Asymptotic behavior of solutions of a class of nonlinear difference systems.

1 Introduction

The problem of asymptotic behavior is one of the most important topic in the qualitative study of nonlinear scalars and systems of difference equations and has been the subject of many investigations. Recently, Li [5] and Agarwal, Li and Pang [1] studied a class of two-dimensional nonlinear difference systems ofthe form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

They provided a classification scheme for positive solutions of the above system and gave conditions for the existence of solutions with designated asymptotic properties. Moreover, Li and Raffoul [6] studied the classification and existence of positive solutions ofthe Volterra nonlinear difference system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this paper, we shall consider the general nonlinear difference system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where f, g : N([n.sub.0]) x R x R [right arrow] R, N([n.sub.0]) = {[n.sub.0], [n.sub.0] + 1, [n.sub.0] + 2, ...} ([n.sub.0] a nonnegative integer, x([n.sub.0]) = [x.sub.0], y([n.sub.0]) = [y.sub.0]), and [DELTA] denotes the forward difference operator, that is, [DELTA]x(n) = x(n +1) - x(n) for a sequence x(n). Next, we state the following definition.

Definition 1.1. We say that (1.1) has an asymptotic equilibrium if

(i) there exist [xi] [member of] R and r > 0 such that any solution x(n, [n.sub.0], [x.sub.0]) of (1.1) with [absolute value [x.sub.0]] < r satisfies

x(n) = [xi] + o(1) as n [right arrow] [infinity]; (1.2)

(ii) corresponding to each [xi] [member of] R, there is a solution of (1.1) satisfying (1.2).

2 Main Results

First, we need the following comparison principle for difference inequalities.

Lemma 2.1 (See [3]). Let [psi](n, r) be a nonnegative nondecreasingfunction in r for any fixed n [member of] N([n.sub.0]). Suppose that for any n [greater than or equal to] [n.sub.0], nonnegative functions u(n) and v(n) defined on N([n.sub.0]) satisfy the inequality

v(n) - [n-1.summation over (s = [n.sub.0])][psi](s, v(s)) < u(n) - [n-1.summation over (s=[n.sub.0])][psi](s, u(s)).

If v([n.sub.0]) < u([n.sub.0]), then v(n) < u(n) for all n [greater than or equal to] [n.sub.0].

Theorem 2.2. Assume that for n [member of] n([n.sub.0]), x, y [member of] r, and the functions f and g satisfy

[absolute value f(n, x, y)] [less than or equal to] V(n [absolute value x]) [absolute value g(n, y, x)] [less than or equal to] [psi](n, [absolute value y] (2.1)

where [psi]: N x [R.sup.+] [right arrow] [R.sup.+] is nondecreasing with respect to the second variable for each n G n([n.sub.0]). Also, suppose that each solution u(n) of the difference equation

[DELTA]u(n) = [psi](n,u(n)), u([n.sub.0]) = [u.sub.0], n [greater than or equal to] [n.sub.0] (2.2)

is bounded on n([n.sub.0]). Then any solution {(x(n), y(n))} of (1.1) with [square root of [[absolute value x].sup.2] + [[absolute value y].sup.2] < r for each n [greater than or equal to] [n.sub.0] satisfies the asymptotic property (1.2).

Proof. Let {(x(n), y(n))} be any solution of (1.1). Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Lemma 2.1, we obtain

[absolute value x(n)] < u(n), [absolute value y(n)] < u(n), n [greater than or equal to] [n.sub.0]. (2.3)

Since every solution u(n) of (2.2) is bounded on N([n.sub.0]), it follows from (2.3) that the solution {(x(n), y(n))} of (1.1) is also bounded on n([n.sub.0]). Furthermore, for any n > m [greater than or equal to] [n.sub.0], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

Since every solution u(n) of (2.2) is nondecreasing and bounded on N([n.sub.0]), this implies that, given any [epsilon] > 0, we can choose a T > 0 sufficiently large so that 0 [less than or equal to] u(n) - u(m) < [epsilon] for all n [greater than or equal to] T. It then follows from (2.4) that [absolute value x(n) - x(m)] < [epsilon], and [absolute value y(n) - y(m)] < [epsilon] for all n [greater than or equal to] m [greater than or equal to] T, which shows that the solution {(x(n), y(n))} of (1.1) converges to {([[xi].sub.1], [[xi].sub.2]} as n [right arrow] [infinity]. This completes the proof.

Theorem 2.3. Let the assumptions of Theorem 2.2 hold. In addition, assume that f(n, x, y) and g(n, y, x) are continuous in x and y for any fixed n [member of] N([n.sub.0]) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then for each [rho] > 0 there is an [n.sub.0] large enough such thatfor every [[xi].sub.i] [member of] R with [absolute value [[xi].sub.i]] < [rho], there exists a solution {(x(n), y(n))} of (1.1) tending to {(-1, -2)} as n [right arrow] [infinity].

Proof. Let [[xi].sub.i] [member of] R with 2[absolute value of [[xi].sub.i]] < [rho]. Define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also, we definite the operator S by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then we can show that the following properties (i)-(iii) hold, which are needed in the application of S. S. Cheng and W. T. Patula [2].

(i) There exists an [n.sub.0] such that S maps [B.sub.[rho]] into itself. In fact, from the proof of Theorem 2.2, we see

[n-1.summation over (s=[n.sub.0])][psi](s, u(s)) [less than or equal to] u(n) - [absolute value [x.sup.0]]

This means [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is possible since u(n) is convergent. Therefore, we take [n.sub.0] large enough so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then (x, y) [member of] [B.sub.[rho]] implies that for n [greater than or equal to] [n.sub.0],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

(ii) The operator S is continuous. Let (x, y) [member of] [B.sub.[rho]], and [{[x.sub.i]}.sup.[infinity].sub.=0], [{[y.sub.i]}.sup.[infinity].sub.=0] be arbitrary sequences of elements of [B.sub.[rho]] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since the solutions [bar.u](n) and u(n) of (2.2) are convergent for any [epsilon] > 0, we can choose [n.sub.1] [member of] N([n.sub.0]) large enough that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

from which, as a result of the continuity of f, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Showing that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is similar and hence we omit it. Hence, S is continuous.

(iii) [SB.sub.[rho]] is relatively compact. It suffices to show that S[B.sub.[rho]] is bounded and equiconvergentto ([[xi].sub.1], [[xi].sub.2]). Since [absolute value x([n.sub.0])] [less than or equal to] [u.sub.0] [less than or equal to] [rho], [absolute value y([n.sub.0])] [less than or equal to] [u.sub.0] [less than or equal to] [rho] for any (x, y) [member of] [B.sub.[rho]]. We have

[absolute value Sx(n)] [less than or equal to] [absolute value [[xi].sub.1]] + [[infinity].summation over (s=[n.sub.0]) [psi](s, u(s)) < [infinity]

and

[absolute value Sx(n)] [less than or equal to] [absolute value [[xi].sub.2]] + [[infinity].summation over (s=[n.sub.0]) [psi](s, u(s)) < [infinity]

Therefore, the set [SB.sub.p] is a uniformly bounded subset of the Banach Space. Moreover, it is equiconvergent to ([[xi].sub.1], [[xi].sub.2]), since for every [epsilon] > 0, there exists a [n.sub.2] = [n.sub.2]([[xi].sub.1], [[xi].sub.2]) such that

[absolute value Sx(n)] [less than or equal to] [absolute value [[xi].sub.1]] + [[infinity].summation over (s=[n.sub.2]) [psi](s, u(s)) < [epsilon]

and

[absolute value Sx(n)] [less than or equal to] [absolute value [[xi].sub.2]] + [[infinity].summation over (s=[n.sub.2]) [psi](s, u(s)) < [epsilon]

for every n [greater than or equal to] [n.sub.2] and all (x, y) [member of] [B.sub.[rho]]. Thus, S[B.sub.[rho]] is relatively compact.

Therefore, by S. S. Cheng and W. T. Patula's fixed-point theorem, there exists (x, y) [member of] [B.sub.[rho]] such that S(x, y) = (x, y). That is, there exists a solution {(x(n), y(n))} of (2.5). Obviously, {(x(n), y(n))} is a solution of the problem (1.1). The proof is complete.

In the following paragraphs, we consider the asymptotic behavior of (1.1) under the following conditions (H):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where f, g : N([n.sub.0]) x R [right arrow] R, f(n, u, v) > 0,g(n, u, v) > 0 for u, v > 0. [[lambda].sub.i](n) and [r.sub.i](n) are two sequences of nonnegative real numbers, and [[omega].sub.i](s) are nondecreasing and strictly positive for s [greater than or equal to] [n.sub.0].

Let F be the set of all sequences and define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

([alpha], [beta] are two positive numbers). Now we have the following result. Theorem 2.4. Suppose the conditions (H). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

hold for some positive constants c and d if and only if any solutions {(x(n), y(n))} [member of] Q of (1.1) belongs to the set K([alpha], [beta]).

Proof. Let {(x(n), y(n))} be a solution in [OMEGA] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, there exists a integer N > 0 and two positive constants, namely, c and d such that d [less than or equal to] x(n) [less than or equal to] [alpha], c [less than or equal to] y(n) [less than or equal to] [beta] for n > N. From system (1.1) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Conversely, suppose that A < [infinity] and B < [infinity]. First notice that for n > [n.sub.0] the first equation of (1.1) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In a similar fashion, we obtain from the second equation of (1.1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Next, we can choose an integer N large enough so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let x be the Banach space of all bounded real valued sequences [{(x(n), y(n))}.sup.[infinity].sub.n=N] with the norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and with the usual pointwise ordering [less than or equal to]. Define a subset [OMEGA] of X by

[OMEGA] = j(x(n),y(n)) [member of] X : c/2 [less than or equal to] x(n) [less than or equal to] c, d/2 [less than or equal to] y(n) [less than or equal to] d, n [greater than or equal to] N}.

It is clear that any subset b of Q, we have inf B [member of] [OMEGA] and sup B [member of] [OMEGA]. Define the operator E: [OMEGA] [right arrow] X by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

First, we claim that E maps [OMEGA] into [OMEGA]. To see this we let x(n) [member of] [OMEGA]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Showing that, for y [member of] [OMEGA], [d/2] [less than or equal to] (Ey)(n) [less than or equal to] d is similar and hence we omit it. It is clear from the definition of the operator E that E is decreasing. Thus, by Knaster's fixed point theorem [4], we conclude that there exists (x, y) [member of] [OMEGA] such that (x, y) = E(x, y), i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In view of the comparison principle

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore, {(x(n), y(n))} is a positive solution of (1.1) which belongs to K([alpha], [beta]). This completes the proof.

Acknowledgement

The second author acknowledges support by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011-A25-007-Z), and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of the Ministry of Education of China.

References

[1] Ravi P. Agarwal, Wan-Tong Li, and Peter Y. H. Pang. Asymptotic behavior of nonlinear difference systems. Appl. Math. Comput., 140(2-3):307-316, 2003.

[2] Sui Sun Cheng and William T. Patula. An existence theorem for a nonlinear difference equation. Nonlinear Anal., 20(3):193-203, 1993.

[3] Sung Kyu Choi, Nam Jip Koo, and Hyun Sook Ryu. Asymptotic equivalence between two difference systems. Comput. Math. Appl., 45(6-9):1327-1337, 2003. Advances in difference equations, IV.

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

[6] Wan-Tong Li and Youssef N. Raffoul. Classification and existence of positive solutions of systems of Volterra nonlinear difference equations. Appl. Math. Comput., 155(2):469-478, 2004.

Xinyuan Liao

University of South China, School of Mathematics and Physics Hengyang, Hunan 421001, China xinyuanliao98@yahoo.com.cn

Wan-Tong Li

Lanzhou University, Department of Mathematics Lanzhou, Gansu, 730000, China wtli@lzu.edu.cn

Youssef N. Raffoul

University of Dayton, Department of Mathematics Dayton, OH 45469-2316, U.S.A. youssef.raffoul@notes.udayton.edu

Received February 13, 2009; Accepted December 4, 2009 Communicated by Martin Bohner