Global Behavior of Two Families of Nonlinear Symmetric Difference Equations

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Volume 2010, Article ID 367492,15pages doi:10.1155/2010/367492

Research Article

Global Behavior of Two Families of Nonlinear Symmetric Difference Equations

Wanping Liu,

¹

Xiaofan Yang,

^{1, 2}

and Luxing Yang

³

1College of Computer Science, Chongqing University, Chongqing 400044, China

2School of Computer and Information, Chongqing Jiaotong University, Chongqing 400074, China

3College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China

Correspondence should be addressed to Wanping Liu,wanping [email protected] Received 14 May 2010; Revised 3 June 2010; Accepted 16 June 2010

Academic Editor: Josef Diblik

Copyrightq2010 Wanping Liu et al. 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.

We mainly investigate the global asymptotic stability and exponential convergence of positive solutions to two families of higher-order difference equations, one of which was recently studied in Stević’s paper 2010. A new concise proof is given to a quite recent result by Stević and analogous parallel result of the other inverse equation, which extend related results of Aloqeili 2009, Berenhaut and Stević2007, and Liao et al.2009.

1. Introduction

The interest in investigating rational diﬀerence equations has a long history; for instance, see 1–24and the references cited therein. More generally, it is meaningful to study not only rational recursive equations but also those with powers of arbitrary positive degrees.

For instance, at many conferences, Stevi´c proposed to study the behavior of positive solutions of the following generic diﬀerence equationsee also25:

x_nAx^p_n−m

x^q_n−k, n∈N0, 1.1

whereA, p, q >0, andk, m∈N, k /m. For some recent results in this area, see26–32and the references therein.

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By a useful transformation method from 4, the authors of 3 confirmed that the unique positive equilibrium of the rational recursive equation

y_n y_n−ky_n−m

1y_n−ky_n−m, n∈N0, 1.2

where 1≤k < m, is globally asymptotically stable for all solutions with positive initial values.

In the meantime, they also remarked that the global asymptotic stability for the unique equilibrium of the diﬀerence equation

yn 1y_n−ky_n−m

y_n−ky_n−m , n∈N0, 1.3

can be shown through analogous calculations. Some particular cases of1.3had already been considered in12,13.

In3were proposed the following two conjectures.

Conjecture 1.1. Suppose that 1≤k < l < mand that{yn}satisfies y_n y_n−ky_n−ly_n−my_n−ky_n−ly_n−m

y_n−ky_n−ly_n−ky_n−my_n−ly_n−m1, n∈N0, 1.4 with positive initial values. Then, the sequence{yn}converges to the unique positive equilibrium point y1.

Some special cases of1.4had been studied by Li9,10with a semicycle analysis method, which is useful for lower-order diﬀerence equations but tedious and complicated to some extentsee the explanation in33. Finally,Conjecture 1.1was also confirmed in2 with the similar transformation method used in3,4. However, it is somewhat harder to prove the following conjecture in the same way.

Conjecture 1.2. Assume thatqis odd and 1≤ k1 < k2 < · · ·< kq, and defineS {1,2, . . . , q}. If {yn}satisfies

yn f₁

y_n−k₁, y_n−k₂, . . . , y_n−k_q f₂

y_n−k₁, y_n−k₂, . . . , y_n−k_q, n∈N0, 1.5

with y_−k_q, y_−k_q₁, . . . , y₋₁ ∈0,∞, where

f₁

x₁, x₂, . . . , x_q

q ii1odd

{t1,t2,...,ti}⊂S t1<t2<···<ti

x_t₁x_t₂· · ·x_t_i,

f₂

x₁, x₂, . . . , xq

1 ^q−1

ii2even

{t1,t2,...,ti}⊂S t1<t2<···<ti

xt1xt2· · ·xti.

1.6

Then the sequence{yn}converges to the unique positive equilibrium pointy1.

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Next, we present two definitions as defined in1.

Definition 1.3. A function ofnvariables is symmetric if it is invariant under any permutation of its variables. That is, a functionϕx1, . . . , xnis called symmetric if

ϕx1, . . . , xn ϕ

x_π1, . . . , x_πn

, 1.7

whereπiis any permutation of the numbers{1,2, . . . , n}.

