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On nonlinear oscillations for a second order delay equation
JOURNAL
OF MATHEMATICAL
ANALYSIS
AND
APPLICATIONS
26, 385-389 (1969)
On Nonlinear Oscillations for a Second Order Delay Equation H. E. GOLLWITZER Department
of Mathematics, The University Knoxville, Tennessee 37916
of Tennessee
Submitted by Richard Bellman
This paper is concerned with certain qualitative features of the second order nonlinear differential delay equation r”(t) + 4WYTW = 0
(1)
and q(t) is continuous and eventually nonwhere y7(t)Y = {y(t - I)>’ negative on some half-line [a, co). The constant y is the ratio of odd integers and satisfies 0 < y < 1 or 1 < y. The delay I is assumed, for convenience, to be a positive, continuous function on [a, co) which, for some constant M, satisfies 0 < 7(t) < M, - M < t - 9-(t). Our main purpose is to give necessary and sufficient conditions for all solutions of equation (1) to be oscillatory. We begin with some definitions and related results. The existence and uniqueness properties of solutions to equation (1) can be found in Chapter 1 of the book by El’sgol’ts [3]. Let b > a. Briefly, a solution of equation (1) for t > b is uniquely determined by a continuous initial function y = (b(t) on an appropriate interval I C [b - &I, 61 and an initial value y’(b) = B. The solution y(t) with this initial data can then be found by using the step method. We call a solution y(t) of equation (1) oscillatory if it has no last zero, i.e., if y(t,,) = 0 for some to then there is some t, > to for which y(t,) = 0. We call a solution y(t) of equation (1) nonoscillatory if it is eventually of constant sign. Equation (1) is called oscillatory if every solution is oscillatory. If T(t) 3 0 equation (1) reduces to the ordinary differential equation r”(t) + 4@)YW = 0.
(2)
This well known equation has been the subject of numerous studies [l], [4], [7], [8]. It is then of interest to study equation (1) as one generalization of equation (2). If y > 1, Atkinson [I] has shown that equation (2) is oscillatory if and only if J” sp = co. If y < 1, LiEko and Svec [7] have shown that equation (2) 385
386
GOLLV'ITZER
is oscillatory if and only if s” ~g = CO.For similar and related results XC PI, [61 and [81. We can now state our major results. THEOREM
1. Let y > 1. Equation (1) is oscillatory if and only if -a
sq(s)ds = co.
(3)
i
THEOREM 2. Let 0 < y < 1. Equation (1) is oscillatory if and only ;f
s
50 syq(s)ds = co.
(4)
REMARK. Although we have restricted the delay T(t) to be positive it is clear that the theorems are true if T(t) satisfies 0 < 7(t) < M and we consider only extendable solutions. PROOF OF THEOREM 1. In order to show that condition (3) is sufficient we assume the existence of some solution y(t) which is not zero for large t. Since -y(t) is again a solution we can assume that y(t) > 0 for large t. By taking t larger if necessary we can assume that the condition q(t) > 0 is satisfied. The following proof is an adaptation of a recent result of J. W. Heidel [S] who gave a simplified proof for the case T(t) = 0. For t sufficiently large we clearly have y”(t) < 0, y’(t) > 0 and y(t) > 0. Multiply both members of equation (1) by tyl(t)-Y and integrate,
j: sy”(s) y7(s)-’ ds + s” sq(s) ds = 0, c
(5)
where c is fixed and sufficiently large. Since y(t) is nondecreasing we see that y(t - M) < y,(t) and hence ,I sy”(s) y(s - AI-v ds + j” sq(s) ds < 0.
(6)
c
If in the first integral of equation (6) we integrate by parts, (6) can be rewritten as I1 + [sy(s - AI)-” y’(s)] 1; + y i” sy(s - M)-y-1 y’(s - M)y’ (s) ds e
+ j" q(s)ds< 0, G where II = - jty'(S)y(S c
- &fey ds a - jt Y'(S - M)~(S e
= (y - 1)~'y(s ~- AI-'+1 1:.
