Applied Mathematics E-Notes, 1(2001), 5-10 \begin{matrix}c \\ ◯ \end{matrix} Available free at mirror sites of http://math2.math.nthu.edu.tw/∼amen/

Oscillation of Nonlinear First Order Neutral Difference Equations ^{∗}

Ying Gao and Guang Zhang^{†}^{‡}

Received 1 June 2000

Abstract

In this note, oscillation criteria are obtained for a class of nonlinear neutral difference equations.

Oscillatory behaviors of neutral difference equations of the form

Δ (x_{n} − p_{n}x_{n−τ}) + q_{n}x_{n−σ} = 0, n = 0, 1, ...  (1)

have been explored to some extent in a number of studies [1-3]. However, relatively few results are known for their nonlinear extensions of the form

Δ (x_{n} − p_{n}x_{n−τ}) + q_{n} max_{s∈[n−σ,n]} x_{s} = 0, n = 0, 1, ... .  (2)

Nonlinear functional equations involving the maximum function are important since they appear naturally in automatic control theory, see e.g. Popov [4]. Some of the qualitative theory of these equations has been developed recently, see e.g. [5-8]. In this note, we will consider their oscillatory behaviors.

We will assume that τ and σ are positive integers, that {p_{n}} and {q_{n}} are nonnegative real sequences, and {q_{n}} has a positive subsequence. Let μ = max {τ, σ}. By a solution of (1) or (2), we mean a real sequence {x_{n}}_{n=−μ}^{∞} which satisfies (1) or (2) respectively for n ≥ 0. Such a solution is said to be oscillatory if it is neither eventually positive nor eventually negative.

It is easily seen that {x_{n}} is an eventually positive solution of equation (1) if, and only if, {−x_{n}} is its eventually negative solution. However, such a property is not valid for equation (2). Instead, {x_{n}} is an eventually positive solution of (2) if, and only if, {−x_{n}} is an eventually negative solution of the equation

Δ(x_{n} − p_{n}x_{n−τ}) + q_{n} min_{s∈[n−σ,n]} x_{s} = 0, n = 0, 1, ... .  (3)

Thus, all solutions of (2) are oscillatory if, and only if, both (2) and (3) do not have any eventually positive solutions.

∗Mathematics Subject Classifications: 39A10

†Department of Mathematics, Yanbei Normal College, Datong, Shanxi 037000, P. R. China

‡Supported by the Science Foundation of Shanxi Province of China

LEMMA 1. Assume that there exists a nonnegative integer N ≥ 0 such that p_{N+jτ} ≤ 1 for j = 0, 1, 2, ... . Let {x_{n}} be an eventually positive solution of (2) or (3). Set

y_{n} = x_{n} − p_{n}x_{n−τ}  (4)

for all large n. Then y_{n} > 0 eventually.

The proof is similar to the proof of Lemma 1 in [2].

THEOREM 1. Assume that there exists a nonnegative integer N ≥ 0 such that p_{N+jτ} ≤ 1 for j = 0, 1, 2, ... . Suppose further that either p_{n} > 0 or q_{n} does not vanish identically over any set of consecutive integers of the form {a, a + 1, .., a + σ}. Then equation (2) has an eventually positive solution if, and only if, the following functional inequality

Δ (x_{n} − p_{n}x_{n−τ}) + q_{n} max_{s∈[n−σ,n]} x_{s} ≤ 0, n = 0, 1, ...  (5)

has an eventually positive solution; and equation (3) has an eventually positive solution if, and only if, the functional inequality

Δ (x_{n} − p_{n}x_{n−τ}) + q_{n} min_{s∈[n−σ,n]} x_{s} ≤ 0, n = 0, 1, ...  (6)

has an eventually positive solution.

The proof of Theorem 1 is similar to that of Theorem 1 in [2], and is thus omitted. THEOREM 2. Assume that there exists a nonnegative integer N ≥ 0 such that

p_{N+jτ} ≤ 1 for j = 0, 1, 2, ... . Suppose further that there exists some positive integer T such that the functional inequality

Δy_{n} + q_{n} min_{s∈[n−σ,n]} ∑_{j=0}^{T −1} ∏_{i=0}^{j} p_{s−iτ} y_{n−τ} ≤ 0

does not have any eventually positive solutions. Then all solutions of (2) oscillate.

PROOF. If {x_{n}} is an eventually positive solution of (2), then Δy_{n} ≤ 0 and y_{n} = x_{n} − p_{n}x_{n−τ} > 0 eventually. Thus,

x_{n} = y_{n} + p_{n}x_{n−τ} = y_{n} + p_{n}y_{n−τ} + p_{n}p_{n−τ} x_{n−2τ}
= ... ≥ y_{n} + p_{n}y_{n−τ} + ... + ∏_{i=0}^{T −1} p_{n−iτ} y_{n−(i+1)τ}

≥ ∑_{j=0}^{T −1} ∏_{i=0}^{j} p_{n−iτ} y_{n−τ} .

