On a Class of Difference Equations with Interlacing Indices of the Fourth Order

The following class of nonlinear difference equations of the fourth order xn+1=axn−1+bxn−1xn−3cxn−1+dxn−3,n∈N0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x_{n+1}=ax_{n-1}+\frac{bx_{n-1}x_{n-3}}{cx_{n-1}+dx_{n-3}},\quad n\in { \mathbb{N}}_{0}, $$\end{document} where the parameters a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a$\end{document}, b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b$\end{document}, c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c$\end{document}, d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d$\end{document} and the initial values x−j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x_{-j}$\end{document}, j=0,3‾\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$j=\overline{0,3}$\end{document}, are positive real numbers, has been considered recently in this journal. Here we give a detailed analysis of the results and claims given therein, give many explanations and remarks related to the results and claims, explain some problems with some of the claims by providing suitable counterexamples, compare the results therein with some previous results in the literature, and present a global convergence result.


Introduction
The standard notations N, Z and R, are used for the sets of natural, integer, and real numbers, respectively. Let N k := {l ∈ Z : l ≥ k}, where k ∈ Z. By R + we denote the open interval (0, +∞). If p, q ∈ Z are such that p ≤ q, then the notation j = p, q matches with somewhat complex notation p ≤ j ≤ q, j ∈ Z.
We also understood that for any r ∈ Z, where c j is a sequence of real or complex numbers and j belongs to a subset of Z. Difference equations have been studied analytically since the beginning of the eighteenth century. One of the first studied problems was their solvability, that is, finding closed-form formulas for their solutions. Many results on the topic were obtained during the eighteenth century (see, e.g., [6,12] and the references therein, as well as some later presentations [4,10]). The closed-form formulas can be used in the study of the behavior of the solutions, which is a topic of a great interest. Results of this type can be found, for example, in [1,5,13,19]. They can be also used in some comparison results/principles dealing with difference inequalities (see, e.g., [3,14,15]). For some other recent results on solvability of difference equations and systems, and related topics, see, e.g., [8,9,[16][17][18][19][20][21] and the related references therein.
Some special cases of the difference equation x n = x n−k αx n−l + βx n−k−l γ x n−l + δx n−k−l , n∈ N 0 , where k, l ∈ N, x −j ∈ R, j = 1, k + l, α, β, γ , δ ∈ R, γ 2 + δ 2 = 0, have appeared in the literature from time to time. One of the special cases appeared, for example, in 1999 as Problem 1572 in the Mathematics Magazine, whereas another special case appeared in 2011 in the American Mathematical Monthly, Problem 11559. But we strongly believe that it is possible to find much older references in books or problem sections in some popular journals.
In our opinion equation (1) is a folklore thing ( [8]), since it is closely related to the bilinear difference equation which have been studied for a few centuries. Some information on the difference equation, as well as on some related ones can be found, for example, in [1,4,5,11,12,16,17]. The difference equation where a, b, c, d ∈ R + , x −j ∈ R + , j = 0, 3, can be found in recent literature in the area (see, e.g., [7]). Bearing in mind that equation (3) can be written in the form we see that it is the special case of equation (1) where the indices and parameters are given by The fact that equation (3) is a special case of a difference equation which should be a matter of folklore suggests that all the results in recent literature on the equation should be known for a long time.
Here we give a detailed analysis of the results and claims presented in [7], give many explanations related to the results and claims given therein, present some counterexamples for the wrong claims, compare the results with the results in the literature and show that some of the results follow from known ones, and complement the previous investigations by giving a global convergence result for positive solutions to equation (3).

On Some Local and Global Stability Results on equation (3)
In this section we show that the results on local stability and global attractivity in [7] are not correct.
We begin with some discussion on the claims on the equilibria of equation (3) stated therein. First, it is tried therein to determine the equilibria of the equation. Letx be one of them. Then it has to satisfy the relation Then it is claimed that it must bex It is assumed that from which and (6) it is concluded thatx However, if relation (8) is used in (5), it is obtained a quotient of the form 0 0 , which is not defined. The contradiction occurs because the relations in (5) and (6) are not equivalent (the relation in (5) is not defined forx = 0, whereas the relation in (6) is defined for allx ∈ R, including the valuex = 0).
In this way is obtained thatx = 0 is a unique equilibrium of equation (3), under the assumption in (7). This conclusion is used in [7] from this point onward. But, in fact, as has just been shown, equation (3) does not have any equilibrium under the assumption, so the conclusion is wrong.
On the other hand, if and from relation (6) we see that any point different from zero is an equilibrium of equation (3), showing the existence of many such points. This fact was not noticed in [7]. Now we turn to the claim on local asymptotic stability stated in [7]. Having dealt with the equilibria of equation (3), the linearized equation of (3) about the equilibriumx is found therein. Bearing in mind that the partial derivatives of the function but under the assumptionx = 0.
Note that the partial derivatives are not defined atx = 0. This means that equation (11) is not the linearized equation of equation (3) about the pointx = 0, what is claimed in [7]. The following claim on local asymptotic stability is Theorem 1 therein.
Then the equilibrium point of (3) is locally asymptotically stable.
Bearing in mind that the equilibrium point in [7] is the wrong pointx = 0, for which equation (11) is not the linearized equation about the point, and since the proof of the claim therein uses the linearized equation, we see that the claim makes no sense.

