Global Behavior of Solutions to Two Classes of Second Order Rational Difference Equations

For nonnegative real numbers $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$ such that $B+C>0$ and $\alpha+\beta+\gamma>0$, the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,... %, \quad x_{-1},x_{0}\in [0,\infty) \end{equation*} has a unique positive equilibrium. A proof is given here for the following statements: \medskip \noindent Theorem 1. {\it For every choice of positive parameters $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$, all solutions to the difference equation \begin{equation*} x_{n+1}=\displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{A+B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in [0,\infty) \end{equation*} converge to the positive equilibrium or to a prime period-two solution.} \medskip \noindent Theorem 2. {\it For every choice of positive parameters $\alpha$, $\beta$, $\gamma$, $A$, $B$ and $C$, all solutions to the difference equation \begin{equation*} x_{n+1}= \displaystyle\frac{\alpha +\beta x_{n}+\gamma x_{n-1}}{B x_{n}+C x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1},x_{0}\in (0,\infty) \end{equation*} converge to the positive equilibrium or to a prime period-two solution.}


Introduction and Main Results
In their book [11], M. Kulenović and G. Ladas initiated a systematic study of the difference equation x n+1 = α + β x n + γ x n−1 A + B x n + C x n−1 , n = 0, 1, 2, . . . (1) for nonnegative real numbers α, β, γ, A, B and C such that B+C > 0 and α+β+γ > 0, and for nonnegative or positive initial conditions x −1 , x 0 . Under these conditions, (1) has a unique positive equilibrium. One of their main ideas in this undertaking was to make the task more manageable by considering separate cases when one or more of the parameters in (1) is zero. The need for this strategy is made apparent by cases such as the well known Lyness Equation [17], [22], [9].
x n+1 = α + x n x n−1 (2) whose dynamics differ significantly from other equations in this class. There are a total of 42 cases that arise from (1) in the manner just discussed, under the hypotheses B + C > 0 and α + β + γ > 0. The recent publications [1], [2] give a detailed account of the progress up to 2007 in the study of dynamics of the class of equations (1). After a sustained effort by many researchers (for extensive references, see [1], [2]), there are some cases that have resisted a complete analysis. We list them below in normalized form, as presented in [1], [2].
Conjecture [Ladas et al.] For equations (9) and (10), every solution converges to the positive equilibrium or to a prime period-two solution.
In this article, we prove this conjecture. Our main results are the following.
A reduction of the number of parameters of Eq. (11) is obtained with the change of variables x n = γ C y n , which yields the equation where r = α C γ 2 , p = β γ , and q = B C . The number of parameters of Eq.  can also be reduced, which we proceed to do next. Consider the following affine change of variables which is helpful to reduce number of parameters and simplify calculations: With (12), Eqn.(3-3) may now be rewritten as y n+1 = r + p y n + y n−1 q y n + y n−1 , n = 0, 1, 2, . . . , y −1 , y 0 ∈ [L, ∞) Theorems 1 and 2 can be reformulated in terms of the parameters p, q and r as follows.
Theorem 3 Let α, β, γ, A, B and C be positive numbers, and let p, q, r and L be given by relations (13). Then every solution to Eqn.(3-2-L) converges to the unique equilibrium or to a prime period-two solution.
Theorem 4 Let p, q, r be positive numbers. Then every solution to Eqn.  converges to the unique equilibrium or to a prime period-two solution.
In this paper we prove Theorems 3 and 4; Theorems 1 and 2 follow as an immediate corollary.
The two main differences between Eq.(3-2-L) and Eq.  are the set of initial conditions, and the possibility of having a negative value of r in Eq.(3-2-L), while only positive values of r are allowed in Eq. . Nevertheless, for both Eq.(3-2-L) and Eq.(3-2) the unique equilibrium has the formula: Although it is not possible to prove Theorem 1 as a simple corollary to Theorem 2, the changes of variables leading to Theorems 3 and 4 will result in proofs to the former theorems that are greatly simplified.
Our main results Theorem 1 and Theorem 2 imply that when prime period-two solutions to Eq.(3-3) or Eq.(3-2) do not exist, then the unique equilibrium is a global attractor. We have not treated here certain questions about the global dynamics of Eq.(3-3) and Eq. , such as the character of the prime period-two solutions to either equation, or even for more general rational second order equations, when such solutions exist. This matter will be treated in an upcoming article of the authors [3].