Definition 1.4. Thekth elementary symmetric functionσ_kof variablesx₁, . . . , x_n, wherek ∈ {1,2, . . . , n}is defined by

σ_kx1, . . . , x_n

i1<i2<···<ik

x_i₁x_i₂· · ·x_i_k, 1.8

where the sum is taken over allCⁿ_k choices of the indicesi₁, . . . , i_k from the set of integers {1,2, . . . , n}.

Obviously, the functions f₁, f₂ defined by1.6and 1.7are symmetric and can be rewritten as

f₁

x₁, x₂, . . . , xq

q ii1odd

σi, f₂

x₁, x₂, . . . , xq

1 ^q−1

ii2even

σi. 1.9

In this paper, we give a new proof of a quite recent result by Stevi´c in34where he, among others, studied the stability of one of the following two diﬀerence equations, which are dual:

yn f₂ y_n−k^r

1, y^r_n−k

2, . . . , y_n−k^r _q f₁

y_n−k^r

1, y^r_n−k

2, . . . , y_n−k^r _q, n∈N0, 1.10

y_n f1

y_n−k^r

1, y^r_n−k

2, . . . , y_n−k^r _q f₂

y_n−k^r

1, y^r_n−k

2, . . . , y_n−k^r _q, n∈N0, 1.11

where 3≤q∈Nis odd,r ∈0,1and 1≤k₁< k₂<· · ·< kq.

Apparently, Equation1.11is the generic form of1.2,1.4, and1.5.

In6,18the authors proved that the main results in some of papers9–12are direct consequences of a result confirmed by Kruse and Nesemann35. For example, in6was showed that the main result in13is also a consequence of Corollary 3 in35. On basis of these works, in 2008, Aloqeili1confirmedConjecture 1.2in the same way.

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Later, Liao et al.14 provedConjecture 1.2 by using a new approach. They used a sort of “frame sequences” methodthe notion suggested by Stevi´c, which has been widely used in5,7,18,36–41. Through careful analysis, we find that the method used in14can be further simplified and applied in proving Stevi´c’s result in a more concise and interesting way. Namely, we give a new proof of the following result, which generalizes related results in1,2,9,10,14.

Theorem 1.5. Assume thaty_−k_q, y_−k_q₁, . . . , y₋₁∈0,∞, r ∈0,1,3≤q∈Nis odd and positive integersk₁, k₂, . . . , k_qare satisfying 1≤k₁< k₂<· · ·< k_q. Then

1the unique positive equilibrium pointy1 of 1.10is globally asymptotically stable;

2the unique positive equilibrium pointy1 of 1.11is globally asymptotically stable.

2. Auxiliary Results and Notation

In this section, we will introduce some useful notation and lemmas. Consider the following notationfor similar ones see14, which play an important role in the paper:

α

x₁, x₂, . . . , xq 1

2 _q

i1

x_i^r1

− q

i1

x_i^r−1 ,

β

x₁, x₂, . . . , x_q 1

2 _q

i1

x_i^r1

q i1

x_i^r−1 .

2.1

Employingαandβ, define a mappingΦ1:R^q → Ras follows:

Φ1

x1, x2, . . . , xq α

x1, . . . , xq β

x₁, . . . , xq q

i1

x^r_i 1

−q i1

x^r_i −1 q

i1

x^r_i 1 q

i1

x^r_i −1. 2.2

Then1.10can be rewritten as

yn q

i1

y^r_n−k_i1

−q i1

y_n−k^r _i−1 q

i1

y^r_n−k_i1 q

i1

y_n−k^r _i−1, n∈N0, 2.3

or

y_n Φ1

y_n−k₁, y_n−k₂, . . . , y_n−k_q

, n∈N0, 2.4

with 3≤q∈Nbeing odd, andr∈R.

