SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Unconditional convergence of difference equations

Daniel Franco and Juan Peran*

Author Affiliations

Departamento de Matemática Aplicada, Universidad Nacional de Educación a Distancia (UNED), C/ Juan del Rosal 12, Madrid, 28040, Spain

For all author emails, please log on.

Boundary Value Problems 2013, 2013:63  doi:10.1186/1687-2770-2013-63


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/63


Received:19 December 2012
Accepted:20 March 2013
Published:28 March 2013

© 2013 Franco and Peran; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

MSC: 39A11.

Keywords:
difference equations; global asymptotic stability

1 Introduction

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M1">View MathML</a>, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M2">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M3">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M4">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M5">View MathML</a>, converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M6">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M7">View MathML</a>. Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8">View MathML</a> is the function satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M9">View MathML</a>. Obviously, the second-order difference equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M10">View MathML</a>

also converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M11">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M12">View MathML</a>. Let us continue to add complexity, by considering the second-order difference equations

(1)

(2)

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M15">View MathML</a>, the sequence defined by Equation (1) converges to the unique fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M16">View MathML</a> of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8">View MathML</a>. However, the behavior of Equation (2) depends on the parameter a:

• For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M18">View MathML</a>, the odd and even index terms converge respectively to some limits, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M20">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M21">View MathML</a> may depend on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M23">View MathML</a> (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M24">View MathML</a>).

• For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M25">View MathML</a>, it converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M16">View MathML</a>, whatever the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M27">View MathML</a> one makes.

No sophisticated tools are needed to reach those conclusions: It suffices to note that the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M28">View MathML</a>

must be either finite or equal to ℕ. As the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M30">View MathML</a> are then both eventually monotone, they converge in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M31">View MathML</a> to some limits, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M33">View MathML</a>, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M34">View MathML</a>

Therefore, one of the following statements holds: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M35">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M24">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M37">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M38">View MathML</a>, then that of the sequences, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M29">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M30">View MathML</a>, which converges to +∞, has to be nondecreasing. Just look at Equation (2) to conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M18">View MathML</a> whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M38">View MathML</a>.

The case we are interested in is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M25">View MathML</a> and we will say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M44">View MathML</a> is an unconditional attractor for the map φ, that is, we would consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M45">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M25">View MathML</a> to observe that, not only (1) and (2), but all the following recursive sequences converge to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M44">View MathML</a>, whatever the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M48">View MathML</a> 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.

2 Preliminaries

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 onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M50">View MathML</a> (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

We consider the two points compactification <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M51">View MathML</a> of ℝ endowed with the usual order and compact topology.

2.1.1 Operations and preorder in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M52">View MathML</a>

We define the operations ‘+’, ‘−’, ‘⋅’ and ‘/’ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M53">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M54">View MathML</a>

where ∗ stands for ‘+’, ‘−’, ‘⋅’ or ‘/’. We also agree to write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M55">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M56">View MathML</a>. The closure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M57">View MathML</a> of the graph of f in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M58">View MathML</a> defines an upper semicontinuous compact-valued map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M59">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M60">View MathML</a>, that is, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M61">View MathML</a> (see [5]). Furthermore, as usual, one writes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M62">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M63">View MathML</a>, thereby obtaining a map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M64">View MathML</a>.

To extend arithmetic operations, consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M66">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M67">View MathML</a>, when f denotes addition, substraction or multiplication, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M68">View MathML</a>, when f denotes division.

Also define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M69">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M70">View MathML</a>) to be true if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M73">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M74">View MathML</a>) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M76">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M77">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M78">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M79">View MathML</a>, that is, in the sequel we consider the fixed injection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M80">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M81">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M82">View MathML</a> and we identify <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M81">View MathML</a> with its image. We must point out that, under this convention, when a is expected to be subset of A, we understand ‘<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M75">View MathML</a>’ as ‘there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M85">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M86">View MathML</a>’. For instance, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M88">View MathML</a>.

