Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 1, 111–130

Frequent Oscillation of a Class of Partial Difference Equations

C. J. Tian and B. G. Zhang

Abstract. This paper is concerned with the partial difference equation

A_{m+1,n} + A_{m,n+1} − a_{m,n}A_{m,n} + p_{m,n}A_{m−k,n−l} = 0

where k and l are non-negative integers, {a_{m,n}} and {p_{m,n}} are real double sequences. Frequent oscillation criteria of this equation are obtained.

Keywords: Partial difference equations, frequent oscillation, frequent positivity AMS subject classification: 39 A 10

1. Introduction

We consider the partial difference equation

A_{m+1,n} + A_{m,n+1} − a_{m,n}A_{m,n} + p_{m,n}A_{m−k,n−l} = 0  (1.1)

where {a_{m,n}} and {p_{m,n}} are real double sequences, m and n are non-negative integers, k and l are positive integers. Equation (1.1) with a_{m,n} ≡ 1 has been studied in [2, 6, 8 - 11]. Partial difference equations arise from considerations of random walk problems, the study of molecular orbits [5], mathematical physics problems [8], and numerical difference approximation problems [1, 4]. Difference equation (1.1) can be obtained from the difference approximation of a class of delay partial differential equation of the form (see [4])

\frac{∂^{2}u}{∂x^{2}} + \frac{∂^{2}u}{∂y^{2}} + g(x, y, u(x, y), u(x, y − σ_{1}), u(x − τ_{1}, y), u(x − τ_{2}, y − σ_{2})) = 0.

The oscillation of the above delay partial differential equation has been investigated by Tramov [7].

By a solution of equation (1.1) we mean a double sequence {A_{m,n}} which is defined for m ≥ −k and n ≥ −l and satisfies (1.1) for m ≥ 0 and n ≥ 0.

C. J. Tian: South China Univ. Techn., Dept. of Elec. & Comm. Eng., Guangzhou 510641, China B. G. Zhang: Ocean Univ. of Qingdao, Dept. Appl. Math., Qingdao 266003, China This work is supported by NNSF of China ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

The usual concept of oscillation of a sequence {A_{m,n}} is the following: A solution {A_{i,j}} of equation (1.1) is said to be eventually positive if A_{i,j} > 0 for all large i and j, and it is said to be oscillatory if it is neither eventually positive nor eventually negative.

However, the above definitions does not catch all the fine details of an oscillatory sequence. A strengthened oscillation which is called frequent oscillation has been posed by Tian et al. in [6]. In the present paper we will discuss frequent oscillation of solutions of equation (1.1).

2. Preparatory lemmas

For the sake of completeness, we list the following definitions which have been given in [6].

The set {1, 2, ⋯} will be denoted by ℕ and the set of integers by ℤ. An element of ℤ^{2} = ℤ × ℤ is called a lattice point. Let Ω be a set of lattice points, i.e. Ω ⊂ ℤ^{2}. The size of such set Ω will be denoted by |Ω|. Given integers m and n, the translation operators X^{m} and Y^{n} are defined by

X^{m}Ω = {(i + m, j) ∈ ℤ^{2}| (i, j) ∈ Ω} and Y^{n}Ω = {(i, j + n) ∈ ℤ^{2}| (i, j) ∈ Ω},

respectively. Let α, β and γ, τ be integers such that α ≤ β and γ ≤ τ. The union ∑_{i=α}^{β} ∑_{j=γ}^{τ} X^{i}Y^{j}Ω is called a derived set of Ω. Thus (see [6])

(i, j) ∈ ℤ^{2}\ ∑_{i=α}^{β} ∑_{j=γ}^{τ} X^{i}Y^{j}Ω

⇕  (2.1)
(i − k, j − l) ∈ ℤ^{2} \ Ω for α ≤ k ≤ β and γ ≤ l ≤ τ.

Definition 2.1. Let Ω be a set of lattice points and

Ω^{(m,n)} = {(i, j) ∈ Ω| i ≤ m and j ≤ n}.

If

lim sup_{m,n→∞} \frac{|Ω^{(m,n)}|}{mn}

exists, then this limit denoted by μ^{∗}(Ω) will be called the upper frequent measure of Ω. Similarly, if

lim inf_{m,n→∞} \frac{|Ω^{(m,n)}|}{mn}

exists, then this limit denoted by μ_{∗}(Ω) will be called the lower frequent measure of Ω. If μ^{∗} = μ_{∗}, then the common limit denoted by μ(Ω) will be called the frequent measure of Ω.

Definition 2.2. Let X = {x_{i,j}} be a real double sequence. If μ^{∗}(X ≤ 0) = 0, then the sequence X is said to be frequently positive, and if μ^{∗}(X ≥ 0) = 0, then X is said to be frequently negative. The sequence X is said to be frequently oscillatory if it is neither frequently positive nor frequently negative.

Obviously, if a double sequence is eventually positive, then it is frequently positive, and if it is eventually negative, then it is also frequently negative. Thus, if the sequence is frequently oscillatory, then it is oscillatory.

Definition 2.3. Let X be a real double sequence. If μ^{∗}(X ≤ 0) ≤ w, then X is said to be frequently positive of upper degree w, and if μ^{∗}(X ≥ 0) ≤ w, then X is said to be frequently negative of upper degree w. The sequence X is said to be frequently oscillatory of upper degree w if it is neither frequently positive nor frequently negative of the same upper degree w. The concept of frequently positivity of lower degree etc. is similarly defined by means of μ_{∗}.

In order to show the difference of frequent oscillation and usual oscillation, let us see the following two examples in one dimension [6].

Example 2.1. Consider the sequence

X = {x_{k}}_{k=1}^{∞} = {1, 1, 1, −1, 1, 1, 1, −1, ⋯ }.  (2.2)

Since μ_{∗}(X ≤ 0) = \frac{1}{4} and μ_{∗}(X ≥ 0) = \frac{3}{4}, thus X is frequently positive of lower degree \frac{1}{4}, frequently negative of lower degree \frac{3}{4}, and frequently oscillatory of lower degree \frac{1}{5}.

Example 2.2. The sequence

x_{k} = \{ \begin{matrix}−1 & if k = 2^{n} (n ∈ ℕ) \\ +1 & otherwise \end{matrix}

is oscillatory, and frequently positive.

To prove our main results we need several preparatory results. The first three lemmas to show basic properties of frequency measures are taken from [6].

Lemma 2.1. Let Ω, Γ ⊂ ℕ^{2}. Then

μ^{∗}(Ω + Γ) ≤ μ^{∗}(Ω) + μ^{∗}(Γ).  (2.3)

Furthermore, if Ω and Γ are disjoint, then

μ_{∗}(Ω) + μ_{∗}(Γ) ≤ μ_{∗}(Ω + Γ) ≤ μ_{∗}(Ω) + μ^{∗}(Γ) ≤ μ^{∗}(Ω + Γ) ≤ μ^{∗}(Ω) + μ^{∗}(Γ).

From Lemma 2.1 we can get

μ_{∗}(Ω) + μ^{∗}(ℕ^{2} \ Ω) = 1  (2.4)

for any subset Ω of ℕ^{2}.

114 C. J. Tian and B. G. Zhang

Lemma 2.2. Let Ω, Γ ⊂ ℕ^{2}. If μ^{∗}(Ω) + μ_{∗}(Γ) > 1, then Ω ∩ Γ is an infinite set. Lemma 2.3. Let Ω ⊂ ℕ^{2} and let α, β and γ, τ be integers such that α ≤ β and

γ ≤ τ. Then

μ_{∗}( ∑_{i=α}^{β} ∑_{j=γ}^{τ} X^{i}Y^{j}Ω) ≤ (β − α + 1)(τ − γ + 1)μ_{∗}(Ω)

and

μ^{∗}( ∑_{i=α}^{β} ∑_{j=γ}^{τ} X^{i}Y^{j}Ω) ≤ (β − α + 1)(τ − γ + 1)μ^{∗}(Ω).

