R E S E A R C H
Open Access
Unconditional convergence of difference
equations
Daniel Franco and Juan Peran
**Correspondence:
[email protected] Departamento de Matemática Aplicada, Universidad Nacional de Educación a Distancia (UNED), C/ Juan del Rosal 12, Madrid, 28040, Spain
Abstract
We put forward the notion ofunconditional convergenceto 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 equa-tions can be extended to some higher-order ones (see, for example, [–]). Consider the following simple example. Ifϕis the mapϕ(x,y) = + (ax/y), the sequenceyndefined by
yn=ϕ(x,yn–), that is,
yn= +
ax yn–
,
withy,a,x> , converges toFϕ(x) = ( + √
+ ax)/ for anyy. Observe thatFϕis the function satisfyingϕ(x,Fϕ(x)) =Fϕ(x). Obviously, the second-order difference equation
yn= +
ax yn–
,
also converges toFϕ(x) for anyy,y,a,x> . Let us continue to add complexity, by
con-sidering the second-order difference equations
yn= +
ayn–
yn–
, ()
yn= +
ayn–
yn–
. ()
For ally,y,a> , the sequence defined by Equation () converges to the unique fixed
pointμϕ=a+ of the functionFϕ. However, the behavior of Equation () depends on the parametera:
• Fora≥, the odd and even index terms converge respectively to some limits,
μ∈[, +∞]andμ/(μ– )∩[, +∞], whereμmay depend ony,y(fora= ).
• For <a< , it converges toμϕ=a+ , whatever the choice ofy,y> one makes.
No sophisticated tools are needed to reach those conclusions: It suffices to note that the set
A=n: (yn+–yn+)(yn+–yn)≥
must be either finite or equal toN. As the sequencesyn+andynare then both eventually
monotone, they converge in [, +∞] to some limits, sayμandμ, satisfying
μ= +
aμ μ
and μ= +
aμ μ
.
Therefore, one of the following statements holds:μ=μ/(μ– )∩[, +∞], witha= ,
or{μ} ∪ {μ} ∈ {{, +∞},{ +a}}.
If {μ} ∪ {μ}={, +∞}, then that of the sequences,yn+ or yn, which converges to
+∞, has to be nondecreasing. Just look at Equation () to conclude thata≥ whenever {μ} ∪ {μ}={, +∞}.
The case we are interested in is <a< and we will say thatμϕ= +ais an uncondi-tional attractor for the mapϕ, that is, we would considerϕ(x,y) = + (ax/y) with <a< to observe that, not only () and (), but all the following recursive sequences converge to
μϕ= +a, whatever the choice ofy, . . . ,ymax{k,m}> we make:
yn= +
ayn–k
yn–m
,
yn= +a
yn–k++yn–k
yn–m++yn–m
,
yn= +a
yn–k+yn–k
yn–m+yn–m
,
yn= +a
max{yn–k+,yn–k}
min{yn–m+,yn–m}
,
. . . .
In this paper, we proceed as follows. The next section is dedicated to notation and a technical result of independent interest. In Section , 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 , 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 func-tion with a multifuncfunc-tion. We noticed that set-valued difference equafunc-tions are not con-cerned 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 homoge-neous way. In the second part, we introduce the families of mapsk
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 compactificationR= [–∞, +∞] ofRendowed with the usual order and compact topology.
.. Operations and preorder inR\ {∅}
We define the operations ‘+’, ‘–’, ‘·’ and ‘/’ in R\ {∅}by
A∗B=lim sup(an∗bn) :an,bn,an∗bn∈Rfor alln∈Nandliman∈A,limbn∈B
,
where∗stands for ‘+’, ‘–’, ‘·’ or ‘/’. We also agree to writeA∗ ∅=∅ ∗A=∅.
Remark We introduce the above notation in order to manage unboundedness and sin-gular situations, but we point out that these are natural set-valued extensions for the arith-metic operations. Let X,Y be compact (Hausdorff ) spaces,U a dense subset ofXand f :U→Y. The closureGr(f) of the graph off inX×Ydefines an upper semicontinuous compact-valued mapf :X→Y byf(x) ={y∈Y: (x,y)∈Gr(f)}, that is, byGr(f) =Gr(f)
(see []). Furthermore, as usual, one writesf(A) =x∈Af(x) forA∈X, thereby obtaining
a mapf: X→Y.