2.1.3 Extension of a function as a multifunction

Consider a map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M89">View MathML</a> and denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M90">View MathML</a> the set formed by those <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M91">View MathML</a> for which there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M92">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M93">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M94">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M95">View MathML</a> is defined to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M96">View MathML</a>. Also, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M97">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M98">View MathML</a> is defined to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M99">View MathML</a>.

For each function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M100">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M101">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M102">View MathML</a>

It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M103">View MathML</a> if and only if x is in the closure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M104">View MathML</a> of U in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M105">View MathML</a>. Also notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M106">View MathML</a>

when φ is continuous. In this case, and if no confusion is likely to arise, we agree to denote also by φ the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M107">View MathML</a>. For example, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M108">View MathML</a>

when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M110">View MathML</a>.

2.2 The maps in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M112">View MathML</a>

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.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M113">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M50">View MathML</a> be the set formed by the maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M115">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M116">View MathML</a>

(3)

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M117">View MathML</a> whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M119">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M112">View MathML</a> be defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M121">View MathML</a>

We note that the functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M112">View MathML</a> 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.

Some trivial examples of functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M123">View MathML</a> are:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M124">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M125">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M127">View MathML</a>.

An important particular case is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M129">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M130">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M131">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M125">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M133">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M134">View MathML</a>.

We refer to this function simply as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M135">View MathML</a>, when it is assumed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M136">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M137">View MathML</a>, where J is a nonempty subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M138">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M139">View MathML</a>, where J is a nonempty subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M138">View MathML</a>.

2.3 A technical result

Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M141">View MathML</a>, in the rest of this section. Recall that a continuous non-increasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M142">View MathML</a> has a unique fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M143">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M144">View MathML</a>.

Lemma 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M145">View MathML</a>be a continuous non-increasing function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M144">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M147">View MathML</a>. Define<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M148">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M150">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M151">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M152">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M153">View MathML</a>.

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M154">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M155">View MathML</a>are, respectively, a nondecreasing and a nonincreasing sequence in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M156">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M157">View MathML</a>for allkand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M158">View MathML</a>.

Proof Since the map F is nonincreasing and taking into account the hypothesis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M159">View MathML</a>, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M154">View MathML</a> is a nondecreasing sequence. Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M161">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M162">View MathML</a> to reach a contradiction

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M163">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M154">View MathML</a> is a nondecreasing sequence, so by definition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M155">View MathML</a> is nonincreasing.

On the other hand, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M166">View MathML</a>, we see by induction that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M157">View MathML</a> for all k,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M168">View MathML</a>

Because of the continuity of F, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M169">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M170">View MathML</a>

 □

Remark 2 Suppose F not to be identically equal to +∞ and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M171">View MathML</a>. The map

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M172">View MathML</a>

is decreasing in the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M173">View MathML</a>

Unless <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M174">View MathML</a>, the map F verifies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M175">View MathML</a> if and only if there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M176">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M177">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M178">View MathML</a>.

Therefore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M179">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M147">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M181">View MathML</a> and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M182">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M183">View MathML</a>

As a consequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M155">View MathML</a> are well defined, without the need of extending F.

3 Unconditional convergence to a point

For a map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M186">View MathML</a>, the difference equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M187">View MathML</a>

(4)

is always well defined whatever the initial points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M188">View MathML</a> are, even though the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M2">View MathML</a> are subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M81">View MathML</a>, rather than points.

A point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M191">View MathML</a> is said to be an equilibrium for the map h if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M192">View MathML</a>. The equilibrium μ is said to be stable if, for each neighborhood V of μ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M193">View MathML</a>, there is a neighborhood W of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M194">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M90">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M196">View MathML</a> for all n, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M197">View MathML</a>.

The equilibrium μ is said to be an attractor in a neighborhood V of μ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M193">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M199">View MathML</a> for all n and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M200">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M201">View MathML</a>, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M202">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M203">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M204">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M205">View MathML</a>.

