 Research
 Open Access
Unconditional convergence of difference equations
 Daniel Franco^{1} and
 Juan Peran^{1}Email author
https://doi.org/10.1186/16872770201363
© Franco and Peran; licensee Springer. 2013
Received: 19 December 2012
Accepted: 20 March 2013
Published: 28 March 2013
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 secondorder 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
For all ${y}_{1},{y}_{2},a>0$, the sequence defined by Equation (1) converges to the unique fixed point ${\mu}_{\phi}=a+1$ of the function ${F}_{\phi}$. However, the behavior of Equation (2) depends on the parameter a:

For $a\ge 1$, the odd and even index terms converge respectively to some limits, ${\mu}_{1}\in [1,+\mathrm{\infty}]$ and ${\mu}_{1}/({\mu}_{1}1)\cap [1,+\mathrm{\infty}]$, where ${\mu}_{1}$ may depend on ${y}_{1}$, ${y}_{2}$ (for $a=1$).

For $0<a<1$, it converges to ${\mu}_{\phi}=a+1$, whatever the choice of ${y}_{1},{y}_{2}>0$ one makes.
Therefore, one of the following statements holds: ${\mu}_{2}={\mu}_{1}/({\mu}_{1}1)\cap [1,+\mathrm{\infty}]$, with $a=1$, or $\{{\mu}_{1}\}\cup \{{\mu}_{2}\}\in \{\{1,+\mathrm{\infty}\},\{1+a\}\}$.
If $\{{\mu}_{1}\}\cup \{{\mu}_{2}\}=\{1,+\mathrm{\infty}\}$, then that of the sequences, ${y}_{2n+1}$ or ${y}_{2n}$, which converges to +∞, has to be nondecreasing. Just look at Equation (2) to conclude that $a\ge 1$ whenever $\{{\mu}_{1}\}\cup \{{\mu}_{2}\}=\{1,+\mathrm{\infty}\}$.
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 setvalued 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 ${\mathrm{\Lambda}}_{m}^{k}$ (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 $\overline{\mathbb{R}}=[\mathrm{\infty},+\mathrm{\infty}]$ of ℝ endowed with the usual order and compact topology.
2.1.1 Operations and preorder in ${2}^{\overline{\mathbb{R}}}\setminus \{\mathrm{\varnothing}\}$
where ∗ stands for ‘+’, ‘−’, ‘⋅’ or ‘/’. We also agree to write $A\ast \mathrm{\varnothing}=\mathrm{\varnothing}\ast A=\mathrm{\varnothing}$.
Remark 1 We introduce the above notation in order to manage unboundedness and singular situations, but we point out that these are natural setvalued extensions for the arithmetic operations. Let X, Y be compact (Hausdorff) spaces, U a dense subset of X and $f:U\to Y$. The closure $\overline{Gr(f)}$ of the graph of f in $X\times Y$ defines an upper semicontinuous compactvalued map $\overline{f}:X\to {2}^{Y}$ by $\overline{f}(x)=\{y\in Y:(x,y)\in \overline{Gr(f)}\}$, that is, by $Gr(\overline{f})=\overline{Gr(f)}$ (see [5]). Furthermore, as usual, one writes $\overline{f}(A)={\bigcup}_{x\in A}\overline{f}(x)$ for $A\in {2}^{X}$, thereby obtaining a map $\overline{f}:{2}^{X}\to {2}^{Y}$.
To extend arithmetic operations, consider $X=\overline{\mathbb{R}}\times \overline{\mathbb{R}}$, $Y=\overline{\mathbb{R}}$ and $U=\mathbb{R}\times \mathbb{R}$, when f denotes addition, substraction or multiplication, and $U=\mathbb{R}\times (\mathbb{R}\setminus \{0\})$, when f denotes division.
Also define $A\le B$ (respectively $A<B$) to be true if and only if $A\ne \mathrm{\varnothing}$, $B\ne \mathrm{\varnothing}$ and $a\le b$ (respectively $a<b$) for all $a\in A$, $b\in B$. Here $A,B\in {2}^{\overline{\mathbb{R}}}$.
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 $a\in \overline{\mathbb{R}}$ with $\{a\}\in {2}^{\overline{\mathbb{R}}}$, that is, in the sequel we consider the fixed injection $a\to \{a\}$ of $\overline{\mathbb{R}}$ into ${2}^{\overline{\mathbb{R}}}$ and we identify $\overline{\mathbb{R}}$ with its image. We must point out that, under this convention, when a is expected to be subset of A, we understand ‘$a\in A$’ as ‘there is $b\in A$ with $a=\{b\}$’. For instance, one has $0\cdot (+\mathrm{\infty})=\overline{\mathbb{R}}$, $1/0=\{\mathrm{\infty},+\mathrm{\infty}\}$.
2.1.3 Extension of a function as a multifunction
Consider a map $h:{\overline{\mathbb{R}}}^{m}\to {2}^{\overline{\mathbb{R}}}$ and denote by $\mathcal{D}(h)$ the set formed by those $x\in {\overline{\mathbb{R}}}^{m}$ for which there is $b\in \overline{\mathbb{R}}$ with $h(x)=\{b\}$.
If $A\in {2}^{({\overline{\mathbb{R}}}^{m})}$, then $h(A)\in {2}^{\overline{\mathbb{R}}}$ is defined to be ${\bigcup}_{a\in A}h(a)$. Also, if $B\in {({2}^{\overline{\mathbb{R}}})}^{m}$, then $h(B)\in {2}^{\overline{\mathbb{R}}}$ is defined to be $h(B)=h({B}_{1}\times \cdots \times {B}_{m})$.
when $U=\mathbb{R}\setminus \{0\}$ and $\phi (x)=sin(1/x)$.
2.2 The maps in ${\mathrm{\Lambda}}_{m}^{k}$ and ${\mathrm{\Lambda}}^{k}$
As we have announced, the unconditional convergence of a difference equation guarantees that there exists a family of difference equations that inherit its asymptotic behavior. Here, we define the set of functions that we employ to construct that family of difference equations.
We note that the functions in ${\mathrm{\Lambda}}^{k}$ 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 ${\mathrm{\Lambda}}^{1}$ are:

