Skip to content

Advertisement

  • Research
  • Open Access

Forced oscillation of higher-order nonlinear neutral difference equations with positive and negative coefficients

Advances in Difference Equations20122012:110

https://doi.org/10.1186/1687-1847-2012-110

  • Received: 22 March 2012
  • Accepted: 2 July 2012
  • Published:

Abstract

In this paper, we study the forced oscillation of the higher-order nonlinear difference equation of the form

Δ m [ x ( n ) p ( n ) x ( n τ ) ] + q 1 ( n ) Φ α ( n σ 1 ) + q 2 ( n ) Φ β ( n σ 2 ) = f ( n ) ,

where m 1 , τ, σ 1 and σ 2 are integers, 0 < α < 1 < β are constants, Φ ( u ) = | u | 1 u , p ( n ) , q 1 ( n ) , q 2 ( n ) and f ( n ) are real sequences with p ( n ) > 0 . By taking all possible values of τ, σ 1 and σ 2 into consideration, we establish some new oscillation criteria for the above equation in two cases: (i) q 1 = q 1 ( n ) 0 , q 2 = q 2 ( n ) > 0 ; (ii) q 1 0 , q 2 < 0 .

MSC:39A10.

Keywords

  • forced oscillation
  • neutral difference equation
  • positive and negative coefficients
  • higher-order

1 Introduction

Qualitative theory of difference equations has received much attention in recent years due to its extensive applications in computer, probability theory, queuing problems, statistical problems, stochastic time series, combinatorial analysis, number theory, electrical networks, genetics in biology, economics, psychology, sociology, and so on [1, 2].

In this paper, we consider the oscillation of the following m th-order forced nonlinear difference equation of the form
Δ m [ x ( n ) p ( n ) x ( n τ ) ] + q 1 ( n ) Φ α ( n σ 1 ) + q 2 ( n ) Φ β ( n σ 2 ) = f ( n ) ,
(1)
where m 1 , τ, σ 1 and σ 2 are integers, Φ ( u ) = | u | 1 u , p ( n ) , q 1 ( n ) , q 2 ( n ) and f ( n ) are real sequences defined on N = { 0 , 1 , 2 , } with p ( n ) > 0 , 0 < α < 1 < β are constants, and
Δ x ( n ) = x ( n + 1 ) x ( n ) , Δ i x ( n ) = Δ ( Δ i 1 x ( n ) ) , 2 i m .

As usual, a solution of Eq. (1) is said to be oscillatory, if for every integer N 0 , there exists n N such that x ( n ) x ( n + 1 ) 0 ; otherwise, it is called nonoscillatory.

For the continuous version of Eq. (1), many authors have studied its oscillation (see monograph [3] and references therein). To the best of our knowledge, little has been known about the forced oscillation of Eq. (1) with positive and negative coefficients ( q 1 0 q 2 > 0 or q 1 0 q 2 < 0 ) and mixed nonlinearities ( 0 < α < 1 β > 1 ). For some particular cases of Eq. (1), there have been many oscillation results in [419], to name a few. Motivated by the work in [2022], we study the forced oscillation of Eq. (1) in this paper.

The main contribution of this paper is that we establish some new oscillation criteria for Eq. (1) with positive and negative coefficients and mixed nonlinearities. Unlike some existing results in the literature, all possible values of delays τ, σ 1 and σ 2 are considered.

2 Main results

Throughout this paper, we denote
ϕ 0 ( n , s ) = ϕ ( n , s ) = ( n s ) ( k ) = ( n s ) ( n s + 1 ) ( n s + k 1 ) , k m ,
(2)
ϕ i ( n , s ) = ( 1 ) i Δ s i ϕ ( n , s ) = C k i ( n s ) ( k i ) , i = 1 , 2 , , m .
(3)
By the straightforward computation, it is not difficult to see that
{ ϕ ( n , s ) = 0 , n s n + k 1 , ϕ i ( n + i + 1 , n + m ) = 0 , i = 0 , 1 , 2 , , m 1 , ϕ m ( n , s ) 0 , 0 s n 1 ,
(4)
and
lim n ϕ i ( n + i + 1 , n 0 ) ϕ ( n + 1 , n 0 ) = o ( 1 ) , i = 1 , 2 , , m ,
(5)

where n 0 0 is an integer. We also denote s = l k = 0 if k < l .

