doi:10.1155/2009/128602
Research Article
Global Behavior of Solutions to Two Classes of Second-Order Rational Difference Equations
Sukanya Basu and Orlando Merino
Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA
Correspondence should be addressed to Orlando Merino,[email protected] Received 9 December 2008; Accepted 7 July 2009
Recommended by Ondrej Dosly
For nonnegative real numbersα,β,γ,A,B, andCsuch thatBC >0 andαβγ >0, the difference equationxn1 αβxnγxn−1/ABxnCxn−1,n0,1,2, . . .has a unique positive equilibrium.
A proof is given here for the following statements:1For every choice of positive parametersα,β,γ, A,B, andC, all solutions to the difference equationxn1 αβxnγxn−1/ABxnCxn−1, n 0,1,2, . . . , x−1, x0 ∈0,∞converge to the positive equilibrium or to a prime period-two solution.
2 For every choice of positive parameters α, β, γ, B, andC, all solutions to the difference equation xn1 αβxnγxn−1/Bxn Cxn−1,n 0,1,2, . . . , x−1, x0 ∈ 0,∞converge to the positive equilibrium or to a prime period-two solution.
Copyrightq2009 S. Basu and O. Merino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and Main Results
In their book1, Kulenovi´c and Ladas initiated a systematic study of the difference equation
xn1 αβxnγxn−1
ABxnCxn−1, n0,1,2, . . . , 1.1 for nonnegative real numbersα, β, γ, A, B,and Csuch that BC > 0 andαβγ > 0, and for nonnegative or positive initial conditionsx−1, x0. Under these conditions,1.1has a unique positive equilibrium. One of their main ideas in this undertaking was to make the task more manageable by considering separate cases when one or more of the parameters in 1.1is zero. The need for this strategy is made apparent by cases such as the well-known Lyness Equation2–4.
xn1 αxn
xn−1 , 1.2
whose dynamics differ significantly from other equations in this class. There are a total of 42 cases that arise from1.1in the manner just discussed, under the hypothesesBC >0 andαβγ > 0. The recent publications5,6give a detailed account of the progress up to 2007 in the study of dynamics of the class of equations1.1. After a sustained effort by many researchersfor extensive references, see5,6, there are some cases that have resisted a complete analysis. We list them as follows in normalized form, as presented in5,6:
xn1 αxn
Axn−1, 1.3
xn1 αxn
xnCxn−1, 1.4
xn1 αβxnγxn−1
xn−1 , 1.5
xn1 αxn
ABxnxn−1, 1.6
xn1 βxnxn−1
ABxnxn−1, 1.7
xn1 αβxnγxn−1
Axn−1 , 1.8
xn1 αxnγxn−1
Bxnxn−1 , 1.9
xn1 αβxnxn−1
ABxnxn−1. 1.10
The dynamics of 1.7has been settled recently in7,8. Global attractivity of the positive equilibrium of 1.3 has been proved recently in 9. Since 1.6 can be reduced to 1.3 through a change of variables 10, global behavior of solutions to 1.6 is also settled.
Equation 1.5 is another equation that can be reduced to 1.3, through the change of variablesxnynγ11.
Ladas and coworkers1,5,6have posed a series of conjectures on these equations.
One of them is the following.
Conjecture 1.1 Ladas et al.. For 1.9 and 1.10, every solution converges to the positive equilibrium or to a prime period-two solution.
In this article, we prove this conjecture. Our main results are the following.
Theorem 1.2. For every choice of positive parameters α, β, γ, A, B,and C, all solutions to the difference equation
xn1 αβxnγxn−1
ABxnCxn−1, n0,1,2, . . . , x−1, x0∈0,∞ 1.11 converge to the positive equilibrium or to a prime period-two solution.
Theorem 1.3. For every choice of positive parametersα, β, γ, B,andC, all solutions to the difference equation
xn1 αβxnγxn−1
BxnCxn−1 , n0,1,2, . . . , x−1, x0∈0,∞ 1.12 converge to the positive equilibrium or to a prime period-two solution.
A reduction of the number of parameters of 1.12 is obtained with the change of variablesxn γ/Cyn, which yields the equation
yn1 rpynyn−1
qynyn−1 , n0,1,2, . . . , y−1, y0∈0,∞, 1.13 whererαC/γ2,pβ/γ, andqB/C.
The number of parameters of 1.11 can also be reduced, which we proceed to do next. Consider the following affine change of variables which is helpful to reduce number of parameters and simplify calculations:
xn γ
C A
BC
yn− A
BC. 1.14
With1.14,1.11may now be rewritten as
yn1 rpynyn−1
qynyn−1 , n0,1,2, . . . , y−1, y0∈L,∞, 1.15 where
p AB BCβ
AC BCγ, q B
C, r CBC
BαCα−Aβ−Aγ γBC AC2 ,
L AC
AC BCγ.
1.16
Theorems1.2and 1.3can be reformulated in terms of the parametersp,q, and r as follows.
Theorem 1.4. Letα, β, γ, A, B,and Cbe positive numbers, and letp, q, r,andL be given by relations 1.16. Then every solution to 1.15 converges to the unique equilibrium or to a prime period-two solution.
Theorem 1.5. Letp, q, andr be positive numbers. Then every solution to1.13converges to the unique equilibrium or to a prime period-two solution.
In this paper we prove Theorems 1.4 and 1.5; Theorems 1.2 and 1.3 follow as an immediate corollary.
The two main differences between1.15and1.13are the set of initial conditions, and the possibility of having a negative value ofrin1.15, while only positive values ofr are allowed in1.13. Nevertheless, for both1.15and1.13the unique equilibrium has the formula:
y p1
p12 4r
q1 2
q1 . 1.17
Although it is not possible to prove Theorem 1.2as a simple corollary toTheorem 1.3, the changes of variables leading to Theorems 1.4 and 1.5 will result in proofs to the former theorems that are greatly simplified.
