# Oscillation of neutral advanced difference equation.

Introduction

We consider the first order neutral advanced difference equation of the form

[DELTA][x(n) - p(n)x([tau](n))] - q(n)x([sigma](n)) = 0, n [greater than or equal to] [n.sub.0] (1)

where {p(n)}, [q(n)} are sequences of real numbers, {[sigma]a(n)} is a sequence of positive integers such that [sigma](n) > n + 1, ([tau](n)} is a nondecreasing sequence of nonnegative integers such that [tau](n) < n and [DELTA] is the forward difference operator defined by the equation [DELTA]x(n) = x(n + 1) - x(n).

We present some sufficient conditions such that every solution of (1) is either oscillatory or tends to zero as n [right arrow] [infinity].

By a solution of equation (1), we mean a real sequence (x(n)}, n [member of] N([tau]([n.sub.0])) = {[tau]([n.sub.0]), [tau]([n.sub.0]) + 1, [tau]([n.sub.0]) + 2, ...} satisfying (1). We consider only such solutions which are non trivial for all large n. A solution of (1) is said to oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory.

The qualitative properties of the solution of the advanced difference equation (1) have been the subject of our investigation. With respect to the oscillation of delay difference equation with variable coefficients, reader can refer to [3,4,8-11]. For the several background on difference equation, we can refer to [1,2,5-7].

The following conditions are assumed to be hold throughout the paper.

([C.sub.1]) {p(n)}, {q(n)} are sequences of nonnegative real numbers and {q(n)} is not identically zero.

([C.sub.2]) {[sigma](n)} is a nondecreasing sequence of positive integers such that [sigma](n) > n + 1 on N([n.sub.0]), [lim.sub.n[right arrow][infinity]] [sigma](n) = [infinity] and [lim.sub.n[right arrow][infinity]] ([tau](n)) = [infinity].

([C.sub.3] {[tau](n)} is a nondecreasing sequence of nonnegative integers such that [tau](n) < n on N([n.sub.0]) = {[n.sub.0], [n.sub.0] + 1, ...} and [lim.sub.n[right arrow][infinity]][tau](n) = [infinity].

([C.sub.4]) There exist a constant p such that 0 [less than or equal to] p(n) [less than or equal to] p < 1.

Before giving the main results, we present some lemmas which will be used in the proofs of Theorems.

Lemma 1.1 Set

z(n) = x(n) - p(n)x([tau](n)). (2)

If {x(n)} is an eventually positive solution of equation (1) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

then z(n) > 0 eventually.

Proof. Let {x(n)} be an eventually positive solution of (1) such that [lim.sub.n[right arrow][infinity]] x(n) > 0. Assume the contrary. That is, z(n) < 0 for large n. If {x(n)} is unbounded, then there exists a sequence {[n.sub.k]} of integers such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then from (2), we have

z([n.sub.k]) = x([n.sub.k]) - p([n.sub.k])x([tau]([n.sub.k])) [greater than or equal to] x([n.sub.k])(1 - p)

and hence [lim.sub.k[right arrow][infinity]]([n.sub.k]) = [infinity].

If {x(n)} is bounded, then there is a sequence {[n.sub.k]} of integers such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [lim.sub.k[right arrow][infinity]]x([tau]([n.sub.k])) [less than or equal to] L, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This is also a contradiction and the proof is complete.

Lemma 1.2 The sequence {[sigma](n)} has the properties

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By ([C.sub.2]), {[sigma](n)} is nondecreasing, and

[sigma](n) > n for n [greater than or equal to] [n.sub.0].

So, we have

[sigma]([sigma](n)) [greater than or equal to] [sigma](n), for n [greater than or equal to] [n.sub.0].

Moreover, by this inequality, we can see easily that

[sigma]([sigma](n)) - n [greater than or equal to] [sigma](n) - n, n [greater than or equal to] [n.sub.0].

Taking limit as n [right arrow] [infinity] on both sides, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By [sigma](n) [right arrow] [infinity] as n [right arrow] [infinity], it is obvious that [sigma]([sigma](n)) [right arrow] [infinity] as n [right arrow] [infinity].