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By the notation defined by2.1, define the other functionΦ2:R^q → Rsuch that:

Φ2

x₁, x₂, . . . , x_q β

x₁, . . . , x_q α

x1, . . . , xq q

i1

x^r_i 1 q

i1

x^r_i −1 q

i1

x^r_i 1

−q i1

x^r_i −1. 2.5

Then1.11can be rewritten as

y_n q

i1

y^r_n−k_i1 q

i1

y_n−k^r _i−1 q

i1

y^r_n−k_i1

−q i1

y_n−k^r _i−1, n∈N0, 2.6

or

yn Φ2

y_n−k₁, y_n−k₂, . . . , y_n−k_q

, n∈N0, 2.7

with 3≤q∈Nbeing odd, andr∈R.

Lemma 2.1. Ifr∈0,1, then both2.4and2.7have the unique positive equilibrium pointy1.

Proof. Suppose thatλ1>0 is an equilibrium of2.4, then

λ1 Φ1λ1, . . . , λ1

λ^r₁1q− λ^r₁−1q

λ^r₁1q

λ^r₁−1q, 2.8

which implies

1−λ₁ λ^r₁1_q

λ₁1 λ^r₁−1_q

. 2.9

Obviouslyλ11, due to the diﬀerent signs of both sides of the last equality for the caseλ1/1.

Likewise, letλ₂>0 be an equilibrium of2.7; then

λ₂ Φ2λ2, . . . , λ₂

λ^r₂1_q

λ^r₂−1_q λ^r₂1_q

−

λ^r₂−1_q, 2.10

which indicates

λ2−1

λ^r₂1q λ₂1

λ^r₂−1q. 2.11

Assume thatλ₂/1. Ifλ₂>1, then by the monotonicity of the maphx x1/x−1,x >1 we have that

λ₂1

λ₂−1 ≤ λ^r₂1 λ^r₂−1 <

λ^r₂1 λ^r₂−1

q

, 2.12

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which contradicts 2.11.Similarly, if 0 < λ2 < 1, then by the monotonicity of the function

lx 1x/1−x,x∈0,1,we have that

1λ₂

1−λ₂ ≤ 1λ^r₂ 1−λ^r₂ <

1λ^r₂ 1−λ^r₂

q

, 2.13

which also contradicts2.11. Thusλ₂1.

The proof is complete.

Lemma 2.2. (1) LetΦ1 be defined by2.2; thenΦ1 is monotonically increasing inx_j if and only ifq

i1,i /jxi−1< 0,and monotonically decreasing inxjif and only ifq

i1,i /jxi−1 > 0,for j1,2, . . . , q.

(2) Let Φ2 be defined by 2.5, then Φ2 is monotonically decreasing in x_j if and only if q

i1,i /jxi−1 < 0,and monotonically increasing in xj if and only ifq

i1,i /jxi−1 > 0,for j1,2, . . . , q.

Proof. The results follow directly from the facts below:

∂Φ1

∂x_j −rx^r−1_j q i1,i /j

x^2r_i −1 β

x1, x2, . . . , xq₂

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

<0, q i1,i /j

x_i−1>0,

>0, q i1,i /j

xi−1<0,

j ∈

1, . . . , q ,

∂Φ2

∂x_j rx_j^r−1q i1,i /j

x^2r_i −1 α

x1, x2, . . . , xq₂

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

>0, q i1,i /j

x_i−1>0,

<0, q i1,i /j

xi−1<0, j∈

1, . . . , q .

2.14

Remark 2.3. The second statementi.e.,2inLemma 2.2can also be found in Stevi´c’s paper 34 see Lemma 1 and Corollary 1.

Forr∈R,3≤q∈Nodd, define a mapΨ:R → Rsuch that

Ψx 1x^r^q−1−x^r^q

1x^r^q 1−x^r^q, 2.15

which has the following simple property:

Ψ 1

x

1

Ψx, x∈R. 2.16

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Lemma 2.4. Suppose that 0 < ξ <1,and letΓ Ψξ,Φ ∈ {Φ1,Φ2}.Ifx1, x2, . . . , xq ∈ξ,1/ξ, then

Γ≤Φ

x₁, x₂, . . . , x_q

≤ 1

Γ. 2.17 Proof. Since Φx1, x₂, . . . , xq is symmetric in x₁, x₂, . . . , xq, without loss of generality, we suppose thatξ ≤ x1 ≤ x2 ≤ · · · ≤ xq ≤ 1/ξ.If there existsj ∈ {1, . . . , q}such thatxj 1, then by2.2and2.5we can easily get thatΦx1, x₂, . . . , x_q 1. Thus, assumex_j/1 for all j∈ {1, . . . , q}.

