Dynamics of a family of two-dimensional difference ... - Semantic Scholar
Applied Mathematics and Computation 219 (2013) 5949–5955
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Dynamics of a family of two-dimensional difference systems Wanping Liu a,b,⇑, Xiaofan Yang a, Xinzhi Liu b a b
College of Computer Science, Chongqing University, Chongqing 400044, China Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
a r t i c l e
i n f o
a b s t r a c t The boundedness and global asymptotic behavior of positive solutions of the nonlinear two-dimensional difference systems xn ¼ Uðxnt1 ; yns1 Þ; yn ¼ Wðyns2 ; xnt2 Þ, where t1 ; s1 ; s2 ; t 2 are all positive integers, are investigated. Some sufficient conditions are established such that the positive solutions converge to the unique equilibrium. Finally, some examples are given to illustrate our results. Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Difference equation System Equilibrium point Global attractivity
1. Introduction There has been a long history of interest in difference equations and difference systems, e.g., see [1–37] and the references therein. In particular, a wide variety of nonlinear difference systems have been studied because they model numerous real life problems in biology, physics, population dynamics, economics and so on. One of the difference equations that has attracted some attention is
xnþ1 ¼ A þ
xpnm ; xrnk
n 2 N0 ;
ð1Þ
where A; r > 0; p P 0 and k; m 2 N0 , and its numerous special cases had been studied in a large number of papers, e.g., see [1–4,7–9,11,12,14,26–34] and the references therein. Note that in all these papers the equations treated therein are of the form xn ¼ f ðxnk ; xnl Þ, for some k; l 2 N where the function f ðx; yÞ is monotone in both variables x and y. For related max-type difference equations see, e.g. [25,29], and the references therein. Usually, many specific difference equations are studied before an abstract one is investigated. For example, Camouzis and Papaschinopoulos [6] investigated a specific system which consists of the following two rational difference equations
xnþ1 ¼ 1 þ
xn ; ynm
ynþ1 ¼ 1 þ
yn ; xnm
n 2 N0 ;
and suggested to study the generalized system
xnþ1 ¼ A þ
xn ; ynm
ynþ1 ¼ B þ
yn ; xnm
n 2 N0 :
For some related papers on systems of difference equations, see, e.g. [5,15–19,21–24]. The following abstract nonlinear difference equation
xnþ1 ¼ f ðxns ; xnt Þ;
⇑ Corresponding author at: College of Computer Science, Chongqing University, Chongqing 400044, China. E-mail address: [email protected] (W. Liu). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.11.058
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where s; t 2 N0 satisfying s < t, was studied by Sun and Xi [37], in which they gave some sufficient conditions such that every positive solution of this equation converges to the unique positive equilibrium point. Furthermore, Sun and Xi [35] investigated the global asymptotic behavior of positive solutions to the following difference system
xnþ1 ¼ f ðxn ; ynk Þ;
ynþ1 ¼ f ðyn ; xnk Þ;
n 2 N0 ;
ð2Þ
with positive initial conditions and k 2 N. Certain appropriate assumptions about the function f in system (2) were given to guarantee that each positive solution to this system converges to a positive equilibrium point. Recently, Sun and Xi [36] have further studied the generic difference equation system
xnþ1 ¼ f ðynq ; xns Þ;
ynþ1 ¼ gðxnt ; ynp Þ;
n 2 N0 ;
where p; q; s; t are all nonnegative integers satisfying s P t and p P q, and the initial values are positive, by posing some sufficient conditions such that the unique positive equilibrium is a global attractor. Inspired by the above works, we study in this paper the dynamics of the following class of difference systems
xn ¼ Uðxnt1 ; yns1 Þ;
yn ¼ Wðyns2 ; xnt2 Þ;
n 2 N0 ;
ð3Þ
where t1 ; s1 ; s2 ; t2 are all positive integers and both Uðx; yÞ and Wðx; yÞ are decreasing in x and increasing in y. For s1 ; s2 ; t 1 ; t 2 in system (3), let t ¼ maxft1 ; t2 g and s ¼ maxfs1 ; s2 g in the sequel. As far as we know, system (3) has not yet been studied. Through careful analysis we have found that all positive solutions of system (3) converge to the unique equilibrium under some assumptions. These conditions are different from those given in [36]. 2. Main results In this section, we first present six basic assumptions about the mappings U and W in system (3), which are necessary for the main result at the end of this section. For the two mappings in system (3), we assume (A1): Let I ¼ ½a; þ1Þ or ða; þ1Þ, where a P 0, and U; W : I 2 ! I are continuous with c ¼ inf ðx;yÞ2I 2 Uðx; yÞ ¼ inf ðx;yÞ2I 2 Wðx; yÞ. Furthermore, all the initial values of system (3) xt ; xtþ1 ; . . . ; x1 ; ys ; ysþ1 ; . . . ; y1 belong to the interval I ; (A2): There exists only one fixed point k 2 ðc; þ1Þ such that Uðc; kÞ ¼ Wðc; kÞ ¼ k; Þ with 2 ½Wðk; cÞ; k; (A3): The system x ¼ Uðx; yÞ; y ¼ Wðy; xÞ has a unique solution ð x; y x 2 ½Uðk; cÞ; k; y (A4): Both the mappings Uðx; yÞ and Wðx; yÞ are strictly decreasing in x for each y 2 I and strictly increasing in y for each x 2 I; (A5): Both Uðc; xÞ=x and Wðc; xÞ=x are nonincreasing on the interval ðc; þ1Þ; (A6): The system M ¼ Uðm; T Þ; m ¼ UðM; tÞ; T ¼ Wðt; MÞ; t ¼ WðT ; mÞ with M; m 2 ½Uðk; cÞ; k; T ; t 2 ½Wðk; cÞ; k has a unique solution with M ¼ m; T ¼ t. Lemma 2.1. Assume that there exist two bounded sequences fan g1 and n¼t an ¼ Uðant1 ; bns1 Þ; bn ¼ Wðbns2 ; ant2 Þ; n 2 N0 , where both U and W satisfy (A1), (A4). Then
fbn g1 n¼s
such
that
lim sup an 6 Uðlim inf ant1 ; lim sup bns1 Þ; n!1
n!1
n!1
lim inf an P Uðlim sup ant1 ; lim inf bns1 Þ; n!1
n!1
n!1
ð4Þ
lim sup bn 6 Wðlim inf bnt2 ; lim sup ans2 Þ; n!1
n!1
n!1
lim inf bn P Wðlim sup bnt2 ; lim inf ans2 Þ: n!1
n!1
n!1
Proof. The proof is simple and known but is given here for the completeness. Since the sequences fan g and fbn g are bounded, denote
a1;k ¼ inf fan g ¼ inffak ; akþ1 ; . . .g; a2;k ¼ inf fbn g ¼ inffbk ; bkþ1 ; . . .g; nPk
nPk
b1;k ¼ supfan g ¼ supfak ; akþ1 ; . . .g; nPk
b2;k ¼ supfbn g ¼ supfbk ; bkþ1 ; . . .g: nPk
Then by (5) and the assumption (A4), we have that for k P maxfs1 ; t1 g,
!
