Oscillatory behaviour of a class of nonlinear systems of first-order difference equations.

1 Introduction

Consider the nonlinear system of first-order difference equations of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where a(n), b(n), c(n) and d(n) are real-valued functions defined for n [member of] N([n.sub.0]) = {[n.sub.0], [n.sub.0] + 1, [n.sub.0] + 2, * * * }, [n.sub.0] [greater than or equal to] 0 such that a(n)d(n) - b(n)c(n) [not equal to] 0 and f, g [member of] C (R, R) with uf (u) > 0, ug(u) > 0 for u [not equal to] 0.

By a solution of (1.1) we mean a real-valued vector function X (n) = [[x(n),y(n)].sup.T] which satisfies (1.1), for n [greater than or equal to] [n.sub.0]. We say that the solution X(n) oscillates componentwise or simply oscillates if each component oscillates. Otherwise, X(n) is nonoscillatory. Therefore, a solution of (1.1) is nonoscillatory if it has a component which is eventually positive or eventually negative. A solution x(n) of (1.1) is said to be disconjugate on [[n.sub.0], [infinity]) if each of the components is disconjugate on [[n.sub.0], [infinity]).

During the last several years, oscillatory, nonoscillatory and asymptotic behaviour of solutions of nonlinear/linear systems of first-order difference equations have been studied extensively and many interesting results have appeared in the literature, see e.g., [1-6]. A close observation reveals that almost all works in this direction are the discrete analogues of differential systems. It seems that not much work has been done on the matrix equation (1.1).

In a recent paper [7], the author has studied the linear system

X (n + 1) = A(n)X

and its associated nonhomogeneous system of equations

X (n + 1) = A(n)X + H (n),

where A(n) is the coefficient matrix of (1.1) and H(n) = [[[h.sub.1](n), [h.sub.2](n)].sup.T], and obtained some oscillation results. It is interesting to note that such results are not the discrete analogues of differential systems. Motivated by the object of the work in [7], an attempt is made here to study the oscillatory behaviour of solutions of the system of equations (1.1) and its corresponding nonhomogeneous system of equations

X(n + 1) = A(n)F(X (n)) + H(n), (1.2)

where F(X(n)) = [[f(x(n)), g(y(n))].sup.T].

In [3, 5, 6], authors have studied the oscillatory and asymptotic behaviour of the vector solutions ofthe two dimensional matrix equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

where {[a.sub.n]} and {[b.sub.n]} are sequences of reals and f,g [member of] C (R, R) with uf (u) > 0, ug(u) > 0 for u [not equal to] 0. Keeping in view the discrete analogue of the two dimensional matrix differential equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

they have obtained similar results for (1.1). However, the purpose of this work is altogether in a different approach.

2 Oscillation of System (1.1)

This section deals with the study of the oscillatory behaviour of solutions of the matrix difference equation (1.1). We use the notation X(n) as any solution of (1.1) such that X(n) = [[x(n),y(n)].sup.T] defined for n [greater than or equal to] [n.sub.0] [greater than or equal to] 0.

Lemma 2.1. Assume that a(n) [equivalent to] 0 [equivalent to] d(n) for all n. If b(n)c(n) > 0, then x(n) is nonoscillatory if and only if y(n) is nonoscillatory, where x(n) and y(n) are the components of the vector X (n).

Proof. The system (1.1) can be written as

x(n +1) = b(n)g(y(n)), (2.1)

y(n +1) = c(n)f (x(n)). (2.2)

Let x(n) be the nonoscillatory component of X(n). Without loss of generality, we may assume that x(n) > 0 for n [greater than or equal to] [n.sub.0]. Hence (2.2) yields that y(n + 1) > 0 for n [greater than or equal to] [n.sub.0]. On the other hand, Equ. (2.1) implies that y(n) > 0 for n [greater than or equal to] [n.sub.0]. Therefore y(n) > 0 for n [greater than or equal to] [n.sub.0] when we consider the case b(n) > 0 and c(n) > 0. It is easy to verify the above fact when we consider the case b(n) < 0 and c(n) < 0 for n [greater than or equal to] [n.sub.0]. The converse part can be dealt with similarly. The proof is complete. []

Now it is immediate to prove the following lemma:

Lemma 2.2. Assume that all the conditions of Lemma 2.1 hold. Then x(n) is oscillatory ifand only ify( n) is oscillatory.

