Special Issues

Refined Wilf-equivalences by Comtet statistics

  • Received: 01 May 2020 Revised: 01 November 2020 Published: 15 March 2021
  • Primary: 05A05, 05A15, 05A19; Secondary: 05C05

  • We launch a systematic study of the refined Wilf-equivalences by the statistics comp and iar, where comp(π) and iar(π) are the number of components and the length of the initial ascending run of a permutation π, respectively. As Comtet was the first one to consider the statistic comp in his book Analyse combinatoire, any statistic equidistributed with comp over a class of permutations is called by us a Comtet statistic over such class. This work is motivated by a triple equidistribution result of Rubey on 321-avoiding permutations, and a recent result of the first and third authors that iar is a Comtet statistic over separable permutations. Some highlights of our results are:

    ● Bijective proofs of the symmetry of the joint distribution (comp,iar) over several Catalan and Schröder classes, preserving the values of the left-to-right maxima.

    ● A complete classification of comp- and iar-Wilf-equivalences for length 3 patterns and pairs of length 3 patterns. Calculations of the (des,iar,comp) generating functions over these pattern avoiding classes and separable permutations.

    ● A further refinement of Wang's descent-double descent-Wilf equivalence between separable permutations and (2413,4213)-avoiding permutations by the Comtet statistic iar.

    Citation: Shishuo Fu, Zhicong Lin, Yaling Wang. Refined Wilf-equivalences by Comtet statistics[J]. Electronic Research Archive, 2021, 29(5): 2877-2913. doi: 10.3934/era.2021018

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  • We launch a systematic study of the refined Wilf-equivalences by the statistics comp and iar, where comp(π) and iar(π) are the number of components and the length of the initial ascending run of a permutation π, respectively. As Comtet was the first one to consider the statistic comp in his book Analyse combinatoire, any statistic equidistributed with comp over a class of permutations is called by us a Comtet statistic over such class. This work is motivated by a triple equidistribution result of Rubey on 321-avoiding permutations, and a recent result of the first and third authors that iar is a Comtet statistic over separable permutations. Some highlights of our results are:

    ● Bijective proofs of the symmetry of the joint distribution (comp,iar) over several Catalan and Schröder classes, preserving the values of the left-to-right maxima.

    ● A complete classification of comp- and iar-Wilf-equivalences for length 3 patterns and pairs of length 3 patterns. Calculations of the (des,iar,comp) generating functions over these pattern avoiding classes and separable permutations.

    ● A further refinement of Wang's descent-double descent-Wilf equivalence between separable permutations and (2413,4213)-avoiding permutations by the Comtet statistic iar.



    In this paper, we study the behavior of the solutions of the following three-dimensional system of difference equations of second order

    xn+1=f(yn,yn1),yn+1=g(zn,zn1),zn+1=h(xn,xn1) (1)

    where nN0, the initial values xi, yi and zi,i=0,1, are positive real numbers, the functions f,g,h:(0,+)2(0,+) are continuous and homogeneous of degree zero. We establish results on local and global stability of the unique positive equilibrium point. To do this we prove some general convergence theorems, that can be applied to generalize a lot of existing systems and to study new ones. Some results on existence of periodic and oscillatory solutions are also proved.

    Now, we explain our motivation for doing this work. Clearly if we take zi=xi,i=0,1, and hg, then the system (1), will be

    xn+1=f(yn,yn1),yn+1=g(xn,xn1). (2)

    Noting also that if we choose zi=yi=xi,i=0,1, and hgf, then System (1) will be

    xn+1=f(xn,xn1). (3)

    In [23], the behavior of the solutions of System (2) has been investigated. System (2) is a generalization of Equation (3), studied in [17]. The present System (1) is the three-dimensional generalization of System (2).

    In the literature there are many studies on difference equations defined by homogeneous functions, see for instance [1,2,5,7,11,16]. Noting that also a lot of studies are devoted to various models of difference equations and systems, not necessary defined by homogeneous functions, see for example [4,9,10,12,14,18,19,20,21,22,24,25,26,27,28,29].

    Before we state our results, we recall the following definitions and results. For more details we refer to the following references [3,6,8,13].

    Let F:(0,+)6(0,+)6 be a continuous function and consider the system of difference equations

    Yn+1=F(Yn),nN0, (4)

    where the initial value Y0(0,+)6. A point ¯Y(0,+)6 is an equilibrium point of (4), if it is a solution of ¯Y=F(¯Y).

    Definition 1.1. Let ¯Y be an equilibrium point of System (4), and let . any convenient vector norm.

    1. We say that the equilibrium point ¯Y is stable (or locally stable) if for every ϵ>0 there exists δ>0 such that for every initial condition Y0: Y0¯Y<δ implies Yn¯Y<ϵ. Otherwise, the equilibrium point ¯Y is unstable.

    2. We say that the equilibrium point ¯Y is asymptotically stable (or locally asymptotically stable) if it is stable and there exists γ>0 such that Y0¯Y<γ implies

    limnYn=¯Y.

    3. We say that the equilibrium point ¯Y is a global attractor if for every Y0,

    limnYn=¯Y.

    4. We say that the equilibrium point ¯Y is globally (asymptotically) stable if it is stable and a global attractor.

    Assume that F is C1 on (0,+)6. To System (4), we associate a linear system, about the equilibrium point ¯Y, given by

    Zn+1=FJ(¯Y)Zn,nN0,Zn=Yn¯Y,

    where FJ is the Jacobian matrix of the function F evaluated at the equilibrium point ¯Y.

    To study the stability of the equilibrium point ¯Y, we need the following theorem.

    Theorem 1.2. Let ¯Y be an equilibrium point of System (4). Then, the following statements are true:

    (i) If all the eigenvalues of the Jacobian matrix FJ lie in the open unit disk |λ|<1, then the equilibrium ¯Y is asymptotically stable.

    (ii) If at least one eigenvalue of FJ has absolute value greater than one, then the equilibrium ¯Y is unstable.

    Definition 1.3. A solution (xn,yn,zn)n1 of System (1) is said to be periodic of period pN if

    xn+p=xn,yn+p=yn,zn+p=zn,n1. (5)

    The solution (xn,yn,zn)n1 is said to be periodic with prime period pN, if it is periodic with period p and p is the least positive integer for which (1.3) holds.

    Definition 1.4. Let (xn,yn,zn)n1 be a solution of System (1). We say that the sequence (xn)n1 (resp. (yn)n1, (zn)n1) oscillates about ¯x (resp. ¯y, ¯z) with a semi-cycle of length one if: (xn¯x)(xn+1¯x)<0,n1 (resp. (yn¯y)(yn+1¯y)<0,n1, (zn¯z)(zn+1¯z)<0,n1).

