TITLE:
Estimating the Lower Bounds on the Eigenvalue of the Smallest Modulus in the Case of Mixed Boundary Conditions
AUTHORS:
Boas Chisha, Mervis Kikonko
KEYWORDS:
Eigenvalue, Eigenfunction, Fredholm Integral Operator, Greens Function, Integral Equations, Non-Definite, Right-Definite, Left-Definite, Mixed Boundary Conditions
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.6,
June
24,
2026
ABSTRACT: We derive a lower bound on the eigenvalue of the smallest modulus associated with a mixed boundary condition problem in the general case of a regular Sturm-Liouville problem. Using the Fredholm integral operator and estimates on its norm, we derive bounds for the eigenvalue of the smallest modulus under the assumption that the coefficient function
q(
x
)
and the weight function
r(
x
)
exhibit no sign restrictions. The results contribute to a broader understanding of eigenvalue problems with mixed boundary conditions in the general case of the regular Sturm-Liouville problems. We also recommend similar studies in higher order Sturm-Liouville problems.