b1;k ¼ sup Uðant1 ; bns1 Þ 6 U inf fant1 g; supfbns1 g nPk
nPk
nPk
¼ Uða1;kt1 ; b2;ks1 Þ;
ð5Þ
W. Liu et al. / Applied Mathematics and Computation 219 (2013) 5949–5955
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!
a1;k ¼ inf Uðant1 ; bns1 Þ P U supfant1 g; inf fbns1 g ¼ Uðb1;kt1 ; a2;ks1 Þ: nPk
nPk
nPk
Therefore by taking limits on both sides of the above two inequalities and in light of the continuity of the function U, we have
lim supan ¼ lim b1;k 6 U lim a1;kt1 ; lim b2;ks1 ¼ U lim inf ant1 ; lim supbns1 ; k!1
n!1
k!1
k!1
n!1
n!1
lim inf an ¼ lim a1;k P U lim b1;kt1 ; lim a2;ks1 ¼ U lim supant1 ; lim inf bns1 : n!1
k!1
k!1
k!1
n!1
The proof of the last two inequalities of (4) is similar and thus omitted.
n!1
h
Lemma 2.2. Under the assumptions (A1), (A2), (A4) and (A5), the following two statements hold: (1) xn 6 k if yns1 6 k, while xn 6 yns1 if yns1 > k; n P maxfs1 ; t1 g; (2) yn 6 k if xnt2 6 k, while yn 6 xnt2 if xnt2 > k; n P maxfs2 ; t 2 g.
Proof. Observe that xn ¼ Uðxnt1 ; yns1 Þ 6 Uðc; yns1 Þ; n P maxfs1 ; t1 g. By the assumptions (A2), (A4) and (A5), we have that if yns1 6 k, then
xn 6 Uðc; yns1 Þ 6 Uðc; kÞ ¼ k; while if yns1 > k then
Uðc; yns1 Þ yns1
6
Uðc; kÞ k
¼ 1;
which implies xn 6 Uðc; yns1 Þ 6 yns1 . Likewise, we get
yn ¼ Wðyns2 ; xnt2 Þ 6 Wðc; xnt2 Þ;
n P maxfs2 ; t2 g:
From this inequality, we know that if xnt2 6 k, then
yn ¼ Wðyns2 ; xnt2 Þ 6 Wðc; kÞ ¼ k; while if xnt2 > k then
Wðc; xnt2 Þ xnt2
6
Wðc; kÞ k
¼ 1;
which indicates that yn 6 Wðc; xnt2 Þ 6 xnt2 . h Lemma 2.3. For each j 2 f0; 1; . . . ; s1 þ t 2 1g, there exists N j 2 N such that xnðt2 þs1 Þþj 6 k and ynðt2 þs1 Þþt2 þj 6 k for all n P N j . Proof. Suppose that for some j 2 f0; 1; . . . ; s1 þ t 2 1g there exists wj 2 N0 such that xwj ðt2 þs1 Þþj 6 k. By using both statements in Lemma 2.2 we have that
xnðt2 þs1 Þþj 6 k for all n P wj and
ynðt2 þs1 Þþt2 þj 6 k for all n P wj : Likewise, if for some j 2 f0; 1; . . . ; t 2 þ s1 1g there exists zj 2 N0 such that yzj ðt2 þs1 Þþj 6 k. Then recursively we get that for all n P zj there hold
ynðt2 þs1 Þþj 6 k and xnðt2 þs1 Þþs1 þj 6 k: Therefore, on the contrary assume that there exists some j 2 f0; 1; . . . ; s1 þt2 1g such that Lemma 2.3 does not hold. Then applying the above two conclusions, we can simply get that
xnðt2 þs1 Þþj > k and ynðt2 þs1 Þþt2 þj > k for all n 2 N0 : Using Lemma 2.2, we have that for all n 2 N0 ,
k < xðnþ1Þðt2 þs1 Þþj 6 xnðt2 þs1 Þþj and k < yðnþ1Þðt2 þs1 Þþt2 þj 6 ynðt2 þs1 Þþt2 þj :
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Denote
lim xnðt2 þs1 Þþj ¼ kj and lim ynðt2 þs1 Þþt2 þj ¼ cj ;
n!1
n!1
then kj P k and cj P k. By Lemma 2.2 we know fxn g; fyn g are bounded. Let
/j ¼ lim inf xnðt2 þs1 Þt1 þj and n!1
lj ¼ lim inf ynðt2 þs1 Þþt2 s2 þj ; n!1
obviously we have /j P c; lj P c. By system (3), we have that
xnðt2 þs1 Þþj ¼ Uðxnðt2 þs1 Þt1 þj ; ynðt2 þs1 Þs1 þj Þ;
ð6Þ
ynðt2 þs1 Þþt2 þj ¼ Wðynðt2 þs1 Þþt2 s2 þj ; xnðt2 þs1 Þþj Þ: Employing Lemma 2.1, it follows from (6) that
kj 6 Uð/j ; cj Þ 6 Uðc; cj Þ ¼ cj Uðc; cj Þ=cj 6 cj Uðc; kÞ=k ¼ cj ;
cj 6 Wðlj ; kj Þ 6 Wðc; kj Þ ¼ kj Wðc; kj Þ=kj 6 kj Wðc; kÞ=k ¼ kj : Thus kj ¼ cj , which plus the assumptions (A2) and (A4) leads to kj ¼ cj ¼ k; /j ¼ lj ¼ c. Let Aj ¼ lim supn!1 xnðt2 þs1 Þþj2t1 and Bj ¼ lim inf n!1 ynðt2 þs1 Þþjt1 s1 , then Aj ; Bj P c. Again by system (3) we get
xnðt2 þs1 Þþjt1 ¼ Uðxnðt2 þs1 Þþj2t1 ; ynðt2 þs1 Þþjt1 s1 Þ:
ð7Þ
It follows by Lemma 2.1 and (7) that c P UðAj ; Bj Þ > UðAj þ 1; Bj Þ P c, which is obviously a contradiction. The proof is complete. h The following lemma follows directly from Lemma 2.3 and shows the explicit bounds of positive solution to system (3) for sufficiently large n. Lemma 2.4. Suppose (A1)–(A5) hold for system (3), then there exists N 2 N such that Uðk; cÞ 6 xn 6 k and
Wðk; cÞ 6 yn 6 k; n > N . Proof. According to Lemma 2.3, let
N ¼ max fN j g ðs1 þ t 2 Þ þ maxfsi ; ti g þ t 2 ; 06j<s1 þt 2
i¼1;2