- M)-~ ds
(61,
ON NONLINEAR OSCILLATIONSFORA SECONDORDERDELAY EQUATION
387
Thus I1 , and consequently the left member of (6), , is bounded from below by terms which are either constant or positive. But since condition (3) holds we eventually have a contradiction and hence equation (1) is oscillatory. The necessity of condition (3) is proved by using a technique found in Atkinson [I]. Assume that j’” $4 < co. If the integral equation r(t)
= 1 - s,’ 0 - 4 cd4 YN’
ds
(8)
has a solution C$which is continuous and uniformly bounded as t approaches infinity then + will be a nonoscillatory solution of equation (1). The existence of such a solution is established by using the method of successive approximation. The details of this computation are essentially the same as those outlined in Atkinson [I] and we omit them. This completes the proof of Theorem 1. Before giving the proof of Theorem 2 we will state a special case of a lemma of Kiguradze [6]. A proof will not be given here. LEMMA 1. Let f(t) be a positive real valued function deJined on [t, , co) for someto and satisfying f ‘(t) > 0 and f “(t) < 0 on [to , 00). Then there is a number L > 0 and a t, > t, such that
f(t> f’(t)
>Lt
(9)
for t > tl . PROOF OF THEOREM 2. We first show that condition (4) is sufficient. As in the proof of Theorem 1 we will assume that y(t) is a positive solution and q(t) > 0 for large t. Since (4) holds we see that for sufficiently large t, y”(t) < 0, y’(t) > 0 and y(t) > 0. Let c be fixed and sufficiently large. If, following Heidel [5], we multiply both members of equation (1) by y:(t)+ and use the lemma we obtain r”(t) Y:(t)-’
+ q(t) [L’(t
- 7(t))‘]
< 0.
(10)
Since y’(t) is nonincreasing and positive we see that y:(t) > y’(t) for t 3 c and hence y’(t)-‘y”(t) + Lyq(t) (t - 7(t))’ < 0. (11) Integrating (11) we see that y’(t)‘-’
- y’(~)l-~ + (1 - r) LV 1” q(s) (s - ~(s))y ds < 0. G
(12)
This inequality eventually leads to a contradiction since (4) holds, T(S) is bounded and y’(t) is positive. Thus equation (1) is oscillatory.
388
GOLLWITZER
The necessity of condition (4) is easily established and is an adaptation of the proof when I = 0 [7]. Suppose that f” syq < co. It is sufficient to construct a nonoscillatory solution on some half-line [to, a). Choose t, so large that m t, syq(s) ds .: 4. , (13) J where t’ = t, - M and r(s) < M. C onsider the solution y(t) defined by the initial data y’(to) = 1;
r(t) = 0,
which is
t < t, .
(14)
We claim that this solution does not vanish on [to , co). Ify(tJ = 0 for some tl > t, then, by Rolle’s theorem, there must be some point .$E (t, , tJ for which y’(f) = 0. However, this will be in contradiction to the following fact: the function y’(t) can never vanish on [to , tl). Since -y”(t) is nonnegative on (to , t r ) we see, after two integrations, that y(t) G 0 - to),
t, < t < t, .
(15)
From equation (l), y’(t) = 1 - s” q(s) y,(s)” ds 3 1 - irn (s - T(S) - to)’ q(s) ds 3 ; . to to
(16)
Hence y’(t) never vanishes and the proof of Theorem 2 is complete. The previous theorems permit us to give necessary and sufficient conditions for all solutions of the equation r”(t) + 41(t) Y,W + %2(t)Y&P = 0
(17)
to be oscillatory. The functions qi(t), i = 1, 2 are continuous and eventually nonnegative on some half line [a, co) and the delays T(t), u(t) have the same properties as before, y and oi are the ratios of odd integers and satisfy O
=03. sm(%+sq*)
(18)
PROOF. The sufficiency of condition (18) is easily established when we observe that if y(t) is a positive nonoscillatory solution then, from equation (17), either r”(t) + mYAt)’ G 0 (19)
ON NONLINEAR
OSCILLATIONS FOR A SECOND ORDER DELAY EQUATION
389
or r”(t) + 42WYoW G 0
(20)
holds if t is sufficiently large. If j” sq2 = co we can use (20) and the argument of Theorem 1 to prove that y(t) is oscillatory. If s” svql = co we can use (19) and the argument of Theorem 2 to prove that y(t) is oscillatory. We omit the obvious details. The necessity of condition (18) is established by constructing a nonoscillatory solution of equation (17) in much the same manner as in [l], [4]. Suppose that s” (syql + sqJ < 00. It is not too difficult to show that there is a t, 3 a and a solution +(t) of the integral equation r(t) = 1 - j-p (s - t> {q,(s) rN’
+ qs(4 y&N
ds
defined on [to, co) such that 4 < + < 1 for t > t, and lim,,,+(t) = 1. The details of this computation are similar to those found in [I], [4] and we omit them. Since 4(t) is a solution of equation (17) we see that the corollary is complete. REMARK. It is noteworthy that the corollary is true if 7 = (T= 0 even though solutions of equation (17) may not be extendable or unique.