This work is organized as follows. The main results are stated in Section 1. Results from the literature which are used here are given in Section 2 for convenience. In Section 3, it is shown that either every solution to Eq.(3-2-L) converges to the equilibrium, or there exists an invariant and attracting interval I with the property that the function f (x, y) associated with the difference equation is coordinate-wise strictly-monotonic on I × I. In Section 4, a global convergence result is obtained for Eq.(3-2) over a specific range of parameters and for initial conditions in an invariant compact interval. Theorem 3 is proved in Section 5, and the proof of Theorem 4 is given in Section 6. Section 7 includes computer algebra system code for performing certain calculations that involve polynomials with a large number of terms (over 365,000 in one case). These computer calculations are used to support certain statements in Section 4. Finally, we refer the reader to [11] for terminology and definitions that concern difference equations.

Results from the literature
The results in this subsection are from the literature, and they are given here for easy reference. The first result is a reformulation of Theorems (1.4.5) -(1.4.8) in [11]. i. f (x, y) is nondecreasing in x, y, and Then y n+1 = f (y n , y n−1 ) has a unique equilibrium in [a, b], and every solution with initial values in [a, b] converges to the equilibrium.
the subsequences {x 2n } and {x 2n+1 } of even and odd terms do exactly one of the following: (i) They are both monotonically increasing.
(ii) They are both monotonically decreasing.
(iii) Eventually, one of them is monotonically increasing and the other is monotonically decreasing.
Theorem 7 has this corollary.

Corollary 1 ([5])
If I is a compact interval, then every solution of Eq. (14) converges to an equilibrium or to a prime period-two solution.
Theorem 8 ( [8]) Assume the following conditions hold: (ii) h(x, y) is decreasing in x and strictly decreasing in y.
(iii) x h(x, x) is strictly increasing in x.
(iv) The equation has a unique positive equilibrium x.
Then x is a global attractor of all positive solutions of Eq.(15).

Existence of an Invariant And Attracting Interval
In this section we prove a proposition which is key for later developments. We will need the function associated to Eq.(3-2-L).
Proposition 1 At least one of the following statements is true: (A) Every solution to (3-2-L) converges to the equilibrium.
The next lemma states that the function f (·, ·) associated to Eq.(3-2-L) is bounded.
Lemma 1 There exist positive constants L and U such that L < L and In particular, and necessarily satisfy m = M . In either case, the hypotheses (i) or (ii) of Theorem 5 are satisfied, and the conclusion of the lemma follows. 2 We will need the following elementary result, which is given here without proof. ii. D 2 f (x, y) = 0 if and only if x = −r p−q , and D 2 f (x, y) > 0 if and only if (q − p) x > r.
We will need to refer to the values K 1 and K 2 where the partial derivatives of f (x, y) change sign.
The proof will be complete when it is shown that m = M . There are a total of four cases to consider: (a) r ≥ 0 and p > q, (b) r < 0 and p < q, (c) r ≥ 0 and p < q, and (d) r < 0 and p > q. We present the proof of case (a) only, as the proof of the other cases is similar. If r ≥ 0 and p > q, then By Lemma 3, the signs of the partial derivatives of f (x, y) are constant on the interior of each of the sets From (20) and (21) one has Combine (22) with relation (19) to obtain the system of equations Eliminating M from system (23) gives the cubic in m which has the roots Only one root in the list (25) is positive, namely Substituting into one of the equations of system (23) one also obtains M = y, which gives the desired relation m = M = y.
