Oscillation Criteria for Damping Quasi-Linear Neutral Differential ...
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JOURNAL OF NETWORKS, VOL. 9, NO. 7, JULY 2014
Oscillation Criteria for Damping Quasi-Linear Neutral Differential Equations with Distributed Deviating Arguments Xuequan Tian, Yi Yang*, and Hongke Wang Department of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing, China *Corresponding author, Email: [email protected], [email protected], [email protected]
Abstract—In this paper, we investigate the oscillation criteria for damp quasi-linear neutral differential equations with distributed deviating arguments. By using the generalized Riccati transformation and integral averaging, several new criteria that ensure the oscillation of solutions are obtained. Our results not only include some previously known results, but apply to some damped differential equations that have not been investigated so far. Index Terms—Half-Linear Differential Oscillatory Behavior; Riccati Transformation
I.
Equation;
Furthermore, O. Dosly et al. [11] deal with the following equation:
r (t ) x
q(t , ) x g (t , ) d ( ) 0 b
a
(1)
Later, F. W. Meng [10] studies the oscillation criteria for even order differential equation: ( n 1) 1
r (t ) x(t ) c(t ) x( t ) q(t ) f x g (t ) 0
© 2014 ACADEMY PUBLISHER doi:10.4304/jnw.9.7.1908-1913
( n 1) x(t ) c(t ) x( t ) (2)
(3)
and X. Yang [12] studies the oscillation criteria for the following equation: r (t ) ( y ( n 1) (t ))
Qualitative analysis for different types of differential equations with damping is a fertile research area and increasingly attracts many concerns, especially in the oscillatory behavior of solutions of solutions of higher order neutral differential equations which are of both theoretical and practical interest. Some applicable examples can be found in the monograph of Hale [18]. There have been some results on the oscillatory and asymptotic behavior of quasi-linear neutral differential equations. To the best of our knowledge, very little has been done with distributed deviating arguments. However, we note that in many area of their actual application, models describing these problems are often effected by such factors as seasonal changes. Therefore it is necessary, either theoretically or practically, to study a type of equation in a more general sense differential equations with distributed deviating arguments. Here we particularly refer the reader to the papers [1–17, 19–25]. In particular, P. G.Wang et al. [3] investigate and establish several oscillatory criteria for the following even order differential equation with distributed deviating arguments: (n)
(t ) p(t ) x ( n 1) (t )
q(t ) x g (t ) 0
INTRODUCTION AND LEMMAS
x(t ) c(t ) x(t )
( n 1)
b
a
q(t , ) x g (t , ) d ( ) 0
(4)
Motivated by the previous works, this paper investigates the oscillation criteria for the following class of even order functional differential equations with damping: r (t ) ( y ( n 1) (t )) p(t ) ( y ( n 1) (t )) q(t , ) f x g (t , ) d ( ) 0 b
(5)
a
where n is an even positive integer and 1
(s) s s, 0 , y(t ) x(t ) c(t ) x(t ) . It is clear that (5) includes (1) – (4), i.e. (5) (1) for r(t) = 1, p(t) = 0, f(x) = x; (5) (2) for p(t) = 0, g(t,ζ) = g(t), q(t,ζ) = q(t),
b
a
d ( ) 1 ;
(5) (3) for c(t) = 0, g(t,ζ) = g(t),
q(t,ζ) = q(t), f(x) = φ(x),
b
a
d ( ) 1 ;
(5) (4) for p(t) = 0, f(x) = φ(x). Our aims to get the criteria for the oscillatory solutions of (5). Throughout this paper, we will set forth conditions to establish the criteria for the oscillatory solutions of (5). (H1) P(t) > 0, 1
(H2)
f ( x) x
1
x
K 0,
t0
1 dt , R(t )
JOURNAL OF NETWORKS, VOL. 9, NO. 7, JULY 2014
1909
1
1
1 lim ds t t0 R ( s ) t
(H3)
Thus,
.
Furthermore,
E (t , t0 )r (t ) ( y ( n 1) (t )) E (t , t0 ) q(t , ) f x g (t , ) d ( ) 0 b
y
The
1
y ( n ) (t ) 0
This plus r’(t)≥0 leads to that y(n)(t) ≤ 0. By Corprolast Lemma, we get y(t ) 0 . (ii) From y’(t) > 0, we have y(t) = x(t) + c(t)x(t - ) > 0 such that x(t ) y(t ) c(t ) x(t ) y(t ) c(t ) y(t ) (1 c(t )) y(t ) . By
f ( x) 1
K 0 and (s) s
1
s, 0 ,
x x we obtain
E (t , t0 ) q(t , ) f x g (t , ) d ( )
E (t , t0 ) Kq(t , ) x g (t , ) d ( ) b
a
E (t , t0 ) Kq(t , ) 1 C ( g (t , )) y( g (t, ))d ( )
b
E (t , t0 ) q(t , ) f x g (t , ) d ( ) b
decreasing
and
a
y ( n 1) (t ) 0 . Now, we claim that
E (t , t0 ) y (t ) Kq(t , ) 1 C ( g (t , )) d ( ) 0
( n 1)
R(t ) y ( n 1) (t )
(t ) 0 , we have
Q(t) y g(t) 0 .
Lemma 2. Suppose x(t ) is a positive solution of (5) and
R(t ) y ( n 1) (t ) 0 ;
R(t ) y ( n 1) (t ) M 0, R(t ) y ( n 1) (t ) M .
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a
b
y ( n 1) (t ) 0 .
Otherwise if y
r (t ) y ( n 1) (t ) r (t ) y ( n 1) (t )
Thus, there exists (t ) such that (t ) g (t , ) and (t ) 1, (t ) 0 which leads to that (t ) for t . It implies that
Proof. (i) Since x(t) > 0, we have R(t ) y ( n 1) (t ) 0 . R(t ) y ( n 1) (t )
and r (t ) y ( n 1) (t ) r (t ) y ( n 1) (t )
a
a
is
t
Working inductively, we get y ( n 3) (t ) 0, , y(t ) 0 which contradicts y(t) > 0. Hence the claim is established. Consequently, we will prove y ( n ) (t ) 0 . It is clear that y ( n 1) (t ) 0 r (t ) y ( n 1) (t ) 0
b
Q(t ) KE (t , t0 ) q(t , )(1 c[ g (t , )]) d ( ) .
