Franco and Peran Boundary Value Problems 2013 2013:63 R E S E A R C H Open Access Unconditional convergence of difference equations Daniel Franco and Juan Peran * * Correspondence: jperan@ind.uned.es 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 of unconditional convergence to an equilibrium of a difference equation. Roughly speaing 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 equations can be extended to some higher-order ones see for example [1 4]). Consider the following simple example. If ϕ is the map ϕx y)=1+ax/y) the sequence y n defined by y n = ϕx y n 1 ) that is y n =1+ ax y n 1 with y 1 a x >0convergestoF ϕ x) =1+ 1+4ax)/2 for any y 1.ObservethatF ϕ is the function satisfying ϕx F ϕ x)) = F ϕ x). Obviously the second-order difference equation y n =1+ ax y n 2 also converges to F ϕ x) for any y 1 y 2 a x > 0. Let us continue to add complexity by considering the second-order difference equations y n =1+ ay n 1 y n 2 1) y n =1+ ay n 2 y n 1. 2) For all y 1 y 2 a > 0 the sequence defined by Equation 1) converges to the unique fixed point μ ϕ = a +1ofthefunctionF ϕ. However the behavior of Equation 2)dependsonthe parameter a: 2013 Franco and Peran; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/2.0) which permits unrestricted use distribution and reproduction in any medium provided the original wor is properly cited.
Franco and Peran Boundary Value Problems 2013 2013:63 Page 2 of 14 For a 1 the odd and even index terms converge respectively to some limits μ 1 [1 + ] and μ 1 /μ 1 1) [1 + ]whereμ 1 may depend on y 1 y 2 for a =1). For0<a <1itconvergestoμ ϕ = a +1 whatever the choice of y 1 y 2 >0one maes. No sophisticated tools are needed to reach those conclusions: It suffices to note that the set A = { n :y n+3 y n+1 )y n+2 y n ) 0 } must be either finite or equal to N.Asthesequencesy 2n+1 and y 2n are then both eventually monotone they converge in [1 + ] to some limits say μ 1 and μ 2 satisfying μ 1 =1+ aμ 1 μ 2 and μ 2 =1+ aμ 2 μ 1. Therefore one of the following statements holds: μ 2 = μ 1 /μ 1 1) [1 + ] with a =1 or {μ 1 } {μ 2 } {{1 + } {1+a}}. If {μ 1 } {μ 2 } = {1 + } then that of the sequences y 2n+1 or y 2n whichconvergesto + has to be nondecreasing. Just loo at Equation 2) toconcludethata 1whenever {μ 1 } {μ 2 } = {1 + }. Thecaseweareinterestedinis0<a <1andwewillsaythatμ ϕ =1+a is an unconditional attractor for the map ϕ that is we would consider ϕx y)=1+ax/y)with0<a <1 to observe that not only 1)and2) but all the following recursive sequences converge to μ ϕ =1+a whatever the choice of y 1...y max{m} >0wemae: y n =1+ ay n y n m y n =1+a y n +1 + y n y n m+1 + y n m yn +1 y n y n =1+a y n m+1 y n m y n =1+a max{y n +1 y n } min{y n m+1 y n m }... 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 framewor. 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 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 m a ind of averages
Franco and Peran Boundary Value Problems 2013 2013:63 Page 3 of 14 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 R =[ + ]ofr endowed with the usual order and compact topology. 2.1.1 Operations and preorder in 2 R \{ } We define the operations + and / in2 R \{ }by A B = { lim supa n b n ):a n b n a n b n R for all n N and lim a n A lim b n B } where stands for + or /.WealsoagreetowriteA = A =. Remar 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 adensesubsetofx and f : U Y.TheclosureGrf )ofthegraphoff in X Y defines an upper semicontinuous compact-valued map f : X 2 Y by f x)={y Y :x y) Grf )} thatisbygrf )=Grf ) see [5]). Furthermore as usual one writes f A)= x A f x)fora 2X thereby obtaining amapf :2 X 2 Y. To extend arithmetic operations consider X = R R Y = R and U = R R whenf denotes addition substraction or multiplication and U = R R \{0}) when f denotes division. Also define A B respectively A < B) tobetrueifandonlyifa B and a b respectively a < b) for all a A b B.HereA B 2 R. Notice that both relations and < are transitive but neither reflexive nor symmetric. 2.1.2 Canonical injections When no confusion is liely to arise we identify a R with {a} 2 R thatisinthesequel we consider the fixed injection a {a} of R into 2 R and we identify R with its image. We must point out that under this convention when a is expected to be subset of A we understand a A as thereisb A with a = {b}. For instance one has 0 + )=R 1/0 = { + }. 2.1.3 Extension of a function as a multifunction Consider a map h : R m 2 R and denote by Dh) the set formed by those x R m for which there is b R with hx)={b}. If A 2 Rm) thenha) 2 R is defined to be a A ha). Also if B 2R ) m thenhb) 2 R is defined to be hb)=hb 1 B m ). For each function ϕ : U R m Rlet ϕ : R m 2 R be defined by ϕx)= { lim sup ϕx n ):x n U for all n N and lim x n = x }.
Franco and Peran Boundary Value Problems 2013 2013:63 Page 4 of 14 It is obvious that ϕx) if and only if x is in the closure U of U in R m.alsonoticethat U D ϕ) U when ϕ is continuous. In this case and if no confusion is liely to arise we agree to denote also by ϕ the map ϕ. For example we write ϕ0) = [ 1 1]; ϕ )=ϕ+ )=0; Dϕ)=R \{0} when U = R \{0} and ϕx)=sin1/x). 2.2 The maps in m and 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 m Nlet m be the set formed by the maps λ : Rm R such that min x j λ i x) max x j for all x R m 1 i. 3) 1 j m 1 j m Notice that λ γ m whenever λ r γ r m.let be defined as follows: = m N m. We note that the functions in 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 1 are: λx 1...x m )= m j=1 α jx j withα j 0 for j =1...m m j=1 α j =1. An important particular case is α j0 =1 α j =0for j j 0. λx 1...x m )= { mj=1 x α j j if min 1 j m x j >0 min 1 j m x j if min 1 j m x j 0 with α j 0 for j =1...m m j=1 α j =1. We refer to this function simply as λx 1...x m )= m j=1 xα j j when it is assumed that λ 1. λx 1...x m )=max j J x j wherej is a nonempty subset of {1...m}. λx 1...x m )=min j J x j wherej is a nonempty subset of {1...m}. 2.3 A technical result Assume a < b + in the rest of this section. Recall that a continuous nonincreasing function F :[a b] [a b] hasauniquefixedpointμ [a b] that is {μ} = FixF). Lemma 1 Let F :[a b] [a b] be a continuous non-increasing function {μ} = FixF) and ɛ >0.Define Fx)=Fa) for x < a Fx)=Fb) for x > band a 0 = a; b 0 = Fa 0 ); a = F b 1 + ɛ ) ; b = F a ɛ ) for 1.
Franco and Peran Boundary Value Problems 2013 2013:63 Page 5 of 14 Then a ) and b ) are respectively a nondecreasing and a nonincreasing sequence in [a b]. Furthermore a μ b for all and {lim a μ lim b } FixF F). Proof Since the map F is nonincreasing and taing into account the hypothesis a 0 a 1 we see that a ) is a nondecreasing sequence. Assume a 1 a and a > a +1 to reach a contradiction a > a +1 F b 1 + ɛ ) > F b + ɛ ) +1 b 1 + ɛ < b + ɛ b 1 < b F a 1 ɛ 1 +1 ) < F a 1 ɛ 1 > a ɛ a ɛ ) a 1 > a. Therefore a ) is a nondecreasing sequence so by definition b )isnonincreasing. On the other hand as b 0 = Fa 0 ) Fμ)=μ a 0 we see by induction that a μ b for all a = F b 1 + ɛ ) Fμ)=μ = Fμ) F a ɛ ) = b for 1. Because of the continuity of Fweconcludethat lim a = Flim b )=F Flim a ) ) and lim b = Flim a )=F Flim b ) ). Remar 2 Suppose F not to be identically equal to + and let x [a μ). The map ɛ F Fx)+ɛ ) ɛ is decreasing in the set { ɛ 0:F Fx)+ɛ ) <+ }. Unless Fx)=b <+ themapf verifies FFx)) > x if and only if there exists ɛ 0 >0such that FFx)+ɛ) ɛ > x for all ɛ [0 ɛ 0 ). Therefore if FFa)) > athereexistsɛ >0suchthatFFa)+ɛ) ɛ a and taing a = a 0 a F Fa)+ɛ ) ɛ = a 1 ɛ a ɛ μ b 1 + ɛ b 0 + ɛ = Fa)+ɛ b. As a consequence a ) b ) are well defined without the need of extending F.