Lemma 2.4. Assume that {A_{m,n}} is a solution of equation (1.1) such that

A_{m,n} > 0  for \{ \begin{matrix}m ∈ \{\overline{m} − 2k, …, \overline{m} + τ\} \\ n ∈ \{n̄ − 2l, …, n̄ + τ\} \end{matrix}
p_{m,n} ≥ p̄ > 0 and 0 < a_{m,n} ≤ ā  for \{ \begin{matrix}m ∈ \{\overline{m} − k, …, \overline{m} + τ\} \\ n ∈ \{n̄ − l, …, n̄ + τ\} \end{matrix}

where \overline{m} ≥ k, n̄ ≥ l and τ ≥ 2 are positive integers. Then

ā^{τ+1}A_{\overline{m},n̄} ≥ ∑_{i=0}^{τ+1} C_{τ+1}^{i}A_{\overline{m}+τ+1−i,n̄+i}

+ (τ + 1)p̄∑_{j=0}^{τ} C_{τ}^{j}A_{\overline{m}+τ−k−j,n̄+j−l}

+ p̄^{2} ∑_{i=1}^{τ} iā^{τ−i} ∑_{j=0}^{i−1} C_{i−1}^{j}A_{\overline{m}+i−1−j−2k,n̄+j−2l}.

Proof. From (1.1), we have

A_{\overline{m}+1,n̄} + A_{\overline{m},n̄+1} + p_{\overline{m},n̄}A_{\overline{m}−k,n̄−l} = a_{\overline{m},n̄}A_{\overline{m},n̄} ≤ āA_{\overline{m},n̄}

and, for m ∈ \{\overline{m} − k, …, \overline{m} + τ\} and n ∈ \{n̄ − l, …, n̄ + τ\},

A_{m+1,n} + A_{m,n+1} + p_{m,n}A_{m−k,n−l} ≤ a_{m,n}A_{m,n} ≤ āA_{m,n}.

Therefore,

\begin{aligned}A_{\overline{m}+2,n̄} + A_{\overline{m}+1,n̄+1} + p_{\overline{m}+1,n̄}A_{\overline{m}+1−k,n̄−l} = a_{\overline{m}+1,n̄}A_{\overline{m}+1,n̄} & ≤ āA_{\overline{m}+1,n̄} \\ A_{\overline{m}+1,n̄+1} + A_{\overline{m},n̄+2} + p_{\overline{m},n̄+1}A_{\overline{m}−k,n̄+1−l} = a_{\overline{m},n̄+1}A_{\overline{m},n̄+1} & ≤ āA_{\overline{m},n̄+1} \\ A_{\overline{m}+1−k,n̄−l} + A_{\overline{m}−k,n̄+1−l} + p_{\overline{m}−k,n̄−l}A_{\overline{m}−2k,n̄−2l} & ≤ āA_{\overline{m}−k,n̄−l}. \end{aligned}

Frequent Oscillation of Difference Equations 115

From the last three inequalities, we have

ā^{2}A_{\overline{m},n̄} ≥ ā(A_{\overline{m}+1,n̄} + A_{\overline{m},n̄+1} + p_{\overline{m},n̄}A_{\overline{m}−k,n̄−l})
≥ A_{\overline{m}+2,n̄} + 2A_{\overline{m}+1,n̄+1} + A_{\overline{m},n̄+2} + (p_{\overline{m}+1,n̄} + p_{\overline{m},n̄})A_{\overline{m}+1−k,n̄−l}
+ (p_{\overline{m},n̄+1} + p_{\overline{m},n̄})A_{\overline{m}−k,n̄+1−l} + p_{\overline{m},n̄}p_{\overline{m}−k,n̄−l}A_{\overline{m}−2k,n̄−2l},

i.e.

ā^{2}A_{\overline{m},n̄} ≥ ∑_{i=0}^{2} C_{2}^{i}A_{\overline{m}+2−i,n̄+i}

+ 2p̄∑_{j=0}^{1} C_{1}^{j}A_{\overline{m}+1−k−j,n̄+j−l}

+ p̄^{2} ∑_{i=1}^{1} iā^{1−i} ∑_{j=0}^{i−1} C_{i−1}^{j}A_{\overline{m}+i−1−j−2k,n̄+j−2l}.

From (1.1) we obtain

\begin{aligned}A_{\overline{m}+3,n̄} + A_{\overline{m}+2,n̄+1} + p_{\overline{m}+2,n̄}A_{\overline{m}+2−k,n̄−l} & ≤ āA_{\overline{m}+2,n̄} \\ A_{\overline{m}+2,n̄+1} + A_{\overline{m}+1,n̄+2} + p_{\overline{m}+1,n̄+1}A_{\overline{m}+1−k,n̄+1−l} & ≤ āA_{\overline{m}+1,n̄+1} \\ A_{\overline{m}+1,n̄+2} + A_{\overline{m},n̄+3} + p_{\overline{m},n̄+2}A_{\overline{m}−k,n̄+2−l} & ≤ āA_{\overline{m},n̄+2} \\ A_{\overline{m}+2−k,n̄−l} + A_{\overline{m}+1−k,n̄+1−l} + p_{\overline{m}+1−k,n̄−l}A_{\overline{m}+1−2k,n̄−2l} & ≤ āA_{\overline{m}+1−k,n̄−l} \\ A_{\overline{m}+1−k,n̄+1−l} + A_{\overline{m}−k,n̄+2−l} + p_{\overline{m}−k,n̄+1−l}A_{\overline{m}−2k,n̄+1−2l} & ≤ āA_{\overline{m}−k,n̄+1−l}. \end{aligned}

Hence

\begin{aligned}ā^{3}A_{\overline{m},n̄} & ≥ A_{\overline{m}+3,n̄} + 3A_{\overline{m}+2,n̄+1} + 3A_{\overline{m}+1,n̄+2} + A_{\overline{m},n̄+3} \\ & + (p_{\overline{m}+2,n̄} + p_{\overline{m}+1,n̄} + p_{\overline{m},n̄})A_{\overline{m}+2−k,n̄−l} \\ & + (2p_{\overline{m}+1,n̄+1} + p_{\overline{m}+1,n̄} + p_{\overline{m},n̄+1} + 2p_{\overline{m},n̄})A_{\overline{m}+1−k,n̄+1−l} \\ & + (p_{\overline{m},n̄+2} + p_{\overline{m},n̄+1} + p_{\overline{m},n̄})A_{\overline{m}−k,n̄+2−l} \\ & + āp̄^{2}A_{\overline{m}−2k,n̄−2l} + 2p̄^{2}A_{\overline{m}+1−2k,n̄−2l} + 2p̄^{2}A_{\overline{m}−2k,n̄+1−2l}. \end{aligned}  (2.5)

In view of the conditions of Lemma 2.4, we have

ā^{3}A_{\overline{m},n̄} ≥ ∑_{i=0}^{3} C_{3}^{i}A_{\overline{m}+3−i,n̄+i}

+ 3p̄∑_{j=0}^{2} C_{2}^{j}A_{\overline{m}+2−k−j,n̄+j−l}

+ p̄^{2} ∑_{i=1}^{2} iā^{2−i} ∑_{j=0}^{i−1} C_{i−1}^{j}A_{\overline{m}+i−1−j−2k,n̄+j−2l}.

