The Distance between Points of a Solution of a Second Order Linear Differential Equation Satisfying General Boundary Conditions

Abstract

This paper presents a method to obtain lower and upper bounds for the minimum distance between points $a$ and $b$ of the solution of the second order linear differential equation $y^{\mathrm{\prime \prime }}+q(x)y=0$ satisfying general separated boundary conditions of the type $a_{11}y(a)+a_{12}{y}^{\prime }(a)=0$ and $a_{21}y(b)+a_{22}{y}^{\prime }(b)=0$. The method is based on the recursive application of a linear operator to certain functions, a recursive application that makes these bounds converge to the exact distance between $a$ and $b$ as the recursivity index grows. The method covers conjugacy and disfocality as particular cases.