Franco and Peran Boundary Value Problems 2013 2013:63 Page 6 of 14 3 Unconditional convergence to a point For a map h : R 2 R thedifference equation y n = hy n 1...y n ) 4) is always well defined whatever the initial points y 1...y R are even though the y n are subsets of R rather than points. Apointμ R is said to be an equilibrium for the map h if hμ...μ)={μ}.theequilibrium μ is said tobe stable if for each neighborhood V of μ in R there is a neighborhood W of μ...μ)indh)suchthaty n V for all nwhenevery 1...y ) W. The equilibrium μ is said to be an attractor in a neighborhood V of μ in Rify n R for all n and y n μ in Rwhenevery n V for n. 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 )ofh λ for all λ. Definition 2 We define the equilibria stableequilibria attractors unconditional equilibria unconditional stable equilibria and unconditional attractors of a continuous function ϕ : U R R to be those of ϕ. 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 < c a < b d + and consider in the sequel a continuous function ϕ :a b) c d) c d) satisfying the following conditions: H1) ϕx 1 y)<ϕx 2 y)whenevera < x 2 < x 1 < b and c < y < d. H2) There exists F ϕ :[a b] [a b] such that F ϕ x) y ϕx y) y 1 whenever y c d) \{F ϕ x)}. The functions ϕ y):[a b] [c d] andϕx ):c d) c d) are defined in the obvious way. Notice that ϕa ) is the limit of a monotone increasing sequence of continuous functions thus it is lower-semicontinuous liewise ϕb ) is an upper-semicontinuous function. Remember that we denote both ϕ and ϕ 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 ϕ. Lemma 2 Let ϕ :a b) c d) c d) where < c a < b d + be a continuous function satisfying H1) and H2). Then the function F ϕ in H2) is unique and it is a continuous nonincreasing map thus it has a unique fixed point μ ϕ. Furthermore i) a < ϕx y)<b for all x y a b) and a ϕx y) b for all x y [a b]. ii) If x [a b] y c d) and ϕx y)=y then y = F ϕ x). iii) ϕx F ϕ x)) = F ϕ x) for all x [a b]. iv) F ϕ is decreasing in F ϕ 1 a b)).
Franco and Peran Boundary Value Problems 2013 2013:63 Page 7 of 14 We are in conditions of presenting and proving our main result. Theorem 1 Let ϕ :a b) c d) c d) where < c a < b d + be a continuous function satisfying H1) and H2). If μ ϕ a b) and FixF ϕ F ϕ )=FixF ϕ ) then μ ϕ is an unconditional stable attractor of ϕ in a b). Proof of Theorem 1 Consider λ 2 m and denote y n = ϕ λy n 1...y n m ) for some y 1...y m a b). Notice that y n a b) for all n as a consequence of i) in Lemma 2. We are going to prove first that μ ϕ is a stable equilibrium of ϕ λ. By iii) in Lemma 2 as λμ ϕ...