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Unconditional convergence of difference equations
Boundary Value Problems volume 2013, Article number: 63 (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.
1 Introduction
It is somewhat frequent that the global asymptotic stability of a family of difference equations can be extended to some higherorder ones (see, for example, [1–4]). Consider the following simple example. If φ is the map $\phi (x,y)=1+(ax/y)$, the sequence ${y}_{n}$ defined by ${y}_{n}=\phi (x,{y}_{n1})$, that is,
with ${y}_{1},a,x>0$, converges to ${F}_{\phi}(x)=(1+\sqrt{1+4ax})/2$ for any ${y}_{1}$. Observe that ${F}_{\phi}$ is the function satisfying $\phi (x,{F}_{\phi}(x))={F}_{\phi}(x)$. Obviously, the secondorder difference equation
also converges to ${F}_{\phi}(x)$ for any ${y}_{1},{y}_{2},a,x>0$. Let us continue to add complexity, by considering the secondorder difference equations
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.
No sophisticated tools are needed to reach those conclusions: It suffices to note that the set
must be either finite or equal to ℕ. As the sequences ${y}_{2n+1}$ and ${y}_{2n}$ are then both eventually monotone, they converge in $[1,+\mathrm{\infty}]$ to some limits, say ${\mu}_{1}$ and ${\mu}_{2}$, satisfying
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 $\{{\mu}_{1}\}\cup \{{\mu}_{2}\}=\{1,+\mathrm{\infty}\}$.
The case we are interested in is $0<a<1$ and we will say that ${\mu}_{\phi}=1+a$ is an unconditional attractor for the map φ, that is, we would consider $\phi (x,y)=1+(ax/y)$ with $0<a<1$ to observe that, not only (1) and (2), but all the following recursive sequences converge to ${\mu}_{\phi}=1+a$, whatever the choice of ${y}_{1},\dots ,{y}_{max\{k,m\}}>0$ 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 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 ${2}^{\overline{\mathbb{R}}}\setminus \{\mathrm{\varnothing}\}$
We define the operations ‘+’, ‘−’, ‘⋅’ and ‘/’ in ${2}^{\overline{\mathbb{R}}}\setminus \{\mathrm{\varnothing}\}$ by
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 $h(B)=h({B}_{1}\times \cdots \times {B}_{m})$.
For each function $\phi :U\subset {\overline{\mathbb{R}}}^{m}\to \overline{\mathbb{R}}$, let $\stackrel{\u02c6}{\phi}:{\overline{\mathbb{R}}}^{m}\to {2}^{\overline{\mathbb{R}}}$ be defined by
It is obvious that $\stackrel{\u02c6}{\phi}(x)\ne \mathrm{\varnothing}$ if and only if x is in the closure $\overline{U}$ of U in ${\overline{\mathbb{R}}}^{m}$. Also notice that
when φ is continuous. In this case, and if no confusion is likely to arise, we agree to denote also by φ the map $\stackrel{\u02c6}{\phi}$. For example, we write
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.
For $k,m\in \mathbb{N}$, let ${\mathrm{\Lambda}}_{m}^{k}$ be the set formed by the maps $\lambda :{\overline{\mathbb{R}}}^{m}\to {\overline{\mathbb{R}}}^{k}$ such that
Notice that $\lambda \circ \gamma \in {\mathrm{\Lambda}}_{m}^{k}$ whenever $\lambda \in {\mathrm{\Lambda}}_{r}^{k}$, $\gamma \in {\mathrm{\Lambda}}_{m}^{r}$. Let ${\mathrm{\Lambda}}^{k}$ be defined as follows:
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, $\{\mu \}=Fix(F)$.
Lemma 1 Let $F:[a,b]\to [a,b]$ be a continuous nonincreasing function, $\{\mu \}=Fix(F)$ and $\u03f5>0$. Define $F(x)=F(a)$ for $x<a$, $F(x)=F(b)$ for $x>b$ and
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 $\{lim{a}_{k},\mu ,lim{b}_{k}\}\subset Fix(F\circ F)$.
Proof Since the map F is nonincreasing and taking into account the hypothesis ${a}_{0}\le {a}_{1}$, we see that $({a}_{k})$ is a nondecreasing sequence. Assume ${a}_{k1}\le {a}_{k}$ and ${a}_{k}>{a}_{k+1}$ to reach a contradiction
Therefore, $({a}_{k})$ is a nondecreasing sequence, so by definition, $({b}_{k})$ is nonincreasing.
