Lower bound estimates of the eigenvalue of the smallest modulus associated with a general weighted regular Sturm-Liouville problem with mixed boundary conditions.

This study aims to obtain a lower bound on the eigenvalue of the smallest modulus in a general weighted regular Sturm-Liouville problem with mixed boundary conditions. The problem is expressed as -u′′(x) + q(x)u(x) = λr(x)u(x) over the interval [a,b] with u′(a) = u(b) = 0. In 1988 Mingarelli [31] and in 2016 Kikonko &Mingarelli [23] focused on finding lower bounds under Dirichlet boundary conditions with assumptions that q ∈ L∞(a,b) and q ∈ L1(a,b), respectively. However, to the best of our knowledge the case of mixed boundary conditions has not been covered. In this study we obtain the lower bound on the eigenvalue of the smallest modulus in the case of mixed boundary conditions. We consider two different assumptions on q, and obtain a bound in each case. To obtain the bounds we use the Fredholm integral operators alongside solutions of related Cauchy problem to compute the bounds. We then give examples to verify the results and compare them to those obtained under Dirichlet boundary conditions to show insights into the eigenvalue behavior. This study uses the ”Root Finding Analytic” package in Maple software for eigenvalue calculations and contributes to the broader understanding of Sturm-Liouville problems with mixed boundary conditions, highlighting key differences from Dirichlet cases. Finally, we raise some open questions for the future studies on general weighted regular Sturm-Liouville problems.