$\lambda ({x}_{1},\dots ,{x}_{m})={\sum}_{j=1}^{m}{\alpha}_{j}{x}_{j}$, with ${\alpha}_{j}\ge 0$ for $j=1,\dots ,m$, ${\sum}_{j=1}^{m}{\alpha}_{j}=1$.
An important particular case is ${\alpha}_{{j}_{0}}=1$, ${\alpha}_{j}=0$ for $j\ne {j}_{0}$.

$\lambda ({x}_{1},\dots ,{x}_{m})=\{\begin{array}{cc}{\prod}_{j=1}^{m}{x}_{j}^{{\alpha}_{j}}\hfill & \text{if}{min}_{1\le j\le m}{x}_{j}0,\hfill \\ {min}_{1\le j\le m}{x}_{j}\hfill & \text{if}{min}_{1\le j\le m}{x}_{j}\le 0,\hfill \end{array}$ with ${\alpha}_{j}\ge 0$ for $j=1,\dots ,m$, ${\sum}_{j=1}^{m}{\alpha}_{j}=1$.
We refer to this function simply as $\lambda ({x}_{1},\dots ,{x}_{m})={\prod}_{j=1}^{m}{x}_{j}^{{\alpha}_{j}}$, when it is assumed that $\lambda \in {\mathrm{\Lambda}}^{1}$.

$\lambda ({x}_{1},\dots ,{x}_{m})={max}_{j\in J}{x}_{j}$, where J is a nonempty subset of $\{1,\dots ,m\}$.

