Integral criteria for oscillation of third order nonlinear differential equations

Nonlinear Analysis 71 (2009) e1496–e1502

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Integral criteria for oscillation of third order nonlinear differential equations M.F. Aktas a,∗ , A. Tiryaki b , A. Zafer c a

Department of Mathematics, Gazi University, Faculty of Arts and Sciences, Teknikokullar, 06500 Ankara, Turkey

b

Department of Mathematics and Computer Sciences, Izmir University, Faculty of Arts and Sciences, 35350 Uckuyular, Izmir, Turkey

c

Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey

article

abstract

info

MSC: 34K11 34C10

In this paper we are concerned with the oscillation of third order nonlinear differential equations of the form

(r2 (t )(r1 (t )y0 )0 )0 + p(t )y0 + q(t )f (y) = 0,

Keywords: Oscillatory Nonoscillatory Third order Nonlinear

t ≥ t0 ,

where t0 > 0 is a fixed real number; r1 , r2 , p, q : I := [t0 , ∞) → R are nonnegative continuous functions with r1 > 0, r2 > 0; f : R → R is a continuous function. By making use of a generalized Riccati transformation we establish some new sufficient integral conditions under which the equation has at least one oscillatory solution. Examples are given to illustrate the importance of our results. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction We are concerned with the oscillation of third order differential equations of the form

(r2 (t )(r1 (t )y0 )0 )0 + p(t )y0 + q(t )f (y) = 0,

t ∈ I := [t0 , ∞),

(1.1)

where t0 > 0 is a fixed real number; r1 , r2 , p, q : I → R are nonnegative continuous functions with r1 (t ) > 0, r2 (t ) > 0, supt ∈I {q(s), s > t } > 0; f : R → R is a continuous function that satisfies f (u)/u ≥ K for some K > 0 and for all u 6= 0. As usual, it is assumed that Eq. (1.1) has solutions which are defined and nontrivial for all sufficiently large values of t. Such a solution is called oscillatory if it has arbitrarily large zeros; otherwise it is called nonoscillatory. The equation is called oscillatory if there is at least one oscillatory solution. Note that contrary to the second order linear equations a third order linear differential equation may have both oscillatory and nonoscillatory solutions in view of the fact that there is no separation of the zeros theorem for the latter. The problem of the oscillation of solutions of differential equations has received the attention of many authors since the well known work of Sturm on second order linear equations. Although there is a large body of literature on the oscillation of second order linear and nonlinear equations, much less is known on third order differential equations. The main reason is the lack of many tools which are available in the study of second order equations. For a sampling of works done on the oscillation of third order equations we may refer to [1–18]. Our motivation stems from the results obtained in [7,13,16], where the authors, by employing the integral averaging technique with weighted averages of coefficients, study the third order linear and nonlinear equations of the form y000 + p(t )y0 + q(t )y = 0,

∗

t ≥ t0 ,

Corresponding author. E-mail addresses: [email protected] (M.F. Aktas), [email protected] (A. Tiryaki), [email protected] (A. Zafer).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.194

(1.2)

M.F. Aktas et al. / Nonlinear Analysis 71 (2009) e1496–e1502

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and y000 + p(t )y0 + q(t )f (y) = 0,

t ≥ t0 ,

(1.3)

where the functions p, q, and f are as given above. One of our objectives in this paper is to establish some integral criteria for the oscillation of Eq. (1.1) which are applicable to Eqs. (1.2) and (1.3) in the case when some known results obtained in the literature, in particular, Theorems A–C below fail to apply. With respect to Eq. (1.2), Lazer [7] defines F [y](t ) = 2y(t )y00 (t ) − y02 (t ) + p(t )y2 (t ). Theorem A ([7, Theorem 3.1]). If 2q(t ) − p0 (t ) ≥ 0 and not identically zero in any subinterval of I and there exists a number m < 1/2 such that the second order linear equation z 00 + (p(t ) + mtq(t ))z = 0 is oscillatory, then Eq. (1.2) is also oscillatory. In fact, any nonzero solution y of Eq. (1.2) satisfying F [y](t1 ) ≤ 0 for some t1 ≥ t0 is oscillatory. Theorem B ([13, Theorem 1]). Let 2q(t ) − p0 (t ) ≥ 0 and not identically zero in any subinterval of I, and t 2 p(t ) ≤ 1/4 for all t ≥ t0 . If