Lemma 2.3. Let a(n) [less than or equal to] 0 and d(n) [less than or equal to] 0 for n [greater than or equal to] [n.sub.0]. Then for any b(n) [not equal to] 0 and c(n) [not equal to] 0, x(n) is oscillatory if and only if y(n) is oscillatory.

Proof. Let x(n) be the oscillatory component of X(n) defined for n [greater than or equal to] [n.sub.0]. We claim that y(n) is oscillatory. If not, there exists [n.sub.1] > [n.sub.0] such that y(n) > 0 or < 0 for n [greater than or equal to] [n.sub.1]. Assume that y(n) > 0 for n [greater than or equal to] [n.sub.1]. Then

0 < y(n + 1) - d(n)g(y(n)) = c(n)f(x(n))

implies that x(n) is nonoscillatory for any c(n) [not equal to] 0, a contradiction. So our claim holds. The case y(n) < 0 for n [greater than or equal to] [n.sub.1] is similar. Next, we assume that y(n) is oscillatory. If x(n) is nonoscillatory for n [greater than or equal to] [n.sub.1] > [n.sub.0], then

0 < x(n + 1) - a(n)f(x(n)) = b(n)g(y(n))

implies that y(n) is nonoscillatory for any b(n) [not equal to] 0, a contradiction. Hence the lemma is proved. []

Lemma 2.4. Assume that any one of the conditions

(a) a(n) [less than or equal to] 0, b(n) [less than or equal to] 0, c(n) [less than or equal to] 0 and d(n) [greater than or equal to] 0,

(b) a(n) [less than or equal to] 0' b(n) [greater than or equal to] 0' c(n) [greater than or equal to] 0 and d(n) [greater than or equal to] 0

holds, where a(n) [not equivalent to] 0 [not equivalent to] b(n) and c(n) [not equivalent to] 0 [not equivalent to] d(n). If x(n) is nonoscillatory, then y(n) is nonoscillatory.

Proof. Suppose that x(n) is a nonoscillatory component of X(n). Without loss of generality, we may assume that x(n) > 0 for n [greater than or equal to] [n.sub.0]. Then

0 < x(n + 1) - a(n)f (x(n)) = b(n)g(y(n))

implies that b(n) and g(y(n)) are of same sign for n [greater than or equal to] [n.sub.0]. If b(n) [greater than or equal to] 0 for all n, then

y(n +1) = c(n)f(x(n)) + d(n)g(y(n)) [greater than or equal to] 0,

if c(n) [greater than or equal to] 0 and d(n) [greater than or equal to] 0 and hence y(n) is a nonoscillatory component of X(n). If b(n) [less than or equal to] 0 for all n, then

y(n + 1) = c(n)f(x(n)) + d(n)g(y(n)) [less than or equal to] 0

when c(n) [less than or equal to] 0 and d(n) [greater than or equal to] 0, that is, y(n) is a nonoscillatory component of X(n). This completes the proof. []

Lemma 2.5. Assume that any one of the conditions

(a) d(n) [less than or equal to] 0, a(n) [greater than or equal to] 0, b(n) [less than or equal to] 0 and c(n) [less than or equal to] 0,

(b) d(n) [less than or equal to] 0, a(n) [greater than or equal to] 0, b(n) [greater than or equal to] 0 and c(n) [greater than or equal to] 0

holds, where a(n) [not equivalent to] 0 [not equivalent to] b(n) and c(n) [not equivalent to] 0 [not equivalent to] d(n). If y(n) is nonoscillatory, then x(n) is nonoscillatory.

Proof. The proof follows from Lemma 2.4. Hence the details are omitted. []

Remark 2.6. When Lemma 2.4(a) holds, then the nonoscillatory solutions X(n) = [[x(n),y(n)].sup.T] lie in the second or fourth open quadrant and when Lemma 2.4(b) or Lemma 2.5(b) holds, then the nonoscillatory solutions X(n) lie in the first or third open quadrant.

Theorem 2.7. Let b(n) [equivalent to] 0 [equivalent to] c(n) for all n. Then the following statements hold.