Remark 1
Claim 1 only makes sense if it is applied to an equilibriumx = 0, which was not the case in [7].

Remark 2
It should be also said that even in the case of an equilibriumx = 0, the results of this type are trivial. Namely, condition (12) is immediately obtained from the condition which is a special case of the well-known sufficient condition where k ∈ N 0 , for the stability of the linear difference equation with constant coefficients y n+1 + a k y n + · · · + a 0 y n−k = 0, for n ∈ N l for some l ∈ Z. Indeed, since min{a, b, c, d} > 0, a simple calculation shows that condition (13) becomes which directly leads to (12). Let us also note that it becomes a recent habit to formulate direct consequences of this well-known result for guarantying local stability of an equilibrium in many papers, but ignoring the other, usually more complex, cases when the condition (14) is not satisfied. Now we turn to the global attractivity result in [7]. Before this, note that well-defined solutions to equation (3) satisfy the condition Indeed, if (x n ) n∈N −3 is a well-defined solution to equation (3) and for some n 0 ∈ N −1 , and then we have The relation is not possible, since then x n 0 +2 will not be defined. Therefore, condition (16) must hold for such a solution. From (15) and (17) it follows that From (18) along with the relation we see that x n 0 +4 is not defined, contradicting the assumption that (x n ) n∈N −3 is a well-defined solution to equation (3).
Without loss of generality we may also assume that Otherwise, we can consider the equation with shifted indices. Therefore the investigation of equation (3) can be reduced to solutions satisfying the condition The main result in [7] should have been the following claim (see Theorem 2 therein).

Remark 3
Note that, as we have shown,x = 0 is not an equilibrium point of equation (3), implying that the claim refers to a wrong equilibrium. Now we show that there are special cases of equation (3) satisfying the conditions in Claim 2, with solutions not converging to any equilibriumx of equation (3), by giving some concrete examples of such difference equations. To do this, we use the fact that equation (3) is solvable in closed form [8]. The fact is a consequence of the solvability of equation (2), which has been known since the end of the eighteenth century ( [12]). Some methods for solving equation (2) and representations of the general solution of the equation can be found, e.g., in [1,4,5,11,12,16,17].
for n ∈ N 0 . First, note that in this case so that equation (20) satisfies condition (7). Using the method of mathematical induction together with recursive relation (20) it is easily shown that the assumption for m ∈ N 0 , j = −1, 0.
Recall that λ 1 and λ 2 are the zeros of the characteristic polynomial associated to equation (25).
Now note that from (22) it follows that Thus, for such chosen initial values the corresponding solutions of equation (20) are unbounded.

Remark 4
To get unbounded solutions to equation (20) it is enough to choose the initial values x j and x j −2 to be positive and to satisfy condition (30) only for one of the values j ∈ {−1, 0}, since for such chosen value of j then will hold relation (31), from which together with relation (32) we will have that one of the subsequences (x 2m ) m∈N −1 , (x 2m−1 ) m∈N −1 is unbounded. This is, among other things, connected to the form of equation (20) (moreover, of equation (3)) what will be discussed in the following section. (20) is only one of many difference equations of the form in (3) with unbounded solutions. In fact, it is one of the typical special cases of the equation. The reason for this is that the general solution to equation (3) is, among other things, determined by the values of the zeros of the corresponding characteristic polynomial (in Example 1 it is polynomial (28)), which is obtained during the process of finding the general solution to the equation conducted in the above example. This means, that many other examples of the equations with unbound solutions can be found in this way. It is not difficult to see that the zeros of the characteristic polynomial and the limits of the subsequences (y 2m+j ) m∈N 0 , j = −1, 0, depend continuously on the values of parameters a, b, c, d, implying that the difference equations obtained from equation (20) with small perturbations of the parameters have also unbounded solutions.

