# Oscillatory behavior of forced neutral difference equations with positive and negative coefficients.

1. Introduction

Consider the forced neutral difference equations with positive and negative coefficients

[DELTA]([X.sub.n] - [c.sub.n][x.sub.n-r]) + [p.sub.n][x.sub.n-k] - [q.sub.n][x.sub.n-l] = [e.sub.n], n [greater than or equal to] [n.sub.0], (1.1)

where [DELTA] denotes the forward difference operator [DELTA][x.sub.n] = [x.sub.n+1] - [x.sub.n]. In the absence of the forcing term en, equation (1.1) becomes

[DELTA]([X.sub.n] - [c.sub.n][x.sub.n-r]) + [p.sub.n][x.sub.n-k] - [q.sub.n][x.sub.n-l] = 0. (1.2)

The oscillation of equation (1.2) have been investigated by several authors, see for example [1-7]. In particular, many authors considered the oscillation results for equation (1.2) with the following conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

which played a very important role in the study of equation (1.1). However, Luo et al. , Tang and Yu  obtained some results for the oscillatory behavior of equation (1.1), which do not require condition (1.3). They used the following condition:

[c.sub.n] + [n-1.summation over (s=n-k+l)][q.sub.s] [equivalent to] 1. (1.4)

In this paper, our aim is to provide some new sufficient conditions for the oscillation of equation (1.1) without the usual condition (1.4). Our results improve the known results in the literature. For any positive integer t [member of]{0, 1,...,k - l} such that

[R.sub.n](t) = [c.sub.n] + [n-1.summation over (s=n-t)][q.sub.s] + [n-t+k-l- 1.summation over (s=n)][p.sub.s] [equivalent to] 1. (1.5)

Let [m.sup.*.sub.f] = max{k, r}. By a solution of equation (1.1), we mean a sequence {[x.sub.n]} of real numbers which is defined for n [greater than or equal to] -[m.sup.*.sub.t] and satisfies equation (1.1) for n [member of] N(0). It is easy to see under the initial conditions [x.sub.n] = [b.sub.n],n = - [m.sup.*.sub.t], -[m.sup.*.sub.t] + 1,..., -1. Then the Equation (1.1) has a unique solution.

As is customary, a nontrivial solution {[x.sub.n]} of equation (1.1) is said to be non-oscillatory if the terms [x.sub.n] are either eventually positive or eventually negative. Otherwise, it is said to be oscillatory. Throughout this paper, we assume

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the following conditions hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

2. Basic Lemmas

In this section, first we shall prove some basic lemmas which will be used in the proofs of the theorems in section 3. For n [greater than or equal to] [n.sub.0], we consider the following difference inequalities

[DELTA]([x.sub.n] - [c.sub.n][x.sub.n-r] + [[infinity].summation over (s=n)][e.sub.s]) + [p.sub.n][x.sub.n-k] - [q.sub.n][x.sub.n-l] [less than or equal to] 0, (2.7)

[DELTA]([x.sub.n] - [c.sub.n][x.sub.n-r] + [[infinity].summation over (s=n)][e.sub.s]) + [p.sub.n][x.sub.n-k] - [q.sub.n][x.sub.n-l] [greater than or equal to] 0. (2.8)

Lemma 2.1. Suppose there exists an integer t [member of]{0,1,...,k - l} such that

[c.sub.n] + [n-1.summation over (s=n-t)][q.sub.s] + [n-t+k-l-1.summation over (s=n)][p.sub.s] [less than or equal to] 1 (2.9)

for large n. Let

[z.sub.n] = [x.sub.n] - [c.sub.n][x.sub.n-r] - [n-1.summation over (s=n- t)][q.sub.s][x.sub.s-l] - [n-t+k-l-1.summation over (s=n)][p.sub.s][x.sub.s-k] + [[infinity].summation over (s=n)][e.sub.s]. (2.10)

Then, the following statements are true.

(i) If {[x.sub.n]} is an eventually positive solution of (2.7), then we have [DELTA][z.sub.n] [less than or equal to] 0 and [z.sub.n] > 0 for large n.

(ii) If {[x.sub.n]} is an eventually negative solution of (2.8), then we have [DELTA][z.sub.n] [greater than or equal to] 0 and [z.sub.n] < 0 for large n.