bFFXENCF.S
1. F. V. ATKINSON. On second-order
nonlinear
oscillations.
643-647. 2. C. V. COFFMAN AND D. F. ULLRICH. On the continuation 3. 4. 5.
6.
7. 8.
Pac$c
J. Math.
5 (1955),
of solutions of a certain nonlinear differential equation. Monatsheftefiir Math. 71 (1967), 385-392. L. E. EL’SGOL’TS. “Introduction to the Theory of Differential Equations with Deviating Arguments.” Holden-Day, San Francisco, 1966. J. W. HEIDEL. A nonoscillation theorem for a nonlinear second-order differential equation. PYOC.Amer. Math. Sot. To appear. J. W. HEIDFZL. The J. Barrett Seminar, University of Tennessee. To appear. I. T. KIGURADZE. Oscillation properties of solutions of certain ordinary differential equations. (Russian). Dokl. Aknd. Nuuk SSSR. 144 (1962), 33-36. [Translated as Soviet Math. Dokl. 3 (1962), 649-652).] IMRICH LIEKO, AND MARKO SVEC. Le caracttre oscillatoire des solutions de 1’Cquation ~(“1 +f(x)ya = 0, n > 1. Czech. Math. J. 13 (1963), 481-491. W. R. UTZ. Properties of Solutions of U” + g(t)@+’ = 0, II. Monatskefte fiir Math. 69 (1965), 353-361.
OF MATHEMATICAL
ANALYSIS
AND
APPLICATIONS
26, 385-389 (1969)
On Nonlinear Oscillations for a Second Order Delay Equation H. E. GOLLWITZER Department
of Mathematics, The University Knoxville, Tennessee 37916
of Tennessee
Submitted by Richard Bellman
This paper is concerned with certain qualitative features of the second order nonlinear differential delay equation r”(t) + 4WYTW = 0
(1)
and q(t) is continuous and eventually nonwhere y7(t)Y = {y(t - I)>’ negative on some half-line [a, co). The constant y is the ratio of odd integers and satisfies 0 < y < 1 or 1 < y. The delay I is assumed, for convenience, to be a positive, continuous function on [a, co) which, for some constant M, satisfies 0 < 7(t) < M, - M < t - 9-(t). Our main purpose is to give necessary and sufficient conditions for all solutions of equation (1) to be oscillatory. We begin with some definitions and related results. The existence and uniqueness properties of solutions to equation (1) can be found in Chapter 1 of the book by El’sgol’ts [3]. Let b > a. Briefly, a solution of equation (1) for t > b is uniquely determined by a continuous initial function y = (b(t) on an appropriate interval I C [b - &I, 61 and an initial value y’(b) = B. The solution y(t) with this initial data can then be found by using the step method. We call a solution y(t) of equation (1) oscillatory if it has no last zero, i.e., if y(t,,) = 0 for some to then there is some t, > to for which y(t,) = 0. We call a solution y(t) of equation (1) nonoscillatory if it is eventually of constant sign. Equation (1) is called oscillatory if every solution is oscillatory. If T(t) 3 0 equation (1) reduces to the ordinary differential equation r”(t) + 4@)YW = 0.
(2)
This well known equation has been the subject of numerous studies [l], [4], [7], [8]. It is then of interest to study equation (1) as one generalization of equation (2). If y > 1, Atkinson [I] has shown that equation (2) is oscillatory if and only if J” sp = co. If y < 1, LiEko and Svec [7] have shown that equation (2) 385
386
GOLLV'ITZER
is oscillatory if and only if s” ~g = CO.For similar and related results XC PI, [61 and [81. We can now state our major results. THEOREM
1. Let y > 1. Equation (1) is oscillatory if and only if -a
sq(s)ds = co.