To extend arithmetic operations, considerX=R×R,Y =RandU=R×R, whenf denotes addition, substraction or multiplication, andU=R×(R\ {}), whenf denotes division.
Also defineA≤B(respectivelyA<B) to be true if and only ifA= ∅,B=∅anda≤b (respectivelya<b) for alla∈A,b∈B. HereA,B∈R.
Notice that both relations≤and < are transitive but neither reflexive nor symmetric.
.. Canonical injections
When no confusion is likely to arise, we identifya∈Rwith{a} ∈R, that is, in the sequel we consider the fixed injectiona→ {a}ofRinto R and we identifyRwith its image. We must point out that, under this convention, whenais expected to be subset ofA, we understand ‘a∈A’ as ‘there isb∈Awitha={b}’. For instance, one has ·(+∞) =R, / ={–∞, +∞}.
.. Extension of a function as a multifunction
Consider a maph:Rm→Rand denote byD(h) the set formed by thosex∈Rmfor which there isb∈Rwithh(x) ={b}.
IfA∈(Rm), thenh(A)∈Ris defined to be
a∈Ah(a). Also, ifB∈(R)m, thenh(B)∈R
is defined to beh(B) =h(B× · · · ×Bm).
For each functionϕ:U⊂Rm→R, letϕ:Rm→Rbe defined by
ϕ(x) =lim supϕ(xn) :xn∈Ufor alln∈Nandlimxn=x
It is obvious thatϕ(x)=∅if and only ifxis in the closureUofUinRm. Also notice that
U⊂D(ϕ)⊂U
whenϕis continuous. In this case, and if no confusion is likely to arise, we agree to denote also byϕthe mapϕ. For example, we write
ϕ() = [–, ]; ϕ(–∞) =ϕ(+∞) = ; D(ϕ) =R\ {}
whenU=R\ {}andϕ(x) =sin(/x).
2.2 The maps in
k
mand
k
As we have announced, the unconditional convergence of a difference equation guaran-tees 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.
Fork,m∈N, letk
mbe the set formed by the mapsλ:R m
→Rksuch that
min
≤j≤mxj≤λi(x)≤max≤j≤mxj for allx∈R m
, ≤i≤k. ()
Notice thatλ◦γ ∈k
mwheneverλ∈kr,γ ∈rm. Letkbe defined as follows:
k=
m∈N km.
We note that the functions inksatisfy that their behavior isenvelopedby 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 inare: • λ(x, . . . ,xm) =
m
j=αjxj, withαj≥forj= , . . . ,m,
m
j=αj= .
An important particular case isαj= ,αj= forj=j.
• λ(x, . . . ,xm) = m
j=x αj
j ifmin≤j≤m xj> ,
min≤j≤m xj ifmin≤j≤m xj≤,withαj≥forj= , . . . ,m,
m
j=αj= .
We refer to this function simply asλ(x, . . . ,xm) =
m
j=x
αj
j , when it is assumed that λ∈.
• λ(x, . . . ,xm) =maxj∈Jxj, whereJis a nonempty subset of{, . . . ,m}.
• λ(x, . . . ,xm) =minj∈Jxj, whereJis a nonempty subset of{, . . . ,m}.
2.3 A technical result
Assume –∞ ≤a<b≤+∞, in the rest of this section. Recall that a continuous non-increasing functionF: [a,b]→[a,b] has a unique fixed pointμ∈[a,b], that is,{μ}=
Fix(F).
Lemma Let F: [a,b]→[a,b]be a continuous non-increasing function,{μ}=Fix(F)and
> .Define F(x) =F(a)for x<a,F(x) =F(b)for x>b and
a=a; b=F(a); ak=F
bk–+
k
; bk=F
ak–
k
Then(ak)and(bk)are,respectively,a nondecreasing and a nonincreasing sequence in
[a,b].Furthermore,ak≤μ≤bkfor all k and{limak,μ,limbk} ⊂Fix(F◦F).
Proof Since the mapFis nonincreasing and taking into account the hypothesisa≤a,
we see that (ak) is a nondecreasing sequence. Assumeak–≤akandak>ak+to reach a
contradiction
ak>ak+ ⇒ F
bk–+
k
>F
bk+
k+
⇒ bk–+
k <bk+
k+ ⇒ bk–<bk
⇒ F
ak––
k–
<F
ak–
k
⇒ ak––
k– >ak–
k ⇒ ak–>ak.