$\lambda ({x}_{1},\dots ,{x}_{m})={min}_{j\in J}{x}_{j}$, where J is a nonempty subset of $\{1,\dots ,m\}$.
2.3 A technical result
Assume $\mathrm{\infty}\le a<b\le +\mathrm{\infty}$, in the rest of this section. Recall that a continuous nonincreasing function $F:[a,b]\to [a,b]$ has a unique fixed point $\mu \in [a,b]$, that is, $\{\mu \}=Fix(F)$.
for $k\ge 1$.
Then $({a}_{k})$ and $({b}_{k})$ are, respectively, a nondecreasing and a nonincreasing sequence in $[a,b]$. Furthermore, ${a}_{k}\le \mu \le {b}_{k}$ for all k and $\{lim{a}_{k},\mu ,lim{b}_{k}\}\subset Fix(F\circ F)$.
Therefore, $({a}_{k})$ is a nondecreasing sequence, so by definition, $({b}_{k})$ is nonincreasing.
□
Unless $F(x)=b<+\mathrm{\infty}$, the map F verifies $F(F(x))>x$ if and only if there exists ${\u03f5}_{0}>0$ such that $F(F(x)+\u03f5)\u03f5>x$ for all $\u03f5\in [0,{\u03f5}_{0})$.
As a consequence, $({a}_{k})$, $({b}_{k})$ are well defined, without the need of extending F.
3 Unconditional convergence to a point
is always well defined whatever the initial points ${y}_{1},\dots ,{y}_{k}\in \overline{\mathbb{R}}$ are, even though the ${y}_{n}$ are subsets of $\overline{\mathbb{R}}$, rather than points.
A point $\mu \in \overline{\mathbb{R}}$ is said to be an equilibrium for the map h if $h(\mu ,\dots ,\mu )=\{\mu \}$. The equilibrium μ is said to be stable if, for each neighborhood V of μ in $\overline{\mathbb{R}}$, there is a neighborhood W of $(\mu ,\dots ,\mu )$ in $\mathcal{D}(h)$ such that ${y}_{n}\in V$ for all n, whenever $({y}_{1},\dots ,{y}_{k})\in W$.
The equilibrium μ is said to be an attractor in a neighborhood V of μ in $\overline{\mathbb{R}}$, if ${y}_{n}\in \overline{\mathbb{R}}$ for all n and ${y}_{n}\to \mu $ in $\overline{\mathbb{R}}$, whenever ${y}_{n}\in V$ for $n\le k$.
Definition 1 The point μ is said to be an unconditional equilibrium of h (respectively, unconditional stable equilibrium, unconditional attractor in V) if it is an equilibrium (respectively, stable equilibrium, attractor in V) of $h\circ \lambda $ for all $\lambda \in {\mathrm{\Lambda}}^{k}$.
Definition 2 We define the equilibria, stable equilibria, attractors, unconditional equilibria, unconditional stable equilibria and unconditional attractors of a continuous function $\phi :U\subset {\overline{\mathbb{R}}}^{k}\to \overline{\mathbb{R}}$ to be those of $\stackrel{\u02c6}{\phi}$.
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 $\mathrm{\infty}<c\le a<b\le d\le +\mathrm{\infty}$ and consider in the sequel a continuous function $\phi :(a,b)\times (c,d)\to (c,d)$, satisfying the following conditions:
(H1) $\phi ({x}_{1},y)<\phi ({x}_{2},y)$, whenever $a<{x}_{2}<{x}_{1}<b$ and $c<y<d$.
whenever $y\in (c,d)\setminus \{{F}_{\phi}(x)\}$.
The functions $\phi (\cdot ,y):[a,b]\to [c,d]$ and $\phi (x,\cdot ):(c,d)\to (c,d)$ are defined in the obvious way. Notice that $\phi (a,\cdot )$ is the limit of a monotone increasing sequence of continuous functions, thus it is lowersemicontinuous, likewise $\phi (b,\cdot )$ is an uppersemicontinuous function. Remember that we denote both φ and $\stackrel{\u02c6}{\phi}$ 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 ${F}_{\phi}$.
 (i)
$a<\phi (x,y)<b$ for all $x,y\in (a,b)$ and $a\le \phi (x,y)\le b$ for all $x,y\in [a,b]$.
 (ii)
If $x\in [a,b]$, $y\in (c,d)$ and $\phi (x,y)=y$, then $y={F}_{\phi}(x)$.
 (iii)
$\phi (x,{F}_{\phi}(x))={F}_{\phi}(x)$ for all $x\in [a,b]$.
 (iv)
${F}_{\phi}$ is decreasing in ${F}_{\phi}^{1}((a,b))$.
We are in conditions of presenting and proving our main result.
Theorem 1 Let $\phi :(a,b)\times (c,d)\to (c,d)$, where $\mathrm{\infty}<c\le a<b\le d\le +\mathrm{\infty}$, be a continuous function satisfying (H1) and (H2). If ${\mu}_{\phi}\in (a,b)$ and $Fix({F}_{\phi}\circ {F}_{\phi})=Fix({F}_{\phi})$, then ${\mu}_{\phi}$ is an unconditional stable attractor of φ in $(a,b)$.
for some ${y}_{1},\dots ,{y}_{m}\in (a,b)$. Notice that ${y}_{n}\in (a,b)$ for all n, as a consequence of (i) in Lemma 2.
we see that ${\mu}_{\phi}$ is an equilibrium.
Therefore, ${F}_{\phi}([{a}^{\prime},{b}^{\prime}])\subset [{a}^{\prime},{b}^{\prime}]$.
whenever ${y}_{n}\in ({a}^{\prime},{b}^{\prime})$ for $n\le m$, thus ${\mu}_{\phi}$ is an unconditional stable equilibrium of φ.
because of the continuity of ${F}_{\phi}$. As ${\mu}_{\phi}\in (a,b)$, this implies $\overline{y}={\mu}_{\phi}$.
for all $n>{n}_{k1}+m$.
As a consequence, the nonincreasing sequence ${w}_{n}$ is bounded below by ${b}_{k1}$. It cannot be the case that $lim{w}_{n}>{b}_{k1}$, because in such a case there is a subsequence ${y}_{{n}_{j}}>{b}_{k1}$ converging to $lim{w}_{n}$ and such that ${\lambda}_{2}({y}_{{n}_{j}1},\dots ,{y}_{{n}_{j}m})$ converges to a point $w\le lim{w}_{n}$.
a contradiction.
□
Proof of Lemma 2

