Oscillation Criteria for Even Order Functional Differential Equations with Damped Term

Mathematical Computation December 2015, Volume 4, Issue 4, PP.77-82

Oscillation Criteria for Even Order Functional Differential Equations with Damped Term Ailing Shi Science School, Beijing University of Civil Engineering and Architecture, Beijing 100044, China Email: shiailing@bucea.edu.cn

Abstract This paper investigates a class of even order functional differential equations with damped term, and derives two new oscillatory criteria of solution.

Keywords: Functional Differential Equation; Even Order; Damped; Oscillation

1 INTRODUCTION Oscillation of even order functional differential equations has been studied extensively. It is caused by the applications such as physics, chemistry, phenomena arising in engineering, economy, and science; for example, [1–6]. Recently, more and more authors have paid their attentions. We refer to the monographs [1, 2]. However, most of the literatures dealing with equations do not contain the damped term. It seems that very little is known about certain equations with damped term, (for example, the literatures [3-5]), especially the case of containing distributed deviating arguments. In this paper, we deal with a class of even order functional differential equations with damped of the form d (r (t )  x(t ) + ∫ c(t ,h ) x [ h(t ,h )]d µ (h )    c

( n −1)

d )' + p (t )  x(t ) + ∫ c(t ,h ) x [ h(t ,h )]d µ (h )    c

( n −1)

(1) + ∫ q(t , x ) f ( x(t ), x , x [ g (t , x ) ])dσ (x ) = 0, t ≥ t0 > 0 d

a

and establish two new oscillatory criteria of solution, where n is an even number, and the following conditions(H) are always assumed to hold: ( H 1) ( pt ) ∈ C ([t0 , ∞), R+ ), r (t ) ∈ C ([t0 , ∞), R+ ), r '(t ) > 0, q(t , ξ ) ∈ C ([t0 , ∞), R+ ), C (t ,η ) ∈ C ([t0 , ∞) × [a, b], R+ ) is not identically zero on any [tµ , ∞) × [a, b], tµ ≥ t0 ;

( H 2) g (t , ξξξ ) ∈ C ([t0 , ∞) × [a, b], R), g (t , ) < t , ∈ [a, b] ,and lim inft →∞ ,ξ ∈[ a ,b ] , g (t , ξ ) = ∞ ； ( H 3) f (u1 , u2 , u3 ) ∈ C ( R × R × R, R) have same sign with u1 , u2 , u3 , when u1 , u2 , u3 have same sign;

( H 4) σ (ξ ) ∈ ([a, b], R) is nondecreasing, the integral of equation (1) is a Stieltjes one.

We restrict out attention to proper solutions of equation (1), i.e to nonconstant solutions existing on [t , ∞) for some T ≥ t0 and satisfying supt ≥T x(t ) > 0 . A proper solution x(t ) of equation (1) is called oscillatory if it does not the largest zero, otherwise, it is called nonoscillatory.

The following three Lemmas will be needed in the proof of our results:

Lemma 1. [6] Let u (t ) be a positive and n times differentiable function on [0, ∞) . If u ( n ) (t ) is a constant sign and not identically zero on any ray [t1 , ∞), t1 > 0 , then there exists a tµ ≥ t1 and an integer l (0 ≤ l ≤ n) , with n + l even for u (t )u ( n ) (t ) ≥ 0 or n + l odd for u (t )u ( n ) (t ) ≤ 0 ; and for t ≥ tµ , u (t )u ( k ) (t ) > 0, 0 ≤ k ≤ l ; (−1)( k −l ) u (t )u ( k ) (t ) > 0, l ≤ k ≤ n

(2)

Lemma 2. [7] Suppose that the conditions of Lemma l are satisfied, and then for any constant θ ∈ (0,1) and sufficiently large t, there exists a constant M satisfying - 77 www.ivypub.org/mc

µ '(θ (t ) ≥ Mt ( n − 2) µ ( n −1) (t )

(3)