(3)
i
THEOREM 2. Let 0 < y < 1. Equation (1) is oscillatory if and only ;f
s
50 syq(s)ds = co.
(4)
REMARK. Although we have restricted the delay T(t) to be positive it is clear that the theorems are true if T(t) satisfies 0 < 7(t) < M and we consider only extendable solutions. PROOF OF THEOREM 1. In order to show that condition (3) is sufficient we assume the existence of some solution y(t) which is not zero for large t. Since -y(t) is again a solution we can assume that y(t) > 0 for large t. By taking t larger if necessary we can assume that the condition q(t) > 0 is satisfied. The following proof is an adaptation of a recent result of J. W. Heidel [S] who gave a simplified proof for the case T(t) = 0. For t sufficiently large we clearly have y”(t) < 0, y’(t) > 0 and y(t) > 0. Multiply both members of equation (1) by tyl(t)-Y and integrate,
j: sy”(s) y7(s)-’ ds + s” sq(s) ds = 0, c
(5)
where c is fixed and sufficiently large. Since y(t) is nondecreasing we see that y(t - M) < y,(t) and hence ,I sy”(s) y(s - AI-v ds + j” sq(s) ds < 0.
(6)
c
If in the first integral of equation (6) we integrate by parts, (6) can be rewritten as I1 + [sy(s - AI)-” y’(s)] 1; + y i” sy(s - M)-y-1 y’(s - M)y’ (s) ds e
+ j" q(s)ds< 0, G where II = - jty'(S)y(S c
- &fey ds a - jt Y'(S - M)~(S e
= (y - 1)~'y(s ~- AI-'+1 1:.
- M)-~ ds
(61,
ON NONLINEAR OSCILLATIONSFORA SECONDORDERDELAY EQUATION
387
Thus I1 , and consequently the left member of (6), , is bounded from below by terms which are either constant or positive. But since condition (3) holds we eventually have a contradiction and hence equation (1) is oscillatory. The necessity of condition (3) is proved by using a technique found in Atkinson [I]. Assume that j’” $4 < co. If the integral equation r(t)
= 1 - s,’ 0 - 4 cd4 YN’
ds
(8)
has a solution C$which is continuous and uniformly bounded as t approaches infinity then + will be a nonoscillatory solution of equation (1). The existence of such a solution is established by using the method of successive approximation. The details of this computation are essentially the same as those outlined in Atkinson [I] and we omit them. This completes the proof of Theorem 1. Before giving the proof of Theorem 2 we will state a special case of a lemma of Kiguradze [6]. A proof will not be given here. LEMMA 1. Let f(t) be a positive real valued function deJined on [t, , co) for someto and satisfying f ‘(t) > 0 and f “(t) < 0 on [to , 00). Then there is a number L > 0 and a t, > t, such that
f(t> f’(t)
>Lt
(9)
for t > tl . PROOF OF THEOREM 2. We first show that condition (4) is sufficient. As in the proof of Theorem 1 we will assume that y(t) is a positive solution and q(t) > 0 for large t. Since (4) holds we see that for sufficiently large t, y”(t) < 0, y’(t) > 0 and y(t) > 0. Let c be fixed and sufficiently large. If, following Heidel [5], we multiply both members of equation (1) by y:(t)+ and use the lemma we obtain r”(t) Y:(t)-’
+ q(t) [L’(t
- 7(t))‘]
< 0.
(10)
Since y’(t) is nonincreasing and positive we see that y:(t) > y’(t) for t 3 c and hence y’(t)-‘y”(t) + Lyq(t) (t - 7(t))’ < 0. (11) Integrating (11) we see that y’(t)‘-’
- y’(~)l-~ + (1 - r) LV 1” q(s) (s - ~(s))y ds < 0. G
(12)
This inequality eventually leads to a contradiction since (4) holds, T(S) is bounded and y’(t) is positive. Thus equation (1) is oscillatory.