Z t lim sup t →∞

t0

2

3/2

t q(t ) + tp(t ) − √ (1 − t p(t )) 3 3t 2

2

 dt = ∞,

then Eq. (1.2) is oscillatory. In fact, any nonzero solution y of Eq. (1.2) satisfying F [y](t1 ) ≤ 0 for some t1 ≥ t0 is oscillatory. Theorem C ([16, Remark 3.4]). Suppose that the second order linear equation z 00 + p(t )z = 0 is nonoscillatory. If

Z t lim sup t →∞

t0



2 Kt 2 q(t ) + tp(t ) − √ (1 − t 2 p(t ))3/2 dt = ∞, 3 3t

then Eq. (1.3) is oscillatory. In fact, any nonzero solution y of Eq. (1.3) having a zero is oscillatory. Remark 1.1. If y is a solution with a zero at a point t1 , i.e., y(t1 ) = 0, then F [y](t1 ) ≤ 0 holds. Therefore, Theorems A and B imply that a nonzero solution with a zero must be oscillatory. Note that if y is a solution of Eq. (1.1), then −y is a solution of

(r2 (t )(r1 (t )y0 )0 )0 + p(t )y0 + q(t )f ∗ (y) = 0, where f ∗ (y) = −f (−y). Since f ∗ and f are of the same class, we may restrict our attention only to a positive solution of Eq. (1.1) whenever a nonoscillatory solution of Eq. (1.1) is concerned. 2. Some preliminary lemmas In this section we provide some lemmas to be needed in proving our theorems. For the sake of brevity, we denote L0 y = y ;

Li y = ri (t )(Li−1 y)0 ,

(i = 1, 2);

L3 y = (L2 y)0 .

With this notation, Eq. (1.1) takes the form L3 y + p(t )y0 + q(t )f (y) = 0. Definition 2.1. A nonoscillatory solution y of Eq. (1.1) is said to have the property V2 if and only if there exists a t1 ≥ t0 such that L0 y(t )Lk y(t ) > 0,

k = 0, 1, 2;

L0 y(t )L3 y(t ) ≤ 0

(2.4)

(L1 y(t ))2 + p(t )y2 (t ).

(2.5)

for all t ≥ t1 . Following Lazer [7] we define

F [y](t ) = 2y(t )L2 y(t ) −

r2 (t ) r1 (t )

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Denote R2 (t , s) =

t

Z

du

for t0 ≤ s ≤ t < ∞.

r2 (u)

s

Lemma 2.1. If y is a nonoscillatory solution of Eq. (1.1) which satisfies y(t )L1 y(t ) ≥ 0 for all t large, and R2 (t , t0 ) → ∞ as t → ∞,

(2.6)

then y has the property V2 . Lemma 2.2. Let (2.6) hold. Suppose that r2 /r1 ∈ C 1 (I ) such that

(r2 /r1 )0 (t ) ≥ 0 for all t ∈ I

(2.7)

2Kq(t ) − p0 (t ) ≥ 0,

(2.8)

and 2Kq(t ) − p0 (t ) 6≡ 0 eventually.

If y is a solution of Eq. (1.1) such that F [y](t1 ) ≤ 0 for some t1 ≥ t0 , then either y is oscillatory or has the property V2 . Lemma 2.3. Let g∗ ∈ C (I ) satisfy g∗ (t ) < t

lim g∗ (t ) = ∞.

and

t →∞

(2.9)

If y is a solution of Eq. (1.1) with the property V2 , then there exists a t1 ≥ t0 such that L1 y(t ) ≥ R2 (t , g∗ (t ))L2 y(t ) for all t ≥ t1 .

