Global Behavior of a New Rational Nonlinear Higher-Order Difference Equation

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Introduction
Difference equations have wide applications in biology, computer science, digital signal processing, and economics.A general solution structure exists for linear difference equations [ ].However, in various situations of nonlinear higher-order difference equations, solution properties can only be observed by numerical simulation, and it is o en exceedingly difficult to give a full mathematical proof for the properties predicted by numerical simulation and the conclusions formed on the basis of guesswork [ ].It is, therefore, important to make qualitative analysis on nonlinear higherorder difference equations, which is the topic of the current study.
ere have been some related studies on rational nonlinear difference equations in the literature (see, e.g., [ -]).Global asymptotic properties of spectral functions are also crucial in determining algebro-geometric solutions of soliton equations (see, e.g., [ , ]) and scattering data in matrix spectral problems (see, e.g., [ ]).
An iterative algorithm to approximate a zero of a given function  reads and an application of this to a quadratic function () =  2 − ,  > 0, gives Let  be a nonnegative integer and  a real number greater than or equal to .We would like to consider a more general rational nonlinear higher-order difference equation with positive initial values  − ,  −+1 , ⋅ ⋅ ⋅ ,  0 , which engender positive solutions.We take a transformation and then obtain another difference equation Obviously, the equilibrium solution of the rational nonlinear difference equation ( ),  = , becomes the equilibrium solution of the transformed difference equation ( ),  = 1.

Complexity
If we further take  = 1, then we obtain the nonlinear difference equation discussed in [ , ]: Introducing   = √  into ( ) yields where  > 0. When  = 1, this gives the nonlinear difference equation in ( ). e equation ( ) in the case of  = 2 was studied in [ ] and its closed-form solution was presented in [ ].In the general case of , the asymptotic stability of the positive equilibrium solution  = 1 of the equation ( ) was proved in [ ].
It is direct to see that the rational nonlinear higher-order difference equation, defined by ( ), possesses three equilibria:  = −1, 0, .In this article, we would like to explore global behavior of solutions to the rational nonlinear higher-order difference equation ( ), show the global asymptotic stability of its positive equilibrium solution  = , and present two illustrative examples of positive solutions.

Global Behavior
. .Classification of Solutions.First of all, based on the rational difference equation ( ), one can have Further from ( ) and ( ), we can easily derive the following solution properties.
=− is a solution to the rational nonlinear difference equation ( ), then one has where  ≥ 0.
Generally, there are three types of solutions to the rational nonlinear higher-order difference equation ( ).
=− is a solution to the rational nonlinear higher-order difference equation ( ), and then (a) it is eventually equal to , more precisely   = ,  ≥ , which occurs when   =  for some  ≥ 0; (b) it is eventually less than , more precisely   <  +1 < ,  ≥  + , which occurs when   ,  +1 , ⋅ ⋅ ⋅ ,  + <  for some  ≥ −; or (c) it oscillates about , possessing at most  consecutive increasing terms less than  and at most  + 1 consecutive decreasing terms greater than .
Proof.Equality ( ) and property ( ) directly tell that we have three types of solutions to the rational nonlinear higher-order difference equation ( ).
e decreasing and increasing characteristics of oscillatory solutions in the third solution situation (c) can be proved as follows.
Suppose that  1 ,  2 ≥ 0 are two integers satisfying  1 <  2 .We express where  can be written as

by ( ).
If   >  for  1 ≤  ≤  2 , then each term in  is less than zero, and so   2 <   1 , due to ( ).If   <  for  1 ≤  ≤  2 , then each term in  is greater than zero, and so   2 >   1 , due to ( ). is completes the proof.
Note that based on ( ), we can see that there is no solution situation that a solution of ( ) is eventually greater than .
. .Global Asymptotic Stability.When  = 0, the equilibrium solution  =  of the first-order rational difference equation ( ) is globally asymptotically stable, since it is a globally attractive equilibrium solution of a first-order difference equation (see [ ] for a general theory).
For a general  ≥ 1, we can show the same global asymptotic stability of the positive equilibrium solution  =  of the rational nonlinear difference equation ( ), by establishing the local asymptotic stability and the global attractivity, which imply the global asymptotic stability [ ]. Instead, we establish a strong negative feedback property [ ] to guarantee the global asymptotic stability of  =  (see [ ] for details on the strong negative feedback property).

Complexity
Theorem 3. e positive equilibrium solution  =  of the rational nonlinear higher-order difference equation ( ) is globally asymptotically stable.
Proof.Based on the rational nonlinear difference equation ( ), one can have From this equality and the equality in ( ), we can obtain which leads to a strong negative feedback property: with equality for all  ≥ 0 if and only if   = ,  ≥ −.
It, therefore, follows from a stability theorem (Corollary of [ ]) that the equilibrium solution  =  of the rational nonlinear difference equation ( ) is globally asymptotically stable.us, the proof is finished.

( )
Since {  } ∞ =− is oscillatory, eorem guarantees that both   and   have infinitely many numbers.A basic open question that we are very interested in is if   is decreasing on   and increasing on   .We point out that through the above two examples, we failed to find any counterexample to this statement, but found that two cases could occur: either  −1 ,  +1 <  but   >  or  −1 ,  +1 >  but   <  for some  > 1.
Question.To illustrate the global properties stated in eorems and , here we present two illustrative examples associated with two special cases: From the plot pictures, we see that the convergence is achieved very fast in both cases.Finally, let  ≥ 1.For an oscillatory solution {  } ∞ =− of the rational nonlinear difference equation ( ), we define   = { |   >  and  ≥ 0} ,   = { |   <  and  ≥ 0} .