    Remark 1. For every term xn0 of the sequence (xn)n1, the notation "+ " means xn0¯x>0 and the notation "" means xn0¯x<0. The same notations will be used for the terms of the sequences (yn)n1 and (zn)n1.

    Definition 1.5. A function Φ:(0,+)2(0,+) is said to be homogeneous of degree mR if we have

    Φ(λu,λv)=λmΦ(u,v)

    for all (u,v)(0,+)2 and for all λ>0.

    Theorem 1.6. Let Φ:(0,+)2(0,+) be a C1 function on (0,+)2.

    1. Then, Φ is homogeneous of degree m if and only if

    uΦu(u,v)+vΦv(u,v)=mΦ(u,v),(u,v)(0,+)2.

    (This statement is usually called Euler's Theorem).

    2. If Φ is homogeneous of degree m on (0,+)2, then Φu and Φv are homogenous of degree m1 on (0,+)2.

    A point (¯x,¯y,¯z)(0,+)3 is an equilibrium point of System (1) if it is a solution of the following system

    ¯x=f(¯y,¯y),¯y=g(¯z,¯z),¯z=g(¯x,¯x).

    Using the fact that f, g and h are homogeneous of degree zero, we get that

    (¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1))

    is the unique equilibrium point of System (1).

    Let F:(0,+)6(0,+)6 be the function defined by

    F(W)=(f1(W),f2(W),g1(W),g2(W),h1(W),h2(W)),W=(u,v,w,t,r,s)

    with

    f1(W)=f(w,t),f2(W)=u,g1(W)=g(r,s),g2(W)=w,h1(W)=h(u,v),g2(W)=r.

    Then, System (1) can be written as follows

    Wn+1=F(Wn),Wn=(xn,xn1,yn,yn1,zn,zn1)t,nN0.

    So, (¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)) is an equilibrium point of system (1) if and only if

    ¯W=(¯x,¯x,¯y,¯y,¯z,¯z)=(f(1,1),f(1,1),g(1,1),g(1,1),h(1,1),h(1,1))

    is an equilibrium point of Wn+1=F(Wn).

    Assume that the functions f, g and h are C1 on (0,+)2. To System (1), we associate about the equilibrium point ¯W the following linear system

    Xn+1=JFXn,nN0

    where JF is the Jacobian matrix associated to the function F evaluated at

    ¯W=(f(1,1),f(1,1),g(1,1),g(1,1),h(1,1),h(1,1)).

    We have

    JF=(00fw(¯y,¯y)ft(¯y,¯y)001000000000gr(¯z,¯z)gs(¯z,¯z)001000hu(¯x,¯x)hv(¯x,¯x)0000000010)

    As f, g and h are homogeneous of degree 0, then using Part 1. of Theorem 1.6, we get

    ¯yfw(¯y,¯y)+¯yft(¯y,¯y)=0

    which implies

    ft(¯y,¯y)=fw(¯y,¯y).

    Similarly we get

    gs(¯z,¯z)=gr(¯z,¯z),hv(¯x,¯x)=hu(¯x,¯x).

    It follows that JF takes the form:

    JF=(00fw(¯y,¯y)fw(¯y,¯y)001000000000gr(¯z,¯z)gr(¯z,¯z)001000hu(¯x,¯x)hu(¯x,¯x)0000000010)

    The characteristic polynomial of the matrix JF is given by

    P(λ)=λ6hu(¯x,¯x)gr(¯z,¯z)fw(¯y,¯y)λ3+3hu(¯x,¯x)gr(¯z,¯z)fw(¯y,¯y)λ23hu(¯x,¯x)gr(¯z,¯z)fw(¯y,¯y)λ+hu(¯x,¯x)gr(¯z,¯z)fw(¯y,¯y).

    Now assume that

    |hu(¯x,¯x)gr(¯z,¯z)fw(¯y,¯y)|<18

    and consider the two functions

    Φ(λ)=λ6,
    Ψ(λ)=hu(¯x,¯x)gr(¯z,¯z)fw(¯y,¯y)λ3+3hu(¯x,¯x)gr(¯z,¯z)fw(¯y,¯y)λ23hu(¯x,¯x)gr(¯z,¯z)fw(¯y,¯y)λ+hu(¯x,¯x)gr(¯z,¯z)fw(¯y,¯y).

    We have

    |Ψ(λ)|8|hu(¯x,¯x)gr(¯z,¯z)fw(¯y,¯y)|<1=|Φ(λ)|,λC:|λ|=1.

    So, by Rouché's Theorem it follows that all roots of P(λ) lie inside the unit disk.

    Hence, by Theorem 1.2, we deduce from the above consideration that the equilibrium point (¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)) is locally asymptotically stable.

    Using Part 2. of Theorem 1.6 and the fact that the function f, g, h are homogeneous of degree zero, we get that fw, gr and hu are homogeneous of degree 1. So, it follows that

    fw(¯y,¯y)=fw(1,1)¯y,gr(¯z,¯z)=gr(1,1)¯z,hu(¯x,¯x)=hu(1,1)¯x.

    In summary, we have proved the following result.

    Theorem 2.1. Assume that f(u,v), g(u,v) and h(u,v) are C1 on (0,+)2. The equilibrium point

    (¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1))

    of System (1) is locally asymptotically stable if

    |fu(1,1)gu(1,1)hu(1,1)|<f(1,1)g(1,1)h(1,1)8.

    Now, we will prove some general convergence results. The obtained results allow us to deal with the global attractivity of the equilibrium point (¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)) and so the global stability.

    Theorem 2.2. Consider System (1). Assume that the following statements are true:

    1. H1: There exist a,b,α,β,λ,γ(0,+) such that

    af(u,v)b,αg(u,v)β,λh(u,v)γ,(u,v)(0,+)2.

    2. H2: f(u,v),g(u,v),h(u,v) are increasing in u for all v and decreasing in v for all u.

    3. H3: If (m1,M1,m2,M2,m3,M3)[a,b]2×[α,β]2×[λ,γ]2 is a solution of the system

    m1=f(m2,M2),M1=f(M2,m2),m2=g(m3,M3),M2=g(M3,m3),m3=h(m1,M1),M3=h(M1,m1)

    then

    m1=M1,m2=M2,m3=M3.

    Then every solution of System (1) converges to the unique equilibrium point

    (¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)).

    Proof. Let

    m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ

    and for each i=0,1,...,

    mi+11:=f(mi2,Mi2),Mi+11:=f(Mi2,mi2),
    mi+12:=g(mi3,Mi3),Mi+12:=g(Mi3,mi3),
    mi+13:=h(mi1,Mi1),Mi+13:=h(Mi1,mi1).