(2.10)

The reader is referred to [12, Lemma 1, Lemma 2] for the proofs of Lemmas 2.1 and 2.2. The proof of Lemma 2.3 is similar to [17, Lemma 3] and hence it is omitted. We also need the next two lemmas. The proof of Lemma 2.4 proceeds as in the lines of that in [17]. Lemma 2.5 is adapted from Nehari [19], where although the special case r1 (t ) = r2 (t ) ≡ 1 was considered, the result can be easily extended. Lemma 2.4. Suppose that the second order linear equation

(r2 (t )z 0 )0 +

p(t ) r1 ( t )

z=0

(2.11)

is nonoscillatory. If y is a nonoscillatory solution of (1.1) on I, then there exists a t1 ≥ t0 such that y(t )L1 y(t ) > 0 or y(t )L1 y(t ) < 0 for t ≥ t1 . Lemma 2.5. If (2.6) is valid and Eq. (2.11) is disconjugate on [t0 , ∞), then for any function v ∈ C 1 [b, c ], t0 ≤ b < c, satisfying v(b) = 0 and v(t ) 6≡ 0 on (b, c ), the strict inequality

Z c

p(s)

2

r2 (s)v 0 (s) −

b

r1 (s)

 v 2 (s) ds > 0

holds. 3. The main results Theorem 3.1. Let (2.6)–(2.9) hold. If there exists an eventually positive function ρ(t ) ∈ C 1 [t0 , ∞) such that for some T ≥ t0 ,

Z t lim sup t →∞

K ρ(s)q(s) −

T

B2 (s)



4A(s)

ds = ∞,

(3.12)

where A(t ) =

R2 (t , g∗ (t )) r1 (t )ρ(t )

,

B(t ) =

ρ 0 (t ) R2 (t , g∗ (t )) − p(t ) , ρ(t ) r1 (t )

then Eq. (1.1) is oscillatory. In fact, any solution y satisfying F [y](t1 ) ≤ 0 for some t1 ≥ t0 is oscillatory, where the functional F is given by (2.5). Proof. Let y be a solution of Eq. (1.1) which satisfies F [y](t1 ) ≤ 0 for some t1 ≥ t0 . By Lemma 2.2 either y is oscillatory or y is nonoscillatory with the property V2 for all large t. Let y have the property V2 for t ≥ t2 for some t2 ≥ t1 . We define

ω(t ) = ρ(t )

L2 y(t ) y(t )

for t ≥ t2 .

M.F. Aktas et al. / Nonlinear Analysis 71 (2009) e1496–e1502

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It follows that ω(t ) > 0. Differentiating the function ω and making use of Eq. (1.1) and Lemma 2.3, we have

ω0 (t ) ≤ −K ρ(t )q(t ) − [ω2 (t )A(t ) − ω(t )B(t )],

(3.13)

and hence by completing the square,

ω0 (t ) < −K ρ(t )q(t ) +

B2 (t ) 4A(t )

.

(3.14)

Integrating (3.14) from t2 to t leads to B2 (s)

Z t

K ρ(s)q(s) −

t2



ds ≤ ω(t2 ),

4A(s)

which contradicts (3.12). The proof is complete.



Example 3.1. Consider the third order differential equation 3t

(t (ty0 )0 )0 + (t 2 − 1)y0 +

2 + sin 2t

(y + y3 ) = 0,

t ≥ 1.

(3.15)

Taking ρ(t ) = 1 and g∗ (t ) = te−1/t we have 2

 (ρ 0 (s)r1 (s) − ρ(s)p(s)R2 (s, g∗ (s)))2 ds 4R2 (s, g∗ (s))ρ(s)r1 (s) t →∞ T  Z t 3s (s2 − 1)2 = lim sup − ds 2 + sin 2s 4s3 t →∞ T   Z t (s2 − 1)2 ≥ lim sup ds = ∞. s− 3 Z t

K ρ(s)q(s) −

lim sup

t →∞

4s

T

Thus, condition (3.12) is satisfied. It is easy to check that the other conditions of Theorem 3.1 are also satisfied. Hence, any solution y of (3.15) with F [y](t1 ) ≤ 0 is oscillatory. Indeed, y(t ) = sin t + cos t is a solution. Example 3.2. Consider the third order differential equation

(e−t (e−t y0 )0 )0 + 7e−2t y0 + 13e−2t y = 0, Taking ρ(t ) = e

2t

t ≥ 1.