The following two facts can be easily proved.

Fact 1. Set F ( x ) = a x b x λ , where x 0 , a 0 and b > 0 . If λ > 1 , F ( x ) obtains its maximum F max = ( λ 1 ) λ λ 1 λ a λ λ 1 b 1 1 λ .

Fact 2. Set G ( x ) = c x d x λ , where x 0 , c > 0 and d 0 . If 0 < λ < 1 , G ( x ) obtains its minimum G min = ( λ 1 ) λ λ 1 λ c λ λ 1 d 1 1 λ .

We now present the main results of this paper as follows.

Theorem 1 Assume that q 1 ( n ) 0 , q 2 ( n ) > 0 , σ 1 m and σ 2 τ m . If
lim n sup 1 ϕ ( n + 1 , n 0 ) { s = n 0 n + m 1 [ ϕ ( n , s ) f ( s ) Q 2 ( n , s ) ] + s = n 0 n σ 1 1 Q 1 ( n , s ) } = + ,
(6)
lim n inf 1 ϕ ( n + 1 , n 0 ) { s = n 0 n + m 1 [ ϕ ( n , s ) f ( s ) + Q 2 ( n , s ) ] s = n 0 n σ 1 1 Q 1 ( n , s ) } = ,
(7)
where
Q 1 ( n , s ) = ( α 1 ) α α 1 α [ ϕ m ( n + m , s ) ] α α 1 [ ϕ ( n , s + σ 1 ) | q 1 ( s + σ 1 ) | ] 1 1 α ,
(8)
Q 2 ( n , s ) = ( β 1 ) β β 1 β [ ϕ m ( n + m , s ) p ( s ) ] β β 1 [ ϕ ( n , s + σ 2 τ ) q 2 ( s + σ 2 τ ) ] 1 1 β ,
(9)

all solutions of Eq. (1) are oscillatory.

Proof Assume to the contrary that there exists a nontrivial solution x ( n ) of Eq. (1) such that x ( n ) is nonoscillatory. That is, x ( n ) does not change sign eventually. Without loss of generality, let x ( n σ 1 ) 0 , x ( n σ 2 ) 0 , x ( n τ ) 0 for n n 0 , where n 0 0 is sufficiently large. By the straightforward computation, we have
F 1 ( n , s ) F 2 ( n , s ) = s = n 0 n + m 1 ϕ ( n , s ) f ( s ) ,
(10)
where
F 1 ( n , s ) = s = n 0 n + m 1 ϕ ( n , s ) Δ m x ( s ) s = n 0 n + m 1 ϕ ( n , s ) | q 1 ( s ) | x α ( s σ 1 ) , F 2 ( n , s ) = s = n 0 n + m 1 ϕ ( n , s ) Δ m [ p ( s ) x ( n τ ) ] s = n 0 n + m 1 ϕ ( n , s ) q 2 ( s ) x β ( s σ 2 ) .
Noting that
ϕ ( n , s ) Δ x ( s ) = Δ [ ϕ ( n , s 1 ) x ( s ) ] + ϕ 1 ( n , s 1 ) x ( s ) = Δ [ ϕ ( n + 1 , s ) x ( s ) ] + ϕ 1 ( n + 1 , s ) x ( s ) ,
we can get from (2), (3) and (4) that
s = n 0 n + m 1 ϕ ( n , s ) Δ m x ( s ) = s = n 0 n + m 1 Δ [ ϕ ( n + 1 , s ) Δ m 1 x ( s ) ] + s = n 0 n + m 1 ϕ 1 ( n + 1 , s ) Δ m 1 x ( s ) = ϕ ( n + 1 , n 0 ) Δ m 1 x ( n 0 ) + s = n 0 n + m 1 ϕ 1 ( n + 1 , s ) Δ m 1 x ( s ) = ϕ ( n + 1 , n 0 ) Δ m 1 x ( n 0 ) + s = n 0 n + m 1 Δ [ ϕ 1 ( n + 2 , s ) Δ m 2 x ( s ) ] + s = n 0 n + m 1 ϕ 2 ( n + 2 , s ) Δ m 2 x ( s ) = ϕ ( n + 1 , n 0 ) Δ m 1 x ( n 0 ) ϕ 1 ( n + 2 , n 0 ) Δ m 2 x ( n 0 ) + s = n 0 n + m 1 ϕ 2 ( n + 2 , s ) Δ m 2 x ( s ) = i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 x ( n 0 ) + s = n 0 n + m 1 ϕ m ( n + m , s ) x ( s ) .
(11)
Since ϕ ( n , s ) = 0 for n s n + m 1 due to (4), we get from (11) that
F 1 ( n , s ) = s = n 0 n + m 1 ϕ m ( n + m , s ) x ( s ) s = n 0 n 1 ϕ ( n , s ) | q 1 ( s ) | x α ( s σ 1 ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 x ( n 0 ) = s = n 0 n + m 1 ϕ m ( n + m , s ) x ( s ) s = n 0 σ 1 n σ 1 1 ϕ ( n , s + σ 1 ) | q 1 ( s + σ 1 ) | x α ( s ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 x ( n 0 ) .
(12)
Noting that σ 1 m , we have that n + m 1 n σ 1 1 . Therefore, we get from (12) that
F 1 ( n , s ) s = n 0 n σ 1 1 [ ϕ m ( n + m , s ) x ( s ) ϕ ( n , s + σ 1 ) | q 1 ( s + σ 1 ) | x α ( s ) ] s = n 0 σ 1 n 0 1 ϕ ( n , s + σ 1 ) | q 1 ( s + σ 1 ) | x α ( s ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 x ( n 0 ) .
(13)
By Fact 2 and (13), it is not difficult to see that
F 1 ( n , s ) s = n 0 n σ 1 1 Q 1 ( n , s ) s = n 0 σ 1 n 0 1 ϕ ( n , s + σ 1 ) | q 1 ( s + σ 1 ) | x α ( s ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 x ( n 0 ) ,
(14)