Therefore, (ak) is a nondecreasing sequence, so by definition, (bk) is nonincreasing.
On the other hand, asb=F(a)≥F(μ) =μ≥a, we see by induction thatak≤μ≤bk
for allk,
ak=F
bk–+
k
≤F(μ) =μ=F(μ)≤F
ak–
k
=bk fork≥.
Because of the continuity ofF, we conclude that
limak=F(limbk) =F
F(limak)
and
limbk=F(limak) =F
F(limbk)
.
Remark SupposeFnot to be identically equal to +∞and letx∈[a,μ). The map
→FF(x) +–
is decreasing in the set
≥ :FF(x) +< +∞.
UnlessF(x) =b< +∞, the mapFverifiesF(F(x)) >xif and only if there exists> such
thatF(F(x) +) –>xfor all∈[,).
Therefore, ifF(F(a)) >a, there exists> such thatF(F(a) +) –≥aand takinga=a
a≤FF(a) +–=a–≤ak–
k≤μ≤bk–+
k≤b+=F(a) +≤b.
3 Unconditional convergence to a point For a maph:Rk→R, thedifference equation
yn=h(yn–, . . . ,yn–k) ()
is always well defined whatever the initial pointsy, . . . ,yk∈Rare, even though theynare
subsets ofR, rather than points.
A pointμ∈Ris said to be anequilibriumfor the maphifh(μ, . . . ,μ) ={μ}. The equilib-riumμis said to bestableif, for each neighborhoodVofμinR, there is a neighborhood W of (μ, . . . ,μ) inD(h) such thatyn∈Vfor alln, whenever (y, . . . ,yk)∈W.
The equilibriumμis said to be anattractor ina neighborhoodVofμinR, ifyn∈Rfor
allnandyn→μinR, wheneveryn∈Vforn≤k.
Definition The pointμis said to be anunconditional equilibriumofh(respectively, unconditional stable equilibrium,unconditional attractor in V) if it is an equilibrium (re-spectively, stable equilibrium, attractor inV) ofh◦λfor allλ∈k.
Definition We define theequilibria,stable equilibria,attractors,unconditional equilib-ria,unconditional stable equilibriaandunconditional attractorsof a continuous function
ϕ:U⊂Rk→Rto be those ofϕ.
3.1 Sufficient condition for unconditional convergence
After giving Definitions and we are going to prove a result guaranteeing that a general second order difference equation as in () has an unconditional stable attractor.
Let –∞<c≤a<b ≤d ≤+∞ and consider in the sequel a continuous function
ϕ: (a,b)×(c,d)→(c,d), satisfying the following conditions: (H) ϕ(x,y) <ϕ(x,y), whenevera<x<x<bandc<y<d.
(H) There existsFϕ: [a,b]→[a,b]such that
Fϕ(x) –y
ϕ(x,y) –y≥
whenevery∈(c,d)\ {Fϕ(x)}.
The functionsϕ(·,y) : [a,b]→[c,d] andϕ(x,·) : (c,d)→(c,d) are defined in the obvi-ous way. Notice thatϕ(a,·) is the limit of a monotone increasing sequence of continuous functions, thus it is lower-semicontinuous, likewise ϕ(b,·) is an upper-semicontinuous function. Remember that we denote bothϕandϕbyϕ.
The next lemma, which we prove at the end of this section, shows that if (H) and (H) holds we can get some information about the behavior and properties ofϕandFϕ.
Lemma Letϕ: (a,b)×(c,d)→(c,d),where–∞<c≤a<b≤d≤+∞,be a continu-ous function satisfying(H)and(H).Then the function Fϕin(H)is unique and it is a continuous nonincreasing map,thus it has a unique fixed pointμϕ.Furthermore,
(i) a<ϕ(x,y) <bfor allx,y∈(a,b)anda≤ϕ(x,y)≤bfor allx,y∈[a,b]. (ii) Ifx∈[a,b],y∈(c,d)andϕ(x,y) =y,theny=Fϕ(x).
We are in conditions of presenting and proving our main result.