(t )
t
a
b
1
E (t , t0 ) Kq(t , ) x g (t , ) d ( )
R(t ) y
1 y (t ) y (T ) M ds T R( s ) Then lim y ( n 2) (t ) . 1
( n 2)
b
Q(t) y g(t) 0
( n 1)
M . Integrating (t ) R(t )
a
t p( s ) R(t ) E (t , t0 )r (t ) r (t ) exp ds t0 r ( s)
where
( n 1)
1
y ( n 1) (t ) 0, y ( n) (t ) 0, y(t ) 0 , (t )
(t ) 0 , we obtain y
( n 2)
a
( n 1)
( n 1)
on both sides, we have
following lemmas play a crucial role in the qualitative theory to proof the main results to be presented in this paper. Lemma 1. Suppose x(t ) is a positive solution of (5) and y (t ) satisfies the following conditions:
R(t) y
from
1
In the sequel, it will be always assumed that solutions of (5) exist for any t0 ≥ 0.A solution x(t) of (5) is called eventually positive solution (or negative solution) if there exists a sufficiently large positive number t1 > t0 such that x(t) > 0 (or x(t) < 0) for all t > t1. A nontrivial solution x(t) of (5) is called oscillatory if it has arbitrary large zeros; otherwise it is called nonoscillatory. Equation (5) is called oscillatory if all its solutions are oscillatory. If not otherwise stated, we assume that the solution of (5) is the eventually positive solution. The fundamental terminologies concerned with differential equations can be found in Ref. [18]. This section presents some basic relations that will be used later. t p( s) ds Let E (t , t0 ) exp t0 r ( s)
Thus
M (t ) R(t )
1
1 s p(u ) lim exp du ds t t0 R ( s ) t0 r (u ) where x(t) is a solution of (5) and x(t ) C1 Tx , , R , Tx t0 . t
and
y
( n 1)
1 1 y ( n 1) (t ) (t ) R (t ) y ( n 1) (t ) w(t ) (t ) R(t ) (6) y[ g (t )] y[ g (t )]
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JOURNAL OF NETWORKS, VOL. 9, NO. 7, JULY 2014
y[ g (t )] Mt n 2 y ( n 1) (t ), (0,1)
1 (s) R(s) ( s ) Q ( s ) ds . T ( 1) 1 ( s) R( s) Hence (5) is oscillatory. Let D (t , s) t s t0 , D0 (t , s) t s t0 . If
(7)
t
then w(t ) (t )Q(t ) A(t ) w(t ) B(t ) w
where A(t )
1
(8) (t )
(t ) G(t ) , B(t ) . 1 (t ) (t ) R[ g (t )]
Proof. From (6) and (7), we get w(t ) (t )Q(t ) y ( n 1) (t )
y 1[ g (t )] y [ g (t )] g (t ) y 2 [ g (t )]
(t )Q(t ) y ( n 1) (t )
g (t )Mg n 2 (t ) y ( n 1) [ g (t )] y1 [ g (t )]
Since y ( n 1) (t ) is decreasing, we have 1
H (t , s) A( s) H (t , s) h(t , s) H 1 (t , s) s where (t , s) D0 ; (B4)
(B3)
G(t ) y ( n 1) [ g (t )] y1 [ g (t )]
w(t ) (t )Q(t ) y ( n 1) (t )
the function H C D, R satisfies the following conditions: (A1) H (t , t ) 0 , t t0 , H (t , s) 0 , (t , s) D0 ; (A2) The second variable of H is the second consecutive non-negative partial derivative. We call the function H satisfies property P. Theorem 4. If the function H has the following properties: (B1) H satisfies the property P , (B2) h(t , s) C D0 , R , (t ) C1 I , R ;
1
1 1 (t ) R (t ) y ( n 1) (t ) G(t ) 1 1 y [ g ( t )] (t ) R (t ) where G(t ) g (t )Mg n 2 (t ) .
limsup t
t 1 H (t , s) ( s)Q( s) h(t , s) H (t , t0 ) t0 ( 1)
where 1
Thus, w(t ) (t )Q(t ) A(t )w(t ) B(t ) w (t ) .
1
1 ds B( s )
b
Q(t ) KE (t , t0 ) q(t , )(1 c[ g (t , )]) d ( ) a
G (t )
B( s)
1
,
.
(t ) R[ g (t )] Then (5) is oscillatory. Proof. Assume x(t ) is a positive solution of (5). Then from Lemma 1, we get y(t ) 0 , , y ( n 1) (t ) 0 y ( n ) (t ) 0
II.
MAIN RESULT
This section addresses Eq. (5). Theorem 3. Assume there exists C1 I , R such that t (s) R g (s) limsup ( s)Q( s) ds 1 t0 t 1 (s)G(s) . Then (5) is oscillatory. Proof. Let x(t) be a positive solution of (5). By Lemma 1 and Lemma 2, we get w(t) satisfies (8) i.e.
1
R(t) y
( 1)
1
A 1 leads to that B
1 (t ) R(t ) (t ) G(t ) 1 w(t ) w (t ) 1 1 (t ) ( 1) (t ) R(t ) (t ) R(t ) 1
1 (t ) R(t ) → w(t ) (t )Q(t ) . ( 1) 1 (t ) R(t ) Then integrating it with respect to t from T to t, we get that 1 t (s) R(s) w(t ) w(T ) (s)Q(s) ds T ( 1) 1 ( s) R( s)
for t which leads to that
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Q(t) y g(t) 0 .