Proof. Arguing by contradiction, suppose m * < M * and for all ∈ N,  4 The Equation (3-2) with r ≥ 0, p > q and q r p−q < p q In this section we restrict our attention to the equation For p > 0, q > 0, and r ≥ 0, Eq.(3-2) has a unique positive equilibrium y = p + 1 + (p + 1) 2 + 4r (q + 1) 2 (q + 1) We note that if I ⊂ (0, ∞) is an invariant compact interval, then necessarily y ∈ I. The goal in this section is to prove the following proposition, which will provide an important part of the proofs of Theorems 1 and 4.  , and substitute the latter into (30) to see that x = M is a solution to the quadratic equation By a symmetry argument, one has that x = m is also a solution to (33). By inspection of the coefficients of the polynomial in the left-hand-side of (33) one sees that two positive solutions are possible only when q < 1. To get the conclusion of the lemma, note that the fact that (30) has no solutions with m = M is just hypothesis (iv) of Theorem 5. Proof. By substituting that is, x n+1 =f (x n , x n−1 , x n−2 ), wheref (x, y, z) = p r + p 2 y + q r y + q y 2 + p z + r z + y z q r + p q y + q x 2 + q z + y z where the x has been kept inf (x, y, z) for bookkeeping purposes. Thusf (x, y, z) is constant in x. We claimf (x, y, z) is decreasing in both y and z. To see that the partial derivative is negative just use p > q and the inequality (p − q) y − q r > 0, which is true by Lemma 3. The remaining partial derivative is D 2f (x, y, z) = − L(y, z) (q r + p q y + q y 2 + q z + y z) 2 where h(y, z) := −q 2 r 2 + 2 p q r y − 2 q 2 r y + p 2 q y 2 − p q 2 y 2 + q 2 r y 2 + p r z − q r z + + p q r z − q 2 r z + 2 p q y z − 2 q 2 y z + 2 q r y z + p z 2 − q z 2 + r z 2 We have, Since q r p−q ≤m, thus we conclude that D 2f (x, y, z) < 0 for x, y, z ∈ [m,M ].
To complete the proof we verify the hypotheses of Theorem 6. We claim that the system of equations Since m = M , we may use the second factor in the left-hand-side term of (43) to solve for M in terms of m, which upon substitution intof (m, m, m) = M and simplification yields the equation where a 0 = r p + 2 p q + p 2 q + q r + 2 q 2 r a 1 = p + p 2 + 2 p q + 3 p 2 q + p 3 q + r − p r + 4 q r + 4 q 2 r + 2 p q 2 r a 2 = (1 + q) (1 + 2 q + p q + q r) By hypothesis (34) we have r p ≤ p 3 q − p 2 < p 3 q, hence p 3 q − r p > 0, which implies a 1 ≥ 0. By direct inspection one can see that a 0 > 0 and a 2 > 0. Thus (44)  Proof. Solving for r in y = r + (p + 1) y (q + 1) y gives r = (q + 1) y 2 − (p + 1) y Then a calculation shows D 1 f (y, y) = p−q y y (q+1) Set t 1 := D 1 f (y, y) and t 2 := D 2 f (y, y). The equilibrium y is locally asymptotically stable if the roots of the characteristic polynomial have modulus less than one [11]. By the Schur-Cohn Theorem, y is L.A.S. if and only if |t 1 | < 1 − t 2 < 2. It can be easily verified that 1 − t 2 < 2 if and only if 0 < q y + 1 which is true regardless of the allowable parameter values. Since p − q y > 0 by the hypothesis, we have |t 1 | = | p−q y y(q+1) | = p−q y y(q+1) , hence some algebra gives |t 1 | < 1 − t 2 if and only if 1 2 But (46) is a true statement by formula (28). We conclude y is L.A.S. Proof. The proof begins with a change of variable in Eq.  to produce a transformed equation with normalized coefficients analogous to those in the standard normalized Lyness' Equation [17], [22], [9] z n+1 =α + x n x n−1 (48) We seek to use an argument of proof similar to the one used in [21], in which one takes advantage of the existence of invariant curves of Lyness' Equation to produce a Lyapunov-like function for Eq. (3-2). Set y n = pz n in Eq.  to obain the equation We shall denote with z the unique equilibrium of Eq.(49). Note that It is convenient to parametrize Eq.(49) in terms of the equilibrium. We will use the symbol u to represent the equilibrium z of Eq.(49). By direct substitution of the equilibrium u = z into Eq.(49) we obtain By (52), r ≥ 0 iff u ≥ g+1 b+1 . Using (52) to eliminate r from Eq.(49) gives the following equation for b > 0, g > 0 and u ≥ g+1 b+1 , equivalent to Eq.(49): z n+1 = (b + 1) u 2 − (g + 1) u + z n + g z n−1 b z n + z n−1 , n = 0, 1, 2, . . . , y −1 , y 0 ∈ (0, ∞) Therefore it suffices to prove that all solutions of Eq.(53) converge to the equilibrium u.
The following statement is crucial for the proof of the proposition.
Claim 1 u > 1 if and only if r > p 2 q − p.