where Q 1 ( n , s ) is defined by (8).

On the other hand, similar to the above analysis, we have that
F 2 ( n , s ) = s = n 0 n + m 1 ϕ m ( n + m , s ) p ( s ) x ( s τ ) s = n 0 n + m 1 ϕ ( n , s ) q 2 ( s ) x β ( s σ 2 ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 [ p ( n 0 ) x ( n 0 τ ) ] = s = n 0 n + m 1 ϕ m ( n + m , s ) p ( s ) x ( s τ ) s = n 0 n 1 ϕ ( n , s ) q 2 ( s ) x β ( s σ 2 ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 [ x ( n 0 τ ) p ( n 0 ) ] = s = n 0 n + m 1 ϕ m ( n + m , s ) p ( s ) x ( s τ ) s = n 0 σ 2 + τ n σ 2 + τ 1 ϕ ( n , s + σ 2 τ ) q 2 ( s + σ 2 τ ) x β ( s τ ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 [ p ( n 0 ) x ( n 0 τ ) ] .
(15)
Since σ 2 τ m , we have that n σ 2 + τ 1 n + m 1 . By (15), we get
F 2 ( n , s ) s = n 0 n + m 1 [ ϕ m ( n + m , s ) p ( s ) x ( s τ ) ϕ ( n , s + σ 2 τ ) q 2 ( s + σ 2 τ ) x β ( s τ ) ] + s = n 0 n 0 σ 2 + τ 1 ϕ ( n , s + σ 2 τ ) q 2 ( s + σ 2 τ ) x β ( s τ ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 [ p ( n 0 ) x ( n 0 τ ) ] .
(16)
By Fact 1 and (16), we have that
F 2 ( n , s ) s = n 0 n + m 1 Q 2 ( n , s ) + s = n 0 n 0 σ 2 + τ 1 ϕ ( n , s + σ 2 τ ) q 2 ( s + σ 2 τ ) x β ( s τ ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 [ p ( n 0 ) x ( n 0 τ ) ] ,
(17)

where Q 2 ( n , s ) is defined by (9).

Multiplying 1 ϕ ( n + 1 , n 0 ) on both sides of (10), by (14), (17) and (5), we have that there exists a constant M 1 such that
1 ϕ ( n + 1 , n 0 ) { s = n 0 n + m 1 [ ϕ ( n , s ) f ( s ) + Q 2 ( n , s ) ] s = n 0 n σ 1 1 Q 1 ( n , s ) } M 1 ,

which contradicts (7). For the case when x ( n ) is eventually negative, we can similarly get a contradiction to (6). This completes the proof of Theorem 1. □