(i) If a(n) > 0 and d(n) > 0 for n [greater than or equal to] 0, then the system (1.1) is disconjugate on [0, [infinity]).

(ii) If a(n) < 0 and d(n) < 0 for n [greater than or equal to] [n.sub.0], then the system (1.1) is oscillatory.

(iii) If a(n) and d(n) changes sign, then the vector solution of (1.1) is oscillatory.

Proof. We first prove (i). The system (1.1) can be written as

x(n + 1) = a(n)f (x(n)), (2.3) y(n +1) = d(n)g(y(n)). (2.4)

Let k [member of] [0, [infinity]) = {0, 1, 2, 3, * * *} be a generalized zero of a solution x(n) of (2.3). If k = 0, then x(0) = 0 and hence x(n) is a trivial solution of (2.3). Assume that k [member of] (0, [infinity]). If x(k) = 0, then x(n) = 0 for n [greater than or equal to] k, that is, x(n) is a trivial solution of (2.3). If x(k) = 0, then x(k - 1)x(k) < 0. If we consider x(k) > 0, then x(k - 1) < 0. Ultimately,

0 < x(k) = a(k - 1)f (x(k - 1)) < 0,

which is absurd. Altering the sign of x(k) and x(k - 1), we have the same observation. Hence x(n) has no generalized zero in [0, [infinity]), that is, (2.3) is disconjugate on [0, [infinity]). Proceeding as above, we can show that a solution y(n) of (2.4) has no generalized zero on [0, [infinity]). Thus the system (1.1) is disconjugate on [0, [infinity]).

Now we show (ii). It is enough to show that (2.3) and (2.4) are oscillatory. Let x(n) be a nonoscillatory solution of (2.3). Without loss of generality, we may assume that x(n) > 0 for n [greater than or equal to] [n.sub.0]. Then

0 < x(n +1) = a(n)f(x(n)) < 0,

a contradiction. Hence (2.3) is oscillatory. Similarly, we can show that (2.4) is oscillatory. Therefore, the vector solution X(n) = [[x(n), y(n)].sup.T] is oscillatory.

Finally, we prove (iii). Without loss of generality, let us assume that x(n) is a nonoscillatory solution of (2.3) such that x(n) > 0 for n [greater than or equal to] [n.sub.0]. Since a(n) changes sign, there exists k [greater than or equal to] [n.sub.1] [greater than or equal to] [n.sub.0] + 1 such that a(k - 1)a(k) < 0. If a(k) < 0, then

0 < x(k + 1) = a(k)f(x(k)) < 0,

a contradiction. If a(k) > 0, then a(k - 1) < 0 and hence

0 < x(k) = a(k - 1 )f(x(k - 1)) < 0,

a contradiction. Thus (2.3) is oscillatory. Proceeding as above, we can show that (2.4) is oscillatory. Consequently, the vector solution X(n) of (1.1) is oscillatory. []

Corollary 2.8. If a(n)d(n) < 0, then the system (1.1) is nonoscillatory. []

Proof. The proof follows from the Theorem 2.7. Hence the details are omitted. =]

Theorem 2.9. Let b(n) [equivalent to] 0 = [equivalent to] for all n. Assume that

([H.sub.1]) f(uv) = f(u)f(v), g(uv) = g(u)g(v) for u, v [member of] R.

Furthermore, assume that x(0) [not equal to] [x.sub.0] [not equal to] 0 = [y.sub.0] = y(0). Then any vector solution X(n) of(1.1) is oscillatory if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are oscillatory, where [f.sup.n](x) = f [f.sup.n-1](x) and [g.sup.n](x) = g([g.sup.n-1](x)) for n [greater than or equal to] 1.

Proof. Using ([H.sub.1]), it is easy to verify that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are the solutions of (2.3) and (2.4) respectively. Hence the proof follows. []

Theorem 2.10. Let a(n) [equivalent to] 0 [equivalent to] d(n) for all n. If b(n)c(n) > 0, then (1.1) is either oscillatory or nonoscillatory.

