We put forward the notion of unconditional convergence to an equilibrium of a difference equation. Roughly speaking, it means that can be constructed a wide family of higher order difference equations, which inherit the asymptotic behavior of the original difference equation. We present a sufficient condition for guaranteeing that a second-order difference equation possesses an unconditional stable attractor. Finally, we show how our results can be applied to two families of difference equations recently considered in the literature.
Keywords:difference equations; global asymptotic stability
It is somewhat frequent that the global asymptotic stability of a family of difference equations can be extended to some higher-order ones (see, for example, [1-4]). Consider the following simple example. If φ is the map , the sequence defined by , that is,
No sophisticated tools are needed to reach those conclusions: It suffices to note that the set
The case we are interested in is and we will say that is an unconditional attractor for the map φ, that is, we would consider with to observe that, not only (1) and (2), but all the following recursive sequences converge to , whatever the choice of we make:
In this paper, we proceed as follows. The next section is dedicated to notation and a technical result of independent interest. In Section 3, we introduce the main definition and the main result in this paper, unconditional convergence and a sufficient condition for having it in a general framework. We conclude, in Section 4, showing how the later theorem can be applied to provide short proofs for some recent convergence results on two families of difference equations and to improve them.
This section is mainly devoted to the notation we employ. In the first part, we establish some operations between subsets of real numbers and we clarify how we identify a function with a multifunction. We noticed that set-valued difference equations are not concerned with us in this paper. The reason for dealing with those set operations and notation is because it allows us to manage unboundedness and singular situations in a homogeneous way. In the second part, we introduce the families of maps (a kind of averages of their variables) that we shall employ in the definition of unconditional convergence. We finish the section with a technical result on monotone sequences converging to the fixed point of a monotone continuous function.
2.1 Basic notations
Remark 1 We introduce the above notation in order to manage unboundedness and singular situations, but we point out that these are natural set-valued extensions for the arithmetic operations. Let X, Y be compact (Hausdorff) spaces, U a dense subset of X and . The closure of the graph of f in defines an upper semicontinuous compact-valued map by , that is, by (see ). Furthermore, as usual, one writes for , thereby obtaining a map .
Notice that both relations ≤ and < are transitive but neither reflexive nor symmetric.
2.1.2 Canonical injections
When no confusion is likely to arise, we identify with , that is, in the sequel we consider the fixed injection of into and we identify with its image. We must point out that, under this convention, when a is expected to be subset of A, we understand ‘’ as ‘there is with ’. For instance, one has , .
2.1.3 Extension of a function as a multifunction
As we have announced, the unconditional convergence of a difference equation guarantees that there exists a family of difference equations that inherit its asymptotic behavior. Here, we define the set of functions that we employ to construct that family of difference equations.
We note that the functions in satisfy that their behavior is enveloped by the maximum and minimum functions of its variables, which is a common hypothesis in studying higher order nonlinear difference equations.
2.3 A technical result
Because of the continuity of F, we conclude that
is decreasing in the set
3 Unconditional convergence to a point
Definition 1 The point μ is said to be an unconditional equilibrium of h (respectively, unconditional stable equilibrium, unconditional attractor inV) if it is an equilibrium (respectively, stable equilibrium, attractor in V) of for all .
3.1 Sufficient condition for unconditional convergence
After giving Definitions 1 and 2 we are going to prove a result guaranteeing that a general second order difference equation as in (4) has an unconditional stable attractor.
The functions and are defined in the obvious way. Notice that is the limit of a monotone increasing sequence of continuous functions, thus it is lower-semicontinuous, likewise is an upper-semicontinuous function. Remember that we denote both φ and by φ.
We are in conditions of presenting and proving our main result.
Now, if we see that
By applying (H2), we see that
Proof of Lemma 2
and , then
• (iii): Since , it is worth considering the following three cases for each : first, , and then (after probing continuity, monotonicity and statement (iv)), we proceed with the case , and finally with , .Case and : Since when and when , we see that
• Monotonicity and (iv): Suppose
, then , or , thus
when . □
4 Examples and applications
The paper  is devoted to prove that every positive solution to the difference equation
Although paper  complements , where the case had been considered, it should be noticed that the case , is not dealt with in . Furthermore, we cannot assure the global attractivity in this case.
The results in  can be easily obtained by applying Theorem 1 above. Furthermore, we slightly improve the results in  by establishing the unconditional stability of the equilibrium , whenever , . We may assume without loss of generality that the initial values are greater than A. Here, and in the sequel .
As for condition
Since the function
By this reasoning, we also get for free, unconditional stable convergence for several difference equations as, for instance:
just considering respectively
Here, . In 2003, three conjectures on the above equation were posed in . In all three cases (, ; , ; and , respectively) it was postulated the global asymptotic stability of the equilibrium. These conjectures have resulted in several papers since then (see [9-12]). Let us see when there is unconditional convergence.
If we choose
Other choices of λ result on the unconditional stable convergence of difference equations such as
The authors declare that they have no significant competing financial, professional, or personal interests that might have influenced the performance or presentation of the work described in this paper.
Both authors contributed to each part of this work equally and read and approved the final version of the manuscript.
Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.
We are grateful to the anonymous referees for their helpful comments and suggestions. This research was supported in part by the Spanish Ministry of Science and Innovation and FEDER, grant MTM2010-14837.
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