Theorem 2 Assume that q 1 ( n ) 0 , q 2 ( n ) < 0 , σ 1 τ m and σ 2 m . If
lim n sup 1 ϕ ( n + 1 , n 0 ) { s = n 0 n + m 1 [ ϕ ( n , s ) f ( s ) P 1 ( n , s ) ] + s = n 0 n σ 2 + τ 1 P 2 ( n , s ) } = + ,
(18)
lim n inf 1 ϕ ( n + 1 , n 0 ) { s = n 0 n + m 1 [ ϕ ( n , s ) f ( s ) + P 1 ( n , s ) ] s = n 0 n σ 2 + τ 1 P 2 ( n , s ) } = ,
(19)
where
P 1 ( n , s ) = ( β 1 ) β β 1 β [ ϕ m ( n + m , s ) ] β β 1 [ ϕ ( n , s + σ 2 ) | q 2 ( s + σ 2 ) | ] 1 1 β ,
(20)
P 2 ( n , s ) = ( α 1 ) α α 1 α [ ϕ m ( n + m , s ) p ( s ) ] α α 1 [ ϕ ( n , s + σ 1 τ ) q 1 ( s + σ 1 τ ) ] 1 1 α ,
(21)

all solutions of Eq. (1) are oscillatory.

Proof Suppose to the contrary that there exists a nontrivial solution x ( n ) of Eq. (1) such that x ( n ) is nonoscillatory. We may let x ( n σ 1 ) 0 , x ( n σ 2 ) 0 , x ( n τ ) 0 for n n 0 , where n 0 0 is sufficiently large. By the straightforward computation, we get from Eq. (1) that
G 1 ( n , s ) G 2 ( n , s ) = s = n 0 n + m 1 ϕ ( n , s ) f ( s ) ,
(22)
where
G 1 ( n , s ) = s = n 0 n + m 1 ϕ ( n , s ) Δ m x ( s ) s = n 0 n + m 1 ϕ ( n , s ) | q 2 ( s ) | x β ( s σ 2 ) , G 2 ( n , s ) = s = n 0 n + m 1 ϕ ( n , s ) Δ m [ p ( s ) x ( n τ ) ] s = n 0 n + m 1 ϕ ( n , s ) q 1 ( s ) x α ( s σ 1 ) .
Noticing that ϕ ( n , s ) = 0 for n s n + m 1 , we get from (11) that
G 1 ( n , s ) = s = n 0 n + m 1 ϕ m ( n + m , s ) x ( s ) s = n 0 n 1 ϕ ( n , s ) | q 2 ( s ) | x β ( s σ 2 ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 x ( n 0 ) = s = n 0 n + m 1 ϕ m ( n + m , s ) x ( s ) s = n 0 σ 2 n σ 2 1 ϕ ( n , s + σ 2 ) | q 2 ( s + σ 2 ) | x β ( s ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 x ( n 0 ) .
(23)
Since σ 2 m , we have that n + m 1 n σ 2 1 . Thus, we can get from (23) that
G 1 ( n , s ) s = n 0 n + m 1 [ ϕ m ( n + m , s ) x ( s ) ϕ ( n , s + σ 2 ) | q 2 ( s + σ 2 ) | x β ( s ) ] + s = n 0 n 0 σ 2 1 ϕ ( n , s + σ 2 ) | q 2 ( s + σ 2 ) | x β ( s ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 x ( n 0 ) .
(24)
By Fact 1 and (24), it is easy to see that
G 1 ( n , s ) s = n 0 n + m 1 P 1 ( n , s ) + s = n 0 n 0 σ 2 1 ϕ ( n , s + σ 2 ) | q 2 ( s + σ 2 ) | x β ( s ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 x ( n 0 ) ,
(25)

where P 1 ( n , s ) is defined by (20).