Uniqueness of ${F}_{\phi}$: Let ${y}_{1}<{y}_{2}$ and x in $[a,b]$ such that$\frac{{y}_{i}y}{\phi (x,y)y}\ge 1>0\phantom{\rule{1em}{0ex}}\text{for}i=1,2,y\in ({y}_{1},{y}_{2}).$
a contradiction.

(i): It suffices to prove the first assertion, because $[a,b]$ is a closed set and, by definition,$\phi (x,y)=\{lim\hspace{0.17em}sup\phi ({x}_{n},{y}_{n}):{x}_{n}\to x,{y}_{n}\to y,{x}_{n}\in (a,b),{y}_{n}\in (c,d)\}$
for all $(x,y)\in [a,b]\times [c,d]$. Assume now that $(x,y)\in (a,b)\times (a,b)$. We consider the following three possible situations. If $\phi (x,y)=y$, it is obvious that $\phi (x,y)\in (a,b)$.On the other hand, if $\phi (x,y)>y$ and ${x}^{\prime}\in (a,x)$, then

(ii): Suppose $y\ne {F}_{\phi}(x)$. Since $\phi (x,y)=y\in \overline{\mathbb{R}}$, then $\phi (x,y)y=0$ or $\phi (x,y)y=\overline{\mathbb{R}}$. In any event, it cannot be the case that$\frac{{F}_{\phi}(x)y}{\phi (x,y)y}\ge 1,$
which contradicts hypothesis (H2), thus $y={F}_{\phi}(x)$.

(iii): Since ${F}_{\phi}([a,b])\subset [a,b]\subset [c,d]$, it is worth considering the following three cases for each $x\in [a,b]$: first, $x\in (a,b)$, ${F}_{\phi}(x)\in (c,d)$ and then (after probing continuity, monotonicity and statement (iv)), we proceed with the case $x\in \{a,b\}$, ${F}_{\phi}(x)\in (c,d)$ and finally with $x\in [a,b]$, ${F}_{\phi}(x)\in \{c,d\}$.Case $x\in (a,b)$ and ${F}_{\phi}(x)\in (c,d)$: Since $\phi (x,y)>y$ when $y<{F}_{\phi}(x)$ and $\phi (x,y)<y$ when $y>{F}_{\phi}(x)$, we see that$\phi (x,{F}_{\phi}(x))={F}_{\phi}(x),$
because of the continuity of $\phi (x,\cdot )$.

