Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 67, 1-16; http://www.math.u-szeged.hu/ejqtde/

On the Oscillatory Behavior for a Certain Class of Third Order Nonlinear Delay Difference Equations

S. H. Saker ^{a,b}, J. O. Alzabut ^{c,}^{∗}, A. Mukheimer ^{c}

aDepartment of Mathematics Skills, PYD, King Saud University Riyadh 11451, Saudi Arabia bDepartment of Mathematics, Faculty of Science, Mansoura University Mansoura 35516, Egypt cDepartment of Mathematics and Physical Sciences, Prince Sultan University P. O. Box 66833, Riyadh 11586, Saudi Arabia

Abstract

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria for a certain class of third order nonlinear delay difference equations. Our results extend and improve some previously obtained ones. An example is worked out to demonstrate the validity of the proposed results.

Key Words and Phrases: Oscillation; Generalized Riccati transformation; Third order nonlinear delay difference equation. 2000 AMS Subject Classification: 34K11, 39A10.

1 Introduction

The oscillation theory and asymptotic behavior of difference equations and their applications have been and still are receiving intensive attention over the last two decades. Indeed, the last few years have witnessed the appearance of several monographs and hundreds of research papers, see for example the references [1, 3, 6, 11]. Determination of oscillatory behavior for solutions of second order difference equations has occupied a great part of researchers’ interest. Compared to this, however, the study of third order difference equations has received considerably less attention in the literature even though such equations often arise in the study of economics, mathematical biology and many other areas of mathematics whose discrete models are used, we refer to [2, 4, 5, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 19]. Some of these results will be briefly stated below. Since we are interested in the oscillatory and asymptotic behavior of solutions near infinity, we make a standing hypothesis that the equation under consideration does possess such solutions. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution x_{n} is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory. We say that an equation is oscillatory if it has at least one oscillatory solution.

Here are some background details that may serve the readers and motivate the contents of this paper.

∗Corresponding Author E-mail Address: jalzabut@psu.edu.sa

For oscillation of linear difference equations: In [14], Smith considered the equation of the form

Δ^{3}x_{n} − p_{n}x_{n+2} = 0, n ≥ n_{0}  (1)

and studied the asymptotic and oscillatory behavior of the solutions subject to the condition p_{n} > 0 for n ≥ n_{0}. Indeed, he proved that if

∑_{n=n_{0}}^{∞} p_{n} = ∞  (2)

then (1) is oscillatory. Further, the author considered the quasi-adjoint difference equation

Δ^{3}x_{n} + p_{n}x_{n+1} = 0, n ≥ n_{0}  (3)

and proved that (1) is oscillatory if and only if (3) is oscillatory. However, one can easily see that the results cannot be applied if p_{n} = n^{−α} for α > 1.

In [12], the authors studied the difference equation of the form

Δ^{3}x_{n} + q_{n}x_{n} = 0, n ≥ n_{0}  (4)

and established some sufficient conditions for (4) to have monotonic and nonoscillatory solutions. They proved that if q_{n} > 1 for n ≥ n_{0} is a positive sequence then (4) is oscillatory.

In [13], it was proved that if

∑_{l=n_{0}}^{∞} [ ∑_{t=n_{0}}^{l−1} ∑_{s=n_{0}}^{t−1} p_{s}] = ∞  (5)

and there exists a positive sequence ρ_{n} such that

lim_{n→∞} sup ∑_{s=n_{0}}^{n} [ρ_{s}p_{s} − \frac{(Δρ_{s})^{2}}{4ρ_{s}(s − n_{0})}] = ∞  (6)

then the solution x_{n} of (3) either oscillates or satisfies lim_{n→∞} x_{n} = 0. Results established in [13] provided substantial improvements for those obtained in [12] and [14].

In [16], the author considered the linear difference equation

Δ^{3}x_{n} − p_{n+1}Δx_{n+1} + q_{n+1}x_{n+1} = 0, n ≥ n_{0},  (7)

where p_{n} and q_{n} are nonnegative real sequences satisfying

Δp_{n} + q_{n+1} > 0  (8)

and proved that if x_{n} is a nonoscillatory solution of (7) then there exists an integer N for which either x_{n}Δx_{n} > 0 or x_{n}Δx_{n} < 0 for all n > N.

In [15], the author investigated the linear difference equation

Δ^{3}x_{n} + p_{n+1}Δx_{n+2} + q_{n}x_{n+2} = 0, n ≥ n_{0},  (9)

where p_{n} and q_{n} are real sequences satisfying

p_{n} ≥ 0, q_{n} < 0 and ∑_{n=n_{0}}^{∞}(Δp_{n} − 2q_{n}) = ∞.  (10)

It was shown that if p_{n+1} + q_{n} ≤ 0 for n ≥ n_{0} then signx_{n} = signΔx_{n} = signΔ^{2}x_{n} and (9) has both oscillatory and nonoscillatory solutions. Further, the author established a sufficient condition for the existence of oscillatory solutions. The main investigation is based on the value of the functional F_{1}(x_{n}) = (Δx_{n})^{2} −2x_{n+1}Δ^{2}x_{n} −p_{n}x_{n+2}^{2}. In particular, it was proved that if there is a solution x_{n} of (9) such that F (x_{n}) > 0 then x_{n} is oscillatory. However, one can easily see that the condition depends on the solution itself whose determination might not be possible.

For oscillation of nonlinear difference equations: The authors in [18] considered the equation

Δ(Δ^{2}x_{n} + p_{n}x_{n+1}) + p_{n}Δx_{n} + f(x_{n+1}) = 0, n ≥ n_{0},  (11)

where f(x)/x ≥ k > 0 and p_{n} is a bounded real sequence such that

∑_{s=n_{0}}^{∞} p_{n} = ∞.  (12)

The authors studied the asymptotic behavior of the solutions and proved that if there exists a solution x_{n} of (11) satisfying F_{2}(x_{n}) < 0, where F_{2}(x_{n}) = 2x_{n}(Δ^{2}x_{n} + p_{n}x_{n+1}) − (Δx_{n})^{2}, then x_{n} is oscillatory. On the other hand, the authors proved that if there exists a solution x_{n} of (11) satisfying F_{2}(x_{n}) > 0 then lim_{n→∞} x_{n} = lim_{n→∞} Δx_{n} = lim_{n→∞} Δ^{2}x_{n} = 0. Nevertheless, due to condition (12) the results are no longer valid if p_{n} = n^{−α} for α > 1.

