Georgian Mathematical Journal Volume 10 (2003), Number 2, 343–352

OSCILLATION THEOREMS OF NONLINEAR DIFFERENCE EQUATIONS OF SECOND ORDER

S. H. SAKER

Abstract. Using the Riccati transformation techniques, we establish some new oscillation criteria for the second-order nonlinear difference equation Δ^{2}x_{n} + F (n, x_{n}, Δx_{n}) = 0 for n ≥ n_{0}. Some comparison between our theorems and the previously known results in special cases are indicated. Some examples are given to illustrate the relevance of our results.

2000 Mathematics Subject Classification: 39A10.

Key words and phrases: Oscillation, second-order difference equations.

1. Introduction

In recent years, the oscillation and asymptotic behavior of second order difference equations has been the subject of investigations by many authors. In fact, in the last few years several monographs and hundreds of research papers have been written, see, e.g., the monographs [1–8].

Following this trend in this paper, we consider the nonlinear difference equation

Δ^{2}x_{n} + F (n, x_{n}, Δx_{n}), n ≥ n_{0},  (1.1)

where n_{0} is a fixed nonnegative integer, Δ denotes the forward difference operator Δx_{n} = x_{n+1} − x_{n}.

Throughout, we shall assume that there exists a real sequence {q_{n}} such that

F (n, u, v) sign u ≥ q_{n} |u|^{β} for n ≥ n_{0} and u, v ∈ R,  (1.2)

where q_{n} ≥ 0 and not identically zero for large n, and β > 0 is a positive integer. We say that equation (1.1) is strictly superlinear if β > 1, strictly sublinear if β < 1 and linear if β = 1.

By a solution of (1.1) we mean a nontrivial sequence {x_{n}} satisfying equation (1.1) for n ≥ n_{0}. A solution {x_{n}} of (1.1) is said to be oscillatory if for every n_{1} > n_{0} there exists n ≥ n_{1} such that x_{n}x_{n+1} ≤ 0, otherwise it is nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

Different kinds of the dynamical behavior of solutions of second order difference equations are possible; here we shall only be concerned with the conditions which are sufficient for all solutions of (1.1) to be oscillatory.

Our concern is motivated by several papers, especially by Hooker and Patula [11], Szmanda [13], Wong and Agarwal [14] and Fu and Tsai [10].

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

Our aim in this paper is to establish some new oscillation criteria for equation (1.1) by using the Riccati transformation techniques. Some comparison between our theorems and the previously known results [10, 11, 13] are indicated. Examples are given to illustrate the relevance of our results.

2. Main Results

In what follows we shall assume that equation (1.1) is strictly superlinear, strictly sublinear or linear.

First, we consider the case where (1.1) is strictly superlinear. As a variant of the Riccati transformation techniques, we shall derive new oscillation criteria which can be considered as a discrete analogue of Philos condition for the oscillation of second-order differential equations [12].

Theorem 2.1. Assume that (1.2) holds, and let {ρ_{n}} be a positive sequence. Furthermore, assume that there exists a double sequence {H_{m,n} : m ≥ n ≥ 0}such that

(i) H_{m,m} = 0 for m ≥ 0,

(ii) H_{m,n} > 0 for m > n > 0,

(iii) Δ_{2}H_{m,n} = H_{m,n+1} − H_{m,n}.

If

lim_{m→∞} sup \frac{1}{H_{m,0}} ∑_{n=0}^{m−1} [H_{m,n}ρ_{n}q_{n} − \frac{(ρ_{n+1})^{2}}{4 ρ^{−}_{n}} (h_{m,n} − \frac{Δρ_{n}}{ρ_{n+1}} \sqrt{H_{m,n}})^{2}] = ∞, (2.1)

where

ρ^{−}_{n}= 2^{1−β}M^{β−1}ρ_{n}, h_{m,n} = −\frac{ΔH_{m,n}}{\sqrt{H_{m,n}}} .  (2.2)

for some positive constant M, then every solution of equation (1.1) oscillates.

