ASYMPTOTIC PROPERTY OF SOLUTIONS OF A CLASS OF THIRD-ORDER DIFFERENTIAL EQUATIONS N. PARHI AND P. DAS

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1 proceedings of the american mathematical society Volume 110, Number 2, October 1990 ASYMPTOTIC PROPERTY OF SOLUTIONS OF A CLASS OF THIRD-ORDER DIFFERENTIAL EQUATIONS N. PARHI AND P. DAS (Communicated by Kenneth R. Meyer) Abstract. It has been shown that the equation (*) y" + a(t)y" + b(t)y' + c(t)y = 0, where a, b, and c are real-valued continuous functions on [a, oo) such that a(t) > 0, b(t) < 0, and c(t) > 0, admits at most one solution y(t) (neglecting linear dependence) with the property y(t)y (t) < 0, y(t)y (/) > 0 for t 6 [a, oo) and lim/_>00y(t) = 0, if (*) has an oscillatory solution. Further, sufficient conditions have been obtained so that (*) admits an oscillatory solution. 1. In this paper, we consider (1) y'" + a(t)y" + b(t)y + c(t)y = D, where a, b, and c e C([a, oo), R), a e R, such that a(t) > 0, b(t) < 0, and c(t) > 0 for t e [a, oo). A solution of ( 1 ) is said to be oscillatory if it has arbitrarily large zeros and is said to be nonoscillatory otherwise. The purpose of this work is to show that ( 1 ) admits at most one solution y(t) (neglecting linear dependence) with the property y(t)y'(t) < 0, y(t)y"(t) > 0 for t e[a, œ) and limi_>oo>'(/) = 0, if (1) has an oscillatory solution. Jones has obtained this result for (1) with a(t) = 0 (see [3, 4]). In his work [1], Erbe has partially generalized the work of Jones to (1). Besides other conditions, he has assumed that a(t) < 0. Moreover, our conditions are comparatively simpler than those of Erbe. We are thankful to Prof. M. Gregus for bringing this problem to our attention. Suppose that a(t), b(t), and c(t) in (1) are constants a, b, and c, respectively. It is well known that ( 1 ) admits an oscillatory solution if 2a ab 2 (a,\ it-t+'-ïtht-*) >0- Received by the editors May 13, 1989 and, in revised form, August 21, Mathematics Subject Classification (1985 Revision). Primary 34C10, 34C11. The work of the second author was done under a scheme supported by the University Grants Commission, New Delhi under grant No. F.8-9/87 (SR-III) American Mathematical Society /90 $1.00+ $.25 per page

2 388 N. PARHI AND P. DAS In this paper we have obtained a result which generalizes the above result to variable coefficients. Lazer's result [5, Theorem 1.3] follows from our result if a(t) = This section deals with asymptotic behavior of nonoscillatory solutions of (1) in the presence of an oscillatory solution. Following two lemmas are easy generalizations of Lazer's results (see [5]). Lemma 1. If y(t) is a solution of (1) satisfying y(ß) > 0, y'(ß) < 0, y"(ß) > 0, ß e (a, oc), then y(t) > 0, y'(t) < 0, y"(t) > 0, and y'"(t) < 0 for te[a,ß). Lemma 2. Equation (1) admits a nonoscillatory solution y(t) suchthat y(t) > 0, y'(t) < 0, y"(t) > 0, y'"(t) < 0 for t e [a, oo) and lim^^/m = 0 = \imt_tooy"(t), \iml^ooy(t) = k ±oo. In the following we state a lemma due to Erbe [ 1 ] for our use in the sequel. Lemma 3. A necessary and sufficient condition for (I) to have an oscillatory solution is that for any nontrivial nonoscillatory solution u(t) of (I) the following conditions hold: u(t)u (t)u"(t)u'"(t) t 0, for t >a and lim u'(t) - lim u"(t) = 0, t >oo l *oo i»oo sgn u(t) = sgn u"(t) ^ sgn u'(t) - sgn u'"(t) lim u(t) - k / ±oo. Theorem 4. Suppose that (1) has an oscillatory solution and lim^^ t b(t) ^ 0. // u(t) is a nonoscillatory solution of (I), then lim/^oo u(t) = 0. Proof. Without any loss of generality, we may assume that u(t) > 0 for t > a. From Lemma 3 we obtain u(t) < 0, u"(t) > 0, u'"(t) < 0 for t > a and l^tnl^oou(t) = 0 = liml^oou"(t), lim;_>oo«(z) = k, where 0 < k < oo. If possible, let k ^ 0. Further, without any loss of generality we may assume that k - j. Since u'(t) < 0, t > a, it is possible to find t0 > a such that {- < u(t) < 1 for t > t0. Equation ( 1 ) may be written as (2) (r(t)u"(t))'+q(t)u(t)+p(t)u(t) = 0, where r(t) = exp(f[a(s)ds), p(t) = c(t)r(t), and q(t) = b(t)r(t). If v(t) is the oscillatory solution of (1), then W-^zz, v)(t) = u(t)v'(t) - u(t)v(t) is an oscillatory solution of (3) (r(t)x'\, (r(t)u"(t) + q(t)u(t)\ n u(t) I \ u2(t) Proceeding as in Jones [3], one may get lim^^ / u (?) = 0. Since r(t) is monotonie increasing, we have r(t) > r(t) > r(a) = 1, for t > t0. u(t).2

