On solutions of 𝑁th order linear differential equations with 𝑁 zeros

Abstract

Left as an open question in that paper was whether the result remains true when the restriction that the interval be open is removed. The answer to that question is yes if the interval is half-open and no if it is closed. In addition, there is the problem of giving a more exact description to the solution with distinct zeros, that is, does there exist a solution with exactly n distinct zeros all of which are simple? The answer to this question is also yes.As an application of these results it is established that the first conjugate point r)i{t) of the point t is a continuous function of the coefficients in (1). It is, by the way, not difficult using the techniques in [3] to establish that rji(t) is a continuous function of /; however, an easier proof is available using the results stated herein. The results of this paper are stated in terms of equation (1)

Conjugate Points, Triangular Matrices, and Riccati Equations

“…Theorem HI. If there exists a differentiable matrix RGt on [tx, t2] such that R(tx)=I and (14) Ç2\\R-\R' +A*R)Wdt<^, then the interval (tx, t2] does not contain a point conjugate to tx with respect to equation (2).…”

“…Let A = (aik) be a 2x2 continuous matrix such that a2X =£ 0 on [tx, t2]. In order that the interval (tx, t2] contain no point conjugate to tx with respect to equation (2) it is necessary and sufficient that there exist a differentiable 2x2 matrix REt which satisfies the inequality (14) and the initial condition R(tl)=I.…”

“…In the terminology used in the theory of equation (21), such a point is called a focal point, and the term "conjugate point" is reserved for a point t2 such that (21) has a nontrival solution j> with y(tl)=y'(tl) = ---=yVc-i\tl)=y(t2)=y'(t2)=-=y(n-k-1\t2) = 0 [4], [5], [7], [8], [9], [14]. In the corresponding vector-matrix equation x' =Ax, the latter type of point is characterized by the existence of a nontrivial…”

Distribution of zeros of solutions of a fourth order differential equation

“…If (2£ 4 ) is disconjugate on [£, oo) we write rjjj) -oo. Otherwise Ύ) γ (t) will denote the first conjugate point of t. For properties of η^t) see [9], [12]. A nontrivial solution of (i<7 4 ) having four zeros on [t, η^t)] is called an extremal solution of (E 4 ) for Ύ) x {t).…”

Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity

“…One advantage of introducing generalized zeros is that the following analogue of a result [12] or [19] on differential equations becomes valid: (1.1) is disconjugate (= r-disconjugate) if and only if u{m) = 0 is the only solution of (1.1) having k (> 0) successive zeros at m = a, . .…”