Proof. Suppose the contrary that {x_{n}} is an eventually positive solution of (1.1), say, x_{n} > 0 for all n ≥ n_{1} ≥ n_{0}. We shall consider only this case, because the proof when x_{n} < 0 is similar. From equations (1.1) and (1.2) we have

Δ^{2}x_{n} ≤ −q_{n}x_{n}^{β} ≤ 0 for n ≥ n_{1}  (2.3)

and so {Δx_{n}} is a nonincreasing sequence. We first have to show that Δx_{n} ≥ 0 for n ≥ n_{1}. Indeed, if there exists an integer n_{2} ≥ n_{1} such that Δx_{n_{2}} = c < 0, then Δx_{n} ≤ c for n ≥ n_{2}, that is

x_{n} ≤ x_{n_{2}} + c(n − n_{0}) → −∞ as n → ∞,  (2.4)

which contradicts the fact that x_{n} > 0 for n ≥ n_{1}. Therefore we have

Δx_{n} ≥ 0 and Δ^{2}x_{n} ≤ 0 for n ≥ n_{1}.  (2.5)

Define the sequence

w_{n} = ρ_{n} \frac{Δx_{n}}{x_{n}^{β}} .  (2.6)

Then w_{n} > 0 and

Δw_{n} = Δx_{n+1}Δ [\frac{ρ_{n}}{x_{n}^{β}} ] + \frac{ρ_{n}Δ^{2}x_{n}}{x_{n}^{β}};

this and (2.4) imply

Δw_{n} ≤ −ρ_{n}q_{n} + \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1} − \frac{ρ_{n}Δx_{n+1}Δ(x_{n}^{β})}{x_{n}^{β}x_{n+1}^{β}} .  (2.7)

But (2.5) implies that x_{n+1} ≥ x_{n}; then from (2.7) we have

Δw_{n} ≤ −ρ_{n}q_{n} + \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1} − \frac{ρ_{n}Δx_{n+1}Δ(x_{n}^{β})}{(x_{n+1}^{β})^{2}} .  (2.8)

Now, by using the inequality

x^{β} − y^{β} ⩾ 2^{1−β}(x − y)^{β} for all x ≥ y > 0 and β ⩾ 1

we find that

Δ(x_{n}^{β}) = x_{n+1}^{β} − x_{n}^{β} ⩾ 2^{1−β}(x_{n+1} − x_{n})^{β} = 2^{1−β} (Δx_{n})^{β} , β ⩾ 1.  (2.9)

Substituting (2.9) in (2.8), we have

Δw_{n} ≤ −ρ_{n}q_{n} + \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1} − ρ_{n} \frac{2^{1−β} (Δx_{n})^{β} Δx_{n+1}}{(x_{n+1}^{β})^{2}} .  (2.10)

From (2.5) and (2.10) we obtain

Δw_{n} ≤ −ρ_{n}q_{n} + \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1} − ρ_{n} \frac{2^{1−β} (Δx_{n+1})^{2β}}{(x_{n+1}^{β})^{2} (Δx_{n+1})^{β−1}} ;

this and (2.6) imply that

Δw_{n} ≤ −ρ_{n}q_{n} + \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1} − \frac{2^{1−β}ρ_{n}}{(ρ_{n+1})^{2}} w_{n+1}^{2} \frac{1}{(Δx_{n+1})^{β−1}} .  (2.11)

Since Δ^{2}x_{n} ≤ 0, from (2.5) it follows that Δx_{n} is a nonincreasing and positive sequence and there exists sufficiently large n_{2} ≥ n_{1} such that Δx_{n} ≤ M for some positive constant M and n ≥ n_{2}, and hence Δx_{n+1} ≤ M so that

\frac{1}{(Δx_{n+1})^{β−1}} ⩾ M^{β−1}.  (2.12)

Now, from (2.11) and (2.12), we have

ρ_{n}q_{n} ≤ −Δw_{n} + \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1} − \begin{matrix}ρ^{−} & n \\ (ρ_{n+1}) & 2 \end{matrix} w_{n+1}^{2}.  (2.13)

346 S. H. SAKER

Therefore

∑_{n=n_{2}}^{m−1} H_{m,n}ρ_{n}q_{n} ≤ − ∑_{n=n_{2}}^{m−1} H_{m,n}Δw_{n} + ∑_{n=n_{2}}^{m−1} H_{m,n} \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1}
− ∑_{n=n_{2}}^{m−1} H_{m,n} \begin{matrix}ρ^{−} & n \\ (ρ_{n+1}) & 2 \end{matrix} w_{n+1}^{2},  (2.14)

which yields after summing by parts

∑_{n=n_{2}}^{m−1} H_{m,n}ρ_{n}q_{n}

≤ H_{m,n_{2}}w_{n_{2}} + ∑_{n=n_{2}}^{m−1} w_{n+1}Δ_{2}H_{m,n} + ∑_{n=n_{2}}^{m−1} H_{m,n} \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1}