Theorem Letϕ: (a,b)×(c,d)→(c,d),where–∞<c≤a<b≤d≤+∞,be a continu-ous function satisfying(H)and(H).Ifμϕ∈(a,b)andFix(Fϕ◦Fϕ) =Fix(Fϕ),thenμϕis an unconditional stable attractor ofϕin(a,b).
Proof of Theorem Considerλ∈
mand denote
yn=ϕ◦λ(yn–, . . . ,yn–m)
for some y, . . . ,ym∈(a,b). Notice that yn∈(a,b) for all n, as a consequence of (i) in
Lemma .
We are going to prove first thatμϕis a stable equilibrium ofϕ◦λ. By (iii) in Lemma , as
λ(μϕ, . . . ,μϕ) = (μϕ,μϕ),
we see thatμϕis an equilibrium.
Let∈(,min{μϕ–a,b–μϕ}). Because of the continuity ofFϕ, there isa∈(μϕ–,μϕ) such that
b≡Fϕ
a∈(μϕ,μϕ+).
AsFix(Fϕ◦Fϕ) =Fix(Fϕ) andFϕ(Fϕ(a))≥a, we have
Fϕ
Fϕ(x)
>x for allx∈[a,μϕ).
Ifx∈[a,b], then
Fϕ(x)≤Fϕ
a=b
and
Fϕ(x)≥Fϕ
b=Fϕ
Fϕ
a>a.
Therefore,Fϕ([a,b])⊂[a,b].
By replacinga,bbya,bin Lemma (i), we see that
yn∈
a,b⊂(μϕ–,μϕ+) for alln,
wheneveryn∈(a,b) forn≤m, thusμϕis an unconditional stable equilibrium ofϕ. Now, if we see that
we are done with the whole proof. Indeed, for each accumulation pointyof (yn), one would
have
Fϕ(μϕ) =μϕ=Fϕ(y),
because of the continuity ofFϕ. Asμϕ∈(a,b), this impliesy=μϕ.
Therefore, as a consequence of Lemma , it suffices to find an increasing sequencenkof
natural numbers such that
ak≤Fϕ(yn)≤bk for allk≥,n≥nk.
Here,akandbkare defined as in Lemma , witha=aand= ,
b=Fϕ(a); ak=Fϕ
bk–+
k
; bk=Fϕ
ak–
k
fork≥.
Letn=m, so that
a≤Fϕ(yn)≤Fϕ(a) =b forn≥n.
Having in mind thatλ∈
msatisfies (), we findnkfromnk–as follows. Denote
zk=Fϕ–(bk–)
and momentarily assumen>nk–+mandbk–<ynin such a way that
bk–< yn=ϕ
λ(yn–, . . . ,yn–m),λ(yn–, . . . ,yn–m)
≤ϕmin{yn–, . . . ,yn–m},λ(yn–, . . . ,yn–m)
≤ϕzk,λ(yn–, . . . ,yn–m)
,
which implies
ϕzk,λ(yn–, . . . ,yn–m)
≤λ(yn–, . . . ,yn–m)
and then
yn≤max
bk–,λ(yn–, . . . ,yn–m)
≤max{bk–,yn–, . . . ,yn–m} ≡wn
for alln>nk–+m.
As a consequence, the nonincreasing sequencewnis bounded below bybk–. It cannot
be the case that limwn>bk–, because in such a case there is a subsequence ynj >bk–
converging tolimwnand such thatλ(ynj–, . . . ,ynj–m) converges to a pointw≤limwn.
Since
ynj≤ϕ
zk,λ(ynj–, . . . ,ynj–m)
one has
ϕ(zk,w) =limwn>bk–=F(zk)
and then
ϕ(zk,w) –w>F(zk) –w.
By applying (H), we see that
limwn=ϕ(zk,w) <w,
a contradiction.
Therefore,limwn=bk–and there existsmk≥nk–such that
yn<bk–+
k for alln≥mk,
that is,
Fϕ(yn)≥ak for alln≥mk.
Analogously, we see that there existsnk≥mksuch that
Fϕ(yn)≤bk for alln≥nk.
Proof of Lemma
•Uniqueness ofFϕ: Lety<yandxin[a,b]such that
yi–y
ϕ(x,y) –y≥ > fori= , ,y∈(y,y).
Then
<ϕ(x,y) –y< ,
a contradiction.