1
1
(t )
y ( n 1) (t ) Let w(t ) (t ) R(t ) , t T . y[ g (t )] Combining with Lemma, we have w(t )
w(t ) (t )Q(t ) A(t )w(t ) B(t ) w (t ) .
This plus Au Bu
( n 1)
and
(t )Q(t ) A(t ) w(t ) B(t ) w
1
for t T and
(t )
1 t H (t , s) ( s)Q( s)ds H (t , s) w( s) A( s) w( s) B( s) w ( s) ds T T
t
H (t , T ) w(T ) 1 t H (t , s ) H (t , s) A( s) w( s) H (t , s) B( s) w ( s) ds T s
H (t , T ) w(T ) 1 t (9) h(t , s) H 1 (t , s) w( s) H (t , s ) B( s) w ( s) ds T 1 M 1 Using inequality Mu Nu for 1 ( 1) N M 0, N 0, u 0 , we have
JOURNAL OF NETWORKS, VOL. 9, NO. 7, JULY 2014
h(t , s) H 1 (t , s) w( s) H (t , s) B( s) w
1
( s)
1911
h(t , s) ( 1)
1
1
t 1 H (t , s ) ( s )Q( s )ds T H (t , T ) W (T ) F (T ) G (T )
B ( s)
Integrating it with respect to t for t T0 T t0 , we get that 1
h(t , s) H ( t , s ) ( s ) Q ( s ) ds T (10) ( 1) 1 B ( s) H (t , T0 ) w(T0 ) H (t , t0 ) w(T0 ) t
and
t
T0
t
T0
t0
Tt0
t0
H (t , s) ( s)Q( s)ds H (t , t0 )w(T0 )
t0
H (t , t0 ) ( s)Q( s)ds w(T0 ) t0 T0
(11)
In view of (11), we have 1 t h(t , s) 1 lim sup H (t , s) ( s)Q( s) ds t H (t , T ) T B ( s)
W (T )
Combing (D3-1) with (13), the following statements are true: (E1) W (T ) (T ) , t 1 H (t , s) ( s)Q(s)ds (T ) . T H (t , T ) By (12) and (E2), we obtain G (T ) F (T ) t
t 1 H (t , s) ( s)Q( s)ds T H (t , T ) which leads to that
W (T )
h(t , s) t 1 ds . (C2) limsup t H (t , t0 ) t0 B ( s) Then (5) is oscillatory. Theorem 6. If the function H satisfies conditions (B1) – (B3) and the following conditions: H (t , s) (D1) 0 lim liminf 0, s t0 t H (t , t0 ) 1
lim inf G (T ) F (T ) t
t 1 H (t , s) ( s)Q( s)ds (14) T H (t , T ) W (T ) (T )
W (T ) lim sup t
Hence we claim that
1
1 t h(t , s) 1 limsup H ( t , s ) ( s ) Q ( s ) ds (T ) t H (t , T ) T B ( s)
T
limsup B( s) ( s)ds (D3-2) t0
where (s) max ( s),0 , then (5) is oscillatory. Proof. From (9), we have t 1 H (t , s) ( s)Q(s)ds H (t , T ) T
1
H (t , T ) T Let t 1 1 F (t ) h ( t , s ) H (t , s) w( s)ds H (t , T ) T
t
H (t , s) B(s)w H (t , T ) T Therefore, we get that G(t )
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(s)ds
1
B( s) w ( s)ds
for t T
and G(t )
,
t
H (t , T ) T
H (t , s) B( s) w
1
( s)ds for t T1
1 s H ( t , s ) d B ( u ) w (u )du T T H (t , T ) 1 s t B(u ) w (u )du H (t , s ) T H (t , T ) T
( s)ds
1
t
T
t 1 1 w(T ) h ( t , s ) H (t , s ) w( s )ds H (t , T ) T
H (t , s) B( s) w
1
B( s) w ( s)ds
(16)
H (t , s) lim liminf 0 . s t0 t H (t , t0 ) In view of (16), there exists T1 T for 0 such that
1
t
(15)
By (D1), there exists 0 such that
(D3-1)
1
B( s) w ( s)ds
Otherwise, if
(D3) there exists a function (t ) C I , R such that
T
t h(t , s ) 1 ds , (D2) limsup t t H (t , t0 ) 0 B ( s)
t
(13)
(E2) limsup
Thus (5) is oscillatory. The following corollary is now a simple exercise. Corollary 5. Let the following conditions hold, t 1 (C1) limsup H (t , s) (s)Q(s) ds t H (t , t0 ) t0
t
(12)
t
1 H (t , s ) s B(u ) w (u )du ds T s T t
H (t , T )
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1
1 H (t , s) s B(u ) w (u )du ds T1 T s
H (t , T )
1 tn h(t n , s ) F (tn ) 1 1 ds T G (tn ) H (tn , T ) B (s)
t
t H (t , s) ds H (t , T ) T s H (t , T ) H (t , T ) which leads to lim G(t ) .
tn n1 T ,
n such that lim G(tn ) F (tn )
1
ds
1
t
Let a sequence
1
tn h(tn , s ) 1 H (tn , t0 ) t0 B ( s) which imply that
1
and tn for
tn h(tn , s ) 1 lim n H (t , t ) t0 B ( s) n 0 and
ds 1
tn h(tn , s ) 1 lim sup t n H (tn , t0 ) 0 B ( s) It contradicts (D2).
t
liminf G(t ) F (t ) t
Therefore there exists K > 0 and sufficiently large n such that G(tn ) F (tn ) K which leads to that F (tn ) K 1 1 G(tn ) G(tn ) 2
(17)
Hence
T
On
T
1
B( s) w ( s)ds holds.
the 1
ds .
other 1
B(s) ( s)ds B(s)w T
hand,
(s)ds
which contradicts (D3-2), then (5) is oscillatory.
