Consider a nonlinear neutral type two-dimensional delay difference system of the form

Δ ( x n + p n x n − k ) = b n y n Δ ( y n ) = − a n x n − l + 1 , n ∈ ℕ ( n 0 ) = 1 , 2 , 3 , ⋯ (1.1)

Subject to the following conditions:

( c 1 ) , { a n } and { b n } are nonnegative real sequences such that

∑ n = 1 ∞ b n 0 = ∞ .

( c 2 ) , { p n } is a positive real sequence.

( c 3 ) , f,g : ℝ → ℝ are continous non-decreasing with u f ( u ) > 0 , u g ( u ) > 0 , for u ≠ 0 and | f ( u ) | ≥ k | u | , where k is a constant.

( c 4 ) , k and l are nonnegative integers.

Let θ = max { k , 1 } . By a solution of the system (1.1), we mean a real sequence { x n , y n } which is defined for all n ≥ n 0 − θ and satisfies (1.1) for all n ∈ ℕ ( n 0 ) .

Let W be the set of all solutions X = { x n , y n } of the system (1.1) which exists for n ∈ ℕ ( n 0 ) and satisfies

sup { | x n | + | y n | ; n ≥ N } > 0 foranyinteger N ≥ N 0 .

A real sequence defined on ℕ ( n 0 ) is said to be oscillatory if it is neither eventually positive nor eventually negative and nonoscillatory otherwise.

A solution X ∈ W is said to be oscillatory if both components are oscillatory and it will be called nonoscillatory otherwise.

Some oscillation results for difference system (1.1) when p n = 0 for n ∈ N ( N 0 ) and n − l + 1 = n have been presented in [1] , In particular when b n > 0 for all n ∈ N ( n 0 ) . The difference system (1.1) reduces to the second order nonlinear neutral difference equation

Δ ( 1 b n Δ ( x n + p n x n − k ) ) = − a n x n − l + 1 . (1.2)

If b n = 1 , in Equation (1.2), we have a second order linear equation

Δ 2 ( x n + p n x n − k ) = − a n x n − l + 1 . (1.3)

For oscillation criteria regarding Equations (1.1)-(1.3), we refer to [2] - [12] and the references cited therein. In Section 2, we present some basic lemmas. In Section 3, we establish oscillation criteria for oscillation of all solutions of the system (1.1). Examples are given in Section 4 to illustrate our theorems.

Denote A n = ∑ s = n 0 n − 1 a s , n ∈ ℕ ( n 0 ) , For any x n , we define z n by

z n = x n + p n x n − k (2.1)

We begin with the following lemma.

2.1. Let ( c 1 ) − ( c 4 ) hold and let { ( x n , y n ) } ∈ W be a solution of system (1.1) with { x n } either eventually positive or eventually negative for n ∈ ℕ ( n 0 ) . Then { ( x n , y n ) } is nonoscillatory and { z n } and { y n } are monotone for n ∈ ℕ ( N ) for N ∈ ℕ ( n 0 ) .

Proof. Let { ( x n , y n ) } ∈ W and let { x n } be nonoscillatory on ℕ ( n 0 ) . Then from the second equation of system (1.1), we have Δ y n ≤ 0 for all n ≥ N 1 ∈ ℕ ( n 0 ) and Δ y n ,and y n are not identically zero for infinitely many values of n. Thus { y n } is monotone for n ≥ N . Hence { y n } is either eventually positive or eventually negative for n ≥ N 1 . Then, { ( x n , y n ) } is nonoscillatory. Further from the first equation of the system (1.1). We have Δ z n > 0 or Δ z n < 0 eventually. Hence { z n } is monotone and nonoscillatory for all n ≥ N ≥ N 1 . The proof is similar when { x n } is eventually negative.

Lemma 2.2. In addition to conditions ( c 1 ) − ( c 2 ) assume that 0 < p n ≤ 1 for all n ∈ ℕ ( n 0 ) . Let { x n } be a nonoscillatory solution of the inequality

x n ( x n + p n x n − k ) ≥ 0 (2.2)

for sufficiently large n. If for n − k for all n ∈ ℕ ( n 0 ) . Then, { x n } is bounded.

Proof. Without loss of generality we may assume that { x n } be an eventually positive solution of the inequality (2.1), the proof for the case { x n } eventually negative is similar. From (2.1) we have

( x n + p n x n − k ) ≥ 0 , for n ≥ ℕ ( n 0 ) .

and 0 < p n ≤ 1 , we have from (2.2), x n − k ≤ p n x n − k ≤ x n for all n ≥ N . Hence { x n } is bounded.

Next, we state a lemma whose proof can be found in [1] .