Proof. From Lemma 2.1, it follows that the components of X(n) are either oscillatory or nonoscillatory and hence any solution X(n) of (1.1) is either oscillatory or nonoscillatory. This completes the proof. []

When a(n) [equivalent to] 0 [equivalent to] d(n) for all n, then the system (1.1) can be written as

x(n +1) = b(n)g(y(n)), (2.5)

y(n +1) = c(n)f(x(n)). (2.6)

Using (2.6), (2.5) yields that

x(n + 2) = b(n + 1)g[c(n)f(x(n))], (2.7)

and using (2.5), (2.6) becomes

y(n + 2) = c(n + 1)f[b(n)g(y(n))]. (2.8)

Theorem 2.11. Let a(n) [equivalent to] 0 [equivalent to] d(n) for all n. Assume that b(n)c(n) < 0 for every n [greater than or equal to] [n.sub.0] > 0 such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Furthermore, suppose that there exist constants M > 0 and N > 0 such that

([H.sub.2]) [f(x)/x] [greater than or equal to] M, [g(x)/x] [greater than or equal to] N, for x [member of] R.

Then the system (1.1) is oscillatory.

Proof. In order to prove that the system (1.1) is oscillatory, it is sufficient to show that the nonlinear second-order difference equations (2.7) and (2.8) are oscillatory. Suppose on the contrary that, x(n) is a nonoscillatory solution of (2.7) such that x(n) > 0 for n [greater than or equal to] [n.sub.0]. Without loss of generality, we may assume that b(n) < 0 and c(n) > 0 for all n [greater than or equal to] [n.sub.0]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Using ([H.sub.2]), (2.7) becomes

x(n + 2) [less than or equal to] MNb(n + 1)c(n)x(n)

for n [greater than or equal to] [n.sub.1] > [n.sub.0]. Hence for n [greater than or equal to] [n.sub.1],

[x(n+2)/x(n)] [less than or equal to] MNb(n +1)c(n),

that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then (2.9) yields that

[[lambda].sup.2] - MN[alpha][beta] [less than or equal to] 0.

Let F([lambda]) = [[lambda].sup.2] - MN[alpha][beta]. Clearly, F([lambda]) attains minimum at [lambda] = 0. Consequently, min F([lambda]) [less than or equal to] F([lambda]) implies that MN[alpha][beta] [greater than or equal to] 0, a contradiction. The proof is similar if we assume that x(n) < 0 for n [greater than or equal to] [n.sub.0]. Hence (2.7) is oscillatory. Proceeding as above, we can show that (2.8) is oscillatory. Hence the system (1.1) is oscillatory. This completes the proof. []

Remark 2.12. Theorem 2.11 generalizes [7, Theorem 2.4].

Remark 2.13. The prototype of f and g satisfying ([H.sub.2]) could be of the type

f(u) = (M + a[[absolute value of u].sup.[lambda]])[absolute value of u]sgnu

and

g(v) = (N + b[[absolute value of v].sup.[mu]])[absolute value of v]sgnv,

where M > 0, N > 0, a [greater than or equal to] 0, b [greater than or equal to] 0, [lambda] [greater than or equal to] 0 and [mu] [greater than or equal to] 0.

Example 2.14. Consider the system of equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

where

f(x(n)) = ([3/16] + [1/16][[absolute value of x(n)].sup.[lambda]]) [absolute value of x(n)]sgnx

and

f(y(n)) = ([2/9] + [1/9][[absolute value of y(n)].sup.[mu]]) [absolute value of y(n)]sgny

Clearly, M = 3/16 and N = 2/9 and (2.10) satisfies all the conditions of Theorem 2.11.

Hence (2.10) is oscillatory. In particular, X(n) = [[(-1).sup.n], [[(-1).sup.n]].sup.T] is such an oscillatory solution of (2.10).

Theorem 2.15. If a(n) [less than or equal to] 0, d(n) [less than or equal to] 0 and b(n) [not equal to] 0 [not equal to] c(n) for every n [greater than or equal to] [n.sub.0], then every vector solution of (1.1) is oscillatory.

