The main goal of this paper is to investigate the global asymptotic behavior of the difference equation xn+1=β1xn/A1+yn, yn+1=β2xn+γ2yn/xn+yn, n=0,1,2,… with β1,β2,γ2,A1∈(0,∞) and the initial value (x0,y0)∈[0,∞)×[0,∞) such that x0+y0≠0. The major conclusion shows that, in the case where γ2<β2, if the unique positive equilibrium (x-,y-) exists, then it is globally asymptotically stable.
1. Introduction and Preliminaries
Our aim in this paper is to investigate the dynamics of the following difference equation:
(1)xn+1=β1xnA1+yn,yn+1=β2xn+γ2ynxn+yn,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin=0,1,2,…
with β1,β2,γ2,A1∈(0,∞) and the initial value (x0,y0)∈[0,∞)×[0,∞) such that x0+y0≠0.
System (1) is a special case of the rational system
(2)xn+1=α1+β1xn+γ1ynA1+B1xn+C1yn,yn+1=α2+β2xn+γ2ynA2+B2xn+C2yn,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin=0,1,2,…,
where all parameters and the initial value (x0,y0) are nonnegative such that denominators are always positive. There is some interest in systems of rational and related difference equations, for example, see [1–9]. In this paper, we will determine the global convergence properties of the system (1) under certain conditions.
When β2=γ2, the first component {xn} of the solution (xn,yn) of the system (1) satisfies the first-order linear difference equation
(3)xn+1=β1A1+β2xn,n=0,1,…
and the second component {yn} is constant and equal to γ2 for n≥1.
If the initial value is given by x0>0, then by simple iteration, it is easy to find that
(4)xn=(β1A1+β2)nx0
is the solution of (3). If β1>A1+β2, then limn→∞xn=∞. If β1=A1+β2, then xn=x0 for all n>0, and for β1<A1+β2, we have limn→∞xn=0.
Therefore, in the remaining part, we will assume that β2≠γ2.
Clearly, (0,γ2) is always an equilibrium, and when
(5)A1+min{β2,γ2}<β1<A1+max{β2,γ2},
(1) also has a unique positive equilibrium
(6)(x-,y-)=((β1-A1)(β1-A1-γ2)A1+β2-β1,β1-A1).
Equation (1) was investigated in [10] and the main result they obtained is the following.
Theorem 1.
(i) Assume that γ2>β2. Then every solution of the system (1) converges to (0,γ2) if and only if β1≤A1+β2, and when β1>A1+β2, the system (1) has unbounded solutions.
(ii) Assume that β2>γ2. Then every positive solution of the system (1) is bounded if and only if β1<A1+β2. In particular, when β1≤A1+γ2, the equilibrium (0,γ2) is a global attractor of all solutions of the system (1).
In [10], the author proposed the following conjecture.
Conjecture 2.
Assume that
(7)γ2<β1-A1<β2.
Show that the unique positive equilibrium (x-,y-) of the system (1) is globally asymptotically stable.
Inspired by Conjecture 2, we investigate the global behavior of the system (1). To start our discussion, some basic results should be presented which will be useful in the sequel.
Consider the system
(8)xn+1=f(xn,yn),yn+1=g(xn,yn),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin=0,1,2,…,
where F=(f,g):D→R2 is continuous and D⊂R2.
A vital tool for dealing with the linearized stability of (8) is the following well-known result which we incorporate in the following lemma (see, e.g., [11, 12]).
Lemma 3.
Let F=(f,g) be a continuously differentiable function defined on an open set D⊂R2.
If the eigenvalues of the Jacobian matrix JF((x-,y-)) have modulus less than one, then the equilibrium of (8) is locally asymptotically stable.
If at least one of the eigenvalues of the Jacobian matrix JF((x-,y-)) has modulus greater than one, then the equilibrium of (8) is unstable.
The equilibrium (x-,y-) of (8) is locally asymptotically stable if every solution of the characteristic equation of the Jacobian matrix
(9)λ2-pλ+q=0
lies inside the unit circle, that is, if
(10)|p|<1+q<2.
In this case, (x-,y-) is also called a sink.
The equilibrium (x-,y-) of (8) is a repeller if every solution of characteristic equation (9) lies outside the unit circle, which is equivalent to the following condition:
(11)|q|>1,|p|<|1+q|.
The equilibrium (x-,y-) of (8) is a saddle point if the Jacobian matrix JF((x-,y-)) has one eigenvalue that lies inside the unit circle and if the other one lies outside the unit circle, that is, if and only if
(12)|p|>|1+q|,p2-4q>0.
The following well-known comparison result will be used in estimating the value of a solution of the system (1).
Lemma 4 (a comparison result).