then xnti ; ynsi 6 k; i ¼ 1; 2, for all n > N. It follows by system (3) xn ¼ Uðxnt1 ; yns1 Þ P Uðk; cÞ; yn ¼ Wðyns2 ; xnt2 Þ P Wðk; cÞ, for all n > N . The proof is complete. h
and
(A4)
that
Theorem 2.1. If (A1)–(A6) hold for system (3), then every positive solution ðxn ; yn Þ to system (3) converges to the unique equiÞ. librium ð x; y 1 1 1 Proof. Let t ¼ maxft1 ; t2 g; s ¼ maxfs1 ; s2 g, and define four sequences fmi g1 i¼t ; fMi gi¼t ; ft i gi¼s ; fT i gi¼s recursively in the following way:
Mi ¼ Uðmit1 ; T is1 Þ; T i ¼ Wðt is2 ; Mit2 Þ;
mi ¼ UðMit1 ; tis1 Þ; ti ¼ WðT is2 ; mit2 Þ;
ð8Þ
where i 2 N0 and the initial values satisfy
mj ¼ Uðk; cÞ; t j ¼ Wðk; cÞ;
Mj ¼ k; j 2 ft; t þ 1; . . . ; 1g; T j ¼ k; j 2 fs; s þ 1; . . . ; 1g:
ð9Þ
Obviously we have that mj1 6 mj ; Mj 6 Mj1 ; j 2 ft þ 1; . . . ; 1g and t j1 6 tj ; T j 6 T j1 ; j 2 fs þ 1; . . . ; 1g. Assume that mj1 6 mj 6 k; Uðk; cÞ 6 Mj 6 Mj1 hold for t þ 1 6 j 6 k, and tj1 6 tj 6 k; Wðk; cÞ 6 T j 6 T j1 hold for s þ 1 6 j 6 k; k P 1. Then employing (A4), for j ¼ k þ 1 we obtain that
mkþ1 ¼ UðMkt1 þ1 ; t ks1 þ1 Þ P UðMkt1 ; tks1 Þ ¼ mk ; Mkþ1 ¼ Uðmkt1 þ1 ; T ks1 þ1 Þ 6 Uðmkt1 ; T ks1 Þ ¼ Mk ; t kþ1 ¼ WðT ks2 þ1 ; mkt2 þ1 Þ P WðT ks2 ; mkt2 Þ ¼ tk ; T kþ1 ¼ Wðtks2 þ1 ; Mkt2 þ1 Þ 6 Wðt ks2 ; Mkt2 Þ ¼ T k : 1 1 Working inductively, we have that fmi g1 i¼t and ft i gi¼s are nondecreasing and bounded from above by k, and fMi gi¼t and 1 fT i gi¼s are nonincreasing and bounded from below by Uðk; cÞ; Wðk; cÞ, respectively.
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Hence fmi g; fMi g; fti g; fT i g are convergent, and denote
m ¼ limmi ; i!1
M ¼ limMi ; i!1
t ¼ limt i ; i!1
T ¼ limT i : i!1
Then m; M 2 ½Uðk; cÞ; k and t; T 2 ½Wðk; cÞ; k. By taking limits on both sides of (8) and in view of the continuity of U and W, we have that
M ¼ Uðm; T Þ;
m ¼ UðM; tÞ;
T ¼ Wðt; MÞ;
t ¼ WðT ; mÞ:
It follows from (A3) and (A6) that m ¼ M ¼ x; t ¼ T ¼ y. 1 Let fxn g1 n¼t and fyn gn¼s be a solution to system (3), then by Lemma 2.4 there exists K 2 N such that
Uðk; cÞ 6 xj 6 k for all j P K t and
Wðk; cÞ 6 yj 6 k for all j P K s: By (8) and (9), we have
mjK 6 xj 6 MjK ; K t 6 j 6 K 1 and
tjK 6 yj 6 T jK ; K s 6 j 6 K 1: If mjK 6 xj 6 MjK holds for K t 6 j 6 n 1 and tjK 6 yj 6 T jK holds for K s 6 j 6 n 1; n P K, then observe that
xn ¼ Uðxnt1 ; yns1 Þ P UðMnKt1 ; tnKs1 Þ ¼ mnK ; xn ¼ Uðxnt1 ; yns1 Þ 6 UðmnKt1 ; T nKs1 Þ ¼ MnK ; yn ¼ Wðyns2 ; xnt2 Þ P WðT nKs2 ; mnKt2 Þ ¼ tnK ; yn ¼ Wðyns2 ; xnt2 Þ 6 Wðt nKs2 ; MnKt2 Þ ¼ T nK : By induction, we derive mnK 6 xn 6 MnK and tnK 6 yn 6 T nK , for all n P K. This plus m ¼ M ¼ x; t ¼ T ¼ y yields . The proof is complete. h limn!1 xn ¼ x and limn!1 yn ¼ y 3. Examples In this section, we will discuss several specific examples to verify the correctness of the main result given in Section 2. Example 3.1. Consider the following difference equation system
xn ¼ p þ
yns1 ; xnt1
yn ¼ p þ
xnt2 ; yns2
where s1 ; t 1 ; s2 ; t 2 2 N; p > 1 and t ¼ maxft1 ; t2 g; s ¼ maxfs1 ; s2 g.
n 2 N0 ; the
initial
ð10Þ values
xi ; yj > 0,
for
i ¼ t; . . . ; 1; j ¼ s; . . . ; 1,
where
Proof. Define Uðx; yÞ ¼ Wðx; yÞ ¼ p þ y=x with x; y 2 Rþ . It can be seen that U; W satisfy assumptions (A1)–(A5) with
c ¼ p; k ¼ p2 =ðp 1Þ; x ¼ y ¼ p þ 1. Hence it suffices to verify the following system
M ¼ p þ T=m;
m ¼ p þ t=M;
T ¼ p þ M=t;
t ¼ p þ m=T
ð11Þ
has a unique solution with m ¼ M; t ¼ T. It follows from (11) that
pðM mÞ ¼ T t and pðT tÞ ¼ M m; which implies T ¼ t; M ¼ m for p > 1. Thus (A6) holds for U; W. In view of Theorem 2.1, for every positive solution limn!1 xn ¼ x ¼ p þ 1; limn!1 yn ¼ y ¼ p þ 1. h
ðxn ; yn Þ
to
system
(10)
we
conclude
Example 3.2. Consider the following difference equation system
xn ¼
pyns1 þ xnt1 ; q þ xnt1
yn ¼
pxnt2 þ yns2 ; q þ yns2
ð12Þ
where s1 ; s2 ; t1 ; t2 2 N; 0 < q < p < q þ 1 with p þ q > 1 and initial values xi ; yj 2 ðq=p; þ1Þ, for i ¼ t; . . . ; 1; j ¼ s; . . . ; 1, where t ¼ maxft 1 ; t 2 g; s ¼ maxfs1 ; s2 g.
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Proof. Let Uðx; yÞ ¼ Wðx; yÞ ¼ ðpy þ xÞ=ðq þ xÞ with x; y 2 ðq=p; þ1Þ. It is straightforward to verify that (A1)–(A5) hold for U; W with c ¼ 1; k ¼ 1=ðq p þ 1Þ; x ¼ y ¼ p q þ 1. Easily, it follows from the system
M¼
pT þ m ; qþm
m¼
pt þ M ; qþM
T¼
pM þ t ; qþt
t¼
pm þ T ; qþT
that ðq þ 1ÞðM mÞ ¼ pðT tÞ and ðq þ 1ÞðT tÞ ¼ pðM mÞ. So we get ðq þ 1Þ2 ðT tÞ ¼ p2 ðT tÞ which implies T ¼ t, and then M ¼ m. Hence (A6) holds for U; W. In light of Theorem 2.1, for every positive solution ðxn ; yn Þ of system (12) we have limn!1 xn ¼ x ¼ p q þ 1; limn!1 yn ¼ y ¼ p q þ 1. h Example 3.3. Consider the system
xn ¼ p þ qynt1 þ
A ; xns1
yn ¼ p þ qxnt2 þ
A ; yns2
n 2 N0 ;
ð13Þ
where all the initial values are positive, and p; A > 0; 0 < q < 1 satisfy the inequality
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1 þ q þ Að1 qÞ=ðA þ p2 Þ > A=ð1 qÞ:
ð14Þ
Proof. Set Uðx; yÞ ¼ Wðx; yÞ ¼ p þ qy þ A=x with x; y 2 Rþ . With c ¼ p; k ¼ ðA þ p2 Þ=ðpð1 qÞÞ; x ¼ y ¼ ðpþ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p þ 4Að1 qÞÞ=ð2ð1 qÞÞ, Uðx; yÞ and Wðx; yÞ satisfy (A1)–(A5). From the following system
M ¼ p þ qT þ A=m; T ¼ p þ qM þ A=t;
m ¼ p þ qt þ A=M;
ð15Þ
t ¼ p þ qm þ A=T;
we can easily get ð1 A=ðmMÞÞðM mÞ ¼ qðT tÞ and ð1 A=ðtTÞÞðT tÞ ¼ qðM mÞ, which indicate that ð1 A=ðmMÞÞð1 A=ðtTÞÞðT tÞ ¼ q2 ðT tÞ. If m; M 2 ½Uðk; cÞ; k; t; T 2 ½Wðk; cÞ; k, then by (14) it is simple to verify ð1 A=ðmMÞÞð1 A=ðtTÞÞ > q2 . Thus T ¼ t; M ¼ m. That is, (A6) holds for mappings U; W. It follows by Theorem 2.1 that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi limn!1 xn ¼ limn!1 yn ¼ ðp þ p2 þ 4Að1 qÞÞ=ð2ð1 qÞÞ for every positive solution ðxn ; yn Þ to system (13). h Example 3.4. Consider the following system (from Riccati equation)
xn ¼
p1 þ yns1 ; q1 þ xnt1
yn ¼
p2 þ xnt2 ; q2 þ yns2
n 2 N0 ;
ð16Þ
with nonnegative initial values, and p1 ; p2 > 0; q1 ; q2 > 1 satisfying
p1 q2 þ p2 ¼ p2 q1 þ p1 ;
q22 2p2 þ q1 P 0;
q1 q2 2p2 q2 1 P 0;
p22 q1 p2 p1 < 0:
ð17Þ
Proof. Define two mappings Uðx; yÞ ¼ ðp1 þ yÞ=ðq1 þ xÞ and Wðx; yÞ ¼ ðp2 þ yÞ=ðq2 þ xÞ; x; y 2 ½0; þ1Þ. It is easy to verify that (A1),(A2) and (A4)–(A6) hold for U; W with c ¼ 0; k ¼ p1 =ðq1 1Þ ¼ p2 =ðq2 1Þ. Hence it suffices to show that the system
x2 þ q1 x ¼ p1 þ y;
y2 þ q2 y ¼ p2 þ x
ð18Þ
has a unique positive solution ðx; yÞ. It follows from system (18) that
y4 þ 2q2 y3 þ ðq1 þ q22 2p2 Þy2 þ ðq1 q2 2p2 q2 1Þy ¼ q1 p2 þ p1 p22 : 4
3
q22
2
ð19Þ p22
Let HðtÞ ¼ t þ 2q2 t þ ðq1 þ 2p2 Þt þ ðq1 q2 2p2 q2 1Þt þ q1 p2 p1 , then H0 ðtÞ ¼ 4t 3 þ 6q2 t2 þ 2ðq1 þ q22 2p2 Þt þ q1 q2 2p2 q2 1, and H00 ðtÞ ¼ 12t 2 þ 12q2 t þ 2ðq1 þ q22 2p2 Þ. . Then by substituting y ¼ y into (18) we get We conclude by (17) that (19) has only one positive root, denoted by y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 =2. It follows from Theorem 2.1 that every positive solution to system (16) converges to the x ¼ q1 þ q1 þ 4p1 þ 4y Þ. h unique positive equilibrium ð x; y Acknowledgement The authors thank the anonymous referees for their valuable comments. This work was supported by the Scholarship Award for Excellent Doctoral Student granted by Ministry of Education (No. 0903005109081-10), the Doctorate Foundation of Educational Ministry of China (Grant No. 20110191110022) as well as the China Scholarship Council (CSC).