On the other hand, similar to the computation of (11), we can get
G 2 ( n , s ) = s = n 0 n + m 1 ϕ m ( n + m , s ) p ( s ) x ( s τ ) s = n 0 n 1 ϕ ( n , s ) q 1 ( s ) x α ( s σ 1 ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 [ p ( n 0 ) x ( n 0 τ ) ] = s = n 0 n + m 1 ϕ m ( n + m , s ) p ( s ) x ( s τ ) s = n 0 σ 1 + τ n σ 1 + τ 1 ϕ ( n , s + σ 1 τ ) q 1 ( s + σ 1 τ ) x α ( s τ ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 [ p ( n 0 ) x ( n 0 τ ) ] .
(26)
Noting that σ 1 τ m implies n + m 1 n σ 1 + τ 1 , we get from (26) that
G 2 ( n , s ) s = n 0 n σ 1 + τ 1 [ ϕ m ( n + m , s ) p ( s ) x ( s τ ) ϕ ( n , s + σ 1 τ ) q 1 ( s + σ 1 τ ) x α ( s τ ) ] s = n 0 σ 1 + τ n 0 1 ϕ ( n , s + σ 1 τ ) q 1 ( s + σ 1 τ ) x α ( s τ ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 [ p ( n 0 ) x ( n 0 τ ) ] .
(27)
By Fact 2 and (27), we have that
G 2 ( n , s ) s = n 0 n σ 1 + τ 1 P 2 ( n , s ) s = n 0 σ 1 + τ n 0 1 ϕ ( n , s + σ 1 τ ) q 1 ( s + σ 1 τ ) x α ( s τ ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 [ p ( n 0 ) x ( n 0 τ ) ] ,
(28)

where P 2 ( n , s ) is defined by (21).

Multiplying 1 ϕ ( n + 1 , n 0 ) on both sides of (22), from (25), (28) and (5), we have that there exists a constant M 2 such that
1 ϕ ( n + 1 , n 0 ) { s = n 0 n + m 1 [ ϕ ( n , s ) f ( s ) P 1 ( n , s ) ] + s = n 0 n σ 2 + τ 1 P 2 ( n , s ) } M 2 .

This is a contradiction to (18). For the case when x ( n ) is eventually negative, we can similarly get a contradiction to (19). This completes the proof of Theorem 2. □

By Theorems 1 and 2, the following two corollaries are immediate.

Corollary 1 Assume that q 1 ( n ) 0 , q 2 ( n ) > 0 and σ 2 τ m . If
lim n sup 1 ϕ ( n + 1 , n 0 ) s = n 0 n + m 1 [ ϕ ( n , s ) f ( s ) Q 2 ( n , s ) ] = + , lim n inf 1 ϕ ( n + 1 , n 0 ) s = n 0 n + m 1 [ ϕ ( n , s ) f ( s ) + Q 2 ( n , s ) ] = ,

where Q 2 ( n , s ) is defined by (9), all solutions of Eq. (1) are oscillatory for any constant σ 1 .

Proof In fact, we have that F 1 ( n , s ) 0 for any constant σ 1 since q 1 ( n , s ) 0 . So, we can drop F 1 ( n , s ) in the estimation of (10). The other proof runs as that of Theorem 1, and hence it is omitted. □

Corollary 2 Assume that q 1 ( n ) 0 , q 2 ( n ) < 0 and σ 2 m . If
lim n sup 1 ϕ ( n + 1 , n 0 ) s = n 0 n + m 1 [ ϕ ( n , s ) f ( s ) P 1 ( n , s ) ] = + , lim n inf 1 ϕ ( n + 1 , n 0 ) s = n 0 n + m 1 [ ϕ ( n , s ) f ( s ) + P 1 ( n , s ) ] = ,

where P 1 ( n , s ) is defined by (20), all nontrivial solutions of Eq. (1) are oscillatory.

Proof For this case, we have that G 2 ( n , s ) 0 for any constant σ 1 since q 1 ( n , s ) 0 . Therefore, we can drop G 2 ( n , s ) in the estimation of (22). The other proof runs as that of Theorem 2. □

For other cases of σ 1 and σ 2 that are not covered by Theorem 1 and Theorem 2, the above method usually does not give sufficient conditions for the oscillation of all solutions of Eq. (1). However, when assuming that the solutions of Eq. (1) satisfy appropriate conditions, sufficient conditions for such solutions can also be derived. In the following, we are focused on the oscillation of all solutions of Eq. (1) satisfying x ( n ) = O ( n r ) for some r > 0 . Here, x ( n ) = O ( n r ) means that there exists a constant c > 0 such that | x ( n ) | c n r for n n 0 .