− ∑_{n=n_{2}}^{m−1} H_{m,n} \begin{matrix}ρ^{−} & n \\ (ρ_{n+1}) & 2 \end{matrix} w_{n+1}^{2}

= H_{m,n_{2}}w_{n_{2}} − ∑_{n=n_{2}}^{m−1} h_{m,n}\sqrt{H_{m,n}}w_{n+1} + ∑_{n=n_{2}}^{m−1} H_{m,n} \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1}

− ∑_{n=n_{2}}^{m−1} H_{m,n} \begin{matrix}ρ^{−} & n \\ (ρ_{n+1}) & 2 \end{matrix} w_{n+1}^{2}

= H_{m,n_{2}}w_{n_{2}} − ∑_{n=n_{2}}^{m−1} [\frac{\sqrt{H_{m,n} ρ^{−}_{n}}}{ρ_{n}} w_{n+1}
+\frac{ρ_{n+1}}{2\sqrt{H_{m,n} ρ^{−}_{n}}} (h_{m,n}\sqrt{H_{m,n}} − \frac{Δρ_{n}}{ρ_{n+1}} H_{m,n})]^{2}

+\frac{1}{4} ∑_{n=n_{2}}^{m−1} \begin{matrix}(ρ_{n+1}) & 2 \\ ρ^{−} & n \end{matrix} (h_{m,n} − \frac{Δρ_{n}}{ρ_{n+1}} \sqrt{H_{m,n}})^{2}

< H_{m,n_{2}}w_{n_{2}} + \frac{1}{4} ∑_{n=n_{2}}^{m−1} \begin{matrix}(ρ_{n+1}) & 2 \\ ρ^{−} & n \end{matrix} (h_{m,n} − \frac{Δρ_{n}}{ρ_{n+1}} \sqrt{H_{m,n}})^{2} .

Therefore

∑_{n=n_{2}}^{m−1} [H_{m,n}ρ_{n}q_{n} − \frac{(ρ_{n+1})^{2}}{4 ρ^{−}_{n}} (h_{m,n} − \frac{Δρ_{n}}{ρ_{n+1}} \sqrt{H_{m,n}})^{2}] < H_{m,n_{2}}w_{n_{2}} ≤ H_{m,0}w_{n_{2}}

which implies that

∑_{n=0}^{m−1} [H_{m,n}ρ_{n}q_{n} − \frac{(ρ_{n+1})^{2}}{4 ρ^{−}_{n}} (h_{m,n} − \frac{Δρ_{n}}{ρ_{n+1}} \sqrt{H_{m,n}})^{2}] < H_{m,0} (w_{n_{2}} + ∑_{n=0}^{n_{2}−1} ρ_{n}q_{n})

OSCILLATION THEOREMS 347

Hence

lim_{m→∞} sup \frac{1}{H_{m,0}} ∑_{n=0}^{m−1} [H_{m,n}ρ_{n}q_{n} − \frac{(ρ_{n+1})^{2}}{4 ρ^{−}_{n}} (h_{m,n} − \frac{Δρ_{n}}{ρ_{n+1}} \sqrt{H_{m,n}})^{2}]

< (w_{n_{2}} + ∑_{n=0}^{n_{2}−1} ρ_{n}q_{n}) < ∞,

which contradicts (2.1). Therefore every solution of (1.1) oscillates. □

Remark 2.1. From Theorem 2.1 we can obtain different conditions for oscillation of all solutions of equation (1.1) when (1.2) holds by different choices of {ρ_{n}} and H_{m,n}.

Let H_{m,n} = 1. By Theorem 2.1 we have the following result.

Corollary 2.1. Assume that (1.2) holds. Furthermore, assume that there exists a positive sequence {ρ_{n}}such that for some positive constant M

lim_{n→∞} sup ∑_{l=n_{0}}^{n} [ρ_{l}q_{l} − \frac{(Δρ_{l})^{2}}{2^{3−β}M^{β−1}ρ_{l}} ] = ∞.  (2.15)

Then every solution of equation (1.1) oscillates.

Let ρ_{n} = n^{λ}, n ≥ n_{0}, λ ⩾ 1 be a constant and H_{m,n} = 1; then from Theorem 2.1 we have the following result.