Proof. Since y = p z = p u, we have u > 1 if and only if y > p, which holds if and only if p + 1 + (p + 1) 2 + 4 r (q + 1) 2 (q + 1) > p After an elementary simplification, the latter inequality can be rewritten as r > p 2 q − p. 2 By the hypotheses of the lemma, by Claim 1, and by (50) and (52) we have b < 1, g < 1, 1 < u < 1 b , and We now introduce a function which is the invariant function for (48) with constantα = u 2 − u (in this case the the equilibrium of (48) is u): Note that g(x, y) > 0 for all x, y ∈ (0, ∞) whenever u > 1. By using elementary calculus, one can show that the function g(x, y) has a strict global minimum at (u, u) [9], [22], i.e., g(u, u) < g(x, y), (x, y) ∈ (0, ∞) 2 (56) We need some elementary properties of the sublevel sets We denote with Q (u, u), = 1, 2, 3, 4 the four regions be the map associated to Eq. (53) (see [16]).
Proof. This proof requires extensive use of a computer algebra system to verify certain inequalities involving rational expressions. Here we give an outline of the steps, and refer the reader to Section 7 for the details. Since b < g < 1 < u < 1 b , and g+1 b+1 < u we may write The expression ∆ 2 := g(x, y) − g(T 2 (x, y)) may be written as a single ratio of polynomials, ∆ 2 = N D with D > 0. The next step is to show N > 0 for (x, y) = (u, u).
Points (x, y) in Q 1 (u, u) may be written in the form . Substituting x, y, u and g in terms of v, w, s and t into the expression for N one obtains a rational expressionÑ D with positive denominator. The numeratorÑ has some negative coefficients. At this points two cases are considered, w ≥ v, and w ≤ v. These can be written as w = v + k and v = w + k for nonnegative k. Substitution of each one of the latter expressions inÑ gives a polynomial with positive coefficients. This proves ∆ 2 (x, y) > 0 for (x, y) ∈ Q 2 (u, u).
If now we assume (x, y) ∈ Q 3 (u, u) with (x, y) = (u, u), we may write The rest of the proof is as in the first case already discussed. Details can be found in Section 7.
The function f (x, y) is assumed to be coordinate-wise monotonic on [m * , M * ], and there are four possible cases in which this can happen: (a) f (x, y) is increasing in both variables, (b) f (x, y) is decreasing in both variables, (c) f (x, y) is decreasing in x and increasing in y, and (d) f (x, y) is increasing in x and decreasing in y.
We present several lemmas before completing the proof of Theorem 3. By considering the restriction of the map T of Eq.(3-2-L) to [m * , M * ] 2 , an application of the Schauder Fixed Point Theorem [7] gives that [m * , M * ] 2 contains the fixed point of T , namely (y, y). Thus we have the following result. Proof. By the standing assumption (SA), p = q. By Lemma 3, the coordinate-wise monotonicity hypothesis, and the fact y ∈ [m * , M * ] from Lemma 10, we have (p − q) y > q r and (q − p) y < r The inequalities in (68) cannot hold simultaneously unless p − q > 0. Proof. Since y ∈ [m * , M * ] by Lemma 10, we have D 1 (y, y) > 0 and D 2 (y, y) < 0. By Lemma 12, p > q, and by Lemma 3, (p − q) y > q r and (q − p) y < r (71) Then, y > q r p − q (72) In addition, by Lemma 3, Lemma 15 Suppose f (x, y) is increasing in x and decreasing in y for (x, y) ∈ [m * , M * ]. If r < 0, then p − q + r > 0.
The proof of Theorem 3 may be reproduced here in its entirety with the only change being the elimination of the case r < 0, which presently does not apply. Everything else in the proof applies to Eq.(3-2). The proof of Theorem 4 is complete.
7 Appendix: Computer Algebra System Code Table 1: Mathematica code needed to do the calculations in Claim 3. Here we define the functions g, f and T , as well as the expression DELTA2. The reparametrizations indicated in the proof of Claim 3 for the case g > b are defined as substitution rules. To verify the positive sign of a polynomial of nonnegative variables z, s, . . . , we form a list with the terms of the polynomial, and then substitute the number 1 for the variables in order to extract the smallest coefficient. This input was tested on Mathematica Version 5.0 [19]. Table 2: Mathematica code needed to do the calculations in Claim 3 when g ≤ b The functions g, f and T are defined as before (not shown). This input was tested on Mathematica Version 5.0 [19].