    We have

    af(α,β)f(β,α)b,
    αg(λ,γ)g(γ,λ)β,
    λh(a,b)h(b,a)γ,

    and so,

    m01=af(m02,M02)f(M02,m02)b=M01,
    m02=αg(m03,M03)g(M03,m03)β=M02,

    and

    m03=λh(m01,M01)g(M01,m01)γ=M03.

    Hence,

    m01m11M11M01,
    m02m12M12M02,

    and

    m03m13M13M03.

    Now, we have

    m11=f(m02,M02)f(m12,M12)=m21f(M12,m12)=M21f(M02,m02)=M11,
    m12=g(m03,M03)g(m13,M13)=m22g(M13,m13)=M22g(M03,m03)=M12,
    m13=h(m01,M01)h(m11,M11)=m23h(M11,m11)=M23h(M01,m01)=M13,

    and it follows that

    m01m11m21M21M11M01,
    m02m12m22M22M12M02,

    and

    m03m13m23M23M13M03.

    By induction, we get for i=0,1,..., that

    a=m01m11...mi11mi1Mi1Mi11...M11M01=b,
    α=m02m12...mi12mi2Mi2Mi12...M12M02=β,

    and

    λ=m03m13...mi13mi3Mi3Mi13...M13M03=γ.

    It follows that the sequences (mi1)iN0, (mi2)iN0, (mi3)iN0 (resp. (Mi1)iN0, (Mi2)iN0, (Mi3)iN0) are increasing (resp. decreasing) and also bounded, so convergent. Let

    m1=limi+mi1,M1=limi+Mi1,
    m2=limi+mi2,M2=limi+Mi2.
    m3=limi+mi3,M3=limi+Mi3.

    Then

    am1M1b,αm2M2β,λm3M3γ.

    By taking limits in the following equalities

    mi+11=f(mi2,Mi2),Mi+11=f(Mi2,mi2),
    mi+12=g(mi3,Mi3),Mi+12=g(Mi3,mi3),
    mi+13=h(mi1,Mi1),Mi+13=h(Mi1,mi1),

    and using the continuity of f, g and h we obtain

    m1=f(m2,M2),M1=f(M2,m2),m2=g(m3,M3),M2=g(M3,m3),m3=h(m1,M1),M3=h(M1,m1)

    so it follows from H3 that

    m1=M1,m2=M2,m3=M3.

    From H1, for n=1,2,, we get

    m01=axnb=M01,m02=αynβ=M02,m03=λznγ=M03.

    For n=2,3,..., we have

    m11=f(m02,M02)xn+1=f(yn,yn1)f(M02,m02)=M11,
    m12=g(m03,M03)yn+1=g(zn,zn1)g(M03,m03)=M12,

    and

    m13=h(m01,M01)zn+1=h(xn,xn1)h(M01,m01)=M13,

    that is

    m11xnM11,m12ynM12,m13znM13,n=3,4,.

    Now, for n=4,5,..., we have

    m21=f(m12,M12)xn+1=f(yn,yn1)f(M12,m12)=M21,

    and

    m22=g(m13,M13)yn+1=g(zn,zn1)g(M13,m13)=M22,
    m23=h(m11,M11)zn+1=h(xn,xn1)h(M11,m11)=M23,

    that is

    m21xnM21,m22ynM22,m23znM23,n=5,6,.

    Similarly, for n=6,7,..., we have

    m31=f(m22,M22)xn+1=f(yn,yn1)f(M22,m22)=M31,
    m32=g(m23,M23)yn+1=g(zn,zn1)g(M23,m23)=M32,

    and

    m33=h(m21,M21)zn+1=h(xn,xn1)h(M21,m21)=M33,

    that is

    m31xnM31,m32ynM32,m33znM33,n=7,8,.

    It follows by induction that for i=0,1,... we get

    mi1xnMi1,mi2ynMi2,mi3znMi3,n2i+1.

    Using the fact that i+ implies n+ and m1=M1,m2=M2,m3=M3, we obtain that

    limn+xn=M1,limn+yn=M2,limn+zn=M3.

    From (1) and using the fact that f, g and h are continuous and homogeneous of degree zero, we get

    M1=f(M2,M2)=f(1,1),M2=g(M3,M3)=g(1,1),M3=h(M1,M1)=h(1,1).

    Theorem 2.3. Consider System (1). Assume that the following statements are true:

    1. H1: There exist a,b,α,β,λ,γ(0,+) such that

    af(u,v)b,αg(u,v)β,λh(u,v)γ,(u,v)(0,+)2.

    2. H2: f(u,v),g(u,v) are increasing in u for all v and decreasing in v for all u and h(u,v) is decreasing in u for all v and increasing in v for all u.

    3. H3: If (m1,M1,m2,M2,m3,M3)[a,b]2×[α,β]2×[λ,γ]2 is a solution of the system

    m1=f(m2,M2),M1=f(M2,m2),m2=g(m3,M3),M2=g(M3,m3),m3=h(M1,m1),M3=h(m1,M1)

    then

    m1=M1,m2=M2,m3=M3.

    Then every solution of System (1) converges to the unique equilibrium point

    (¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)).

    Proof. Let

    m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ

    and for each i=0,1,...,

    mi+11:=f(mi2,Mi2),Mi+11:=f(Mi2,mi2),
    mi+12:=g(mi3,Mi3),Mi+12:=g(Mi3,mi3),
    mi+13:=h(Mi1,mi1),Mi+13:=h(mi1,Mi1).

    We have

    af(α,β)f(β,α)b,
    αg(λ,γ)g(γ,λ)β,
    λh(b,a)h(a,b)γ,

    and so,

    m01=af(m02,M02)f(M02,m02)b=M01,
    m02=αg(m03,M03)g(M03,m03)β=M02,

    and

    m03=λh(M01,m01)h(m01,M01)γ=M03.

    Hence,

    m01m11M11M01,
    m02m12M12M02,

    and

    m03m13M13M03.

    Now, we have

    m11=f(m02,M02)f(m12,M12)=m21f(M12,m12)=M21f(M02,m02)=M11,
    m12=g(m03,M03)g(m13,M13)=m22g(M13,m13)=M22g(M03,m03)=M12,
    m13=h(M01,m01)h(M11,m11)=m23h(m11,M11)=M23h(m01,M01)=M13,

    and it follows that

    m01m11m21M21M11M01,
    m02m12m22M22M12M02,

    and

    m03m13m23M23M13M03.

    By induction, we get for i=0,1,..., that

    a=m01m11...mi11mi1Mi1Mi11...M11M01=b,
    α=m02m12...mi12mi2Mi2Mi12...M12M02=β,

    and

    λ=m03m13...mi13mi3Mi3Mi13...M13M03=γ.

