On a Fourth Order Boundary Value Problem with Periodic and Transmission Conditions

Öz

This study investigates the asymptotic expressions of eigenvalues and eigenfunctions for a fourth-order boundary value problem subject to periodic boundary conditions. It is also examined in the problem with transmission boundary conditions at zero. At t=0, one of the transmission boundary conditions have jump discontinuity. Firstly, asymptotic formulas of fundamental solutions are found. The asymptotic formulas of the eigenvalues are computed by the aid of Rouche method. Finally corresponding eigenfunctions to these eigenvalues are presented.

Anahtar Kelimeler

• periodic boundary conditions
• Fourth order problem
• trasmission conditions

Kaynakça

1. Agarwal, R. P. (1989). On fourth order boundary value problems arising in beam analysis. Differential and Integral Equations, 2(1), 91-110.
2. Amster, P., & Mariani, M. C. (2007). Oscillating solutions of a nonlinear fourth order ordinary differential equation. Journal of Mathematical Analysis and Applications, 325, 1133-1141.
3. Cabri, O. (2019). On the Riesz basis property of the root functions of a discontinuous boundary problem. Mathematical Methods in Applied Sciences, 6733-6740.
4. Cabri, O., & Mamedov, Kh. R. (2020). Riesz basisness of root functions of a Sturm-Liouville operator with transmission conditions. Lobachevskii Journal of Mathematics, 41(1), 1-6.
5. Cabri, O., & Mamedov, Kh. R. (2020). On the Riesz basisness of root functions of a Sturm–Liouville operator with transmission conditions. Lobachevskii Journal of Mathematics, 41(9), 1784-1790.
6. Graef, J. R., Qian, C., & Yang, B. (2003). A three point boundary value problem for nonlinear fourth order differential equations. Journal of Mathematical Analysis and Applications, 287(1), 217-233.
7. Gupta, C. (1988). Solvability of a fourth order boundary value problem with periodic boundary conditions. International Journal of Mathematics and Mathematical Sciences, 11(2), 275-284.
8. Kerimov, N. B., & Kaya, U. (2013). Spectral asymptotics and basis properties of fourth order differential operators with regular boundary conditions. Mathematical Methods in the Applied Sciences, 36. https://doi.org/10.1002/mma.2827.

9. Korman, P. (1989). A maximum principle for fourth-order ordinary differential equations. Applied Analysis, 33, 267-373.
10. Li, Y., & Wang, D. (2023). An existence result of positive solutions for the bending elastic beam equations. Symmetry, 15(2), 405. DOI:10.3390/sym15020405.
11. Menken, H. (2010). Accurate asymptotic formulas for eigenvalues and eigenfunctions of a boundary value problem of fourth order. Boundary Value Problems.
12. Muravei, L. A. (1967). Riesz bases in L2(−1; 1). Proceedings of the Steklov Institute of Mathematics, 91, 113-131. Naimark, M. A. (1967). Linear differential operators, Part I. New York: Frederick Ungar.
13. Yao, Q. (2004). Positive solutions for eigenvalue problems of fourth-order elastic beam equations. Applied Mathematics Letters, 17, 237-243.