Then we have the followingq1 cases to consider:

1 ξ≤x₁≤ · · · ≤x_q−1≤x_q<1< 1 ξ; 2 ξ≤x₁≤ · · · ≤x_q−1<1< xq ≤ 1

ξ; 3 ξ≤x1≤ · · ·<1< x_q−1≤xq ≤ 1

ξ; ...

q

ξ≤x₁<1<· · · ≤x_q−1≤xq ≤ 1 ξ; q1

ξ <1< x₁≤ · · · ≤x_q−1≤x_q ≤ 1 ξ.

2.18

ByLemma 2.2, for the above cases, we have that

1 1<Φ1

x₁, . . . , x_q−1, x_q

≤ 1 Γ; 2 Γ≤Φ1

x₁, . . . , x_q−1, x_q

<1;

3 1<Φ1

x1, . . . , x_q−1, xq

≤ 1 Γ; ...

q

1<Φ1

x1, . . . , x_q−1, xq

≤ 1 Γ; q1

Γ≤Φ1

x1, . . . , x_q−1, xq

<1.

2.19

Obviously,Γ≤Φ1x1, x2, . . . , xq≤1/Γfollows directly from the above inequalities.

The proof of the caseΦ Φ2is analogous and hence omitted.

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Lemma 2.5. Suppose that 0< r ≤1, ξ∈0,1is fixed and letΓ Ψξ.Then we have

Γ> ξ. 2.20

Proof. By the monotonicity of the functionhx 1x/1−x, x∈0,1,we have that 1ξ

1−ξ ≤ 1ξ^r 1−ξ^r <

1ξ^r 1−ξ^r

_q

, 2.21

which implies

1−ξ1ξ^r^q>1ξ1−ξ^r^q. 2.22 Therefore,1ξ^r^q−1−ξ^r^q/1ξ^r^q 1−ξ^r^q> ξ, that is,Γ> ξ.

The following corollary follows directly fromLemma 2.4andLemma 2.5.

Corollary 2.6. Assume that 0 < ξ < 1.If any positive solution yn^∞_n−k_q to2.4or2.7has the initial values

y_−k_q, y_−k_q₁, . . . , y₋₁∈

ξ,1 ξ

, 2.23

then we havey_n∈ξ,1/ξ,forn∈N0.

Define two sequencesξi^∞_i0 andηi^∞_i0 as follows:

ξ_i1 Ψξ_i, η_i1 Ψ η_i

, i∈N0, 2.24

with initial valuesξ₀, η₀>0.

Lemma 2.7. For the sequencesξi^∞_i0 andηi^∞_i0 defined by2.24, if 0< ξ₀ <1,andξ₀η₀1, then

ilim→ ∞ξ_i lim

i→ ∞η_i1. 2.25

Proof. Inductively, we can simply obtain that 0 < ξ_i <1 < η_i < ∞, i∈N0.Through simple calculations, by2.16, we have that

ξ_iη_i≡1, ∀i∈N0. 2.26

Therefore byLemma 2.4andLemma 2.5, we get that

0< ξi≤ξ_i1<1< η_i1 ≤ηi, i∈N0, 2.27

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which implies that the sequencesξi^∞_i0 andηi^∞_i0 converge to some limitsdenoted byξ^∗ andη^∗, resp., that is,

ξ^∗ lim

i→∞ξi∈ξ0,1, η^∗ lim

i→∞ηi∈ 1, η0

. 2.28

By taking limits on both sides of the first identity of2.24, we get

ξ^∗

1 ξ^∗^r_q

−

1−ξ^∗^r_q 1 ξ^∗^rq

1−ξ^∗^rq, 2.29

which implies

1−ξ^∗

1 ξ^∗^r_q

1ξ^∗1−ξ^∗^r^q. 2.30

Suppose thatξ^∗/1; then by the monotonicity of the functionfx 1x/1−x,x∈0,1, we have that

1ξ^∗

1−ξ^∗ ≤ 1 ξ^∗^r 1−ξ^∗^r <

1 ξ^∗^r 1−ξ^∗^r

q

, 2.31

which contradicts2.29. Hence, we have thatξ^∗ 1 and then obviously it follows by2.26 and2.28thatξ^∗η^∗1.

