Advertisement

Advances in Difference Equations

, 2009:985161 | Cite as

Convergence Results on a Second-Order Rational Difference Equation with Quadratic Terms

  • D. M. Chan
  • C. M. Kent
  • N. L. Ortiz-Robinson
Open Access
Research Article

Abstract

We investigate the global behavior of the second-order difference equation Open image in new window , where initial conditions and all coefficients are positive. We find conditions on Open image in new window under which the even and odd subsequences of a positive solution converge, one to zero and the other to a nonnegative number; as well as conditions where one of the subsequences diverges to infinity and the other either converges to a positive number or diverges to infinity. We also find initial conditions where the solution monotonically converges to zero and where it diverges to infinity.

Keywords

Differential Equation Partial Differential Equation Ordinary Differential Equation Functional Analysis Functional Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction and Preliminaries

There are a number of studies published on second-order rational difference equations (see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9]). We investigate the global behavior of the second-order difference equation
where the numerator is quadratic and the denominator is linear with Open image in new window . Under various hypotheses on the parameters, we establish the existence of different behaviors of even and odd subsequences of solutions of (1.1). Our results are summarized below.
  1. (i)

    Let Open image in new window and Open image in new window , then we have the following.

    1. (a)

      There are infinitely many solutions, Open image in new window , such that for each, one of its subsequences, Open image in new window , Open image in new window , converges to zero and the other diverges to infinity.

       
    2. (b)

      There exist solutions, Open image in new window , which

      1. (1)

        converge to zero if Open image in new window ;

         
      2. (2)

        diverge to infinity if Open image in new window ;

         
      3. (3)

        are constant if Open image in new window .

         
       
     
  1. (i)

    Let Open image in new window and Open image in new window . Then for each positive solution Open image in new window , one of the subsequences, Open image in new window , Open image in new window , diverges to infinity and the other to a positive number that can be arbitrarily large depending on initial values. Further there, are positive initial values for which the corresponding solution, Open image in new window , increases monotonically to infinity.

     
  2. (ii)

    Let Open image in new window and Open image in new window . Then for each positive solution Open image in new window , one of the subsequences, Open image in new window , Open image in new window , converges to zero and the other to a nonnegative number. Further, there are positive initial values for which the corresponding solution, Open image in new window , decreases monotonically to zero.

     

We note that the following results address and solve the first five conjectures posed by Sedaghat in [10].

2. Results

In order to establish this first result, we reduce (1.1) to a first-order equation by means of the substitution Open image in new window This transforms (1.1) to

Theorem 2.1.

Let Open image in new window and Open image in new window in (1.1). Then one has the following.
  1. (i)

    There are infinitely many solutions, Open image in new window , such that for each, one of its subsequences, Open image in new window , Open image in new window , converges to zero and the other to infinity.

     
  2. (ii)

    There exist solutions, Open image in new window , which

    1. (a)

      converge to zero if Open image in new window ;

       
    2. (b)

      diverge to infinity if Open image in new window ;

       
    3. (c)

      are constant if Open image in new window .

       
     

Proof.

Starting with (2.1), let the function Open image in new window be defined as Open image in new window . Note that for Open image in new window , Open image in new window is a decreasing function since Open image in new window . Also note that Open image in new window and Open image in new window . Hence Open image in new window has a unique positive fixed point Open image in new window .

We next compute the expression Open image in new window and simplify, it including canceling the common factor Open image in new window from the numerator and denominator, thereby obtaining the following:

Note that since Open image in new window , Open image in new window and Open image in new window . Thus the numerator of Open image in new window has one and only one sign change. Therefore, by Descartes' rule of signs, the numerator of Open image in new window has exactly one positive root, Open image in new window .

In addition, we see that Open image in new window and so, given that Open image in new window is the only positive root of the numerator of Open image in new window , we have Open image in new window for Open image in new window . Thus, since Open image in new window and Open image in new window is continuous, we must have Open image in new window for Open image in new window . Therefore,

We consider two cases depending on the initial value Open image in new window for (2.1).

Case 1 ( Open image in new window ).

Using induction and the fact that Open image in new window is a decreasing function so that Open image in new window is an increasing function, we have

Thus, Open image in new window Since Open image in new window is the only positive fixed point of Open image in new window , then we must have Open image in new window and Open image in new window

Case 2 ( Open image in new window ).

The argument is similar to that in Case 1 in showing Open image in new window and Open image in new window In both cases, the solution, Open image in new window , of (2.1) is divided into even and odd subsequences, Open image in new window and Open image in new window , where one subsequence converges monotonically to zero and the other to infinity.

We now go back to (1.1) by inferring the behavior of Open image in new window from Open image in new window . To do this we first consider Open image in new window . Without loss of generality, we will assume that Open image in new window and so Open image in new window and Open image in new window .

Next, observe that

From this and our assumption with Open image in new window , we have
and by induction, for Open image in new window ,
This, in turn, implies that
The argument is similar in showing that Open image in new window since

Hence, result (i) is true.

Now consider Open image in new window . Then Open image in new window for all Open image in new window , and so Open image in new window for all Open image in new window . Induction then gives us Open image in new window for all Open image in new window . We thus have one of the following:

Thus the result (ii) is true and this completes the proof.

For the next couple of results we rewrite (1.1) in the form

Lemma 2.2.