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Papaschinopoulos, M. Radin, C. Schinas, On the system of two difference equations of exponential form: xnþ1 ¼ a þ bxn1 eyn ; ynþ1 ¼ c þ dyn1 exn , Math. Comput. Model. 54 (2011) 2969–2977. [22] S. Stevic´, On a system of difference equations, Appl. Math. Comput. 218 (2011) 3372–3378. [23] S. Stevic´, On a system of difference equations with period two coefficients, Appl. Math. Comput. 218 (2011) 4317–4324. [24] S. Stevic´, On some solvable systems of difference equations, Appl. Math. Comput. 218 (2011) 5010–5018. [25] S. Stevic´, On a nonlinear generalized max-type difference equation, J. Math. Anal. Appl. 376 (2011) 317–328. [26] S. Stevic´, On a class of higher-order difference equations, Chaos Solitons Fract. 42 (1) (2009) 138–145. [27] S. Stevic´, Boundedness character of a class of difference equations, Nonlinear Anal.: Theory Methods Appl. 70 (2009) 839–848. [28] S. Stevic´, Boundedness character of two classes of third-order difference equations, J. Differ. Equ. 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Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Dynamics of a family of two-dimensional difference systems Wanping Liu a,b,⇑, Xiaofan Yang a, Xinzhi Liu b a b
College of Computer Science, Chongqing University, Chongqing 400044, China Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
a r t i c l e
i n f o
a b s t r a c t The boundedness and global asymptotic behavior of positive solutions of the nonlinear two-dimensional difference systems xn ¼ Uðxnt1 ; yns1 Þ; yn ¼ Wðyns2 ; xnt2 Þ, where t1 ; s1 ; s2 ; t 2 are all positive integers, are investigated. Some sufficient conditions are established such that the positive solutions converge to the unique equilibrium. Finally, some examples are given to illustrate our results. Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Difference equation System Equilibrium point Global attractivity
1. Introduction There has been a long history of interest in difference equations and difference systems, e.g., see [1–37] and the references therein. In particular, a wide variety of nonlinear difference systems have been studied because they model numerous real life problems in biology, physics, population dynamics, economics and so on. One of the difference equations that has attracted some attention is
xnþ1 ¼ A þ
xpnm ; xrnk
n 2 N0 ;
ð1Þ
where A; r > 0; p P 0 and k; m 2 N0 , and its numerous special cases had been studied in a large number of papers, e.g., see [1–4,7–9,11,12,14,26–34] and the references therein. Note that in all these papers the equations treated therein are of the form xn ¼ f ðxnk ; xnl Þ, for some k; l 2 N where the function f ðx; yÞ is monotone in both variables x and y. For related max-type difference equations see, e.g. [25,29], and the references therein. Usually, many specific difference equations are studied before an abstract one is investigated. For example, Camouzis and Papaschinopoulos [6] investigated a specific system which consists of the following two rational difference equations
xnþ1 ¼ 1 þ
xn ; ynm
ynþ1 ¼ 1 þ
yn ; xnm
n 2 N0 ;
and suggested to study the generalized system
xnþ1 ¼ A þ
xn ; ynm
ynþ1 ¼ B þ
yn ; xnm
n 2 N0 :
For some related papers on systems of difference equations, see, e.g. [5,15–19,21–24]. The following abstract nonlinear difference equation
xnþ1 ¼ f ðxns ; xnt Þ;
⇑ Corresponding author at: College of Computer Science, Chongqing University, Chongqing 400044, China. E-mail address: [email protected] (W. Liu). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.11.058
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where s; t 2 N0 satisfying s < t, was studied by Sun and Xi [37], in which they gave some sufficient conditions such that every positive solution of this equation converges to the unique positive equilibrium point. Furthermore, Sun and Xi [35] investigated the global asymptotic behavior of positive solutions to the following difference system
xnþ1 ¼ f ðxn ; ynk Þ;
ynþ1 ¼ f ðyn ; xnk Þ;
n 2 N0 ;
ð2Þ
with positive initial conditions and k 2 N. Certain appropriate assumptions about the function f in system (2) were given to guarantee that each positive solution to this system converges to a positive equilibrium point. Recently, Sun and Xi [36] have further studied the generic difference equation system
xnþ1 ¼ f ðynq ; xns Þ;
ynþ1 ¼ gðxnt ; ynp Þ;
n 2 N0 ;
where p; q; s; t are all nonnegative integers satisfying s P t and p P q, and the initial values are positive, by posing some sufficient conditions such that the unique positive equilibrium is a global attractor. Inspired by the above works, we study in this paper the dynamics of the following class of difference systems
xn ¼ Uðxnt1 ; yns1 Þ;
yn ¼ Wðyns2 ; xnt2 Þ;
n 2 N0 ;
ð3Þ
where t1 ; s1 ; s2 ; t2 are all positive integers and both Uðx; yÞ and Wðx; yÞ are decreasing in x and increasing in y. For s1 ; s2 ; t 1 ; t 2 in system (3), let t ¼ maxft1 ; t2 g and s ¼ maxfs1 ; s2 g in the sequel. As far as we know, system (3) has not yet been studied. Through careful analysis we have found that all positive solutions of system (3) converge to the unique equilibrium under some assumptions. These conditions are different from those given in [36]. 2. Main results In this section, we first present six basic assumptions about the mappings U and W in system (3), which are necessary for the main result at the end of this section. For the two mappings in system (3), we assume (A1): Let I ¼ ½a; þ1Þ or ða; þ1Þ, where a P 0, and U; W : I 2 ! I are continuous with c ¼ inf ðx;yÞ2I 2 Uðx; yÞ ¼ inf ðx;yÞ2I 2 Wðx; yÞ. Furthermore, all the initial values of system (3) xt ; xtþ1 ; . . . ; x1 ; ys ; ysþ1 ; . . . ; y1 belong to the interval I ; (A2): There exists only one fixed point k 2 ðc; þ1Þ such that Uðc; kÞ ¼ Wðc; kÞ ¼ k; Þ with 2 ½Wðk; cÞ; k; (A3): The system x ¼ Uðx; yÞ; y ¼ Wðy; xÞ has a unique solution ð x; y x 2 ½Uðk; cÞ; k; y (A4): Both the mappings Uðx; yÞ and Wðx; yÞ are strictly decreasing in x for each y 2 I and strictly increasing in y for each x 2 I; (A5): Both Uðc; xÞ=x and Wðc; xÞ=x are nonincreasing on the interval ðc; þ1Þ; (A6): The system M ¼ Uðm; T Þ; m ¼ UðM; tÞ; T ¼ Wðt; MÞ; t ¼ WðT ; mÞ with M; m 2 ½Uðk; cÞ; k; T ; t 2 ½Wðk; cÞ; k has a unique solution with M ¼ m; T ¼ t. Lemma 2.1. Assume that there exist two bounded sequences fan g1 and n¼t an ¼ Uðant1 ; bns1 Þ; bn ¼ Wðbns2 ; ant2 Þ; n 2 N0 , where both U and W satisfy (A1), (A4). Then
fbn g1 n¼s
such
that
lim sup an 6 Uðlim inf ant1 ; lim sup bns1 Þ; n!1
n!1
n!1
lim inf an P Uðlim sup ant1 ; lim inf bns1 Þ; n!1
n!1
n!1
ð4Þ
lim sup bn 6 Wðlim inf bnt2 ; lim sup ans2 Þ; n!1
n!1
n!1
lim inf bn P Wðlim sup bnt2 ; lim inf ans2 Þ: n!1
n!1
n!1
Proof. The proof is simple and known but is given here for the completeness. Since the sequences fan g and fbn g are bounded, denote
a1;k ¼ inf fan g ¼ inffak ; akþ1 ; . . .g; a2;k ¼ inf fbn g ¼ inffbk ; bkþ1 ; . . .g; nPk
nPk
b1;k ¼ supfan g ¼ supfak ; akþ1 ; . . .g; nPk
b2;k ¼ supfbn g ¼ supfbk ; bkþ1 ; . . .g: nPk
Then by (5) and the assumption (A4), we have that for k P maxfs1 ; t1 g,
!