Proof. (i). Let [n.sub.1] be a positive integer such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for n [greater than or equal to] [n.sub.1]. Using (2.10), (2.7) becomes

[DELTA][z.sub.n] + [h.sub.n-t+k-l][x.sub.n-t-l] [less than or equal to] 0

which implies [DELTA][z.sub.n] [less than or equal to] 0,n [greater than or equal to] [n.sub.1]. Hence, [z.sub.n] is non increasing for n [greater than or equal to] [n.sub.1]. If [z.sub.n] > 0 does not hold, then eventually [z.sub.n] < 0, so there exist an integer [n.sub.2] [greater than or equal to] [n.sub.1] and [alpha] > 0 such that [z.sub.n] [right arrow] [alpha] for n [greater than or equal to] [n.sub.2],

[x.sub.n] [less than or equal to] -[alpha] + [c.sub.n][x.sub.n-r] + [n- 1.summation over (s=n-t)][q.sub.s][x.sub.s-k] - [[infinity].summation over (s=n)][e.sub.s], n [greater than or equal to] [n.sub.2]. (2.11)

We consider the following two possible cases.

Case (i). Assume that {[x.sub.n]} is unbounded, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, there exists a sequence of points [{[s.sub.i]}.sup.[infinity].sub.i=1] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] From (2.9) and (2.11), we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let i [right arrow] 0 and one has [alpha] [less than or equal to] 0, which is a contradiction.

Case (ii). Assume that {[x.sub.n]} is bounded. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Choose a sequence [{[[bar.s].sub.i]}.sup.[infinity].sub.i=1] such that [[bar.s].sub.i] [right arrow] [infinity] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [[xi].sub.i] be such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [[xi].sub.i] - 0 as i [right arrow] [infinity] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. From (2.11), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which yields on applying (2.9), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Taking the limit superior as i [right arrow] 0, we obtain [alpha] [less than or equal to] 0, which is a contradiction. Thus the proof of (i) is complete. (ii). Let [n.sub.1] be a positive integer such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for n [greater than or equal to] [n.sup.1]. Using (2.10), (2.8) becomes [DELTA][z.sub.n] [greater than or equal to] -[h.sub.n][x.sub.n-k] [greater than or equal to] 0, which implies [DELTA][z.sub.n] [greater than or equal to] 0, n > [n.sub.1]. Hence, [z.sub.n] is non decreasing for n [greater than or equal to] [n.sub.1]. If [z.sub.n] < 0 does not hold, then eventually [z.sub.n] > 0 and so there exist an integer [n.sub.2] [greater than or equal to] [n.sub.1] and [mu] > 0 such that [z.sub.n] [greater than or equal to] [mu] for n [greater than or equal to] [n.sub.2],

[x.sub.n] [greater than or equal to] [mu] + [c.sub.n][x.sub.n-r] + [n- 1.summation over (s=n-k+l)][q.sub.s][x.sub.s-l] + [n-t+k-l-1.summation over (s=n)][p.sub.s][x.sub.s-k] - [[infinity].summation over (s=n)][e.sub.s], n [greater than or equal to] [n.sub.2]. (2.12)

We consider the following two possible cases.

Case(i). Assume that {[x.sub.n]} is unbounded, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, there exists a sequence of points [{[s.sub.i]}.sup.[infinity].sub.i=1] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as From (2.9) and (2.12), we find that [mu] [less than or equal to] 0, which is a contradiction.

Case (ii). Assume that {[x.sub.n]} is bounded. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Choose a sequence [{[[bar.s].sub.i]}.sup.[infnity].sub.i=1] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [[xi].sub.i] be such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [xi]i [right arrow] [infinity] as i [right arrow] [infinity] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. From (2.9) and (2.12), we get [mu] [less than or equal to] 0, which is a contradiction. Thus the proof of (ii) is complete.

Lemma 2.2. Assume that (1.6) holds, and there exists an integer t [member of] {0, 1,...,k - l} such that

[c.sub.n] + [n-1.summation over (s=n-t)][q.sub.s] + [n-t+k-l-1.summation over (s=n)][p.sub.s] [greater than or equal to] 1 (2.13)

and [[infinity].summation over (s=n)][e.sub.s] is non decreasing and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.14)

Let {[z.sub.n]} be defined by (2.10). Then we have,

(i) if {[x.sub.n]} is an eventually positive solution of inequality (2.7) and the following second order difference inequality

[[DELTA].sup.2][y.sub.n] + [1/[m.sup.*.sub.t]][h.sub.n-t+k-l] [y.sub.n] [less than or equal to] 0 (2.15)

has no eventually positive solution, then eventually [DELTA][z.sub.n] [less than or equal to] 0 and [z.sub.n] < 0 for large n.