Theorem 3 Assume that q 1 ( n ) 0 , q 2 ( n ) > 0 , and (6) and (7) hold. All solutions satisfying x ( n ) = O ( n r ) are oscillatory if one of the following conditions holds:
  1. (i)
    σ 1 < m , σ 2 τ m , and
    lim n sup 1 ϕ ( n + 1 , n 0 ) s = n + m n σ 1 [ ϕ m ( n + m , s ) s r ] < ,
    (29)
     
  2. (ii)
    σ 1 m , σ 2 τ > m , and
    lim n sup 1 ϕ ( n + 1 , n 0 ) s = n σ 2 + τ n + m 1 [ ϕ ( n , s + σ 2 τ ) q 2 ( s + σ 2 τ ) s r β ] < ,
    (30)
     
  3. (iii)

    σ 1 < m , σ 2 τ > m , (29) and (30) hold.

     
Proof Assume that there exists a nontrivial solution x ( n ) of Eq. (1) such that x ( n ) is nonoscillatory. Without loss of generality, let x ( n σ 1 ) 0 , x ( n σ 2 ) 0 , x ( n τ ) 0 for n n 0 , where n 0 0 is sufficiently large.
  1. (i)
    For the case σ 1 < m , we have that n + m 1 < n σ 1 1 . Therefore, we get from (12) that
    F 1 ( n , s ) = s = n 0 n σ 1 1 [ ϕ m ( n + m , s ) x ( s ) ϕ ( n , s + σ 1 ) | q 1 ( s + σ 1 ) | x α ( s ) ] s = n + m n σ 1 1 ϕ m ( n + m , s ) x ( s ) + s = n 0 n 0 σ 1 1 ϕ ( n , s + σ 1 ) | q 1 ( s + σ 1 ) | x α ( s ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 x ( n 0 ) .
    (31)
     
By Fact 2 and (31), and noting that x ( n ) c n r for n n 0 and some constant c > 0 , we get
F 1 ( n , s ) s = n 0 n σ 1 1 Q 1 ( n , s ) c s = n + m n σ 1 1 ϕ ( n , s + σ 1 ) | q 1 ( s + σ 1 ) | s r i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 x ( n 0 ) ,
(32)
where Q 1 ( n , s ) is defined by (8). Multiplying 1 ϕ ( n + 1 , n 0 ) on both sides of (10), from (32), (17), (5) and (29), we get a contradiction to (7).
  1. (ii)
    For the case σ 2 τ > m , we have that n σ 2 + τ 1 < n + m 1 . By (15), we get
    F 2 ( n , s ) = s = n 0 n + m 1 [ ϕ m ( n + m , s ) p ( s ) x ( s τ ) ϕ ( n , s + σ 2 τ ) q 2 ( s + σ 2 τ ) x β ( s τ ) ] + s = n σ 2 + τ n + m 1 ϕ ( n , s + σ 2 τ ) q 2 ( s + σ 2 τ ) x β ( s τ ) s = n 0 σ 2 + τ n 0 1 ϕ ( n , s + σ 2 τ ) q 2 ( s + σ 2 τ ) x β ( s τ ) i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 [ p ( n 0 ) x ( n 0 τ ) ] .
    (33)
     
By Fact 1 and (33), and noting that x ( n ) c n r for n n 0 , we have that
F 2 ( n , s ) s = n 0 n + m 1 Q 2 ( n , s ) + c β s = n σ 2 + τ n + m 1 ϕ ( n , s + σ 2 τ ) q 2 ( s + σ 2 τ ) ( s τ ) β r i = 0 m 1 ϕ i ( n + i + 1 , n 0 ) Δ m i 1 [ p ( n 0 ) x ( n 0 τ ) ] ,
(34)
where Q 2 ( n , s ) is defined by (9). Multiplying 1 ϕ ( n + 1 , n 0 ) on both sides of (10), from (14), (34) and (5), we can get a contradiction to (7).
  1. (iii)

    Multiplying 1 ϕ ( n + 1 , n 0 ) on both sides of (10), from (32), (34), (29) and (30), we derive a contradiction. The proof of Theorem 3 is complete. □

     
Theorem 4 Assume that q 1 ( n ) 0 , q 2 ( n ) < 0 , (18) and (19) hold. All solutions satisfying x ( n ) = O ( n r ) are oscillatory if one of the following conditions holds:
  1. (i)
    σ 1 τ m , σ 2 > m , and
    lim n sup 1 ϕ ( n + 1 , n 0 ) s = n σ 2 n + m 1 [ ϕ ( n , s + σ 2 ) | q 2 ( s + σ 2 ) | s r β ] < ,
    (35)
     
  2. (ii)
    σ 1 τ m , σ 2 m , and
    lim n sup 1 ϕ ( n + 1 , n 0 ) s = n + m n σ 1 + τ 1 [ ϕ m ( n + m , s ) p ( s ) ( s τ ) r ] < ,
    (36)
     
  3. (iii)

    σ 1 τ < m , σ 2 > m , (35) and (36) hold.