Similar to the proof of literature [8], we have

Lemma 3. Suppose that x(t ) is a nonoscillatory solution of equation (1). If

Then

t

s

t →∞ t1

t1

lim ∫ exp(− ∫ p(t )dt )ds = +∞, t ≥ t0

(4)

x(t ) x ( n −1) (t ) > 0 , for any large t

(5)

2 MAIN RESULTS Theorem 1. Suppose that ( H 5) There exists a function σ (t ) ∈ C '([t − 0, ∞), (0, ∞) satisfying σ (t ) ≤ g (t , ξ ) ≤ t , 0 < c ≤ σ '(t ) ≤ 1 , and limt →∞ σ (t ) = ∞ ; ( H 6)

∂f ∂f exists, and ≥ ci > 0 , where ci > 0 are some constants, i = 1,1,3 . If for any large T ∂ui ∂ui lim sup t →∞

1 t

m −1

∫

t

T

b

(t − u ) m −3 u k [ ∫ (t − u )2 q(u , ξsξ )d ( ) a

pr (u ) k − + m − 1]2 γ (u ) u − ∞ ]du = 1 σ ( n − 2) (u ) 4M 1 γ (u ) in which M 1 is a constant, then every solution of equation (1) is oscillatory. [(t − u )

(6)

Proof. We assume that equation (1) has a nonoscillatory solution x(t ) . Without loss of generality, we may suppose that x(t ) > 0 for all large t. The case of x(t ) < 0 can be considered by the same method. From (H1), (H2), and (H3), there exists a t1 ≥ t0 such that x[h(t ,h )] > 0 , f ( x(t ), x , x[ g (t , x )] > 0 for t ≥ t1 and ξ ∈ [a, b] .

Let d

z= (t ) x(t ) + ∫ c(t ,h ) x[h(t ,h )]d µ (h ) c

(7)

Thus equation (1) turns into b

(r (t ) z ( n −1) (t ))' + p(t )(z ( n −1) (t )) + ∫ q(t , x ) f ( x(t ), x , x[ g (t , x )])dσ (x ) = 0, t ≥ t0 > 0 a

Then, From (H1), we obtain

z(t ) ≥ 0, z(t ) ≥ x(t ) . Using

z(t ) ≥ x(t ) , and from (H6), we can get

f ( x(t ), x , z[ g (t , x )]) ≥ f ( x(t ), x , x[ g (t , x )]) Thus '

b

(r (t ) z ( n −1) (t ))' + p(t )(z ( n −1) (t )) + ∫ q(t , x ) f ( x(t ), x , x[ g (t , x )])dσ (x ) ≤ 0, t ≥ t0 > 0 a

(8)

From the assumption of (H1), we have z[t ]) ≥ x(t ) > 0, (r (t ) z ( n −1) (t ))' ≤ 0 , for t ≥ t1 , and (r (t ) z ( n −1) (t ))' is not eventually zero, thus, we have (r (t ) z ( n −1) (t ))' = r ' (t ) z ( n −1) (t ) + r (t ) z ( n ) (t ) ≤ 0

(9)

Similar to the proof of Lemma 3, we have z ( n −1) (t ) > 0 . Thus from (9), we obtain z ( n ) (t ) ≤ 0 for t ≥ t1 .

From Lemma 1, there exists a t2 ≥ t1 and an odd number (0 < l < n) , such that for t ≥ t2 z ( k ) (t ) > 0, 0 ≤ k ≤ l ;(−1)( k −1) z ( k ) (t ) > 0, l ≤ k ≤ n

By choosing k = 1, we have z'(t ) > 0, t ≥ t2 - 78 www.ivypub.org/mc

(10)

It follows that Lemma 2, there exists a constant M > 0 and t3 ≥ t2 such that t σ (t ) z'( ) ≥ Mt ( n − 2) z ( n −1) (t ), z'( ) ≥ M σ ( n − 2) (t ) z ( n −1) (t ) 2 2 From ( H 6), z (t ) ≥ x(t ) , we have f ( z (t ), x , z[ g (t , x )] ≥ f ( x(t ), x , x[ g (t , x )]) , thus b