•(i): It suffices to prove the first assertion, because[a,b]is a closed set and, by definition,
ϕ(x,y) =lim supϕ(xn,yn) :xn→x,yn→y,xn∈(a,b),yn∈(c,d)
for all(x,y)∈[a,b]×[c,d]. Assume now that(x,y)∈(a,b)×(a,b). We consider the following three possible situations. Ifϕ(x,y) =y, it is obvious thatϕ(x,y)∈(a,b). On the other hand, ifϕ(x,y) >yandx∈(a,x), then
a≤y<ϕ(x,y) <ϕx,y≤Fϕ
x≤b.
Finally, ifϕ(x,y) <yandx∈(x,b), then
b≥y>ϕ(x,y) >ϕx,y≥Fϕ
•(ii): Supposey=Fϕ(x). Sinceϕ(x,y) =y∈R, thenϕ(x,y) –y= orϕ(x,y) –y=R. In any event, it cannot be the case that
Fϕ(x) –y
ϕ(x,y) –y≥,
which contradicts hypothesis (H), thusy=Fϕ(x).
•(iii): SinceFϕ([a,b])⊂[a,b]⊂[c,d], it is worth considering the following three cases for eachx∈[a,b]: first,x∈(a,b),Fϕ(x)∈(c,d)and then (after probing continuity, mono-tonicity and statement (iv)), we proceed with the casex∈ {a,b},Fϕ(x)∈(c,d)and finally withx∈[a,b],Fϕ(x)∈ {c,d}.
Casex∈(a,b)andFϕ(x)∈(c,d): Sinceϕ(x,y) >ywheny<Fϕ(x)andϕ(x,y) <ywhen y>Fϕ(x), we see that
ϕx,Fϕ(x)
=Fϕ(x),
because of the continuity ofϕ(x,·). •Monotonicity and (iv): Suppose
Fϕ(x)≤Fϕ(x) fora≤x<x≤b.
Ify∈[Fϕ(x),Fϕ(x)], theny=a,y=bory≤ϕ(x,y) <ϕ(x,y)≤y, thus
Fϕ(x),Fϕ(x)
⊂ {a,b}.
•Continuity: Ifx∈[a,b]and
w∈I=
lim inf
z→x Fϕ(z),lim supy→x Fϕ(y)
,
then there exist two sequencesyn,zn→xwith
Fϕ(zn) <w<Fϕ(yn).
Thus,
ϕ(zn,w) <w<ϕ(yn,w)
and, by (ii), one hasϕ(x,w) =w. Sincew∈(c,d), this would implyFϕ(x) =wfor allw∈I, which is impossible.
•(iii) Casex∈ {a,b}andFϕ(x)∈(c,d): Since
Fϕ(t)≤ϕ
t,Fϕ(a)
≤Fϕ(a) for allt∈(a,b),
and because of the continuity ofFϕ, we have
Fϕ(a) =sup
t
Fϕ(t)≤sup
t
ϕt,Fϕ(a)
but
sup
t
ϕt,Fϕ(a)
=ϕa,Fϕ(a)
.
Analogously, it can be seen thatϕ(b,Fϕ(b)) =Fϕ(b).
•(iii) CaseFϕ(x)∈ {c,d}: First assumeFϕ(x) =cand recall that, by definition,
ϕ(x,c) =lim supϕ(xn,yn) :xn→x,yn→c,xn∈(a,b),yn∈(c,d)
.
Supposeϕ(xn,yn)≥c>cfor alln. Then
≤ Fϕ(xn) –yn
ϕ(xn,yn) –yn≤
Fϕ(xn) –yn
c–yn
,
which impliesFϕ(xn)≥c>c, eventually for alln.
SinceFϕ(xn)→c, we reach a contradiction. Therefore,
ϕx,Fϕ(x)
={c}=Fϕ(x)
.
Analogously, we see thatϕ(x,Fϕ(x)) ={d}whenFϕ(x) =d.
4 Examples and applications
4.1 The difference equationyn=A+ (ynyn––kq)pwith 0 <p< 1
The paper [] is devoted to prove that every positive solution to the difference equation
yn=A+
yn–k
yn–q
p
converges to the equilibriumA+ , whenever
A∈(, +∞) and p∈,min, (A+ )/.
Here,k,q∈ {, , , . . .}are fixed numbers.