μ ϕ )=μ ϕ μ ϕ ) we see that μ ϕ is an equilibrium. Let ɛ 0 min{μ ϕ a b μ ϕ }). Because of the continuity of F ϕ thereisa μ ϕ ɛ μ ϕ ) such that b F ϕ a ) μ ϕ μ ϕ + ɛ). As FixF ϕ F ϕ )=FixF ϕ )andf ϕ F ϕ a)) awehave F ϕ Fϕ x) ) > x for all x [a μ ϕ ). If x [a b ] then F ϕ x) F ϕ a ) = b and F ϕ x) F ϕ b ) = F ϕ Fϕ a )) > a. Therefore F ϕ [a b ]) [a b ]. By replacing a b by a b in Lemma 2i) we see that y n a b ) μ ϕ ɛ μ ϕ + ɛ) for all n whenever y n a b )forn mthusμ ϕ is an unconditional stable equilibrium of ϕ. Now if we see that lim F ϕ y n )=μ ϕ
Franco and Peran Boundary Value Problems 2013 2013:63 Page 8 of 14 we are done with the whole proof. Indeed for each accumulation point y of y n ) one would have F ϕ μ ϕ )=μ ϕ = F ϕ y) because of the continuity of F ϕ.asμ ϕ a b) this implies y = μ ϕ. Therefore as a consequence of Lemma 1 it suffices to find an increasing sequence n of natural numbers such that a F ϕ y n ) b for all 0 n n. Here a and b are defined as in Lemma 1witha 0 = a and ɛ =1 b 0 = F ϕ a); a = F ϕ b 1 + 1 ); b = F ϕ a 1 ) for 1. Let n 0 = msothat a 0 F ϕ y n ) F ϕ a)=b 0 for n n 0. Having in mind that λ 2 m satisfies 3) we find n from n 1 as follows. Denote z = F 1 ϕ b 1) and momentarily assume n > n 1 + m and b 1 < y n in such a way that b 1 < y n = ϕ λ 1 y n 1...y n m ) λ 2 y n 1...y n m ) ) ϕ min{y n 1...y n m } λ 2 y n 1...y n m ) ) ϕ z λ 2 y n 1...y n m ) ) which implies ϕ z λ 2 y n 1...y n m ) ) λ 2 y n 1...y n m ) and then y n max { b 1 λ 2 y n 1...y n m ) } max{b 1 y n 1...y n m } w n for all n > n 1 + m. As a consequence the nonincreasing sequence w n is bounded below by b 1. It cannot be the case that lim w n > b 1 becauseinsuchacasethereisasubsequencey nj > b 1 converging to lim w n and such that λ 2 y nj 1...y nj m) converges to a point w lim w n. Since y nj ϕ z λ 2 y nj 1...y nj m) ) w nj
Franco and Peran Boundary Value Problems 2013 2013:63 Page 9 of 14 one has and then ϕz w)=lim w n > b 1 = Fz ) ϕz w) w > Fz ) w. By applying H2) we see that lim w n = ϕz w)<w a contradiction. Therefore lim w n = b 1 and there exists m n 1 such that y n < b 1 + 1 for all n m that is F ϕ y n ) a for all n m. Analogously we see that there exists n m such that F ϕ y n ) b for all n n. Proof of Lemma 2 Uniqueness of F ϕ :Lety 1 < y 2 and x in [a b] such that Then y i y ϕx y) y 1>0 for i =12y y 1 y 2 ). 0<ϕx y) y <0 a contradiction. i): It suffices to prove the first assertion because [a b] is a closed set and by definition ϕx y)= { lim sup ϕx n y n ):x n x y n y x n a b) y n 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. If ϕx y) =y it is obvious that ϕx y) a b). On the other hand if ϕx y)>y and x a x)then a y < ϕx y)<ϕ x y ) F ϕ x ) b. Finally if ϕx y)<y and x x b)then b y > ϕx y)>ϕ x y ) F ϕ x ) a.