Proof. The proof is an immediate consequence of Lemma 2.3, and hence the details are omitted. []

Example 2.16. Consider the system of equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

where -4 [less than or equal to] a(n) [less than or equal to] -2, -5 [less than or equal to] d(n) [less than or equal to] -3, 1 [less than or equal to] b(n) [less than or equal to] 3, 2 [less than or equal to] c(n) [less than or equal to] 4, f(x(n)) = [x.sup.3](n) and g(y(n)) = [y.sup.5](n). Applying Theorem 2.15, every vector solution of (2.11) is oscillatory. In particular, X(n) = [[[(-1).sup.n], [(-1).sup.n]].sup.T] is such an oscillatory solution of (2.11).

Theorem 2.17. Assume that a(n) [less than or equal to] 0, b(n) [less than or equal to] 0, c(n) [less than or equal to] 0 and d(n) [less than or equal to] 0 for all n [greater than or equal to] [n.sub.0]. If one of the conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

holds, then the system (1.1) is oscillatory.

Proof. Let X(n) = [[x(n),y(n)].sup.T] be the nonoscillatory solution of (1.1), for n [greater than or equal to] [n.sub.0], where at least one vector component is nonoscillatory. Assume that x(n) > 0 for n [greater than or equal to] [n.sub.0]. Then y(n) could be nonoscillatory or oscillatory for n [greater than or equal to] [n.sub.0]. Let y(n) < 0 for n [greater than or equal to] [n.sub.0]. Clearly,

x(n + 1) - a(n)f(x(n)) = b(n)g(y(n)) (2.12)

implies that

-a(n)f(x(n)) < b(n)g(y(n)), n [greater than or equal to] [n.sub.0].

Consequently,

y(n + 1) = c(n)f(x(n)) + d(n)g(y(n))

yields that

y(n +1) >(d(n) - [b(n)c(n)/a(n)]) g(y(n)),

that is,

0 < [y(n +1)/g(y(n))] < [detA(n)/a(n)],

a contradiction to ([H.sub.3]). If y(n) > 0 or y(n) is oscillatory for n [greater than or equal to] [n.sub.0], then (2.12) gives a contradiction.

Next, we assume that x(n) < 0 for n [greater than or equal to] [n.sub.0]. We verify the above three cases for y(n). If y(n) > 0 for n [greater than or equal to] [n.sub.0]. then

0 < y(n + 1) - d(n)g(y(n)) = c(n)f(x(n))

implies that,

-d(n)g(y(n)) < c(n)f(x(n)), n [greater than or equal to] [n.sub.0].

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

yields that

0 < [x(n + 1)/f(x(n))] < [detA(n)/d(n)],

a contradiction to ([H.sub.4]). If y(n) < 0 or y(n) is oscillatory for n [greater than or equal to] [n.sub.0], then (2.12) gives a contradiction. Hence x(n) is oscillatory.

If our supposition is such that y(n) is nonoscillatory, then we can proceed as above to find similar contradictions keeping in view the three cases of x(n). Hence the proof is complete. []

Theorem 2.18. Suppose that b(n)c(n) [greater than or equal to] 0, a(n)d(n) [less than or equal to] 0 and detA(n) [not equal to] 0 for all n [greater than or equal to] [n.sub.0]. Then there exists a nonoscillatory solution to the system (1.1).

Proof. The proof follows from Lemma 2.4 and Lemma 2.5. Hence the details are omitted. []

Example 2.19. Theorem 2.17 is applicable to the system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where f (x(n)) = [x.sup.3](n) and g(y(n)) = [y.sup.3](n). Clearly, detA(n) = [1/150][15 + 11[(-1).sup.n]] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

implies that the above system is oscillatory and in particular, X(n) = [[[(-1).sup.n], [(-1).sup.n]].sup.T] is such an oscillatory solution.

3 Oscillation of System (1.2)

In this section, sufficient conditions are obtained for the oscillation of the vector solutions ofthe system ofdifference equations (1.2).

Theorem 3.1. Assume that b(n) [equivalent to] 0 [equivalent to] c(n) for all n.

(i) If a(n) < 0, d(n) < 0 and [h.sub.i](n) changes sign for i = 1, 2, then (1.2) is oscillatory.

(ii) If a(n) > 0, d(n) > 0 and [h.sub.i](n) [greater than or equal to] 0 for i = 1, 2, then there exists a nonoscillatory solution to (1.2).