(3.16)

and g∗ (t ) = t − 1 we have

  Z t (ρ 0 (s)r1 (s) − ρ(s)p(s)R2 (s, g∗ (s)))2 (2 − 7(1 − e−1 ))2 ds = lim sup ds = ∞. K ρ(s)q(s) − 13 − 4R2 (s, g∗ (s))ρ(s)r1 (s) 4(1 − e−1 ) t →∞ T

Z t lim sup t →∞

T

Again, condition (3.12) is satisfied. Hence by Theorem 3.1, any solution y of (3.16) with F [y](t1 ) ≤ 0 is oscillatory. It is easy to verify that y(t ) = e2t cos 3t is a solution. Letting r1 (t ) = r2 (t ) ≡ 1, f (y) = y, and ρ(t ) = t 2 in Theorem 3.1, we obtain the following corollary. Corollary 3.1. Let the hypotheses of Theorem 3.1 hold. If

Z t

s2 q(s) + sp(s) −

lim sup t →∞

T

1 s − g∗ (s)

− (s − g∗ (s))

(sp(s))2 4

 ds = ∞,

then Eq. (1.2) is oscillatory. Example 3.3. Consider the third order differential equation y000 +

1 4t 2

y0 +

25 4t 3

y = 0,

t ≥ 1.

(3.17)

Taking g∗ (t ) = t /2, we see that all conditions of Corollary 3.1 are satisfied. Therefore, Eq. (3.17) is oscillatory. An example of an oscillatory solution is y(t ) = t 2 sin( 32 ln t ). Note that since t 2 p(t ) = 14 (m = 0), Theorem A fails. Example 3.4. Consider the third order differential equation y000 +

2 0 8 y + 3 y = 0, t

t2

t ≥ 1.

(3.18)

Take g∗ (t ) = t /2 and applying Corollary 3.1, we may conclude that Eq. (3.18) is oscillatory. In fact, y(t ) = t 2 sin(2 ln t ) is such a solution. In this example, since t 2 p(t ) = 2, both Theorems B and C fail to apply.

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Example 3.5. Consider the third order differential equation sin2 t 0 y + y = 0, t2

y000 +

t ≥ 1.

(3.19)

With g∗ (t ) = t − 1t , we easily verify that

Z t lim sup t →∞

s2 q(s) + sp(s) −

T

1 s − g∗ (s)

− (s − g∗ (s))

(sp(s))2

Z t



4

s2 +

ds = lim sup t →∞

T

Z ≥ lim sup t →∞

sin2 s s

−s−

sin4 s



4s3

ds

t

(s2 − s) = ∞.

T

By Corollary 3.1, we see that Eq. (3.19) is oscillatory. It is easy to check that Theorems A–C are not applicable. Next, we present further new oscillation results for Eq. (1.1), by deriving integral averaging conditions. Following Philos [20], we introduce a class of functions P . Let D0 = {(t , s) : t > s > t0 }

and D = {(t , s) : t ≥ s > t0 }.

A function H ∈ C (D, R) is said to belong to a class P if (i) H (t , t ) = 0 for all t ≥ t0 ; H (t , s) > 0 for all (t , s) ∈ D0 ; (ii) H (t , s) has a continuous and nonpositive partial derivative on D0 with respect to the second variable, and for a positive continuous function h(t , s)

−

p ∂ H (t , s) = h(t , s) H (t , s) ∂s

for all (t , s) ∈ D0 .

