On monotone solutions of some classes of difference equations, Discrete Dyn

We describe a method for finding monotone solutions of some
classes of difference equations converging to the corresponding
equilibria. The method enables us to confirm three conjectures
posed by the present author in a talk, which are extensions of
three conjectures by M. R. S. Kulenovic and G. Ladas,
Dynamics of Second Order Rational Difference Equations.
With Open Problems and Conjectures. Chapman and Hall/CRC, 2002.
It is interesting that the method, in some cases, can be applied
also when the parameters are variable.


Introduction
Recently there has been a great interest in studying nonlinear difference equations of order greater than one.Many of these equations stem from mathematical biology, economy, population dynamics , and so forth (see, e.g., [5, 7-9, 11, 14] and the references therein).An interesting problem in the theory of difference equations is finding monotone solutions.This paper is devoted to this problem.
Motivated by [8,Conjectures 5.4.6 and 6.10.3] in a talk (see, [16]) we posed the following three conjectures.The first one concerns a generalization of (1.2).
In [6] DeVault et al. investigate the behavior of the positive solutions of the difference equation , n = 0,1,..., (1.2) where p > 0 and k ∈ N is fixed.Among other things they have proved that all nonoscillatory solutions of (1.2) converge to the positive equilibrium x = p + 1.
Based on this observation they have posed the following open problem.
Our aim in this paper is to confirm the above mentioned conjectures.The linearized equation for (1.1), respectively, (1.3) and (1.4), about the corresponding positive equilibrium xi , i ∈ {1, 2,3}, is The characteristic polynomial associated with (1.5), respectively, (1.6) and (1.7), is , it follows that for each p > −1, there is a unique positive root t 1 of the polynomial (1.8) belonging to the interval (0,1).

Stevo Stević 3
Similarly, it can be shown that (1.9) and (1.10) have also a unique positive roots t 2 and t 3 in the interval (0,1).
We solve the open problem, showing that such solutions exist, developing Berg's idea in [2] which are based on asymptotics.Asymptotics for solutions of difference equations has been investigated for a long time by L. Berg and S. Stević, see, for example, [1][2][3][4][10][11][12][13][14][15] and the reference therein.We solve it by constructing two appropriate sequences y n and z n with for sufficiently large n.In [1,2], some methods can be found for the construction of these bounds, see, also [3,4].
From (1.11) and results in Berg's paper [2], we expect that for k ≥ 2 such solutions have the first four members in their asymptotics in the following form: (1.13)

The inclusion theorem
We need the following result in the proof of the main theorem.The proof of the result is similar to that of [2, Theorem 1].
Theorem 2.1.Let f : I k+2 → I be a continuous and nondecreasing function in each argument on the interval I ⊂ R, and let (y n ) and (z n ) be sequences with y n < z n for n ≥ n 0 and such that ) Then there is a solution of the following difference equation: with property (1.12) for n ≥ n 0 .
Proof.Let N be an arbitrary integer such that N > n 0 + k − 1.The solution (x n ) of (2.2) with given initial values x N ,x N+1 ,...,x N+k satisfying (1.12) for n ∈ {N ,N + 1,...,N + k} can be continued by (2.2) to all n < N. Inequalities (2.1) and the monotonic character of f imply that (1.12) holds for all n ∈ {n 0 ,...,N + k}.Let A N be the set of all (k + 1)tuples (x n0 ,...,x n0+k ) such that there exist solutions (x n ) of (2.2) with these initial values satisfying (1.12) for all n ∈ {n 0 ,...,N + k}.It is clear that A N is a closed nonempty set for every N > n 0 + k − 1, and that A N+1 ⊂ A N .It follows that the set A = ∩ ∞ n=n0+k A N is a nonempty subset of R k+1 and that if (x n0 ,...,x n0+k ) ∈ A, then the corresponding solutions of (2.2) satisfy (1.12) for all n ≥ n 0 , as desired.

The main result
In this section we prove the main result of this paper, which confirms Conjectures 1.1, 1.3, and 1.4.
Theorem 3.1.The following statements are true:

1) has a positive solution which remains above the equilibrium x1
3) has a nontrivial positive solution which decreases to the equilibrium x2 ; Then (1.4) has a nontrivial positive solution which decreases to the equilibrium x3 = √ α.
With the notations we get (3.10) These relations show that the inequalities in (1.12) are satisfied for sufficiently large n, where f = F + x n−k and F is given by (3.1).Applying Theorem 2.1 it follows that there is a solution of (1.1) with the asymptotics x n = ϕ n + o t 2n 1 , in particular, the solution of (1.1) converges monotonically to the positive equilibrium x1 = p + 1, when p > −1 and n ≥ n 0 .Hence, the solution x n+n0+k converges monotonically for n ≥ −k.
With the notations These relations show that the inequalities in (1.12) are satisfied for sufficiently large n, where f = F + x n−k and F is given by (3.11).Applying Theorem 2.1 it follows that there is a solution of (1.3) with the asymptotics x n = ϕ n + o(t 2n 2 ).This solution obviously converges monotonically to the positive equilibrium x2 = ( √ 5 + 1)/2, for n ≥ n 1 .A suitable shift of x n is decreasing for all n ≥ −k.
With the notations we get F y n−k ,..., y n , y n+1 ∼ H t3 q 5 t 2n 3 > 0, F z n−k ,...,z n ,z n+1 ∼ H t3 q 6 t 2n 3 < 0. (3.28) These relations show that the inequalities in (1.12) are satisfied for sufficiently large n, where f = F + x n−k and F is given by (3.20).Hence, there is a solution of (1.4) with the asymptotics x n = ϕ n + o t 2n 3 .The result follows similarly to the above mentioned cases.
Remark 3.4.Note that using (1.13) better asymptotics for these solutions can be obtained, that is, x n = ϕ n + o(t 3n i ), i ∈ {1, 2,3}, where b is given by (3.4), (3.14), or (3.24), and c can be found equating to zero the coefficient nearby t 3n .Remark 3.5.From the proof of Theorem 3.1, we see that we can assume that the parameter p in (1.1) can be replaced by a nondecreasing sequence with the following asymptotics: p n = p + o(t 2n 1 ).