Corollary 2.2. Assume that all the assumptions of Theorem 2.1 hold except that condition (2.1) is replaced by

lim_{n→∞} sup ∑_{s=n_{0}}^{n} [s^{λ}q_{s} − \frac{((s + 1)^{λ} − s^{λ})^{2}}{2^{3−β}M^{β−1}s^{λ}} ] = ∞.  (2.16)

Then every solution of equation (1.1) oscillates.

Remark 2.2. Note that when F (n, u, v) = q_{n}u, equation (1.1) reduces to the linear difference equation

Δ^{2}x_{n} + q_{n}x_{n} = 0, n = 0, 1, 2, …,  (2.17)

and condition (2.15) reduces to

lim_{n→∞} sup ∑_{l=n_{0}}^{n} [ρ_{l}q_{l} − \frac{(Δρ_{l})^{2}}{4ρ_{l}} ] = ∞.  (2.18)

Then Theorem 2.1 is an extension of Theorem 4 in [13] and improves Theorem A in [10].

Remark 2.3. If F (n, u, v) = q_{n}u^{β}, then equation (1.1) reduces to the equation Δ^{2}x_{n} + q_{n}x_{n}^{β} = 0, n = 0, 1, 2, …, and condition (2.1) reduces to

lim_{n→∞} sup ∑_{l=n_{0}}^{n} [ρ_{l}q_{l} − \frac{(Δρ_{l})^{2}}{2^{3−β}M^{β−1}ρ_{l}} ] = ∞,

which improves Theorem 4.1 in [11].

The following example is illustrative.

Example 2.1. Consider the discrete Euler equation

Δ^{2}x_{n} + \frac{μ}{n^{2}} x_{n} = 0, n ≥ 1.  (2.19)

Here β = 1,

F (n, x_{n}, Δx_{n}) = \frac{μ}{n^{2}} x_{n},

where μ > \frac{1}{4}. Thus q_{n} = \frac{μ}{n^{2}} . Therefore if ρ_{n} = n, then (2.18) becomes

lim_{n→∞} sup ∑_{l=n_{0}}^{n} [ρ_{l}q_{l} − \frac{(Δρ_{l})^{2}}{4ρ_{l}} ] = lim_{n→∞} sup ∑_{s=n_{0}}^{n} [\frac{μ}{s} − \frac{1}{4s}]

= lim_{n→∞} sup ∑_{s=n_{0}}^{n} \frac{4μ − 1}{s} → ∞.

By Corollary 2.1, every solution of the discrete Euler equation oscillates. It is known [15] that when μ ≤ \frac{1}{4}, the discrete Euler equation has a nonoscillatory solution. Hence Theorem 2.1 and Corollary 2.1 are sharp.

Remark 2.4. We can use a general class of double sequences {H_{m,n}} as the parameter sequences in Theorem 2.1 to obtain different conditions for oscillation of equation (1.1). By choosing specific sequence {H_{m,n}}, we can derive several oscillation criteria for equation (1.1). Let us consider the double sequence {H_{m,n}} defined by

\begin{matrix}H_{m,n} = (m − n)^{λ}, & m ≥ n ≥ 0, λ ≥ 1, \\ H_{m,n} = (log \frac{m+1}{n+1})^{λ} , & m ≥ n ≥ 0, λ ≥ 1. \end{matrix}  (2.20)

Then H_{m,m} = 0 for m ≥ 0, and H_{m,n} > 0 and Δ_{2}H_{m,n} ≤ 0 for m > n ≥ 0. Hence we have the following results.

Corollary 2.3. Assume that all the assumptions of Theorem 2.2 hold except that condition (2.1) is replaced by

lim_{m→∞} sup \frac{1}{m^{λ}} ∑_{n=0}^{m} [(m − n)^{λ}ρ_{n}q_{n}

− \frac{ρ_{n+1}^{2}}{4 ρ^{−}_{n}} (λ(m − n)^{\frac{λ−2}{2}} − \frac{Δρ_{n}}{ρ_{n+1}} (m − n)^{\frac{λ}{2}} )^{2} ]= ∞.  (2.21)

Then every solution of equation (1.1) oscillates.