     

Proof The proof is similar to that of Theorem 2 and Theorem 3, and hence it is omitted. □

3 Examples

We here work out two simple examples to illustrate the importance of Theorem 1 and Theorem 2.

Example 1 Consider the following third-order neutral difference equation:
Δ 3 [ x ( n ) x ( n 1 ) ] Φ 1 / 2 ( n σ 1 ) + Φ 2 ( n σ 2 ) = n k sin n , n 0 ,
(37)
where k > 0 is a constant. It is evident that m = 3 , σ 1 = τ = 1 , σ 2 = 2 , α = 1 / 2 , β = 2 , p ( n ) 1 , q 1 ( n ) 1 , q 2 ( n ) 1 and f ( n ) = n k sin n . If we choose ϕ ( n , s ) = ( n s ) ( 3 ) , we have ϕ 3 ( n , s ) = 6 . By the straightforward computation, we have that
Q 1 ( n , s ) = [ ( n s 1 ) ( 2 ) ] 2 / 24 , Q 2 ( n , s ) = [ ( n s + 3 ) ( 3 ) ] 1 .
By Theorem 1, we have that every solution of Eq. (37) is oscillatory if
lim n sup 1 ( n + 1 ) ( n + 2 ) ( n + 3 ) [ s = 0 n + 2 ( n s ) ( 3 ) s k sin s + s = 0 n 2 Q 1 ( n , s ) ] = + , lim n inf 1 ( n + 1 ) ( n + 2 ) ( n + 3 ) [ s = 0 n + 2 ( n s ) ( 3 ) s k sin s s = n 0 n 2 Q 1 ( n , s ) ] = .

It is not difficult to see that the above two inequalities hold for appropriate k > 0 .

Example 2 Consider the following third-order neutral difference equation:
Δ 3 [ x ( n ) x ( n 1 ) ] + Φ 1 / 2 ( n σ 1 ) Φ 2 ( n σ 2 ) = n k cos n , n 0 ,
(38)
where k > 0 is a constant. It is obvious that m = 3 , σ 1 = 2 , σ 2 = τ = 1 , α = 1 / 2 , β = 2 , p ( n ) 1 , q 1 ( n ) 1 , q 2 ( n ) 1 and f ( n ) = n k cos n . We also choose ϕ ( n , s ) = ( n s ) ( 3 ) . By the straightforward computation, we have that
P 1 ( n , s ) = [ ( n s + 3 ) ( 3 ) ] 1 , P 2 ( n , s ) = [ ( n s 1 ) ( 3 ) ] 2 / 24 .
By Theorem 2, we have that every solution of Eq. (38) is oscillatory if
lim n sup 1 ( n + 1 ) ( n + 2 ) ( n + 3 ) [ s = 0 n + 2 ( n s ) ( 3 ) s k cos s + s = 0 n 1 P 2 ( n , s ) ] = + , lim n inf 1 ( n + 1 ) ( n + 2 ) ( n + 3 ) [ s = 0 n + 2 ( n s ) ( 3 ) s k cos s s = 0 n 1 P 2 ( n , s ) ] = .

It is not difficult to see that the above two inequalities hold for appropriate k > 0 .

Declarations

Acknowledgements

The authors thank the reviewers for their helpful and valuable comments on this paper. This work was supported by the Natural Science Foundation of Shandong Province (grant nos. ZR2010AL002 and JQ201119) and the National Natural Science Foundation of China (grant no. 61174217).