Assume that α∈(0,∞) and β∈R. Let {un}n=0∞ and {vn}n=0∞ be sequences of real numbers such that u0≤v0 and
(13)un+1≤αun+β,vn+1=αvn+β,n=0,1,2,….
Then un≤vn for n≥0.
Consider the following difference equation:
(14)un+1=f(un),n=0,1,2,….
The following result of Hautus and Bolis [13] (see also [11, 12]) deals with the global attractivity of (14).
Lemma 5.
Let I⊆[0,∞) be some interval and assume that f∈C[I,(0,∞)] satisfies the following conditions:
f(u) is nondecreasing in u;
equation (14) has a unique positive equilibrium u-∈I and the function f(u) satisfies the negative feedback condition:
(15)(u-u-)(f(u)-u)<0foreveryu∈I∖{u-}.
Then every positive solution of (14) with initial conditions in I converges to u-.
To prepare for our major investigation, we consider the following equation:
(16)un+1=g(un,un-1),n=0,1,2,…,
and the following lemma should be mentioned which is from [12].
Lemma 6.
Let [a,b] be an interval of real numbers and assume that g:[a,b]2→[a,b] is a continuous function satisfying the following properties:
g(x,y) is nondecreasing in each of its arguments;
the function g(m,m)=m has a unique positive solution.
Then (16) has a unique equilibrium u-∈[a,b] and every solution of (16) converges to u-.
2. Linearized Stability
In this section, we will make some conclusions about linearized stability. Consider the map T on R2 associated with the system (1), that is,
(17)T(x,y)=(f1(x,y)f2(x,y))=(β1xA1+yβ2x+γ2yx+y).
Calculating the partial derivatives of the functions f1(x,y) and f2(x,y) shows that
(18)∂f1∂x=β1A1+y,∂f1∂y=-β1x(A1+y)2,∂f2∂x=(β2-γ2)y(x+y)2,∂f2∂y=(γ2-β2)x(x+y)2.
The Jacobian matrix of T evaluated at (0,γ2) is
(19)JF((0,γ2))=(β1A1+γ20β2-γ2γ20),
and its eigenvalues are λ1=0 and λ2=β1/(A1+γ2).
Another equilibrium (x-,y-), namely, (6), exists if and only if (5) holds. Using the equality A1+y-=β1, the Jacobian matrix of T evaluated at (x-,y-) is
(20)JF((x-,y-))=(1-x-β1(β2-γ2)y-(x-+y-)2(γ2-β2)x-(x-+y-)2),
and its characteristic equation associated with (x-,y-) is given by
(21)λ2-pλ+q=0,
where
(22)p=1+(γ2-β2)x-(x-+y-)2=(β1-A1)(β2-γ2)-(β1-A1-γ2)(A1+β2-β1)(β1-A1)(β2-γ2),q=(β2-γ2)x-(y--β1)β1(x-+y-)2=-A1(β1-A1-γ2)(A1+β2-β1)β1(β1-A1)(β2-γ2).
When β2>γ2, we find that γ2<β1-A1<β2 and A1/β1<1. Thus p>0 and
(23)1+q=1-A1(β1-A1-γ2)(A1+β2-β1)β1(β1-A1)(β2-γ2)>1-(β1-A1-γ2)(A1+β2-β1)(β1-A1)(β2-γ2)=|p|,(24)0>q=-A1(β1-A1-γ2)(A1+β2-β1)β1(β1-A1)(β2-γ2)=-A1β1β1-A1-γ2β1-A1·β2-(β1-A1)β2-γ2>-1.
When γ2>β2, A1/β1<1 holds and by simple computation, we have
(25)p2-4q=[1+(γ2-β2)x-(x-+y-)2]2-4(β2-γ2)x-(y--β1)β1(x-+y-)2>[1+(γ2-β2)x-(x-+y-)2]2-4(γ2-β2)x-(x-+y-)2=[1-(γ2-β2)x-(x-+y-)2]2≥0.
Furthermore, p,q>0 and
(26)|1+q|=1+A1(γ2-β2)x-β1(x-+y-)2<1+(γ2-β2)x-(x-+y-)2=|p|.
Employing Lemma 3, we formulate the results in the following.
Theorem 7.
(i) The equilibrium (0,γ2) of the system (1) is locally asymptotically stable when β1-A1<γ2, and it is unstable (a saddle point) when β1-A1>γ2, and it is nonhyperbolic when β1-A1=γ2.
(ii) Assume that γ2<β2 and (7) holds. Then the unique positive equilibrium (x-,y-) of the system (1) is locally asymptotically stable.
(iii) Assume that β2<γ2 and β2<β1-A1<γ2. Then the unique positive equilibrium (x-,y-) of the system (1) is unstable; further, it is a saddle point.