Hence,

max_{s∈[n−σ,n]} x_{s} ≥ max_{s∈[n−σ,n]} ∑_{j=0}^{T −1} ∏_{i=0}^{j} p_{s−iτ} y_{s−τ} ≥ max_{s∈[n−σ,n]} ∑_{j=0}^{T −1} ∏_{i=0}^{j} p_{s−iτ} y_{n−τ} .

Substituting the last inequality into (2), we have

Δy_{n} + q_{n} max_{s∈[n−σ,n]} ∑_{j=0}^{T −1} ∏_{i=0}^{j} p_{s−iτ} y_{n−τ} ≤ 0,

which is a contradiction. If {z_{n}} is an eventually negative solution of (2), then x_{n} = −z_{n} is an eventually positive solution of equation (3). Similarly, we have

Δy_{n} + q_{n} min_{s∈[n−σ,n]} ∑_{j=0}^{T −1} ∏_{i=0}^{j} p_{s−iτ} y_{n−τ} ≤ Δy_{n} + q_{n} max_{s∈[n−σ,n]} ∑_{j=0}^{T −1} ∏_{i=0}^{j} p_{s−iτ} y_{n−τ} ≤ 0,

This is also a contradiction. The proof is complete.

For the equation

Δ (x_{n} − x_{n−τ}) + q_{n} max_{s∈[n−σ,n]} x_{s} = 0, n = 0, 1, ...,  (7)

we have the following result.

THEOREM 3. Equation (7) has nonoscillatory solutions if, and only if,

Δ^{2}z_{n−1} + \frac{1}{τ} q_{n}z_{n} = 0  (8)

also has nonoscillatory solutions.

PROOF. Assume that {x_{n}} is an eventually positive solution of (7). In view of Lemma 1, we see that there is an integer N_{1} such that x_{n−τ} > 0, y_{n} = x_{n} − x_{n−τ} > 0, and Δy_{n} ≤ 0 for n ≥ N_{1}. Set m = min{x_{N_{1}−τ} , x_{N_{1}−τ+1}, ..., x_{N_{1}−1}}. For n ≥ N_{1} + τ, there exists a positive integer k such that

N_{1} + kτ ≤ n < N_{1} + (k + 1)τ

and

x_{n} = x_{n−kτ} + ∑_{j=0}^{k−1} y_{n−jτ} ≥ m + ∑_{j=0}^{k−1} y_{n−jτ} .

Furthermore, since Δy_{n} ≤ 0 for n ≥ N_{1}, and since n − kτ + τ ≤ N_{1} + 2τ = N_{2},

τ ∑_{j=0}^{k−1} y_{n−jτ} = τy_{n−(k−1)τ} + τy_{n−(k−2)τ} + ... + τy_{n}
≥ (y_{n−(k−1)τ} + y_{n−(k−1)τ+1} + ... + y_{n−(k−2)τ−1})
+(y_{n−(k−2)τ} + y_{n−(k−2)τ+1} + ... + y_{n−(k−3)τ−1}
+... + (y_{n} + y_{n+1} + ... + y_{n+τ−1})
≥ ∑_{i=N_{2}}^{n} y_{i},

we have

x_{n} ≥ m + \frac{1}{τ} ∑_{i=N_{2}}^{n} y_{i}.

Let

z_{n} = m + \frac{1}{τ} ∑_{i=N_{2}}^{n} y_{i},

then z_{n} > 0 and

τΔ^{2}z_{n−1} + q_{n}z_{n} = Δy_{n} + q_{n} (m + \frac{1}{τ} ∑_{i=N_{2}}^{n} y_{i})

= Δy_{n} + q_{n} max_{s∈[n−σ,n]} (m + \frac{1}{τ} ∑_{i=N_{2}}^{s} y_{i})
≤ Δy_{n} + q_{n} max_{s∈[n−σ,n]} x_{s} = 0.

In view of Lemma 1, we know that equation (8) has an eventually positive solution. If {x_{n}} is an eventually positive solution of the equation

Δ(x_{n} − x_{n−τ}) + q_{n} min_{s∈[n−σ,n]} x_{s} = 0, n = 0, 1, ...,  (9)

we can also prove that

τΔ^{2}z_{n−1} + q_{n}z_{n−σ} ≤ 0

has an eventually positive solution. In view of Theorem 2 in [3], we know that equation (8) has an eventually positive solution. We now show that the converse holds. Let {z_{n}} be an eventually positive solution of (8), then it is positive and concave for all large n, so that {Δz_{n}} is eventually positive and nonincreasing. Thus it is easy to see that there exists a sufficiently large integer N such that 0 < τΔz_{n−1} ≤ z_{n−τ} for n ≥ N. Let

H_{n} = \{ \begin{matrix}τΔz_{n−1} & n ≥ N \\ (n − N + τ)Δz_{N−1} & N − τ ≤ n < N \\ 0 & n < N − τ \end{matrix} ,

and let

x_{n} = z_{N−τ} − τΔz_{N−1} + ∑_{i=0}^{∞} H_{n−iτ} , n ≥ 0.

