Unconditional convergence of difference equations

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.

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 \phi (x,y)=1+(ax/y), the sequence y_n defined by y_n=\phi (x,y_{n-1}), 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 second-order 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 second-order difference equations

(1)

(2)

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 , 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 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 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.

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 {\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 set-valued 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 compact-valued 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 ) to be true if and only if A\ne \mathrm{\varnothing }, B\ne \mathrm{\varnothing } and a\le b (respectively ) 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 \hat{\phi }:{\overline{\mathbb{R}}}^m\to 2^{\overline{\mathbb{R}}} be defined by

It is obvious that \hat{\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 \hat{\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 , 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).

Lemma 1 Let F:[a,b]\to [a,b] be a continuous non-increasing function, \{\mu \}=Fix(F) and ϵ>0. Define F(x)=F(a) for , 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_{k-1}\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 , 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).

Therefore, if F(F(a))>a, there exists ϵ>0 such that F(F(a)+ϵ)-ϵ\ge a and taking a=a_0

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

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 \hat{\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 and consider in the sequel a continuous function \phi :(a,b)\times (c,d)\to (c,d), satisfying the following conditions:

(H1) , whenever and .

(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 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 }.

Lemma 2 Let \phi :(a,b)\times (c,d)\to (c,d), where , 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,

1. (i)

   for all x,y\in (a,b) and a\le \phi (x,y)\le b for all x,y\in [a,b].

2. (ii)

   If x\in [a,b], y\in (c,d) and \phi (x,y)=y, then y=F_{\phi }(x).

3. (iii)

   \phi (x,F_{\phi }(x))=F_{\phi }(x) for all x\in [a,b].

4. (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 , 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 ϵ\in (0,min\{{\mu }_{\phi }-a,b-{\mu }_{\phi }\}). Because of the continuity of F_{\phi }, there is a^{\prime }\in ({\mu }_{\phi }-ϵ,{\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 ϵ=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_{k-1} as follows. Denote

and momentarily assume n>n_{k-1}+m and in such a way that

which implies

and then

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.

Since

one has

and then

By applying (H2), we see that

a contradiction.

Therefore, lim{w}_n=b_{k-1} and there exists m_k\ge n_{k-1} 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 and x in [a,b] such that

  \frac{y_i-y}{\phi (x,y)-y}\ge 1>0\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 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 and 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 , 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)\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. □

4.1 The difference equation y_n=A+{(\frac{y_{n-k}}{y_{n-q}})}^p with

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 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_{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.

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.

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.

Affiliations

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

   • Daniel Franco
   •  & Juan Peran

Corresponding author

Correspondence to Juan Peran.

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.

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