Franco and Peran Boundary Value Problems 2013 2013:63 Page 10 of 14 ii): Suppose y F ϕ x).sinceϕx y)=y Rthenϕx y) y =0or ϕx y) y = R.Inany event it cannot be the case that F ϕ x) y ϕx y) y 1 which contradicts hypothesis H2) thus y = F ϕ x). iii): Since F ϕ [a b]) [a b] [c d] it is worth considering the following three cases for each x [a b]: firstx a b) F ϕ x) c d) and then after probing continuity monotonicity and statement iv)) we proceed with the case x {a b} F ϕ x) c d) and finally with x [a b] F ϕ x) {c d}. Case x a b) and F ϕ x) c d): Sinceϕx y)>y when y < F ϕ x) and ϕx y)<y when y > F ϕ x)weseethat ϕ x F ϕ x) ) = F ϕ x) because of the continuity of ϕx ). Monotonicity and iv): Suppose F ϕ x 1 ) F ϕ x 2 ) for a x 1 < x 2 b. If y [F ϕ x 1 ) F ϕ x 2 )]theny = a y = b or y ϕx 2 y)<ϕx 1 y) ythus [ Fϕ x 1 ) F ϕ x 2 ) ] {a b}. Continuity: If x [a b] and w I = lim inf z x ) F ϕ z) lim sup F ϕ y) y x then there exist two sequences y n z n x with F ϕ z n )<w < F ϕ y n ). Thus ϕz n w)<w < ϕy n w) and by ii) one has ϕx w)=w.sincew c d) this would imply F ϕ x)=w for all w I which is impossible. iii) Case x {a b} and F ϕ x) c d): Since F ϕ t) ϕ t F ϕ a) ) F ϕ a) for all t a b) and because of the continuity of F ϕ wehave F ϕ a)=sup t F ϕ t) sup ϕ t F ϕ a) ) F ϕ a) t
Franco and Peran Boundary Value Problems 2013 2013:63 Page 11 of 14 but sup ϕ t F ϕ a) ) = ϕ a F ϕ a) ). t Analogouslyitcanbeseenthatϕb F ϕ b)) = F ϕ b). iii) Case F ϕ x) {c d}: First assume F ϕ x)=c and recall that by definition ϕx c)= { lim sup ϕx n y n ):x n x y n c x n a b) y n c d) }. Suppose ϕx n y n ) c > c for all n.then 1 F ϕx n ) y n ϕx n y n ) y n F ϕx n ) y n c y n which implies F ϕ x n ) c > c eventually for all n. Since F ϕ x n ) c we reach a contradiction. Therefore ϕ x F ϕ x) ) = {c} = { F ϕ x) }. Analogously we see that ϕx F ϕ x)) = {d} when F ϕ x)=d. 4 Examples and applications 4.1 The difference equation y n = A + y n y n q ) p with 0 < p <1 The paper [6] is devoted to prove that every positive solution to the difference equation y n = A + ) p yn y n q converges to the equilibrium A +1whenever A 0 + ) and p 0 min { 1 A +1)/2 }). Here q {123...} are fixed numbers. Although paper [6] complements [7] where the case p = 1 had been considered it should be noticed that the case p =1A 1isnot 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 +1wheneverA 0 + ) p 0 min{1 A +1)/2}). We may assume without loss of generality that the initial values y 1...y m are greater than A. Here and in the sequel m = max{ q}. Let A 0 + ) p 0 1) a = c = A b = d =+ ) y p ϕx y)=a + x
Franco and Peran Boundary Value Problems 2013 2013:63 Page 12 of 14 and λx 1...x m )=x q x ). Define F ϕ + )=A consider for the moment a fixed x [A ) anddefinef ϕ x) tobe the unique positive zero of the function f x given by f x y)=ϕx y) y. Notice that f x is concave f x 0) = A >0andf x + )=. Clearly F ϕ x) is also the unique zero of the increasing function j x y)=ϕx y) F ϕ x). Since j x A)= Ap F ϕ x)) p x p 0 and j x + )=+ >0 we see that condition H2) holds and μ ϕ = A +1. As for condition FixF ϕ F ϕ )=FixF ϕ ) 5) if A x < y <+ with A + ) y p = y x then y > A +1and ϕy x) x = A + Since the function hz)=a + 1 z A z z A) 1/p ) x p x = A + 1 y y A y y A). 1/p has a unique critical point in A+ ) andha +1)=0h+ ) =A the necessary and sufficient condition for 5)toholdisthath A +1) 0 that is p A +1)/2. By this reasoning we also get for free unconditional stable convergence for several difference equations as for instance: or ) yn q y p n r y n = A + with 0 < p < min { 1/2 A +1)/4 } y n s y n t ) yn q + y p n r y n = A + with 0 < p < min { 1 A +1)/2 } y n s + y n t