In [16], the author investigated equation of form

Δ(Δ^{2}x_{n} − p_{n+1}x_{n+1}) − q_{n+2}x_{n+2} = 0, n ≥ n_{0},  (13)

where p_{n} and q_{n} are nonnegative real sequences and satisfying (8). It was shown that there exists a solution x_{n} of (13) such that x_{n}Δx_{n}Δ^{2}x_{n} ≠ 0, x_{n} > 0, Δx_{n} > 0 and Δ^{2}x_{n} > 0 for n ≥ n_{0} and if x_{n} is a nonoscillatory solution then there exists an integer N for which either x_{n}Δx_{n} > 0 or x_{n}Δx_{n} < 0 for all n > N. Furthermore, the author investigated the same result for equation (7) and proved that if v_{n} is a nonoscillatory solution of (13) then the two independent solutions of (7) satisfy the self-adjoint second order equation

Δ (\frac{Δx_{n}}{v_{n}} ) + (\frac{Δ^{2}v_{n−1} − p_{n}v_{n}}{v_{n}v_{n+1}} ) x_{n+1} = 0.  (14)

In [8], the authors studied the oscillatory behavior of

Δ(c_{n}Δ(d_{n}Δ(x_{n}))) + q_{n}f(x_{n−σ+1}) = 0, n ≥ n_{0},  (15)

where σ is a nonnegative integer and f ∈ C(ℝ, ℝ) such that uf(u) > 0 for u ≠ 0 and satisfies

f(u) − f(v) = g(u, v)(u − v), for u, v ≠ 0 and g(u, v) ⩾ μ > 0  (16)

and q_{n}, c_{n}, d_{n} are positive sequences of real numbers such that

∑_{n=n_{0}}^{∞} (\frac{1}{c_{n}} ) = ∑_{n=n_{0}}^{∞} (\frac{1}{d_{n}} ) = ∞ and Δc_{n} ⩾ 0.  (17)

For the linear case, they used the Riccati transformation technique and established a sufficient condition for oscillation of equation (15). For the nonlinear case, however, some oscillation criteria were provided by reducing the oscillation of the equation to the existence of positive solution of a Riccati difference inequality. Nevertheless, one can easily see that condition (16) might not be satisfied when f(u) = u^{γ} for γ > 0 and the results are valid only when Δc_{n} ⩾ 0. Therefore, one of our aims in this paper is to establish some sufficient conditions for oscillation bypassing condition (16) and removing the restriction in (17).

In [2], the authors considered the nonlinear delay difference equation

Δ^{3}x_{n} = p_{n}Δ^{2}x_{n+m} + q_{n}F (x_{n−g}, x_{n−h}) = 0, n ≥ n_{0},  (18)

where p_{n} and q_{n} are positive real sequences, p_{n} is nonincreasing, m, g, h are nonnegative integers and F (x, y) = signx ≥ |x|^{c_{1}} |y|^{c_{2}} where c_{1} and c_{2} are nonnegative constants such that c_{1} + c_{2} > 0. They established some sufficient conditions for the existence of oscillatory solutions. The main results are proved by reducing the order of the equation under consideration. Indeed, the oscillation of equation (18) reduces to the oscillation of a first order delay or advanced difference equations.

In [5], the authors considered the nonlinear difference equation

Δ(c_{n}Δ(d_{n}Δ(x_{n}))) + q_{n}f(x_{n+σ}) = 0, n ≥ n_{0},  (19)

where c_{n}, d_{n}, q_{n} are sequences of nonneagtive real numbers and the function f ∈ C(ℝ, ℝ) such that uf(u) > 0 for u ≠ 0. The main result in [5] was the classification of the nonoscillatory solutions with respect to the sign of their quasi differences.

In [7], the authors considered the nonlinear delay difference equation

Δ(c_{n} (Δ^{2}x_{n})^{γ}) + q_{n}f(x(σ_{n})) = 0, n ≥ n_{0},  (20)

where c_{n}, σ_{n}, q_{n} are sequences of nonneagtive real numbers, σ_{n} < n, γ is quotient of odd positive integers, f ∈ C(ℝ, ℝ) such that uf(u) > 0 for u ≠ 0, f′(x) > 0, −f(−xy) ≥ f(xy) ≥ f(x)f(y) for xy > 0 and

∑_{n=n_{0}}^{∞} (\frac{1}{c_{n}} )^{γ} < ∞.

The main approach of proving the results in [7] was also based on the reduction of the oscillation of (20) to the oscillation of first order delay difference equation. However, the results can only be applied in the case when σ_{n} < n. Further, the restriction f′(x) > 0 might not be satisfied. Indeed, if f(x) = x (\frac{1}{9} + \frac{1}{1+x^{2}} ) then f′(x) = \frac{(x^{2}−2)(x^{2}−5)}{9(1+x^{2})^{2}} changes sign four times.

Following this trend, we are concerned with the oscillation and the asymptotic behavior of solutions of the nonlinear delay difference equation of form

Δ(c_{n}Δ(d_{n}Δx_{n})^{γ}) + q_{n}f(x_{n−σ}) = 0, n ≥ n_{0},  (21)

where γ > 0 is quotient of odd positive integers, σ ∈ ℕ and

(h_{1}) c_{n}, d_{n}, q_{n} are positive sequences of real numbers;
(h_{2}) f ∈ C(ℝ, ℝ) such that uf(u) > 0 for u ≠ 0 and f(u)/u^{γ} ⩾ K > 0.

Our attention is restricted to those solutions of (21) which exist on [n_{x}, ∞) and satisfy sup{|x(n)| : n > n_{1}} > 0 for any n_{1} ≥ n_{x}. It is to be noted that the results of the above mentioned papers provided several oscillation criteria under the conditions

∑_{n=n_{0}}^{∞} (\frac{1}{c_{n}} )^{γ} = ∞ and ∑_{n=n_{0}}^{∞} (\frac{1}{d_{n}} ) = ∞.  (22)

Therefore, it will be of great interest to establish oscillation criteria when

∑_{n=n_{0}}^{∞} (\frac{1}{c_{n}} )^{γ} < ∞ and ∑_{n=n_{0}}^{∞} (\frac{1}{d_{n}} ) < ∞.  (23)

The aim of the paper is to employ Riccati transformation technique to establish some new oscillation criteria for equation (21) under assumptions (22). We will prove our results bypassing condition (16) and removing the restriction Δc_{n} ⩾ 0. Unlike previously obtained results, new oscillation criteria are also obtained under assumptions (23). We will complement and improve the results in [8] and extend those in [13]. Some comparison between our theorems and those previously known ones are indicated throughout the paper.

The paper is organized as follows: In Section 2, we present some fundamental lemmas that will be useful in proving our main results. In Section 3, we will state and prove the main oscillation theorems. An example is given to demonstrate the validity of the results.

2 Some Fundamental Lemmas

In this section, we present some fundamental lemmas that will be used in the proofs of the main results. For equation (21), we define the quasi differences by

x_{n}^{[0]} = x_{n}, x_{n}^{[1]} = d_{n}Δx_{n}, x_{n}^{[2]} = c_{n}Δ (x_{n}^{[1]})^{γ} and x_{n}^{[3]} = Δx_{n}^{[2]}.  (24)

It is to be noted that if x_{n} is a solution of (21) then z = −x is also a solution of (21) since uf(u) > 0 for u ≠ 0. Thus, concerning nonoscillatory solutions of (21), we will only restrict our attention to the positive ones.

We start with the following lemma which provides the signs of the quasi differences of the solution x_{n} of (21).