Lemma 1.3 Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Since q(n) [greater than or equal to] 0 for n [greater than or equal to] [n.sub.0], by assumption, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

On the other hand, by discrete mean value theorem, we have

[[sigma](n)-1.summation over (s=n+1)] q(s) [greater than or equal to] ([sigma](n) - n - 1)q([bar.n]),

where [bar.n] [member of] {n + 2, n + 3, ..., [sigma](n) - 1}. Thus [bar.n] [right arrow] [infinity] as n [right arrow] [infinity]. Then it is clear that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We are now in a position to state and prove our main results.

Main Results

Theorem 2.1 Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then every solution of (1) is either oscillatory or tends to zero.

Proof. Without loss of generality, we may assume that {x(n)} is an eventually positive solution of (1) such that [lim.sub.n[right arrow][infinity]]x(n) > 0. Then by Lemma 1.1 , z(n) > 0 eventually, where z(n) is defined by (2). From (1) and (2), we obtain

[DELTA]z(n) - q(n)z([sigma](n)) [greater than or equal to] 0, n [greater than or equal to] [n.sub.1] [greater than or equal to] [n.sub.0]. (4)

Since [DELTA]z(n) [greater than or equal to] 0, n [greater than or equal to] [n.sub.1], we have {z(n)} is nondecreasing on N([n.sub.1]). From (4), we can obtain

z(a(n)) ^ z(a(5))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Set

z(a(n))

w(n) = z([sigma](n))/z(n + 1), n [greater than or equal to] [n.sub.1]. (6)

It is clear that w(n) [greater than or equal to] 1, n [greater than or equal to] [n.sub.1].

So by the last inequality, we have

lnw(n) [greater than or equal to] [[sigma](n)-1.summation over (s=n+1)] q(s)w(s). (7)

Let l = [lim.sub.n[right arrow]+[infinity]] w(n), then 1 [less than or equal to] l [less than or equal to] +[infinity]. Now we divide our discussion into the following two cases i) l [not equal to] +[infinity] ii) l = +[infinity]

l is finite

There exists a sequence {nk} of integers such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand, since {[sigma](n)} is nondecreasing and {q(n)} is nonnegative, so it follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By Lemma 1.3, we see that it is a contradiction.

(ii) l = +[infinity].

Thus

.. z(a(n))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Summing (4) on both sides from [sigma](n) + 1 to [sigma]([sigma](n)) -- 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Dividing both sides of this inequality by z([sigma]([sigma](n))), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

And by (8), we know

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Taking limit on both sides of inequality (10), in view of Lemmas 1.2 and 1.3, we have a contradiction.

The proof is complete.

Corollary 2.2 Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then every solution of the difference inequality

[DELTA][x(n) - p(n)x([tau](n))] - q(n)x([sigma](n)) [greater than or equal to] 0, (or [less than or equal to] 0), n [greater than or equal to] [n.sub.0], (11)

is either not positive (or negative) or tends to zero.

Corollary 2.3 Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Then every solution of (1) is either oscillatory or tends to zero and the every solution of the difference inequality (11) is either not positive or tends to zero.

Proof. For Corollary 2.3, we can see that in the proof of Theorem 2.1, if {x(n)} is an eventually positive solution of (1), when l is finite, then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts (12). When l = +[infinity], in view of (10), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [sigma](n) [right arrow] +[infinity] as n [right arrow] +[infinity], it follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which contradicts (12). Thus the result of Corollary 2.3 holds.

Note that the condition (12) is much weaker than the condition in Theorem 2.1. We can see this from Lemma 1.3.

Consider the following difference equation

[DELTA][x(n) - p(n)x([tau](n))] - f(n,x([sigma](n))) = 0, n [greater than or equal to] [n.sub.0] (13)

and the inequality

[delta][x(n) - p(n)x([tau](n))] - f(n, x([sigma](n))) [greater than or equal to] 0, (or [less than or equal to] 0), n [greater than or equal to] [n.sub.0] (14)

The function f: N x R [right arrow] R satisfies the following conditions:

a) f(n, v)v > 0, for v [not equal to] 0, v [member of] R,

b) f(n,0) = 0,

c) [absolute value of f(n,v)] [greater than or equal to] q(n)[absolute value of v], n [member of] N, v [member of] R,

where {p(n)}, {q(n)}, {[tau](n)} and {[sigma](n)} are sequences appeared in the equation (1) and satisfies all the conditions mentioned at the beginning of this paper.

