Unconditional convergence of difference equations

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.

2.2 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:

An important particular case is ${\alpha }_{j_0}=1$, ${\alpha }_j=0$ for $j\ne j_0$.

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 $\lambda \in {\mathrm{\Lambda }}^1$.

2.3 A technical result

Assume $-\mathrm{\infty }\le a<b\le +\mathrm{\infty }$, in the rest of this section. Recall that a continuous non-increasing 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.

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Unless $F(x)=b<+\mathrm{\infty }$, the map F verifies $F(F(x))>x$ if and only if there exists ${ϵ}_0>0$ such that $F(F(x)+ϵ)-ϵ>x$ for all $ϵ\in [0,{ϵ}_0)$.

As a consequence, $(a_k)$, $(b_k)$ are well defined, without the need of extending F.

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 lower-semicontinuous, likewise $\phi (b,\cdot )$ is an upper-semicontinuous function. Remember that we denote both φ and $\hat{\phi }$ 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 $F_{\phi }$.

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 $n>n_{k-1}+m$.

As a consequence, the nonincreasing sequence $w_n$ is bounded below by $b_{k-1}$. It cannot be the case that $lim{w}_n>b_{k-1}$, because in such a case there is a subsequence $y_{n_j}>b_{k-1}$ 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.

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Proof of Lemma 2

a contradiction.

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

which contradicts hypothesis (H2), thus $y=F_{\phi }(x)$.

because of the continuity of $\phi (x,\cdot )$.

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.

but

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,

4.1 The difference equation $y_n=A+{(\frac{y_{n-k}}{y_{n-q}})}^p$ 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 $m=max\{k,q\}$.

Let

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, $p\le (A+1)/2$.

just considering respectively

where $m=max\{q,r,s,t\}$.

4.2 The difference equation $y_n=\frac{\alpha +\beta y_{n-1}}{A+B{y}_{n-1}+C{y}_{n-2}}$

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.