Lemma 1. Let x_{n} be a nonoscillatory solution of (21). Assume that (h_{1}) − (h_{2}) hold. Then there exists N > n_{0} such that x_{n}^{[i]} ≠ 0 for i = 0, 1, 2 and n ≥ N.

Proof. Without loss of generality, we assume that x_{n} is an eventually positive solution of (21) and there exists n_{1} ≥ n_{0} such that x_{n} and x_{n−σ} > 0 for n ≥ n_{1}. Since q_{n} > 0, then x_{n}^{[3]} < 0. Thus, there exists n_{2} ≥ n_{1} such that x_{n}^{[2]} is either positive or negative for n ≥ n_{2}. It follows that x_{n}^{[1]} is either increasing or decreasing for n ≥ n_{2} and so there exists N ≥ n_{2} such that x_{n}^{[0]} is either positive or negative for n ≥ N.

In view of Lemma 1, we deduce that all nonoscillatory solutions of (21) belong to the following classes:

C_{0} = {x_{n} : ∃ N such that x_{n}x_{n}^{[1]} < 0, x_{n}x_{n}^{[2]} > 0 for n ≥ N},
C_{1} = {x_{n} : ∃ N such that x_{n}x_{n}^{[1]} > 0, x_{n}x_{n}^{[2]} < 0 for n ≥ N},
C_{2} = {x_{n} : ∃ N such that x_{n}x_{n}^{[1]} > 0, x_{n}x_{n}^{[2]} > 0 for n ≥ N},
C_{3} = {x_{n} : ∃ N such that x_{n}x_{n}^{[1]} < 0, x_{n}x_{n}^{[2]} < 0 for n ≥ N}.

Lemma 2. Let x_{n} be a nonoscillatory solution of (21). Assume that (h_{1}) − (h_{2}) hold. If

∑_{n=n_{0}}^{∞} \frac{1}{d_{n}} ∑_{s=n_{0}}^{n−1} \frac{1}{(c_{s})^{\frac{1}{γ}}} = ∞.  (25)

Then C_{3} is empty.

Proof. To prove that C_{3} is empty, it is sufficient to show that if there is a positive solution x_{n} of (21), then the case x_{n}x_{n}^{[1]} < 0 and x_{n}x_{n}^{[2]} < 0 for n ≥ N is impossible. For the sake of contradiction, assume that there exists n_{1} > n_{0} such that x_{n}^{[1]} < 0 and x_{n}^{[2]} < 0 for n ≥ n_{1}. Denote a_{0} = x_{n_{1}}^{[2]} < 0. Then, since x_{n}^{[2]} is decreasing we have c_{n} (Δx_{n}^{[1]})^{γ} < a_{0} for n ≥ n_{1}. Thus by summing from n_{1} to n − 1, we have

x_{n}^{[1]} < x_{n_{1}}^{[1]} + a0^{\frac{1}{γ}} ∑_{s=n_{1}}^{n−1} \frac{1}{(c_{s})^{\frac{1}{γ}}} .

Using that x_{n_{1}}^{[1]} < 0, we get

x_{n}^{[1]} < a0^{\frac{1}{γ}} ∑_{s=n_{1}}^{n−1} \frac{1}{(c_{s})^{\frac{1}{γ}}} .

Summing up from n_{1} to n − 1, we obtain

x_{n} < x_{n_{1}} + a0^{\frac{1}{γ}} ∑_{s=n_{1}}^{n−1} \frac{1}{d_{s}} ∑_{u=n_{1}}^{s−1} \frac{1}{(c_{u})^{\frac{1}{γ}}} .

Letting n → ∞, then by (25) we deduce that lim_{n→∞} x_{n} = −∞ which contradicts that x_{n} > 0. The proof is complete.

Lemma 3. Let x_{n} be a nonoscillatory solution of (21). Assume that (h_{1}) − (h_{2}) hold. If (22) holds. Then x_{n} ∈ C_{0} ∪ C_{2}.

Proof. Without loss of generality, we assume that x_{n} is an eventually positive solution of (21) and there exists n_{1} ⩾ n_{0} such that x_{n} and x_{n−σ} > 0 for n ⩾ n_{1}. In virtue of Lemma 1, we deduce that x_{n}^{[0]}, x_{n}^{[1]} and x_{n}^{[2]} are monotone and eventually of one sign. Therefore to complete the proof, we show that there are only the following two cases for n ⩾ n_{0} sufficiently large:

(I) x_{n}^{[0]} > 0, x_{n}^{[1]} > 0 and x_{n}^{[2]} > 0;
(II) x_{n}^{[0]} > 0, x_{n}^{[1]} < 0 and x_{n}^{[2]} > 0.

In view of (h_{2}) and (21), we see that x_{n}^{[3]} < 0 for n ⩾ n_{1}. We claim that there is n_{2} ⩾ n_{1} such that for n ⩾ n_{2}, x_{n}^{[2]} > 0. Suppose to the contrary that x_{n}^{[2]} ≤ 0 for n ⩾ n_{2}. Since x_{n}^{[2]} is nonincreasing, there exists a negative constant L and n_{3} ⩾ n_{2} such that x_{n}^{[2]} ≤ L for n ⩾ n_{3}. Dividing by c_{n} and summing from n_{3} to n − 1, we obtain

x_{n}^{[1]} ≤ x_{n_{3}}^{[1]} + L^{\frac{1}{γ}} ∑_{s=n_{3}}^{n−1} \frac{1}{(c_{s})^{\frac{1}{γ}}} .

Letting n → ∞, then by ( 22) we deduce that x_{n}^{[1]} → −∞. Thus, there is an integer n_{4} ⩾ n_{3} such that for n ⩾ n_{4}, x_{n}^{[1]} ≤ x_{n_{4}}^{[1]} < 0. Dividing by d_{n} and summing from t_{4} to t, we have

x_{n} − x_{n_{4}} ≤ x_{n_{4}}^{[1]} ∑_{s=n_{4}}^{n−1} \frac{1}{d_{s}} ,

which implies that x_{n} → −∞ as n → ∞. This contradicts the fact that x_{n} > 0. Then x_{n}^{[2]} > 0.

Lemma 4. Let x_{n} be a nonsocillatory solution of (21) that belongs to C_{0}. Assume that (h_{1}) − (h_{2}) and n − σ ≤ n hold. If

∑_{n=n_{0}}^{∞} \frac{1}{d_{n}} [ ∑_{t=n_{0}}^{n−1} \frac{1}{c_{t}} ∑_{s=n_{0}}^{t−1} q_{s}]^{\frac{1}{γ}} = ∞.  (26)

Then lim_{n→∞} x_{n} = 0.

Proof. Without loss of generality, we assume that x_{n−σ} > 0 for n ≥ n_{1} where n_{1} is chosen sufficiently large. In view of (h_{2}) and (21), we obtain

x_{n}^{[3]} + Kq_{n}x_{n−σ}^{γ} ≤ 0, n ≥ n_{1}.  (27)

Since x_{n} is positive and decreasing it follows that lim_{n→∞} x_{n} = b ≥ 0. Now we claim that b = 0. If b ≠ 0 then x_{n−σ}^{γ} → b^{γ} > 0 as n → ∞. Hence there exists n_{2} ≥ n_{1} such that x_{n−σ}^{γ} ≥ b^{γ}. Therefore from (27), we have

x_{n}^{[3]} + Kq_{n}b^{γ} ≤ 0, n ≥ n_{2}.