3. Stability

In this section, we give a new, concise and clear proof of Stevi´c’sTheorem 1.5, by the lemmas inSection 2.

Proof ofTheorem 1.5. EmployingLemma 2.2, the linearized equations of2.4and2.7about the equilibriumy1 are both

z_n0·z_n−k₁0·z_n−k₂· · ·0·z_n−k_q ≡0, n∈N0. 3.1

Then by the Linearized Stability Theorem,y1 is locally stable.

Thus it suﬃces to confirm thaty1 is also a global attractor for all positive solutions of2.4and2.7.

Letyn^∞_n−k_qbe a positive solution to2.4or2.7with initial values

y_−kq, y_−kq1, . . . , y₋₁ ∈0,∞. 3.2

We need to prove that lim_n_{→ ∞}y_n1.

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Apparently, there existsξ0∈0,1such that y_i∈

ξ₀, η₀

, i−kq,−kq1, . . . ,−1, 3.3

whereη₀1/ξ₀.EmployingCorollary 2.6, we have yn∈

ξ0, η0

, n−kq,−kq1, . . . . 3.4

Let sequences ξi^∞_i0 and ηi^∞_i0 be defined by2.24. Let Φ ∈ {Φ1,Φ2}; then in light of Lemma 2.4,3.4, and2.26, we get

Ψξ0≤Φ

y_n−k₁, y_n−k₂, . . . , y_n−kq

≤ 1

Ψξ₀ Ψ η0

, n∈N0. 3.5

That is,

y_n∈ ξ₁, η₁

, n∈N0. 3.6

In view of3.6,2.26andLemma 2.4, we have that Ψξ1≤Φ

y_n−k₁, y_n−k₂, . . . , y_n−k_q

≤ 1

Ψξ₁ Ψ η1

, n≥kq. 3.7

That is,

yn∈ ξ2, η2

, n≥kq. 3.8

Reasoning inductively, we can get y_n∈

ξ_i1, η_i1

, n≥ik_q, i∈N0. 3.9

ByLemma 2.7and3.9, we obtain 1 lim

i→ ∞ξ_i1≤ lim

n→ ∞yn≤ lim

i→ ∞η_i11,

3.10

which implies

nlim→ ∞yn1. 3.11

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4. Exponential Convergence

In this section, we will prove that all positive solutions to2.4and2.7with 0 < r ≤ 1 are exponentially convergent, by using an approach from paper42.

Theorem 4.1. Ifr∈0,1,then every positive solution to2.4and2.7exponentially converges to 1.

Proof. Letyn^∞_n−k_qbe a positive solution to2.4or2.7; then byTheorem 1.5, there exists a suﬃciently large natural numberNsuch that for arbitrary fixedε >0 we have|yn−1|< εfor alln≥N.

DenoteT_n|yn−1|, n≥ −kq; thenT_n< εfor alln≥N.

(1). For2.4.

Let 0< ε≤1−^r

1/q; then by2.4, we have

T_ny_n−1 2q

j1y_n−k^r _j−1 q

j1

y_n−k^r _j1 q

j1

y^r_n−k_j−1

≤ 2q

j1y^r_n−k_j−1 2q

j1y_n−k^r _j ≤ q

j1y^r_n−k_j−1 q1−ε^r

≤ q

j1

y_n−k^r _j−1≤ q

j1

y_n−k_j−1

q j1

T_n−k_j ≤ε^q−1T_n−k_q, n≥Nk_q.

4.1

(2). For2.7.

Let 0< ε <1 be fixed; then by2.7, we get

Tnyn−1 2q

j1y_n−k^r _j−1 q

j1

y_n−k^r _j1

−q j1

y^r_n−k_j−1

≤ q

j1

y_n−k^r _j−1≤ q

j1

y_n−k_j−1

q j1

T_n−k_j ≤ε^q−1T_n−k_q, n≥Nrq.