In view of (17), we have
ACKNOWLEDGMENT
F (tn ) 1 F (tn ) 1 , G (tn ) 2 G (tn ) 2 and 1
F (tn ) 1 F (tn ) . 2 G(tn ) Combing with Holder inequality, we obtain tn 1 1 F (tn ) h ( t , s ) H (tn , s) w( s)ds n H (tn , T ) T
tn h(tn , s ) 1 1 1 B ( s ) H (tn , s) w( s)ds H (tn , T ) T 1 B ( s) 1
1 1 tn h(tn , s ) 1 ds H (tn , T ) T B ( s ) tn 1 * B( s) H (tn , s) w T H (tn , T )
F
→
1
1
( s)ds 1
tn h(tn , s ) 1 1 (tn ) T H (tn , T ) B ( s )
1
1
ds .
tn * B( s) H (tn , s) w ( s)ds T H ( t , T ) n Clearly, 1
1
1
1 tn h(tn , s ) F (tn ) 1 1 ds G(tn ) H (tn , T ) T B ( s) and
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This work was supported in part by a grant from Natural Science Foundation of Chongqing (No. cstc2013jcyjA10049) and Research Foundation of Chongqing University of Science and Technology (No.CK2011B36). REFERENCES [1] W. Peiguang, W. Yonghong and L. Caccetta, “Oscillation criteria for boundary value problems of high-order partial functional differential equations”, Journal of Computational and Applied Mathematics, vol. 206, pp. 567 – 577, Sept. 2007. [2] W. Peiguang, K. L. Teo and L. Yanqun, “Oscillation properties for even order neutral equations with distributed deviating arguments”, Journal of Computational and Applied Mathematics, vol.182, pp. 290 – 303, Oct. 2005. [3] W. Peiguang, “Oscillation properties for even order equation with distributed deviating arguments”, Journal of Computational and Applied Mathematics, vol. 182, pp. 209 – 303, 2005. [4] X. Zhiting, “Averaging technique and oscillation for even order damped delay differential equations”, Portugaliae Mathematica, vol. 63, pp. 132 – 142, 2006. [5] X. Zhiting and W. Peixuan, “Oscillation of second order neutral equations with distributed deviating argument”, Journal of Computational and Applied Mathematics, vol. 202, pp. 460–477, 2007. [6] W. Qiru, “Oscillation and asymptotics for second-order half-linear differential equations”, Applied Mathematics and Computation, vol. 122, pp. 253–266, 2001. [7] Z. Quanxin, L. Gao and Y. Yuanhong, “Oscillation of even order half-linear neutral differential equations with distributed delay”, Acta Mathematicae Applicatae Sinica, vol. 34, pp. 895-905, 2011. [8] Z. Quanxin, L. Gao and W. Shaoying, “Oscillatory and Asymptotic Behavior of a Second-Order Nonlinear Functional Differential Equations”, Communications in Theoretical Physics, vol. 232, pp. 342 – 356, 2012.
JOURNAL OF NETWORKS, VOL. 9, NO. 7, JULY 2014
[9] R. Xu and M. Fanwei, “Oscillation criteria for second order quasi-linear neutral delay differential equations”, Applied Mathematics and Computation, vol.192, pp. 216– 222, 2007. [10] M. Fanwei, “Oscillation criteria for certain even order quasi-linear neutral differential equations with distributed deviating arguments”, Applied Mathematics and Computation, vol. 19, pp. 458-464, 2007. [11] O. Dosly and A. Lomtatidze, “Oscillation and nonoscillation criteria for half-liner second order differential equation”, Hirshima Math, vol. 36, pp.203-219, 2006. [12] X. Yang, “Oscillation criteria for a class of quasi-linear, differential equation”, Appl. Math. .Comput, vol.15, pp. 225 – 229, 2004. [13] R. Xu and F. Meng, “Some new oscillation criteria for second order quasi-linear neutral delays differential equation”, Appl. Math. Comput, vol. 182, pp. 797-803, 2006. [14] C. G. Philos, “Oscillation theorems for linear differential equations of second order”, Arch Math, vo.53, 482-492, 1989. [15] P. G. Wang and Y. H. Wu, “Further results on differential inequality of a class of second Order neutral type”, Tamkng J Math, vol. 35, pp. 43 – 51, 2004. [16] Z. Xu and Y. Xia, “Integral averaging technique and oscillation criteria even order delay differential equations”, J. Math. anal. Appl.,vol. 292, pp. 238 – 246, 2004. [17] R. P. Agarwal, S. R. Grace and O. D. Regan, “Oscillation criteria for certain n th order differential equations with
© 2014 ACADEMY PUBLISHER
1913
[18] [19]
[20]
[21]
[22] [23] [24] [25]
deviating arguments”, J.math anal.appl., vol. 262, pp. 601 – 622, 2001. H. J. Hale, Theory of functional differential equations, Dover Publications Inc, 2003. E. Tun and H. Avc, “New oscillation theorems for a class of second-order damped nonlinear differential equations”, Ukrainian Mathematical Journal, vol. 63, pp. 1441 – 1457, 2012. G. Chunxia and W. Peiguang, “Oscillation theorems for even order damped equations with distributed deviating arguments”, Abstract and Applied Analysis, vol. 2013, Article ID 393892. S. Fisharova and R. Marik, “Oscillation of half-linear differential equations with delay”, Abstract and Applied Analysis, vol. 2013, Article ID 583147. Z. Oplustil and J. Sremr, “On oscillations of solutions to second-order linear delay differential equations”, Georgian Mathematical Journal, vol. 20, pp. 65 – 94, 2013. B. Baculikova, T. Li and J. Dzurina, “Oscillation theorems for second-order superlinear neutral differential equations”, Mathematica Slovaca, vol. 63, pp. 123 – 134, 2013. W. Zhongtang, W. Mengjiao, J. Jianxiu and F. Jiuchao, “A novel strange attractor and its dynamic analysis”, Journal of Multimedia, vol. 9, pp. 408 – 415, Mar 2014. S. Shanhui, L. Zhuangzhuang, W. Yimin and X.Yan, “Contracting similarity fixed point of general sierpinski gasket”, Journal of Multimedia, vol. 8, pp. 816 – 822, Dec 2013.