    It follows that the sequences (mi1)iN0, (mi2)iN0, (mi3)iN0 (resp. (Mi1)iN0, (Mi2)iN0, (Mi3)iN0) are increasing (resp. decreasing) and also bounded, so convergent. Let

    m1=limi+mi1,M1=limi+Mi1,
    m2=limi+mi2,M2=limi+Mi2.
    m3=limi+mi3,M3=limi+Mi3.

    Then

    am1M1b,αm2M2β,λm3M3γ.

    By taking limits in the following equalities

    mi+11=f(mi2,Mi2),Mi+11=f(Mi2,mi2),
    mi+12=g(mi3,Mi3),Mi+12=g(Mi3,mi3),
    mi+13=h(Mi1,mi1),Mi+13=h(mi1,Mi1),

    and using the continuity of f, g and h we obtain

    m1=f(m2,M2),m2=f(M2,m2),m2=g(m3,M3),M2=g(M3,m3),m3=h(M1,m1),M3=h(m1,M1)

    so it follows from H3 that

    m1=M1,m2=M2,m3=M3.

    From H1, for n=1,2,, we get

    m01=axnb=M01,m02=αynβ=M02,m03=λznγ=M03.

    For n=2,3,..., we have

    m11=f(m02,M02)xn+1=f(yn,yn1)f(M02,m02)=M11,
    m12=g(m03,M03)yn+1=g(zn,zn1)g(M03,m03)=M12,

    and

    m13=h(M01,m01)zn+1=h(xn,xn1)h(m01,M01)=M13,

    that is

    m11xnM11,m12ynM12,m13znM13,n=3,4,.

    Now, for n=4,5,..., we have

    m21=f(m12,M12)xn+1=f(yn,yn1)f(M12,m12)=M21,

    and

    m22=g(m13,M13)yn+1=g(zn,zn1)g(M13,m13)=M22,
    m23=h(M11,m11)zn+1=h(xn,xn1)h(m11,M11)=M23,

    that is

    m21xnM21,m22ynM22,m23znM23,n=5,6,.

    Similarly, for n=6,7,..., we have

    m31=f(m22,M22)xn+1=f(yn,yn1)f(M22,m22)=M31,
    m32=g(m23,M23)yn+1=g(zn,zn1)g(M23,m23)=M32,

    and

    m33=h(M21,m21)zn+1=h(xn,xn1)h(m21,M21)=M33,

    that is

    m31xnM31,m32ynM32,m33znM33,n=7,8,.

    It follows by induction that for i=0,1,... we get

    mi1xnMi1,mi2ynMi2,mi3znMi3,n2i+1.

    Using the fact that i+ implies n+ and m1=M1,m2=M2,m3=M3, we obtain that

    limn+xn=M1,limn+yn=M2,limn+zn=M3.

    From (1) and using the fact that f, g and h are continuous and homogeneous of degree zero, we get

    M1=f(M2,M2)=f(1,1),M2=g(M3,M3)=g(1,1),M3=h(M1,M1)=h(1,1).

    Theorem 2.4. Consider System (1). Assume that the following statements are true:

    1. H1: There exist a,b,α,β,λ,γ(0,+) such that

    af(u,v)b,αg(u,v)β,λh(u,v)γ,(u,v)(0,+)2.

    2. H2: f(u,v) is increasing in u for all v and decreasing in v for all u and g(u,v), h(u,v) are decreasing in u for all v and increasing in v for all u.

    3. H3: If (m1,M1,m2,M2,m3,M3)[a,b]2×[α,β]2×[λ,γ]2 is a solution of the system

    m1=f(m2,M2),M1=f(M2,m2),m2=g(M3,m3),M2=g(m3,M3),m3=h(M1,m1),M3=h(m1,M1)

    then

    m1=M1,m2=M2,m3=M3.

    Then every solution of System (1) converges to the unique equilibrium point

    (¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)).

    Proof. Let

    m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ

    and for each i=0,1,...,

    mi+11:=f(mi2,Mi2),Mi+11:=f(Mi2,mi2),
    mi+12:=g(Mi3,mi3),Mi+12:=g(mi3,Mi3),
    mi+13:=h(Mi1,mi1),Mi+13:=h(mi1,Mi1).

    We have

    af(α,β)f(β,α)b,
    αg(γ,λ)g(λ,γ)β,
    λh(b,a)h(a,b)γ,

    and so,

    m01=af(m02,M02)f(M02,m02)b=M01,
    m02=αg(M03,m03)g(m03,M03)β=M02,

    and

    m03=λh(M01,m01)h(m01,M01)γ=M03.

    Hence,

    m01m11M11M01,
    m02m12M12M02,

    and

    m03m13M13M03.

    Now, we have

    m11=f(m02,M02)f(m12,M12)=m21f(M12,m12)=M21f(M02,m02)=M11,
    m12=g(M03,m03)g(M13,m13)=m22g(m13,M13)=M22g(m03,M03)=M12,
    m13=h(M01,m01)h(M11,m11)=m23h(m11,M11)=M23h(m01,M01)=M13,

    and it follows that

    m01m11m21M21M11M01,
    m02m12m22M22M12M02,

    and

    m03m13m23M23M13M03.

    By induction, we get for i=0,1,..., that

    a=m01m11...mi11mi1Mi1Mi11...M11M01=b,
    α=m02m12...mi12mi2Mi2Mi12...M12M02=β,

    and

    λ=m03m13...mi13mi3Mi3Mi13...M13M03=γ.

    It follows that the sequences (mi1)iN0, (mi2)iN0, (mi3)iN0 (resp. (Mi1)iN0, (Mi2)iN0, (Mi3)iN0) are increasing (resp. decreasing) and also bounded, so convergent. Let

    m1=limi+mi1,M1=limi+Mi1,
    m2=limi+mi2,M2=limi+Mi2.
    m3=limi+mi3,M3=limi+Mi3.

    Then

    am1M1b,αm2M2β,λm3M3γ.

    By taking limits in the following equalities

    mi+11=f(mi2,Mi2),Mi+11=f(Mi2,mi2),
    mi+12=g(Mi3,mi3),Mi+12=g(mi3,Mi3),
    mi+13=h(Mi1,mi1),Mi+13=h(mi1,Mi1),

    and using the continuity of f, g and h we obtain

    m1=f(m2,M2),M1=f(M2,m2),m2=g(M3,m3),M2=g(m3,M3),m3=h(M1,m1),M3=h(m1,M1)

    so it follows from H3 that

    m1=M1,m2=M2,m3=M3.

    From H1, for n=1,2,, we get

    m01=axnb=M01,m02=αynβ=M02,m03=λznγ=M03.

