34C10

A Simple Proof for Sturm’s Separation Theorem

Géza Makay

Bolyai Institute, Aradi vértanúk tere 1., H-6720 Szeged, Hungary

Consider the second order linear differential equation

y′′ + f(x)y′ + g(x)y = 0,  (1)

where f, g : R → R are continuous. Sturm’s theorem says that the zeros of two linearly independent solutions of (1) separate one another. The standard proof of this theorem is based upon some properties of the Wronsky determinant (see e.g. 1, p. 124 ). In this note we present a proof using only the elementary calculus.

Theorem A. Let y_{1}, y_{2} be linearly independent solutions of equation (1). If ξ, η (ξ < η) are succesive zeros of y_{1}, then y_{2} has one and only one zero in the interval (ξ, η).

Proof. Suppose the contrary. Then we can assume, without loss of generality, that y_{1}(x) > 0 and y_{2}(x) > 0 for all x ∈ (ξ, η) (if y is a solution, then −y is also a solution). Since y_{1}, y_{2} are linearly independent solutions, y_{2}(x) > 0 for all x ∈ ξ, η , too. Define the set

C = {c ∈ R : y_{1}(x) ≤ cy_{2}(x) for every x ∈ ξ, η }

Since y_{1} is bounded above and y_{2} is bounded away from zero, C is not empty. Let c_{0} = inf C.It is easy to see that y_{1}(x) ≤ c_{0}y_{2}(x) on the interval ξ, η and there is an x_{0} ∈ (ξ, η) with y_{1}(x_{0}) = c_{0}y_{2}(x_{0}). Obviously, y′_{1}(x_{0}) = c_{0}y′_{2}(x_{0}); therefore, solutions y_{1}, c_{0}y_{2} satisfy the same initial conditions at x_{0}. By uniqueness we obtain y_{1} = c_{0}y_{2}, a contradiction to the linear independence of y_{1}, y_{2}.

Since the role of y_{1} and y_{2} can be changed, the zero of y_{2} on (ξ, η) is unique.

This proof works also for some nonlinear equations.

Theorem B. Suppose that the second order equation

F (y′′, y′, y, x) = 0  (F : R^{4} → R continuous)  (2)

satisfies the following conditions:

(a) if y is a solution of (2), then cy is also a solution for all c ∈ R;

(b) the solutions of initial value problems for (2) are unique.

Then Theorem A is true for the equation (2).

Example: Consider the second order nonlinear equation

y′′(y′)^{2} + y^{3} = 0.  (3)

Let the function S_{0} be defined by

x = ∫_{0}^{S_{0}(x)} \frac{ds}{\sqrt[4]{1 − s^{4}}}

for x ∈ [0, π̂/2], where

π̂ := \frac{π/2}{sin(π/4)}.

It can be seen 3 that the 2π̂-periodic function

S(x) := \{ \begin{matrix}S_{0}(x) & , if 0 ≤ x < π̂/2 \\ S_{0}(π̂ − x) & , if π̂/2 ≤ x < π̂ \\ −S_{0}(x − π̂) & , if π̂ ≤ x < 3π̂/2 \\ −S_{0}(2π̂ − x) & , if 3π̂/2 ≤ x < 2π̂ \end{matrix}

is the solution of equation (3) satisfying the initial conditions S(0) = 0, S′(0) = 1. The function S can be considered as a generalization of function sin. Since the equation is autonomous and it satisfies conditions (a) and (b) in Theorem B (see 2 ), every solution is of the form

y(x) = cS(x + α) (c, α ∈ R).

According to Theorem B, the zeros of two linearly independent solutions of equation (3) separate one another.

REFERENCES

1. E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen, Akademische Verlagsgesellschaft, Becker & Erler KOM.-GES., Leipzig, 1943.

2. Á. Elbert, On the half-linear second order differential equations, Acta Math. Hungar., 49 (1987) 487-508.

3. Á. Elbert, A half-linear second order differential equation, Coll. Math. Soc. J. Bolyai, 30 (1981) 153-180.