On the other hand, as ${b}_{0}=F({a}_{0})\ge F(\mu )=\mu \ge {a}_{0}$, we see by induction that ${a}_{k}\le \mu \le {b}_{k}$ for all k,
Because of the continuity of F, we conclude that
and
□
Remark 2 Suppose F not to be identically equal to +∞ and let $x\in [a,\mu )$. The map
is decreasing in the set
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 $\u03f5\in [0,{\u03f5}_{0})$.
Therefore, if $F(F(a))>a$, there exists $\u03f5>0$ such that $F(F(a)+\u03f5)\u03f5\ge a$ and taking $a={a}_{0}$
As a consequence, $({a}_{k})$, $({b}_{k})$ are well defined, without the need of extending F.
3 Unconditional convergence to a point
For a map $h:{\overline{\mathbb{R}}}^{k}\to {2}^{\overline{\mathbb{R}}}$, the difference equation
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 $c<y<d$.
(H2) There exists ${F}_{\phi}:[a,b]\to [a,b]$ such that
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 ${F}_{\phi}$.
Lemma 2 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). Then the function ${F}_{\phi}$ in (H2) is unique and it is a continuous nonincreasing map, thus it has a unique fixed point ${\mu}_{\phi}$. Furthermore,

(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 $(a,b)$.
Proof of Theorem 1 Consider $\lambda \in {\mathrm{\Lambda}}_{m}^{2}$ and denote
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 are going to prove first that ${\mu}_{\phi}$ is a stable equilibrium of $\phi \circ \lambda $. By (iii) in Lemma 2, as
we see that ${\mu}_{\phi}$ is an equilibrium.
Let $\u03f5\in (0,min\{{\mu}_{\phi}a,b{\mu}_{\phi}\})$. Because of the continuity of ${F}_{\phi}$, there is ${a}^{\prime}\in ({\mu}_{\phi}\u03f5,{\mu}_{\phi})$ such that
As $Fix({F}_{\phi}\circ {F}_{\phi})=Fix({F}_{\phi})$ and ${F}_{\phi}({F}_{\phi}(a))\ge a$, we have
If $x\in [{a}^{\prime},{b}^{\prime}]$, then
and
Therefore, ${F}_{\phi}([{a}^{\prime},{b}^{\prime}])\subset [{a}^{\prime},{b}^{\prime}]$.
By replacing a, b by ${a}^{\prime}$, ${b}^{\prime}$ in Lemma 2(i), we see that
whenever ${y}_{n}\in ({a}^{\prime},{b}^{\prime})$ for $n\le m$, thus ${\mu}_{\phi}$ is an unconditional stable equilibrium of φ.
Now, if we see that
we are done with the whole proof. Indeed, for each accumulation point $\overline{y}$ of $({y}_{n})$, one would have
because of the continuity of ${F}_{\phi}$. As ${\mu}_{\phi}\in (a,b)$, this implies $\overline{y}={\mu}_{\phi}$.
Therefore, as a consequence of Lemma 1, it suffices to find an increasing sequence ${n}_{k}$ of natural numbers such that
Here, ${a}_{k}$ and ${b}_{k}$ are defined as in Lemma 1, with ${a}_{0}=a$ and $\u03f5=1$,
Let ${n}_{0}=m$, so that
Having in mind that $\lambda \in {\mathrm{\Lambda}}_{m}^{2}$ satisfies (3), we find ${n}_{k}$ from ${n}_{k1}$ as follows. Denote
and momentarily assume $n>{n}_{k1}+m$ and ${b}_{k1}<{y}_{n}$ in such a way that
which implies
and then
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 $w\le lim{w}_{n}$.
Since
one has
and then
By applying (H2), we see that
a contradiction.