Proof.

Statements 1 and 2 follow from the fact that
by properties (P2) and (P3). Statement 3 follows from the fact that either Open image in new window , and so Open image in new window and Open image in new window are undefined by property (P1); or Open image in new window and

where Statements 1 and 2 and the continuity of Open image in new window (Property (P1) hold. Finally, Statement 4 follows immediately from Statement 3 and Property (P4).

In the first three results, we characterize the convergence of the odd and even subsequences of solutions of (1.1).

Theorem 2.3.

Let Open image in new window and Open image in new window in (1.1). Then for each positive solution, Open image in new window , one of the subsequences, Open image in new window , Open image in new window , converges to zero and the other to a nonnegative number.

Proof.

Consider (1.1) with Open image in new window , Open image in new window , and Open image in new window . Then it follows from Lemma 2.2 that for each positive solution of (1.1), Open image in new window , one of the subsequences, Open image in new window , Open image in new window , converges to zero and the other to a nonnegative number.

Theorem 2.4.

Let Open image in new window and Open image in new window in (1.1). Then for each positive solution Open image in new window , one of the subsequences, Open image in new window , Open image in new window , diverges to infinity and the other to a positive number or diverges to infinity.

Proof.

Consider (1.1) with Open image in new window and Open image in new window . Using the transformation Open image in new window convert (1.1) to the equation

Then Open image in new window , and so it follows from Lemma 2.2 that for each positive solution of (2.16), Open image in new window , one of the subsequences, Open image in new window , Open image in new window , converges to zero and the other to a nonnegative number. Hence, for each positive solution of (1.1), Open image in new window , one of the subsequences, Open image in new window , Open image in new window , diverges to infinity and the other to a positive number or diverges to infinity.

In the following results, we show the existence of monotonic solutions for (1.1). As with Theorem 2.1 we use the substitution Open image in new window

Theorem 2.5.

Let Open image in new window and Open image in new window in (1.1). Then there are positive initial values for which the corresponding solutions, Open image in new window , decrease monotonically to zero.

Proof.

Note that an equilibrium equation for (2.1) satisfies,

Set Open image in new window Given Descartes' rule of signs, we have that there exists a unique positive equilibrium, Open image in new window , where Open image in new window and Open image in new window Recall that Open image in new window and let Open image in new window for all Open image in new window . Then Open image in new window for all Open image in new window . It follows from induction that Open image in new window for all Open image in new window . Since Open image in new window , Open image in new window , with Open image in new window , decreases monotonically to zero.

Theorem 2.6.

Let Open image in new window and Open image in new window in (1.1). Then there are positive initial values for which the corresponding solution, Open image in new window , increases monotonically to infinity.

Proof.

As in the previous proof, an equilibrium equation for (2.1) satisfies (2.17). Setting Open image in new window we obtain from Descartes' rule of signs, a unique positive equilibrium, Open image in new window , where Open image in new window and Open image in new window Recall that Open image in new window and let Open image in new window for all Open image in new window . Then Open image in new window for all Open image in new window . It follows from induction that Open image in new window for all Open image in new window . Since Open image in new window , Open image in new window , with Open image in new window , increases monotonically to infinity.

References

  1. 1.
    Amleh AM, Camouzis E, Ladas G: On second-order rational difference equation—I. Journal of Difference Equations and Applications 2007,13(11):969-1004. 10.1080/10236190701388492MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Amleh AM, Camouzis E, Ladas G: On second-order rational difference equation—II. Journal of Difference Equations and Applications 2008,14(2):215-228. 10.1080/10236190701761482MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Huang YS, Knopf PM: Boundedness of positive solutions of second-order rational difference equations. Journal of Difference Equations and Applications 2004,10(11):935-940. 10.1080/10236190412331285360MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Kosmala WA, Kulenović MRS, Ladas G, Teixeira CT:On the recursive sequence Open image in new window. Journal of Mathematical Analysis and Applications 2000,251(2):571-586. 10.1006/jmaa.2000.7032MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kulenović MRS, Ladas G: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2002:xii+218.MATHGoogle Scholar
  6. 6.
    Kulenović MRS, Ladas G, Prokup NR:On the recursive sequence Open image in new window. Journal of Difference Equations and Applications 2000,6(5):563-576. 10.1080/10236190008808246MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kulenović MRS, Ladas G, Sizer WS:On the recursive sequence Open image in new window. Mathematical Sciences Research Hot-Line 1998,2(5):1-16.MathSciNetMATHGoogle Scholar
  8. 8.
    Kulenović MRS, Merino O:Global attractivity of the equilibrium of Open image in new window for Open image in new window. Journal of Difference Equations and Applications 2006,12(1):101-108. 10.1080/10236190500410109MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ladas G:On the recursive sequence Open image in new window. Journal of Difference Equations and Applications 1995,1(3):317-321. 10.1080/10236199508808030MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Sedaghat H: Open problems and conjectures. Journal of Difference Equations and Applications 2008,14(8):889-897. 10.1080/10236190802054118MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© D. M. Chan et al. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • D. M. Chan
    • 1
  • C. M. Kent
    • 1
  • N. L. Ortiz-Robinson
    • 1
  1. 1.Department of Mathematics and Applied MathematicsVirginia Commonwealth UniversityRichmondUSA

Personalised recommendations