Our main results Theorems1.2and1.3imply that when prime period-two solutions to 1.11or 1.13do not exist, then the unique equilibrium is a global attractor. We have not treated here certain questions about the global dynamics of 1.11and 1.13, such as the character of the prime period-two solutions to either equation, or even for more general rational second-order equations, when such solutions exist. This matter has been treated in 12.
This work is organized as follows. The main results are stated inSection 1. Results from literature which are used here are given inSection 2 for convenience. InSection 3, it is shown that either every solution to1.15converges to the equilibrium or there exists an invariant and attracting intervalIwith the property that the functionfx, yassociated with the difference equation is coordinatewise strictly monotonic onI ×I. InSection 4, a global convergence result is obtained for1.13over a specific range of parameters and for initial conditions in an invariant compact interval.Theorem 1.4is proved inSection 5, and the proof ofTheorem 1.5is given inSection 6. Tables1and2include computer algebra system code for performing certain calculations that involve polynomials with a large number of termsover 365 000 in one case. These computer calculations are used to support certain statements inSection 4. Finally, we refer the reader to1for terminology and definitions that concern difference equations.
2. Results from Literature
The results in this subsection are from literature, and they are given here for easy reference.
The first result is a reformulation of1, Theorems 1.4.5–1.4.8.
Theorem 2.1see1,13. Suppose that a continuous functionf :a, b2 → a, bsatisfies one of (i)–(iv):
ifx, yis nondecreasing inx,y, and
∀m, M∈a, b2,
fm, m m, fM, M M
⇒mM; 2.1
iifx, yis nonincreasing inx,y, and
∀m, M∈a, b2,
fm, m M, fM, M m
⇒mM; 2.2
iiifx, yis nonincreasing inxand nondecreasing iny, and
∀m, M∈a, b2,
fm, M M, fM, m m
⇒mM; 2.3
ivfx, yis nondecreasing inxand nonincreasing iny, and
∀m, M∈a, b2,
fM, m M, fm, M m
⇒mM. 2.4
Thenyn1 fyn, yn−1has a unique equilibrium ina, b, and every solution with initial values in a, bconverges to the equilibrium.
The following result is1, Theorem A.0.8.
Theorem 2.2. Suppose that a continuous functionf : a, b3 → a, bis nonincreasing in all variables, and
∀m, M∈a, b3,
fm, m, m M, fm, m, m M
⇒mM. 2.5
Thenyn1 fyn, yn−1, yn−2has a unique equilibrium in a, b, and every solution with initial values ina, bconverges to the equilibrium.
Theorem 2.3see14. LetIbe a set of real numbers, and letF:I×I → Ibe a functionFu, v which decreases inuand increases inv. Then for every solution{xn}∞n−1of the equation
xn1Fxn, xn−1, n0,1, . . ., 2.6
the subsequences{x2n}and{x2n1}of even and odd terms do exactly one of the following.
iThey are both monotonically increasing.
iiThey are both monotonically decreasing.
iiiEventually, one of them is monotonically increasing and the other is monotonically decreas- ing.
Theorem 2.3has this corollary.
Corollary 2.4see14. IfI is a compact interval, then every solution of 2.6converges to an equilibrium or to a prime period-two solution.
Theorem 2.5see15. Assume the following conditions hold.
ih∈C0,∞×0,∞,0,∞.
iihx, yis decreasing inxand strictly decreasing iny.
iiixhx, xis strictly increasing inx.
ivThe equation
xn1xnhxn, xn−1, n0,1, . . . 2.7 has a unique positive equilibriumx.
Thenxis a global attractor of all positive solutions of 2.7.
3. Existence of an Invariant and Attracting Interval
In this section we prove a proposition which is key for later developments. We will need the function
f x, y
: rpxy
qxy , x, y∈L,∞, 3.1
associated to1.15.
Proposition 3.1. At least one of the following statements is true.
AEvery solution to1.15converges to the equilibrium.
BThere existm∗,M∗withL < m∗< M∗such that the following is true.
i m∗, M∗is an invariant interval for 1.15, that is, fm∗, M∗×m∗, M∗ ⊂ m∗, M∗.
iiEvery solution to1.15eventually entersm∗, M∗. iiifx, yis coordinatewise strictly monotonic onm∗, M∗2.
The proof of Proposition 3.1will be given at the end of the section, after we prove several lemmas. The next lemma states that the functionf·,·associated to1.15is bounded.
Lemma 3.2. There exist positive constantsLandUsuch thatL <Land L ≤f
x, y
≤ U, x, y∈L,∞. 3.2
In particular,
fL,U×L,U⊂L,U. 3.3
Proof. The function
f x, y
αβxγy ABxCy,
x, y
∈0,∞2, 3.4
associated to1.11is bounded:
min α, β, γ
max{A, B, C} ≤ αβxγy
ABxCy ≤ max α, β, γ min{A, B, C},
x, y
∈0,∞2. 3.5
SetL:min{α, β, γ}/max{A, B, C}andU :max{α, β, γ}/min{A, B, C}. The affine change of coordinates1.14maps the rectangular regionL,U 2 onto a rectangular regionL,U2 which satisfies3.2and3.3.
Lemma 3.3. Ifpq, then every solution to1.15converges to the unique equilibrium.
