Note on difference equations with the right-hand side function nonincreasing in each variable

We present an example of a difference equation of arbitrary order, possessing the right-hand side function that is homogeneous to a certain degree and nonincreasing in each variable, which has a unique positive equilibrium, as well as solutions that do not converge to the equilibrium. The example shows that the main result in the paper: O. Moaaz, Dynamics of difference equation xn+1 = f (xn–l , xn–k) (Adv. Differ. Equ. 2018:447, 2018), is incorrect.


Introduction
Let, as usual, N be the set of positive integers, Z the set of integers, R the set of reals, and C the set of the complex numbers. By N k , where k ∈ Z, we denote the set of all j ∈ Z such that j ≥ k. If p, q ∈ Z and such that p ≤ q, then the notation j = p, q means that j takes the values of all integers between p and q (including p and q).
Of the many classes of difference equations and systems of difference equations solvable in closed form, here we mention an important class, which is used in this note. In addition, we also mention a simple method for getting a sequence of difference equations or systems of difference equations from a given one, which can sometimes be used to obtain some counterexamples in the theory of difference equations.

Product-type difference equations
Some recent investigations on solvability are devoted to product-type difference equations and systems of difference equations, or to some difference equations and systems that can be reduced to them using some suitable transformations (see, e.g., [22][23][24] and the related references therein).
General product-type difference equation is where the sequences (a n ) n∈N 0 , (α (j) n ) n∈N 0 , j = 0, k -1, as well as the initial values x j , j = 0, k -1, are real or complex.
In some cases, equation (1) can be solved by taking the logarithm, but in the case of complex initial values x j , j = 0, k -1, coefficients and exponents, some other methods can be used (see, e.g., [22][23][24] and the related reference therein). Since the equation is related to the general linear difference equation, it is of great importance.

Difference equations with interlacing indices
There is a simple method enabling to construct a family of 'cloned' difference equations from a given one. The equations are called the difference equations with interlacing indices [27]. The equations appear from time to time in the literature, and it seems that some authors are not aware that they are obtained by cloning some simpler difference equations (see the examples and analyses conducted in [25][26][27][28]). However, the equations can be useful in providing some counterexamples in the theory of difference equations (see, e.g., [6]). Systems of difference equations with interlacing indices can also be constructed by the cloning method. Now we briefly describe the method.
The general form of the difference equation with interlacing indices is the following where l ∈ N, and k ≥ 2.
If we define k sets of indices by we obviously have then from (2), we have This means that the sequences (y with unrelated initial values. Using this procedure in the reverse direction, from equation (3), one can obtain a family of cloned equations in (2), i.e., a family of difference equations with interlacing indices.

Global convergence results and a claim
One of the main problems in the theory of difference equations is the convergence of the solutions to the equations. There are many results on convergence in the literature, some of which can be found in the literature mentioned above.
The following difference equation where k, m ∈ N 0 , has been recently studied in [14].
The following claim is the main result concerning the difference equations appearing in [14] (Theorem 3.3 therein).
Theorem A Assume that f has non-positive partial derivatives and is homogeneous with degree s. Then equation (4) has a unique positive equilibrium x * , and every solution to the equation converges to x * .
In this note, we show that the claim in Theorem A is not true by giving an example of equation (4) such that the function f is homogeneous (with a degree to be chosen appropriately) and has non-positive partial derivatives, and equation (4) has a unique positive equilibrium x * and solutions that do not converge to the equilibrium.

A counterexample to Theorem A
In this section, we give a counterexample to Theorem A. To construct the counterexample, we are looking for a difference equation belonging to the above classes of equations; that is, we are looking for a product-type difference equation with interlacing indices that is solvable in the closed form.
Example 1 Consider the difference equation where k ∈ N, and α > 1, which is a special case of the general difference equation of higher order where k ∈ N, and f is an arbitrary function.
Here we are interested in positive solutions to equation (5). Hence, we assume that For the case of equation (5), we have that is the right-hand side function, which generates their solutions along with the initial values. Since ∂f ∂t j (t 1 , . . . , t k ) = 0, j = 1, k -1 and ∂f ∂t k (t 1 , . . . , t k ) = -α t α+1 k < 0, when t k > 0, we have that the function (7) has non-positive partial derivatives on the set (0, +∞) k , from which we choose our initial values.
On the other hand, since holds for every λ > 0, we see that the function defined in (7) is homogeneous with degree -α.
Further, if is an equilibrium solution to equation (5), then it must be from which it immediately follows that x * = 1, which means that equation (5) has a unique positive equilibrium. Now note that equation (5) is a difference equation with interlacing indices, which is obtained by cloning the following product-type difference equation of first order that is, equation (5) consists of k copies of equation (8) with initial values not connected to each other. Equation (8) obviously has the same equilibrium. Hence, to show that the claim of Theorem A is not true, it is enough to show that equation (8) has solutions, which do not converge to the equilibrium. Now note that by iterating equation (8), we get and y 2n+3 = y α 2 2n+1 , for n ∈ N 0 . By a simple inductive argument from relations (9), (10), and equation (8) with n = 1, one can easily obtain y 2n = y α 2n 0 (11) and for n ∈ N 0 . Taking α > 1, we have Hence, if y 0 ∈ (0, 1), then letting n → +∞ in (11) and (12) and using (13), it follows that lim n→+∞ y 2n = 0 (14) and lim n→+∞ y 2n+1 = +∞.
The above analysis shows that the claim of Theorem A is not true.
Remark 1 Note that the only bounded positive solution to equation (5) is the one which is generated by the initial values The solution is, in fact, the equilibrium solution x n ≡ 1, n ∈ N 0 , which is easily proved using the initial values (18) in equation (5), together with a simple inductive argument.
Remark 2 The above consideration also shows that the equilibrium solution is the unique positive solution to equation (5) that converges, which indicates to what extent the claim in Theorem A fails.
Remark 3 Note also that the linearized equation associated with equation (8) about the equilibrium point is z n+1 = -αz n , n ∈ N 0 , from which, along with the assumption α > 1, it follows that the equilibrium is a repeller. Note that from (19), we have |z n | = α n |z 0 |, n ∈ N 0 , from which together with (13), we see that for each z 0 = 0, the solution to (19) goes to infinity as n → +∞.
Given that equation (8) is solvable and has a closed-form formula for its solutions, from which their long-term behavior is easily described, linearization is not required. However, the linearization argument also suggests in which direction counterexamples should be sought.