JOURNAL OF NETWORKS, VOL. 9, NO. 7, JULY 2014
Oscillation Criteria for Damping Quasi-Linear Neutral Differential Equations with Distributed Deviating Arguments Xuequan Tian, Yi Yang*, and Hongke Wang Department of Mathematics and Physics, Chongqing University of Science and Technology, Chongqing, China *Corresponding author, Email: [email protected], [email protected], [email protected]
Abstract—In this paper, we investigate the oscillation criteria for damp quasi-linear neutral differential equations with distributed deviating arguments. By using the generalized Riccati transformation and integral averaging, several new criteria that ensure the oscillation of solutions are obtained. Our results not only include some previously known results, but apply to some damped differential equations that have not been investigated so far. Index Terms—Half-Linear Differential Oscillatory Behavior; Riccati Transformation
I.
Equation;
Furthermore, O. Dosly et al. [11] deal with the following equation:
r (t ) x
q(t , ) x g (t , ) d ( ) 0 b
a
(1)
Later, F. W. Meng [10] studies the oscillation criteria for even order differential equation: ( n 1) 1
r (t ) x(t ) c(t ) x( t ) q(t ) f x g (t ) 0
© 2014 ACADEMY PUBLISHER doi:10.4304/jnw.9.7.1908-1913
( n 1) x(t ) c(t ) x( t ) (2)
(3)
and X. Yang [12] studies the oscillation criteria for the following equation: r (t ) ( y ( n 1) (t ))
Qualitative analysis for different types of differential equations with damping is a fertile research area and increasingly attracts many concerns, especially in the oscillatory behavior of solutions of solutions of higher order neutral differential equations which are of both theoretical and practical interest. Some applicable examples can be found in the monograph of Hale [18]. There have been some results on the oscillatory and asymptotic behavior of quasi-linear neutral differential equations. To the best of our knowledge, very little has been done with distributed deviating arguments. However, we note that in many area of their actual application, models describing these problems are often effected by such factors as seasonal changes. Therefore it is necessary, either theoretically or practically, to study a type of equation in a more general sense differential equations with distributed deviating arguments. Here we particularly refer the reader to the papers [1–17, 19–25]. In particular, P. G.Wang et al. [3] investigate and establish several oscillatory criteria for the following even order differential equation with distributed deviating arguments: (n)
(t ) p(t ) x ( n 1) (t )
q(t ) x g (t ) 0
INTRODUCTION AND LEMMAS
x(t ) c(t ) x(t )
( n 1)
b
a
q(t , ) x g (t , ) d ( ) 0
(4)
Motivated by the previous works, this paper investigates the oscillation criteria for the following class of even order functional differential equations with damping: r (t ) ( y ( n 1) (t )) p(t ) ( y ( n 1) (t )) q(t , ) f x g (t , ) d ( ) 0 b
(5)
a
where n is an even positive integer and 1
(s) s s, 0 , y(t ) x(t ) c(t ) x(t ) . It is clear that (5) includes (1) – (4), i.e. (5) (1) for r(t) = 1, p(t) = 0, f(x) = x; (5) (2) for p(t) = 0, g(t,ζ) = g(t), q(t,ζ) = q(t),
b
a
d ( ) 1 ;
(5) (3) for c(t) = 0, g(t,ζ) = g(t),
q(t,ζ) = q(t), f(x) = φ(x),
b
a
d ( ) 1 ;
(5) (4) for p(t) = 0, f(x) = φ(x). Our aims to get the criteria for the oscillatory solutions of (5). Throughout this paper, we will set forth conditions to establish the criteria for the oscillatory solutions of (5). (H1) P(t) > 0, 1
(H2)
f ( x) x
1
x
K 0,
t0
1 dt , R(t )
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1909
1
1
1 lim ds t t0 R ( s ) t
(H3)
Thus,
.
Furthermore,
E (t , t0 )r (t ) ( y ( n 1) (t )) E (t , t0 ) q(t , ) f x g (t , ) d ( ) 0 b
y
The
1
y ( n ) (t ) 0
This plus r’(t)≥0 leads to that y(n)(t) ≤ 0. By Corprolast Lemma, we get y(t ) 0 . (ii) From y’(t) > 0, we have y(t) = x(t) + c(t)x(t - ) > 0 such that x(t ) y(t ) c(t ) x(t ) y(t ) c(t ) y(t ) (1 c(t )) y(t ) . By
f ( x) 1
K 0 and (s) s
1
s, 0 ,
x x we obtain
E (t , t0 ) q(t , ) f x g (t , ) d ( )
E (t , t0 ) Kq(t , ) x g (t , ) d ( ) b
a
E (t , t0 ) Kq(t , ) 1 C ( g (t , )) y( g (t, ))d ( )
b
E (t , t0 ) q(t , ) f x g (t , ) d ( ) b
decreasing
and
a
y ( n 1) (t ) 0 . Now, we claim that
E (t , t0 ) y (t ) Kq(t , ) 1 C ( g (t , )) d ( ) 0
( n 1)
R(t ) y ( n 1) (t )
(t ) 0 , we have
Q(t) y g(t) 0 .
Lemma 2. Suppose x(t ) is a positive solution of (5) and
R(t ) y ( n 1) (t ) 0 ;
R(t ) y ( n 1) (t ) M 0, R(t ) y ( n 1) (t ) M .
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a
b
y ( n 1) (t ) 0 .
Otherwise if y
r (t ) y ( n 1) (t ) r (t ) y ( n 1) (t )
Thus, there exists (t ) such that (t ) g (t , ) and (t ) 1, (t ) 0 which leads to that (t ) for t . It implies that
Proof. (i) Since x(t) > 0, we have R(t ) y ( n 1) (t ) 0 . R(t ) y ( n 1) (t )
and r (t ) y ( n 1) (t ) r (t ) y ( n 1) (t )
a
a
is
t
Working inductively, we get y ( n 3) (t ) 0, , y(t ) 0 which contradicts y(t) > 0. Hence the claim is established. Consequently, we will prove y ( n ) (t ) 0 . It is clear that y ( n 1) (t ) 0 r (t ) y ( n 1) (t ) 0
b
Q(t ) KE (t , t0 ) q(t , )(1 c[ g (t , )]) d ( ) .
