Oscillation of Solutions of a Linear Second-Order Discrete-Delayed Equation

Abstract

A linear second-order discrete-delayed equation with a positive coefficient is considered for . This equation is known to have a positive solution if fulfils an inequality. The goal of the paper is to show that, in the case of the opposite inequality for , all solutions of the equation considered are oscillating for .

1. Introduction

The existence of a positive solution of difference equations is often encountered when analysing mathematical models describing various processes. This is a motivation for an intensive study of the conditions for the existence of positive solutions of discrete or continuous equations. Such analysis is related to an investigation of the case of all solutions being oscillating (for relevant investigation in both directions, we refer, e.g., to [1–15] and to the references therein). In this paper, sharp conditions are derived for all the solutions being oscillating for a class of linear second-order delayed-discrete equations.

We consider the delayed second-order linear discrete equation

(1.1)

where , is fixed, , and . A solution of (1.1) is positive (negative) on if () for every . A solution of (1.1) is oscillating on if it is not positive or negative on for arbitrary .

Definition 1.1.

Let us define the expression , , by , where and , , and instead of , , we will only write and .

In [2] a delayed linear difference equation of higher order is considered and the following result related to (1.1) on the existence of a positive solution is proved.

Theorem 1.2.

Let be sufficiently large and . If the function satisfies

(1.2)

for every , then there exist a positive integer and a solution , of (1.1) such that holds for every .

Our goal is to answer the open question whether all solutions of (1.1) are oscillating if inequality (1.2) is replaced by the opposite inequality

(1.3)

assuming and is sufficiently large. Below we prove that if (1.3) holds and , then all solutions of (1.1) are oscillatory. The proof of our main result will use a consequence of one of Domshlak's results [8, Corollary , page 69].

Lemma 1.3.

Let and be fixed natural numbers such that . Let be a given sequence of positive numbers and a positive number such that there exists a number satisfying

(1.4)

Then, if and for

(1.5)

holds, then any solution of the equation

(1.6)

has at least one change of sign on .

Moreover, we will use an auxiliary result giving the asymptotic decomposition of the iterative logarithm [7]. The symbols "" and "" used below stand for the Landau order symbols.

Lemma 1.4.

For fixed and fixed integer , the asymptotic representation

(1.7)

holds for .

2. Main Result

In this part, we give sufficient conditions for all solutions of (1.1) to be oscillatory as .

Theorem 2.1.

Let be sufficiently large, , and . Assuming that the function satisfies inequality (1.3) for every , all solutions of (1.1) are oscillating as .

Proof.

We set

(2.1)

and consider the asymptotic decomposition of when is sufficiently large. Applying Lemma 1.4 (for , , and ), we get

(2.2)

Finally, we obtain

(2.3)

Similarly, applying Lemma 1.4 (for , , and ), we get

(2.4)

Using the previous decompositions, we have

(2.5)

Recalling the asymptotical decomposition of when : , we get (since )

(2.6)

as . Due to (2.3) and (2.4) we have and as . Then it is easy to see that, for the right-hand side of the inequality (1.5), we have

(2.7)

where

(2.8)

Moreover, for , we will get an asymptotical decomposition as . We represent in the form

(2.9)

As the asymptotical decompositions for

(2.10)

have been derived above (see (2.3)–(2.5)), after some computation, we obtain

(2.11)

Thus we have

(2.12)

Finalizing our decompositions, we see that

(2.13)

It is easy to see that inequality (1.5) becomes

(2.14)

and will be valid if (see (1.3))

(2.15)

or

(2.16)

for . If where is sufficiently large, then (2.16) holds for sufficiently small with fixed because . Consequently, (2.14) is satisfied and the assumption (1.5) of Lemma 1.3 holds for . Let in Lemma 1.3 be fixed and let be so large that inequalities (1.4) hold. This is always possible since the series is divergent. Then Lemma 1.3 holds and any solution of (1.1) has at least one change of sign on . Obviously, inequalities (1.4) can be satisfied for another couple of , say with and sufficiently large, and by Lemma 1.3 any solution of (1.1) has at least one change of sign on . Continuing this process, we get a sequence of intervals with such that any solution of (1.1) has at least one change of sign on . This fact concludes the proof.