4.2

From this inequality and Lemma 1 in43 see also Corollary 1 therein, the result directly follows.

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5. Other Simple Results

In this section, we will present some elementary results of2.3and2.6withr >1.

Proposition 5.1. Ifr >1, then there is no positive solutionyn^∞_n−k_qto2.3such that limn→ ∞yn

∞.

Proof. Supposeyn^∞_n−k_qis a positive solution to2.3such that

nlim→ ∞y_n ∞. 5.1

Then for some fixedM >1, there existsN∈Nsuch that

∀n≥N, y_n> M. 5.2

Employing2.3and5.2, we can simply get that

y_Nk_q q

j1

y^r_Nk_q_−k_j1

−q j1

y_Nk^r _q_−k_j−1 q

j1

y_Nk^r _q_−k_j1 q

j1

y^r_Nk_q_−k_j−1 <1 5.3

which contradicts5.2. The proof is complete.

Proposition 5.2. We have the following simple statements:

1if r > 1, then 2.6 has nonoscillatory positive solutions with all initial values yi ≥ 1,

−kq≤i≤ −1,or 0< y_i<1,−kq ≤i≤ −1;

2letH {yi | y_i/1,−kq ≤ i ≤ −1}and denote byHthe cardinality of the setH. If H< q, then for any positive solutionyn^∞_n−k_qto2.3or2.6, we gety_n ≡1,for all n≥0.

6. Conclusions

In the following, leta^∗max{a,1/a}for anya∈Ras defined in20and firstly we present 20, Theorem 1.

Theorem 6.1see20. Letf, g satisfy the following two conditions:

H1 fu1, u₂, . . . , uk^∗fu^∗₁, u^∗₂, . . . , u^∗_kandgu1, . . . , ul^∗gu^∗₁, . . . , u^∗_l; H2fu^∗₁, u^∗₂, . . . , u^∗_k≤u^∗₁.

Thenx1 is the unique positive equilibrium for equation (1) which is globally asymptotically stable.

The Equation (1) mentioned in Theorem 6.1 is the following diﬀerence equation:

x_n1 fxn−r1, . . . , x_n−r_kgxn−m1, . . . , x_n−m_l 1

fx_n−r₁, . . . , x_n−r_k gx_n−m₁, . . . , x_n−m_l , n∈N, 6.1

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wheref∈CR^k, Randg ∈CR^l, Rwithk, l∈ {1,2, . . .},0≤r1<· · ·< rk, and 0≤m1<· · ·<

m_l, and the initial values are positive real numbers.

Remark 6.2. Equation1.10is a special case of equation1in20.

Proof. Letk≥3, f1u u^ru >0,r∈R,and define a recursive equation

fj

u1, u2, . . . , uj

f_j−1

u₁, u₂, . . . , u_j−1 u^r_j1 f_j−1

u₁, u₂, . . . , u_j−1

u^r_j , 6.2

for all 2≤j ≤q.Then the following diﬀerence equation:

ynfq

y_n−r₁, . . . , y_n−r_q

, n∈N0, 6.3

where 1 ≤ r₁ < r₂ < · · · < r_q and the initial valuesy_−r_q, y_−r_q₁, . . . , y₋₁ ∈ 0,∞, is the very Equation1.10in this paper.

Remark 6.3. Letfu u^r,r ∈0,1, andgu1, u2, . . . , uk fku1, u2, . . . , uk. Then through simple calculations, we have

H1 fu^∗fu^∗andgu1, u₂, . . . , u_k^∗gu^∗₁, u^∗₂, . . . , u^∗_k; H2 H2fu^∗≤u^∗.

Thus the conditionsH1and H2of20, Theorem 1hold. By20, Theorem 1, we know that the unique positive equilibrium y 1 of1.10 also6.3is globally asymptotically stable.

Remark 6.4. Although the stability of1.10can be also obtained as a corollary from Theorem 1 of the paper by Sun and Xi20, the method of proof ofTheorem 1.5in this paper is distinct.

Acknowledgments

The authors are grateful to the referees for their huge number of valuable suggestions, which considerably improved the presentation of this paper. This work was financially supported by National Natural Science Foundation of Chinano. 10771227.

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