(t )
t
a
b
1
E (t , t0 ) Kq(t , ) x g (t , ) d ( )
R(t ) y
1 y (t ) y (T ) M ds T R( s ) Then lim y ( n 2) (t ) . 1
( n 2)
b
Q(t) y g(t) 0
( n 1)
M . Integrating (t ) R(t )
a
t p( s ) R(t ) E (t , t0 )r (t ) r (t ) exp ds t0 r ( s)
where
( n 1)
1
y ( n 1) (t ) 0, y ( n) (t ) 0, y(t ) 0 , (t )
(t ) 0 , we obtain y
( n 2)
a
( n 1)
( n 1)
on both sides, we have
following lemmas play a crucial role in the qualitative theory to proof the main results to be presented in this paper. Lemma 1. Suppose x(t ) is a positive solution of (5) and y (t ) satisfies the following conditions:
R(t) y
from
1
In the sequel, it will be always assumed that solutions of (5) exist for any t0 ≥ 0.A solution x(t) of (5) is called eventually positive solution (or negative solution) if there exists a sufficiently large positive number t1 > t0 such that x(t) > 0 (or x(t) < 0) for all t > t1. A nontrivial solution x(t) of (5) is called oscillatory if it has arbitrary large zeros; otherwise it is called nonoscillatory. Equation (5) is called oscillatory if all its solutions are oscillatory. If not otherwise stated, we assume that the solution of (5) is the eventually positive solution. The fundamental terminologies concerned with differential equations can be found in Ref. [18]. This section presents some basic relations that will be used later. t p( s) ds Let E (t , t0 ) exp t0 r ( s)
Thus
M (t ) R(t )
1
1 s p(u ) lim exp du ds t t0 R ( s ) t0 r (u ) where x(t) is a solution of (5) and x(t ) C1 Tx , , R , Tx t0 . t
and
y
( n 1)
1 1 y ( n 1) (t ) (t ) R (t ) y ( n 1) (t ) w(t ) (t ) R(t ) (6) y[ g (t )] y[ g (t )]
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y[ g (t )] Mt n 2 y ( n 1) (t ), (0,1)
1 (s) R(s) ( s ) Q ( s ) ds . T ( 1) 1 ( s) R( s) Hence (5) is oscillatory. Let D (t , s) t s t0 , D0 (t , s) t s t0 . If
(7)
t
then w(t ) (t )Q(t ) A(t ) w(t ) B(t ) w
where A(t )
1
(8) (t )
(t ) G(t ) , B(t ) . 1 (t ) (t ) R[ g (t )]
Proof. From (6) and (7), we get w(t ) (t )Q(t ) y ( n 1) (t )
y 1[ g (t )] y [ g (t )] g (t ) y 2 [ g (t )]
(t )Q(t ) y ( n 1) (t )
g (t )Mg n 2 (t ) y ( n 1) [ g (t )] y1 [ g (t )]
Since y ( n 1) (t ) is decreasing, we have 1
H (t , s) A( s) H (t , s) h(t , s) H 1 (t , s) s where (t , s) D0 ; (B4)
(B3)
G(t ) y ( n 1) [ g (t )] y1 [ g (t )]
w(t ) (t )Q(t ) y ( n 1) (t )
the function H C D, R satisfies the following conditions: (A1) H (t , t ) 0 , t t0 , H (t , s) 0 , (t , s) D0 ; (A2) The second variable of H is the second consecutive non-negative partial derivative. We call the function H satisfies property P. Theorem 4. If the function H has the following properties: (B1) H satisfies the property P , (B2) h(t , s) C D0 , R , (t ) C1 I , R ;
1
1 1 (t ) R (t ) y ( n 1) (t ) G(t ) 1 1 y [ g ( t )] (t ) R (t ) where G(t ) g (t )Mg n 2 (t ) .
limsup t
t 1 H (t , s) ( s)Q( s) h(t , s) H (t , t0 ) t0 ( 1)
where 1
Thus, w(t ) (t )Q(t ) A(t )w(t ) B(t ) w (t ) .
1
1 ds B( s )
b
Q(t ) KE (t , t0 ) q(t , )(1 c[ g (t , )]) d ( ) a
G (t )
B( s)
1
,
.
(t ) R[ g (t )] Then (5) is oscillatory. Proof. Assume x(t ) is a positive solution of (5). Then from Lemma 1, we get y(t ) 0 , , y ( n 1) (t ) 0 y ( n ) (t ) 0
II.
MAIN RESULT
This section addresses Eq. (5). Theorem 3. Assume there exists C1 I , R such that t (s) R g (s) limsup ( s)Q( s) ds 1 t0 t 1 (s)G(s) . Then (5) is oscillatory. Proof. Let x(t) be a positive solution of (5). By Lemma 1 and Lemma 2, we get w(t) satisfies (8) i.e.
1
R(t) y
( 1)
1
A 1 leads to that B
1 (t ) R(t ) (t ) G(t ) 1 w(t ) w (t ) 1 1 (t ) ( 1) (t ) R(t ) (t ) R(t ) 1
1 (t ) R(t ) → w(t ) (t )Q(t ) . ( 1) 1 (t ) R(t ) Then integrating it with respect to t from T to t, we get that 1 t (s) R(s) w(t ) w(T ) (s)Q(s) ds T ( 1) 1 ( s) R( s)
for t which leads to that
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Q(t) y g(t) 0 .
1
1
(t )
y ( n 1) (t ) Let w(t ) (t ) R(t ) , t T . y[ g (t )] Combining with Lemma, we have w(t )
w(t ) (t )Q(t ) A(t )w(t ) B(t ) w (t ) .