Definition 2 We define the equilibria, stable equilibria, attractors, unconditional equilibria, unconditional stable equilibria and unconditional attractors of a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M206">View MathML</a> to be those of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M107">View MathML</a>.

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.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M208">View MathML</a> and consider in the sequel a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M209">View MathML</a>, satisfying the following conditions:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M210">View MathML</a>, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M211">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M212">View MathML</a>.

(H2) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M213">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M214">View MathML</a>

whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M215">View MathML</a>.

The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M216">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M217">View MathML</a> are defined in the obvious way. Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M218">View MathML</a> is the limit of a monotone increasing sequence of continuous functions, thus it is lower-semicontinuous, likewise <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M219">View MathML</a> is an upper-semicontinuous function. Remember that we denote both φ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M107">View MathML</a> by φ.

The next lemma, which we prove at the end of this section, shows that if (H1) and (H2) holds we can get some information about the behavior and properties of φ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8">View MathML</a>.

Lemma 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M222">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M208">View MathML</a>, be a continuous function satisfying (H1) and (H2). Then the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8">View MathML</a>in (H2) is unique and it is a continuous nonincreasing map, thus it has a unique fixed point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M225">View MathML</a>. Furthermore,

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M226">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M227">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M228">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M229">View MathML</a>.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M231">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M232">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M233">View MathML</a>.

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M9">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M230">View MathML</a>.

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8">View MathML</a>is decreasing in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M237">View MathML</a>.

We are in conditions of presenting and proving our main result.

Theorem 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M222">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M208">View MathML</a>, be a continuous function satisfying (H1) and (H2). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M240">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M241">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M242">View MathML</a>is an unconditional stable attractor ofφin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M243">View MathML</a>.

Proof of Theorem 1 Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M244">View MathML</a> and denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M245">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M246">View MathML</a>. Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M247">View MathML</a> for all n, as a consequence of (i) in Lemma 2.

We are going to prove first that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M225">View MathML</a> is a stable equilibrium of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M249">View MathML</a>. By (iii) in Lemma 2, as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M250">View MathML</a>

we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M225">View MathML</a> is an equilibrium.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M252">View MathML</a>. Because of the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M254">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M255">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M256">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M257">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M258">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M259">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M260">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M261">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M262">View MathML</a>.

By replacing a, b by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M263">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M264">View MathML</a> in Lemma 2(i), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M265">View MathML</a>

whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M266">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M267">View MathML</a>, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M225">View MathML</a> is an unconditional stable equilibrium of φ.

Now, if we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M269">View MathML</a>

we are done with the whole proof. Indeed, for each accumulation point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M270">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M271">View MathML</a>, one would have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M272">View MathML</a>

because of the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8">View MathML</a>. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M240">View MathML</a>, this implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M275">View MathML</a>.

Therefore, as a consequence of Lemma 1, it suffices to find an increasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M276">View MathML</a> of natural numbers such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M277">View MathML</a>

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M278">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M279">View MathML</a> are defined as in Lemma 1, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M280">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M281">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M282">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M283">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M284">View MathML</a>

Having in mind that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M285">View MathML</a> satisfies (3), we find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M276">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M287">View MathML</a> as follows. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M288">View MathML</a>

and momentarily assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M289">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M290">View MathML</a> in such a way that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M291">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M292">View MathML</a>

and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M293">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M289">View MathML</a>.

As a consequence, the nonincreasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M295">View MathML</a> is bounded below by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M296">View MathML</a>. It cannot be the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M297">View MathML</a>, because in such a case there is a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M298">View MathML</a> converging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M299">View MathML</a> and such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M300">View MathML</a> converges to a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M301">View MathML</a>.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M302">View MathML</a>

one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M303">View MathML</a>

and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M304">View MathML</a>

By applying (H2), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M305">View MathML</a>

a contradiction.