Corollary 2.4. Assume that all the assumptions of Theorem 2.2 hold except that condition (2.1) is replaced by

lim_{m→∞} sup \frac{1}{(log(m + 1))^{λ}} ∑_{n=0}^{m} [(log \frac{m + 1}{n + 1})^{λ} ρ_{n}q_{n} − \frac{ρ_{n+1}^{2}A_{m,n}}{4 ρ^{−}_{n}} ] = ∞, (2.22)

where

A_{m,n} = (\frac{λ}{n + 1} (log \frac{m + 1}{n + 1})^{\frac{λ−2}{2}} − \frac{Δρ_{n}}{ρ_{n+1}} (log \frac{m + 1}{n + 1})^{\frac{λ}{2}} )^{2} .

Then every solution of equation (1.1) oscillates.

Another H_{m,n} may be chosen as

H_{m,n} = ϕ(m − n), m ≥ n ≥ 0,

or

H_{m,n} = (m − n)^{(λ)}, λ > 2, m ≥ n ≥ 0,

where ϕ : [0, ∞) → [0, ∞) is a continuously differentiable function which satisfies ϕ(0) = 0 and ϕ(u) > 0, ϕ′(u) ≥ 0 for u > 0, and (m − n)^{(λ)} = (m − n)(m − n + 1) ⋯ (m − n + λ − 1) and

Δ_{2}(m − n)^{(λ)} = (m − n − 1)^{(λ)} − (m − n)^{(λ)} = −λ(m − n)^{(λ−1)}.

The corresponding corollaries can also be stated.

Now, we consider the case where (1.1) is strictly sublinear.

Theorem 2.2. Assume that (1.2) holds, and let {ρ_{n}} be a positive sequence. Furthermore, assume that there exists a double sequence {H_{m,n} : m ≥ n ≥ 0} such that

(i) H_{m,m} = 0 for m ≥ 0,

(ii) H_{m,n} > 0 for m > n > 0,

(iii) Δ_{2}H_{m,n} = H_{m,n+1} − H_{m,n}.

If

lim_{m→∞} sup \frac{1}{H_{m,0}} ∑_{n=0}^{m−1} [H_{m,n}ρ_{n}q_{n} − \frac{ρ_{n+1}^{2}}{4P_{n}} (h_{m,n} − \frac{Δρ_{n}}{ρ_{n+1}} \sqrt{H_{m,n}})^{2}] = ∞, (2.23)

where P_{n} = \frac{βρ_{n}}{b^{1−β}(n+1)^{1−β}}, then every solution of equation (1.1) oscillates.

Proof. Proceeding as in the proof of Theorem 2.1, we assume that equation (1.1) has a nonoscillatory solution x_{n} > 0 for all n≥ n_{0}. Defining again w_{n} by (2.6), we obtain (2.8). Now, using the inequality (cf. [9, p. 39]),

x^{β} − y^{β} ⩾ βx^{β−1}(x − y) for all x ≠ y > 0 and 0 < β ≤ 1

we find that

Δ(x_{n}^{β}) = x_{n+1}^{β} − x_{n}^{β} ⩾ β (x_{n+1})^{β−1} (x_{n+1} − x_{n}) = β (x_{n+1})^{β−1} (Δx_{n}) . (2.24)

Substituting (2.24) in (2.8), we have

Δw_{n} ≤ −ρ_{n}q_{n} + \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1} − ρ_{n}a_{n} \frac{β (x_{n+1})^{β−1} (Δx_{n}) (Δx_{n+1})}{(x_{n+1}^{β})^{2}} .

From (2.5) and the last inequality we obtain

Δw_{n} ≤ −ρ_{n}q_{n} + \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1} − \frac{βρ_{n}}{(ρ_{n+1})^{2} (x_{n+1})^{1−β}} \begin{matrix}(ρ_{n+1}) & 2 & (Δx_{n+1}) & 2 \\ (x_{n+1}^{β}) & 2 \end{matrix} .

Hence

Δw_{n} ≤ −ρ_{n}q_{n} + \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1} − \frac{βρ_{n}}{(ρ_{n+1})^{2} (x_{n+1})^{1−β}} w_{n+1}^{2} .  (2.25)

From (2.5) we conclude that

x_{n} ≤ x_{n_{0}} + Δx_{n_{0}}(n − n_{0}), n ≥ n_{0},

and consequently there exists n_{1} ≥ n_{0} and an appropriate constant b ≥ 1 such that

x_{n} ≤ bn for n ≥ n_{1}.