    For n=2,3,..., we have

    m11=f(m02,M02)xn+1=f(yn,yn1)f(M02,m02)=M11,
    m12=g(M03,m03)yn+1=g(zn,zn1)g(m03,M03)=M13,

    and

    m13=h(M01,m01)zn+1=h(xn,xn1)h(m01,M01)=M13,

    that is

    m11xnM11,m12ynM12,m13znM13,n=3,4,.

    Now, for n=4,5,..., we have

    m21=f(m12,M12)xn+1=f(yn,yn1)f(M12,m12)=M21,

    and

    m22=g(M13,m13)yn+1=g(zn,zn1)g(m13,M13)=M22,
    m23=h(M11,m11)zn+1=h(xn,xn1)h(m11,M11)=M23,

    that is

    m21xnM21,m22ynM22,m23znM23,n=5,6,.

    Similarly, for n=6,7,..., we have

    m31=f(m22,M22)xn+1=f(yn,yn1)f(M22,m22)=M31,
    m32=g(M23,m23)yn+1=g(zn,zn1)g(m23,M23)=M32,

    and

    m33=h(M21,m21)zn+1=h(xn,xn1)h(m21,M21)=M33,

    that is

    m31xnM31,m32ynM32,m33znM33,n=7,8,.

    It follows by induction that for i=0,1,... we get

    mi1xnMi1,mi2ynMi2,mi3znMi3,n2i+1.

    Using the fact that i+ implies n+ and m1=M1,m2=M2,m3=M3, we obtain that

    limn+xn=M1,limn+yn=M2,limn+zn=M3.

    From (1) and using the fact that f, g and h are continuous and homogeneous of degree zero, we get

    M1=f(M2,M2)=f(1,1),M2=g(M3,M3)=g(1,1),M3=h(M1,M1)=h(1,1).

    Theorem 2.5. Consider System (1). Assume that the following statements are true:

    1. H1: There exist a,b,α,β,λ,γ(0,+) such that

    af(u,v)b,αg(u,v)β,λh(u,v)γ,(u,v)(0,+)2.

    2. H2: f(u,v), h(u,v) are increasing in u for all v and decreasing in v for all u and g(u,v) is decreasing in u for all v and increasing in v for all u.

    3. H3: If (m1,M1,m2,M2,m3,M3)[a,b]2×[α,β]2×[λ,γ]2 is a solution of the system

    m1=f(m2,M2),M1=f(M2,m2),m2=g(M3,m3),M2=g(m3,M3),m3=h(m1,M1),M3=h(M1,m1)

    then

    m1=M1,m2=M2,m3=M3.

    Then every solution of System (1) converges to the unique equilibrium point

    (¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)).

    Proof. Let

    m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ

    and for each i=0,1,...,

    mi+11:=f(mi2,Mi2),Mi+11:=f(Mi2,mi2),
    mi+12:=g(Mi3,mi3),Mi+12:=g(mi3,Mi3),
    mi+13:=h(mi1,Mi1),Mi+13:=h(Mi1,mi1).

    We have

    af(α,β)f(β,α)b,
    αg(γ,λ)g(λ,γ)β,
    λh(a,b)h(b,a)γ,

    and so,

    m01=af(m02,M02)f(M02,m02)b=M01,
    m02=αg(M03,m03)g(m03,M03)β=M02,

    and

    m03=λh(m01,M01)h(M01,m01)γ=M03.

    Hence,

    m01m11M11M01,
    m02m12M12M02,

    and

    m03m13M13M03.

    Now, we have

    m11=f(m02,M02)f(m12,M12)=m21f(M12,m12)=M21f(M02,m02)=M11,
    m12=g(M03,m03)g(M13,m13)=m22g(m13,M13)=M22g(m03,M03)=M12,
    m13=h(m01,M01)h(m11,M11)=m23h(M11,m11)=M23h(M01,m01)=M13,

    and it follows that

    m01m11m21M21M11M01,
    m02m12m22M22M12M02,

    and

    m03m13m23M23M13M03.

    By induction, we get for i=0,1,..., that

    a=m01m11...mi11mi1Mi1Mi11...M11M01=b,
    α=m02m12...mi12mi2Mi2Mi12...M12M02=β,

    and

    λ=m03m13...mi13mi3Mi3Mi13...M13M03=γ.

    It follows that the sequences (mi1)iN0, (mi2)iN0, (mi3)iN0 (resp. (Mi1)iN0, (Mi2)iN0, (Mi3)iN0) are increasing (resp. decreasing) and also bounded, so convergent. Let

    m1=limi+mi1,M1=limi+Mi1,
    m2=limi+mi2,M2=limi+Mi2.
    m3=limi+mi3,M3=limi+Mi3.

    Then

    am1M1b,αm2M2β,λm3M3γ.

    By taking limits in the following equalities

    mi+11=f(mi2,Mi2),Mi+11=f(Mi2,mi2),
    mi+12=g(Mi3,mi3),Mi+12=g(mi3,Mi3),
    mi+13=h(mi1,Mi1),Mi+13=h(Mi1,mi1),

    and using the continuity of f, g and h we obtain

    m1=f(m2,M2),M1=f(M2,m2),m2=g(M3,m3),M2=g(m3,M3),m3=h(m1,M1),M3=h(M1,m1)

    so it follows from H3 that

    m1=M1,m2=M2,m3=M3.

    From H1, for n=1,2,, we get

    m01=axnb=M01,m02=αynβ=M02,m03=λznγ=M03.

    For n=2,3,..., we have

    m11=f(m02,M02)xn+1=f(yn,yn1)f(M02,m02)=M11,
    m12=g(M03,m03)yn+1=g(zn,zn1)g(m03,M03)=M13,

    and

    m13=h(m01,M01)zn+1=h(xn,xn1)h(M01,m01)=M13,

    that is

    m11xnM11,m12ynM12,m13znM13,n=3,4,.

    Now, for n=4,5,..., we have

    m21=f(m12,M12)xn+1=f(yn,yn1)f(M12,m12)=M21,

    and

    m22=g(M13,m13)yn+1=g(zn,zn1)g(m13,M13)=M22,
    m23=h(m11,M11)zn+1=h(xn,xn1)h(M11,m11)=M23,

    that is

    m21xnM21,m22ynM22,m23znM23,n=5,6,.

    Similarly, for n=6,7,..., we have

    m31=f(m22,M22)xn+1=f(yn,yn1)f(M22,m22)=M31,
    m32=g(M23,m23)yn+1=g(zn,zn1)g(m23,M23)=M32,

    and

    m33=h(m21,M21)zn+1=h(xn,xn1)h(M21,m21)=M33,

    that is

    m31xnM31,m32ynM32,m33znM33,n=7,8,.

    It follows by induction that for i=0,1,... we get

    mi1xnMi1,mi2ynMi2,mi3znMi3,n2i+1.