Proof. If p q, then D1fx, y −pr/pxy2 and D2fx, y −r/pxy2. Thus, depending on the sign ofr, the functionfx, yis either nondecreasing in both coordinates, or nonincreasing in both coordinates onL,∞. ByLemma 3.2, all solutions{yn}∞n−1satisfy yn ∈ L,Uforn≥ 1. A direct algebraic calculation may be used to show that all solutions m, M∈L,Uof either one of the systems of equations
MfM, M,
mfm, m, 3.6
and
Mfm, m,
mfM, M 3.7
necessarily satisfymM. In either case, the hypothesesioriiofTheorem 2.1are satisfied, and the conclusion of the lemma follows.
We will need the following elementary result, which is given here without proof.
Lemma 3.4. Supposeq /p. The functionfx, yhas continuous partial derivatives onL,∞2, and iD1fx, y 0 if and only ifyqr/q−p, andD1fx, y>0 if and only ifp−qy > qr;
iiD2fx, y 0 if and only ifx−r/p−q, andD2fx, y>0 if and only ifq−px > r.
We will need to refer to the valuesK1andK2where the partial derivatives offx, y change sign.
Definition 3.5. Ifp /q, set
K1 : qr
p−q, K2 : −r
p−q. 3.8
m, M
m, m
M, K1 f↑,↓
f↓,↑
Figure 1: The arrows indicate type of coordinatewise monotonicity offx, yon each region.
Definition 3.6. ForL≤m≤M, let
φm, M:min f
x, y :
x, y
∈m, M2 , Φm, M:max
f x, y
: x, y
∈m, M2 .
3.9
Lemma 3.7. Supposep /q. Ifm, M⊂L,Uis an invariant interval for1.15withm≤K1 ≤M orm≤K2≤M, thenm < φm, MorΦm, M< MormMy.
Proof. By definition ofφandΦ,m≤φm, MandΦm, M≤M. Suppose
mφm, M, Φm, M M. 3.10
The proof will be complete when it is shown thatm M. There are a total of four cases to consider:ar ≥0 andp > q,br <0 andp < q,cr≥0 andp < q, anddr <0 andp > q.
We present the proof of caseaonly, as the proof of the other cases is similar.
Ifr≥0 andp > q, thenK1∈m, MandK2/∈m, M. Note that m, M×m, M m, M×m, K1
m, M×K1, M. 3.11
ByLemma 3.4, the signs of the partial derivatives offx, yare constant on the interior of each of the setsm, M×m, K1andm, M×K1, M, as shown inFigure 1.
Sincefx, yis nonincreasing in bothxandyonm, M×m, K1, fM, K1≤f
x, y
≤ fm, m for x, y
∈m, M×m, K1. 3.12
Similarlyfx, yis nondecreasing inxand nonincreasing inyonm, M×K1, M, hence fm, M ≤ f
x, y
≤ fM, K1 for x, y
∈m, M×K1, M. 3.13
From3.12and3.13one has
φm, M fm, M, Φm, M fm, m. 3.14
Combine3.14with relation3.10to obtain the system of equations fm, M m,
fm, m M. 3.15
EliminatingMfrom system3.15gives the cubic inm q
q1 m3
1−pq m2
−1−p−qr
m−r0, 3.16
which has the roots
−1
q, 1−p− 1p2
4r 1q 2
1q , 1−p
1p2 4r
1q 2
1q . 3.17
Only one root in the list3.17is positive, namely,
m 1−p 1p2
4r 1q 2
1q y. 3.18
Substituting into one of the equations of system3.15one also obtainsM y, which gives the desired relationmMy.
Definition 3.8. Let m0 : L,M0 : U, and for 0,1,2, . . . let m : φm, M, M : Φm, M.
By the definitions ofm,M,φ·,·andΦ·,·, we have thatm1, M1⊂m, M for 0,1,2, . . .. Thus the sequence{m}is nondecreasing, and{M}is nonincreasing. Let m∗:limm andM∗:limM.
Lemma 3.9. Supposep /q. Either there existsN∈Nsuch that{K1, K2} ∩mN, MN ∅, or there existsm∗M∗y.
Proof. Arguing by contradiction, supposem∗< M∗and for all∈N,{K1, K2} ∩m, M/∅.
Since the intervalsm, Mare nested and∩m, M m∗, M∗, it follows that{K1, K2} ∩ m∗, M∗/∅. ByLemma 3.7, we have
m∗< φm∗, M∗ or Φm∗, M∗< M∗. 3.19 Continuity of the functionsφandΦimplies
φm∗, M∗ limφm, M limm1m∗,
or Φm∗, M∗ limΦm, M limM1M∗. 3.20 Statements3.19and3.20give a contradiction.
Proof ofProposition 3.1. Suppose that statementAis not true. ByLemma 3.3, one must have p /q. Note that if {y} is a solution to 1.15, then y1 ∈ m, M for 0,1,2, . . .. If m∗ M∗, since m → m∗ and M → M∗, we havey → y, but this is statementA which we are negating. Thus m∗ < M∗, and byLemma 3.9there exists N ∈ Nsuch that {K1, K2} ∩mN, MN ∅; so fx, y is coordinatewise monotonic on mN, MN. The set mN, MNis invariant, and every solution entersmN, MNstarting at least with the term with subindexN1. We have shown that if statementAis not true, then statementBis necessarily true. This completes the proof of the proposition.
4. Equation 1.13 with r ≥ 0, p > q and qr/p − q < p/q
In this section we restrict our attention to the equation
xn1fxn, xn−1, n0,1, . . . , x−1, x0∈0,∞, 4.1
where
f x, y
: rpxy
qxy . 4.2
Forp >0,q >0, andr≥0,1.13has a unique positive equilibrium
y p1
p124r q1 2
q1 . 4.3
We note that ifI⊂0,∞is an invariant compact interval, then necessarilyy∈I.
The goal in this section is to prove the following proposition, which will provide an important part of the proofs of Theorems1.2and1.5.