(r (t ) z ( n −1) (t )) '+ p(t )(z ( n −1) (t )) + ∫ q(t , ξξξ ) f ( z (t ), , z[ g (t , )])dσ (ξ ) ≤ 0, t ≥ t0 > 0 a

(11)

(12)

Let t k z ( n −1) (t )  t  a  σ (t )  f (z   , , z  )  2  2  2 

y (t ) =

(13)

Then we conclude that t k (r (t ) z ( n −1) (t ))' kt ( k −1) r (t ) z ( n −1) (t ) 1 t k r (t ) z ( n −1) (t ) + −  t  a  σ (t )   t  a  σ (t )  2 2  t  a  σ (t )  f (z   , , z  ) f (z   , , z  ) f (z   , , z  )  2  2  2   2  2  2   2  2  2   t  a  σ (t )  ) ∂f ( z   , , z  ( k −1) ( n −1) t k r (t ) z ( n −1) (t ) (t ) 1  2  2  2  + kt r (t ) z ×( − t   t  a  σ (t )  2 2  t  a  σ (t )  f (z   , , z  f (z   , , z  ) ) ∂z   2  2  2  2   2  2  2  In view of (H3) and (H4), and noting that y ' (t ) =

b

b

) f ( z (t ), , z[ g (t , )])dσ (ξξ ) − ∫ q(t , ) f ( z (t ), a, z (σ (t ))dσ (ξ ) − ∫ q(t , ξξξ a ≤ a  t  a  σ (t )   t  a  σ (t )  ) ) f (z   , , z  f (z   , , z   2 2  2   2  2  2  b

≤ − ∫ q(t , ξ )dσ (ξ ) a

(14)

We conclude that b

y ' (t ) ≤ −t k ∫ q (t , ξ )dσ (ξ ) − ( a

p(t ) k − ) y (t ) r (t ) t

t k r (t ) z ( n −1) (t )  t  a  σ (t )  f 2 (z   , , z  )  2  2  2  b p(t ) k t −k ≤ −t k ∫ q(t , ξ )dσ (ξ ) − ( − ) y (t ) − M 1σ ( n − 2) (t ) y 2 (t ) a r (t ) t r (t )

1 − (c1 Mt ( n − 2) + c3cM σ ( n − 2) (t )) z ( n −1) (t ) × 2

in which = M1

(15)

1 M (c1 + c3c) . From (14), for any large t > t3 , we conclude that 2 t t b ( m −1) ' ( m −1) k )du ≤ −( ) − ( ) ∫ (t − u ) y (u )du ≤ −∫ ∫ (t − u ) u q(u, ξ )dσ (ξξξ t3

t3

a

t

− ∫ (t − u )( m −1) − ( t3

t

p(u ) k − ) y (u )du r (u ) u

− ∫ M 1 (t − u )( m −1) σ ( n − 2) (u ) t3

u −k 2 y (u )du r (u )

By part integrating, we conclude that

∫

t

t3

t

(t − u )( m −1) y ' (u )du = (m − 1) ∫ (t − u ) m − 2 y (u )du − y (t3 )(t − t3 )( m −1) t3

Thus t

b

t3

a

∫∫

(t − u ) m −1 u k q(u , ξ )dσ (ξ )du ≤ y (t3 )(t − t3 )( m −1) - 79 www.ivypub.org/mc

(16)

t

t

t3

t3

−(m − 1) ∫ (t − u )m − 2 y (u )du − ∫ (t − u )m −1 − (

p(u ) k − ) y (u )du r (u ) u

u −k 2 y (u )du t3 r (u ) t − t3 m −1 1 t b (t − u ) m −1 u k q(u , ξ )dσ (ξ )du ≤ y (t3 )( ) m −1 ∫t ∫a 3 t t p(u ) k 1  t − m −1  ∫ [(m − 1)(t − u ) m − 2 + (t − u ) m −1 − ( − )] y (u )du t  t3 r (u ) u t