    Using the fact that i+ implies n+ and m1=M1,m2=M2,m3=M3, we obtain that

    limn+xn=M1,limn+yn=M2,limn+zn=M3.

    From (1) and using the fact that f, g and h are continuous and homogeneous of degree zero, we get

    M1=f(M2,M2)=f(1,1),M2=g(M3,M3)=g(1,1),M3=h(M1,M1)=h(1,1).

    Theorem 2.6. Consider System (1). Assume that the following statements are true:

    1. H1: There exist a,b,α,β,λ,γ(0,+) such that

    af(u,v)b,αg(u,v)β,λh(u,v)γ,(u,v)(0,+)2.

    2. H2: f(u,v),g(u,v),h(u,v) are decreasing in u for all v and increasing in v for all u.

    3. H3: If (m1,M1,m2,M2,m3,M3)[a,b]2×[α,β]2×[λ,γ]2 is a solution of the system

    m1=f(M2,m2),M1=f(m2,M2),m2=g(M3,m3),M2=g(m3,M3),m3=h(M1,m1),M3=h(m1,M1)

    then

    m1=M1,m2=M2,m3=M3.

    Then every solution of System (1) converges to the unique equilibrium point

    (¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)).

    Proof. Let

    m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ

    and for each i=0,1,...,

    mi+11:=f(Mi2,mi2),Mi+11:=f(mi2,Mi2),
    mi+12:=g(Mi3,mi3),Mi+12:=g(mi3,Mi3),
    mi+13:=h(Mi1,mi1),Mi+13:=h(mi1,Mi1).

    We have

    af(β,α)f(α,β)b,
    αg(γ,λ)g(λ,γ)β,
    λh(b,a)h(a,b)γ

    and so,

    m01=af(M02,m02)f(m02,M02)b=M01,
    m02=αg(M03,m03)g(m03,M03)β=M02,
    m03=λh(M01,m01)h(m01,M01)γ=M03.

    Hence,

    m01m11M11M01,
    m02m12M12M02,

    and

    m03m13M13M03.

    Now, we have

    m11=f(M02,m02)f(M12,m12)=m21f(m12,M12)=M21f(m02,M02)=M11,
    m12=g(M03,m03)g(M13,m13)=m22g(m13,M13)=M22g(m03,M03)=M12,
    m13=h(M01,m01)h(M11,m11)=m23h(m11,M11)=M23h(m01,M01)=M13

    and it follows that

    m01m11m21M21M11M01,
    m02m12m22M22M12M02,
    m03m13m23M23M13M03.

    By induction, we get for i=0,1,..., that

    a=m01m11...mi11mi1Mi1Mi11...M11M01=b,
    α=m02m12...mi12mi2Mi2Mi12...M12M02=β,

    and

    λ=m03m13...mi13mi3Mi3Mi13...M13M03=γ.

    It follows that the sequences (mi1)iN0, (mi2)iN0 (mi3)iN0(resp. (Mi1)iN0, (Mi2)iN0, (Mi3)iN0) are increasing (resp. decreasing) and also bounded, so convergent. Let

    m1=limi+mi1,M1=limi+Mi1,
    m2=limi+mi2,M2=limi+Mi2.
    m3=limi+mi3,M3=limi+Mi3.

    Then

    am1M1b,αm2M2β,λm3M3γ.

    By taking limits in the following equalities

    mi+11=f(Mi2,mi2),Mi+11=f(mi2,Mi2),
    mi+12=g(Mi3,mi3),Mi+12=g(mi3,Mi3),
    mi+13=h(Mi1,mi1),Mi+13=h(mi1,Mi1)

    and using the continuity of f, g, and h we obtain

    m1=f(M2,m2),M1=f(m2,M2),m2=g(M3,m3),M2=g(m3,M3),m3=h(M1,m1),M3=h(m1,M1)

    so it follows from H3 that

    m1=M1,m2=M2,m3=M3.

    From H1, for n=1,2,, we get

    m01=axnb=M01,m02=αynβ=M02,m03=λznγ=M03.

    For n=2,3,..., we have

    m11=f(M02,m02)xn+1=f(yn,yn1)f(m02,M02)=M11,
    m12=g(M03,m03)yn+1=g(zn,zn1)g(m03,M03)=M12,
    m13=h(M01,m01)zn+1=h(xn,xn1)h(m01,M01)=M13,

    that is

    m11xnM11,m12ynM12,m13znM13,n=3,4,.

    Now, for n=4,5,..., we have

    m21=f(M12,m12)xn+1=f(yn,yn1)f(m12,M12)=M21,
    m22=g(M13,m31)yn+1=g(zn,zn1)g(m13,M31)=M22,
    m23=h(M11,m11)zn+1=h(xn,xn1)h(m11,M11)=M23,

    that is

    m21xnM21,m22ynM22,m23znM23,n=5,6,.

    Similarly, for n=6,7,..., we have

    m31=f(M22,m22)xn+1=f(yn,yn1)f(m22,M22)=M31,
    m32=g(M23,m23)yn+1=g(zn,zn1)g(m23,M23)=M32,
    m33=h(M21,m21)zn+1=h(xn,xn1)h(m21,M21)=M33,

    that is

    m31xnM31,m32ynM32,m33znM33,n=7,8,.

    It follows by induction that for i=0,1,... we get

    mi1xnMi1,mi2ynMi2,mi3znMi3,n2i+1.

    Using the fact that i+ implies n+ and m1=M1,m2=M2,m3=M3, we obtain that

    limn+xn=M1,limn+yn=M2,limn+zn=M3.

    From (1) and using the fact that f, g and h are continuous and homogeneous of degree zero, we get

    M1=f(M2,M2)=f(1,1),M2=g(M3,M3)=g(1,1),M3=h(M1,M1)=h(1,1).

    Theorem 2.7. Consider System (1). Assume that the following statements are true:

    1. H1: There exist a,b,α,β,λ,γ(0,+) such that

    af(u,v)b,αg(u,v)β,λh(u,v)γ,(u,v)(0,+)2.

    2. H2: f(u,v),g(u,v) are decreasing in u for all v and increasing in v for all u, however h(u,v) is increasing in u for all v and decreasing in v for all u.

    3. H3: If (m1,M1,m2,M2,m3,M3)[a,b]2×[α,β]2×[λ,γ]2 is a solution of the system

    m1=f(M2,m2),M1=f(m2,M2),m2=g(M3,m3),M2=g(m3,M3),m3=h(m1,M1),M3=h(M1,m1)

    then

    m1=M1,m2=M2,m3=M3.

    Then every solution of System (1) converges to the unique equilibrium point

    (¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)).