116 C. J. Tian and B. G. Zhang

Assume that

ā^{τ} A_{\overline{m},n̄} ≥ ∑_{i=0}^{τ} C_{τ}^{i} A_{\overline{m}+τ−i,n̄+i}

+ τp̄∑_{j=0}^{τ−1} C_{τ−1}^{j}A_{\overline{m}+τ−1−k−j,n̄+j−l}

+ p̄^{2} ∑_{i=1}^{τ−1} iā^{τ−i−1} ∑_{j=0}^{i−1} C_{i−1}^{j}A_{\overline{m}+i−1−j−2k,n̄+j−2l}.

Then, for 0 ≤ i ≤ τ and 0 ≤ j ≤ τ − 1, we have

A_{\overline{m}+τ+1−i,n̄+i} + A_{\overline{m}+τ−i,n̄+1+i} + p̄A_{\overline{m}+τ−k−i,n̄+i−l} ≤ āA_{\overline{m}+τ−i,n̄+i}
A_{\overline{m}+τ−k−j,n̄+j−l}
+A_{\overline{m}+τ−1−k−j,n̄+1+j−l} + p̄A_{\overline{m}+τ−1−2k−j,n̄+j−2l} ≤ āA_{\overline{m}+τ−1−k−j,n̄+j−l}.

Thus

ā^{τ+1}A_{\overline{m},n̄}

≥ ∑_{i=0}^{τ} C_{τ}^{i} (A_{\overline{m}+τ+1−i,n̄+i} + A_{\overline{m}+τ−i,n̄+1+i} + p̄A_{\overline{m}+τ−k−i,n̄+i−l})

+ τp̄∑_{j=0}^{τ−1} C_{τ−1}^{j}(A_{\overline{m}+τ−k−j,n̄+j−l}

+ A_{\overline{m}+τ−1−k−j,n̄+1+j−l} + p̄A_{\overline{m}+τ−1−2k−j,n̄+j−2l})

+ p̄^{2} ∑_{i=1}^{τ−1} iā^{τ−i} ∑_{j=0}^{i−1} C_{i−1}^{j}A_{\overline{m}+i−1−j−2k,n̄+j−2l}

= A_{\overline{m}+τ+1,n̄}

+ ∑_{i=1}^{τ} C_{τ}^{i} A_{\overline{m}+τ+1−i,n̄+i}

+ ∑_{i=0}^{τ−1} C_{τ}^{i} A_{\overline{m}+τ−i,n̄+1+i} + A_{\overline{m},n̄+τ+1}

+ p̄∑_{i=0}^{τ} C_{τ}^{i} A_{\overline{m}+τ−k−i,n̄+i−l} + τp̄(A_{\overline{m}+τ−k,n̄−l}

+ ∑_{j=1}^{τ−1} C_{τ−1}^{j}A_{\overline{m}+τ−k−j,n̄+j−l}

+ ∑_{j=0}^{τ−2} C_{τ−1}^{j}A_{\overline{m}+τ−1−k−j,n̄+1+j−l} + A_{\overline{m}−k,n̄+τ−l})

Frequent Oscillation of Difference Equations 117

+ τp̄^{2} ∑_{j=0}^{τ−1} C_{τ−1}^{j}A_{\overline{m}+τ−1−2k−j,n̄+j−2l}

+ p̄^{2} ∑_{i=1}^{τ−1} iā^{τ−i} ∑_{j=0}^{i−1} C_{i−1}^{j}A_{\overline{m}+i−1−j−2k,n̄+j−2l}

= A_{\overline{m}+τ+1,n̄}

+ ∑_{i=1}^{τ}(C_{τ}^{i} + C_{τ}^{i−1})A_{\overline{m}+τ+1−i,n̄+i} + A_{\overline{m},n̄+τ+1}

+ p̄∑_{i=0}^{τ} C_{τ}^{i} A_{\overline{m}+τ−k−i,n̄+i−l} + τp̄(A_{\overline{m}+τ−k,n̄−l}

+ ∑_{j=1}^{τ−1}(C_{τ−1}^{j} + C_{τ−1}^{j−1})A_{\overline{m}+τ−k−j,n̄+j−l} + A_{\overline{m}−k,n̄+τ−l})

+ p̄^{2} ∑_{i=1}^{τ} iā^{τ−i} ∑_{j=0}^{i−1} C_{i−1}^{j}A_{\overline{m}+τ−1−2k−j,n̄+j−2l}

= ∑_{i=0}^{τ+1} C_{τ+1}^{i}A_{\overline{m}+τ−1−i,n̄+i}

+ (τ + 1)p̄∑_{j=0}^{τ} C_{τ}^{i} A_{\overline{m}+τ−k−j,n̄+j−l}

+ p̄^{2} ∑_{i=1}^{τ} iā^{τ−i} ∑_{j=0}^{i−1} C_{i−1}^{j}A_{\overline{m}+τ−1−2k−j,n̄+j−2l}.

The proof is complete

The following two lemmas can easily be derived from Lemma 2.4 and their proofs are thus omitted.

Lemma 2.5. Assume that {A_{m,n}} is a solution of equation (1.1) such that

A_{m,n} > 0  for \{ \begin{matrix}m ∈ \{\overline{m} − 3k, …, \overline{m} + l\} \\ n ∈ \{n̄ − 3l, …, n̄ + k\} \end{matrix}
p_{m,n} ≥ 0 and 0 < a_{m,n} ≤ ā  for \{ \begin{matrix}m ∈ \{\overline{m} − 2k, …, \overline{m} + l − 1\} \\ n ∈ \{n̄ − 2l, …, n̄ + k − 1\} \end{matrix}

where \overline{m} ≥ 2k and n̄ ≥ 2l are positive integers. Then C_{k+l}^{l}A_{\overline{m},n̄} ≤ ā^{k+l}A_{\overline{m}−k,n̄−l}.

118 C. J. Tian and B. G. Zhang

Lemma 2.6. Assume that {A_{m,n}} is a solution of equation (1.1) such that

A_{m,n} > 0  for \{ \begin{matrix}m ∈ \{\overline{m} − 2k − l, …, \overline{m} + 2k + l + 1\} \\ n ∈ \{n̄ − 2l − k, …, n̄ + k + 2l + 1\} \end{matrix}
p_{m,n} ≥ p̄ and 0 < a_{m,n} ≤ ā  for \{ \begin{matrix}m ∈ \{\overline{m} − k − l, …, \overline{m} + 2k + l\} \\ n ∈ \{n̄ − k − l, …, n̄ + k + 2l\} \end{matrix}

where \overline{m}, n̄ ≥ k + l are positive integers. Then

ā^{k+l+1}A_{\overline{m}−h,n̄+h} ≥ (k + l + 1)p̄C_{k+l}^{l−h}A_{\overline{m},n̄}.

for −k ≤ h ≤ l.

Lemma 2.7. Assume that the conditions of Lemma 2.4 hold and p̄ = 0. Then

ā^{τ+1}A_{\overline{m},n̄} ≥ ∑_{i=0}^{τ+1} C_{τ+1}^{i}A_{\overline{m}+τ+1−i,n̄+i} + ∑_{j=0}^{τ} ( ∑_{s=0}^{τ−j} ∑_{t=0}^{j} p_{\overline{m}+s,n̄+t})A_{\overline{m}+τ−k−j,n̄+j−l}.

Proof. As in the proof of Lemma 2.1, we know that inequality (2.5) holds. Assume that

ā^{τ} A_{\overline{m},n̄} ≥ ∑_{i=0}^{τ} C_{τ}^{i} A_{\overline{m}+τ−i,n̄+i} + ∑_{j=0}^{τ−1} ( ∑_{s=0}^{τ−1−j} ∑_{t=0}^{j} p_{\overline{m}+s,n̄+t})A_{\overline{m}+τ−k−j−1,n̄+j−l}.

