Approximability of the zeros of a function by the zeros of one of its asymptotic representations. Application to the solutions of linear second order differential equations
Given a \(C^ 1\) function f(x) defined on \(0\leq x<\infty\), which is ''regularly oscillatory'' (the zeros of f(x) and of f'(x) alternate), with bounded derivative, satisfying a condition of separation of zeros, having a differentiable asymptotic expansion, it is proved that the zeros of this asymptotic expansion approximate the zeros of f(x). This result is then applied to the solutions of differential equations of the form \((p(t)y'(t))'+q(t)y(t)=0\), satisfying \(\int^{\infty}\sqrt{q(t)/p(t)} dt=+\infty\). Several counterexamples show that the assumptions in the corresponding theorems are needed.
scientific article; zbMATH DE number 3886341