3. Global Attractivity
In this section, we will commence global asymptotic stability analysis. Let (xn,yn) be a solution of the system (1), then it is easy to obtain the following result from the second equation of the system (1).
Theorem 8.
(i) Assume that β2>γ2. Then every solution (xn,yn) of the system (1) satisfies γ2≤yn≤β2 for n≥1.
(ii) Assume that γ2>β2. Then every solution (xn,yn) of the system (1) satisfies β2≤yn≤γ2 for n≥1.
Theorem 9.
Every solution of the system (1) with x0=0 converges to (0,γ2).
Proof.
Since x0=0 implies that xn=0 for n≥1, thus limn→∞yn=γ2, finishing the proof.
Theorem 10.
Assume that γ2<β2≤β1-A1. Then every solution of the system (1) with x0>0 satisfies limn→∞xn=∞, limn→∞yn=β2.
Proof.
Using Theorem 8, we get that when γ2<β2<β1-A1,
(27)xn+1=β1A1+ynxn>β1A1+β2xn⟶∞,
and when γ2<β2=β1-A1,
(28)xn+1=β1A1+ynxn>β1A1+β2xn=xn⟶∞,
since the only equilibrium of the system (1) is (0,γ2) when β1=A1+β2.
Further, using the boundedness of yn, we have
(29)yn+1=β2xn+γ2ynxn+yn=β2+γ2(yn/xn)1+(yn/xn)⟶β2.
The proof is complete.
For the case where β1≤A1+γ2, the authors had obtained that the unique positive equilibrium (0,γ2) is a global attractor of all solutions of the system (1) in [10], see Theorem 1 (ii). Moreover, in view of Theorem 7 (i), we may formulate the result in the following theorem.
Theorem 11.
Assume that β1-A1<γ2<β2. Then the unique equilibrium (0,γ2) of the system (1) is globally asymptotically stable.
Now, we pay attention to dealing with the global attractivity of the unique positive equilibrium (x-,y-), namely, (6), under the condition that γ2<β2. In this case, (x-,y-) exists if and only if (7) holds. To obtain the global attractivity of (x-,y-), the following useful lemma should first be established.
Consider the following difference equation:
(30)un+1=aun(un+1)un+b,n=1,2,…,
where 0<b<a<1 and the initial value u1=x1/y1=β1x0(x0+y0)/(A1+y0)(β2x0+γ2y0). Equation (30) possesses two equilibria, namely, zero and u-=(a-b)/(1-a).
Lemma 12.
Every positive solution of (30) converges to the unique positive equilibrium u-.
Proof.
Clearly, u0>0 implies that un>0 for n≥1. Let f(u)=au(u+1)/(u+b), then f(u) is increasing in u for u>0 and
(31)(u-u-)[au(u+1)u+b-u]=u(u-u-)[a-b-(1-a)uu+b]=-(1-a)u(u-u-)2u+b<0.
Thus limn→∞un=u- for u0>0 by applying Lemma 5.
The proof is complete.
Theorem 13.
Assume that (7) holds. Then the unique positive equilibrium (x-,y-) of the system (1) is globally asymptotically stable.
Proof.
In view of Theorem 7, it is sufficient to show that (x-,y-) is a global attractor of all positive solutions of the system (1).
In this case, yn≥γ2>0 holds for n≥1 and thus the system (1) yields
(32)xn+1yn+1=β1(xn/yn)1+(A1/yn)(xn/yn)+1β2(xn/yn)+γ2=β1β2(1+(A1/yn))(xn/yn)((xn/yn)+1)(xn/yn)+(γ2/β2),yn+1=β2(xn/yn)+γ2(xn/yn)+1,
for n≥1. Let un=xn/yn, vn=yn, then the system (1) becomes
(33)un+1=β1β2(1+(A1/vn))un(un+1)un+(γ2/β2),vn+1=β2un+γ2un+1,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin=1,2,….
Further, the system (33) may reduce to the following second-order difference equation:
(34)un+1=β1un(un+1)β2un+γ2β2un-1+γ2β2un-1+γ2+A1(un-1+1),n=2,3,….
Clearly, zero is always the equilibrium of (34) and when (7) holds, (34) also possesses a unique positive equilibrium
(35)u-=β1-A1-γ2A1+β2-β1.
Notice that γ2≤vn=yn≤β2 for n≥1, and we get
(36)β1γ2β2(A1+γ2)≤β1β2(1+(A1/vn))≤β1A1+β2,
and thus
(37)β1γ2β2(A1+γ2)un(un+1)un+(γ2/β2)≤un+1=β1β2(1+(A1/vn))un(un+1)un+(γ2/β2)≤β1A1+β2un(un+1)un+(γ2/β2),n≥1.