For the choice H (t , s) = (t − s)n , (n ≥ 1), the Philos-type conditions reduce to the Kamenev-type ones. Other choices of H include

 ln

t

n

s

√ √ ( t − s) n ,

,

(t 3 − s3 )n ,

(et −s − es−t )n .

Theorem 3.2. Let (2.6)–(2.9) hold. If there exist an eventually positive function ρ ∈ C 1 [t0 , ∞) and a function H ∈ P such that lim sup t →∞

1 H (t , T )

Z t

K ρ(s)q(s)H (t , s) −

Q 2 (t , s) 4A(s)

T

 ds = ∞

(3.20)

for every T ≥ t0 , where A(t ) =

R2 (t , g∗ (t )) r1 (t )ρ(t )

Q (t , s) = h(t , s) −

;

 0  ρ ( s) R2 (s, g∗ (s)) H ( t , s) − p(s) , ρ(s) r1 (s)

p

then Eq. (1.1) is oscillatory. In fact, any solution y satisfying F [y](t1 ) ≤ 0 for some t1 ≥ t0 is oscillatory, where the functional F is given by (2.5). Proof. Proceeding as in the proof of Theorem 3.1 we arrive at the inequality (3.13). Then we see that

Z

t

ρ(s)q(s)H (t , s)ds ≤ t2

Z

t

H (t , s)[−ω0 (s) + B(s)ω(s) − A(s)ω2 (s)]ds t2

 ∂ H (t , s) 2 = −H (t , s)ω(s) | + ω(s) + H (t , s)[B(s)ω(s) − A(s)ω (s)] ds ∂s t2   Z t p 2 = H (t , t2 )ω(t2 ) − ω (s)A(s)H (t , s) + ω(s) h(t , s) H (t , s) − H (t , s)B(s) ds Z t

t t2

t2

≤ H (t , t2 )ω(t2 ) +

Z

t t2

where B(t ) =

R2 (t , g∗ (t )) ρ 0 (t ) − p(t ) . ρ(t ) r1 (t )

Q 2 ( t , s) 4A(s)

ds,

M.F. Aktas et al. / Nonlinear Analysis 71 (2009) e1496–e1502

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Thus we obtain that 1

Z t

H (t , t2 )

t2

s2 q(s)H (t , s) −

which clearly contradicts (3.20).

Q 2 ( t , s) 4A(s)



ds ≤ ω(t2 ),



Corollary 3.2. Let (2.6)–(2.9) hold. If there exist an eventually positive function ρ ∈ C 1 [t0 , ∞) and a function H ∈ P such that lim sup t →∞

lim sup t →∞

t

Z

1 H (t , T )

ρ(s)q(s)H (t , s)ds = ∞;

T

H (t , T )

Q 2 (t , s)

t

Z

1

A(s)

T

ds < ∞,

then Eq. (1.1) is oscillatory. The proof of Theorem 3.3 below is similar to that of [21, Theorem 5.2] and hence it is omitted. Theorem 3.3. Let all the assumptions except (3.20) of Theorem 3.2 hold. Suppose that for every t ≥ t0 ,



0 < inf lim inf t →∞

s≥T

H (t , s)

 ≤ ∞;

H (t , T )

lim sup t →∞

t

Z

1 H (t , T )

Q 2 ( t , s) A(s)

T

ds < ∞,

and that there exists a function Ψ ∈ C [t0 , ∞) such that ∞

Z

Ψ+2 (s)A(s)ds = ∞, Ψ+ (s) = max{Ψ (t ), 0};  Z t Q 2 ( t , s) 1 lim sup K ρ(s)q(s)H (t , s) − ds ≥ sup Ψ (t ). 4A(s) t →∞ H (t , T ) T t ≥T T

Then Eq. (1.1) is oscillatory. In fact, any solution y satisfying F [y](t1 ) ≤ 0 for some t1 ≥ t0 is oscillatory, where the functional F is given by (2.5). Example 3.6. Consider the third order differential equation



1 0 y t2

00

1

+

t2

y = 0,

t > 1.