− ∫ M 1 (t − u ) m −1σ ( n − 2) (u )

t

+ M 1 ∫ (t − u ) m −1σ ( n − 2) (u ) t3

= y (t3 )(

 u −k 2 y (u )du  r (u ) 

t − t3 m −1 1 t u −k ) − m −1 ∫ [ M 1 (t − u ) m −1σ ( n − 2) (u ) y (u ) t3 t t r (u )

(t − u ) m − 2 [(t − u ) +

p(u ) k − + m − 1] r (u ) u

2 M 1 (t − u ) m −1σ ( n − 2) (u )

+

(t − u ) m −3 u k [(t − u )

1

t

t m −1 ∫t3

4M 1

−k

]2 du

u r (u )

p(u ) k − + m − 1]2 r (u ) u

1 ( n − 2) σ (u ) r (u )

du

Furthermore, we conclude that 1 t

m −1

∫

t

t3

b

(t − u ) m −3 u k [ ∫ (t − u )2 q(u , ξ )dσ (ξ ) a

p(u ) k − + m − 1]2 t − t3 m −1 r (u ) u ]du ≤ y (t3 )( ) 1 ( n − 2) t 4M 1 σ (u ) r (u )

[(t − u ) −

Letting t → ∞ in (16), we find that 1 t

m −1

∫

t

t3

b

(t − u )m −3 u k [ ∫ (t − u )2 q(u , ξ )dσ (ξ ) a

p(u ) k − + m − 1]2 r (u ) u − ]du ≤ y (t3 ) 1 ( n − 2) σ 4M 1 (u ) r (u ) Which contradicts with (6). This completes the proof of Theorem 1. [(t − u )

(17)

Theorem 2. Suppose that (H7) There exist constant N and α , such that lim inf y →∞

f ( x, y , z ) ≥ α > 0, x > N z

(H8) There exists a function ϕ (ξ ) ∈ C ([a, b], (0, ∞)) such that ϕ (ξξ ) ≤ q(t , ).t ≥ t0 , and for any ξξ ∈ [a, b], limt →∞ q (t , ) exists.

If for any large T limsup t →∞

1 t

m −1

∫

t

T

b

)d ( ) b (t − u )m −3 u k [(t − u )2 ∫ ϕ (ξsξ a

- 80 www.ivypub.org/mc

p(u ) k − γ + m − 1]2 r (u ) u ]du = −γ ∞ 1 ( n − 2) (u ) σ r (u ) in which β and γ are some constants, then every solution of equation (1) is oscillatory. [(t − u )

(18)

Proof. Suppose that equation (1) has a nonoscillatory solution x(t ) > 0 . By using the same arguments as in the proof of Theorem 1, there exists a t1 ≥ t0 such that x[h(t ,h )] > 0, x[ g (t ,η )] > 0, f ( x(t )x , x[ g (t , x )] > 0 and ξ ∈ [a, b], z ' (t ) > 0, z ( n −1) (t ) > 0 , for t ≥ t1 , and there exist constant M > 0

and t2 ≥ t1 such that t σ (t ) z ' ( ) ≥ Mt ( n − 2) z ( n −1) (t ), z ' ( ) ≥ M σ ( n − 2) (t ) z ( n −1) (t ) 2 2

ω (t ) =

t k r (t ) z ( n −1) (t ) ] σ (t ) z[ 2

(19) (20)

Then,

ω ' (t ) =

t k (r (t ) z ( n −1) (t ))' kt k −1r (t ) z ( n −1) (t ) 1 t k r (t ) z ( n −1) (t ))' ' σ (t ) ' + − z( )σ (t ) σ (t ) σ (t ) 2 z 2 [σ (t ) ] 2 z[ z[ ] ] 2 2 2 b

