Global asymptotic stability for quadratic fractional difference equation

Consider the difference equation xn+1=α+∑i=0kaixn−i+∑i=0k∑j=ikaijxn−ixn−jβ+∑i=0kbixn−i+∑i=0k∑j=ikbijxn−ixn−j,n=0,1,…,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{n+1} = \frac{\alpha+ \sum_{i=0}^{k} a_{i} x_{n-i} + \sum_{i=0}^{k} \sum_{j=i}^{k} a_{i j} x_{n-i} x_{n-j} }{\beta+ \sum_{i=0}^{k} b_{i} x_{n-i} + \sum_{i=0}^{k} \sum_{j=i}^{k} b_{ij} x_{n-i} x_{n-j}}, \quad n=0,1, \ldots, $$\end{document} where all parameters α, β, ai\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{i}$\end{document}, bi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{i}$\end{document}, aij\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{ij}$\end{document}, bij\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{ij}$\end{document}, i,j=0,1,…,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i,j=0,1,\ldots, k$\end{document}, and the initial conditions xi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x_{i}$\end{document}, i∈{−k,…,0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i \in\{-k, \ldots, 0 \}$\end{document}, are nonnegative. We investigate the asymptotic behavior of the solutions of the considered equation. We give simple explicit conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.


Introduction
Consider the difference equation the second order linear fractional difference equation , n = , , . . . , () and the third order linear fractional difference equation that we get from Eq. () for k =  and a ij = b ij =  for all i, j. The global behavior and the exact solutions of Eq. () even for real parameters were found in []. The global behavior of solutions of Eq. (), in many subcases when one or more parameters are zero, was established in []. There is still one conjecture left whose answer will complete the global picture of the asymptotic behavior of solutions of Eq. (). As far as the third order linear fractional difference equation is concerned, there is a large number of sporadic results that are systemized in a book []. The characterization of the global asymptotic behavior of solutions of Eq. () for k =  seems to be much harder than for the second order Eq. (). Consequently an attempt at giving the characterization of the global asymptotic behavior of solutions of Eq. () seems to be a formidable task at this time. However, by using some known global attractivity results, we can describe the global asymptotic behavior of solutions of Eq. () in some subspaces of the parametric space and the space of initial conditions. See where a, b >  and the initial conditions A kth order generalization of Eq. () with the same property is where a, b i >  and the initial conditions x -k+ , . . . , x  ≥ , x -k+ + · · · + x  > , when a ≤ . Another special case of Eq. () with quadratic terms is where B, d, e >  and the initial conditions Another interesting case of Eq. () with quadratic terms is the equation where a >  and x  ∈ R. When a >  every solution of Eq. () converges either to the zero equilibrium or to the bigger positive equilibrium is the smaller positive equilibrium.
None of these asymptotic behaviors which are present in the cases of Eqs. ()-() are possible in the case of the linear fractional equation

Preliminaries
The following general global results will be applied to Eq. (), see [].
Consider the difference equation As we have observed in [], condition () is actually a contraction condition in the Banach contraction principle.
In addition, we will need the following stability result from [].
Theorem  Suppose that Eq. () can be linearized into the form wherex is an equilibrium of Eq. () and the functions g i : The next result follows from Theorems  and .
Then m ≤ x N+ ≤ M.

Main results
In this section we investigate the stability of the unique positive equilibriumx of Eq. () by using Theorems  and . Observe that Eq. () has a zero equilibrium if and only if α =  and β > , in which case Eq. () becomes j=i a ij , then there is no positive equilibrium. The following result shows that there is an interval in which every solution of Eq. () converges to the zero equilibrium. For convenience of notation, let Q denote the denominator of Eq. (), that is, Proof Observe that Eq. () can be linearized into the form () where l =  as follows for n ≥  Then, for i = , . . . , k, By Lemma  withx = , h i = |g i |, i = , . . . , k, and N = , we get that Again using Lemma  withx = , h i = |g i |, i = , . . . , k, and N = , we get that x  ≤ M. Hence By induction we get that for n ≥  k i= and so the result follows from Corollary  wherex =  and the interval is [, M).
All equilibrium solutions of Eq. () satisfy the equilibrium equation which can be rewritten as Equilibrium Eq. () has at least one nonnegative zero and it may have between  and  positive zeros. When α >  and either k Ifx >  is an equilibrium, then for n ≥  where in view of Eqs. (), () Now applying the identity we get that for n ≥  Thus for n ≥  Therefore for n ≥  We can now obtain easy-to-check conditions which show when the positive equilibrium of Eq. () is globally asymptotically stable. We will then apply these conditions to various cases of Eq. ().
Theorem  Assume that Eq. () has a unique positive equilibriumx and there exist L ≥  and U, N >  such that for every solution {x n } of Eq. () L ≤ x n ≤ U for all n ≥ N and where β + L > . Then the unique positive equilibriumx of Eq. () is globally asymptotically stable on the interval [, ∞).
Proof As we have seen Eq. () can be written in the form of the linearized Eq. (), where the coefficients g i are given as (). We have for n ≥  and so for i = , . . . , k and n ≥  By rearranging the terms we can show that for n ≥  In view of () and (), we obtain On the other hand, if for every i, j ∈ {, . . . , k} such that a i , a ij >  we have b i , b ij > , then the uniform lower bound L for all solutions of Eq. () for n ≥  is The next result follows from Lemma  and can be used to find the part of the basin of attraction of a positive equilibrium in the case when there are several positive equilibrium points. The proof of this result is similar to the proof of Theorem  and it will be omitted. The following result is a consequence of Theorem  in some special cases when the unique positive equilibrium satisfies some specific conditions.
Proof The positive equilibriumx of Eq. () satisfies () In view of our assumption for i, j ∈ {, . . . , k}, we have |a i -xb i | = a i -xb i , |a ij -xb ij | = a ij -xb ij . By using () we obtain In view of our assumption, and so the result follows from Theorem .
Many cases of Eq. () have some combination of a i <xb i , a i >xb i and a i =xb i . In view of this we will adopt the following notations, where I > = {i| such that a i >xb i }, I = = {i| such that a i =xb i }, I < = {i| such that a i <xb i }: Then the positive equilibriumx of Eq. () is globally asymptotically stable on the interval [, ∞).
Proof In view of the equilibrium Eq. () and by assumption (c), we have that Now, the conclusion follows from Theorem .
In the case of general second order quadratic fractional equation of the form x n+ = Ax  n + Bx n x n- + Cx  n- + Dx n + Ex n- + F ax  n + bx n x n- + cx  n- + dx n + ex n- + f , n = , , . . . , with nonnegative parameters and initial conditions such that A + B + C > , and ax  n + bx n x n- + cx  n- + dx n + ex n- + f > , n = , , . . . , the obtained results take the following form. In the special case of second order equation with quadratic terms only, we obtain the following result.

Corollary  Consider the following equation:
x n+ = Ax  n + Bx n x n- + Cx  n- ax  n + bx n x n- + cx  n- , n = , , . . . , with all positive parameters and nonnegative initial conditions such that ax  n + bx n x n- + cx  n- >  for all n ≥ . If the following condition holds: