二阶线性方程的振动解与非振动解

Oscillatory Solutions and Nonoscillatory Solutions for Second Order Linear Equations

本文通过引入一个积分方程,给出了方程y″(x)+P(x)y(x)=f(x)存在振动解与非振动解的充分条件;并给出了使该方程最多只有一个非振动解的条件。

Consider an ordinary differential equationy' + P(x)y(x) =f(x), (1)Where P(x),f(x) ∈C1[0, +∞). By introducing an integral equation, We can obtain the following results: Theorem 1. Assume that on [0, +∞) P(x)>0,f(x)≥0, P'(x) ≥0 and f'(x)≤0, then the solutions for Eq. (1) that satisfy the initial valuecondition (y'(0))2 -y(0)(f (0) - P(0)/2y(0) )≥0 would be oscillatory. Theorem 2. Suppose the assumption of Theorem 1 holds and if limx →+∞P(x) = +∞ or limx →+∞ f(x) = 0, thenEq. (1) has at most one nonoscillatory solution. Theorem 3. Assume that P'(x)and f'(x)≥0 on [0, +∞) , then the solutions for Eq. (1) that satisfy the initialvalue condition y(0)>0 and (y'(x))2 -y(0)(f(0) - P(0)/2 y(0)) <0 would be positiveon [0, +∞). The asymptotic behaviours of nonoscillatory solutions for Eq. (1) are also discussed.