    Proof. Let

    m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ

    and for each i=0,1,...,

    mi+11:=f(Mi2,mi2),Mi+11:=f(mi2,Mi2),
    mi+12:=g(Mi3,mi3),Mi+12:=g(mi3,Mi3),
    mi+13:=h(mi1,Mi1),Mi+13:=h(Mi1,mi1).

    We have

    af(β,α)f(α,β)b,
    αg(γ,λ)g(λ,γ)β,
    λh(a,b)h(b,a)γ,

    and so,

    m01=af(M02,m02)f(m02,M02)b=M01,
    m02=αg(M03,m03)g(m03,M03)β=M02,

    and

    m03=λh(m01,M01)h(M01,m01)γ=M03.

    Hence,

    m01m11M11M01,
    m02m12M12M02,

    and

    m03m13M13M03.

    Now, we have

    m11=f(M02,m02)f(M12,m12)=m21f(m12,M12)=M21f(m02,M02)=M11,
    m12=g(M03,m03)g(M13,m13)=m22g(m13,M13)=M22g(m03,M03)=M12,
    m13=h(m01,M01)h(m11,M11)=m23h(M11,m11)=M23h(M01,m01)=M13,

    and it follows that

    m01m11m21M21M11M01,
    m02m12m22M22M12M02,

    and

    m03m13m23M23M13M03.

    By induction, we get for i=0,1,..., that

    a=m01m11...mi11mi1Mi1Mi11...M11M01=b,
    α=m02m12...mi12mi2Mi2Mi12...M12M02=β,

    and

    λ=m03m13...mi13mi3Mi3Mi13...M13M03=γ.

    It follows that the sequences (mi1)iN0, (mi2)iN0 (mi3)iN0(resp. (Mi1)iN0, (Mi2)iN0, (Mi3)iN0) are increasing (resp. decreasing) and also bounded, so convergent. Let

    m1=limi+mi1,M1=limi+Mi1,
    m2=limi+mi2,M2=limi+Mi2.
    m3=limi+mi3,M3=limi+Mi3.

    Then

    am1M1b,αm2M2β,λm3M3γ.

    By taking limits in the following equalities

    mi+11=f(Mi2,mi2),Mi+11=f(mi2,Mi2),
    mi+12=g(Mi3,mi3),Mi+12=g(mi3,Mi3),
    mi+13=h(mi1,Mi1),Mi+13=h(Mi1,mi1),

    and using the continuity of f, g, and h we obtain

    m1=f(M2,m2),M1=f(m2,M2),m2=g(M3,m3),M2=g(m3,M3),m3=h(m1,M1),M3=h(M1,m1)

    so it follows from H3 that

    m1=M1,m2=M2,m3=M3.

    From H1, for ,n=1,2,, we get

    m01=axnb=M01,m02=αynβ=M02,m03=λznγ=M03.

    For n=2,3,..., we have

    m11=f(M02,m02)xn+1=f(yn,yn1)f(m02,M02)=M11,
    m12=g(M03,m03)yn+1=g(zn,zn1)g(m03,M03)=M12,

    and

    m13=h(m01,M01)zn+1=h(xn,xn1)h(M01,m01)=M13,

    that is

    m11xnM11,m12ynM12,m13znM13,n=3,4,.

    Now, for n=4,5,..., we have

    m21=f(M12,m12)xn+1=f(yn,yn1)f(m12,M12)=M21,
    m22=g(M13,m31)yn+1=g(zn,zn1)g(m13,M31)=M22,

    and

    m23=h(m11,M11)zn+1=h(xn,xn1)h(M11,m11)=M23,

    that is

    m21xnM21,m22ynM22,m23znM23,n=5,6,.

    Similarly, for n=6,7,..., we have

    m31=f(M22,m22)xn+1=f(yn,yn1)f(m22,M22)=M31,
    m32=g(M23,m23)yn+1=g(zn,zn1)g(m23,M23)=M32,

    and

    m33=h(m21,M21)zn+1=h(xn,xn1)h(M21,m21)=M33,

    that is

    m31xnM31,m32ynM32,m33znM33,n=7,8,.

    It follows by induction that for i=0,1,... we get

    mi1xnMi1,mi2ynMi2,mi3znMi3,n2i+1.

    Using the fact that i+ implies n+ and m1=M1,m2=M2,m3=M3, we obtain that

    limn+xn=M1,limn+yn=M2,limn+zn=M3.

    From (1) and using the fact that f, g and h are continuous and homogeneous of degree zero, we get

    M1=f(M2,M2)=f(1,1),M2=g(M3,M3)=g(1,1),M3=h(M1,M1)=h(1,1).

    Theorem 2.8. Consider System (1). Assume that the following statements are true:

    1. H1: There exist a,b,α,β,λ,γ(0,+) such that

    af(u,v)b,αg(u,v)β,λh(u,v)γ,(u,v)(0,+)2.

    2. H2: f(u,v) is decreasing in u for all v and increasing in v for all u, however g(u,v),h(u,v) are increasing in u for all v and decreasing in v for all u.

    3. H3: If (m1,M1,m2,M2,m3,M3)[a,b]2×[α,β]2×[λ,γ]2 is a solution of the system

    m1=f(M2,m2),M1=f(m2,M2),m2=g(m3,M3),M2=g(M3,m3),m3=h(m1,M1),M3=h(M1,m1)

    then

    m1=M1,m2=M2,m3=M3.

    Then every solution of System (1) converges to the unique equilibrium point

    (¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)).

    Proof. Let

    m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ

    and for each i=0,1,...,

    mi+11:=f(Mi2,mi2),Mi+11:=f(mi2,Mi2),
    mi+12:=g(mi3,Mi3),Mi+12:=g(Mi3,mi3),
    mi+13:=h(mi1,Mi1),Mi+13:=h(Mi1,mi1).

    We have

    af(β,α)f(α,β)b,
    αg(λ,γ)g(γ,λ)β,
    λh(a,b)h(b,a)γ,

    and so,

    m01=af(M02,m02)f(m02,M02)b=M01,
    m02=αg(m03,M03)g(M03,m03)β=M02,

    and

    m03=λh(m01,M01)g(M01,m01)γ=M03.

    Hence,

    m01m11M11M01,
    m02m12M12M02,

    and

    m03m13M13M03.

    Now, we have

    m11=f(M02,m02)f(M12,m12)=m21f(m12,M12)=M21f(m02,M02)=M11,

    and it follows that

    and

    By induction, we get for that

    and

    It follows that the sequences , , (resp. , , ) are increasing (resp. decreasing) and also bounded, so convergent. Let

    Then

    By taking limits in the following equalities

    and using the continuity of , and we obtain

    so it follows from that

    From , for , we get

    For we have

    and

    that is

    Now, for we have

    and

    that is

    Similarly, for we have

    and

    that is

    It follows by induction that for we get

    Using the fact that implies and , we obtain that

    From (1) and using the fact that , and are continuous and homogeneous of degree zero, we get

    Theorem 2.9. Consider System . Assume that the following statements are true:

    1. : There exist such that

    2. : are decreasing in for all and increasing in for all , however is increasing in for all and decreasing in for all .