Then, for 0 ≤ i ≤ τ and 0 ≤ j ≤ τ − 1, we have

\begin{matrix}A_{\overline{m}+τ−k−j,n̄+j−l} + A_{\overline{m}+τ−k−1−j,n̄+1+j−l} ≤ āA_{\overline{m}+τ−1−k−j,n̄+j−l} \\ A_{\overline{m}+τ+1−i,n̄+i} + A_{\overline{m}+τ−i,n̄+1+i} + p_{\overline{m}+τ−i,n̄+i}A_{\overline{m}+τ−i−k,n̄+i−l} ≤ āA_{\overline{m}+τ−i,n̄+i}. \end{matrix}

Thus

ā^{τ+1}A_{\overline{m},n̄}

≥ ∑_{i=0}^{τ} C_{τ}^{i} (A_{\overline{m}+τ+1−i,n̄+i} + A_{\overline{m}+τ−i,n̄+1+i} + p_{\overline{m}+τ−i,n̄+i}A_{\overline{m}+τ−i−k,n̄+i−l})

+ ∑_{j=0}^{τ−1} ( ∑_{s=0}^{τ−1−j} ∑_{t=0}^{j} p_{\overline{m}+s,n̄+t})(A_{\overline{m}+τ−k−j,n̄+j−l} + A_{\overline{m}+τ−k−1−j,n̄+1+j−l})

= ∑_{i=0}^{τ+1} C_{τ+1}^{i}A_{\overline{m}+τ+1−i,n̄+i} + ∑_{i=0}^{τ} C_{τ}^{i} p_{\overline{m}+τ−i,n̄+i}A_{\overline{m}+τ−i−k,n̄+i−l}

+ ∑_{s=0}^{τ−1} p_{\overline{m}+s,n̄}A_{\overline{m}+τ−k,n̄−l} + ∑_{j=1}^{τ−1} ( ∑_{s=0}^{τ−1−j} ∑_{t=0}^{j} p_{\overline{m}+s,n̄+t})A_{\overline{m}+τ−k−j,n̄+j−l}

+ ∑_{j=0}^{τ−2} ( ∑_{s=0}^{τ−1−j} ∑_{t=0}^{j} p_{\overline{m}+s,n̄+t})A_{\overline{m}+τ−k−1−j,n̄+1+j−l} + ∑_{t=0}^{τ−1} p_{\overline{m},n̄+t}A_{\overline{m}−k,n̄+τ−l}

≥ ∑_{i=0}^{τ+1} C_{τ+1}^{i}A_{\overline{m}+τ+1−i,n̄+i} + ∑_{i=1}^{τ−1} p_{\overline{m}+τ−i,n̄+i}A_{\overline{m}+τ−i−k,n̄+i−l}

+ p_{\overline{m}+τ,n̄}A_{\overline{m}+τ−k,n̄−l} + p_{\overline{m},n̄+τ} A_{\overline{m}−k,n̄+τ−l}

+ ∑_{s=0}^{τ−1} p_{\overline{m}+s,n̄}A_{\overline{m}+τ−k,n̄−l} + ∑_{t=0}^{τ−1} p_{\overline{m},n̄+t}A_{\overline{m}−k,n̄+τ−l}

+ ∑_{j=1}^{τ−1} ( ∑_{s=0}^{τ−1−j} ∑_{t=0}^{j} p_{\overline{m}+s,n̄+t} + ∑_{s=0}^{τ−j} ∑_{t=0}^{j−1} p_{\overline{m}+s,n̄+t})A_{\overline{m}+τ−k−j,n̄+j−l}

≥ ∑_{i=0}^{τ+1} C_{τ+1}^{i}A_{\overline{m}+τ+1−i,n̄+i} + ∑_{s=0}^{τ} p_{\overline{m}+s,n̄}A_{\overline{m}+τ−k,n̄−l}

+ ∑_{t=0}^{τ} p_{\overline{m},n̄+t}A_{\overline{m}−k,n̄+τ−l} + ∑_{j=1}^{τ−1} (p_{\overline{m}+τ−j,n̄+j}

+ ∑_{s=0}^{τ−1−j} ∑_{t=0}^{j} p_{\overline{m}+s,n̄+t} + ∑_{s=0}^{τ−j} ∑_{t=0}^{j−1} p_{\overline{m}+s,n̄+t})A_{\overline{m}+τ−k−j,n̄+j−l}

≥ ∑_{i=0}^{τ+1} C_{τ+1}^{i}A_{\overline{m}+τ+1−i,n̄+i} + ∑_{j=0}^{τ} ( ∑_{s=0}^{τ−j} ∑_{t=0}^{j} p_{\overline{m}+s,n̄+t})A_{\overline{m}+τ−k−j,n̄+j−l}.

The proof is complete

Lemma 2.8. If 0 < a_{m,n} ≤ 1 and p_{m,n} ≥ 0 for all large m and n, then every non-oscillatory solution of equation (1.1) tends to zero as m, n → ∞.

Proof. Assume that {A_{m,n}} is an eventually positive solution of equation (1.1). Then there exist positive integers \overline{m} ≥ −3k and n̄ ≥ −3l such that

A_{m,n} > 0  for \{ \begin{matrix}m ≥ \overline{m} − 3k \\ n ≥ n̄ − 3l \end{matrix}
0 < a_{m,n} ≤ 1 and p_{m,n} ≥ 0  for \{ \begin{matrix}m ≥ \overline{m} − 2k \\ n ≥ n̄ − 2l. \end{matrix}

By Lemma 2.4, for positive integers τ_{1} and τ_{2} we have

A_{\overline{m},n̄} ≥ C_{τ_{1}+τ_{2}}^{τ_{2}}A_{\overline{m}+τ_{1},n̄+τ_{2}}.

Thus

A_{\overline{m}+τ_{1},n̄+τ_{2}} ≤ \frac{A_{\overline{m},n̄}}{C_{τ_{1}+τ_{2}}^{τ_{2}}} → 0 as τ_{1}, τ_{2} → ∞.

The proof is complete

Lemma 2.9. Assume that for some positive integers \overline{m} and n̄, equation (1.1) has a solution {A_{m,n}} such that

A_{m,n} > 0 for \{ \begin{matrix}\overline{m} − 3k ≤ m ≤ \overline{m} + l + k + 1 \\ n̄ − 3l ≤ n ≤ n̄ + k + l + 1. \end{matrix}

120 C. J. Tian and B. G. Zhang

If

p_{m,n} ≥ 0 and 0 < a_{m,n} ≤ ā  for \{ \begin{matrix}\overline{m} − 2k ≤ m ≤ \overline{m} + l + k \\ n̄ − 2l ≤ n ≤ n̄ + k + l \end{matrix}
∑_{i=m−k}^{m} ∑_{j=n−l}^{n} p_{i,j} ≥ B > 0  for \{ \begin{matrix}\overline{m} ≤ m ≤ \overline{m} + k \\ n̄ ≤ n ≤ n̄ + l, \end{matrix}

then

\frac{A_{\overline{m}−k,n̄−l}}{A_{\overline{m},n̄}} < (\frac{2}{B} )^{4}ā^{3k+3l+4}.

Proof. It is easy to see that there exists m^{∗} such that \overline{m} ∈ \{m^{∗} − k, …, m^{∗}\} and

∑_{i=m^{∗}−k}^{\overline{m}} ∑_{j=n̄−l}^{n̄} p_{i,j} ≥ \frac{B}{2} and ∑_{i=\overline{m}}^{m^{∗}} ∑_{j=n̄−l}^{n̄} p_{i,j} ≥ \frac{B}{2} .