Franco and Peran Boundary Value Problems 2013 2013:63 Page 13 of 14 just considering respectively λx 1...x m )= x s x t x q x r ) xs + x t λx 1...x m )= x ) q + x r 2 2 where m = max{q r s t}. 4.2 The difference equation y n = α+βy n 1 A+By n 1 +Cy n 2 Here α β A B C y 0 y 1 0. In 2003 three conjectures on the above equation were posed in [8]. In all three cases B =0α β A C >0;A =0α β B C >0;andα β A B C >0 respectively) it was postulated the global asymptotic stability of the equilibrium. These conjectures have resulted in several papers since thensee[9 12]). Let us see when there is unconditional convergence. Consider a = c =0b = c =+ and ϕx y)= α + βy A + Dx with α β A 0 D >0.Wesolveiny the equation y = α+βy to obtain A+Dx y = α A β + Dx so we consider A > β α >0todefine F ϕ x)= α A β + Dx. A simple calculation shows that H2) holds μ ϕ a b)andfixf ϕ F ϕ )=FixF ϕ ): F ϕ x) y ϕx y) y =1+ β A β + Dx 1 μ ϕ = 1 A A + β + β)2 +4Dα )) 2D F ϕ F ϕ x)) x F ϕ x) x = A β + Dx)A β) A β) 2 + DxA β)+dα >0. Therefore μ ϕ is an unconditional stable attractor of ϕ in a b)=0+ )whenevera > β 0 D >0andα >0. If we choose ) Bx1 + Cx 2 λx 1 x 2 )= B + C x 1 and D = B + C we obtain unconditional stable convergence for the equation y n = α + βy n 1 A + By n 1 + Cy n 2 whenever A > β 0 B + C >0C 0andα >0.
Franco and Peran Boundary Value Problems 2013 2013:63 Page 14 of 14 Other choices of λ result on the unconditional stable convergence of difference equations such as y n = α + βy n 1 + γ y n 2 A + By n 1 + Cy n 2 with A > β + γ B + C >0γ 0 α >0β 0 C 0. Or y n = α + β max{y n 1 y n 2 } A + D min{y n 1 y n 2 } with A > β 0 D >0α >0. 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 wor described in this paper. Authors contributions Both authors contributed to each part of this wor equally and read and approved the final version of the manuscript. Acnowledgements 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 1. Agarwal RP: Difference Equations and Inequalities. Theory Methods and Applications 2nd edn. Monographs and Textboos in Pure and Applied Mathematics vol. 228. Deer New Yor 2000) 2. El-Morshedy HA Gopalsamy K: Oscillation and asymptotic behaviour of a class of higher-order non-linear difference equations. Ann. Mat. Pura Appl. 182143-159 2003) 3. El-Morshedy HA Liz E: Globally attracting fixed points in higher order discrete population models. J. Math. Biol. 53 365-384 2006) 4. Kocić VL Ladas G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Mathematics and Its Applications vol. 256. Kluwer Academic Dordrecht 1993) 5. Beer G: Topologies on Closed and Closed Convex Sets. Mathematics and Its Applications vol. 268. Kluwer Academic Dordrecht 1993) 6. Berenhaut KS Foley JD Stević S: The global attractivity of the rational difference equation y n = A + y n yn m )p.proc.am. Math. Soc. 136 103-110 2008) 7. Berenhaut KS Foley JD Stević S: The global attractivity of the rational difference equation y n =1+ y n.proc.am. yn m Math. Soc. 135 1133-1140 2007) α+βxn 8. Kulenović MRS Ladas G Martins LF Rodrigues IW: The dynamics of x n+1 = : facts and conjectures. A+Bxn+Cx n 1 Comput. Math. Appl. 45 1087-1099 2003) 9. Merino O: Global attractivity of the equilibrium of a difference equation: an elementary proof assisted by computer algebra system. J. Differ. Equ. Appl. 17 33-41 2011) 10. Su Y-H Li W-T Stević S: Dynamics of a higher order nonlinear rational difference equation. J. Differ. Equ. Appl. 11 133-150 2005) 11. Su Y-H Li W-T: Global asymptotic stability of a second-order nonlinear difference equation. Appl. Math. Comput. 168 981-989 2005) 12. Su Y-H Li W-T: Global attractivity of a higher order nonlinear difference equation. J. Differ. Equ. Appl. 11 947-958 2005) doi:10.1186/1687-2770-2013-63 Cite this article as: Franco and Peran: Unconditional convergence of difference equations. Boundary Value Problems 2013 2013:63.