In view of the definition of H_{n}, it is clear that 0 < x_{n} < ∞ for all n ≥ 0, that

max {x_{N−τ} , x_{N−τ+1}, ..., x_{N−1}} = z_{N−τ} − τΔz_{N−1} + (τ − 1)Δz_{N−1} ≤ z_{N−τ} ,

and that

x_{n} − x_{n−τ} = H_{n} = τΔz_{n−1}

for n ≥ N. For any n which satisfies N ≤ n ≤ N + τ − 1, we see that

x_{n} = τΔz_{n−1} + x_{n−τ} ≤ ∑_{i=n−τ}^{n−1} Δz_{i} + x_{n−τ} ≤ ∑_{i=N−τ}^{n−1} Δz_{i} + z_{N−τ} = z_{n}.

By induction, it is easy to prove that for any n which satisfies N +kτ ≤ n < N +(k+1)τ where k = 0, 1, 2, ..., x_{n} ≤ z_{n} is still valid. Thus

Δ(x_{n} − x_{n−τ}) + q_{n} max_{s∈[n−σ,n]} x_{s} ≤ τΔ^{2}z_{n−1} + q_{n}z_{n} = 0

as desired. Similarly, we have also

x_{n} ≤ ∑_{i=N−τ}^{n−1} Δz_{i} + z_{N−τ} ≤ ∑_{i=N−τ}^{n−1+σ} Δz_{i} + z_{N−τ} = z_{n+σ}.

Thus, we have

Δ(x_{n} − x_{n−τ}) + q_{n} min_{s∈[n−σ,n]} x_{s} ≤ τΔ^{2}z_{n−1} + q_{n} min_{s∈[n−σ,n]} z_{s+σ}

= τΔ^{2}z_{n−1} + q_{n}z_{n} = 0.

The proof is complete.

As an important corollary, the equation

Δ(x_{n} − x_{n−τ}) + \frac{α}{(n + 1)^{2}} min_{s∈[n−σ,n]} x_{s} = 0, n = 0, 1, 2, ....

in view of Theorem 3, is oscillatory if, and only if, α/τ > 1/4.

We make several additional remarks. Let us say that a solution of (1) or (2) is strongly oscillatory if for any given nonnegative integer N, there is a corresponding integer m ≥ N such that x_{m}x_{m+1} < 0. Assume that q_{n} > 0 for n ≥ 0. In this case, let {x_{n}} be a nonnegative solution of (1) or (2), and let y_{n} be defined by (4). We have already seen that y_{n} > 0 and Δy_{n} ≤ 0 eventually. Note that x_{n} ≥ y_{n} for all large n.

THEOREM 4. Assume that there exists a nonnegative integer N ≥ 0 such that p_{N+jτ} ≤ 1 for j = 0, 1, 2, ... . Suppose further that q_{n} > 0 for n ≥ 0. Then every nontrivial solution of equation (1) is strongly oscillatory if, and only if, the inequality

Δ (x_{n} − p_{n}x_{n−τ}) + q_{n}x_{n−σ} ≤ 0,: n = 0, 1, ...

does not have any eventually nonnegative solutions, and equation (2) has a nonnegative solution if, and only if, (5) has a nonnegative solution.

THEOREM 5. Assume that there exists a nonnegative integer N ≥ 0 such that p_{N+jτ} ≤ 1 for j = 0, 1, 2, ... . Assume further that q_{n} > 0 for n ≥ 0, If there exists some integer T such that the functional inequality

Δy_{n} + q_{n} max_{s∈[n−σ,n]} ∑_{j=0}^{T −1} ∏_{i=0}^{j} p_{s−iτ} y_{n−τ} ≤ 0

does not have an eventually positive solution, then equation (2) does not have an eventually nonnegative solution. If there exists some integer T such that the functional inequality

Δy_{n} + q_{n} ∑_{j=0}^{T −1} ∏_{i=0}^{j} p_{n−iτ} y_{n−τ} ≤ 0

does not have an eventually positive solution, then every solution of equation (1) is strongly oscillatory.

THEOREM 6. Assume that q_{n} > 0 for n ≥ 0. Then all solutions of the equation

Δ (x_{n} − x_{n−τ}) + q_{n}x_{n−σ} = 0,

are strongly oscillatory if, and only if, equation (8) is oscillatory; and equation (7) has an eventually nonnegative solution if, and only if, (8) has an eventually positive solution.

The proofs of Theorem 4, Theorem 5 and Theorem 6 are similar to the proofs of Theorem 1, Theorem 2 and Theorem 3 respectively. They will be omitted. Results analogous to Theorem 4, Theorem 5 and Theorem 6 are not valid for equation (3). For example, the sequence {sin(nπ/2) − 1} is a nonpositive solution of the equation

Δ(x_{n} − x_{n−4}) + q_{n} max_{s∈[n−6,n]} x_{s} = 0,

where {q_{n}} is any real sequence.

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