By Lemma 2.7, we have

ā^{m^{∗}−\overline{m}+l+1}A_{\overline{m},n̄−l} ≥ ( ∑_{s=0}^{m^{∗}−\overline{m}} ∑_{t=0}^{l} p_{\overline{m}+s,n̄+t−l})A_{m^{∗}−k,n̄−l} ≥ \frac{B}{2} A_{m^{∗}−k,n−l}

and

ā^{\overline{m}−m^{∗}+k+l+1}A_{m^{∗}−k,n̄−l} ≥ ( ∑_{s=0}^{\overline{m}−m^{∗}+k} ∑_{t=0}^{l} p_{m^{∗}−k+s,n̄−l+t})A_{\overline{m}−k,n̄−l} ≥ \frac{B}{2} A_{\overline{m}−k,n̄−l}.

Thus

A_{\overline{m},n̄−l} ≥ (\frac{B}{2} )^{2}ā^{−k−2l−2}A_{\overline{m}−k,n̄−l}.

Similarly, there exists n^{∗} such that n̄ ∈ {n^{∗} − l, …, n^{∗}} and

∑_{i=\overline{m}}^{\overline{m}+k} ∑_{j=n^{∗}−l}^{n̄} p_{i,j} ≥ \frac{B}{2} and ∑_{i=\overline{m}}^{\overline{m}+k} ∑_{j=n̄}^{n^{∗}} p_{i,j} ≥ \frac{B}{2} .

By Lemma 2.7, we have

ā^{n^{∗}−n̄+k+1}A_{\overline{m},n̄} ≥ ( ∑_{s=0}^{k} ∑_{t=0}^{n^{∗}−n̄} p_{\overline{m}+s,n̄+t})A_{\overline{m},n^{∗}−l} ≥ \frac{B}{2} A_{\overline{m},n^{∗}−l}

and

ā^{n̄−n^{∗}+k+l+1}A_{\overline{m},n^{∗}−l} ≥ ( ∑_{s=0}^{k} ∑_{t=0}^{n̄−n^{∗}+l} p_{\overline{m}+s,n^{∗}−l+t})A_{\overline{m},n̄−l} ≥ \frac{B}{2} A_{\overline{m},n̄−l}.

Hence

A_{\overline{m},n̄} ≥ (\frac{B}{2} )^{2}ā^{−l−2k−2}A_{\overline{m},n̄−l}.

Therefore

\frac{A_{\overline{m}−k,n̄−l}}{A_{\overline{m},n̄}} = \frac{A_{\overline{m}−k,n̄−l}}{A_{\overline{m},n̄−l}} \frac{A_{\overline{m},n̄−l}}{A_{\overline{m},n̄}} < (\frac{2}{B} )^{4}ā^{3k+3l+4}.

The proof is complete

The following fact is also an easy consequence of Lemma 2.4 and its proof is thus omitted.

Lemma 2.10. Assume that for positive integers \overline{m}, n̄ and τ equation (1.1) has a solution {A_{m,n}} such that

A_{m,n} > 0  for \{ \begin{matrix}m ∈ \{\overline{m} − k, …, \overline{m} + τ\} \\ n ∈ \{n̄ − l, …, n̄ + τ\} \end{matrix}

p_{m,n} ≥ 0 and 0 < a_{m,n} ≤ ā  for \{ \begin{matrix}m ∈ \{\overline{m}, …, \overline{m} + τ\} \\ n ∈ \{n̄, …, n̄ + τ\}. \end{matrix}

Then

ā^{τ} A_{\overline{m},n̄} > A_{\overline{m}+τ,n̄}  and  ā^{τ} A_{\overline{m},n̄} > A_{\overline{m},n̄+τ} .

3. Main results

Let a = {a_{m,n}} and p = {p_{m,n}} be real double sequences.

Theorem 3.1. Assume that p̄ and ā are two positive real numbers and w a nonnegative number. Let μ^{∗}(a > ā or a ≤ 0) = Ā and μ^{∗}(p < p̄) = B̄. Suppose further that

p̄ \frac{(k + l + 1)!}{k!l! ā^{k+l+1}} ≥ 1

and

(2k + l + 1)(2l + k + 1)(Ā + B̄) + (3k + l + 2)(3l + k + 2)w < 1.

Then every solution of equation (1.1) is frequently oscillatory of lower degree w.

Proof. Suppose to the contrary there is a frequently positive solution A = {A_{m,n}} of equation (1.1) with lower degree w such that μ_{∗}(A ≤ 0) ≤ w. In view of Lemmas 2.1 - (2.4) we have

μ_{∗}(ℕ^{2}\ ∑_{i=−k−l}^{k} ∑_{j=−k−l}^{l} X^{i}Y^{j}(a > ā or p < p̄))

+ μ^{∗}(ℕ^{2}\ ∑_{i=−k−l−1}^{2k} ∑_{j=−k−l−1}^{2l} X^{i}Y^{j}(A ≤ 0))

= 2 − μ^{∗}( ∑_{i=−k−l}^{k} ∑_{j=−k−l}^{l} X^{i}Y^{j}(a > ā or p < p̄))

− μ_{∗}( ∑_{i=−k−l−1}^{2k} ∑_{j=−k−l−1}^{2l} X^{i}Y^{j}(A ≤ 0))

≥ 2 − (2k + l + 1)(2l + k + 1)μ^{∗}(a > ā or p < p̄)

− (3k + l + 2)(3l + k + 2)μ_{∗}(A ≤ 0)

\begin{matrix}≥ 2 − (2k + l + 1)(2l + k + 1)(Ā + B̄) − (3k + l + 2)(3l + k + 2)w \\ > 1. \end{matrix}

Hence, by Lemma 2.2,

\begin{matrix}(ℕ^{2}\ ∑_{i=−k−l}^{k} ∑_{j=−k−l}^{l} X^{i}Y^{j}(a > ā or p < p̄)) \\ ⋂ (ℕ^{2}\ ∑_{i=−k−l−1}^{2k} ∑_{j=−k−l−1}^{2l} X^{i}Y^{j}(A ≤ 0)) \end{matrix}

is an infinite subset of ℕ^{2}, which together with (2.1), imply that there is a lattice point (\overline{m}, n̄) such that

A_{m,n} > 0  for \{ \begin{matrix}\overline{m} − 2k ≤ m ≤ \overline{m} + k + l + 1 \\ n̄ − 2l ≤ n ≤ n̄ + k + l + 1 \end{matrix}

0 < a_{m,n} ≤ ā and p_{m,n} ≥ p̄ for \{ \begin{matrix}\overline{m} − k ≤ m ≤ \overline{m} + k + l \\ n̄ − l ≤ n ≤ n̄ + k + l. \end{matrix}

By Lemma 2.4, we obtain ā^{k+l+1}A_{\overline{m},n̄} > (k + l + 1)p̄C_{k+l}^{l}A_{\overline{m},n̄}. Hence

p̄ \frac{(k + l + 1)!}{k!l! ā^{k+l+1}} < 1

which is a contradiction. The proof is complete

Lemma 3.1. Let k > 0, l > 0, Ψ > 0, ā > 0 and
Γ = ā^{k+l+1}λ^{λ}2^{−λ}(1 + λ)^{−1−λ} where λ = 2kl(k + l)^{−1}.