Monotonicity and (iv): Suppose

Continuity: If $x\in [a,b]$ and$w\in I=(\underset{z\to x}{lim\hspace{0.17em}inf}{F}_{\phi}(z),\underset{y\to x}{lim\hspace{0.17em}sup}{F}_{\phi}(y)),$
and, by (ii), one has $\phi (x,w)=w$. Since $w\in (c,d)$, this would imply ${F}_{\phi}(x)=w$ for all $w\in I$, which is impossible.

(iii) Case $x\in \{a,b\}$ and ${F}_{\phi}(x)\in (c,d)$: Since${F}_{\phi}(t)\le \phi (t,{F}_{\phi}(a))\le {F}_{\phi}(a)\phantom{\rule{1em}{0ex}}\text{for all}t\in (a,b),$
but

(iii) Case ${F}_{\phi}(x)\in \{c,d\}$: First assume ${F}_{\phi}(x)=c$ and recall that, by definition,$\phi (x,c)=\{lim\hspace{0.17em}sup\phi ({x}_{n},{y}_{n}):{x}_{n}\to x,{y}_{n}\to c,{x}_{n}\in (a,b),{y}_{n}\in (c,d)\}.$
which implies ${F}_{\phi}({x}_{n})\ge {c}^{\prime}>c$, eventually for all n.Since ${F}_{\phi}({x}_{n})\to c$, we reach a contradiction. Therefore,
4 Examples and applications
4.1 The difference equation ${y}_{n}=A+{(\frac{{y}_{nk}}{{y}_{nq}})}^{p}$ with $0<p<1$
Here, $k,q\in \{1,2,3,\dots \}$ are fixed numbers.
Although paper [6] complements [7], where the case $p=1$ had been considered, it should be noticed that the case $p=1$, $A\ge 1$ 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 $A+1$, whenever $A\in (0,+\mathrm{\infty})$, $p\in (0,min\{1,(A+1)/2\})$. We may assume without loss of generality that the initial values ${y}_{1},\dots ,{y}_{m}$ are greater than A. Here, and in the sequel $m=max\{k,q\}$.
Notice that ${f}_{x}$ is concave, ${f}_{x}(0)=A>0$, and ${f}_{x}(+\mathrm{\infty})=\mathrm{\infty}$.
we see that condition (H2) holds and ${\mu}_{\phi}=A+1$.
has a unique critical point in $(A,+\mathrm{\infty})$ and $h(A+1)=0$, $h(+\mathrm{\infty})=A$, the necessary and sufficient condition for (5) to hold is that ${h}^{\prime}(A+1)\ge 0$, that is, $p\le (A+1)/2$.
where $m=max\{q,r,s,t\}$.
4.2 The difference equation ${y}_{n}=\frac{\alpha +\beta {y}_{n1}}{A+B{y}_{n1}+C{y}_{n2}}$
Here, $\alpha ,\beta ,A,B,C,{y}_{0},{y}_{1}\ge 0$. In 2003, three conjectures on the above equation were posed in [8]. In all three cases ($B=0$, $\alpha ,\beta ,A,C>0$; $A=0$, $\alpha ,\beta ,B,C>0$; and $\alpha ,\beta ,A,B,C>0$, 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.
Therefore, ${\mu}_{\phi}$ is an unconditional stable attractor of φ in $(a,b)=(0,+\mathrm{\infty})$ whenever $A>\beta \ge 0$, $D>0$ and $\alpha >0$.
whenever $A>\beta \ge 0$, $B+C>0$, $C\ge 0$ and $\alpha >0$.
with $A>\beta \ge 0$, $D>0$, $\alpha >0$.
Declarations
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 MTM201014837.
Authors’ Affiliations
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