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M306">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M307">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M308">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M309">View MathML</a>

Analogously, we see that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M310">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M311">View MathML</a>

 □

Proof of Lemma 2

• Uniqueness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8">View MathML</a>: Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M313">View MathML</a> and x in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M314">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M315">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M316">View MathML</a>

a contradiction.

• (i): It suffices to prove the first assertion, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M156">View MathML</a> is a closed set and, by definition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M318">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M319">View MathML</a>. Assume now that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M320">View MathML</a>. We consider the following three possible situations. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M232">View MathML</a>, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M322">View MathML</a>.On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M323">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M324">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M325">View MathML</a>

Finally, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M326">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M327">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M328">View MathML</a>

• (ii): Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M329">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M330">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M331">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M332">View MathML</a>. In any event, it cannot be the case that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M333">View MathML</a>

which contradicts hypothesis (H2), thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M334">View MathML</a>.

• (iii): Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M335">View MathML</a>, it is worth considering the following three cases for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M336">View MathML</a>: first, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M337">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M338">View MathML</a> and then (after probing continuity, monotonicity and statement (iv)), we proceed with the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M339">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M338">View MathML</a> and finally with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M336">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M342">View MathML</a>.Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M337">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M344">View MathML</a>: Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M323">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M346">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M326">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M348">View MathML</a>, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M349">View MathML</a>

because of the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M350">View MathML</a>.

• Monotonicity and (iv): Suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M351">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M352">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M353">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M354">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M355">View MathML</a>, thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M356">View MathML</a>

• Continuity: If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M357">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M358">View MathML</a>

then there exist two sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M359">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M360">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M361">View MathML</a>

and, by (ii), one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M362">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M363">View MathML</a>, this would imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M364">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M365">View MathML</a>, which is impossible.

• (iii) Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M366">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M344">View MathML</a>: Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M368">View MathML</a>

and because of the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M8">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M370">View MathML</a>

but

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M371">View MathML</a>

Analogously, it can be seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M372">View MathML</a>.

• (iii) Case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M373">View MathML</a>: First assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M374">View MathML</a> and recall that, by definition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M375">View MathML</a>

Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M376">View MathML</a> for all n. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M377">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M378">View MathML</a>, eventually for all n.Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M379">View MathML</a>, we reach a contradiction. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M380">View MathML</a>

Analogously, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M381">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M382">View MathML</a>. □

4 Examples and applications

4.1 The difference equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M383">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M384">View MathML</a>

The paper [6] is devoted to prove that every positive solution to the difference equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M385">View MathML</a>

converges to the equilibrium <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M386">View MathML</a>, whenever

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M387">View MathML</a>

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M388">View MathML</a> are fixed numbers.

Although paper [6] complements [7], where the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M389">View MathML</a> had been considered, it should be noticed that the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M389">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M391">View MathML</a> is not dealt with in [6]. Furthermore, we cannot assure the global attractivity in this case.

The results in [6] can be easily obtained by applying Theorem 1 above. Furthermore, we slightly improve the results in [6] by establishing the unconditional stability of the equilibrium <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M386">View MathML</a>, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M393">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M394">View MathML</a>. We may assume without loss of generality that the initial values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M395">View MathML</a> are greater than A. Here, and in the sequel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M396">View MathML</a>.

Let

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M398">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M399">View MathML</a>, consider for the moment a fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M400">View MathML</a> and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M11">View MathML</a> to be the unique positive zero of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M402">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M403">View MathML</a>

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M402">View MathML</a> is concave, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M405">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M406">View MathML</a>.

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M11">View MathML</a> is also the unique zero of the increasing function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M408">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M409">View MathML</a>

we see that condition (H2) holds and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M410">View MathML</a>.

As for condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M411">View MathML</a>

(5)

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M412">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M413">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M414">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M415">View MathML</a>

Since the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M416">View MathML</a>

has a unique critical point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M417">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M418">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M419">View MathML</a>, the necessary and sufficient condition for (5) to hold is that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M420">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M421">View MathML</a>.