b1;k ¼ sup Uðant1 ; bns1 Þ 6 U inf fant1 g; supfbns1 g nPk
nPk
nPk
¼ Uða1;kt1 ; b2;ks1 Þ;
ð5Þ
W. Liu et al. / Applied Mathematics and Computation 219 (2013) 5949–5955
5951
!
a1;k ¼ inf Uðant1 ; bns1 Þ P U supfant1 g; inf fbns1 g ¼ Uðb1;kt1 ; a2;ks1 Þ: nPk
nPk
nPk
Therefore by taking limits on both sides of the above two inequalities and in light of the continuity of the function U, we have
lim supan ¼ lim b1;k 6 U lim a1;kt1 ; lim b2;ks1 ¼ U lim inf ant1 ; lim supbns1 ; k!1
n!1
k!1
k!1
n!1
n!1
lim inf an ¼ lim a1;k P U lim b1;kt1 ; lim a2;ks1 ¼ U lim supant1 ; lim inf bns1 : n!1
k!1
k!1
k!1
n!1
The proof of the last two inequalities of (4) is similar and thus omitted.
n!1
h
Lemma 2.2. Under the assumptions (A1), (A2), (A4) and (A5), the following two statements hold: (1) xn 6 k if yns1 6 k, while xn 6 yns1 if yns1 > k; n P maxfs1 ; t1 g; (2) yn 6 k if xnt2 6 k, while yn 6 xnt2 if xnt2 > k; n P maxfs2 ; t 2 g.
Proof. Observe that xn ¼ Uðxnt1 ; yns1 Þ 6 Uðc; yns1 Þ; n P maxfs1 ; t1 g. By the assumptions (A2), (A4) and (A5), we have that if yns1 6 k, then
xn 6 Uðc; yns1 Þ 6 Uðc; kÞ ¼ k; while if yns1 > k then
Uðc; yns1 Þ yns1
6
Uðc; kÞ k
¼ 1;
which implies xn 6 Uðc; yns1 Þ 6 yns1 . Likewise, we get
yn ¼ Wðyns2 ; xnt2 Þ 6 Wðc; xnt2 Þ;
n P maxfs2 ; t2 g:
From this inequality, we know that if xnt2 6 k, then
yn ¼ Wðyns2 ; xnt2 Þ 6 Wðc; kÞ ¼ k; while if xnt2 > k then
Wðc; xnt2 Þ xnt2
6
Wðc; kÞ k
¼ 1;
which indicates that yn 6 Wðc; xnt2 Þ 6 xnt2 . h Lemma 2.3. For each j 2 f0; 1; . . . ; s1 þ t 2 1g, there exists N j 2 N such that xnðt2 þs1 Þþj 6 k and ynðt2 þs1 Þþt2 þj 6 k for all n P N j . Proof. Suppose that for some j 2 f0; 1; . . . ; s1 þ t 2 1g there exists wj 2 N0 such that xwj ðt2 þs1 Þþj 6 k. By using both statements in Lemma 2.2 we have that
xnðt2 þs1 Þþj 6 k for all n P wj and
ynðt2 þs1 Þþt2 þj 6 k for all n P wj : Likewise, if for some j 2 f0; 1; . . . ; t 2 þ s1 1g there exists zj 2 N0 such that yzj ðt2 þs1 Þþj 6 k. Then recursively we get that for all n P zj there hold
ynðt2 þs1 Þþj 6 k and xnðt2 þs1 Þþs1 þj 6 k: Therefore, on the contrary assume that there exists some j 2 f0; 1; . . . ; s1 þt2 1g such that Lemma 2.3 does not hold. Then applying the above two conclusions, we can simply get that
xnðt2 þs1 Þþj > k and ynðt2 þs1 Þþt2 þj > k for all n 2 N0 : Using Lemma 2.2, we have that for all n 2 N0 ,
k < xðnþ1Þðt2 þs1 Þþj 6 xnðt2 þs1 Þþj and k < yðnþ1Þðt2 þs1 Þþt2 þj 6 ynðt2 þs1 Þþt2 þj :
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Denote
lim xnðt2 þs1 Þþj ¼ kj and lim ynðt2 þs1 Þþt2 þj ¼ cj ;
n!1
n!1
then kj P k and cj P k. By Lemma 2.2 we know fxn g; fyn g are bounded. Let
/j ¼ lim inf xnðt2 þs1 Þt1 þj and n!1
lj ¼ lim inf ynðt2 þs1 Þþt2 s2 þj ; n!1
obviously we have /j P c; lj P c. By system (3), we have that
xnðt2 þs1 Þþj ¼ Uðxnðt2 þs1 Þt1 þj ; ynðt2 þs1 Þs1 þj Þ;
ð6Þ
ynðt2 þs1 Þþt2 þj ¼ Wðynðt2 þs1 Þþt2 s2 þj ; xnðt2 þs1 Þþj Þ: Employing Lemma 2.1, it follows from (6) that
kj 6 Uð/j ; cj Þ 6 Uðc; cj Þ ¼ cj Uðc; cj Þ=cj 6 cj Uðc; kÞ=k ¼ cj ;
cj 6 Wðlj ; kj Þ 6 Wðc; kj Þ ¼ kj Wðc; kj Þ=kj 6 kj Wðc; kÞ=k ¼ kj : Thus kj ¼ cj , which plus the assumptions (A2) and (A4) leads to kj ¼ cj ¼ k; /j ¼ lj ¼ c. Let Aj ¼ lim supn!1 xnðt2 þs1 Þþj2t1 and Bj ¼ lim inf n!1 ynðt2 þs1 Þþjt1 s1 , then Aj ; Bj P c. Again by system (3) we get
xnðt2 þs1 Þþjt1 ¼ Uðxnðt2 þs1 Þþj2t1 ; ynðt2 þs1 Þþjt1 s1 Þ:
ð7Þ