Proposition 4.1. Letp, q,andrbe real numbers such that
p > q >0, r≥0, qr p−q< p
q, 4.4
and letm, M ⊂qr/p−q, p/qbe a compact invariant interval for1.13. Then every solution to1.13withx−1, x0∈m, M converges to the equilibrium.
Proposition 4.1follows from Lemmas4.2,4.3, and4.5, which are stated and proved next.
Lemma 4.2. Assume the hypotheses toProposition 4.1. If eitherq≥1 orp ≤1, then every solution to1.13withx−1, x0∈m, M converges to the equilibrium.
Proof. We verify that hypothesis ivof Theorem 2.1is true. Since x > rq/p−q forx ∈ m, M, the function fx, yis increasing in xand decreasing inyforx, y ∈ m, M 2 by Lemma 3.4. Letm, M∈m, M be such thatm /Mand
fM, m−M0,
fm, M−m0. 4.5
We show first that system4.5 has no solutions if eitherq ≥ 1 or p ≤ 1. By eliminating denominators in both equations in4.5,
m−mMMp−M2qr 0,
M−mMmp−m2qr 0, 4.6
and by subtracting terms in4.6one obtains M−m
1−pqmM
0. 4.7
Sincem /M, we haveqmM p−1, which implies that forp≤1 there are no solutions to system4.5which have both coordinates positive. Now assumep > 1; from4.7,m p−1−Mq/q, and substitute the latter into4.5 to see thatx M is a solution to the quadratic equation
1−q x2
−1p
−1q
q x−1pqr
q 0. 4.8
By a symmetry argument, one has thatxmis also a solution to4.8. By inspection of the coefficients of the polynomial in the left-hand side of4.8one sees that two positive solutions are possible only whenq <1. To get the conclusion of the lemma, note that the fact that4.5 has no solutions withm /Mis just hypothesisivofTheorem 2.1.
Lemma 4.3. Assume the hypotheses toProposition 4.1. If
r ≤ p2q−p, 4.9
then every solution to1.13withx−1, x0 ∈m, M converges to the equilibrium.
Proof. By substitutingxnfxn−1, xn−2intoxn1fxn, xn−1we obtain
xn1 rpxnxn−1
qxnxn−1 rp
rpxn−1xn−2 /
qxn−1xn−2 xn−1 q
rpxn−1xn−2 /
qxn−1xn−2
xn−1 , n0,1, . . ., 4.10
that is,
xn1fx n, xn−1, xn−2, where f x, y, z
prp2yqryqy2pzrzyz
qrpqyqy2qzyz , 4.11 where thexhas been kept infx, y, z for bookkeeping purposes. Thusfx, y, z is constant inx. We claim thatfx, y, z is decreasing in bothyandz. To see that the partial derivative
D3f x, y, z
− r
p−q y
−qr p−q
y
qrpqyqy2qzyz2 4.12
is negative, just usep > qand the inequalityp−qy−qr >0, which is true by the hypotheses ofProposition 4.1. The remaining partial derivative is
D2f x, y, z
− h y, z
qrpqyqy2qzyz2, 4.13
where h
y, z
:−q2r22pqry−2q2ryp2qy2−pq2y2q2ry2prz
−qrzpqrz−q2rz2pqyz−2q2yz2qryzpz2−qz2rz2. 4.14 We have,
D1h y, z
2 p−q
qr2q
p2−pqqr y2q
p−qr z >0, D2h
y, z
p−q 1q
r2q
p−qr y2
p−qr z >0.
4.15
Sinceqr/p−q≤m,
h y, z
≥h qr
p−q, qr p−q
q
1q2
r2
p2−pqqr
p−q2 >0, y, z ∈ m,M
, 4.16
thus we conclude thatD2fx, y, z<0 forx, y, z∈m, M.
To complete the proof we verify the hypotheses of Theorem 2.2. We claim that the system of equations
fm, m, m −M0, fM, M, M −m0
4.17
has no solutionsm, Mwithm /Mwhenever hypothesis4.9holds.
By eliminating denominators in both equations in4.17one obtains
−m2m2M−mp−mp2−m2qmMqm2MqmMpq−mr−pr−mqrMqr0,
−M2mM2−Mp−Mp2mMq−M2qmM2qmMpq−Mr−prmqr−Mqr0, 4.18
and by subtracting terms in4.18one obtains
m−M
−m−MmM−p−p2−mq−MqmMq−r−2qr
0. 4.19
Sincem /M, we may use the second factor in the left-hand side term of 4.19to solve for Min terms ofm, which upon substitution intofm, m, m Mand simplification yields the equation
a2m2a1ma0
−1m 1q
m2mqm2qmpqqr 0, 4.20
where
a0 r
p2pqp2qqr2q2r ,
a1pp22pq3p2qp3qr−pr4qr4q2r2pq2r a2
1q
12qpqqr .
, 4.21
By hypothesis4.9we haverp ≤p3q−p2 < p3q, hencep3q−rp > 0, which impliesa1 ≥0.
By direct inspection one can see thata0>0 anda2>0. Thus4.20has no positive solutions, and we conclude that4.17has no solutionsm, M∈m, M withm /M. We have verified the hypotheses ofTheorem 2.2, and the conclusion of the lemma follows.
Lemma 4.4. Letp > 0,q >0 andr ≥0. If the positive equilibriumyof 1.13satisfiesy < p/q, thenyis locally asymptotically stable (L.A.S.).
Proof. Solving forrin
y r p1 y
q1
y 4.22
gives
r q1
y2− p1
y. 4.23
Then a calculation shows
D1f y, y
p−qy y
q1, D2f
y, y
− y−1 y
q1.