Assume that for some positive integers \overline{m}, n̄ and τ equation (1.1) has a solution \{A_{m,n}\} such that

A_{i,j} > 0  for \{ \begin{matrix}\overline{m} − (τ + 3)k ≤ i ≤ \overline{m} + l \\ n̄ − (τ + 3)l ≤ j ≤ n̄ + k \end{matrix}

0 < a_{i,j} ≤ ā and p_{i,j} ≥ 0  for \{ \begin{matrix}\overline{m} − (τ + 2)k ≤ i ≤ \overline{m} + l \\ n̄ − (τ + 2)l ≤ j ≤ n̄ + k \end{matrix}

\frac{1}{kl} ∑_{i=m−k}^{m−1} ∑_{j=n−l}^{n−1} p_{i,j} ≥ Ψ  for \{ \begin{matrix}\overline{m} − (τ − 1)k ≤ m ≤ \overline{m} \\ n̄ − (τ − 1)l ≤ n ≤ n̄. \end{matrix}

Then

\frac{A_{\overline{m},n̄}}{A_{\overline{m}−k,n̄−l}} ≤ \frac{ā^{k+l}}{C_{k+l}^{l}} (\frac{Γ}{Ψ})^{τ} .

Frequent Oscillation of Difference Equations 123

Proof. From (1.1) and Lemma 2.5, we have

A_{m+1,n} + A_{m,n+1} − āA_{m,n} ≤ A_{m+1,n} + A_{m,n+1} − a_{m,n}A_{m,n}

= −p_{m,n}A_{m−k,n−l}

≤ −p_{m,n} \frac{C_{k+l}^{l}}{ā^{k+l}} A_{m,n}

for \overline{m} − τk ≤ m ≤ \overline{m} and n̄ − τl ≤ n ≤ n̄. Thus

p_{m,n} ≤ \frac{ā^{k+l}}{C_{k+l}^{l}} (ā − \frac{A_{m+1,n} + A_{m,n+1}}{A_{m,n}} ) ≤ \frac{ā^{k+l}}{C_{k+l}^{l}} (ā − \frac{2(A_{m+1,n}A_{m,n+1})^{\frac{1}{2}}}{A_{m,n}} ) .

Hence, for \overline{m} − (τ − 1)k ≤ m ≤ \overline{m} and n̄ − (τ − 1)l ≤ n ≤ n̄ we have

Ψ ≤ \frac{1}{kl} ∑_{i=m−k}^{m−1} ∑_{j=n−l}^{n−1} p_{i,j}

≤ \frac{ā^{k+l}}{C_{k+l}^{l}} \frac{1}{kl} ∑_{i=m−k}^{m−1} ∑_{j=n−l}^{n−1} (ā − \frac{2(A_{i+1,j}A_{i,j+1})^{\frac{1}{2}}}{A_{i,j}} )

= \frac{ā^{k+l}}{C_{k+l}^{l}} (ā − \frac{2}{kl} ∑_{i=m−k}^{m−1} ∑_{j=n−l}^{n−1} \frac{(A_{i+1,j}A_{i,j+1})^{\frac{1}{2}}}{A_{i,j}} ).

Note that by the inequality between the arithmetic and geometric means and Lemma 2.10 we have

∑_{i=m−k}^{m−1} ∑_{j=n−l}^{n−1} \frac{(A_{i+1,j}A_{i,j+1})^{\frac{1}{2}}}{A_{i,j}}

≥ ∑_{i=m−k}^{m−1} l( ∏_{j=n−l}^{n−1} \frac{(A_{i+1,j}A_{i,j+1})^{\frac{1}{2}}}{A_{i,j}} )^{\frac{1}{l}}

= l ∑_{i=m−k}^{m−1} (\frac{A_{i,n}}{A_{i,n−l}} ∏_{j=n−l}^{n−1} \frac{A_{i+1,j}}{A_{i,j}} )^{\frac{1}{2l}}

≥ kl ∏_{i=m−k}^{m−1} (\frac{A_{i,n}}{A_{i,n−l}} )^{\frac{1}{2kl}} ∏_{i=m−k}^{m−1} ∏_{j=n−l}^{n−1} (\frac{A_{i+1,j}}{A_{i,j}} )^{\frac{1}{2kl}}

= kl ∏_{i=m−k}^{m−1} (\frac{A_{i,n}}{A_{i,n−l}} )^{\frac{1}{2kl}} ∏_{j=n−l}^{n−1} (\frac{A_{m,j}}{A_{m−k,j}} )^{\frac{1}{2kl}}

≥ kl ∏_{i=m−k}^{m−1} (\frac{A_{m,n}}{ā^{m−i}} \frac{1}{ā^{i−m+k}A_{m−k,n−l}} )^{\frac{1}{2kl}} ∏_{j=n−l}^{n−1} (\frac{1}{ā^{l}} \frac{A_{m,n}}{A_{m−k,n−l}} )^{\frac{1}{2kl}}

\begin{matrix}= kl ā^{− \frac{k^{2}+l^{2}}{2kl}} (\frac{A_{m,n}}{A_{m−k,n−l}} )^{\frac{k+l}{2kl}} \\ = kl ā^{− \frac{k^{2}+l^{2}}{2kl}} (\frac{A_{m,n}}{A_{m−k,n−l}} )^{\frac{1}{λ}} . \end{matrix}

Thus

Ψ ≤ \frac{ā^{k+l+1}}{C_{k+l}^{l}} (1 − 2ā^{− \frac{(k+l)^{2}}{2kl}} (\frac{A_{m,n}}{A_{m−k,n−l}} )^{\frac{1}{λ}} ),

i.e.

\frac{A_{m,n}}{A_{m−k,n−l}} ≤ ā^{k+l}2^{−λ}(1 − \frac{ΨC_{k+l}^{l}}{ā^{k+l+1}} )^{λ} ≤ \frac{ā^{k+l}}{C_{k+l}^{l}} \frac{Γ}{Ψ}.

Hence in view of (1.1) we have

A_{m+1,n} + A_{m,n+1} − āA_{m,n} ≤ −p_{m,n}A_{m−k,n−l} ≤ −\frac{ΨC_{k+l}^{l}}{Γā^{k+l}} p_{m,n}A_{m,n}.

Thus

p_{m,n} ≤ \frac{ā^{k+l}Γ}{C_{k+l}^{l}Ψ}(ā − \frac{2(A_{m+1,n}A_{m,n+1})^{\frac{1}{2}}}{A_{m,n}} ) for \{ \begin{matrix}\overline{m} − τk ≤ m ≤ \overline{m} \\ n̄ − τl ≤ n ≤ n̄. \end{matrix}

For \overline{m} − (τ − 1)k ≤ m ≤ \overline{m} and n̄ − (τ − 1)l ≤ n ≤ n̄ we have

Ψ ≤ \frac{1}{kl} ∑_{i=m−k}^{m−1} ∑_{j=n−l}^{n−1} p_{i,j}

≤ \frac{ā^{k+l}Γ}{C_{k+l}^{l}Ψ}(ā − \frac{2}{kl} ∑_{i=m−k}^{m−1} ∑_{j=n−l}^{n−1} \frac{(A_{i+1,j}A_{i,j+1})^{\frac{1}{2}}}{A_{i,j}} )

≤ \frac{ā^{k+l+1}Γ}{C_{k+l}^{l}Ψ}(1 − 2ā^{− \frac{(k+l)^{2}}{2kl}} (\frac{A_{m,n}}{A_{m−k,n−l}} )^{\frac{1}{λ}} ).

Hence

\frac{A_{m,n}}{A_{m−k,n−l}} ≤ \frac{ā^{k+l}}{2^{λ}} (1 − \frac{C_{k+l}^{l}Ψ^{2}}{ā^{k+l+1}Γ})^{λ} ≤ \frac{ā^{k+l}}{C_{k+l}^{l}} (\frac{Γ}{Ψ})^{2}.