This plus Au Bu
( n 1)
and
(t )Q(t ) A(t ) w(t ) B(t ) w
1
for t T and
(t )
1 t H (t , s) ( s)Q( s)ds H (t , s) w( s) A( s) w( s) B( s) w ( s) ds T T
t
H (t , T ) w(T ) 1 t H (t , s ) H (t , s) A( s) w( s) H (t , s) B( s) w ( s) ds T s
H (t , T ) w(T ) 1 t (9) h(t , s) H 1 (t , s) w( s) H (t , s ) B( s) w ( s) ds T 1 M 1 Using inequality Mu Nu for 1 ( 1) N M 0, N 0, u 0 , we have
JOURNAL OF NETWORKS, VOL. 9, NO. 7, JULY 2014
h(t , s) H 1 (t , s) w( s) H (t , s) B( s) w
1
( s)
1911
h(t , s) ( 1)
1
1
t 1 H (t , s ) ( s )Q( s )ds T H (t , T ) W (T ) F (T ) G (T )
B ( s)
Integrating it with respect to t for t T0 T t0 , we get that 1
h(t , s) H ( t , s ) ( s ) Q ( s ) ds T (10) ( 1) 1 B ( s) H (t , T0 ) w(T0 ) H (t , t0 ) w(T0 ) t
and
t
T0
t
T0
t0
Tt0
t0
H (t , s) ( s)Q( s)ds H (t , t0 )w(T0 )
t0
H (t , t0 ) ( s)Q( s)ds w(T0 ) t0 T0
(11)
In view of (11), we have 1 t h(t , s) 1 lim sup H (t , s) ( s)Q( s) ds t H (t , T ) T B ( s)
W (T )
Combing (D3-1) with (13), the following statements are true: (E1) W (T ) (T ) , t 1 H (t , s) ( s)Q(s)ds (T ) . T H (t , T ) By (12) and (E2), we obtain G (T ) F (T ) t
t 1 H (t , s) ( s)Q( s)ds T H (t , T ) which leads to that
W (T )
h(t , s) t 1 ds . (C2) limsup t H (t , t0 ) t0 B ( s) Then (5) is oscillatory. Theorem 6. If the function H satisfies conditions (B1) – (B3) and the following conditions: H (t , s) (D1) 0 lim liminf 0, s t0 t H (t , t0 ) 1
lim inf G (T ) F (T ) t
t 1 H (t , s) ( s)Q( s)ds (14) T H (t , T ) W (T ) (T )
W (T ) lim sup t
Hence we claim that
1
1 t h(t , s) 1 limsup H ( t , s ) ( s ) Q ( s ) ds (T ) t H (t , T ) T B ( s)
T
limsup B( s) ( s)ds (D3-2) t0
where (s) max ( s),0 , then (5) is oscillatory. Proof. From (9), we have t 1 H (t , s) ( s)Q(s)ds H (t , T ) T
1
H (t , T ) T Let t 1 1 F (t ) h ( t , s ) H (t , s) w( s)ds H (t , T ) T
t
H (t , s) B(s)w H (t , T ) T Therefore, we get that G(t )
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(s)ds
1
B( s) w ( s)ds
for t T
and G(t )
,
t
H (t , T ) T
H (t , s) B( s) w
1
( s)ds for t T1
1 s H ( t , s ) d B ( u ) w (u )du T T H (t , T ) 1 s t B(u ) w (u )du H (t , s ) T H (t , T ) T
( s)ds
1
t
T
t 1 1 w(T ) h ( t , s ) H (t , s ) w( s )ds H (t , T ) T
H (t , s) B( s) w
1
B( s) w ( s)ds
(16)
H (t , s) lim liminf 0 . s t0 t H (t , t0 ) In view of (16), there exists T1 T for 0 such that
1
t
(15)
By (D1), there exists 0 such that
(D3-1)
1
B( s) w ( s)ds
Otherwise, if
(D3) there exists a function (t ) C I , R such that
T
t h(t , s ) 1 ds , (D2) limsup t t H (t , t0 ) 0 B ( s)
t
(13)
(E2) limsup
Thus (5) is oscillatory. The following corollary is now a simple exercise. Corollary 5. Let the following conditions hold, t 1 (C1) limsup H (t , s) (s)Q(s) ds t H (t , t0 ) t0
t
(12)
t
1 H (t , s ) s B(u ) w (u )du ds T s T t
H (t , T )
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1
1 H (t , s) s B(u ) w (u )du ds T1 T s
H (t , T )
1 tn h(t n , s ) F (tn ) 1 1 ds T G (tn ) H (tn , T ) B (s)
t
t H (t , s) ds H (t , T ) T s H (t , T ) H (t , T ) which leads to lim G(t ) .
tn n1 T ,
n such that lim G(tn ) F (tn )
1
ds
1
t
Let a sequence
1
tn h(tn , s ) 1 H (tn , t0 ) t0 B ( s) which imply that
1
and tn for
tn h(tn , s ) 1 lim n H (t , t ) t0 B ( s) n 0 and
ds 1
tn h(tn , s ) 1 lim sup t n H (tn , t0 ) 0 B ( s) It contradicts (D2).
t
liminf G(t ) F (t ) t
Therefore there exists K > 0 and sufficiently large n such that G(tn ) F (tn ) K which leads to that F (tn ) K 1 1 G(tn ) G(tn ) 2
(17)
Hence
T
On
T
1
B( s) w ( s)ds holds.
the 1
ds .
other 1
B(s) ( s)ds B(s)w T
hand,
(s)ds
which contradicts (D3-2), then (5) is oscillatory.