(3.21)

We take H (t , s) = (t − s)2 , ρ (s) ≡ 1, g∗ (s) = s − 1, and Ψ (t ) = 1/t. In view of Q (t , s) ≡ 2 and A(s) = s2 , we see that lim sup t →∞

1 H (t , T )

Z

Q 2 (t , s)

t

A (s)

T

∞

Z

t →∞

t

(t − T )

2

T

4 s2

ds = 0 < ∞,

∞

Z

Ψ+2 (s) A (s) ds =

ds = lim sup

Z

1

ds = ∞,

T

T

and lim sup t →∞

1 H (t , T )

Z t

K ρ (u) q (u) H (t , u) −

T

Q 2 (t , u) 4A(u)

 du = lim sup t →∞

(t − T )

t

Z

1 2

(t − u)2 − 1

T

u2

du =

1 T

= sup Ψ (t ). t ≥T

Since the conditions of Theorem 3.3 hold, Eq. (3.21) is oscillatory. Finally, we give a theorem which guarantees that every solution of Eq. (1.1) with at least one zero must be oscillatory. Theorem 3.4. Suppose that Eq. (2.11) is nonoscillatory, and that (2.6) and (2.9) hold. If R1 (t , t0 ) → ∞ as t → ∞

(3.22)

and if there exists an eventually positive function ρ ∈ C [t0 , ∞) such that 1

Z t

K ρ(s)q(s) −

lim sup t →∞

T

B2 (s)



4A(s)

ds = ∞

for some T ≥ t0 , where A(t ) =

R2 (t , g∗ (t )) r1 (t )ρ(t )

,

B(t ) =

ρ 0 (t ) R2 (t , g∗ (t )) − p(t ) , ρ(t ) r1 ( t )

then Eq. (1.1) is oscillatory. In fact, any nonzero solution y of Eq. (1.1) having a zero is oscillatory.

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Proof. Let y be a solution of Eq. (1.1) defined on I with y(t1 ) = 0 for some t1 ≥ t0 . Suppose also that y is nonoscillatory. Without loss of generality, we may take y(t ) > 0 for t > t2 , where t2 ≥ t1 is the last zero of y(t ). From Lemma 2.4, it follows that there is a t20 ≥ t2 such that L1 y(t ) < 0

or L1 y(t ) > 0

for t ≥ t20 .

Suppose that L1 y(t ) < 0 for t ≥ t20 . Since y(t2 ) = 0, we must have L1 y(t ) > 0 for t ∈ (t2 , t2 + δ) for some δ > 0. Then, there exists a t3 ∈ (t2 , t20 ) such that L1 y(t3 ) = 0 and L1 y(t ) < 0 for t > t3 . We may claim that L2 y(t ) is eventually nonnegative. For if L2 y(t4 ) < 0 for some t4 ≥ t3 , then from y0 (t ) ≤ L1 y(t4 )/r1 (t ), t ≥ t4 and (3.22), we see that y(t ) takes on negative values for t sufficiently large, which is a contradiction. So, there exists a t5 > t3 such that L2 y(t ) ≥ 0 for t ≥ t5 . Integrating Eq. (1.1) multiplied by L1 y(t ) from t3 to t5 and employing Lemma 2.5 with v = L1 y we have 0 ≥ L1 y(t5 )L2 y(t5 )

Z

t5

= t3

Z

t5

≥ t3



1 r2 (s)

(L2 y(s))2 −

p(s)

 (L1 y(s))2 − L1 y(s)q(s)f (y(s)) ds

r1 (s)   p(s) r2 (s) ([L1 y(s)]0 )2 − (L1 y(s))2 ds > 0, r1 (s)



a contradiction. Let L1 y(t ) > 0 for t ≥ t20 . Set

ω(t ) = ρ(t )

L2 y(t ) y(t )

> 0,

t ≥ t20 .

Proceeding as in the proof of Theorem 3.1, we again obtain a contradiction. The proof is complete.



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