By this reasoning, we also get for free, unconditional stable convergence for several difference equations as, for instance:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M422">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M423">View MathML</a>

just considering respectively

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M425">View MathML</a>.

4.2 The difference equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M426">View MathML</a>

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M427">View MathML</a>. In 2003, three conjectures on the above equation were posed in [8]. In all three cases (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M428">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M429">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M430">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M431">View MathML</a>; and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M432">View MathML</a>, 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.

Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M433">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M434">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M435">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M436">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M437">View MathML</a>. We solve in y the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M438">View MathML</a> to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M439">View MathML</a>

so we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M440">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M441">View MathML</a> to define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M442">View MathML</a>

A simple calculation shows that (H2) holds, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M240">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M256">View MathML</a>:

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M225">View MathML</a> is an unconditional stable attractor of φ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M447">View MathML</a> whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M448">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M437">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M441">View MathML</a>.

If we choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M451">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M452">View MathML</a>, we obtain unconditional stable convergence for the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M453">View MathML</a>

whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M448">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M455">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M456">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M441">View MathML</a>.

Other choices of λ result on the unconditional stable convergence of difference equations such as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M458">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M459">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M455">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M461">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M441">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M463">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M456">View MathML</a>. Or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M465">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M448">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M437">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M441">View MathML</a>.

Competing interests

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.

Authors’ contributions

Both authors contributed to each part of this work equally and read and approved the final version of the manuscript.

Acknowledgements

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.

References

  1. Agarwal, RP: Difference Equations and Inequalities. Theory, Methods, and Applications, Dekker, New York (2000)

  2. El-Morshedy, HA, Gopalsamy, K: Oscillation and asymptotic behaviour of a class of higher-order non-linear difference equations. Ann. Mat. Pura Appl.. 182, 143–159 (2003). Publisher Full Text OpenURL

  3. El-Morshedy, HA, Liz, E: Globally attracting fixed points in higher order discrete population models. J. Math. Biol.. 53, 365–384 (2006). PubMed Abstract | Publisher Full Text OpenURL

  4. Kocić, VL, Ladas, G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic, Dordrecht (1993)

  5. Beer, G: Topologies on Closed and Closed Convex Sets, Kluwer Academic, Dordrecht (1993)

  6. Berenhaut, KS, Foley, JD, Stević, S: The global attractivity of the rational difference equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M469">View MathML</a>. Proc. Am. Math. Soc.. 136, 103–110 (2008). Publisher Full Text OpenURL

  7. Berenhaut, KS, Foley, JD, Stević, S: The global attractivity of the rational difference equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M471">View MathML</a>. Proc. Am. Math. Soc.. 135, 1133–1140 (2007). Publisher Full Text OpenURL

  8. Kulenović, MRS, Ladas, G, Martins, LF, Rodrigues, IW: The dynamics of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/63/mathml/M473">View MathML</a>: facts and conjectures. Comput. Math. Appl.. 45, 1087–1099 (2003). Publisher Full Text OpenURL

  9. Merino, O: Global attractivity of the equilibrium of a difference equation: an elementary proof assisted by computer algebra system. J. Differ. Equ. Appl.. 17, 33–41 (2011). Publisher Full Text OpenURL

  10. Su, Y-H, Li, W-T, Stević, S: Dynamics of a higher order nonlinear rational difference equation. J. Differ. Equ. Appl.. 11, 133–150 (2005). Publisher Full Text OpenURL

  11. Su, Y-H, Li, W-T: Global asymptotic stability of a second-order nonlinear difference equation. Appl. Math. Comput.. 168, 981–989 (2005). Publisher Full Text OpenURL

  12. Su, Y-H, Li, W-T: Global attractivity of a higher order nonlinear difference equation. J. Differ. Equ. Appl.. 11, 947–958 (2005). Publisher Full Text OpenURL