Properties of solutions of quasi-differential equations

Conjugate points and simple zeros for ordinary linear differential equations

“…This follows from Theorem 6 of [5] after the change of variables r= -x. Theorem 2. r¡y(a) is a continuous function of a.…”

“…Hence </>2 has no other zeros on [a, T)y(d)). \f4>2 did not vanish at 771(a) then, by Theorem 1 of [5], there is a linear combination of <j>y and <f>2 with (ii) Yfcí has a zero at a of order at least n -j. These properties will be established by induction on k.…”

“…Using Theorem 1, it shall be shown that r¡y(a) is a continuous function of a and the coefficients in (1). First a lemma which follows from the results of [5] will be established.…”

“…In [5] it was established that r)y(t) is an increasing function of t. The techniques used there could be used to show r¡y(t) is a continuous function of t, however an easier proof is presented here using the results of §2 (see also [3]). …”

“…Let </>(x) be a solution of (1) with a zero of order ky, (1 Sky<n), at a and a zero of order k2 = n-ky atr¡y(a)<b. It is known that for some ky such a solution exists [5]. It can further be supposed that </>(x)>0 on (a, r,y(a)).…”