(ii) if {[x.sub.n]} is an eventually negative solution of inequality (2.8) and the following second order difference inequality

[[DELTA].sup.2][y.sub.n] + [1/[m.sup.*.sub.t]][h.sub.n-t+k-l][y.sub.n] [greater than or equal to] 0 (2.16)

has no eventually negative solution, then eventually [DELTA][z.sub.n] [greater than or equal to] 0 and [z.sub.n] > 0 for large n.

Proof. (i). We give the proof only in the case when [q.sub.n] [not equivalent to] 0. From (2.7) and (2.10), we have eventually

[DELTA][z.sub.n] [less than or equal to] -[h.sub.n-t+k-l][x.sub.n-t-l] [less than or equal to] 0. (2.17)

Assume on the contrary, [z.sub.n] [greater than or equal to] 0. Let [n.sub.1] be a positive integer such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then for [n.sub.1] [less than or equal to] n [less than or equal to] [n.sub.1] + [m.sub.t], from (2.10) and (2.13) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In general, we get

[x.sub.n] [greater than or equal to] M - i[[infinity].summation over (s=n)][e.sub.s], [n.sub.1] + (i - 1)[m.sub.t] [less than or equal to] n [less than or equal to] [n.sub.1] + [im.sub.t], i [member of] N.

It follows from (2.14) that

[x.sub.n] [greater than or equal to] M, n [greater than or equal to] [n.sub.1] - [m.sup.*.sub.t]. (2.18)

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then there exist two possible cases.

Case (i). Assume that [beta] = 0. Let [n.sup.2] > [n.sub.1] be an integer such that [z.sub.n] < M/2 for n > [n.sub.2]. Then for any integer[n'.sub.2] > [n.sub.2], we have

[x.sub.n] [greater than or equal to] [1/[m.sup.*.sub.t]][n+[m.sup.*.sub.t]- 1.summation over (s=[n'.sub.2])][z.sub.s], [n'.sub.2] [less than or equal to] n [less than or equal to] [n'.sub.2] + [m.sup.*.sub.t]

Case (ii). Assume that [beta] > 0. We have [DELTA][z.sub.n] [less than or equal to] 0 for n [greater than or equal to] [n.sub.1], we have [z.sub.n] [greater than or equal to] [beta] for n [greater than or equal to] [n.sub.1]. From (2.10), (2.13) and (2.18), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By induction, we have

[x.sub.n] [greater than or equal to] i[beta] + M - i[[infinity].summation over (s=n)][e.sub.s], n [greater than or equal to] [n.sub.1] (i - 1) [m.sup.*.sub.t], i [member of] N

and so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which implies that there exist an integer [n.sub.3] [greater than or equal to] [n.sub.2] such that

[x.sub.n] [greater than or equal to] [1/[m.sup.*.sub.t]][n+[m.sup.*.sub.t]- 1.summation over (s=[n.sub.3])][z.sub.s], [n.sub.3] [less than or equal to] n [less than or equal to] [n.sub.3] + [m.sup.*.sub.t].

Combining the cases 1 and 2, we know that there is an integer N > [n.sub.2] such that

[x.sub.n] [greater than or equal to] [n+[m.sup.*.sub.t].summation over (s=N)][z.sub.s], N [less than or equal to] n [less than or equal to] N + [m.sup.*.sub.t]. (2.19)

For N + [m.sup.*.sub.t] < n [less than or equal to] N + [m.sup.*.sub.t] + [m.sub.t], by (2.10), (2.13) and (2.19), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By induction, we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since from (2.14), we obtain that [x.sub.n] [greater than or equal to] [1/[m.sup.*.sub.t][n+[m.sup.*.sub.t]-1.summation over (s=N)][z.sub.s], for n [greater than or equal to] N, which implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.20)