Let a=β1γ2/β2(A1+γ2), b=γ2/β2, then a<(β1/β2)(β2/(A1+β2))=β1/(A1+β2)<1 and b<a<1. Hence by Lemma 12, we get that every positive solution of the following difference equation
(38)u˘n+1=β1γ2β2(A1+γ2)u˘n(u˘n+1)u˘n+(γ2/β2),n=1,2,…
converges to its unique positive equilibrium u˘=γ2(β1-(A1+γ2))/(β2A1+γ2(β2-β1)).
Let a=β1/(A1+β2), b=γ2/β2, then γ2/β2<(A1+γ2)/(A1+β2)<β1/(A1+β2)<1, which means that b<a<1. Similarly, by Lemma 12, we know that every positive solution of the following difference equation
(39)u^n+1=β1A1+β2u^n(u^n+1)u^n+(γ2/β2),n=1,2,…
converges to its unique positive equilibrium u^=(β1β2-γ2(A1+β2))/β2(A1+β2-β1).
Applying Lemma 4 and (37), we find that every solution of (34) with initial value u˘1=u^1=u1=(x1/y1)>0 satisfies
(40)u˘n≤un≤u^n,forn≥1.
Hence for 0<ϵ<u˘/2, there exists an integer N such that for n>N,
(41)u˘2<u˘-ϵ<u˘n≤un≤u^n<u^+ϵ<u^+u˘2.
Moreover,
(42)u˘2≤liminfn→∞un≤limsupn→∞un≤u^+u˘2.
Let I=u˘/2, S=u^+(u˘/2), then S>I>0 and every solution of (34) eventually enters the invariant interval [I,S].
Denote the function
(43)g(u,v)=β1u(u+1)β2u+γ2β2v+γ2β2v+γ2+A1(v+1),
and simple computation shows that
(44)∂g∂u=β1(β2u2+2γ2u+γ2)(β2u+γ2)2β2v+γ2β2v+γ2+A1(v+1)>0,(45)∂g∂v=β1u(u+1)β2u+γ2A1(β2-γ2)[β2v+γ2+A1(v+1)]2>0.
Applying Lemma 6, to establish the global attractivity of the equilibrium u- of (34), it is sufficient to confirm that the following equation
(46)g(m,m)=β1m(m+1)β2m+γ2β2m+γ2β2m+γ2+A1(m+1)=m
has a unique positive solution.
Solving (46), we get
(47)β1(m+1)=β2m+γ2+A1(m+1),
from which it follows that
(48)m=β1-A1-γ2A1+β2-β1=u-.
Therefore, limn→∞un=u-, and hence,
(49)limn→∞yn=limn→∞vn=β2u-+γ2u-+1=β1-A1=y-.
Furthermore,
(50)limn→∞xn=limn→∞un·limn→∞yn=u-·y-=(β1-A1)(β1-A1-γ2)A1+β2-β1=x-,
and thus the result follows.
The proof is complete.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
CamouzisE.KulenovićM. R. S.LadasG.MerinoO.Rational systems in the plane200915330332310.1080/10236190802125264MR24987762-s2.0-61449200503IričaninB. D.StevićS.Some systems of nonlinear difference equations of higher order with periodic solutions2006133-4a499508LugoG.PalladinoF. J.On the boundedness character of rational systems in the plane201117121801181110.1080/10236198.2010.491513MR2854825ZBL1243.390102-s2.0-84855814627PapaschinopoulosG.SchinasC. J.On a system of two nonlinear difference equations1998219241542610.1006/jmaa.1997.5829MR16063502-s2.0-0000475245StevićS.On a system of difference equations with period two coefficients201121884317432410.1016/j.amc.2011.10.005MR28621012-s2.0-81855164677StevićS.On some solvable systems of difference equations201221895010501810.1016/j.amc.2011.10.068MR28700252-s2.0-83555172687StevićS.On the difference equation xn=xn-2/bn+cnxn-1xn-2201121884507451310.1016/j.amc.2011.10.032MR28621222-s2.0-81855207255StevićS.DiblíkJ.IričaninB.ŠmardaZ.On a periodic system of difference equations201220125258718MR295973810.1155/2012/258718StevićS.DiblíkJ.IričaninB.ŠmardaZ.On some solvable difference equations and systems of difference equations2012201211541761MR299101410.1155/2012/541761CamouzisE.LadasG.Global results on rational systems in the plane, part 12010168975101310.1080/10236190802649727MR27231712-s2.0-77954844814KocicV. L.LadasG.1993Dordrecht, The NetherlandsKluwer Academic10.1007/978-94-017-1703-8MR1247956KulenovićM. R. S.LadasG.2002Boca Raton, Fla, USAChapman Hall/CRCHautusM. L. J.BolisT. S.Solution to problem E2721197986865866