    3. : If is a solution of the system

    then

    Then every solution of System converges to the unique equilibrium point

    Proof. Let

    and for each

    We have

    and so,

    and

    Hence,

    and

    Now, we have

    and it follows that

    and

    By induction, we get for that

    and

    It follows that the sequences , , (resp. , , ) are increasing (resp. decreasing) and also bounded, so convergent. Let

    Then

    By taking limits in the following equalities

    and using the continuity of , and we obtain

    so it follows from that

    From , for , we get

    For we have

    and

    that is

    Now, for we have

    and

    that is

    Similarly, for we have

    and

    that is

    It follows by induction that for we get

    Using the fact that implies and , we obtain that

    From (1) and using the fact that , and are continuous and homogeneous of degree zero, we get

    The following theorem is devoted to global stability of the equilibrium point .

    Theorem 2.10. Under the hypotheses of Theorem 2.1 and one of Theorems 2.2– 2.9, the equilibrium point is globally stable.

    Now as an application of the previous results, we give an example.

    Example 1. Consider the following system of difference equations

    (6)

    where , , , , , and

    Assume that are positive. For all , we have

    It follows from Theorem 2.1 that the unique equilibrium point

    of System (6) will be locally stable if

    which is equivalent to

    Also, we have conditions and of Theorem 2.2 are satisfied. So, to prove the global stability of the equilibrium point it suffices to check condition of Theorem 2.2.

    For this purpose, let such that

    (7)
    (8)
    (9)

    From (7)-(9), we get

    (10)
    (11)
    (12)

    Now, from (10)-(12), we get

    (13)

    where is equal to

    It is not hard to see that

    Using the fact

    we get

    If we choose the parameters such that

    then we get that

    and so it follows from (13) that

    Using this and (10)-(12), we obtain . That is the condition is satisfied.

    In summary we have the following result.

    Theorem 2.11. Assume that that parameters are such that

    : .

    :

    :

    Then the equilibrium point is globally stable.

    We visualize the solutions of System (6) in Figures 1-3. In Figure 1 and Figure 2, we give the solution and corresponding global attractor of System (6) for and , respectively. Note that the conditions is satisfied for these values.

    Figure 1.   .
    Figure 2.   .
    Figure 3.   .

    However Figure 3 shows the unstable solution corresponding to the values and , which do not satisfy the conditions .

    Here, we are interested in existence of periodic solutions for System (1). In the following result we will established a necessary and sufficient condition for which there exist prime period two solutions of System (1).

    Theorem 3.1. Assume that , and are positive real numbers such that . Then, System has a prime period two solution in the form of

    if and only if

    Proof. Let , , be positive real numbers such that and assume that

    is a solution for System (1). Then, we have

    (14)
    (15)
    (16)
    (17)
    (18)
    (19)

    From (14)-(19), it follows that

    Now, assume that

    and let

    We have

    By induction we get

    In the following, we apply our result in finding prime period two solutions of two special Systems.

    Consider the three dimensional system of difference equations

    (20)

    where , the initial values and the are positive real numbers.

    System (20) can be seen as a generalization of the system

    studied in [23]. This last one is also a generalization of the equation

    studied in [7] and [15].

    Corollary 1. Assume that , then System has prime period two solution of the form

    if and only if

    (21)

    Proof. System (20) can be written as

    where

    So, from Theorem 3.1,

    will be a period prime two solution of System (20) if and only if

    Clearly this condition is equivalent to (21).

    Example 2. If we choose then, condition (21) will be

    The last condition is satisfied for the choice

    of the parameters. The corresponding prime period two solution will be

    and

    that is

    Now, consider the following system of difference equations

    (22)

    where the initial values and are positive real numbers.

    Corollary 2. Assume that , then System has prime period two solution of the form

    if and only if

    (23)

    Proof. System (22) can be written as

    where

    So, from Theorem 3.1,

    will be a period prime two solution of System (20) if and only if

    Clearly this condition is equivalent to

    Example 3. For then, condition (23) will be

    The last condition is satisfied for the choice

    of the parameters. The corresponding prime period two solution will be

    and

    that is

    Here, we are interested in the oscillation of the solutions of System (1) about the equilibrium point .

    Theorem 4.1. Let be a solution of System and assume that , are decreasing in for all and are increasing in for all .

    1. If

    then we get

    That is for the sequences , and we have semi-cycles of length one of the form

    2. If

    then we get

    That is for the sequences , and we have semi-cycles of length one of the form

    Proof. 1. Assume that

    We have

    By induction, we get

    2. Assume that

    We have

    By induction, we get

    So, the proof is completed. Now, we will apply the results of this section on the following particular system.

    Example 4. Consider the system of difference equations

    (24)

    where , the initial values and the parameters are positive real numbers.

    Let , and be the functions defined by

    It is not hard to see that

    System (24) has the unique equilibrium point .

    Corollary 3. Let be a solution of System . The following statements holds true:

    1. Let

    Then the sequences (resp. , ) oscillates about (resp. about , ) with semi-cycle of length one and every semi-cycle is in the form

    2. Let

    Then the sequences (resp. , ) oscillates about (resp. about , ) with semi-cycle of length one and every semi-cycle is in the form

    Proof. 1. Let

    We have

    which implies that

    Using the fact that

    we get

    Also as,

    we get

    Now, as

    we obtain

    Similarly,

    and

    and by induction we get that

    for . That is, the sequences (resp. , ) oscillates about (resp. about , ) with semi-cycle of length one and every semi-cycle is in the form

    and this is the statement of Part 1. of Theorem 4.1.

    Let

    We have

    which implies that

    Using the fact that

    we get

    Also as,

    we get

    Now, as

    we obtain

    Similarly,

    and

    Thus, by induction we get that

    for . That is the sequences (resp. , ) oscillates about (resp. about , ) with semi-cycle of length one and every semi-cycle is in the form

    and this is the statement of Part 2. of Theorem 4.1.

    In this study, the global stability of the unique positive equilibrium point of a three-dimensional general system of difference equations defined by positive and homogeneous functions of degree zero was studied. For this, general convergence theorems were given considering all possible monotonicity cases in arguments of functions , and . In addition, the periodic nature and oscillation of the general system considered was also discussed and successful results were obtained. It is noteworthy that the results obtained on our general three-dimensional system have high applicability.

    The authors thanks the two referees for their comments and suggestions. The work of N. Touafek and Y. Akrour was supported by DGRSDT (MESRS-DZ).



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