The Asymptotic Behavior of a Solution of the Third order Linear Differential Equation

Abstract

We shall consider the differential equation (1) x"' + a(t)x" + b(t)x' + c(t)x = 0, where x is real and a(t), b(t), and c(t) are real valued and continuous for 0^t< oo, and we shall prove:Theorem. If A(r, t) =r3+a(t)r2+b(t)r+c(t) = 0 is the characteristic equation of (I) and if the characteristic roots are real and satisfy \i(t) <a<\2(t)<^<\z (t) for all t^O, where a and /3 are constants, then (1) has a solution of the form x(t) =x(0) exp fl^(s)ds where a<£(t) </3 for 0^t<oo.C. Olech [2] proved a similar theorem for the second order equation. As an alternate means of proof, he suggested the topological method of T. Wazewski and that approach is used here. The details of Wazewski's method may be found in [3] or [l, pp. 179-182].Proof of the theorem. Write (1) as the systemWe seek a solution (x(t), y(t), z(t)) of (2) such that x(t)>0 and a<y(t)/x(t)<0 for 0^< 00, or (x(t), y(t), z(t), t)ET={(x, y, z, t): x>0, ax<y<fix, -=0 <2< w, OgK°o } for all t^O. We will use the theorem of Wazewski: if all points of egress of T are points of strict egress and if there exists a set ZETVJS (S is the set of egress) such that ZC\S is a retract of S but not of Z, then there is at least one point P = (xQ, y0, z0, t0)EZ such that if (x(t), y(t), z(t)) is the solution of (2) with (x(t0), y(t0), z(t0), t0) =P, then (x(t), y(t), z(t), t)ET for all t^t0. Now the boundary of T is F=A\JB\JC where A = {(x, y, z, l): x > 0, y = fix, -«= < z < w, 0 ^ t < » }, B = {(x, y, z, t): x > 0, y = ax, -» < 2 < co,0gi< 00 }, and C = {(x, y, z, t): x = y = 0, -°o <z< °°,0g/< 00}.Let (x(t), y(t), z(t)) be a solution of (2) such that (x(t), y(r), z(t), t)EA. Then (y-px)(r)=0 and (y-fix)'(r) = (z-p2x)(t 391License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Principal Solutions of Disconjugate n-th Order Linear Differential Equations

“…where (10) and (11) imply that Cn (t) * x (t) =_/n (t)n (cn -c4t C ) x(t = p(), Thus (6) follows. In order to prove the existence of Cn'1, we can suppose that the basis x1 (t),-.…”

“…or is near c,. This contradicts the fact that e(t;c*) :0 for t near 13 and proves (11). In order to verify (6), note that if Cn(t) = t (t; cn), then $l(t) =cnX(t) ==c"(t) X…”

“…For n 2, see Olech [9]. If n =-3 and I = [0, oo), Schuur [11] proved the existence of solutions x2 (but not xi, x3) satisfying (11. 8) in Corollary 11.…”

“…If Sj > 0 for j =2, , n 1, that is, Case (ii). If (9) is replaced by (11) a . =* * k < ak+l < * < Xfor some , 1 < c < n -1, and if Pn(j) OiP Pn/Oxi then (5) holds provided that (-1)?L+I+lP,n(n)(t,a,) ?0 for j 0, *,k -1, (17) is written in vector-matrix notation as Y= -A (t) y, then the entries in A (t) are non-negative.…”