It follows a similar way to prove the following results.

Theorem 2.4 Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then every solution of equation (13) is either oscillatory or tends to zero and every solution of the inequality (14) is either not positive (or negative) or tends to zero.

Proof. As a matter of fact, if there exists an eventually positive solution {x(n)} of the equation (13), then by equation (13) and the conditions on f, we have

[DELTA][x(n) - p(n)x([tau](n))] - q(n)x([sigma]([sigma](n))) [greater than or equal to] 0, n [greater than or equal to] [n.sub.1] [greater than or equal to] [n.sub.0].

Then the rest proof can follow the one that we have done in the proof of Theorem 2.1. It has similar steps if we have an eventually negative solution {x(n)} of equation (13). Indeed, if x(n) < 0, for n [greater than or equal to] [n.sub.1] [greater than or equal to] [n.sub.0], we have

[DELTA][x(n) - p(n)x([tau](n))] - q(n)x([sigma](n)) [less than or equal to] 0, n [greater than or equal to] [n.sub.1].

Let y(n) = -x(n), then y(n) > 0, n [greater than or equal to] [n.sub.1], it follows

[DELTA][y(n) - p(n)y([tau](n))] - q(n)y([sigma](n)) [greater than or equal to] 0, n [greater than or equal to] [n.sub.1].

In the following, we investigate existence of positive solutions of (1) and equation (13).

Theorem 2.5 Assume that

p(n) [equivalent to] 0 on N([n.sub.0])

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then equation (1) has a positive solution.

Proof. For the convenience, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

where {x(n)} is a solution of equation (1). By this form, from equation (1), we have the following equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

If we can prove that equation (16) has a solution {[lambda](n)}, then by the form of x(n) in (15), we see that equation (1) has a positive solution. Consider a sequence {[[lambda].sub.k](n)} as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Using the induction, we can prove that {[lambda]k(n)} is a nonincreasing sequence, namely,

[[lambda].sub.k-1](n) [greater than or equal to] [[lambda].sub.k](n), k = 1,2,3, ...

And so we have

1 + eq(n) [greater than or equal to] [[lambda].sub.k](n) [greater than or equal to] 0, n [member of] N([n.sub.0]) for k = 1,2,3,...

It follows that there exists a sequence {[lambda](n)} such that [[lambda].sub.k](n) [right arrow] [lambda](n) as n [right arrow] [infinity], and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It concludes that {[lambda](n)} is a solution of equation (16).

Theorem 2.6 Assume that the function f(n, v) is nondecreasing in v and

p(n) [equivalent to] 0 on N([n.sub.0]).

Suppose that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then equation (13) has a positive solution.

Proof. We can prove this result by a similar way as we have done in the proof of Theorem 2.5. Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where {x(n)} is a solution of (13). Then by (13) and the form of x(n), we have the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

If equation (17) has a solution {[lambda](n)}, then it follows that equation (13) has a positive solution. Construct a sequence {[[lambda].sub.k](n)} as a follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ...

In view of the assumption, we see that [[lambda].sub.k](n) > 0 for k = 0,1,2,3,.... Furthermore by using the induction, we can prove that

1+ f(n, e) [greater than or equal to] [[lambda].sub.k-1](n) [greater than or equal to] [[lambda].sub.k](n) [greater than or equal to] 0, k = 1,2,3,...

It follows that there exists a sequence {[[lambda].sub.k](n)} such that [[lambda].sub.k](n) [right arrow] [lambda](n) as k [right arrow] [infinity]. So there exists a solution {[lambda](n)} of equation (17).

The proof is complete.

References

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[6] W.G. Kelly and A.C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, New York, 1991.

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[11] B.G. Zhang and S.S. Cheng, Oscillation criteria for a neutral difference equations with delay, Appl.Math.Lett., 8(1995), 13-17.

A. Murugesan

Department of Mathematics, Government Arts College (Autonomous), Salem--636 007, Tamil Nadu, India.

Email: amurugesan3@gmail.com
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