Proof. We first prove (i). Suppose on the contrary that (1.2) has a nonoscillatory solution X(n) = [[x(n), y(n)].sup.T] for n [greater than or equal to] [n.sub.0], where the components are the solutions of the equations

x(n + 1) = a(n)f(x(n)) + [h.sub.1](n), (3.1)

y(n +1) = d(n)g(y(n)) + [h.sub.2](n), (3.2)

respectively. If x(n) is a nonoscillatory solution of (3.1), then for n [greater than or equal to] [n.sub.0],

[h.sub.1](n) = x(n + 1) - a(n)f (x(n)) > 0 or < 0,

depending on x(n) whether x(n) > 0 or < 0, a contradiction. Similarly, if y(n) is a nonoscillatory solution of (3.2), then for n [greater than or equal to] [n.sub.0],

[h.sub.2](n) = y(n + 1) - d(n)g(y(n)) > 0 or < 0,

depending on y(n) whether y(n) > 0 or < 0, a contradiction. Hence X(n) is oscillatory.

Now we show (ii). For any x(0) = [x.sub.0] [not equal to] 0 and y(0) = [y.sub.0] [not equal to] 0, it is easy to verify that x(n) and y(n) are the nonoscillatory solutions of (3.1) and (3.2) respectively. []

Theorem 3.2. Let a(n) [equivalent to] 0 [equivalent to] d(n) for all n. Assume that b(n)c(n) < 0 for every n [greater than or equal to] [n.sub.0] > 0 such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [h.sub.i](n), i = 1, 2, n [greater than or equal to] [n.sub.0] is eventually of one sign such that

c(n + 1)f([h.sub.1](n)) + [h.sub.2](n + 1) [less than or equal to] 0, b(n + 1)g([h.sub.2](n)) + [h.sub.1](n + 1) [greater than or equal to] 0].

Furthermore, suppose that ([H.sub.2]) and

([H.sub.5]) there exist [lambda], [mu] > 0 such that f(x) + f(y) [greater than or equal to] [lambda]f (x + y), g(x) + g(y) [greater than or equal to] [mu]g(x + y) hold for all x, y [member of] R.

Then every solution of (1.2) oscillates.

Proof. When a(n) [equivalent to] 0 [equivalent to] d(n) for all n, then (1.2) becomes

x(n + 1) = b(n)g(y(n)) + [h.sub.1](n), y(n + 1) = c(n)f(x(n)) + [h.sub.2](n).

It is easy to verify that x(n) and y(n) are the solutions of the nonlinear second-order difference equations

x(n + 2) = b(n + 1)g[c(n)f(x(n)) + [h.sub.2](n)] + [h.sub.1](n + 1), (3.3) y(n + 2) = c(n + 1)f [b(n)g(y(n)) + [h.sub.1](n)] + [h.sub.2](n + 1), (3.4)

respectively. Suppose on the contrary that X(n) = [[x(n), y(n)].sup.T] is a nonoscillatory solution of (1.2). Without loss of generality, we may assume that x(n) > 0 for n [greater than or equal to] [n.sub.0]. Then x(n) is a solution of(3.3). Using ([H.sub.5]) and then ([H.sub.2]), (3.3) yields that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for n [less than or equal to] [n.sub.0]. Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and define f([sigma]) = [[sigma].sup.2] - [MN[alpha][beta]/[mu]]. Then f([sigma]) attains its minimum at [sigma] = 0, and hence min f([sigma]) [less than or equal to] f([sigma]) implies that-[MN[alpha][beta]/[mu]] [less than or equal to] 0, a contradiction. Ultimately, x(n) is oscillatory.

If y(n) is a nonoscillatory solution of (3.4), then using ([H.sub.4]) and ([H.sub.2]), we get

y(n + 2) [less than or equal to] [MN/[lambda]]c(n + 1)f(b(n))y(n) + c(n + 1)f([h.sub.1](n)) + [h.sub.2](n + 1),

for n [greater than or equal to] [n.sub.0]. Proceeding as above, we have a contradiction. Hence the proof is complete. []

Theorem 3.3. Let a(n) [equivalent to] 0 [equivalent to] d(n) for all n. Assume that f and g are Lipschitzian on a compact interval of the form [a,b], 0 < a < b < [infinity]. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

holds, then there exists a nonoscillatory vector solution to the system (1.2).