≤

) f ( z (t ), , z[ g (t , )])dσ (ξ ) t k (− p(t ) z ( n −1) (t ) − ∫ q(t , ξξξ a

z[

σ (t ) 2

]

k 1 t k r (t ) z ( n −1) (t ) ' σ (t ) ' + ω (t ) − z( )σ (t ) t 2 z 2 [σ (t ) ] 2 2 1 t k r (t ) z ( n −1) (t ) ' σ (t ) ' z( )σ (t ) 2 z 2 [σ (t ) ] 2 2 ' From x (t ) > 0 , we conclude that limt →∞ x(t ) = L exists. −

(21)

(22)

(a) If L < ∞ , then

∫ liminf

b

a

) f ( z (t ), , z[ g (t , )])dσ (ξ ) q(t , ξξξ

t →∞

z[

∫ ≥ liminf

b

a

σ (t ) 2

]

) f ( z (t ), , z[ g (t , )])dσ (ξ ) ϕ (ξξξ

t →∞

z[

σ (t ) 2

]

f ( L, a , L ) b ∫a ϕ (ξ )dσ (ξ ) > 0 L

=

(b) If L = ∞ , then we conclude that from (H2), (H7) and (H8)

∫ liminf

b

a

q(t , ξξξ ) f ( z (t ), , z[ g (t , )])dσ (ξ )

t →∞

b

≥ liminf ∫ q (t , ξ ) t →∞

a

z[

σ (t ) 2

]

f ( z (t ), ξξξ , z[ g (t , )]) g (t , ) [ ]dσ (ξ ) 2 z - 81 www.ivypub.org/mc

(23)

≥ liminf t →∞

a

2∫

b

a

ϕ (ξ )dσ (ξ ) > 0

(24)

 f ( L, a , L ) a  Let β = min ( ), ( )  , for any large t3 ≥ t2 , we conclude that L 2  

∫

b

a

q(t , ξξξ ) f ( z (t ), , z[ g (t , )])dσ (ξ ) z[

σ (t ) 2

b

≥ b ∫ ϕ (ξ )dσ (ξ ) a

]

(25)

Furthermore, from (H4), we conclude that t −k n−2 1 t k r (t ) z ( n −1) (t ) ' σ (t ) ' 1 σ (t )ω 2 (t ) z( )σ (t ) ≥ cM σ t ( ) 2 z2[ 2 2 r (t ) ] 2

in which M 2 =

(26)

1 Mc , then 2

p(t ) k 1 t −k n−2 t −k 2 − )ω (t ) − M 2 σ ω (t ) a r (t ) t 2 r (t ) r (t ) The remainder proof is as same as proof of Theorem 1, we omit it. This completes the proof of Theorem 2. b

ω ' (t ) ≤ − b t k ∫ ϕ (ξ )dσ (ξ ) − (

(27)

ACKNOWLEDGMENT This research was supported by 2013 Science and Technology Research Project of Beijing Municipal Education Commission (KM201310016001）and the National Natural Science Foundation of China (11371364).

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[2]

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P.G.Wang, Oscillations of a class of higher order functional Differential Equations with damped term, Applied Mathematics and Computation. 148(2004): 351-358

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I.T. Kiguradze, On the oscillations of the Eq. ( d u / dt ) + a (t ) u

[7]

Ch.G. Philos, A new criterion for oscillatory and asymptotic behavior of delay differential equations, Bull. Acad. Pol. Sci. Ser. Sci.

[8]

A.G. Kartsatos, Recent results on oscillation of solutions of forced and perturbed nonlinear Differential Equations of even order,

m

m

n

sgn u = 0 , Mat. Sb. 65(1964): 172-187 (in Russian)

Mat., 39(1981): 61-64 Stability of Dynamical Systems: Theory and Applications, Lecture Notes in Pure and Applied Mathematics. Springer, New York, 1977

AUTHORS Ailing Shi was born in Gansu, China, in 1970. She received her Master of Master of Mathematics degree from the Department of Mathematics. She carried out research in the Department of Applied Mathematics, Beijing University of Civil Engineering and Architecture and served since 1995, and became an Associate Professor in 2012. Professor Shi’s current research interests include Differential Equations.

- 82 www.ivypub.org/mc