formulas for the Roots of a Class of Entire Functions. Sbornik: Mathematics, 1968, vol. 75, no. 4, pp. 558-566.