Proof. It is enough to show that either (3.3) or (3.4) has a nonoscillatory solution. As a result, it is easy to verify that if u(n) is a solution of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then u(n) is a solution of (3.3). This can be understood by writing (3.3) as

x(n + 2) - 2x(n + 1) + x(n) = x(n) - 2x(n + 1) + [h.sub.1](n + 1) + b(n +1)g[c(n)f(x(n)) + [h.sub.2](n)]

for all n. It is possible to find N > 0 large enough such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where K is the Lipschitz constant of g on [[1/2], [3/2]] and [K.sup.*] = max {[1/2], Kf (3/2)} due to ([H.sub.6]). Let X be the set of all bounded real-valued functions u(n) with sup norm defined by

[parallel]u[parallel] = sup{[absolute value of u(n)] : n [greater than or equal to] N}.

Then X isa Banach space. Define a subset Q of X as

[OMEGA] = {u [member of] X : [1/2] [less than or equal to] u(n) [less than or equal to] [3/2], n [greater than or equal to] N}.

Clearly, [OMEGA] is abounded, convex and closed subset of X. Furthermore, define an operator T : [OMEGA] [right arrow] X by putting [(Tu).sub.n] equal to

1 + [infinity.summation over (i=n)](i - n + 1) [u(i) - 2u(i + 1) + hi(i + 1) + b(i + 1)g {c(i)f (u(i)) + [h.sub.2](i)}]

for n [greater than or equal to] N. The mapping T has the following properties. First of all, T maps [OMEGA] into [OMEGA]. Indeed, for u [member of] [OMEGA],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

implies that 1/2 [less than or equal to] [(Tu).sub.n] [less than or equal to] 3/2.

Next, we show that T is continuous. Let [u.sup.(k)] [member of] [OMEGA] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [OMEGA] is closed, then u [member of] [OMEGA] and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By the continuity of f and g and using Lebesgue's dominated convergence theorem, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus T is continuous.

In order to apply Schauder's fixed point theorem we need to show that T[OMEGA] is precompact. Let u [member of] [OMEGA] and m, n [greater than or equal to] N. Then for m > n

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

due to ([H.sub.6]). Hence T[OMEGA] is precompact.

By Schauder's fixed point theorem, there exists u [member of] [OMEGA] such that Tu = u. This completes the proof.

4 Summary

Systems of equations like (1.1) and (1.2) naturally apply to various fields of scientific endeavor, like biology (the study of competitive species in population dynamics), physics (the study of the motions of interacting bodies), the study of control systems, neurology and electricity. The results established here may be helpful to study specially the nonlinear biological systems for the discrete time intervals. Keeping in view the above fact, the present work may initiate further study in this direction.

Acknowledgment

The author is thankful to the referee for helpful remarks.

References

[1] Ravi P. Agarwal. Difference equations and inequalities, volume 228 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York, second edition, 2000. Theory, methods, and applications.

[2] Ravi P. Agarwal and Patricia J. Y. Wong. Advanced topics in difference equations, volume 404 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1997.

[3] John R. Graef and Ethiraju Thandapani. Oscillation of two-dimensional difference systems. Comput. Math. Appl., 38(7-8):157-165, 1999.

[4] Istvan Gyori and Gerasimos Ladas. Oscillation theory of delay differential equations. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1991. With applications, Oxford Science Publications.

[5] Jianchu Jiang and Xianhua Tang. Oscillation and asymptotic behavior of two-dimensional difference systems. Comput. Math. Appl., 54(9-10):1240-1249, 2007.

[6] Wan-Tong Li. Classification schemes for nonoscillatory solutions of two-dimensional nonlinear difference systems. Comput. Math. Appl., 42(3-5) 341-355, 2001. Advances in difference equations, III.

[7] Arun Kumar Tripathy. Oscillation criteria for first-order systems of linear difference equations. Electron. J. Differential Equations, pages No. 29, 11, 2009.

A. K. Tripathy

Kakatiya Institute of Technology and Science

Department of Mathematics

Warangal-506015, India

arun_tripathy70@rediffmail.com

Received October 5, 2009; Accepted November 24, 2009 Communicated by Allan Peterson