4.24
Sett1 :D1fy, yandt2 : D2fy, y. The equilibriumyis locally asymptotically stable if the roots of the characteristic polynomial
ρx x2−t1x−t2 4.25
have modulus less than one1. By the Schur-Cohn Theorem,yis L.A.S. if and only if|t1|<
1−t2 < 2. It can be easily verified that 1−t2 < 2 if and only if 0 < qy1 which is true regardless of the allowable parameter values. Sincep−qy > 0 by the hypothesis, we have
|t1||p−q y/yq1| p−q y/yq1; hence some algebra gives|t1|<1−t2if and only if
1 2
p1
q1 < y. 4.26
But4.26is a true statement by formula4.3. We conclude thatyis L.A.S.
Lemma 4.5. Assume the hypotheses toProposition 4.1. If
p >1, q <1, r > p2q−p, 4.27
then every solution to1.13withx−1, x0 ∈m, M converges to the equilibrium.
Proof. The proof begins with a change of variable in1.13to produce a transformed equation with normalized coefficients analogous to those in the standard normalized Lyness’ Equation 2–4
xn1 αxn
xn−1 . 4.28
We seek to use an argument of proof similar to the one used in 9, in which one takes advantage of the existence of invariant curves of Lyness’ Equation to produce a Lyapunov-like function for1.13.
Setynpznin1.13to obain the equation
zn1 azngzn−1
bznzn−1 , n0,1, . . . , z−1, z0 ∈0,∞, 4.29
where
a r
p2, g 1
p, bq. 4.30
We will denote withzthe unique equilibrium of4.29. Note that
ypz. 4.31
It is convenient to parametrize4.29in terms of the equilibrium. We will use the symbolu to represent the equilibriumzof4.29. By direct substitution of the equilibriumu zinto 4.29we obtain
a b1u2− g1
u. 4.32
By4.32,a≥0 if and only ifu≥g1/b1. Using4.32to eliminateafrom4.29gives the following equation forb >0,g >0,andu≥g1/b1, equivalent to4.29:
zn1 b1u2− g1
uzngzn−1
bznzn−1 , n0,1,2, . . . , y−1, y0∈0,∞. 4.33 Therefore it suffices to prove that all solutions of4.33converge to the equilibriumu.
The following statement is crucial for the proof of the proposition.
Claim 1. u >1 if and only ifr > p2q−p.
Proof. Sinceypzpu, we haveu >1 if and only ify > p, which holds if and only if
p1
p124r q1 2
q1 > p . 4.34
After an elementary simplification, the latter inequality can be rewritten asr > p2q−p.
By the hypotheses of the lemma, byClaim 1, and by4.30and4.32we have
b <1, g <1, 1< u < 1
b, g1
b1 ≤u. 4.35
We now introduce a function which is the invariant function for4.28with constant
αu2−uin this case the the equilibrium of4.28isu:
g x, y
1x 1y
u2−uxy
xy . 4.36
Note thatgx, y>0 for allx, y∈0,∞wheneveru >1. By using elementary calculus, one can show that the functiongx, yhas a strict global minimum atu, u 3,4, that is,
gu, u< g x, y
,
x, y
∈0,∞2. 4.37
We need some elementary properties of the sublevel sets
Sc:
s, t∈0,∞:gs, t≤c , c >0. 4.38
We denote withQu, u,1,2,3,4 the four regions Q1u, u:
x, y
∈0,∞×0,∞:u≤x, u≤y , Q2u, u:
x, y
∈0,∞×0,∞:x≤u, u≤y , Q3u, u:
x, y
∈0,∞×0,∞:x≤u, y≤u , Q4u, u:
x, y
∈0,∞×0,∞:u≤x, y≤u .
4.39
Let
T x, y
:
y,
g1
u2−b1uygx byx
,
x, y
∈0,∞×0,∞, 4.40
be the map associated to4.33 see16.
Claim 2. Ifx, y∈Q2u, u∪Q4u, u\ {u, u}, thengTx, y< gx, y.
Proof. Set
Δ1
x, y :g
x, y
−g T
x, y
. 4.41
A calculation yields
Δ1
x, y
−1xF1
x, y F2
x, y xy
bxy F3
x, y , 4.42
where
F1
x, y
:bx−u
y−1 b
y−u
buyu−g , F2
x, y
:bx−u2bx−uubx−uu2 u−y
bu2yg , F3
x, y
: b1u2− 1g
uxgy.
4.43
By4.35, forx, y ∈Q4u, u\ {u, u}we haveu≤xandy≤ u <1/bwithx, y/ u, u, thereforeF1x, y< 0,F2x, y >0 andF3x, y> 0. ConsequentlyΔ1x, y >0 forx, y ∈ Q4u, u\ {u, u}. To see thatΔ1x, y > 0 forx, y ∈ Q2u, u\ {u, u} as well, rewrite F1x, yandF2x, yas follows:
F1 x, y
bu−x 1
b−u
y−u2 y−u
u y−u
u−g b
y−u x, F2
x, y
bx−u xu2
−b y−u
u2− y−u2
g− y−u
ug.
4.44
For x, y ∈ Q2u, u\ {u, u}we have x ≤ u ≤ y and x, y/ u, u. ThusF1x, y > 0, F2x, y<0, andF3x, y>0, which implyΔ1x, y>0.
Claim 3. Supposeg > b. Ifx, y∈Q1u, u∪Q3u, u\ {u, u}, thengT2x, y< gx, y.
Proof. This proof requires extensive use of a computer algebra system to verify certain inequalities involving rational expressions. Here we give an outline of the steps, and refer the reader to Tables1and2for the details.
Sinceb < g <1< u <1/bandg1/b1< u,we may write
u g1
b1 t, t >0, g bs/b
1s , s >0.
