OSCILLATION OF HIGHER ORDER STRONGLY SUPERLINEAR AND STRONGLY SUBLINEAR DIFFERENCE EQUATIONS

Abstract

We establish some new criteria for the oscillation of mth order nonlinear difference equations. We study the case of strongly superlinear and the case of strongly sublinear equations subject to various conditions. We also present a sufficient condition for every solution to be asymptotic at ${\infty}$ to a factorial expression $(t)^{(m-1)}$.

1. Introduction

In what follows, we shall denote ℕ = {0, 1, ...}, ℕ(a) = {a, a+1, ...} where a ∈ ℕ and ℕ(a, b) = {a, a+1, ..., b}, b ∈ ℕ(a).

Consider the mth order nonlinear difference equation

where △ is the forward difference operator defined by △x(t) = x(t + 1) − x(t), m is a positive even integer. We shall assume that

By a solution of equation (1), we mean a nontrivial sequence {x(t)} satisfying equation (1) for all t ∈ ℕ(t0), where t0 ∈ ℕ. A solution {x(t)} is said to be oscillatory if it is neither eventually positive nor eventually negative and it is nonoscillatory otherwise. An equation is said to be oscillatory if all its solutions are oscillatory.

Equation (1) (or the function ƒ) is said to be strongly superlinear if there exists a constant β > 1 such that

and it is said to be strongly sublinear if there exists a constant γ ∈ (0, 1) such that

(3) holds with β = 1, then equation (1) is called superlinear and (4) holds with γ = 1 is called sublinear.

The literature on oscillation of solutions of difference equantions is almost devoted to study of equation (1) when m = 1 and 2 and for recent contribution we refer to Agarwal et.al. [1,2,3]. Only few results are available for the oscillation of equation (1) when m > 2, see Agarwal et.al.[2,4,5].

Therefore, the purpose of this paper is to establish some new results for the oscillation of strongly superlinear and strongly sublinear difference equations. We also provide conditions, which guarantee that every solution defined for all large t ∈ ℕ(t0) is asymptotic at ∞ to (t)(m-1).

The obtained results improve and unify these which have appeared in the recent literature.

 

2. Preliminaries

We shall need the following lemmas given in [2].

Lemma 2.1. (Discrete Toylors Formula) Let x(t) be defined on ℕ(t0). Then for all t ∈ ℕ(t0) and 0 ≤ n ≤ j − 1

Further, for all t ∈ ℕ(t0, z), where z ∈ t ∈ ℕ(t0) and 0 ≤ n ≤ j − 1

Lemma 2.2. (Discrete Kneser’s Theorem) Let x(t) be defined on ℕ(t0), x(t) > 0 and Δmx(t) be eventually of one sign on ℕ(t0). Then there exists an integer k, 0 ≤ k ≤ m with m + k odd for Δmx(t) ≤ 0 and (m + k) even for Δmx(t) ≥ 0 such that

Lemma 2.3. Let x(t) be defined on ℕ(t0) and x(t) > 0 with Δmx(t) ≤ 0 for t ∈ ℕ(t0) and not identically zero. Then there exists large t1 ∈ ℕ(t0) such that

where k is as in Lemma 2.2. Furthermore, if x(t) is increasing, then

Lemma 2.4. (Gronwal Inequality) Let for all t ∈ ℕ(t0) the following inequality be satisfied

where {p(t)}, {q(t)}, {ƒ(t)} and {x(t)} are non-negative real-valued sequence de-fined on ℕ(t0). Then for all t ∈ ℕ(t0),

 

3. Oscillation Criteria

We shall study the oscillatory behavior of all solutions of equation (1) when it is either strongly superlinear or strongly sublinear.

We begin with strongly superlinear case of equation(1).

Theorem 3.1. Suppose that equation (1) is strongly superlinear. If

for some c ≠ 0 and k ∈ {1, 3, ..., m − 1}, then equation (1) is oscillatory.

Proof. Let {x(t)} be a non-oscillatory solution of equation (1), say x(t) > 0 for t ≥ t0 ∈ ℕ(t0). By Lemma 2.2, there exists an integer k ∈ {1, 3, ..., m − 1} such that (7) holds for t ≥ t1 ≥ t0.