Define the sequence u_{n} = x_{n}^{[2]} for n ≥ n_{2}. Then Δx_{n}^{[2]} ≤ −Aq_{n} where A = Kb^{γ} > 0.

Summing the last inequality from n_{2} to n − 1, we get x_{n}^{[2]} ≤ x_{n_{2}}^{[2]} − A ∑_{s=n_{2}}^{n−1} q_{s}. In view of (26),

it is possible to choose an integer n_{3} sufficiently large such that x_{n}^{[2]} ≤ −\frac{A}{2} ∑_{s=n_{2}}^{n−1} q_{s} for all

n ≥ n_{3}. Hence Δ (x_{n}^{[1]})^{γ} ≤ −\frac{A}{2} \frac{1}{c_{n}} ∑_{s=n_{2}}^{n−1} q_{s}. Summing the last inequality from n_{3} to n − 1, we

obtain

(x_{n}^{[1]})^{γ} ≤ (x_{n_{3}}^{[1]})^{γ} − \frac{A}{2} ∑_{t=n_{3}}^{n−1} (\frac{1}{c_{t}} ∑_{s=n_{2}}^{t−1} q_{s}) .

Since Δx_{n} < 0 for n ≥ n_{0}, the last inequality implies that

Δx_{n} ≤ − (\frac{A}{2})^{\frac{1}{γ}} \frac{1}{d_{n}} [ ∑_{t=n_{3}}^{n−1} \frac{1}{c_{t}} ∑_{s=n_{2}}^{t−1} q_{s}]^{\frac{1}{γ}} .

Summing from n_{4} to n − 1, we have

x_{n} ≤ x_{n_{4}} − (\frac{A}{2})^{\frac{1}{γ}} ∑_{l=n_{4}}^{n−1} \frac{1}{d_{l}} [ ∑_{t=n_{3}}^{l−1} \frac{1}{c_{t}} ∑_{s=n_{2}}^{t−1} q_{s}]^{\frac{1}{γ}} .

Condition (26) implies that x_{n} → −∞ as n → ∞ which is a contradiction with the fact that x_{n} > 0. Then b = 0 and this completes the proof.

Lemma 5. Let x_{n} be a nonoscillatory solution of (21) that belongs to C_{2}. Then there exists n_{1} ≥ n_{0} such that

(x_{n−σ}^{[1]})^{γ} ≥ δ_{n−σ}x_{n}^{[2]}, for n ≥ n_{1},

where δ_{n} := ∑_{s=n_{0}}^{n−1} \frac{1}{c_{s}} .

Proof. Since x_{n} ∈ C_{2}, then without loss of generality we can assume that there exists N > n_{0} such that

x_{n} > 0, x_{n}^{[1]} > 0, x_{n}^{[2]} > 0 and x_{n}^{[3]} ≤ 0 for n ≥ N.

Hence

(x_{n}^{[1]})^{γ} = (x_{n_{1}}^{[1]})^{γ} + ∑_{s=n_{1}}^{n−1} \frac{c_{s}Δ(x_{s}^{[1]})^{γ}}{c_{s}} ≥ δ_{n}x_{n}^{[2]}, n ≥ n_{1}.  (28)

Since x_{n}^{[3]} ≤ 0, we have x_{n−σ}^{[2]} ≥ x_{n}^{[2]}. This and (28) imply that

(x_{n−σ}^{[1]})^{γ} ≥ δ_{n−σ}x_{n−σ}^{[2]} ≥ δ_{n−σ}x_{n}^{[2]}, n ≥ N_{1} = N + σ.

Thus

(x_{n−σ}^{[1]})^{γ} ≥ δ_{n−σ}x_{n}^{[2]}, for n ≥ N_{1}.

3 Oscillation Criteria

In this section, we will establish some new sufficient conditions which guarantee that every solution x_{n} of (21) either oscillates or satisfies lim_{n→∞} x_{n} = 0. In our analysis, we will present the proofs of our results under conditions (22) and (23) in two separate investigations.

3.1 Oscillation under condition (22)

Throughout this subsection we assume that there exists a double sequences {H_{m,n} : m ≥ n ≥ 0} and h_{m,n} 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} ≤ 0 for m ≥ n ≥ 0;
(iv) h_{m,n} = −Δ_{2}H_{m,n}(H_{m,n})^{− \frac{1}{γ+1}} , m > n ≥ 0.

For a given sequence ρ_{n}, we define

ψ_{n} : = Kρ_{n}q_{n} − ρ_{n}Δ(c_{n}α_{n}) + \frac{ρ_{n}δn−σ^{\frac{1}{γ}}(c_{n+1})^{1+ \frac{1}{γ}} (α_{n+1})^{1+ \frac{1}{γ}}}{d_{n−σ}}
ξ_{n} : = Δρ_{n} + γρ_{n}(1 + \frac{1}{γ})(c_{n+1}α_{n+1}δ_{n−σ})^{\frac{1}{γ}} d_{n−σ}^{−1}

and

ϕ_{m,n} : = \frac{ρ_{n+1}^{1+γ}}{(1 + γ)^{1+γ}ρ_{n}^{γ}δ_{n−σ}d_{n−σ}^{−γ}H_{m,n}^{γ}} (\frac{ξ_{n}H_{m,n}}{ρ_{n+1}} − h_{m,n}Hm,n^{\frac{1}{γ+1}} )^{1+γ} .

Theorem 6. Let x_{n} be a solution of (21) and ρ_{n} be a given positive sequence. Assume that (h_{1}) − (h_{2}), (22) and (26) hold. If

lim_{m→∞} sup \frac{1}{H_{m,n_{0}}} ∑_{n=n_{0}}^{m−1} [H_{m,n}ψ_{n} − ϕ_{m,n}] = ∞.  (29)

Then x_{n} either oscillates or satisfies lim_{n→∞} x_{n} = 0.

Proof. Suppose to the contrary that x_{n} is a nonoscillatory solution. Without loss of generality, we assume that x_{n} > 0 and x_{n−σ} > 0 for n ≥ n_{1} where n_{1} is chosen so large. In view of Lemma 3, we deduce that condition (22) implies that x_{n} ∈ C_{0} ∪ C_{2}. If x_{n} ∈ C_{0}, then we are back to the proof of Lemma 4 to show that lim_{n→∞} x_{n} = 0. We assume that the solution x_{n} ∈ C_{2} and define the sequence ω_{n} by the generalized Riccati substitution

ω_{n} := ρ_{n} [\frac{x_{n}^{[2]}}{x_{n−σ}^{γ}} + c_{n}α_{n}] , n ≥ n_{1}.  (30)

It follows that

Δω_{n} = Δ(ρ_{n}c_{n}α_{n}) + x_{n+1}^{[2]}Δ [\frac{ρ_{n}}{x_{n−σ}^{γ}} ] + \frac{ρ_{n}x_{n}^{[3]}}{x_{n−σ}^{γ}} .