Lemma 2.3. Assume that { a n } is a non negative real sequence and not identically zero for infinitely many values of n and l is a positive integer. If

lim inf n → ∞ ∑ s = n − l + 1 n − 1 a s > ( l l + 1 ) l + 1

Then the difference inequality

Δ y n + a n x n − l + 1 ≤ 0 n ∈ ℕ ( n 0 )

cannot have an eventually positive solution and

Δ y n + a n x n − l + 1 ≥ 0 n ∈ ℕ ( n 0 )

cannot have an eventually negative solution.

Theorem 3.1. Assume that { p n } is bounded and there exists an integer j such that l > j + k + 2 . If

lim sup n → ∞ A n ∑ s = n − l + 1 ∞ a s > 1 k β (3.1)

and

lim n → ∞ inf ∑ s = n − ( l − j − k ) n − 1 k β b s ( ∑ t = s s + j a t p t − l + k + 1 ) > ( l − j − k l − j − k + 2 ) l − j − k + 2 (3.2)

Then every solution { ( x n , y n ) } ∈ W is a nonoscillatory solution of system (1.1), with { x n } bounded. Without loss of generality we may assume that { x n } is eventually positive and bounded for all n ≥ n 1 ∈ N ( n 0 ) . From the second equation of (1.1), we obtain Δ y n ≤ 0 for sufficiently large n ≥ n 2 ∈ N ( n 1 ) . In view of Lemma 2.1, we have two cases for sufficiently large n 3 ∈ N ( n 2 ) :

1) y n < 0 for n ≥ n 3 ;

2) y n > 0 for n ≥ n 3 .

Case (1). Because { y n } is negative and nonincreasing there is constant L > 0. Such that

y n ≤ − L forall n ≥ n 3 (3.3)

Since { x n } and { p n } are bounded. { z n } defined by (2.1) is bounded. Summing the first equation of (1.1) from n 3 to n − 1 and then using (3.3), we obtain

z n − z n 0 ≤ − L ∑ s = n 3 n − 1 a s , n ≥ n 3 . (3.4)

From (3.3), we see that lim n → ∞ z n = − ∞ which contradicts the fact that { z n } is bounded. Case (1) cannot occur.

Case (2). Let z n > 0 for n ≥ n 4 where n 4 ∈ N ( n 3 ) is sufficiently large. Because { z n } is nondecreasing there is a positive constant M, such that

z n ≥ M , forall n ≥ n 4 . (3.5)

From (2.1), we have z n > x n , and by hypothesis, we obtain

a n z n − l + 1 ≥ a n x n − l + 1 k , n ≥ n 5 ∈ N ( n 4 ) (3.6)

summing the second equation of (1.1) from n to i, using (3.5) and then letting i → ∞ , we obtain

y n ≥ k ∑ s = n ∞ a s z s − l + 1 , n ≥ n 5 . (3.7)

From condition (3.1), we have

1 k β < lim n → ∞ sup A s a s (3.8)

we claim that the condition (3.1) implies

∑ n = N ∞ A n a n = ∞ , N ∈ N ( n 0 ) . (3.9)

Otherwise, if ∑ n = N ∞ A n a n < ∞ , we can choose an integer N 1 ≥ N . So large that ∑ n = N 1 ∞ A n a n < 1 k β which contradicts (3.6).

Using a summation by parts formula, we have

∑ s = N n − 1 A s + 1 Δ g ( y s ) = A n y n − A N y N − z n − z N . (3.10)

From (3.3), (3.4) and (3.6) and the second equation of (1.1), we have

∑ s = N n − 1 A s + 1 Δ g ( y s ) ≤ β ∑ s = N n − 1 A s + 1 Δ y s ≤ − M k β ∑ s = N n − 1 A s + 1 y s ≤ − M k β ∑ s = N n − 1 A s y s

M k β ∑ s = N n − 1 A s y s = − A n y n + A N y N + z n − z N , n ≥ N .

combining (3.6) with (3.8), we obtain

lim n → ∞ ( z n − A n y n ) = ∞ .

and

z n ≥ A n g ( y n ) ≥ β A n y n , n ≥ n 6 ∈ ℕ ( n 5 ) .

The last inequality together with (3.4) and the monotonocity of { z n } implies

z n ≥ k β A n ∑ s = n ∞ a s z s − l + 1 ≥ k β A n ∑ s = n − l + 1 ∞ a s z s − l + 1       ≥ k β A n z n ∑ s = n + l − 1 ∞ a s

and 1 ≥ k β A n z n ∑ s = n + l − 1 ∞ a s , n ∈ ℕ ( n 6 ) which contradicts (1.1). This case cannot occur. The proof is complete.