388
GOLLWITZER
The necessity of condition (4) is easily established and is an adaptation of the proof when I = 0 [7]. Suppose that f” syq < co. It is sufficient to construct a nonoscillatory solution on some half-line [to, a). Choose t, so large that m t, syq(s) ds .: 4. , (13) J where t’ = t, - M and r(s) < M. C onsider the solution y(t) defined by the initial data y’(to) = 1;
r(t) = 0,
which is
t < t, .
(14)
We claim that this solution does not vanish on [to , co). Ify(tJ = 0 for some tl > t, then, by Rolle’s theorem, there must be some point .$E (t, , tJ for which y’(f) = 0. However, this will be in contradiction to the following fact: the function y’(t) can never vanish on [to , tl). Since -y”(t) is nonnegative on (to , t r ) we see, after two integrations, that y(t) G 0 - to),
t, < t < t, .
(15)
From equation (l), y’(t) = 1 - s” q(s) y,(s)” ds 3 1 - irn (s - T(S) - to)’ q(s) ds 3 ; . to to
(16)
Hence y’(t) never vanishes and the proof of Theorem 2 is complete. The previous theorems permit us to give necessary and sufficient conditions for all solutions of the equation r”(t) + 41(t) Y,W + %2(t)Y&P = 0
(17)
to be oscillatory. The functions qi(t), i = 1, 2 are continuous and eventually nonnegative on some half line [a, co) and the delays T(t), u(t) have the same properties as before, y and oi are the ratios of odd integers and satisfy O
=03. sm(%+sq*)
(18)
PROOF. The sufficiency of condition (18) is easily established when we observe that if y(t) is a positive nonoscillatory solution then, from equation (17), either r”(t) + mYAt)’ G 0 (19)
ON NONLINEAR
OSCILLATIONS FOR A SECOND ORDER DELAY EQUATION
389
or r”(t) + 42WYoW G 0
(20)
holds if t is sufficiently large. If j” sq2 = co we can use (20) and the argument of Theorem 1 to prove that y(t) is oscillatory. If s” svql = co we can use (19) and the argument of Theorem 2 to prove that y(t) is oscillatory. We omit the obvious details. The necessity of condition (18) is established by constructing a nonoscillatory solution of equation (17) in much the same manner as in [l], [4]. Suppose that s” (syql + sqJ < 00. It is not too difficult to show that there is a t, 3 a and a solution +(t) of the integral equation r(t) = 1 - j-p (s - t> {q,(s) rN’
+ qs(4 y&N
ds
defined on [to, co) such that 4 < + < 1 for t > t, and lim,,,+(t) = 1. The details of this computation are similar to those found in [I], [4] and we omit them. Since 4(t) is a solution of equation (17) we see that the corollary is complete. REMARK. It is noteworthy that the corollary is true if 7 = (T= 0 even though solutions of equation (17) may not be extendable or unique.
bFFXENCF.S
1. F. V. ATKINSON. On second-order
nonlinear
oscillations.
643-647. 2. C. V. COFFMAN AND D. F. ULLRICH. On the continuation 3. 4. 5.
6.
7. 8.
Pac$c
J. Math.
5 (1955),
of solutions of a certain nonlinear differential equation. Monatsheftefiir Math. 71 (1967), 385-392. L. E. EL’SGOL’TS. “Introduction to the Theory of Differential Equations with Deviating Arguments.” Holden-Day, San Francisco, 1966. J. W. HEIDEL. A nonoscillation theorem for a nonlinear second-order differential equation. PYOC.Amer. Math. Sot. To appear. J. W. HEIDFZL. The J. Barrett Seminar, University of Tennessee. To appear. I. T. KIGURADZE. Oscillation properties of solutions of certain ordinary differential equations. (Russian). Dokl. Aknd. Nuuk SSSR. 144 (1962), 33-36. [Translated as Soviet Math. Dokl. 3 (1962), 649-652).] IMRICH LIEKO, AND MARKO SVEC. Le caracttre oscillatoire des solutions de 1’Cquation ~(“1 +f(x)ya = 0, n > 1. Czech. Math. J. 13 (1963), 481-491. W. R. UTZ. Properties of Solutions of U” + g(t)@+’ = 0, II. Monatskefte fiir Math. 69 (1965), 353-361.