Electronic Journal of Qualitative Theory of Differential Equations

We present a representation of well-defined solutions to the following nonlinear second-order difference equation xn+1 = a + b xn + c xnxn−1 , n ∈N0, where parameters a, b, c, and initial values x−1 and x0 are complex numbers such that c 6= 0, in terms of the parameters, initial values, and a special solution to a thirdorder homogeneous linear difference equation with constant coefficients associated to the nonlinear difference equation, generalizing a recent result in the literature, completing the proof therein by using an essentially constructive method, and giving some theoretical explanations related to the method for solving the difference equation. We also give a more concrete representation of the solutions to the nonlinear difference equation by calculating the special solution to the third-order homogeneous linear difference equation in terms of the zeros of the characteristic polynomial associated to the linear difference equation.


Introduction
In this paper, by N, N 0 , Z and C we denote the set of positive, nonnegative, whole and complex numbers, respectively.
As it is well-known nonhomogeneous linear difference equation with constant coefficients of kth order, that is, the following difference equation c k x n+k + c k−1 x n+k−1 + · · · + c 0 x n = f n , n ≥ l, (1.1) where k ∈ N, l ∈ Z, c j , j = 0, k are given constants, such that and ( f n ) n≥l is a given sequence of real or complex numbers, is a basic example of a difference equation solvable in closed-form (see, for example, books [5,7,10,13,15,16,19]). If one of the conditions in (1.2) does not hold then the equation is of order less than k, which is why these conditions are posed. If f n = 0 for n ≥ l, the equation is called homogeneous linear difference equation with constant coefficients of kth order. Several methods for finding general solution to difference equation (1.1) have been known yet in the 18th century (for example, the method of generating functions which has been first used for solving the difference equation [8], and the method of guessing solution in the form of a geometric progression).
To find a concrete solution to the equation, initial values x l , x l+1 , . . . , x l+k−1 , must be given. Value of index l depends on a specific problem which is studied. Frequently is l = 0 or l = 1, but it can be any other integer.
Since c k = 0, by dividing difference equation (1.1) by c k we obtain the following equation x n+k + c k−1 x n+k−1 + · · · + c 0 x n = f n , (1.3) for n ≥ l, where c j = c j /c k , j = 0, 1, . . . , k − 1, and f n = f n /c k , for n ≥ l, which is a difference equation of the form in (1.1). Hence, it is usually assumed that For linear difference equations which satisfy condition (1.4), we will say that they are normalized.
Many nonlinear difference equations are solved by transforming them by using some suitable changes of variables to linear difference equations with constant coefficients. Somewhat bigger recent interest in solving some concrete nonlinear difference equations in this way, appeared after publication of a note by S. Stević in 2004, in which a special case of the following nonlinear difference equation of second order was solved where n ∈ N 0 (see, e.g., [29] and the references therein).
The study was, among other papers, continued in [30,46], where the following nonlinear difference equation with variable coefficients was studied x n = a n x n−k b n + c n x n−1 · · · x n−k , (see, also [26]). For some related systems see, e.g., [31] and the references therein. Many different ideas and methods for solving nonlinear difference equations, can be found, also in the representative paper [36]. Motivated by some investigations of usually symmetric systems of difference equations (see, e.g., [21][22][23][24]), approximately at the same time started a renewed considerable interest in investigation of some related classes of solvable systems of nonlinear difference equations, usually those which are nowadays called close-to-symmetric, or close-to-cyclic systems of nonlinear difference equations (see, for example, [4,20,45] and the related references cited therein).
As it is also well-known, the bilinear difference equation where parameters α, β, γ, δ, and the initial value z 0 are complex numbers, is one of the first examples of nonlinear difference equations solvable in closed-form (see, for example, books [5,14,15,19,27] where c 1 and c 2 are two undetermined constants, which are chosen such that the change of variables transforms difference equation (1.6) to a linear difference equation (see, for example, [1,5,6,18,19,27]). Equation (1.6) can be also solved by using a system of linear difference equations [14,15]. Since the first method is closer to the topic of this paper and serves as a good motivation for the method which will be used here, we will give a few words on it.
Using (1.7) in (1.6) gives for n ∈ N 0 , that is, Multiplying the last equality by y n , we obtain γ c 1 2 y n+2 + c 1 (γ c 2 − α)y n+1 + (γ c 2 2 + c 2 (δ − α) − β)y n + c 1 (γ c 2 + δ) y n+2 y n y n+1 = 0, n ∈ N 0 , from which it is easy to see that the equation will be linear with constant coefficients if γ c 2 + δ = 0, that is, if To make difference equation (1.9) normalized, then, clearly, it should be chosen Using (1.8) and (1.10) in (1.7), we see that the following change of variables for n ∈ N 0 . Anyone who see the change of variables (1.7) for the first time should be certainly intrigued by the choice. So, let us say that one of the points is that equation (1.6) can be written as [1,18,32] from which it follows that the following equation should be solved, or equivalently, the following one where a ∈ C and b ∈ C \ {0}. What the above procedure says is that difference equation (1.13) is solved by the change of variables x n = y n+1 y n , n ∈ N 0 . (1.14) Motivated by some representations of general solutions to some nonlinear difference equations (for example, those ones in [50]), in terms of the Fibonacci sequence (for some basics of the sequence, see, e.g., [2,51]), recently in [32], S. Stević has given a representation of solutions to equation (1.6) in terms of the solution to a linear second-order homogeneous difference equation with constant coefficients, satisfying the following initial conditions Soon after that it turned out that such solutions play some interesting and important roles in solving some other nonlinear difference equations and systems of nonlinear difference equations, for example, the product-type ones, and frequently appear (see, e.g., [34,37,41,43,44,48,49] and the references therein on such systems).