Therefore, $lim{w}_{n}={b}_{k1}$ and there exists ${m}_{k}\ge {n}_{k1}$ such that
that is,
Analogously, we see that there exists ${n}_{k}\ge {m}_{k}$ such that
□
Proof of Lemma 2

Uniqueness of ${F}_{\phi}$: Let ${y}_{1}<{y}_{2}$ and x in $[a,b]$ such that
$$\frac{{y}_{i}y}{\phi (x,y)y}\ge 1>0\phantom{\rule{1em}{0ex}}\text{for}i=1,2,y\in ({y}_{1},{y}_{2}).$$
Then
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
Finally, if $\phi (x,y)<y$ and ${x}^{\prime}\in (x,b)$, 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
If $y\in [{F}_{\phi}({x}_{1}),{F}_{\phi}({x}_{2})]$, then $y=a$, $y=b$ or $y\le \phi ({x}_{2},y)<\phi ({x}_{1},y)\le y$, thus

Continuity: If $x\in [a,b]$ and
$$w\in I=(\underset{z\to x}{lim\hspace{0.17em}inf}{F}_{\phi}(z),\underset{y\to x}{lim\hspace{0.17em}sup}{F}_{\phi}(y)),$$
then there exist two sequences ${y}_{n},{z}_{n}\to x$ with
Thus,
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
$${F}_{\phi}(t)\le \phi (t,{F}_{\phi}(a))\le {F}_{\phi}(a)\phantom{\rule{1em}{0ex}}\text{for all}t\in (a,b),$$
and because of the continuity of ${F}_{\phi}$, we have
but
Analogously, it can be seen that $\phi (b,{F}_{\phi}(b))={F}_{\phi}(b)$.

(iii) Case ${F}_{\phi}(x)\in \{c,d\}$: First assume ${F}_{\phi}(x)=c$ and recall that, by definition,
$$\phi (x,c)=\{lim\hspace{0.17em}sup\phi ({x}_{n},{y}_{n}):{x}_{n}\to x,{y}_{n}\to c,{x}_{n}\in (a,b),{y}_{n}\in (c,d)\}.$$
Suppose $\phi ({x}_{n},{y}_{n})\ge {c}^{\prime}>c$ for all n. Then
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,
Analogously, we see that $\phi (x,{F}_{\phi}(x))=\{d\}$ when ${F}_{\phi}(x)=d$. □
4 Examples and applications
4.1 The difference equation ${y}_{n}=A+{(\frac{{y}_{nk}}{{y}_{nq}})}^{p}$ with $0<p<1$
The paper [6] is devoted to prove that every positive solution to the difference equation
converges to the equilibrium $A+1$, whenever
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 $m=max\{k,q\}$.
Let
and
Define ${F}_{\phi}(+\mathrm{\infty})=A$, consider for the moment a fixed $x\in [A,\mathrm{\infty})$ and define ${F}_{\phi}(x)$ to be the unique positive zero of the function ${f}_{x}$ given by
Notice that ${f}_{x}$ is concave, ${f}_{x}(0)=A>0$, and ${f}_{x}(+\mathrm{\infty})=\mathrm{\infty}$.
Clearly, ${F}_{\phi}(x)$ is also the unique zero of the increasing function
Since
we see that condition (H2) holds and ${\mu}_{\phi}=A+1$.
As for condition
if $A\le x<y<+\mathrm{\infty}$ with
then $y>A+1$ and
Since the function
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, $p\le (A+1)/2$.
By this reasoning, we also get for free, unconditional stable convergence for several difference equations as, for instance:
or
just considering respectively
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.
Consider $a=c=0$, $b=c=+\mathrm{\infty}$, and
with $\alpha ,\beta ,A\ge 0$, $D>0$. We solve in y the equation $y=\frac{\alpha +\beta y}{A+Dx}$ to obtain
so we consider $A>\beta $, $\alpha >0$ to define
A simple calculation shows that (H2) holds, ${\mu}_{\phi}\in (a,b)$ and $Fix({F}_{\phi}\circ {F}_{\phi})=Fix({F}_{\phi})$:
Therefore, ${\mu}_{\phi}$ is an unconditional stable attractor of φ in $(a,b)=(0,+\mathrm{\infty})$ whenever $A>\beta \ge 0$, $D>0$ and $\alpha >0$.
If we choose
and $D=B+C$, we obtain unconditional stable convergence for the equation
whenever $A>\beta \ge 0$, $B+C>0$, $C\ge 0$ and $\alpha >0$.
Other choices of λ result on the unconditional stable convergence of difference equations such as
with $A>\beta +\gamma $, $B+C>0$, $\gamma \ge 0$, $\alpha >0$, $\beta \ge 0$, $C\ge 0$. Or
with $A>\beta \ge 0$, $D>0$, $\alpha >0$.
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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.
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Franco, D., Peran, J. Unconditional convergence of difference equations. Bound Value Probl 2013, 63 (2013). https://doi.org/10.1186/16872770201363
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Keywords
 difference equations
 global asymptotic stability