By induction, we can get

\frac{A_{m,n}}{A_{m−k,n−l}} ≤ \frac{ā^{k+l}}{C_{k+l}^{l}} (\frac{Γ}{Ψ})^{τ} for \{ \begin{matrix}\overline{m} − k ≤ m ≤ \overline{m} \\ n̄ − l ≤ n ≤ n̄. \end{matrix}

The proof is complete

Frequent Oscillation of Difference Equations 125

In view of Lemma 2.9 and Lemma 3.1, we see that if we let ā, τ, Ψ and Γ = \frac{ā^{k+l+1}λ^{λ}}{2^{λ}(1+λ)^{1+λ}} be positive such that

(\frac{klΨ}{2ā^{k+l+1}} )^{4} ≥ \frac{1}{C_{k+l}^{l}} (\frac{Γ}{Ψ})^{τ}  (3.1)

and \overline{m}, n̄ be positive integers such that

p_{i,j} ≥ 0 and 0 < a_{i,j} ≤ ā  for \{ \begin{matrix}\overline{m} − (τ + 2)k ≤ i ≤ \overline{m} + k + l \\ n̄ − (τ + 2)l ≤ j ≤ n̄ + k + l \end{matrix}
\frac{1}{kl} ∑_{i=m−k}^{m−1} ∑_{j=n−l}^{n−1} p_{i,j} ≥ Ψ  for \{ \begin{matrix}\overline{m} − (τ − 1)k ≤ m ≤ \overline{m} + k \\ n̄ − (τ − 1)l ≤ n ≤ n̄, \end{matrix}

then equation (1.1) has no solution {A_{i,j}}, which satisfies

A_{i,j} > 0 for \{ \begin{matrix}\overline{m} − (τ + 3)k ≤ i ≤ \overline{m} + k + l + 1 \\ n̄ − (τ + 3)l ≤ j ≤ n̄ + k + l + 1. \end{matrix}

The follwing theorem now follows easily.

Theorem 3.2. Assume that λ = \frac{2kl}{k+l}, Γ = \frac{ā^{k+l+1}λ^{λ}}{2^{λ}(1+λ)^{1+λ}}, ā > 0, Ψ > 0, τ > 0 and (3.1) holds. Let q = {q_{m,n}} be a double sequence defined by

q_{m,n} = \frac{1}{kl} ∑_{i=m−k}^{m−1} ∑_{j=n−l}^{n−1} p_{i,j} for \{ \begin{matrix}m = k + 1, k + 2, … \\ n = l + 1, l + 2, …. \end{matrix}

Assume further that

μ^{∗}(p < 0) = B̄, μ^{∗}(a > ā or a ≤ 0) = Ā, μ_{∗}(q < Ψ) = D̄

and

((τ + 3)k + l + 1)((τ + 3)l + k + 1)(Ā + B̄ + D̄) < 1.

Then equation (1.1) has no frequently positive solutions.

Proof. Supposing the contrary, let A = {A_{i,j}} be a solution of equation (1.1) such that μ^{∗}(A ≤ 0) = 0. By Lemmas 2.1 - 2.4 we get

\begin{matrix}μ_{∗}(ℕ^{2}\ ∑_{i=−k−l−1}^{(τ+3)k} ∑_{j=−k−l−1}^{(τ+3)l} X^{i}Y^{j}(A ≤ 0)) \\ + μ^{∗}(ℕ^{2}\ ∑_{i=−k−l}^{(τ+2)k} ∑_{j=−k−l}^{(τ+2)l} X^{i}Y^{j}(a ≤ 0 or a > ā or p < 0 or q < Ψ)) \end{matrix}

= 2 − μ_{∗}( ∑_{i=−k−l}^{(τ+2)k} ∑_{j=−k−l}^{(τ+2)l} X^{i}Y^{j}(a ≤ 0 or a > ā or p < 0 or q < Ψ))

− μ^{∗}( ∑_{i=−k−l−1}^{(τ+3)k} ∑_{j=−k−l−1}^{(τ+3)l} X^{i}Y^{j}(A ≤ 0))

≥ 2 − ((τ + 3)k + l + 1)((τ + 3)l + k + 1) × μ_{∗}(a ≤ 0 or a > ā or p < 0 or q < Ψ) − ((τ + 4)k + l + 2)((τ + 4)l + k + 2)μ^{∗}(A ≤ 0) ≥ 2 − ((τ + 3)k + l + 1)((τ + 3)l + k + 1)(Ā + B̄ + D̄) > 1.

Thus

\begin{matrix}(ℕ^{2}\ ∑_{i=−k−l−1}^{(τ+3)k} ∑_{j=−k−l−1}^{(τ+3)l} X^{i}Y^{j}(A ≤ 0)) \\ ⋂ (ℕ^{2}\ ∑_{i=−k−l}^{(τ+2)k} ∑_{j=−k−l}^{(τ+2)l} X^{i}Y^{j}(a ≤ 0 or a > ā or p < 0 or q < Ψ)) \end{matrix}

is an infinite subset of ℕ^{2}. In view of (2.1), we see that there are positive integers \overline{m} and n̄ such that

\begin{matrix}A_{m,n} > 0 & for \{ \begin{matrix}\overline{m} − (τ + 3)k ≤ m ≤ \overline{m} + k + l + 1 \\ n̄ − (τ + 3)l ≤ n ≤ n̄ + k + l + 1 \end{matrix} \\ a_{m,n} ∈ (0, ā], p_{m,n} ≥ 0, q_{m,n} ≥ Ψ & for \{ \begin{matrix}\overline{m} − (τ + 2)k ≤ m ≤ \overline{m} + k + l \\ n̄ − (τ + 2)l ≤ n ≤ n̄ + k + l \end{matrix} \end{matrix}

which is a contradiction. The proof is complete Corollary 3.1. Assume that

λ = \frac{2kl}{k + l}  and  Γ = \frac{ā^{k+l+1}λ^{λ}}{2^{λ}(1 + λ)^{1+λ}}

and a_{m,n} ≡ ā > 0 for m, n ≥ 0. If p_{m,n} ≥ 0 and

lim inf_{m,n→∞} \frac{1}{kl} ∑_{i=m−k}^{m−1} ∑_{j=n−l}^{n−1} p_{i,j} > Γ,

then every solution of equation (1.1) is frequently oscillatory.

Proof. It is easy to see that there exist τ > 0 and Ψ > Γ such that for all large m and n

q_{m,n} = \frac{1}{kl} ∑_{i=m−k}^{m−1} ∑_{j=n−l}^{n−1} p_{i,j} ≥ Ψ > Γ

Frequent Oscillation of Difference Equations 127

and (3.1) holds, i.e.

(\frac{klΨ}{2ā^{k+l+1}} )^{4} ≥ \frac{1}{C_{k+l}^{l}} (\frac{Γ}{Ψ})^{τ} .

Since

μ^{∗}(a > ā or a ≤ 0) = 0, μ^{∗}(p < 0) = 0, μ_{∗}(q < Ψ) = 0,

then

((τ + 3)k + l + 1)((τ + 3)l + k + 1)(0 + 0 + 0) = 0 < 1.

By Theorem 3.2, every solution of equation (1.1) is frequently oscillatory From Corollary 3.1, the following result is obvious. Corollary 3.2. If a_{m,n} ≡ 1 and p_{m,n} ≥ 0, and

lim inf_{m,n→∞} \frac{1}{kl} ∑_{i=m−k}^{m−1} ∑_{j=n−l}^{n−1} p_{i,j} > \frac{λ^{λ}}{2^{λ}(1 + λ)^{1+λ}}

where λ = \frac{2kl}{k+l}, then equation (1.1) has no frequently positive solutions, and hence it has no eventually positive or eventually negative solutions.