4.45
The expressionΔ2 : gx, y−gT2x, ymay be written as a single ratio of polynomials, Δ2N/DwithD >0. The next step is to showN >0 forx, y/ u, u.
Points x, y in Q1u, umay be written in the form x uv, y uw, where v, w ∈ 0,∞. Substitutingx,y,u,and g in terms ofv,w,s,and tinto the expression for None obtains a rational expressionN/ D with positive denominator. The numeratorNhas some negative coefficients. At this points two cases are considered,w≥v, andw≤v. These can be written asw vkandvwkfor nonnegativek. Substitution of each one of the latter expressions inNgives a polynomial with positive coefficients. This provesΔ2x, y>0 forx, y∈Q2u, u.
If now we assumex, y∈Q3u, uwithx, y/ u, u, we may write
x u
v1, v∈0,∞,
y u
w1, w∈0,∞.
4.46
The rest of the proof is as in the first case already discussed. Details can be found in Tables1 and2.
Claim 4. Supposeg < bandu >1. Ifx, y∈Q1u, u∪Q3u, u\ {u, u}, thengT3x, y<
gx, y.
Table 1: Mathematica code needed to do the calculations inClaim 3. Here we define the functionsg,f andT, as well as the expression DELTA2. The reparametrizations indicated in the proof ofClaim 3for the caseg > bare defined as substitution rules. To verify the positive sign of a polynomial of nonnegative variablesz,s, . . ., we form a list with the terms of the polynomial and then substitute the number 1 for the variables in order to extract the smallest coefficient. This input was tested on Mathematica Version 5.0 18.
g{x , y} 1x1yu2−uxy
xy ;
fx , y b1u2−g1uxgy
bxy ;
T{x , y} {fx, y, x};
DELTA2g{x, y}−gTT{x, y};
rulebb → z z1;
rulegg → Factor1/bsb
s1 /.ruleb; ruleuu→ Factorg1
b1t/.ruleg/.ruleb;
numD2NumeratorTogetherDELTA2;
num1D2vkNumeratorTogethernumD2/.{x → uvk, y → uv};
list1D2vkNumeratorTogethernum1D2vk/.{ruleu,ruleg,ruleb}/.Plus → List;
min 21Minlist1D2vk/.{z → 1, s → 1, t → 1, v → 1, k → 1};
num2D2wkNumeratorTogethernumD2/.{x→ uv, y → uvk};
list2D2wkNumeratorTogethernum2D2wk/.{ruleu,ruleg,ruleb}/.Plus → List;
min 22Minlist2D2wk/.{z → 1, s→ 1, t → 1, v → 1, k → 1};
num22NumeratorTogethernumD2/.{x→ u/w1, y → u/v1};
num2D2vkExpandnum22/.{v → wk};
list2D2vkNumeratorTogethernum2D2vk/.{ruleu,ruleg,ruleb}/.Plus → List;
min 23Minlist2D2vk/.{z → 1, s → 1, t → 1, w → 1, k → 1};
num2D2wkExpandnum22/.{w → vk};
list2D2wkNumeratorTogethernum2D2wk/.{ruleu,ruleg,ruleb}/.Plus → List;
min 24Minlist2D2wk/.{z → 1, s→ 1, t → 1, v → 1, k → 1}; Print“Minimal coefficients:”,{min 21, min 22, min 23, min 24}
Proof. The proof is analogous to the proof ofClaim 3. We provide an outline. More details can be found in Tables1and2.
Since 1 < u, we may writeu 1twitht > 0. Also,u < 1/bimpliesb < 1/u, and b1/1tsfors >0. Sinceg < b,we may writeg 1/1tsfor >0.
The expression Δ3 : gx, y − gT3x, y may be written as a single ratio of polynomials, Δ2 N/D with D > 0. The next step is to show N > 0 for x, y/ u, u.
This is done in a way similar to the procedure described inClaim 3.
Table 2: Mathematica code needed to do the calculations inClaim 4wheng≤b. The functionsg,fandT are defined as beforenot shown. This input was tested on Mathematica Version 5.018.
Tz{x , y} TogetherT{x, y}/.{B → 1
1st, g → 1
1str, u → 1s};
DELTA3Togetherg{x, y}−gTzTzTz{x, y};
numDELTA3Numeratorstep3;
numDELTA32NumeratorTogethernumDELTA3/.{x → 1sv, y → 1sw}; numDELTA33ExpandnumDELTA32;
numDELTA33vkExpandnumDELTA33/.w → vk;
min 31MinnumDELTA33vk/.Plus → List/.{s → 1, t → 1, r → 1, v → 1, k → 1};
numDELTA33wkExpandnumDELTA33/.v → wk;
min 32MinnumDELTA33wk/.Plus → List/.{s → 1, t→ 1, r → 1, w → 1, k → 1}; numDELTA32NumeratorTogethernumDELTA3/.{x → 1s
1v, y → 1s 1w};
numDELTA33ExpandnumDELTA32;
numDELTA33vkExpandnumDELTA33/.w → vk;
min 33MinnumDELTA33vk/.Plus → List/.{s → 1, t → 1, r → 1, v → 1, k → 1};
numDELTA33wkExpandnumDELTA33/.v → wk;
min 34MinnumDELTA33wk/.Plus → List/.{s → 1, t→ 1, r → 1, w → 1, k → 1};
Print“Minimal coefficients :, {min 31, min 32, min 33, min 34};
To complete the proof of the lemma, let φ, ψ ∈ 0,∞ ×0,∞. Let {yn}n≥−1 be the solution to4.33with initial condition y−1, y0 φ, ψ, and let{Tnφ, ψ}n≥0 be the corresponding orbit ofT. The following argument is essentially the same as the one found in 9; we provided here for convenience. Define
c:lim inf
n g Tn
φ, ψ
. 4.47
Note thatc <∞, which can be shown by applying Claims2,3, and4repeatedly as needed to obtain a nonincreasing subsequence of{gTnφ, ψ}n≥0that is bounded below bygu, u.
