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Dynamic behavior of a nonlinear rational difference equation and generalization
Advances in Difference Equations volume 2011, Article number: 36 (2011)
Abstract
This paper is concerned about the dynamic behavior for the following high order nonlinear difference equation x n = (x n-k + x n-m + x n-l )/(x n-k x n-m + x n-m x n-l +1) with the initial data and 1 ≤ k ≤ m ≤ l. The convergence of solution to this equation is investigated by introducing a new sequence, which extends and includes corresponding results obtained in the references (Li in J Math Anal Appl 312:103-111, 2005; Berenhaut et al. Appl. Math. Lett. 20:54-58, 2007; Papaschinopoulos and Schinas J Math Anal Appl 294:614-620, 2004) to a large extent. In addition, some propositions for generalized equations are reported.
1 Introduction
Our aim in this paper is to study the dynamical behavior of the following equation
where the initial data and 1 ≤ k ≤ m ≤ l..
The study of properties of similar difference equations has been an area of intense interest in recent years [1–3]. There have been a lot of work concerning the behavior of the solution. In particular, Çinar [4] studied the properties of positive solution to
Yang et al. [5] investigated the qualitative behavior of the recursive sequence
Li et al. [6] studied the global asymptotic of the following nonlinear difference equation
with a ≥ 0.
For more similar work, one can refer to [7–9] and references therein. Investigation of the equation (1.1) is motivated by the above studies. However, due to the special nonlinear relation, the methods mentioned in the references [4, 5, 7] do not always work for the equation (1.1). In fact, equation (1.1) has lost the perfect symmetry. To this end, we introduce a simple transformed sequence to construct a contraction to prove the convergence of solutions, and apply this way solving a class of general equation.
The rest of this paper proceeds as follows. In Sect. 2, we introduce some definitions and preliminary lemmas. Section 3 contains the main results and their proofs. In Sect. 4, we prove the stability for generalized rational difference equations and present our conjectures for similar equations.
2 Preliminaries
In this section, we introduce some basic but important preliminary lemmas and notation. For any x i ∈ ℝ+, we define a new sequence as . With the help of the transformed sequence , we can deduce the following conclusion.
Lemma 1. Suppose the function f is defined by
then f is decreasing in x and z if and only if y > 1 and increasing in x and z if and only if y < 1. Similarly, f is decreasing in y if and only if x + z > 1, conversely, it is increasing in y.
Proof. This conclusion follows directly from the fact
and
Since x and z is symmetrical, then the proposition is obvious. □
Moreover, we can also prove the following contraction lemma which is useful in showing convergence of solutions in the transformed space mentioned in first paragraph of this section.
Lemma 2. Suppose x n satisfying the equation (1.1), for any n ≥ l and , we have
Proof. Noticed that
and hence from (1.1), x n ≤ 1 whenever x n -m - 1 and x n-k + x n-l - 1 are of the same signs, otherwise, x n ≥ 1. Let x n-k = u, x n-m = v, x n-l = w. The RHS of (2.4) is obvious. Next we prove the LHS part. Indeed we have eight cases to consider. when (1 - v)(u + w - 1) ≥ 0, then
Case (1) (u ≤ 1, v ≤ 1, w ≥ 1, u + w ≥ 1). Here, by lemma 1, note that v* ≥ 1, we have
Case (2) (u ≥ 1, v ≤ 1, w ≤ 1, u + w ≥ 1). Here, since v* ≥ 1, w* ≥ 1, we have
Case (3) (u ≥ 1, v ≤ 1, w ≥ 1, u + w ≥ 1). Similarly,
Case (4) (u ≤ 1, v ≥ 1, w ≤ 1, u + w ≤ 1). Here,
Oppositely, if (1 - v)(u + w - 1) ≤ 0, from the definition of x*, it is obvious that
Case (5) (u ≤ 1, v ≤ 1, w ≤ 1, u + w ≤ 1). By definition of , Lemma 1 and the fact v* ≥ 1, we have
Case (6) (u ≥ 1, v ≥ 1, w ≤ 1, u + w ≥ 1). Here, we have
Case (7) (u ≤ 1, v ≥ 1, w ≥ 1, u + w ≥ 1). Similarly, we have
Case (8) (u ≥ 1, v ≥ 1, w ≥ 1, u + w ≥ 1). By the same way, we have
Inequalities (2.6)-(2.13) suggest our claim. □
Remark. In fact, by Lemma 1 and in view of u* ≥ 1 and w* ≥ 1, the result can also be derived from the argument for front eight different cases.
Now let for all n ≥ l. By Lemma 2, we can deduce the following consequence.
Lemma 3. The sequence {X i } is monotonically non-increasing in i which is much greater than l.
Since X i ≥ 1 for i ≥ l, Lemma 3 implies that as i tends to infinity, the sequence {X i } convergence to some limit, denote X, where X ≥ 1.