3 Further, implies that ASYMPTOTIC PROPERTY OF THIRD-ORDER DIFFERENTIAL EQUATIONS 389 lim t2[u"(t) + b(t)u(t)]<0 t»oo lim lin t2[r(t)u"(t) + q(t)u(t)] = lim t2r(t)[u"(t) + b(t)u(t)] < 0. í»o t roo Hence the equation II Z + t»oo r(t)u"(t) + q(t)u(t) u2(t) z = 0 is nonoscillatory. From Sturm's comparison theorem it follows that (3) is nonoscillatory, a contradiction. Hence the theorem is proved. Remark. The condition lim^^ t b(t) ± 0 in Theorem 4 may be replaced by the assumption r(t) is bounded. Theorem 5. Suppose that ( 1 ) has an oscillatory solution and fq c(t) dt = oo. If u(t) is a nonoscillatory solution of (I), then lim^^ u(t) = 0. Proof. Let u(t) > 0 for t > a. From Lemma 3, we get u'(t) < 0, u"(t) > 0, u'"(t) < 0 for t > a and limi^oo u'(t) = 0 = lim;^oo u"(t), lim^^ u(t) = k, where 0 < k < oo. If possible, let k > 0. Now integrating (2) from a to t, we obtain r(t)u"(t) - u"(a) - q(s)u'(s)ds- p(s)u(s)ds < u"(a) - u(t) / r(s)c(s) ds < u (a) - u(t) / c(s)ds il, This in turn implies that u (t) < 0 for large /, a contradiction. This completes the proof of the theorem. Remark. From Lemma 2 it follows that if the hypotheses of either Theorem 4 or 5 are satisfied, then equation (1) admits a nonoscillatory solution which tends to zero. From the next theorem it follows that every other nonoscillatory solution of (1) is a constant multiple of this solution. Theorem 6. Suppose that ( 1 ) admits an oscillatory solution. If y(t) is a nonoscillatory solution of (I) with lim ^ooy(t) = 0, then every nonoscillatory solution of (I) is a constant multiple of y(t). Proof. Equation ( 1 ) may be written as (4) (r(t)y")' + q(t)y'+p(t)y = 0, where r(t) = exp(f[ a(s)ds), p(t) = c(t)r(t), and q(t) = b(t)r(t). Let ux(t) and u2(t) be two solutions of (4) on [a, oo) with initial conditions ux(a) = ux(a) = 0, r(a)ux(a) - I