Remark. Corollary 3.2 improves [10: Theorem 3.4]. From Lemma 2.6 we can obtain the following result. Theorem 3.3. Assume that p̄ and ā are two positive numbers and ω is a non-

negative number. Let μ^{∗}(a ≤ 0 or a > ā) = Ā and μ_{∗}(p < p̄) = B̄. Further assume that

p̄(k + l + 1)\sqrt{C_{2k+2l}^{2l}} ≥ ā^{k+l+1}

and

(3k + 2l + 1)(3l + 2k + 1)(Ā + B̄) + (4k + 2l + 2)(4l + 2k + 2)ω < 1.

Then every solution of equation (1.1) is frequently oscillatory of upper degree ω.

Proof. Without loss of generality, assume that min(k, l) = l. Suppose to the contrary there is a frequently positive solution A = {A_{m,n}} with upper degree ω such that μ^{∗}(A ≤ 0) ≤ ω. In view of Lemmas 2.1 - 2.3 and (2.4) we have

μ^{∗}(ℕ^{2}\ ∑_{i=−2k−l}^{k+l} ∑_{j=−k−2l}^{k+l} X^{i}Y^{j}(a ≤ 0 or a > ā or p < p̄))

+ μ_{∗}(ℕ^{2}\ ∑_{i=−2k−l−1}^{2k+l} ∑_{j=−k−2l−1}^{k+2l} X^{i}Y^{j}(A ≤ 0))

= 2 − μ_{∗}( ∑_{i=−2k−l}^{k+l} ∑_{j=−k−2l}^{k+l} X^{i}Y^{j}(a ≤ 0 or a > ā or p < p̄)

− μ^{∗}( ∑_{i=−2k−l−1}^{2k+l} ∑_{j=−k−2l−1}^{2l+k} X^{i}Y^{j}(A ≤ 0))

≥ 2 − (3k + 2l + 1)(2k + 3l + 1)μ_{∗}(a ≤ 0 or a > ā or p < p̄)
− (4k + 2l + 2)(2k + 4l + 2)μ^{∗}(A ≤ 0)

≥ 2 − (3k + 2l + 1)(2k + 3l + 1)(Ā + B̄) − (4k + 2l + 2)(2k + 4l + 2)ω

> 1.

Hence by Lemma 2.2,

\begin{matrix}(ℕ^{2}\ ∑_{i=−2k−l}^{k+l} ∑_{j=−k−2l}^{k+l} X^{i}Y^{j}(a ≤ 0 or a > ā or p < p̄)) \\ ⋂ (ℕ^{2} \ ∑_{i=−2k−l−1}^{2k+l} ∑_{j=−k−2l−1}^{2l+k} X^{i}Y^{j}(A ≤ 0)) \end{matrix}

is an infinite subset of ℕ^{2}, which together with (2.1) implies that there is a lattice point (\overline{m}, n̄) such that

A_{m,n} > 0  for \{ \begin{matrix}m ∈ \{\overline{m} − 2k − l, …, \overline{m} + 2k + l + 1\} \\ n ∈ \{n̄ − 2l − k, …, n̄ + k + 2l + 1\} \end{matrix}

0 < a_{m,n} ≤ ā and p_{m,n} ≥ p̄ for \{ \begin{matrix}m ∈ \{\overline{m} − k − l, …, \overline{m} + 2k + l\} \\ n ∈ \{n̄ − k − l, …, n̄ + k + 2l\}. \end{matrix}

By Lemmas 2.4 and 2.6 we obtain

ā^{k+l+1}A_{\overline{m},n̄} > (k + l + 1)p̄∑_{j=0}^{k+l} C_{k+l}^{j}A_{\overline{m}+l−j,n̄+j−l}

≥ (k + l + 1)p̄∑_{j=0}^{2l} C_{k+l}^{j}A_{\overline{m}+l−j,n̄+j−l}

≥ \frac{p̄^{2}(k + l + 1)^{2}}{ā^{k+l+1}} ∑_{j=0}^{2l} C_{k+l}^{j}C_{k+l}^{2l−j}A_{\overline{m},n̄}.

By the inequality ∑_{j=0}^{2l} C_{k+l}^{j}C_{k+l}^{2l−j} = C_{2k+2l}^{2l} we have

ā^{k+l+1} > (k + l + 1)p̄\sqrt{C_{2k+2l}^{2l}}

which is a contradiction. The proof is complete

Example 3.1. Consider the partial difference equation

A_{m+1,n} + A_{m,n+1} − 2A_{m,n} + p_{m,n}A_{m−1,n−1} = 0  (3.2)

Frequent Oscillation of Difference Equations 129

where

p_{m,n} = \{ \begin{matrix}−1 & if m = 8k and n = 8l (k, l ∈ ℕ_{0}) \\ \frac{4}{3} & otherwise. \end{matrix}

Take 0 ≤ ω < \frac{1}{48}. Since

μ^{∗}(a > 2 or a ≤ 0) = 0, μ^{∗}(p < \frac{4}{3}) = \frac{1}{64}, 16 × \frac{1}{64} + 36ω < 1, \frac{4}{3} × \frac{3!}{2^{3}} = 1,

then by Theorem 3.1 every solution of equation (3.2) is frequently oscillatory of lower degree ω, and hence oscillatory.

Example 3.2. Consider the difference equation

A_{m+1,n} + A_{m,n+1} − A_{m,n} + P_{m,n}A_{m−1,n−2} = 0  (3.3)

where

P_{m,n} = \{ \begin{matrix}0 & if m = 9k and n = 9l (k, l ∈ ℕ_{0}) \\ \frac{1}{12} & otherwise. \end{matrix}

Then

μ^{∗}(a ≤ 0 or a > 1) = 0  and  μ_{∗}(p < \frac{1}{12}) = \frac{1}{81}.

Taking 0 ≤ ω < \frac{1}{1080}, then 72 × \frac{1}{81} + 120ω < 1 and \frac{1}{12} × 4 × \sqrt{C_{6}^{2}} > 1. Therefore by Theorem 3.3 every solution of equation (3.3) is frequently oscillatory of upper degree ω and hence oscillatory. But Theorem 3.1 is not available here.

References

[1] Agarwal, R. P.: Difference Equations and Inequalities. New York: Marcel Dekker 1992.

[2] Cheng, S. S. and B. G.Zhang: Qualitative theory of partial difference equations. Part I: Oscillation of nonlinear partial difference equations. Tamkang J. Math. 25 (1994), 279 – 288.

[3] Courant, R., Friedrichs, K. and H. Lewy: On partial difference equations of mathematical physics. IBM J. 11 (1967), 215 – 234.

[4] Kelley, W. G. and A. C. Peterson: Difference Equations. New York: Acad. Press 1991.

[5] Li, X.-P.: Partial difference equations used in the study of molecular orbits (in Chinese). Acta Chimica SINICA 40 (1982), 688 – 698.

[6] Tian, C. J., Xie, S. L. and S. S. Cheng: Measures for oscillatory sequences. Comp. Math. Appl. (to appear).

[7] Tramov, M. I.: Oscillation of partial diffferential equations with deviating arguments (in Russian). Diff. Uravn. 20 (1984), 721 – 723.

[8] Zhang, B. G. and S. T. Liu: Oscillation of partial difference equations. PanAmer. Math. J. 5 (1995)2, 61 – 70.

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[10] Zhang, B. G., Liu, S. T. and S. S. Cheng: Oscillation of a class of delay partial difference equations. J. Difference Equ. Appl. 1 (1995), 215 – 226.

[11] Zhang, B. G. and J. S. Yu: Linearized oscillation theorems for certain nonlinear delay partial difference equations. Comp. Math. Applic. 35 (1998)4, 111 – 116.

Received 29.04.1998: in revised form 16.12.1998