3 Convergence of solutions
In what follows, we state and prove our main result in the sequence space.
Theorem 1. Suppose the initial data of equation (1.1) . Then the solution sequence {x i } converges to the unique positive equilibrium.
Proof. Note that it suffices to show that the transformed sequence converges to 1. By the definition of X i , the values of X i are taken on by entries in the sequence , and as well, by Lemma 2, for i ≥ m. Suppose X > 1, then for any ε ∈ (0, X), we can find an N such that , and for i ≥ N - l, .
Next we consider the eight possible cases again, and show that X = 1. From the definitions of , X i and X, the result follows.
Case (1) (u ≤ 1, v ≤ 1, w ≥ 1, u + w ≥ 1). Here, by lemma 1, we have
Hence
Case (2) (u ≥ 1, v ≤ 1, w ≤ 1, u + w ≥ 1). The argument is identical to that in Case (1).
Case (3) (u ≥ 1, v ≤ 1, w ≥ 1, u + w ≥ 1). Here,
Therefore
Case (4) (u ≤ 1, v ≤ 1, w ≤ 1, u + v ≤ 1). Here,
From this, we have
Namely,
Case (5) (u ≤ 1, v ≤ 1, w ≤ 1, u + w ≤ 1). We have
which also implies
Case (6) (u ≥ 1, v ≥ 1, w ≤ 1, u + w ≥ 1). Here,
We have
Case (7) (u ≤ 1, v ≥ 1, w ≥ 1, u + w ≥ 1). Here, it follows
By the same argument with Case(6), we have
Case (8) (u ≥ 1, v ≥ 1, w ≥ 1, u + w ≥ 1). It here derives
Hence
Collecting all above inequalities which imply X = 1 since ε > 0 is arbitrary, we complete the proof. □
4 Generalization
As mentioned above, the global asymptotic stability of positive solutions to the various equation listed above suggests that the same potentially holds for similar rational equations. We can deduce the following natural generalization of (1.1) and (1.4).
Corollary. Let s ∈ ℕ+ and Z s denote the set Z s = {1, 2,..., s}. Suppose that {x i } satisfies the form
with initial value x -k , x -k+1,..., x -1 ∈ ℝ+, here . Then the sequence {x i } converges to the unique equilibrium 1.
Remark. If we consider the equation which is added a constant a onto numerator and denominator of (4.1), the result is still viable. Indeed this corollary covers the results in [6].
Moreover, consulting the results of article [6, 7, 10], by the similar way to Lemma 2, we have the following generalization.
Theorem 2. Suppose and satisfying . Then the equation
with the corresponding positive initial data has a unique positive equilibrium , and every solution of (4.2) converges to this point.
Proof. Let be a solution sequence of equation (4.2) with initial data x -p , x -p+1, x 0 ∈ ℝ+, where p = max{k r , m s , l t }. By the definition of , From the equation (4.2), the arguments in Lemma 2 and the hypothesis, it follows that for any n ≥ 0,
from which we get that for any n ≥ 0 and 0 ≤ i ≤ m 1, . Hence the sequence with 0 ≤ i ≤ m 1 is convergent. Denote the limit as , then A i ≥ 1. Write and for any integer n. Then there exists some 0 ≤ j ≤ m 1 such that . From (4.3), it suggests .
Combining the facts 1 + ab ≥ a + b and , where a ≥ 1, b ≥ 1 and c ≥ 1, for the different situation in Theorem 1, we have
Therefore
from which it follows M = 1. This implies A i = 1 for 0 ≤ i ≤ m 1 and . Since , we obtain . □
Remark. The stability of solution to equation (4.2) is ever proposed to consider as a conjecture by K.S.Berenhaut etc. in [7]. Indeed, Theorem 1 proved the conjecture partially.
In addition, gathering lots of relevant work listed in reference, we put forward the following conjecture.
Conjecture. Let s ∈ ℕ+, Z s = {1, 2,..., s} and l ij ≥ 0. Suppose that {x i } satisfies
with x -k , x -k+1,..., x -1 ∈ ℝ+, , then the sequence converges to the unique equilibrium 1.
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Acknowledgements
The authors would like to thank the reviewers and the editors for their valuable suggestions and comments; The authors wish to express their deep gratitude to Professor C.Y. Wang for his valuable advice and constant encouragement for this work supported in part by Natural Science Foundation for Colleges and Universities in Hebei Province(Z2011111, Z2011162) and Human Resources and Social Security Subject of Hebei Province(JRS-2011-1042).
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QS completed the main part of this paper, QX and GY corrected the main theorems. XL participated in the design and coordination. All authors read and approved the final manuscript.
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Shi, Q., Xiao, Q., Yuan, G. et al. Dynamic behavior of a nonlinear rational difference equation and generalization. Adv Differ Equ 2011, 36 (2011). https://doi.org/10.1186/1687-1847-2011-36
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DOI: https://doi.org/10.1186/1687-1847-2011-36