Let [y.sub.n] = [n-1.summation over (s=N)][z.sub.s], then [DELTA][y.sub.n] = [z.sub.n], [[DELTA].sup.2][y.sub.n] = [DELTA][z.sub.n], n [greater than or equal to] N. From (2.17) and (2.20), we obtain [DELTA][z.sub.n] + [h.sub.n-t+k-l][x.sub.n-t-l] [less than or equal to] 0

[[DELTA].sup.2][y.sub.n] + [1/[m.sup.*.sub.t]][h.sub.n-t+k-l][y.sub.n] [less than or equal to] 0, n [greater than or equal to] N + [m.sup.*.sub.t]. (2.21)

But every solution of (2.15) is oscillatory if and only if the inequality (2.21) has no eventually positive solution. Thus (2.21) contradicts the assumption that every solution of (2.15) oscillates. Therefore, the proof of (i) is complete.

(ii). Suppose that {[x.sub.n]} is an eventually negative solution of (2.8). Let {[z.sub.n]} be as in (2.10), then we have eventually

[DELTA][z.sub.n] [greater than or equal to] -[h.sub.n-t+k-l][x.sub.n-t-l] [greater than or equal to] 0. (2.22)

Assume on the contrary, [z.sub.n] < 0. Let [n.sub.4] be a positive integer such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then for [n.sub.4] [less than or equal to] n [less than or equal to] [n.sub.4] + [m.sub.t], from (2.10) and (2.13) we obtain

[x.sub.n] [less than or equal to] -M - [[infinity].summation over (s=n)][e.sub.s].

Therefore, similar to the proof of (i), one can obtain in general that

[x.sub.n] [less than or equal to] -M - i[[infinity].summation over (s=n)][e.sub.s], [n.sub.4] + (i - 1)[m.sub.t] [less than or equal to] n [less than or equal to] [n.sub.4] + [im.sub.t], i [member of] N,

Now, similar to proof of (i), one can obtain that there exists N [greater than or equal to] [n.sub.4] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [y.sub.n] = [n-1.summation over (s=N)][z.sub.s], then [DELTA][y.sub.n] = [z.sub.n], [[DELTA].sup.2][y.sub.n] = [DELTA][z.sub.n], n [greater than or equal to] N. Thus, we obtain

[[DELTA].sup.2][y.sub.n] + [1/[m.sup.*.sub.t]][h.sub.n-t+k-l][y.sub.n] [greater than or equal to] 0. (2.23)

But every solution of (2.16) is oscillatory if and only if the inequality (2.23) has no eventually negative solution. Thus (2.23) contradicts the assumption that every solution of (2.16) oscillates. Therefore, the proof of (ii) is complete.

3. Main Theorems

In this section, we will give sufficient conditions on oscillation of every solution of equation (1.1).

Theorem 3.1. Assume there exist two integers t,[t.sub.1] [member of]{0, 1,...,k - l} such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.24)

[R.sub.n](t) = [c.sub.n] + [n-1.summation over (s=n-t)][q.sub.s] + [n-t+l-l- 1.summation over (s=n)][p.sub.s] [greater than or equal to] 1. (3.25)

for all large n. Further assume that the inequality (2.15) does not have any eventually positive solution and the inequality (2.16) does not have any eventually negative solution. Then every solution of equation (1.1) oscillates.

Proof. In fact, if {[x.sub.n]} is a positive solution of equation (1.1), then Lemma 2.1 imply eventually that [z.sub.n] > 0, while Lemma 2.2 imply that [z.sub.n] < 0. This contradiction shows that {[x.sub.n]} cannot eventually be a positive solution of equation (1.1).

On the other hand, if {[x.sub.n]} is a negative solution of (1.1), then Lemma 2.1 implies that eventually [z.sub.n] < 0, while Lemma 2.2 implies that eventually [z.sub.n] > 0. This contradiction shows that {[x.sub.n]} cannot be a negative solution of equation (1.1). Therefore, every solution of equation equation (1.1) oscillates.

We remark that there are many sufficient conditions which guarantee the nonexistence of eventually positive solutions of (2.15). For instance, the following result is taken from . If {[d.sub.n]} is a nonnegative sequence and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.26)

then the following difference inequality

[[DELTA].sup.2][y.sub.n] + [d.sub.n][y.sub.n] [less than or equal to] 0 (3.27)

has no eventually positive solutions. Consequently, if (3.24) and (3.25) hold for t,[t.sub.1] and all large n and if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.28)

then every solution of (1.1) oscillates.