It follows by Lemma 2.1 and (7) that c P UðAj ; Bj Þ > UðAj þ 1; Bj Þ P c, which is obviously a contradiction. The proof is complete. h The following lemma follows directly from Lemma 2.3 and shows the explicit bounds of positive solution to system (3) for sufficiently large n. Lemma 2.4. Suppose (A1)–(A5) hold for system (3), then there exists N 2 N such that Uðk; cÞ 6 xn 6 k and
Wðk; cÞ 6 yn 6 k; n > N . Proof. According to Lemma 2.3, let
N ¼ max fN j g ðs1 þ t 2 Þ þ maxfsi ; ti g þ t 2 ; 06j<s1 þt 2
i¼1;2
then xnti ; ynsi 6 k; i ¼ 1; 2, for all n > N. It follows by system (3) xn ¼ Uðxnt1 ; yns1 Þ P Uðk; cÞ; yn ¼ Wðyns2 ; xnt2 Þ P Wðk; cÞ, for all n > N . The proof is complete. h
and
(A4)
that
Theorem 2.1. If (A1)–(A6) hold for system (3), then every positive solution ðxn ; yn Þ to system (3) converges to the unique equiÞ. librium ð x; y 1 1 1 Proof. Let t ¼ maxft1 ; t2 g; s ¼ maxfs1 ; s2 g, and define four sequences fmi g1 i¼t ; fMi gi¼t ; ft i gi¼s ; fT i gi¼s recursively in the following way:
Mi ¼ Uðmit1 ; T is1 Þ; T i ¼ Wðt is2 ; Mit2 Þ;
mi ¼ UðMit1 ; tis1 Þ; ti ¼ WðT is2 ; mit2 Þ;
ð8Þ
where i 2 N0 and the initial values satisfy
mj ¼ Uðk; cÞ; t j ¼ Wðk; cÞ;
Mj ¼ k; j 2 ft; t þ 1; . . . ; 1g; T j ¼ k; j 2 fs; s þ 1; . . . ; 1g:
ð9Þ
Obviously we have that mj1 6 mj ; Mj 6 Mj1 ; j 2 ft þ 1; . . . ; 1g and t j1 6 tj ; T j 6 T j1 ; j 2 fs þ 1; . . . ; 1g. Assume that mj1 6 mj 6 k; Uðk; cÞ 6 Mj 6 Mj1 hold for t þ 1 6 j 6 k, and tj1 6 tj 6 k; Wðk; cÞ 6 T j 6 T j1 hold for s þ 1 6 j 6 k; k P 1. Then employing (A4), for j ¼ k þ 1 we obtain that
mkþ1 ¼ UðMkt1 þ1 ; t ks1 þ1 Þ P UðMkt1 ; tks1 Þ ¼ mk ; Mkþ1 ¼ Uðmkt1 þ1 ; T ks1 þ1 Þ 6 Uðmkt1 ; T ks1 Þ ¼ Mk ; t kþ1 ¼ WðT ks2 þ1 ; mkt2 þ1 Þ P WðT ks2 ; mkt2 Þ ¼ tk ; T kþ1 ¼ Wðtks2 þ1 ; Mkt2 þ1 Þ 6 Wðt ks2 ; Mkt2 Þ ¼ T k : 1 1 Working inductively, we have that fmi g1 i¼t and ft i gi¼s are nondecreasing and bounded from above by k, and fMi gi¼t and 1 fT i gi¼s are nonincreasing and bounded from below by Uðk; cÞ; Wðk; cÞ, respectively.
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Hence fmi g; fMi g; fti g; fT i g are convergent, and denote
m ¼ limmi ; i!1
M ¼ limMi ; i!1
t ¼ limt i ; i!1
T ¼ limT i : i!1
Then m; M 2 ½Uðk; cÞ; k and t; T 2 ½Wðk; cÞ; k. By taking limits on both sides of (8) and in view of the continuity of U and W, we have that
M ¼ Uðm; T Þ;
m ¼ UðM; tÞ;
T ¼ Wðt; MÞ;
t ¼ WðT ; mÞ:
It follows from (A3) and (A6) that m ¼ M ¼ x; t ¼ T ¼ y. 1 Let fxn g1 n¼t and fyn gn¼s be a solution to system (3), then by Lemma 2.4 there exists K 2 N such that
Uðk; cÞ 6 xj 6 k for all j P K t and
Wðk; cÞ 6 yj 6 k for all j P K s: By (8) and (9), we have
mjK 6 xj 6 MjK ; K t 6 j 6 K 1 and
tjK 6 yj 6 T jK ; K s 6 j 6 K 1: If mjK 6 xj 6 MjK holds for K t 6 j 6 n 1 and tjK 6 yj 6 T jK holds for K s 6 j 6 n 1; n P K, then observe that
xn ¼ Uðxnt1 ; yns1 Þ P UðMnKt1 ; tnKs1 Þ ¼ mnK ; xn ¼ Uðxnt1 ; yns1 Þ 6 UðmnKt1 ; T nKs1 Þ ¼ MnK ; yn ¼ Wðyns2 ; xnt2 Þ P WðT nKs2 ; mnKt2 Þ ¼ tnK ; yn ¼ Wðyns2 ; xnt2 Þ 6 Wðt nKs2 ; MnKt2 Þ ¼ T nK : By induction, we derive mnK 6 xn 6 MnK and tnK 6 yn 6 T nK , for all n P K. This plus m ¼ M ¼ x; t ¼ T ¼ y yields . The proof is complete. h limn!1 xn ¼ x and limn!1 yn ¼ y 3. Examples In this section, we will discuss several specific examples to verify the correctness of the main result given in Section 2. Example 3.1. Consider the following difference equation system
xn ¼ p þ
yns1 ; xnt1
yn ¼ p þ
xnt2 ; yns2
where s1 ; t 1 ; s2 ; t 2 2 N; p > 1 and t ¼ maxft1 ; t2 g; s ¼ maxfs1 ; s2 g.