Let{gTnkφ, ψ}k≥0be a subsequence convergent toc. Therefore there exists c >0 such that
g Tnk
φ, ψ
≤c, ∀k≥0, 4.48
that is,
Tnk φ, ψ
∈Sc:
s, t:gs, t≤c , for k≥0. 4.49
The setScis closed by continuity ofgx, y. Boundedness ofScfollows from
0< x, y < 1x 1y
u2−uxy
xy g
x, y
≤c, for x, y
∈Sc. 4.50
Thus Scis compact, and there exists a convergent subsequence {Tnkφ, ψ} with limit x, y. Note that
c lim
→ ∞g Tnk
φ, ψ g
x,y
. 4.51
We claim thatx, y u, u. If not, then by Claims 2,3, and4, min
g T
x,y
, g T2
x,y
, g T3
x,y
<c. 4.52
Let · denotes the Euclidean norm. By4.52and continuity, there existsδ >0 such that s, t−
x,y< δ⇒min
gTs, t, g
T2s, t , g
T3s, t
<c. 4.53 ChooseL∈Nlarge enough so that
TnkL φ, ψ
−
x,y< δ. 4.54
But then4.53and4.54imply min
g TnkL1
φ, ψ , g
TnkL2s, t , g
TnkL3s, t
<c, 4.55
which contradicts the definition4.47ofc. We conclude x, y u, u. From this and the definition of convergence of sequences we have that for every >0 there existsL ∈Nsuch thatTnkLφ, ψ−u, u< . Finally, since
maxynkL−1−u,ynkL−u
≤
ynkL−1−u, ynkL−uTnkL φ, ψ
−u, u, 4.56 we have that for every > 0 there existsL ∈ Nsuch that|ynkL −u| < and|ynkL−1−u| < . Sinceuis a locally asymptotically stable equilibrium for4.33byLemma 4.4, it follows that yn → u. This completes the proof of the lemma.
5. Proof of Theorem 1.4
To prove Theorem 1.4 it is enough to assume statement B of Proposition 3.1. Also by Lemma 3.3we may assume p /q without loss of generality. Thus we make the following standing assumption valid throughout the rest of this section for1.15.
Standing Assumption (SA)
Assumep /qand that there existm∗,M∗withL ≤m∗< M∗ ≤ Usuch that for1.15and its associated functionfx, y,
i m∗, M∗is an invariant interval;
iievery solution eventually entersm∗, M∗;
iiifx, yis coordinatewise strictly monotonic onm∗, M∗2.
The function fx, yis assumed to be coordinatewise monotonic on m∗, M∗, and there are four possible cases in which this can happen: a fx, y is increasing in both variables, b fx, y is decreasing in both variables, c fx, y is decreasing in x and increasing iny, anddfx, yis increasing inxand decreasing iny.
We present several lemmas before completing the proof ofTheorem 1.4.
By considering the restriction of the mapT of1.15tom∗, M∗2, an application of the Schauder Fixed Point Theorem17 gives thatm∗, M∗2 contains the fixed point ofT, namely,y, y. Thus we have the following result.
Lemma 5.1. One hasy∈m∗, M∗.
Lemma 5.2. Neither one of the systems of equations
MfM, M,
mfm, m, S1
and
Mfm, m,
mfM, M, S2
has solutionsm, M∈m∗, M∗2withm < M.
Proof. Sincexyis the only solution tofx, x x, it is clear that onlyy, ysatisfiesS1.
Now let m, Mbe a solution to S2. From straightforward algebra applied to M−m fm, m−fM, Mone arrives atp1M−m 0, which impliesmM.
Lemma 5.3. Suppose thatfx, yis increasing inxand decreasing inyforx, y∈m∗, M∗. Then p−q >0.
Proof. By the standing assumptionSA,p /q. ByLemma 3.4, the coordinatewise monotonic- ity hypothesis, and the facty∈m∗, M∗fromLemma 5.1, we have
p−q
y > qr, q−p
y < r. 5.1
The inequalities in5.1cannot hold simultaneously unlessp−q >0.
Lemma 5.4. If fx, y is increasing in x and decreasing in y for x, y ∈ m∗, M∗, then fm∗, M∗2⊂1, p/q.
Proof. For x, y ∈ m∗, M∗, the function f is well defined and is componentwise strictly monotonic on the setx,∞×y,∞. Then,
f x, y
< lim
s→ ∞f s, y
lim
s→ ∞
rpsy qsy p
q, f
x, y
> lim
t→ ∞fx, t lim
t→ ∞
rpxt qxt 1.
5.2
Lemma 5.5. Letp > 0, q > 0 and r ≥ 0. If fx, yis increasing in x and decreasing in y on m∗, M∗, then
qr p−q < p
q. 5.3
Proof. Since y ∈ m∗, M∗ by Lemma 5.1, we have D1y, y > 0 and D2y, y < 0. By Lemma 5.3,p > q, and byLemma 3.4,
p−q
y > qr, q−p
y < r. 5.4
Then,
y > qr
p−q. 5.5
In addition, byLemma 3.4,
yf y, y
< lim
s→ ∞f s, y
lim
s→ ∞
rpsy qsy p
q. 5.6
Lemma 5.6. Suppose thatfx, yis increasing inxand decreasing inyforx, y∈ m∗, M∗. If r <0, thenp−qr >0.