Theorem 3.2. Assume that (1.6),(2.14) and (3.28) hold. Further assume that {[q.sub.n]/([p.sub.n+k-l] - [q.sub.n])} and {[p.sub.n]/([p.sub.n] - [q.sub.n- k+l])} are nondecreasing and

[c.sub.n-k]([P.sub.n] - [q.sub.n-k+l]) [less than or equal to] [[lambda].sub.1](.[p.sub.n-r] - [q.sub.n-k+l-r]), (3.29)

[q.sub.n-k]([p.sub.n] - [q.sub.n-k+l]) [less than or equal to] [[lambda].sub.2]([P.sub.n-l] - [q.sub.n-k]), (3.30)

[p.sub.n]([P.sub.n+t+l] - [q.sub.n+t-k+2l]) [less than or equal to] [[lambda].sub.2]([p.sub.n] - [q.sub.n-k+l]), (3.31)

for large n and [[lambda].sub.1], [[lambda].sub.2] are nonnegative constants with [[lambda].sub.1] + [[lambda].sub.2] (k - l) = 1. Then every solution of equation (1.1) oscillates.

Proof.. If the above conclusion does not hold, equation(1.1) has an eventually positive solution {[x.sub.n]} and let [z.sub.n] be defined by (2.10). From Lemma 2.2, we have [DELTa][z.sub.n] [less than or equal to] 0 and [z.sub.n] < 0 eventually. From equation (2.10), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which implies {-[z.sub.n]} is a positive solution of the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which yields a contradiction by Lemmas 2.1 and 2.2. The proof is complete.

Theorem 3.3. Suppose there exists a number t [member of] {0, 1,...,k - l} such that (1.5)holds eventually. Suppose further that (3.28) holds and that

[c.sub.n-k]([p.sub.n] - [q.sub.n-k+l]) [greater than or equal to] ([P.sub.n-r] - - [q.sub.n-k+l-r]) (3.32)

for large n. Then every solution of equation (1.1) oscillates.

Proof. Suppose to contrary that {[x.sub.n]} is an eventually positive solution of (1.1). Then, in view of Lemma 2.1, we have [z.sub.n] > 0 and [DELTA][z.sub.n] [less than or equal to] 0 for large n. From equation (2.10), we obtain [DELTA][z.sub.n] = -[h.sub.n-t+k-l][x.sub.n-t-l]. Which implies

[DELTA][z.sub.n] + [h.sub.n-t+k-l]([z.sub.n-t-l] + [c.sub.n-t-l][x.sub.n-t-l-r]) [less than or equal to] 0 (3.33)

for large n. Using (3.32), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which gives {[z.sub.n]} is an eventually positive solution of the inequality

[DELTA]([z.sub.n] - [z.sub.n-r]) + [h.sub.n-t+k-l][z.sub.n-t-l] [less than or equal to] 0,

we have a contradiction by Lemmas 2.1 and 2.2. The proof is now complete.

If the assumption (3.32) is not satisfied, the following result is available. Theorem 3.4. Suppose there exists a number t [member of] {0, 1,...,k - l} such that (1.5) holds eventually. Further, assume that there exists a constant c [member of][0, 1) such that

[c.sub.n-t-l]([p.sub.n-t-l] - [q.sub.n-t-k]) [greater than or equal to] c([p.sub.n-t-l-r] - [q.sub.n-t-r-k]) (3.34)

for large n. Then every solution of equation (1.1) is oscillatory provided that thefollowing inequality

[DELTA][u.sub.n] + [c(1 - [c.sup.i+1])/1 - c][h.sub.n-t-l-r][u.sub.n-k-r] [less than or equal to] 0, n [member of] N(0) (3.35)

does not have an eventually positive solution.