Some classical applications of solvable difference equations can be found, for example, in [15,17,28]. For some other recent results on finding closed-form formulas or invariants for solutions to other linear or nonlinear difference equations and systems of nonlinear difference equations, and their applications, see, for example [3,11,12,[23][24][25][38][39][40].
Having solved equation (1.13) long time ago, mathematicians started looking for some generalizations of the equation which can be also solved by using the change of variables in (1.14).
The following nonlinear second-order difference equation is a natural extension of equation (1.13) which is solved by the change of variables (1.14) which transforms it to the following third-order linear difference equation with constant coefficients y n+2 = ay n+1 + by n + cy n−1 , (1. 16) for n ∈ N 0 . A special case of equation (1.15), that is, the one with a = 0, has been recently studied in [11]. Instead of (1.14), in [11] was used the following (backward shifted) change of variables so that equation (1.15) is transformed to y n+1 = ay n + by n−1 + cy n−2 , n ∈ N 0 , (1.18) and, similar to [32], a representation of general solution to equation (1.15), with a = 0, was given in terms of the solution (s n ) n≥−2 of equation (1.18) satisfying the following initial conditions The shifted change of variables is usually used so that the initial values of the transformed equation (1.18) are y −2 , y −1 and y 0 . The initial values seem more natural than, for example, the following ones: y −1 , y 0 , y 1 , which appear for the case of equation (1.16).
The following theorem was proved in [11].
Then, it has the following representation The theorem is true, but the proof given in [11] is not complete, since it omits an inductive argument, which is a small deficiency of the paper. A bigger deficiency of paper [11] is of theoretical type, since it does not give or use a specific constructive or half-constructive method for getting representation of solutions to equations and systems studied therein. The solutions therein are simply guessed based on some calculations.
Here, we present a more direct half-constructive approach in getting a representation of solutions to difference equation (1.15) and extend Theorem 1.1. One of the motivations for the approach stems from some recent papers by S. Stević and his collaborators, mostly on producttype difference equations and systems of difference equations [34,37,41,43,44,48,49]. The idea is to iterate some associated third-order homogeneous linear difference equations in a suitable chosen way to get representation of solutions to the original difference equation. We also give a more concrete representation of the solutions to the nonlinear difference equation by calculating the special solution to the third-order homogeneous linear difference equation in terms of the zeros of the characteristic polynomial associated to the linear difference equation.

Main results
The main results in this paper are proved in this section. The first one generalizes Theorem 1.1.
Theorem 2.1. Let parameters a, b, c be complex numbers such that c = 0, and let (s n ) n≥−2 be the solution to equation (1.18) satisfying initial conditions (1.19). Then, every well-defined solution to equation (1.15) has the following representation for every n ∈ N 0 .
Proof. As it has been said in the previous section, by using the change of variables (1.17), equation (1.15) is transformed to equation (1.18). Now we define initial values of three sequences which will be recursively defined and used in the rest of the proof, in the way as it was done, for example, in [33,37]. Let In what follows the following three cases will be considered separately: 1) a = 0, 2) a = 0, b = 0, 3) a = b = 0, since the proofs, although follow the same idea, are not the same.
Case a = 0. We use an iterative procedure which has been recently applied, for example, in papers [34] and [41]. By using the equality (1.18) where n is replaced by n − 2, in the equation (1.18) where n is replaced by n − 1, as well as the notation in (2.2), we have y n = a 1 y n−1 + b 1 y n−2 + c 1 y n−3 = a 1 (ay n−2 + by n−3 + cy n−4 ) + b 1 y n−2 + c 1 y n−3 = (aa 1 + b 1 )y n−2 + (ba 1 + c 1 )y n−3 + ca 1 y n−4 = a 2 y n−2 + b 2 y n−3 + c 2 y n−4 , 3), it follows that y n = a 2 y n−2 + b 2 y n−3 + c 2 y n−4 = a 2 (ay n−3 + by n−4 + cy n−5 ) + b 2 y n−3 + c 2 y n−4 = (aa 2 + b 2 )y n−3 + (ba 2 + c 2 )y n−4 + ca 2 y n−5 = a 3 y n−3 + b 3 y n−4 + c 3 y n−5 , where Based on relations (2.3)-(2.6), we assume that for some k ∈ N such that 2 ≤ k ≤ n − 1, we have that y n = a k y n−k + b k y n−k−1 + c k y n−k−2 , (2.7) and Using further the equality (1.18) where n is replaced by n − k − 1 in (2.7), it follows that Now we prolong sequences a k , b k and c k for some non-positive values of index k, as it was done, for example, in [43,44]. Note that since c = 0, the recurrent relations in (2.8) can be really used for calculating values of sequences a k , b k and c k for every k ≤ 0.
