# The extrema of the first eigenvalue of the Sturm--Liouville problem with third-type boundary conditions

## TL;DR

This paper determines the minimum and maximum possible values of the first eigenvalue for a Sturm--Liouville problem with third-type boundary conditions, considering a class of nonnegative functions with a specific integral constraint.

## Contribution

It provides the extremal bounds of the first eigenvalue for a Sturm--Liouville problem under a new class of integral constraints on the potential function.

## Key findings

- Derived explicit infimum and supremum of the first eigenvalue.
- Characterized extremal potentials achieving these bounds.
- Extended understanding of eigenvalue behavior under integral constraints.

## Abstract

We get the infima and suprema of the first eigenvalue of the problem $y'' + qy + \lambda y = 0$, $y'(0) - k_0^2 y(0) = y'(1) + k_1^2 y(1) = 0$, where $q$ belongs to the set of nonnegative summable functions on [0,1] such that $\int_0^1 q^\gamma dx = 1$, where $\gamma \in R\setminus \{0\}$.