4 390 N. PARHI AND P. DAS and u) = u7(a) = 0, u2(a) = A From Lemma 1 of this paper and Theorem 3.4 due to Hanan [2], it follows that both ux(t) and u2(t) are oscillatory solutions of (4). Clearly, Wx(t) = W(ux, u2)(f) = ux(t)u'2(t) - u\(t)u2(t) is a solution of the adjoint equation (5) (r(t)y')" + (q(t)y)' p(t)y = 0 with Wx(a) = W'x(a) = 0 and (rw[)'(a) > 0. It is easy to see that Wx(t) > 0 for / > a. Indeed, (rw'x)'(a) > 0 and (rw[)'(t) continuous imply that (rw'x)'(t) > 0, t e[a, a + S), for some ô > 0. This in turn implies that W[(t) > 0 for t e (a, a + ô) and hence Wx(t) > 0 for t e (a, a + ô). We claim that Wx(t) > 0 for t > a. If not, there is a ß > a such that Wx(ß) = 0 and Wx(t) > 0 for t e (a, ß). Since Wx(t) is a solution of (5), (rw'x)'(t) + q(t)wx(t) is nondecreasing in [a, ß). Hence (rw[)'(t) > 0 for t e [a, ß]. Consequently, W[(t) > 0 for t e (a, ß], a contradiction. Hence our claim holds, Further, ux(t) and u2(t) are linearly independent oscillatory solutions of the second-order differential equation that is, (6) x WAt) ux(t) u\(t) u"x(t) + u2(t) u'2(t) u'2\t) X I X h X = 0, (rw[)'(t) + q(t)wx(t) r(t)wx2(t) x = 0. So any nontrivial linear combination of ux(t) and u2(t) is oscillatory. Clearly, {«, (t), u2(t), y(t)} forms a basis of solution space of (4). Without any loss of generality, we may assume y(t) > 0 for t>t*^>a. If possible, let w(t) be a nonoscillatory solution of (1) on [a, oo) such that w(t) and y(t) are linearly independent. So w(t) kxux(t) + k2u2(t) + k3y(t), where kx, k2, A3 are constants. k3 0 implies that w(t) is oscillatory. So /L / 0. Dividing A,, we get z(t) y(t) + c,ux(t) + c2u2(t), where z(t) = w(t)/k3, cx = kx/k3 and c2 = k2/k3. Clearly, cx and c2 cannot be equal to zero simultaneously. Since cxux(t) + c2u2(t) is oscillatory, then y(t) and z(t) are of the same sign for t > t0 > t$ > a. Setting yx(t) - -cxux(t) - c2u2(t), we obtain z(t) = y(t) - yx(t). Clearly, yx(t) is an oscillatory solution of (6) and (4). Let z, > t0 be a zero of yx(t) such that y'x(tx) > 0. Let y2(t) be a solution of (4) on [tx, oo) such that y2(tx) = y'2(tx) = 0 and y2(tx) = 1. From Hanan's result [2, Theorem 3.4] it follows that y2(t) is oscillatory. Clearly, W(t) = W(yx, y2)(t) ^ yx(t)y'2(t) - y'x(t)y2(t) isa solution of (5) with W(tx) = 0 = W'(tx) and (rw)'(tx) > 0. Hence W(t) > 0

5 ASYMPTOTIC PROPERTY OF THIRD-ORDER DIFFERENTIAL EQUATIONS 391 for t > tx. Consequently, it follows from (5) that (rw1)' + qw is increasing in [i,, oo). So, for t > tx, (7) (rw)'(t) > (rw')'(t) + q(t)w(t) > (rw')'(tx) > 0. This in turn implies that W'(t) > 0 for r > r,. Clearly, {yx(t), y2(t), y(t)} forms a basis of solution space of (4), because yx(t) and y2(t) are linearly independent oscillatory solutions of the second-order differential equation. \W{t)J V r(t)w2(t) ) So y,w y2(t) y(t) y[(t) y'2(t) y'(t) =k, r(t)y"(t) r(t)y'2\t) r(t)y"(t) a nonzero constant, that is, zc = W(t)r(t)y"(t) - W'(t)r(t)y'(t) + ((rw')'(t) + q(t)w(t))y(t). From Lemma 3 it follows that y'(t) < 0 and y"(t) > 0, t > tx. So (7) yields that k > 0 and, for t > tx, 0<((rW')'(t) + q(t)w(t))y(t)<k. Let {on} be an increasing sequence of maximum points of yx(t) such that an > tx. So, yx(cr ) > 0 and y\(on) = 0. Since z(t) > 0 for t > t0, it follows from (7) that 0<[(rW)'(cJn) + q(on)w(on)]yx(on) <[(rrv')'(an) + q(on)w(on)]y(on)<k. Further, since limn_^ooy(on) = 0, then hin^^y,^) = 0. Consequently, (9) }^[(rw')'(an) + q(on)w(on)\y2x(on) = 0. On the other hand, if then H(t) = r(t)w(t)(y'x(t))2 + ((rw')'(t) + q(t)w(t))y2(t), H'(t) = 2r(t)W(t)y'x(t)y'x\t) + (rw)'(t)(y'x(t))2 + p(t)w(t)y2(t) + 2((rW')'(t) + q(t)w(t))yx(t)y'x(t). Since yx(t) is a solution of (8), we have r(t)w(t)y''x(t) = r(t)w'(t)y'x(t)-[(rw')'(t) and hence, for t > tx, + q(t)w(t)]yx(t) H'(t) = 2z-(/)^'(0(/,(0)2 + (rw)'(t)(y\(t))2 + p(t)w(t)y2(t) > 0,