This implies that

x_{n+1} ≤ b(n + 1) for n ≥ n_{2} = n_{1} − 1

and hence

\frac{1}{(x_{n+1})^{1−β}} ≥ \frac{1}{b^{1−β}(n + 1)^{1−β}} .  (2.26)

From (2.25) and (2.26) we obtain

ρ_{n}q_{n} ≤ −Δw_{n} + \frac{Δρ_{n}}{ρ_{n+1}} w_{n+1} − \frac{P_{n}}{(ρ_{n+1})^{2}} w_{n+1}^{2}.  (2.27)

The remainder of the proof is similar to that of the proof of Theorem 2.1 and hence is omitted. □

From Theorem 2.2 we can obtain different conditions for oscillation of all solutions of equation (1.1) when (1.2) holds by different choices of {ρ_{n}} and H_{m,n}. Let H_{m,n} = 1. By Theorem 2.2 we have the following result.

Corollary 2.5. Assume that (1.2) holds. Furthermore, assume that there exists a positive sequence {ρ_{n}} such that for every b ≥ 1

lim_{n→∞} sup ∑_{l=0}^{n} [ρ_{l}q_{l} − \frac{b^{1−β}(l + 1)^{1−β}(Δρ_{n})^{2}}{4βρ_{l}} ] = ∞.  (2.28)

Then every solution of equation (1.1) oscillates.

Remark 2.5. Corollary 2.5 improves Theorem 4.3 in [11].

Let H_{m,n} = 1 and ρ_{n} = n^{λ}, n ≥ n_{0} and λ ⩾ 1 be a constant; then we have the following result.

Corollary 2.6. Assume that all the assumptions of Theorem 2.3 hold except that condition (2.28) is replaced by

lim_{n→∞} sup ∑_{s=n_{0}}^{n} [s^{λ}q_{s} − \frac{b^{1−β}(s + 1)^{1−β}((s + 1)^{λ} − s^{λ})^{2}}{4βs^{λ}} ] = ∞.  (2.29)

Then every solution of equation (1.1) oscillates.

Example 2.2. Consider the difference equation

Δ^{2}x_{n} + \frac{2n + 1}{[n(n + 1)^{2}]^{\frac{1}{3}}} (x_{n})^{\frac{1}{3}} = 0, n ≥ 1.

Since β = \frac{1}{3}, we have

q_{n} = \frac{1 + 2n}{[n(n + 1)^{2}]^{\frac{1}{3}}} .

By choosing ρ_{n} = n + 1 and b = 1, we have

∑_{s=1}^{n} [ρ_{s}q_{s} − \frac{(s + 1)^{1−β}(Δρ_{l})^{2}}{4βρ_{l}} ] = ∑_{s=1}^{n} [(1 + 2s) − \frac{3(s + 1)^{\frac{2}{3}}}{4(s + 1)} ]

≥ ∑_{s=1}^{n} [(1 + 2s) − \frac{3(s + 1)^{2}}{4(s + 1)} ] → ∞

as n → ∞. So according to Corollary 2.6 every solution of this equation oscillates.

The following corollaries follow immediately from Theorem 2.2.

Corollary 2.7. Assume that all the assumptions of Theorem 2.2 hold except that condition (2.23) is replaced by

lim_{m→∞} sup \frac{1}{m^{λ}} ∑_{n=0}^{m} [(m − n)^{λ}ρ_{n}q_{n}− \frac{ρ_{n+1}^{2}}{4P_{n}} (λ(m−n)^{\frac{λ−2}{2}} − \frac{Δρ_{n}}{ρ_{n+1}} (m − n)^{\frac{λ}{2}} )^{2}]=∞.

Then every solution of equation (1.1) oscillates.

Corollary 2.8. Assume that all the assumptions of Theorem 2.2 hold except that condition (2.23) is replaced by

lim_{m→∞} sup \frac{1}{(log(m + 1))^{λ}} ∑_{n=0}^{m} [(log \frac{m + 1}{n + 1})^{λ} ρ_{n}q_{n} − \frac{ρ_{n+1}^{2}A_{m,n}}{4P_{n}} ] = ∞,

where A_{m,n} is as defined in Corollary 2.4. Then, every solution of equation (1.1) oscillates.

Acknowledgement

The results in this paper have finished when the author was in Faculty of Mathematics and Computer Science, Adam Mickiewicz University.

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(Received 7.02.2002; revised 4.11.2002)

Author’s address:

Mathematics Department Faculty of Science Mansoura University Mansoura, 35516 Egypt E-mail: shsaker@mum.mans.eun.eg