On a solvable class of nonlinear difference equations of fourth order

. We consider a class of nonlinear difference equations of the fourth order, which extends some equations in the literature. It is shown that the class of equations is solvable in closed form explaining theoretically, among other things, solvability of some previously considered very special cases. We also present some applications of the main theorem through two examples, which show that some results in the literature are not correct.


Introduction
Let N, Z, R stand for the sets of natural, integer and real numbers respectively, and let N k = {n ∈ Z : n ≥ k} where k ∈ Z. If l ∈ Z, then, as usual, we regard that ∏ l−1 j=l c j = 1. To obtain some information on solutions of difference equations and systems of difference equations scientists first tried to find some closed form formulas for their solutions. The first important results can be found, for example, in [7,10,11,17,18], as well as in the books [15,16] where many results up to the end of the eighteenth century can be found.
If a 2 = 4b, then the polynomial has two equal roots and the general solution to equation (1.1) in this case is given by the closed-form formula See [7], where, among other things, the method for finding solutions to linear homogeneous difference equations with constant coefficients of arbitrary order in the form x n = λ n , n ∈ N 0 , was described. Closed-form formulas for solutions to linear homogeneous difference equations with constant coefficients of the third order were presented by Euler in [11]. For some later presentations of results in the topic see, for example, the books [8,13,19,20,22].
One of the difference equations which, by using some changes of variables, reduces to equation (1.1) is  [1, 2, 6, 8, 9, 14-16, 19, 21, 22, 31, 32, 34, 35, 37]). Many other classes of difference equations can be reduced to linear difference equations with constant coefficients. It is of some interest to find such classes, as well as some which reduces to equation (1.4). By using some changes of variables it is easy to form many such classes.
There have been some investigations on solvability and behaviour of solutions to the difference equation where a, b, c, d ∈ R.
Here we show that a more general class of difference equations can be solved in closed form, extending some of the results on equation (1.5) in the literature. We use some methods and ideas related to the ones, e.g., in [12,31,32,34,37]. By using obtained closed-form formulas for solutions to equation (1.5), we present some applications of our main theorem by giving two examples which show that some results in [3] are not correct.

Main results
This section presents the main result in the paper. It shows the solvability of a generalization of equation (1.5), by finding closed form formulas for their solutions.
g is a strictly monotone and continuous function, g(R) = R and g(0) = 0. Then, the equation is solvable in closed form.
Proof. By a well known theorem in real analysis we see that the conditions posed on function g imply the existence of the inverse function g −1 which satisfies the same conditions as the function g ( [38]).
Assume that x n 0 = 0 for some n 0 ∈ N 0 . Then from (2.1) we have x n 0 +1 = 0. These facts along with (2.1) imply that x n 0 +4 is not defined. Thus, of interest are the solutions of equation (2.1) such that x n ̸ = 0, n ∈ N 0 . We may also assume that x −j ̸ = 0, j = 1, 3, otherwise the equation can be considered only on the domain N 0 . Hence, we suppose x n ̸ = 0, for n ∈ N −3 .
From above obtained closed-form formulas for solutions to equation (2.1) the theorem follows.

Some applications and discussions
A part of recent literature on difference equations contains many claims which are not established and/or explained. In some of our papers we discussed some aspects of the phenomena (see, e.g., [32][33][34]36]). Here we discuss some incorrect claims on long-term behaviour of solutions to equation (1.5) given in [3].
Note that equation (1.5) can be written in the form In [3] was first tried to find the equilibria of the equation. After some simple algebraic manipulations it was concluded thatx = 0 is a unique equilibrium point of equation (1.5), Assume thatx is an equilibrium of equation (1.5). Then it must satisfy the algebraic equation From (3.2) we see that it must bex This eliminates the possibilityx = 0.
and consequently Therefore, under the last condition anyx ̸ = 0 is an equilibrium of the difference equation. This means that the claim in [3, Theorem 1] that, under a condition, the zero equilibrium point of equation (1.5) is locally asymptotically stable is not correct, since it is not an equilibrium at all.
Further, Theorem 2 in [3] claims the following:  that is, the equation This is the equation (2.1) with g(x) = x, x ∈ R, The associated characteristic polynomial to the corresponding linear equation in (2.10) is and its roots are Since in this case we have Employing the formulas in (2.14)-(2.19), where g(x) = x, x ∈ R, and the coefficients α, β, γ, δ are as in (3.5), we have for m ∈ N 0 . Now note that lim m→+∞ x 0 , i = 0, 2. This means that the solution is not convergent, which is a counterexample to the claim in Theorem 3.1.
Bearing in mind that in [3] is stated that it considers equation (1.5) for the case when all the coefficients a, b, c and d are positive, and that one of the coefficients in Example 3.2 is negative (see (3.3)), in the following example we also give a counterexample to the statement in Theorem 3.1 for the case of positive coefficients. that is, the equation x n+1 = x n x n−2 + 2x n−3 x n−2 + x n−3 , n ∈ N 0 . (3.14) This is the equation (2.1) with g(x) = x, x ∈ R, α = γ = δ = 1, β = 2. Hence, the solutions are not convergent, which is a counterexample to the claim in Theorem 3.1 in the case min{a, b, c, d} > 0.
Remark 3.4. The closed-form formulas for some special cases of equation (1.5) presented in [3] easily follow from the ones in Theorem 2.1. We leave the verification of the fact to the interested reader as some simple exercises. Hence, our Theorem 2.1 gives a theoretical explanation for the closed-form formulas therein.