6 392 N. PARHI AND P. DAS because (rw)'(t) = r'(t)w(t) + r(t)w'(t) > 0. So, H(t) is a positive increasing function. But from (9) we obtain lim^^ H(on) = 0, a contradiction. So y(t) and z(t) are linearly dependent. Consequently, y(t) and w(t) are linearly dependent. Hence the theorem is proved. 3. In this section we obtain sufficient conditions for the existence of an oscillatory solution of (1). The following lemma due to Erbe [1] is needed in the next theorem. Lemma 7. // y(t) is a nonoscillatory solution of (I), then there exists a t0 e [a, oo) such that either (10) or (11) y(t)y (t) < 0 far t > t. y(t)y'(t) > 0 far t > t. Further, if (10) holds, then y(t)y'(t)y"(t) 0, sgny(t) = sgny"(t) / sgny'(t), for t>a and lim,^ y'(t) = lim,^ y"(t) = 0, lirn,^ y(t) = k ±oo. Theorem 8. Suppose that a(t) > 0 and a'(t) < 0. If f 2a\t) a(t)b(t) c(t) then ( 1 ) admits an oscillatory solution. 3^ f 0 a2(t) b{t) 3/2' dt = oo, Proof. Let y(t) be a nonoscillatory solution of (1). From Lemma 7 it follows that there exists a t0 e [a, oo) such that (10) holds or (11) holds. In view of Lemma 3 and the second part of Lemma 7, it is sufficient to prove that (11) does not hold. Suppose that (11) holds. Setting u(t) = y'(t)/y(t) > 0, t > t0, we see that u(t) is a solution of the second-order Riccati equation :i2) z +3zz +a(t)z =-F(u(t), t), where F(u(t), t) = u\t) + a(t)u2(t) + b(t)u(t) + c(t). Clearly, F(u(t),t) attains minimum value for u(t) > 0 at u(t) = i[-a(t) + (a2(t)-3b(t))1/2]. So u (t) + 3u(t)u (t) + a(t)u (t) < - 2a\t) 27 a(t)b(t) + c(t)- 3^ a(t)-3b(t) 3/2'

7 ASYMPTOTIC PROPERTY OF THIRD-ORDER DIFFERENTIAL EQUATIONS 393 Integrating the above inequality from tn to /, we obtain '» v 3,2, u (t) < u (t0) + w (z0) + a(í0)u(í0) IJ t 2a\t) a(t)b(t) 2 /V(i) - 3ft(/) V^ 27 3 w 3\/3 dt. This in turn implies that limi_>00 u (t) = -oo. contradiction. This completes the proof of the theorem. Hence w(?) < 0 for large t, a Each of the following three examples admits only one nonoscillatory solution (neglecting linear dependence) which goes to zero as t goes to infinity. Example 1. Consider III 3 II I t n *^1 y +-y -^y + ey = o, t>\. t This example satisfies conditions of Theorems 5, 6, and 8. Example 2. Consider t m I h I i t.,.. y + -*y?y+ey = o, t>\. r r This example satisfies conditions of Theorems 5 and 8. However, Theorem 6 cannot be applied to this example. Example 3. Consider /. i \ '" 1 " 1/1. 13 y +-^73^ -7173^ +-y = 0, t>l. By Theorem 8, Equation (13) admits an oscillatory solution. satisfies conditions of Theorem 6 but not of Theorem 5. This example ACKNOWLEDGMENT The authors are thankful to the referee for many helpful remarks. References 1. L. Erbe, Existence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations. Pacific J. Math. 64 (1976), M. Hanan, Oscillation criteria for third order linear differential equations, Pacific J. Math. 11 (1961), G. D. Jones, An asymptotic property of solutions of y ' + py + qy = 0, Pacific J. Math. 48 (1973), _, Oscillatory behavior of third order differential equations, Proc. Amer. Math. Soc. 43 (1974), A. C. Lazer, The behaviour of solutions of the differential equation y +p(x)y +q(x)y = 0, Pacific J. Math. 17 (1966), C. A. Swanson, Comparison and oscillation theory of linear differential equations. Academic Press, New York, Department of Mathematics, Berhampur University, Berhampur , India