If we take k = n in (2.7), we obtain y n = a n y 0 + b n y −1 + c n y −2 , (2.23) for n ∈ N 0 . From (2.21)-(2.23), we get y n = a n y 0 + (a n+1 − aa n )y −1 + ca n−1 y −2 , (2.24) for n ∈ N 0 . Using (2.24) in (1.17), we obtain x n = a n y 0 + (a n+1 − aa n )y −1 + ca n−1 y −2 a n−1 y 0 + (a n − aa n−1 )y −1 + ca n−2 y −2 , Now we use a modified procedure, which has been used in the case a = 0 (the modification is necessary since the coefficient at y n is zero and the same procedure does not have an effect in the case).
As above, since c = 0, by using all the relations in (2.33), it is not difficult to see that the sequences b k , c k and d k , can be also calculated for all nonpositive values of index k.
By using the three recurrent relations in (2.33) with indices k = 1, k = 0, k = −1 and k = −2, respectively, after some calculations and repeating use of already calculated terms, we have (2.48) If we choose k = n − 1 in (2.32), we obtain for n ∈ N 0 . By using (2.33) in (2.49), it follows that for n ∈ N 0 . From initial conditions (2.48) we see that sequence b n is the solution to the equation (1.18) with a = 0, b = 0, with the backward shifted initial conditions of the sequence a n defined in (2.2) and (2.8).
Hence, b n−1 = a n , so from (2.50) we see that (2.24) also holds in the case a = 0, b = 0, and consequently the formula in (2.25).
Case a = b = 0. In this case, equation (1.18) becomes y n+1 = cy n−2 , (2.51) for n ∈ N 0 . From (2.51) it easily follows that for m ∈ N 0 and i = 0, 2, from which it is easily verified that (2.24) also holds in this case, and consequently formula (2.25).
Using this fact in (2.52), we immediately obtain formula (2.1), completing the proof of the theorem.
Remark 2.2. The method used here is inductive, since in proving hypotheses (2.7), (2.8), (2.32) and (2.33) the method was used. However, it is also a constructive one, since we define sequences a k , b k , c k , b k , c k , d k , appearing in the proof of Theorem 2.1 in a clear constructive way, by using some initial conditions and recurrent relations. Hence, the above method is half-constructive.
To conduct further investigations we need a lemma, which follows, for example, from the Lagrange interpolation formula (see, e.g., [9]), or by using the residue theorem (see, e.g., [47]).

Lemma 2.3.
Let p(t) = a k t k + a k−1 t k−1 + · · · + a 1 t + a 0 , and t j , j = 1, 2, . . . , k, be the zeros of p(t), which are distinct, that is, Since linear difference equation (1.18) is of the third-order it is practically solvable, since the characteristic equation From (2.60) and (2.61), we see that u 3 and v 3 are the zeros of the following polynomial that is, they are equal to Hence, the zeros of polynomial (2.53) are where ε is a complex zero different from 1, of the polynomial equation The character of zeros λ j , j = 1, 2, 3, depends on the sign of the following quantity where g j ∈ C, j = 1, 2, 3. If we apply Lemma 2.3 to the following polynomial which can be also written in the following form for n ≥ −2 (see, e.g., [42] Case ∆ = 0, a 2 = −3b. In this case all zeros of equation (2.53) are real, but two of them are equal. We may assume that λ 1 = λ 2 = λ 3 . General solution to equation (1.18) has the form y n = g 1 λ n 1 + ( g 2 + g 3 n)λ n 2 , n ∈ N, (2.71) where g j ∈ C, j = 1, 2, 3. The solution to (1.18) satisfying the initial conditions in (1.19) in this case can be obtained by a limiting argument (see [35]), and it is given by 1 − (n + 2)λ 1 λ n+1 2 + (n + 1)λ n+2 2 (λ 2 − λ 1 ) 2 , for n ≥ −2, that is, for n ≥ −2.
From the consideration and Theorem 2.1 we get the following corollary in the case. Then, every well-defined solution to equation (1.15) has the representation given by formula (2.1), where the sequence (s n ) n≥−2 therein is given by (2.72), where λ j , j = 1, 2, 3, are given by while q is given in (2.57).
From the consideration and Theorem 2.1 we get the following corollary in this case.
Corollary 2.7. Let parameters a, b, c be complex numbers such that c = 0, a 2 = −3b and that condition (2.73) holds. Then, every well-defined solution to equation (1.15) has the representation given by formula (2.1), where the sequence (s n ) n≥−2 therein is given by (2.77).