In view of (27) and (30), the above equation can be written in the form

Δω_{n} ≤ −Kρ_{n}q_{n} + ρ_{n}Δ(c_{n}α_{n}) + \frac{Δρ_{n}}{ρ_{n+1}} ω_{n+1} − \frac{ρ_{n}x_{n+1}^{[2]}}{x_{n−σ}^{γ}x_{n−σ+1}^{γ}} Δ(x_{n−σ}^{γ}).  (31)

First: we consider the case when γ ≥ 1. By using the inequality ([9, see p. 39])

x^{γ} − y^{γ} ≥ γy^{γ−1}(x − y) for all x ≠ y > 0 and γ ≥ 1,

we may write

Δ(x_{n−σ}^{γ}) = x_{n−σ+1}^{γ} − x_{n−σ}^{γ} ≥ γx_{n−σ}^{γ−1}Δx_{n−σ}, γ ≥ 1.

Substituting in (31), we find out

Δω_{n} ≤ −Kρ_{n}q_{n} + ρ_{n}Δ(c_{n}α_{n}) + \frac{Δρ_{n}}{ρ_{n+1}} ω_{n+1} − \frac{γρ_{n}x_{n+1}^{[2]}Δx_{n−σ}}{x_{n−σ}x_{n−σ+1}^{γ}} .

Since x_{n} ∈ C_{2}, it follows from Lemma 5 that there exists n_{2} ≥ n_{1} such that

(Δx_{n−σ})^{γ} ≥ \frac{δ_{n−σ}}{d_{n−σ}^{γ}} x_{n}^{[2]} for n ≥ n_{2}.  (32)

Using the fact that x_{n−σ+1} ≥ x_{n−σ}, we obtain

Δω_{n} ≤ −Kρ_{n}q_{n} + ρ_{n}Δ(c_{n}α_{n}) + \frac{Δρ_{n}}{ρ_{n+1}} ω_{n+1} − \frac{γρ_{n}δn−σ^{\frac{1}{γ}}x_{n+1}^{[2]}[x_{n}^{[2]}]^{\frac{1}{γ}}}{d_{n−σ}(x_{n−σ+1})^{γ+1}} .  (33)

Since x_{n}^{[3]} < 0, it follows that x_{n+1}^{[2]} ≤ x_{n}^{[2]} and thus [x_{n+1}^{[2]}]^{\frac{1}{γ}} ≤ [x_{n}^{[2]}]^{\frac{1}{γ}} . This yields that

Δω_{n} ≤ −Kρ_{n}q_{n} + ρ_{n}Δ(c_{n}α_{n}) + \frac{Δρ_{n}}{ρ_{n+1}} ω_{n+1} − \frac{γρ_{n}δn−σ^{\frac{1}{γ}}}{d_{n−σ}} (\frac{x_{n+1}^{[2]}}{x_{n−σ+1}^{γ}} )^{\frac{1+γ}{γ}} .  (34)

Second: we consider the case when 0 < γ < 1. By using the inequality

x^{γ} − y^{γ} ≥ γx^{γ−1}(x − y) for all x ≠ y > 0,

we may write

Δ(x_{n−σ}^{γ}) ≥ γx_{n−σ+1}^{γ−1}Δx_{n−σ}.

Substituting in (31), we have

Δω_{n} ≤ −Kρ_{n}q_{n} + ρ_{n}Δ(c_{n}α_{n}) + \frac{Δρ_{n}}{ρ_{n+1}} ω_{n+1} − \frac{γρ_{n}x_{n+1}^{[2]}Δx_{n−σ}}{x_{n−σ}^{γ}x_{n−σ+1}} .

By using the fact that x_{n} is increasing, we have

−\frac{γρ_{n}x_{n+1}^{[2]}Δx_{n−σ}}{x_{n−σ}^{γ}x_{n−σ+1}} ≤ −\frac{γρ_{n}δn−σ^{\frac{1}{γ}}}{d_{n−σ}} (\frac{x_{n+1}^{[2]}}{x_{n−σ+1}^{γ}} )^{\frac{1+γ}{γ}} .  (35)

Thus, we again obtain (34). However, from (30) we see that

(\frac{x_{n+1}^{[2]}}{x_{n−σ+1}^{γ}} )^{1+ \frac{1}{γ}} = (\frac{ω_{n+1}}{ρ_{n+1}} − c_{n+1}α_{n+1})^{1+ \frac{1}{γ}} .  (36)

Then, by using the inequality [19, see p. 534]

(v − u)^{1+ \frac{1}{γ}} ≥ v^{1+ \frac{1}{γ}} + \frac{1}{γ} u^{1+ \frac{1}{γ}} − (1 + \frac{1}{γ})u^{\frac{1}{γ}} v, γ = \frac{odd}{odd} ≥ 1,

we may write equation (36) as follows

(\frac{ω_{n+1}}{ρ_{n+1}} − c_{n+1}α_{n+1})^{1+ \frac{1}{γ}} ≥ (\frac{ω_{n+1}}{ρ_{n+1}} )^{1+ \frac{1}{γ}} + \frac{(c_{n+1}α_{n+1})^{1+ \frac{1}{γ}}}{γ} − \frac{(1 + \frac{1}{γ})(c_{n+1}α_{n+1})^{\frac{1}{γ}}}{ρ_{n+1}} ω_{n+1}.

Substituting back in (34), we have

Δω_{n} ≤ −Kρ_{n}q_{n} + ρ_{n}Δ(c_{n}α_{n}) − \frac{ρ_{n}δn−σ^{\frac{1}{γ}}(c_{n+1})^{1+ \frac{1}{γ}} (α_{n+1})^{1+ \frac{1}{γ}}}{d_{n−σ}}

+ (\frac{Δρ_{n}}{ρ_{n+1}} + \frac{γρ_{n}(1 + \frac{1}{γ})(c_{n+1}δ_{n−σ}α_{n+1})^{\frac{1}{γ}}}{d_{n−σ}ρ_{n+1}} ) ω_{n+1}

− (\frac{γρ_{n}δn−σ^{\frac{1}{γ}}}{d_{n−σ}(ρ_{n+1})^{1+ \frac{1}{γ}}} ) (ω_{n+1})^{1+ \frac{1}{γ}} .  (37)