Theorem 3.2. Assume that 0 < p n ≤ 1 , then there exists an integer j such that l > j + k and the conditions (3.1) and (3.2) are satisfied. Then all solutions of (1.1) are oscillatory.

Proof . Let { ( x n , y n ) } ∈ W be a nonoscillatory solution of (1.1). Without loss of generality we may assume that { x n } is positive for n n ∈ ℕ ( n 1 ) . As in the proof of above theorem we have two cases.

Case (1). Analogus to the proof of case (1) of above theorem, we can show that lim n → ∞ z n = − ∞ . By Lemma 2.2, { x n } is bounded and hence { z n } is bounded which is a contradiction. Hence case (1) cannot occur.

Case (2). The proof of case (2) is similar to that of the above theorem and hence the details are omitted. The proof is now complete.

Theorem 3.3. Assume that 0 < p n ≤ 1 and

lim sup n → ∞ ∑ s = n − k − l + 1 n − 1 k β ( A n − A s + 1 ) p s − l − k + 1 a s > 1 . (3.14)

∑ n = N ∞ b n ( ∑ s = n ∞ a s ) = ∞ , N ∈ N ( n 0 ) (3.15)

lim n → ∞ sup ( k β A n ∑ s = n ∞ a s ) > 1. (3.16)

Then all solutions of (1.1) are oscillatory.

Proof. Let { ( x n , y n ) } ∈ W be a nonoscillatory solution of (1.1). Without loss of generality we may assume that { x n } is positive for n ∈ ℕ ( n 1 ) . As in the proof of above theorem we have two cases.

Case 1. From (2.1), we have

z n > p n x n − k , n ≥ n 3 ∈ N ( n 0 )

and

f ( x n − l + 1 ) ≥ k x n − l + 1 > k z n − l − k + 1 p n − l − k + 1 , n ≥ n 4 (3.17)

where n 4 ∈ N ( n 3 ) is sufficiently large. Then the following equality

z n = z i + ( A n − A i ) y i + ∑ s = i n − 1 ( A n − A s + 1 ) Δ y s

z n < ∑ s = i n − 1 ( A n − A s + 1 ) Δ y s , n > i ≥ n 5 .

Combining the last inequality with the second equations of (1.1) and (3.17), we have

z n < β ∑ s = i n − 1 ( A n − A s + 1 ) ( − a s f ( x s − l ) ) < k β ∑ s = i n − 1 ( A n − A s + 1 ) a s z s − l − k + 1 p s − l − k + 1 , n > i ≥ n 5 .

Let i = n − l + k + 1 and using the monotonocity of { z n } , from the last inequality, we obtain

z n < z n ∑ s = n − l − k + 1 n − 1 k β ( A n − A s + 1 ) a s p s + l + k − 1

and

1 > z n ∑ s = n − l − k + 1 n − 1 k β ( A n − A s + 1 ) a s p s + l + k − 1

which contradicts the condition (3.14).

Case 2. The proof for this case is similar to that of Theorem (3.1). Here we use condition (3.16) instead of condition (2.1). The proof is complete.

Example 4.1. Consider the difference system

Δ ( x n + 1 2 x n − 3 ) = 1 n y n Δ y n = − n x n − 2 , n ≥ 1. (4.1)

The conditions (3.1) and (3.2) are

lim sup n → ∞ 1 n ∑ s = n + 2 ∞ s = ∞ .

lim inf n → ∞ ∑ s = n − 4 n − 3 1 s ( ∑ t = s − 1 s 2 t ) = 4.

All conditions of Theorem 3.2 are satisfied and so all solutions of the system (4.1) are oscillatory.

Example 4.2. Consider the difference systems

Δ ( x n + 1 4 x n − 2 ) = ( n + 1 ) y n Δ y n = − c n + 1 x n − 1 , n ≥ 1 , (4.2)

where c is a positive constant. The conditions (3.1) and (3.2) are

lim n → ∞ sup ( n + 1 ) ∑ s = n + 1 ∞ c s + 1 = ∞

and

lim inf n → ∞ ∑ s = n − 3 n − 2 ( s + 1 ) ( ∑ t = s s + 1 − 4 c t + 1 ) = 12 c .

For c > 1 12 , all conditions of Theorem 3.2 are satisfied and so all solutions of

the system (4.2) are oscillatory.

Thangavelu, K. and Saraswathi, G. (2017) Oscillation and Asymptotic Behaviour of Solutions of Nonlinear Two-Dimensional Neutral Delay Di- fference Systems. Journal of Applied Mathe- matics and Physics, 5, 1215-1221. https://doi.org/10.4236/jamp.2017.56104