Although paper [] complements [], where the casep= had been considered, it should be noticed that the casep= ,A≥ is not dealt with in []. Furthermore, we cannot assure the global attractivity in this case.
The results in [] can be easily obtained by applying Theorem above. Furthermore, we slightly improve the results in [] by establishing the unconditional stability of the equilib-riumA+ , wheneverA∈(, +∞),p∈(,min{, (A+ )/}). We may assume without loss of generality that the initial valuesy, . . . ,ym are greater thanA. Here, and in the sequel
m=max{k,q}. Let
A∈(, +∞), p∈(, ), a=c=A, b=d= +∞,
ϕ(x,y) =A+
y x
p
and
λ(x, . . . ,xm) = (xq,xk).
DefineFϕ(+∞) =A, consider for the moment a fixedx∈[A,∞) and defineFϕ(x) to be the unique positive zero of the functionfxgiven by
fx(y) =ϕ(x,y) –y.
Notice thatfxis concave,fx() =A> , andfx(+∞) = –∞.
Clearly,Fϕ(x) is also the unique zero of the increasing function
jx(y) =ϕ(x,y) –Fϕ(x).
Since
jx(A) =
Ap– (F
ϕ(x))p
xp ≤ and jx(+∞) = +∞> ,
we see that condition (H) holds andμϕ=A+ . As for condition
Fix(Fϕ◦Fϕ) =Fix(Fϕ), ()
ifA≤x<y< +∞with
A+
y x
p
=y,
theny>A+ and
ϕ(y,x) –x=A+
x y
p
–x=A+ y–A–
y (y–A)/p.
Since the function
h(z) =A+ z–A–
z (z–A)/p
has a unique critical point in (A, +∞) andh(A+ ) = ,h(+∞) =A, the necessary and sufficient condition for () to hold is thath(A+ )≥, that is,p≤(A+ )/.
By this reasoning, we also get for free, unconditional stable convergence for several dif-ference equations as, for instance:
yn=A+
yn–qyn–r
yn–syn–t
p
with <p<min/, (A+ )/
or
yn=A+
yn–q+yn–r
yn–s+yn–t
p
just considering respectively
λ(x, . . . ,xm) = (√xsxt,√xqxr),
λ(x, . . . ,xm) =
xs+xt
, xq+xr
,
wherem=max{q,r,s,t}.
4.2 The difference equationyn=A+Bynα+β–1yn+Cyn–1–2
Here,α,β,A,B,C,y,y≥. In , three conjectures on the above equation were posed
in []. In all three cases (B= ,α,β,A,C> ;A= ,α,β,B,C> ; andα,β,A,B,C> , respectively) it was postulated the global asymptotic stability of the equilibrium. These conjectures have resulted in several papers since then (see [–]). Let us see when there is unconditional convergence.
Considera=c= ,b=c= +∞, and
ϕ(x,y) = α+βy A+Dx
withα,β,A≥,D> . We solve inythe equationy=Aα++Dxβy to obtain
y= α
A–β+Dx,
so we considerA>β,α> to define
Fϕ(x) =
α
A–β+Dx.
A simple calculation shows that (H) holds,μϕ∈(a,b) andFix(Fϕ◦Fϕ) =Fix(Fϕ):
Fϕ(x) –y
ϕ(x,y) –y= +
β
A–β+Dx≥,
μϕ= D
–A+β+(A–β)+ Dα,
Fϕ(Fϕ(x)) –x Fϕ(x) –x
= (A–β+Dx)(A–β) (A–β)+Dx(A–β) +Dα> .
Therefore,μϕis an unconditional stable attractor ofϕin (a,b) = (, +∞) wheneverA>
β≥,D> andα> . If we choose
λ(x,x) =
Bx+Cx
B+C ,x
andD=B+C, we obtain unconditional stable convergence for the equation
yn=
α+βyn–
A+Byn–+Cyn–
,
Other choices ofλresult on the unconditional stable convergence of difference equa-tions such as
yn=
α+βyn–+γyn–
A+Byn–+Cyn–
withA>β+γ,B+C> ,γ≥,α> ,β≥,C≥. Or
yn=
α+βmax{yn–,yn–}
A+Dmin{yn–,yn–}
withA>β≥,D> ,α> .
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.
Received: 19 December 2012 Accepted: 20 March 2013 Published: 28 March 2013 References
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doi:10.1186/1687-2770-2013-63