Authors’ Affiliations

(1)
School of Mathematical Sciences, University of Jinan, Jinan, Shandong, 250022, China

References

  1. Agarwal RP: Difference Equations and Inequalities: Theory, Methods, and Applications. Dekker, New York; 1992.Google Scholar
  2. Agarwal RP, Wong PJY: Advanced Topics in Difference Equations. Kluwer Academic, Dordrecht; 1997.View ArticleGoogle Scholar
  3. Györi I, Ladas G: Oscillation Theory of Delay Differential Equations. Clarendon, Oxford; 1991.Google Scholar
  4. Sun YG, Saker SH: Oscillation for second-order nonlinear neutral delay difference equations. Appl. Math. Comput. 2005, 163: 909–918. 10.1016/j.amc.2004.04.017MathSciNetView ArticleGoogle Scholar
  5. Agarwal RP, Manuel MMS, Thandapani E: Oscillatory and nonoscillatory behavior of second-order neutral delay difference equations. Math. Comput. Model. 1996, 24: 5–11. 10.1016/0895-7177(96)00076-3MathSciNetView ArticleGoogle Scholar
  6. Agarwal RP, Manuel MMS, Thandapani E: Oscillatory and nonoscillatory behavior of second-order neutral delay difference equations II. Appl. Math. Lett. 1997, 10: 103–109.MathSciNetView ArticleGoogle Scholar
  7. Thandapani E, Kavitha N, Pinelas S: Oscillation criteria for second-order nonlinear neutral difference equations of mixed type. Adv. Differ. Equ. 2012., 2012:Google Scholar
  8. Agarwal RP, Thandapani E, Wong PJY: Oscillations of higher-order neutral difference equations. Appl. Math. Lett. 1997, 10: 71–78.MathSciNetView ArticleGoogle Scholar
  9. Jiang J: Oscillation of second order nonlinear neutral delay difference equations. Appl. Math. Comput. 2003, 146: 791–801. 10.1016/S0096-3003(02)00631-8MathSciNetView ArticleGoogle Scholar
  10. Lou JW, Bainov DD: Oscillatory and asymptotic behavior of second-order neutral difference equations with maximum. J. Comput. Appl. Math. 2001, 131: 331–341.Google Scholar
  11. Saker SH: New oscillation criteria for second-order nonlinear neutral delay difference equations. Appl. Math. Comput. 2003, 142: 99–111. 10.1016/S0096-3003(02)00286-2MathSciNetView ArticleGoogle Scholar
  12. Yildiz MK, Öcalan Ö: Oscillation results for higher order nonlinear neutral delay difference equations. Appl. Math. Lett. 2007, 20: 243–247. 10.1016/j.aml.2006.05.001MathSciNetView ArticleGoogle Scholar
  13. Liu B, Yan J: Oscillation theorems for nonlinear neutral difference equations. J. Differ. Equ. Appl. 1995, 1: 307–315. 10.1080/10236199508808029MathSciNetView ArticleGoogle Scholar
  14. Lalli BS, Zhang BG, Li JZ: On the oscillation of solutions and existence of positive solutions of neutral difference equations. J. Math. Anal. Appl. 1991, 158: 213–233. 10.1016/0022-247X(91)90278-8MathSciNetView ArticleGoogle Scholar
  15. Zafer A, Dahiya RS: Oscillation of a neutral difference equations. Appl. Math. Lett. 1993, 6: 71–74.MathSciNetView ArticleGoogle Scholar
  16. Koo NJ: Oscillation of neutral difference equations. J. Chungcheong Math. Soc. 1999, 12: 1–7.Google Scholar
  17. Chen MP, Lalli BS, Yu JS: Oscillation in neutral difference equations with variable coefficients. Comput. Math. Appl. 1995, 29: 5–11.MathSciNetView ArticleGoogle Scholar
  18. Yu JS, Wang ZC: Asymptotic behavior and oscillation in neutral delay difference equations. Funkc. Ekvacioj 1994, 37: 241–248.Google Scholar
  19. Tang XH, Yu JS, Peng DH: Oscillation and nonoscillation of neutral difference equations with positive and negative coefficients. Comput. Math. Appl. 2000, 39: 169–181.MathSciNetView ArticleGoogle Scholar
  20. Agarwal RP, Grace SR: Forced oscillation of n th order nonlinear differential equations. Appl. Math. Lett. 2000, 13: 53–57.MathSciNetView ArticleGoogle Scholar
  21. Ou CH, Wong JSW: Forced oscillation of n th-order functional differential equations. J. Math. Anal. Appl. 2001, 262: 722–732. 10.1006/jmaa.2001.7614MathSciNetView ArticleGoogle Scholar
  22. Yang X: Forced oscillation of n th nonlinear differential equations. Appl. Math. Comput. 2003, 134: 301–305. 10.1016/S0096-3003(01)00284-3MathSciNetView ArticleGoogle Scholar

Copyright

© Gao et al.; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement