ASYMPTOTIC BEHAVIOR OF A CLASS OF NONLINEAR DIFFERENCE EQUATIONS

Motivated by some results of L. Berg (2002), in this paper we find the second member in the asymptotic development of some of the positive solutions of a class of difference equations of second and third orders. The main result in this paper partially solves an open problem by S. Stević (2003), and it is applied to some classes of mathematical biology models, for example, generalized Beverton-Holt stock recruitment model, flour beetle population model, mosquito population equations, and discrete delay logistic difference equation.


Introduction
Recently there has been a great interest in studying nonlinear difference equations and systems (cf.and the references therein).One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models describing real-life situations in population biology, economy, probability theory, genetics, psychology, sociology, and so forth.Such equations also appear naturally as discrete analogues of differential equations which model various biological and economical systems (see, e.g, [7, 13, 16-18, 20, 24, 28, 29] and the references therein).
The following theorem was established in [21,Problem I,page 174].The proof appearing there is attributed to Jacobsthal.
Theorem 1.1.Let f : (0,α) → (0,α), where α > 0, be a continuous function such that 0 < f (x) < x for every x ∈ (0,α) and f (x) = x − ax k + bx k+p + o(x k+p ), when x → +0, where k > 1, p, a, and b are positive numbers.Let x 0 ∈ (0,α) and Our version of this proof is somewhat different to its original form, and the idea and structure of this version was the starting point and inspiration for our further investigations, see [23].In [24] we have noted, by an example, that there are sequences defined by difference equations of second order which behave similarly to those sequences in Theorem 1.1.Motivated by the example in [29], we have found the asymptotics of all positive solutions of a nonlinear difference equation of second order.
A natural question is to find the first term in the asymptotic behavior of the solutions of (1.3)-(1.5)for a + b = 1 and of (1.6) when α + β = 1.
In [29,32] we have generalized Theorem 1.1 in the case of sequences defined by a difference equations of order more than one, where among other results we completely described the asymptotic behavior of solutions of (1.3)-(1.6) in these cases.
Stevo Stević 3 In [32] we have proved the following theorem.
Then the sequence defined by with initial conditions x 0 ,x 1 ,...,x k−1 ∈ [0,α) satisfies the following asymptotic formula: where From the proof of Theorem 1.2, it is easy to see that condition (b) can be replaced by the following condition: , where m > 1, p 1 ,..., p k ∈ [0,1), and K m (z 1 ,...,z k ) is a homogeneous polynomial of order m.Remark 1.4.Note that if p 1 ,..., p k ∈ (0,1), then condition (d) is automatically satisfied, since the partial derivatives of the function f are positive in a neighborhood of the origin.
Equation (1.4).If a,b ∈ (0,1) and a + b = 1, then Equation (1.5).If a,b ∈ (0,1) and a + b = 1, then (1.12) Equation (1.6).If α,β ∈ (0,1) and α + β = 1, then For the class of difference equations of first order defined in Theorem 1.1, we described in [23], under some additional conditions, a method for finding the next members in asymptotic developments of its solutions.It is a natural question to find the other members in the asymptotic development of the solutions of (1.3)-(1.6).Hence in [32] we have offered the following problem.
Open problem 1.5.Find the second member in the asymptotic development of the solutions of (1.3)-(1.6),that is, find the sequence y n such that The main object in the paper is to find the second member in the asymptotic development of some of the positive solutions of a class of nonlinear difference equations generalizing (1.2)-(1.6).

The inclusion theorem
In order to prove the open problem, we need the following result.(2.1) Let further x n be a solution of the following difference equation: such that ) Proof.We prove the theorem by induction.If n = n 0 + k, then using (2.1) and the monotonicity of the function f we have that is, (2.6) Stevo Stević 5 Assume now that it has been proved that where m ≥ n 0 + k, then using again (2.1), induction hypothesis, and the monotonicity of the function f we have that is, y m+1 ≤ x m+1 ≤ z m+1 , finishing the proof.
Remark 2.2.Some other results of this type and their applications can be found, for example, in [2-6, 34, 35].
Remark 2.3.It suffices that the hypotheses concerning f are satisfied in the strip (2.4).

Main result
In this section, we formulate and prove the main result in this paper.As a consequence of this result, we find the second member in the asymptotics of some positive solutions of (1.3)-(1.6).
(a) For k = 2 and p ∈ (0,1), there is a positive solution of (1.7) with the following asymptotics: as n → ∞, where rxy + sy 2 is a positive definite form, and K 3 (x, y) is a homogeneous polynomial in variables x and y of third order.(b) For k = 3 and p, q, p + q ∈ (0,1), there is a positive solution of (1.7) with the following asymptotics: as n → ∞, where 6 Asymptotic behavior of some difference equations (3.6) K 2 (x, y,z)=rx 2 +sy 2 +tz 2 +uxy +vxz +wyz is a positive definite form, and K 3 (x, y) is a homogeneous polynomial in variables x, y, and z of third order.
Proof.In both cases, we suppose that there are solutions which have the following asymptotics: as n → ∞.
We show this by finding the values of the coefficients a and b.
(a) Let Choosing ) and b arbitrary, comparing coefficients in by some calculations, we obtain where and from (3.10), we get Stevo Stević 7 These relations show that inequalities (2.1) are satisfied for sufficiently large n, where f is defined in Theorem 3.1(a) and F is given by (3.8).Thus, since for sufficiently large n we can chose b arbitrary close to b 0 , in view of Theorem 2.1, it follows that in this case there is a solution of (1.7) which has asymptotics (3.4).(b) Let F(x, y,z,w) = x − f (y,z,w).(3.14) Choosing a = (3 − q − 2p)/(K 2 (1,1,1)) and b arbitrary, comparing coefficients in the DERIVE system yields where (3.17 (3.18) These relations show that inequalities (2.1) are satisfied for sufficiently large n, where f is given in Theorem 3.1(b) and F is given by (3.14).Applying Theorem 2.1, we obtain that in this case there is a solution of (1.7) which has asymptotics (3.4).

Case of discrete delay logistic difference equation.
In the case of a general k, we only consider the discrete delay logistic difference equation (1.2) with α = 1, that is, From [29, Theorem 2], it follows that We assume that (3.19) has a positive solution x n with the asymptotics and also that β = 1, otherwise we consider the sequence y n = βx n .Let Then by well-known asymptotics formulae, we have Now note that from this and by Theorem 2.1 we could conclude that there is a positive solution of (3.19) with the following asymptotics: as n → ∞, if the monotonicity conditions were not violated.Hence, for the readers interested in this research area, we leave the following conjecture.

Stevo Stević 9
Remark 3.3.It is interesting that Conjecture 3.2 cannot be confirmed also by [4,Theorem 2.1] (see also [5]), since for the case of (3.19), two coefficients with the largest moduli in [4, formula (2.3)] have the same moduli so that [4, condition (2.4)] is not satisfied.We would like to point out that [4, Theorem 2.1] was applied with a success at many points, for example, in [4,34].

Theorem 2 . 1 .
Let f : I k → I be a continuous and nondecreasing function in each argument on the interval I ⊆ R, and let (y n ) and (z n ) be sequences in I, with y n < z n for n ≥ n 0 and such that y n+1 ≤ f y n ,..., y n−k+1 , f z n ,...,z n−k+1 ≤ z n+1 , n ≥ n 0 + k − 1.

. 11 )
Let b 1 and b 2 be such that b 1 > b 0 and b 2 < b 0 .With the notations

) Let b 1
and b 2 be such that b 1 > b 0 and b 2 < b 0 .With the notations (3.12) and from (3.16), we get