On the Closed-Form Formulas in [7]
Closed-form formulas for solutions to the following special cases of equation (3) x n+1 =x n−1 + x n−1 x n−3 were given in [7]. Paper [7] claims that solutions to equation (34) are given by the formulas for m ∈ N 0 , where the sequences (A n ) n∈N −1 and (B n ) n∈N −1 are the solutions to the equation with the initial values and respectively, that the solutions to equation (35) are given by the formulas for m ∈ N 0 , that the solutions to equation (36) are given by the formulas for m ∈ N 0 , and that the solutions to equation (37) are given by the formulas for m ∈ N 0 . Formulas (38), (39), (43)-(46) were proved by induction, whereas for the formulas in (47)-(52) was stated that they are proved similarly.
Here we show that all the closed-form formulas follow form some previous ones in the literature. Before this we explain the notion of difference equations with interlacing indices [18,20].
The following equation where l ∈ N 0 and k, s ∈ N, is the difference equation with interlacing indices. Let If we use the notation where j ∈ {0, 1, . . . , k}, then equation (53) consists of k + 1 difference equations of the form for j = 0, k, implying that (x (j ) m ) m∈N 0 , j = 0, k, are the k + 1 solutions to the difference equation with the initial values x (j ) −i , i = 1, s, for a fixed j ∈ {0, 1, . . . , k}, respectively.

Remark 6
Note that since equation (53) consists of k + 1 copies of equation (58), it is completely determined by equation (58). From this it follows that if we know some properties of solutions to equation (58), then we immediately know the properties of solutions to equation (53). This means, that it almost makes no sense to study such difference equations. However, they can be useful in getting some counterexamples as it was the case in [9]. Now note that (3) is a special case of equation (53). Indeed, the equation consists of two copies of the equation where a, b, c, d ∈ R + , x −j ∈ R + , j = 0, 1. Equation (59) was previously considered in: E M. Elsayed, Indag. Mathem. (N.S.), 19 (2), 189-201, where among other things, were considered the following difference equations It is claimed therein that solutions to equation (60) are given by the formula for n ∈ N 0 , where (A n ) n∈N −1 and (B n ) n∈N −1 are the solutions to equation (40) with the initial values in (41) and (42), respectively, that the solutions to equation (61) are given by the formulas that the solutions to equation (62) are given by the formula and that the solutions to equation (63) are given by the formulas for n ∈ N 0 . Since equation (3) is a difference equation with interlacing indices which consists of two copies of equation (59), it follows that each of the equations in (34), (35), (36) and (37), consists of two copies of the equations (60), (61), (62) and (63), respectively. This implies that formula (64) can be directly applied to the sequences x 2n−1 and x 2n to get formulas (38) and (39), that formulas (65) and (66) can be directly applied to the sequences x 2n−1 and x 2n to get formulas (43)-(46), that formula (67) can be directly applied to the sequences x 2n−1 and x 2n to get formulas (47) and (48), and that formulas (68) and (69) can be directly applied to the sequences x 2n−1 and x 2n to get formulas (49)-(52). This shows that all the closed-form formulas in [7] easily follow form some results in the previous literature.

Remark 7
It seems that many authors are not aware of the obvious problem with difference equations with interlacing indices, whereas some seem ignore the problem for some reasons. In the last two decades have appeared many papers of this type. We noted the problem a long time ago, discussed it at some talks and conferences, and later presented concrete examples in some of our papers such as it was the case in [18] and [20].

Remark 8
Note that [7] does not give any theoretical explanation for the closed-form formulas therein. Some methods for solving the difference equation where α, β, γ , δ ∈ R, γ 2 + δ 2 = 0, f is a homeomorphism of R such that f (0) = 0, can be found in [21]. For some basics on homeomorphisms on R see, for example, [22].

Global Convergence of Positive Solutions to equation (3)
Beside above mentioned claims and closed-form formulas, in [7] was also proved the following result on the boundedness.
Unlike the previous claims and results, Theorem 1 is correct. However, the theorem is trivial, bearing in mind that it is an immediate consequence of the obvious estimate condition (71), and positivity of the sequence (x n ) n∈N −3 . Some other basic results on boundedness and their applications can be found, e.g., in [2,14,15] (see also the related references therein). Now let us turn to Claim 1 again. Althoughx = 0 is not an equilibrium of equation (3), some of its solutions still can converge to zero. An interesting question is if the claim can be modified so that it makes sense. An answer to the question gives the following theorem.
Since (x n ) n∈N −3 is an arbitrary positive solution to equation (3), the theorem follows.