A REPRESENTATION THEOREM FOR THE LINEAR QUASI-DIFFERENTIAL EQUATION ( py ′ ) ′ + qy = 0

We establish a representation for q in the second-order linear quasi-differential equation (py′)′ +qy = 0. We give a number of applications, including a simple proof of Sturm’s comparison theorem.


Introduction.
We are concerned with the quasi-differential equation p(x)y + q(x)y = 0 (1.1) over a half-line [a, ∞), where 1/p, q are locally Lebesgue integrable over (a, ∞) and p is a positive function.
By the term solution we mean a nontrivial real valued function φ that satisfies (1.1) almost everywhere in (a, ∞) and φ and pφ are locally absolutely continuous over [a, ∞).For a discussion of existence and uniqueness properties of the solutions of (1.1), see Naimark [3].
It is well known that there is a strong relationship between the oscillatory behaviour of the solutions of (1.1) and the existence of solutions to the corresponding Riccati equation.A great deal has been written about this connection, see, Reid [7,Chapter 4] or Willet [8].
We find the following notation convenient: let Ω a denote the space of functions positive and With the above assumptions, either a solution of (1.1) has infinitely many zeros, then every solution of (1.1) has this property, or every solution has at most a finite number of zeros.In the former case, we say that (1.1) is oscillatory, writing (p, q) is oscillatory, and in the latter case, we say that (1.1) is nonoscillatory, writing (p, q) is nonoscillatory.
In the following section, we extend the relationship between the Riccati equation and the quasi-differential equation (1.1).We then show how this extension can be used to distinguish between the mutually exclusive oscillatory behaviour of the solutions of (1.1).Finally, we see how this extension may be interpreted as a representation result, giving a simple proof to Sturm's comparison theorem and a result of Hartman, under the less stringent hypothesis of integrably (mentioned above) rather than the classical conditions of continuity.We also show how to construct oscillatory quasi-differential equations of the form (1.1) without explicitly finding the solutions of such equations.
Remark 2.2.The function ω is not unique and may be chosen so that ω(a) has any prescribed value.
The proposition fails when k = 0 then we have Wintner's result [9], which may also be regarded as a representation result.

Proposition 2.3. There exists
is given by where A and α are constants.
Proof.Define a function y by where λ is a real or a complex constant.Then Differentiating (2.6) and using pω = −2vω, we have (2.8) A straightforward calculation shows that y = c 1 θ 1 + c 2 θ gives (2.4).
The following fundamental result which is a direct consequence of Propositions 2.1 and 2.4 gives necessary and sufficient conditions for the oscillation and nonoscillation of solution of (1.1).
Corollary 2.5.Suppose ω ∈ Ω a and c > 0 be chosen, so that (2.9) Remark 2.6.Using techniques similar to those outlined in O'Hara [5] and O'Hara and Payne [6] and the above results, we can distinguish very effectively between oscillation and nonoscillation.Remark 2.7.For the sake of completeness, we briefly consider the case c = 0. On this occasion, we take φ(x) = over the same half-line as (1.1),where p 1 is a positive function and 1/p 1 ,q 1 ∈ L loc [a, ∞).
We now give an alternate proof of Sturm's comparison theorem.
Proof.By Proposition 2.4, we can find ω ∈ Ω a and a constant c > 0 such that equation (2.9) holds, and ∞ ω/p = ∞.On the contrary, suppose that (p 1 ,q 1 ) is nonoscillatory.Then, by Proposition 2.3, we can find a u ∈ Ω a such that Define a function ψ by u = v + ωψ.It follows that Then (3.4) By hypothesis, q 1 > q, hence by (2.9) and (3.4) we have since p 1 < p and ω is positive.Rearranging inequality (3.6) gives Integrating inequality (3.7), we have This leads to a contradiction since the function tan −1 is bounded and yet we know that ∞ ω/p = ∞.This completes the proof.
An immediate consequence of Theorem 3.1 is the following.

2 .Proposition 2 . 1 .
Main results.We begin by quoting a representation result from O'Hara [4,Lemma 2].There is a ω ∈ Ω a and a real number k ≠ 0 such that

3 .
Applications.Consider another quasi-differential equation of the form (1.1)

Remark 3 . 3 .Example 3 . 4 .
It is possible to construct oscillatory quasi-differential equations of the form (1.1), without having to solve the equation.Consider the differential equation