Thus,

ψ_{n} ≤ −Δω_{n} + \frac{ξ_{n}}{ρ_{n+1}} ω_{n+1} − \frac{ρ̄_{n}}{(ρ_{n+1})^{1+ \frac{1}{γ}}} (ω_{n+1})^{1+ \frac{1}{γ}} , n ≥ n_{3},

where ρ̄_{n} = γρ_{n}δn−σ^{\frac{1}{γ}}d_{n−σ}^{−1}. Therefore, we have

∑_{n=n_{3}}^{m−1} H_{m,n}ψ_{n} ≤ − ∑_{n=n_{3}}^{m−1} H_{m,n}Δω_{n} + ∑_{n=n_{3}}^{m−1} \frac{ξ_{n}H_{m,n}}{ρ_{n+1}} ω_{n+1} − ∑_{n=n_{3}}^{m−1} \frac{ρ̄_{n}H_{m,n}}{(ρ_{n+1})^{1+ \frac{1}{γ}}} (ω_{n+1})^{1+ \frac{1}{γ}} ,

which yields after summing by parts

∑_{n=n_{3}}^{m−1} H_{m,n}ψ_{n} ≤ H_{m,n_{3}}ω_{n_{3}} + ∑_{n=n_{3}}^{m−1} ω_{n+1}Δ_{2}H_{m,n} + ∑_{n=n_{3}}^{m−1} \frac{ξ_{n}H_{m,n}}{ρ_{n+1}} ω_{n+1}
− ∑_{n=n_{3}}^{m−1} \frac{ρ̄_{n}H_{m,n}}{(ρ_{n+1})^{1+ \frac{1}{γ}}} (ω_{n+1})^{1+ \frac{1}{γ}} .

Hence

∑_{n=n_{3}}^{m−1} H_{m,n}ψ_{n} ≤ H_{m,n_{3}}ω_{n_{3}}+ ∑_{n=n_{3}}^{m−1} (\frac{ξ_{n}H_{m,n}}{ρ_{n+1}} − h_{m,n}Hm,n^{\frac{1}{γ+1}} ) ω_{n+1}− ∑_{n=n_{3}}^{m−1} \frac{ρ̄_{n}H_{m,n}}{(ρ_{n+1})^{1+ \frac{1}{γ}}} (ω_{n+1})^{1+ \frac{1}{γ}} .

Using the fact that

Bu − Au^{1+ \frac{1}{β}} ≤ \frac{β^{β}}{(1 + β)^{1+β}} \frac{B^{1+β}}{A^{β}}

EJQTDE, 2010 No. 67, p. 10

for A = _{\frac{ρ̄_{n}H_{m,n}}{(ρ_{n+1})^{1+ \frac{1}{γ}}}} and B = (\frac{ξ_{n}H_{m,n}}{ρ_{n+1}} − h_{m,n}Hm,n^{\frac{1}{γ+1}} ) , we obtain

∑_{n=n_{3}}^{m−1} [H_{m,n}ψ_{n} − ϕ_{m,n}] < H_{m,n_{3}}ω_{n_{3}} ≤ H_{m,n_{0}}ω_{n_{3}},

which implies that

∑_{n=n_{0}}^{m−1} [H_{m,n}ψ_{n} − ϕ_{m,n}] < H_{m,n_{0}} (ω_{n_{3}} + ∑_{n=n_{0}}^{n_{3}−1} ψ_{n}) .

Hence

lim_{m→∞} sup \frac{1}{H_{m,n_{0}}} ∑_{n=n_{0}}^{m−1} [H_{m,n}ψ_{n} − ϕ_{m,n}] < ∞,

which contradicts (29). The proof is complete.

The following result is an immediate consequence of Theorem 6.

Corollary 7. Let x_{n} be a solution of (21) and assume that all the assumptions of Theorem 6 hold, except that the condition (29) is replaced by

lim_{m→∞} sup \frac{1}{H_{m,n_{0}}} ∑_{n=n_{0}}^{m−1} H_{m,n}ψ_{n} = ∞ and lim_{m→∞} sup \frac{1}{H_{m,n_{0}}} ∑_{n=n_{0}}^{m−1} ϕ_{m,n} < ∞.  (38)

Then x_{n} either oscillates or satisfies lim_{n→∞} x_{n} = 0.

In view of Theorem 6, if we choose H_{m,n} = 1 and

(α_{n+1})^{\frac{1}{γ}} := −\frac{γΔρ_{n}}{(γ + 1)ρ_{n}} d_{n−σ}c_{n+1}^{− \frac{1}{γ}}δ_{n−σ}^{− \frac{1}{γ}}  (39)

we deduce that ξ_{n} = 0 and we have the following result.

Theorem 8. Let x_{n} be a solution of (21) and ρ_{n} be a given positive sequence. Assume that (h_{1}) − (h_{2}), (22) and (26) hold. If

lim_{n→∞} sup ∑_{s=n_{0}}^{n} ψ_{s} = ∞.  (40)

Then x_{n} either oscillates or satisfies lim_{n→∞} x_{n} = 0.

Theorem 8 improves Theorem 1 of Graef and Thandapani [8] in the sense that our results are proved for the nonlinear case and do not require condition (16) and that Δc_{n} ≥ 0 for n ≥ n_{0}. Moreover, we note that if γ = 1 and ρ_{n} = 1 then condition (40) reduces to condition 3 of Theorem 1 in [8]. This implies that Theorem 8 is an extension of Theorem 1 in [8].

Theorem 8 might provide different conditions for oscillation of all solutions of equation (21). This occurs upon choosing different values for ρ_{n}. For instance, let ρ_{n} = n^{λ}, n ≥ n_{0} where λ > 1 is a constant. Then, the next result follows.

Corollary 9. Let x_{n} be solution of equation (21) and assume that all the assumptions of Theorem 6 hold, except that condition (40) is replaced by

lim_{n→∞} sup ∑_{s=n_{0}}^{n} [Ks^{λ}q_{s} − s^{λ}Δ(c_{s}α_{s}) + \frac{s^{λ}δs−σ^{\frac{1}{γ}}(c_{s+1})^{1+ \frac{1}{γ}} (α_{s+1})^{1+ \frac{1}{γ}}}{d_{s−σ}} ] = ∞.  (41)

Then x_{n} either oscillates or satisfies lim_{n→∞} x_{n} = 0.

By choosing the sequence H_{m,n} in an appropriate form, one can derive several oscillation criteria for (21). Let us consider the double sequence H_{m,n} defined by

H_{m,n} := (m − n)^{λ} or H_{m,n} := (log \frac{m + 1}{n + 1})^{λ} , λ ≥ 1, m ≥ n ≥ 0,

or

H_{m,n} := (m − n)^{(λ)} λ ≥ 1, m ≥ n ≥ 0,

where

(m − n)^{(λ)} = (m − n)(m − n + 1) … (m − n + λ − 1), (m − n)^{(0)} = 1

and

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

We observe that H_{m,m} = 0 for m ≥ 0 and H_{m,n} > 0 and Δ_{2}H_{m,n} ≤ 0 for m > n ≥ 0. Then, the following results can be formulated.

Corollary 10. Let x_{n} be a solution of (21) and assume that all the assumptions of Theorem 6 hold, except that the condition (29) is replaced by

lim_{m→∞} sup \frac{1}{m^{λ}} ∑_{n=0}^{m−1} [(m − n)^{λ}ψ_{n} − φ_{m,n}] = ∞,  (42)

where

φ_{m,n} = \frac{ρ_{n+1}^{1+γ} (\frac{ξ_{n}(m−n)^{λ}}{ρ_{n+1}} − λ(m − n)^{λ−1})^{1+γ}}{(1 + γ)^{1+γ}ρ_{n}^{γ}δ_{n−σ}d_{n−σ}^{−γ}(m − n)^{λγ}} .