Proof. Suppose to contrary that {[x.sub.n]} is an eventually positive solution of equation (1.1). Then, in view of Lemma 2.1, we have [z.sub.n] > 0 and [DELTA][z.sub.n] [less than or equal to] 0 for large n. From equation (2.10) [DELTA][z.sub.n] = -[h.sub.n-t+k-l][x.sub.n-t-l], proceeding as in the proof of Theorem 3.3, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.36)

Let [u.sub.n] = [z.sub.n] - [cz.sub.n-r]. Similar to the proof of Lemma 2.1, we have [u.sub.n] > 0 and [DELTA][u.sub.n] [less than or equal to] 0 for all large n. Hence, there exists an integer N > 0 such that [z.sub.n] > 0 and [DELTA][z.sub.n] [less than or equal to] 0 and [u.sub.n] > 0 and [DELTA][u.sub.n] [less than or equal to] 0 for n [greater than or equal to] N. Thus

[z.sub.n] = [u.sub.n] + [c.sub.zn-r]

= [u.sub.n] + [cu.sub.n-r] + [c.sup.2][z.sub.n-2r]

In general, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for n [greater than or equal to] (i + 1)r + N. From (3.36), we obtain

[DELTA][u.sub.n] + [c(1 - [c.sup.i+1])/1 - c][h.sub.n-t+k-l][u.sub.n-k-r] [less than or equal to] 0, n [member of](0)

holds for large n, which is contrary to the hypothesis that (3.35) has no eventually positive solution. The proof is now complete.

[FIGURE 1 OMITTED]

4. Numerical Simulations

In this section, we display a numerical example which supports our results in sections 2 and 3. To achieve this we use the MATLAB programming. Example 4.1. Consider the difference equation

[DELTA]([x.sub.n] - [1/2][x.sub.n-1]) + ([1/2][x.sub.n-2]) - [(1/2 - (n - 1)- [beta]).sup.xn-1] = [(-1).sup.n+1]/[square root of n], n > 0. (4.37)

Comparing with equation (1.1), we get r = 1, k = 2,l = 1, [c.sub.n] = 1/2, [p.sub.n] = 1/2, [q.sub.n] = 1/2 - [(n - 1).sup.-[beta]] and [e.sub.n] = [(-1).sup.n+1]/[square root of n]. When r = 1. [R.sub.n](1) = 1 - [(n - 2).sup.-[beta]] [less than or equal to] 0. When t = 0, we get [R.sup.n](0) = 1 and [h.sub.n] = [(n - 2).sup.-[beta]] [greater than or equal to] 0.

(i) Here [m.sup.*.sub.t] = 2. It is easy to see that all conditions of Theorem 3.1 are satisfied when -[infinity] < [beta] < 2 and [m.sup.*.sub.t] < 4. Thus every solution of equation (4.37) oscillates.

(ii) When [beta] - 2, a solution of equation (4.37) is non-oscillatory and negative.

(iii) Expanding spiral type trajectories indicates that the solution produces a rapid growth..

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

References

 R. P. Agarwal, S. R. Grace and Donald O'Regan, Oscillation Theory for Difference Equations and Differential Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, (2000).

 Chuan-Jun Tian, Sui Sun Cheng, Oscillation criteria for delay neutral difference equations with positive and negative coefficients, Bol.Soc.Paran.Mat, (3s) v.21 1/2 (2003); 1-12.

 X. H. Tang, J. S. Yu and D. H. Peng, Oscillation and non oscillation of neutral difference equations with positive and negative coefficients, Comput. Math. Applic., 39, (2000), 169-181.

 Z. G. Luo, J. H. Shen, X. Z. Liu, Oscillation criteria for a class of forced neutral equations, Dyn. Contin. Discrete Impuls. Syst. 7 (2000) 489-501.

 Ozkan Ocalan, Existence of positive solutions for a neutral differential equation with positive and negative coefficients. Applied Mathematics Letters 22(2009) 84-90.

 Ozkan Ocalan, Oscillation of forced neutral differential equations with positive and negative coefficients, Computers and Mathematics with Applications 54(2007) 1411-1421.

 M. M. S. Manuel, A. G. M. Selvam, M. P. Loganathan, Matlab Applications of the oscillations of the Forced Neutral Difference equations with positive and negative coefficients IJPAM 54, No. 4, 521-541 (2009).

S.Lourdu Marian Department of Master of Computer Applications, Saveetha Engineering College, Thandalam, Chennai -600 105.

e-mail: lmjerrome@rediffmail.com

M.Paul Loganathan Dravidian University, Kuppam, India. e-mail: paul.loganathan@gmail.com

A.George Maria Selvam Department of Mathematics, Sacred Heart College, Tirupattur-635 601, Vellore Dist, S.India. e-mail: agm_shc @ rediffmail.com