n 2 N0 ; the
initial
ð10Þ values
xi ; yj > 0,
for
i ¼ t; . . . ; 1; j ¼ s; . . . ; 1,
where
Proof. Define Uðx; yÞ ¼ Wðx; yÞ ¼ p þ y=x with x; y 2 Rþ . It can be seen that U; W satisfy assumptions (A1)–(A5) with
c ¼ p; k ¼ p2 =ðp 1Þ; x ¼ y ¼ p þ 1. Hence it suffices to verify the following system
M ¼ p þ T=m;
m ¼ p þ t=M;
T ¼ p þ M=t;
t ¼ p þ m=T
ð11Þ
has a unique solution with m ¼ M; t ¼ T. It follows from (11) that
pðM mÞ ¼ T t and pðT tÞ ¼ M m; which implies T ¼ t; M ¼ m for p > 1. Thus (A6) holds for U; W. In view of Theorem 2.1, for every positive solution limn!1 xn ¼ x ¼ p þ 1; limn!1 yn ¼ y ¼ p þ 1. h
ðxn ; yn Þ
to
system
(10)
we
conclude
Example 3.2. Consider the following difference equation system
xn ¼
pyns1 þ xnt1 ; q þ xnt1
yn ¼
pxnt2 þ yns2 ; q þ yns2
ð12Þ
where s1 ; s2 ; t1 ; t2 2 N; 0 < q < p < q þ 1 with p þ q > 1 and initial values xi ; yj 2 ðq=p; þ1Þ, for i ¼ t; . . . ; 1; j ¼ s; . . . ; 1, where t ¼ maxft 1 ; t 2 g; s ¼ maxfs1 ; s2 g.
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Proof. Let Uðx; yÞ ¼ Wðx; yÞ ¼ ðpy þ xÞ=ðq þ xÞ with x; y 2 ðq=p; þ1Þ. It is straightforward to verify that (A1)–(A5) hold for U; W with c ¼ 1; k ¼ 1=ðq p þ 1Þ; x ¼ y ¼ p q þ 1. Easily, it follows from the system
M¼
pT þ m ; qþm
m¼
pt þ M ; qþM
T¼
pM þ t ; qþt
t¼
pm þ T ; qþT
that ðq þ 1ÞðM mÞ ¼ pðT tÞ and ðq þ 1ÞðT tÞ ¼ pðM mÞ. So we get ðq þ 1Þ2 ðT tÞ ¼ p2 ðT tÞ which implies T ¼ t, and then M ¼ m. Hence (A6) holds for U; W. In light of Theorem 2.1, for every positive solution ðxn ; yn Þ of system (12) we have limn!1 xn ¼ x ¼ p q þ 1; limn!1 yn ¼ y ¼ p q þ 1. h Example 3.3. Consider the system
xn ¼ p þ qynt1 þ
A ; xns1
yn ¼ p þ qxnt2 þ
A ; yns2
n 2 N0 ;
ð13Þ
where all the initial values are positive, and p; A > 0; 0 < q < 1 satisfy the inequality
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1 þ q þ Að1 qÞ=ðA þ p2 Þ > A=ð1 qÞ:
ð14Þ
Proof. Set Uðx; yÞ ¼ Wðx; yÞ ¼ p þ qy þ A=x with x; y 2 Rþ . With c ¼ p; k ¼ ðA þ p2 Þ=ðpð1 qÞÞ; x ¼ y ¼ ðpþ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p þ 4Að1 qÞÞ=ð2ð1 qÞÞ, Uðx; yÞ and Wðx; yÞ satisfy (A1)–(A5). From the following system
M ¼ p þ qT þ A=m; T ¼ p þ qM þ A=t;
m ¼ p þ qt þ A=M;
ð15Þ
t ¼ p þ qm þ A=T;
we can easily get ð1 A=ðmMÞÞðM mÞ ¼ qðT tÞ and ð1 A=ðtTÞÞðT tÞ ¼ qðM mÞ, which indicate that ð1 A=ðmMÞÞð1 A=ðtTÞÞðT tÞ ¼ q2 ðT tÞ. If m; M 2 ½Uðk; cÞ; k; t; T 2 ½Wðk; cÞ; k, then by (14) it is simple to verify ð1 A=ðmMÞÞð1 A=ðtTÞÞ > q2 . Thus T ¼ t; M ¼ m. That is, (A6) holds for mappings U; W. It follows by Theorem 2.1 that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi limn!1 xn ¼ limn!1 yn ¼ ðp þ p2 þ 4Að1 qÞÞ=ð2ð1 qÞÞ for every positive solution ðxn ; yn Þ to system (13). h Example 3.4. Consider the following system (from Riccati equation)
xn ¼
p1 þ yns1 ; q1 þ xnt1
yn ¼
p2 þ xnt2 ; q2 þ yns2
n 2 N0 ;
ð16Þ
with nonnegative initial values, and p1 ; p2 > 0; q1 ; q2 > 1 satisfying
p1 q2 þ p2 ¼ p2 q1 þ p1 ;
q22 2p2 þ q1 P 0;
q1 q2 2p2 q2 1 P 0;
p22 q1 p2 p1 < 0:
ð17Þ
Proof. Define two mappings Uðx; yÞ ¼ ðp1 þ yÞ=ðq1 þ xÞ and Wðx; yÞ ¼ ðp2 þ yÞ=ðq2 þ xÞ; x; y 2 ½0; þ1Þ. It is easy to verify that (A1),(A2) and (A4)–(A6) hold for U; W with c ¼ 0; k ¼ p1 =ðq1 1Þ ¼ p2 =ðq2 1Þ. Hence it suffices to show that the system
x2 þ q1 x ¼ p1 þ y;
y2 þ q2 y ¼ p2 þ x
ð18Þ
has a unique positive solution ðx; yÞ. It follows from system (18) that
y4 þ 2q2 y3 þ ðq1 þ q22 2p2 Þy2 þ ðq1 q2 2p2 q2 1Þy ¼ q1 p2 þ p1 p22 : 4
3
q22
2
ð19Þ p22
Let HðtÞ ¼ t þ 2q2 t þ ðq1 þ 2p2 Þt þ ðq1 q2 2p2 q2 1Þt þ q1 p2 p1 , then H0 ðtÞ ¼ 4t 3 þ 6q2 t2 þ 2ðq1 þ q22 2p2 Þt þ q1 q2 2p2 q2 1, and H00 ðtÞ ¼ 12t 2 þ 12q2 t þ 2ðq1 þ q22 2p2 Þ. . Then by substituting y ¼ y into (18) we get We conclude by (17) that (19) has only one positive root, denoted by y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 =2. It follows from Theorem 2.1 that every positive solution to system (16) converges to the x ¼ q1 þ q1 þ 4p1 þ 4y Þ. h unique positive equilibrium ð x; y Acknowledgement The authors thank the anonymous referees for their valuable comments. This work was supported by the Scholarship Award for Excellent Doctoral Student granted by Ministry of Education (No. 0903005109081-10), the Doctorate Foundation of Educational Ministry of China (Grant No. 20110191110022) as well as the China Scholarship Council (CSC).
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