Proof. Since D1fx, y > 0 forx, y ∈ m∗, M∗, and by Lemmas3.4,5.1and 5.3, we have y >−r/p−q, that is,
p124r q1
> −2r q1
p−q −p−1. 5.7
If the right-hand side of inequality5.7is nonnegative, then, after squaring both sides of 5.7, we have
p12 4r
q1
>
−2rq1 p−q
2
4r q1
p1
p−q
p12
. 5.8
Further simplification of5.8and the hypothesisr <0 yield
1< r q1
p−q2 p1
p−q, 5.9
which, after some elementary algebra, impliesp−qr >0. Now assume that the right-hand side of inequality5.7is negative relation that we may rewrite as
−r p−q < 1
2 p1
q1. 5.10
If1/2p1/q1≤1, then−r/p−q<1, which gives the conclusionp−qr >0. If 1/2p1/q1>1, that is,p >2q1, then
p−qr > qr1. 5.11
Therefore if qr 1 ≥ 0,the conclusion of the lemma follows from this and from5.11.
Assume
qr1<0. 5.12
From relations1.16we have
qr1 bc
a2c2bc2αc3α−ac2β2abcγac2γb2γ22bcγ2c2γ2 c
acbγcγ2 , 5.13
hence assumption5.12and relation5.13imply
R:a2c2bc2αc3α−ac2β2abcγac2γb2γ22bcγ2c2γ2<0. 5.14
Further algebra gives γ acR−
−c2αacγ−cβγbγ2 c2αbcαγ
a c2αγ
a bγ2cγ22bγ3 a b2γ3
ac cγ3 a >0.
5.15
SinceR <0 by5.14, from inequality5.15we have
−c2αacγ−cβγbγ2<0. 5.16
Finally, from1.16we have
p−qr −bc2
−c2αacγ−cβγbγ2 c
acbγ cγ2 . 5.17
Combining5.16with5.17we obtainp−qr >0.
Lemma 5.7. Ifr <0 andfx, yis increasing inxand decreasing inyforx, y∈m∗, M∗, then every solution converges to the equilibrium.
Proof. Since p −qr > 0 byLemma 5.6, we have K2 −r/p−q < 1, which together with Lemma 5.4implies that 1, p/q is an invariant, attracting compact interval such that fx, yis increasing inxand decreasing inyon1, p/q2. Sincef1, p/q2 ⊂ 1, p/q, we see that every solution to1.15eventually enters the invariant interval1, p/q. The change of variables
yn 1 p/q
zn
1zn , or zn yn−1
p/q−yn 5.18
transforms the equation
yn1 rpynyn−1
qynyn−1 , n0,1, . . . , y−1, y0∈
1,p q
5.19
into the equivalent equation
zn1gzn, zn−1, n0,1, . . . , z−1, z0∈0,∞, 5.20 where
gw, v: q1v
− q
p−qr
−p2pq−qr w 1w
−q
p−q−qr
−p2pqq2r
v . 5.21
We claim that forw, v ∈0,∞,agw, wis increasing inw,bgw, v/w is decreasing inw, andcgw, v/w is decreasing inv. Indeed, since p > q,r < 0,p−qr > 0, and
−r/p−q< p/q,we have d
dw
gw, w
−rq2 1q
−pq2
−pqq2q2r−p2wpqwq2rw2 >0,
∂
∂v
gw, v w
− q
−pq2 q
p−qr
p2−pqqr w −pqq2q2r−p2vpqvq2rv2w1w <0,
∂
∂w
gw, v w
−q1v q
p−qr 2q
p−qr
w
p2−pqqr w2 q
p−q−qr
p2−pq−q2r v
w21w2 <0.
5.22
Also, note that5.20has a unique equilibriumz. Therefore hypothesesi–ivofTheorem 2.5are satisfied; so every solution{zn}to5.20converges toz. By reversing the change of variables, one can conclude that every solution to5.19converges to the equilibrium.
Proof ofTheorem 1.4. The four parts of the proof are as follows.
afx, y is increasing in both x and y on m∗, M∗2. By Lemma 5.2the hypotheses of Theorem 2.1 part i. are satisfied; hence every solution converges to the equilibriumy.
bfx, y is decreasing in bothx and y on m∗, M∗2. By Lemma 5.2the hypotheses of Theorem 2.1 part ii. are satisfied; hence every solution converges to the equilibriumy.
cfx, y is decreasing in x and increasing in y on m∗, M∗2. By the corollary to Theorem 2.3we conclude that every solution converges to the unique equilibrium or to a prime period-two solution.
dfx, yis increasing inxand decreasing inyonm∗, M∗2. By Lemmas3.4,5.3, and 5.4, there is no loss of generality in assuming m∗, M∗ ⊂ K, p/q, whereK : max{−r/p−q, qr/p−q}, which we do. We consider two subcases. Ifr≥0, then Lemmas 5.3, 5.5, and Proposition 4.1 imply that every solution converges to the unique equilibrium. Ifr <0, thenLemma 5.7implies that every solution converges to the unique equilibrium.
This completes the proof ofTheorem 1.4. SinceTheorem 1.4is just a version ofTheorem 1.2 obtained by an affine change of coordinates, we have also provedTheorem 1.2as well.
6. Proof of Theorem 1.5
The first lemma guarantees solutions to1.13to be bounded.
Lemma 6.1. Letp > 0,q > 0 andr ≥ 0. There exist positive constantsLandUsuch that every solution{xn}∞n−1to1.13satisfiesxn∈L,Uforn≥2, and the function
f x, y
rpxy qxy ,
x, y
∈0,∞2 6.1
satisfies
fL,U×L,U⊂L,U. 6.2
Proof. Set
L:min p
q,1
, U:max
p
q,1,r p1 L
q1 L
. 6.3