In view of (17), we have
ACKNOWLEDGMENT
F (tn ) 1 F (tn ) 1 , G (tn ) 2 G (tn ) 2 and 1
F (tn ) 1 F (tn ) . 2 G(tn ) Combing with Holder inequality, we obtain tn 1 1 F (tn ) h ( t , s ) H (tn , s) w( s)ds n H (tn , T ) T
tn h(tn , s ) 1 1 1 B ( s ) H (tn , s) w( s)ds H (tn , T ) T 1 B ( s) 1
1 1 tn h(tn , s ) 1 ds H (tn , T ) T B ( s ) tn 1 * B( s) H (tn , s) w T H (tn , T )
F
→
1
1
( s)ds 1
tn h(tn , s ) 1 1 (tn ) T H (tn , T ) B ( s )
1
1
ds .
tn * B( s) H (tn , s) w ( s)ds T H ( t , T ) n Clearly, 1
1
1
1 tn h(tn , s ) F (tn ) 1 1 ds G(tn ) H (tn , T ) T B ( s) and
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This work was supported in part by a grant from Natural Science Foundation of Chongqing (No. cstc2013jcyjA10049) and Research Foundation of Chongqing University of Science and Technology (No.CK2011B36). REFERENCES [1] W. Peiguang, W. Yonghong and L. Caccetta, “Oscillation criteria for boundary value problems of high-order partial functional differential equations”, Journal of Computational and Applied Mathematics, vol. 206, pp. 567 – 577, Sept. 2007. [2] W. Peiguang, K. L. Teo and L. Yanqun, “Oscillation properties for even order neutral equations with distributed deviating arguments”, Journal of Computational and Applied Mathematics, vol.182, pp. 290 – 303, Oct. 2005. [3] W. Peiguang, “Oscillation properties for even order equation with distributed deviating arguments”, Journal of Computational and Applied Mathematics, vol. 182, pp. 209 – 303, 2005. [4] X. Zhiting, “Averaging technique and oscillation for even order damped delay differential equations”, Portugaliae Mathematica, vol. 63, pp. 132 – 142, 2006. [5] X. Zhiting and W. Peixuan, “Oscillation of second order neutral equations with distributed deviating argument”, Journal of Computational and Applied Mathematics, vol. 202, pp. 460–477, 2007. [6] W. Qiru, “Oscillation and asymptotics for second-order half-linear differential equations”, Applied Mathematics and Computation, vol. 122, pp. 253–266, 2001. [7] Z. Quanxin, L. Gao and Y. Yuanhong, “Oscillation of even order half-linear neutral differential equations with distributed delay”, Acta Mathematicae Applicatae Sinica, vol. 34, pp. 895-905, 2011. [8] Z. Quanxin, L. Gao and W. Shaoying, “Oscillatory and Asymptotic Behavior of a Second-Order Nonlinear Functional Differential Equations”, Communications in Theoretical Physics, vol. 232, pp. 342 – 356, 2012.
JOURNAL OF NETWORKS, VOL. 9, NO. 7, JULY 2014
[9] R. Xu and M. Fanwei, “Oscillation criteria for second order quasi-linear neutral delay differential equations”, Applied Mathematics and Computation, vol.192, pp. 216– 222, 2007. [10] M. Fanwei, “Oscillation criteria for certain even order quasi-linear neutral differential equations with distributed deviating arguments”, Applied Mathematics and Computation, vol. 19, pp. 458-464, 2007. [11] O. Dosly and A. Lomtatidze, “Oscillation and nonoscillation criteria for half-liner second order differential equation”, Hirshima Math, vol. 36, pp.203-219, 2006. [12] X. Yang, “Oscillation criteria for a class of quasi-linear, differential equation”, Appl. Math. .Comput, vol.15, pp. 225 – 229, 2004. [13] R. Xu and F. Meng, “Some new oscillation criteria for second order quasi-linear neutral delays differential equation”, Appl. Math. Comput, vol. 182, pp. 797-803, 2006. [14] C. G. Philos, “Oscillation theorems for linear differential equations of second order”, Arch Math, vo.53, 482-492, 1989. [15] P. G. Wang and Y. H. Wu, “Further results on differential inequality of a class of second Order neutral type”, Tamkng J Math, vol. 35, pp. 43 – 51, 2004. [16] Z. Xu and Y. Xia, “Integral averaging technique and oscillation criteria even order delay differential equations”, J. Math. anal. Appl.,vol. 292, pp. 238 – 246, 2004. [17] R. P. Agarwal, S. R. Grace and O. D. Regan, “Oscillation criteria for certain n th order differential equations with
© 2014 ACADEMY PUBLISHER
1913
[18] [19]
[20]
[21]
[22] [23] [24] [25]
deviating arguments”, J.math anal.appl., vol. 262, pp. 601 – 622, 2001. H. J. Hale, Theory of functional differential equations, Dover Publications Inc, 2003. E. Tun and H. Avc, “New oscillation theorems for a class of second-order damped nonlinear differential equations”, Ukrainian Mathematical Journal, vol. 63, pp. 1441 – 1457, 2012. G. Chunxia and W. Peiguang, “Oscillation theorems for even order damped equations with distributed deviating arguments”, Abstract and Applied Analysis, vol. 2013, Article ID 393892. S. Fisharova and R. Marik, “Oscillation of half-linear differential equations with delay”, Abstract and Applied Analysis, vol. 2013, Article ID 583147. Z. Oplustil and J. Sremr, “On oscillations of solutions to second-order linear delay differential equations”, Georgian Mathematical Journal, vol. 20, pp. 65 – 94, 2013. B. Baculikova, T. Li and J. Dzurina, “Oscillation theorems for second-order superlinear neutral differential equations”, Mathematica Slovaca, vol. 63, pp. 123 – 134, 2013. W. Zhongtang, W. Mengjiao, J. Jianxiu and F. Jiuchao, “A novel strange attractor and its dynamic analysis”, Journal of Multimedia, vol. 9, pp. 408 – 415, Mar 2014. S. Shanhui, L. Zhuangzhuang, W. Yimin and X.Yan, “Contracting similarity fixed point of general sierpinski gasket”, Journal of Multimedia, vol. 8, pp. 816 – 822, Dec 2013.