Then x_{n} either oscillates or satisfies lim_{n→∞} x_{n} = 0.

Corollary 11. Let x_{n} be a solution of (21) and assume that all the assumptions of Theorem 6 hold, except that the condition (29) is replaced by

lim_{m→∞} sup \frac{1}{(log(m + 1))^{λ}} ∑_{n=0}^{m−1} [(log \frac{m + 1}{n + 1})^{λ} ψ_{n} − ϑ_{m,n}]

where

ϑ_{m,n} = \frac{ρ_{n+1}^{1+γ} (\frac{ξ_{n}(log \frac{m+1}{n+1} )^{λ}}{ρ_{n+1}} − [(log \frac{m+1}{n+2})^{λ} − (log \frac{m+1}{n+1})^{λ}])}{(1 + γ)^{1+γ}ρ_{n}^{γ}δ_{n−σ}d_{n−σ}^{−γ}(log \frac{m+1}{n+})^{γλ}}.

Then x_{n} either oscillates or satisfies lim_{n→∞} x_{n} = 0.

Corollary 12. Let x_{n} be a solution of (21) and assume that all the assumptions of Theorem 6 hold, except that the condition (29) is replaced by

lim_{m→∞} sup \frac{1}{m^{λ}} ∑_{n=0}^{m−1}(m − n)^{λ} [ψ_{n} − \frac{ρ_{n+1}^{1+γ} (\frac{ξ_{n}}{ρ_{n+1}} − λ(m − n)^{−1})^{1+γ}}{(1 + γ)^{1+γ}ρ_{n}^{γ}δ_{n−σ}d_{n−σ}^{−γ}(m − n)^{λγ}} ] = ∞.

Then x_{n} either oscillates or satisfies lim_{n→∞} x_{n} = 0.

Example 13. Consider the equation

Δ(\frac{1}{n}Δ(\sqrt[3]{n}Δx_{n})) + nx_{n−1} = 0, n ≥ 1,  (43)

where γ = 1, c_{n} = \frac{1}{n}, d_{n} = \sqrt[3]{n}, q_{n} = n and n − σ = n − 1. It follows that δ_{n} = ∑_{s=1}^{n−1} \frac{1}{c_{s}} = \frac{n(n−1)}{2}. It is clear that the sequences c_{n}, d_{n}, q_{n} and the function f satisfy conditions (h_{1}) − (h_{2}) and (22). It remains to check conditions (26) and (40). From the above assumptions, it follows that

∑_{n=1}^{∞} \frac{1}{n^{\frac{1}{3}}} ∑_{t=1}^{n−1} t ∑_{s=1}^{t−1} s = ∞.

This shows that condition (26) is satisfied. By choosing ρ_{n} = n, one can easily see that

lim sup_{n→∞} ∑_{l=1}^{n} [Kl^{2} − l(\frac{1}{(l − 1)(l − 2)^{\frac{2}{3}} (l − 3)} − \frac{1}{l(l − 1)^{\frac{2}{3}} (l − 2)}) + \frac{1}{2l(l − 1)^{\frac{1}{3}} (l − 2)}] = ∞.

Thus, condition (40) holds. Therefore, by Theorem 8 we conclude that every solution x_{n} of equation (43) either oscillates or satisfies lim_{n→∞} x_{n} = 0.

Remark 14. It is obvious that results obtained in [8] can not be applied to equation (43).

3.2 Oscillation under condition (23)

Throughout this subsection, the sequences ρ_{n}, ψ_{n} and (α_{n+1})^{\frac{1}{γ}} are assumed in similar manner. In addition, we assume that (25) holds and thus in view of Lemma 2, we deduce that the class C_{3} is empty. Therefore, if x_{n} is a solution of (21) then x_{n} ∈ C_{0} ∪ C_{1} ∪ C_{2}.

We define the sequence Q_{n} by

Q_{n} := Kq_{n} (∑_{s=N}^{n−σ} \frac{1}{d_{s}} )^{γ} ,

where n − σ > N for N > n_{0}.

Theorem 15. Let x_{n} be a solution of (21) and ρ_{n} be a given positive sequence such that (40) holds. Assume that (h_{1}) − (h_{2}), (23), (25)and (26) hold. If

lim_{n→∞} sup ∑_{u=n_{6}}^{n−1} \frac{1}{d_{u}} [ ∑_{s=n_{5}}^{u−1} \frac{1}{c_{s}} ∑_{t=n_{4}}^{s−1} Q_{t} ∑_{τ=t−σ}^{∞} \frac{1}{c_{τ}} ]^{\frac{1}{γ}} = ∞.  (44)

Then x_{n} either oscillates or satisfies lim_{n→∞} x_{n} = 0.

Proof. Suppose to the contrary that x_{n} is a nonoscillatory solution of equation (21). Without loss of generality we may assume that x_{n} > 0 and x_{n−σ} > 0 for n ≥ n_{1} where n_{1} is chosen so large. Condition (25) implies that the solution x_{n} belongs to the space C_{0}∪C_{1}∪C_{2}. If x_{n} ∈ C_{0}, then we are back to the proof of Lemma 4 to show that lim_{n→∞} x_{n} = 0. If x_{n} ∈ C_{2}, then we are back to the proof of Theorem 6 to get a contradiction. To complete the proof, it is sufficient to show that under condition (44) there is no solution x_{n} ∈ C_{1}. Therefore, we suppose to the contrary that there exists N > n_{1} such that x_{n}^{[1]} > 0 and x_{n}^{[2]} < 0 for n ≥ N. In view of the quasi differences (24), we observe that

Δx_{n} = \frac{x_{n}^{[1]}}{d_{n}} .

Summing up from N to n − 1, we have

x_{n} − x_{N} = ∑_{s=N}^{n−1} \frac{x_{s}^{[1]}}{d_{s}} ≥ x_{n}^{[1]} ∑_{s=N}^{n−1} \frac{1}{d_{s}} .  (45)

Hence, there exists n_{3} > N such that

x_{n−σ} ≥ x_{n−σ}^{[1]} ∑_{s=N}^{n−σ} \frac{1}{d_{s}}, for n ≥ n_{3}.

Using this in (21), we get

Δ (c_{n}Δ (x_{n}^{[1]})^{γ}) + Kq_{n} (∑_{s=N}^{n−σ} \frac{1}{d_{s}} )^{γ} (x_{n−σ}^{[1]})^{γ} ≤ 0, n ≥ n_{3}.  (46)

Setting y_{n} = (x_{n}^{[1]})^{γ} > 0, we deduce that Δy_{n} < 0 and y_{n} satisfies the difference inequality

Δ(c_{n} (Δy_{n})) + Q_{n}y_{n−σ} ≤ 0, for n ≥ n_{3}.  (47)

Since n − σ → ∞ as n → ∞, we can choose n_{4} > n_{3} such that n − σ ≥ n_{4} for n ≥ n_{4} and thus

y_{∞} − y_{n−σ} = ∑_{s=n−σ}^{∞} Δy_{s} = ∑_{s=n−σ}^{∞} c_{s}Δy_{s} \frac{1}{c_{s}}

< c_{n−σ}Δy_{n−σ} ∑_{s=n−σ}^{∞} \frac{1}{c_{s}} < c_{n_{4}}Δy_{n_{4}} ∑_{s=n−σ}^{∞} \frac{1}{c_{s}} .