Clearly, Δk−1x(t) is positive and increasing for t ≥ t1. Thus from (5), we find for s ≥ t ≥ t0

On the other hand, there exists a constant c > 0 such that

From (6) with n = k and j = m, and equation (1), we have

Using the strong superlinearity of ƒ we obtain

Using (13) in (16) and the fact that m − 1 ≥ k, we have

for t ≥ t1. Or

where

Now, since Δkx(t) is positive and decreasing for t ≥ t1 + 1, we find

Summing this inequality from t1 + 1 to T ≥ t1 + 1 we get

which contradicts condition (12). This completes the proof.

When k = 1, condition (12) is reduced to

For the case when m = 2, we obtain

Corollary 3.2. Suppose that equation (1) with m = 2 is strongly superlinear. If

then equation (1) with m = 2 is oscillatory.

For strongly sublinear equation (1), we have

Theorem 3.3. Let equation (1) be strongly sublinear. If

for some constant c ≠ 0, then equation (1) is oscillatory.

Proof. Let {x(t)} be a non-oscillatory solution of equation (1), say x(t) > 0 for t ∈ ℕ(t0). By Lemma 2.2, these exists a t1 ≥ t0 and constants c1 and c2 such that

and by Lemma 2.3, we find

Summing equation (1) from t ≥ t2 to u ≥ t and letting u → ∞, we get

Using the strong sublinearity of ƒ in the above inequality, we obtain

By applying (23) in the (24), we find

Denoting the right-hand side of (25) by z(t), we find

or

Summing this inequality from t2 + 1 to T ≥ t2 + 1, we get

which contradicts condition (20). This completes the proof.

Remark 3.1. One can easily see that equation (1) is oscillatory if

for some constants c ≠ 0.

Remark 3.2. The results of this section are presented in a form which is essentially new. It extend and improve many of the existing results appeared in the literature, see [1,2,3,4,5].

Remark 3.3. When m = 2, the results obtained include many of the known oscillation results for related second order nonlinear difference equations, see [1,2,3].

Remark 3.4. The results of this section can be extended to mth order nonlinear difference equation with deviating arguments of the form

where ƒ is as in equation (1) and g ∈ {g : ℕ(t∗) → ℕ for some t∗ ∈ ℕ : g(t) ≤ t, limt→∞ g(t) = 0}, {g(t)} is a nondecreasing sequence.

In fact, we may replace s in conditions (12) and (20) by g(s). The details are left to the reader.

 

4. Asymptotic Behavior

In this section we give a sufficient condition for every solution x defined for all large t ∈ ℕ(t0) of equation (1) to satisfy

where c is some real number (depending on solution {x(t)}.

We assume that

where γ ∈ (0, 1], {a(t)} and {b(t)} are nonnegative real-valued sequences.

Theorem 4.1. If

then every solution {x(t)}, t ∈ ℕ(t0) of equation (1) satisfies

and

where c is some constant (depending on solution {x(t)}.

Proof. Let {x(t)} be a solution for t ≥ t0 ∈ ℕ(t0) of equation (1). Then (1) gives

Thus, by (29), we obtain for t ≥ t0,

or

Using the elementary inequality

we find

By the assumption (29), these exists constat C > 0 such that

Applying Lemma 2.4 and using condition (29) we can conclude that there exists a positive constant M such that

Now, by using (28) and (32), we derive

Thus, because of (29), it follows that

But, (1) gives

Therefore,

i.e. (30) holds. Finally, by L’Hospital rule, we obtain

and consequently the solution {x(t)} satisfies (31). This completes the proof.

Of course, Theorem 4.1 remains valid for the equations of the form

where ƒ satisfies conditions (28) and (29).

We may note that the second part of the condition (29) can be replaced by

and also the conclusion (31) can be replace by

As illustrative example, we consider the equation

where the {p(t)} and {q(t)} are non negative real sequence and γ ∈ (0, 1] is a constant. Now, if

then by Theorem 4.1, we conclude that every solution of equation (36) satisfies

where C is some real number depending on the solution {x(t)} .

 

5. General Remarks

1. We many note that conditions (12) and (18) can be replaced by the stronger condition

while condition (20) takes the form

2. Theorem 4.1 when m = 2 is a discrete analog of the results in [6,7,8,9]. Moreover, it improve and unify some of them.

 

6. Example

Consider the following mth order nonlinear difference equation

Let m > 4 is a positive even integer, we have

so ƒ(t,x(t)) is strong sublinear.

Since c ≠ 0,

Then,

Because m > 4 is a positive even integer, let t0 = m − 2, so

and then (40) is divergence, so Theorem 3.3, the solution of function (39) is oscillatory.