Thus

−y_{n−σ} < c_{n_{4}}Δy_{n_{4}} ∑_{s=n−σ}^{∞} \frac{1}{c_{s}} .

Substituting back in (47), we have

Δ(c_{n} (Δy_{n})) < LQ_{n} ( ∑_{s=n−σ}^{∞} \frac{1}{c_{s}} ) , for n ≥ n_{4},  (48)

where L = c_{n_{4}}Δy_{n_{4}} < 0. Summing this inequality from n_{4} to n − 1, we see that

c_{n} (Δy_{n}) < c_{n} (Δy_{n}) − c_{n_{4}} (Δy_{n_{4}}) < L ∑_{s=n_{4}}^{n−1} Q_{s} ∑_{τ=s−σ}^{∞} \frac{1}{c_{τ}} .

where Δy_{n} < 0. Summing again from n_{5} to n − 1, we have

y_{n} < L ∑_{s=n_{5}}^{n−1} \frac{1}{c_{s}} ∑_{t=n_{4}}^{s−1} Q_{t} ∑_{τ=t−σ}^{∞} \frac{1}{c_{τ}}

or equivalently

Δx_{n} < (L)^{\frac{1}{γ}} (\frac{1}{d_{n}} ) [ ∑_{s=n_{5}}^{n−1} \frac{1}{c_{s}} ∑_{t=n_{4}}^{s−1} Q_{t} ∑_{τ=t−σ}^{∞} \frac{1}{c_{τ}} ]^{\frac{1}{γ}} .

Summing from n_{6} to n − 1, we have

x_{n} < x_{n_{6}} + (L)^{\frac{1}{γ}} ∑_{u=n_{6}}^{n−1} \frac{1}{d_{u}} [ ∑_{s=n_{5}}^{u−1} \frac{1}{c_{s}} ∑_{t=n_{4}}^{s−1} Q_{t} ∑_{τ=t−σ}^{∞} \frac{1}{c_{τ}} ]^{\frac{1}{γ}} .

By condition (44), we have lim_{n→∞} x_{n} = −∞ which contradicts the fact that x_{n} > 0. The proof is complete.

Theorem 16. Let x_{n} be a solution of (21). Let ρ_{n} be a positive sequence. Assume that (h_{1}) − (h_{2}), (23), (25) and (26) hold. If (44) holds and there exist double sequences H_{m,n} and h_{m,n} satisfy (29), then x_{n} either oscillates or satisfies lim_{n→∞} x_{n} = 0.

Proof. Suppose to the contrary that x_{n} is a nonoscillatory solution of equation (21). Without loss of generality we may assume that x_{n} > 0 and x_{n−σ} > 0 for n ≥ n_{1} where n_{1} is chosen so large. Condition (25) implies that the solution x_{n} belongs to the space C_{0}∪C_{1}∪C_{2}. If x_{n} ∈ C_{0}, then we are back to the proof of Lemma 4 to show that lim_{n→∞} x_{n} = 0. If x_{n} ∈ C_{1}, then we are back to the proof of Theorem 15 to get a contradiction. To complete the proof, it is sufficient to show that under condition (44) there is no solution x_{n} ∈ C_{1}. Thus, we proceed as in the proof of Theorem 15 to get a contradiction. The proof is complete.

The following results are an immediate consequences of Theorem 16.

Corollary 17. Let x_{n} be solution of equation (21) and assume that all the assumptions of Theorem 16 hold, except that condition (29) is replaced by (41). Then x_{n} either oscillates or satisfies lim_{n→∞} x_{n} = 0.

Corollary 18. Let x_{n} be a solution of (21) and assume that all the assumptions of Theorem 16 hold, except that the condition (29) is replaced by (42). Then x_{n} either oscillates or satisfies lim_{n→∞} x_{n} = 0.

References

[1] R. P. Agarwal, Difference Equations and Inequalities, Theory, Methods and Applications, Second Edition, Revised and Expanded, Marcel Dekker, New York 2000.

[2] R. P. Agarwal and S. R. Grace, Oscillation of certain third-order difference equations, Comp. Math. Appl. 42 (2001) 379–384.

[3] R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, Kluwer Academic Publishers, Dordrecht 1997.

[4] M. Artzroumi, Generalized stable population theory, J. Math. Biology, 21 (1985) 363– 381.

[5] Z. Došlá and A. Kobza, Global asymptotic properties of third-order difference equations, Comp. Math. Appl. 48 (2004) 191–200.

[6] S. Elaydi, An Introduction to Difference Equations, Third Edition, Springer, New York, 2005.

[7] S. R. Grace, R. P. Agarwal and J. Graef, Oscillation criteria for certain third order nonlinear difference equations, Appl. Anal. Discrete Math. 3 (2009), 27–38.

[8] J. Graef and E. Thandapani, Oscillatory and asymptotic behavior of solutions of third order delay difference equations, Funk. Ekvac. 42 (1999) 355–369.

[9] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd Ed. Cambridge Univ. Press 1952.

[10] G. D. Jones, Oscillatory behavior of third order differential equations, Proc. Amer. Math. Soc. 43 (1974) 133–135.

[11] W.G. Kelley and A. C. Peterson, Difference Equations; An Introduction with Applications, Academic Press, New York 1991.

[12] J. Popenda and E. Schmeidel, Nonoscillatory solutions of third order difference equations, Port. Math. 49 (1992) 233–239.

[13] S. H. Saker, Oscillation of third-order difference equations, Port. Math. 61 (2004) 249– 257.

[14] B. Smith, Oscillatory and asymptotic behavior in certain third order difference equations, Rocky Mountain J. Math. 17 (1987) 597–606.

[15] B. Smith, Linear third-order difference equations: Oscillatory and asymptotic behavior, Rocky Mountain J. Math. 22 (1992) 1559–1564.

[16] B. Smith, Oscillation and nonoscillation theorems for third order quasi-adjoint difference equation, Port. Math. 45 (1988), 229–234.

[17] B. Smith and Jr. W. E. Taylor, Asymptotic behavior of solutions of a third order difference equations, Port. Math. 44 (1987) 113–117.

[18] B. Smith and Jr. W. E. Taylor, Nonlinear third order difference equation: Oscillatory and asymptotic behavior, Tamakang J. Math. 19 (1988) 91–95.

[19] J. Jiang, X. Li, Oscillation of second order nonlinear neutral differential equations, App. Math. Comput. 135 (2–3) (2003), 531–540.

(Received May 2, 2010)