Global Behavior of Two Families of Nonlinear Symmetric Difference Equations

Abstract

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 Stevic's paper (2010). A new concise proof is given to a quite recent result by Stevic and analogous parallel result of the other inverse equation, which extend related results of Aloqeili (2009), Berenhaut and Stevic (2007), and Liao et al. (2009).

(ProQuest: ... denotes non-US-ASCII text omitted.)

Wanping Liu 1 and Xiaofan Yang 1,2 and Luxing Yang 3

Recommended by Josef Diblik

College of Computer Science, Chongqing University, Chongqing 400044, China
, School of Computer and Information, Chongqing Jiaotong University, Chongqing 400074, China
, College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China

Received 14 May 2010; Revised 3 June 2010; Accepted 16 June 2010

1. Introduction

The interest in investigating rational difference equations has a long history; for instance, see [1-24] and 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, Stevic proposed to study the behavior of positive solutions of the following generic difference equation (see also [25]): [figure omitted; refer to PDF] where A,p,q>0 , and k,m∈...,k≠m . For some recent results in this area, see [26-32] and the references therein.

By a useful transformation method from [4], the authors of [3] confirmed that the unique positive equilibrium of the rational recursive equation [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] can be shown through analogous calculations. Some particular cases of (1.3) had already been considered in [12, 13].

In [3] were proposed the following two conjectures.

Conjecture 1.1.

Suppose that 1≤k<l<m and that {_yn } satisfies [figure omitted; refer to PDF] with positive initial values. Then, the sequence {_yn } converges to the unique positive equilibrium point y¯=1 .

Some special cases of (1.4) had been studied by Li [9, 10] with a semicycle analysis method, which is useful for lower-order difference equations but tedious and complicated to some extent(see the explanation in [33]). Finally, Conjecture 1.1 was also confirmed in [2] with the similar transformation method used in [3, 4]. However, it is somewhat harder to prove the following conjecture in the same way.

Conjecture 1.2.

Assume that q is odd and 1≤_k1 <_k2 <...<_kq , and define S={1,2,...,q} . If {_yn } satisfies [figure omitted; refer to PDF] with _y-kq ,_{y-kq +1} ,...,_y-1 ∈(0,+∞) , where [figure omitted; refer to PDF] Then the sequence {_yn } converges to the unique positive equilibrium point y¯=1 .

Next, we present two definitions as defined in [1].

Definition 1.3.

A function of n variables is symmetric if it is invariant under any permutation of its variables. That is, a function [straight phi](_x1 ,...,_xn ) is called symmetric if [figure omitted; refer to PDF] where π(i) is any permutation of the numbers {1,2,...,n} .

Definition 1.4.

The k th elementary symmetric function _σk of variables _x1 ,...,_xn , where k∈{1,2,...,n} is defined by [figure omitted; refer to PDF] where the sum is taken over all ^Ckn choices of the indices _i1 ,...,_ik from the set of integers {1,2,...,n} .

Obviously, the functions _f1 ,_f2 defined by (1.6) and (1.7) are symmetric and can be rewritten as [figure omitted; refer to PDF]

In this paper, we give a new proof of a quite recent result by Stevic in [34] where he, among others, studied the stability of one of the following two difference equations, which are dual: [figure omitted; refer to PDF] [figure omitted; refer to PDF] where 3≤q∈... is odd, r∈(0,1] and 1≤_k1 <_k2 <...<_kq .

Apparently, Equation (1.11) is the generic form of (1.2), (1.4), and (1.5).

In [6, 18] the authors proved that the main results in some of papers [9-12] are direct consequences of a result confirmed by Kruse and Nesemann [35]. For example, in [6] was showed that the main result in [13] is also a consequence of Corollary 3 in [35]. On basis of these works, in 2008, Aloqeili [1] confirmed Conjecture 1.2 in the same way.

Later, Liao et al. [14] proved Conjecture 1.2 by using a new approach. They used a sort of "frame sequences" method(the notion suggested by Stevic), which has been widely used in [5, 7, 18, 36-41]. Through careful analysis, we find that the method used in [14] can be further simplified and applied in proving Stevic's result in a more concise and interesting way. Namely, we give a new proof of the following result, which generalizes related results in [1, 2, 9, 10, 14].

Theorem 1.5.

Assume that _y-kq ,_{y-kq +1} ,...,_y-1 ∈(0,+∞),r∈(0,1],3≤q∈... is odd and positive integers _k1 ,_k2 ,...,_kq are satisfying 1≤_k1 <_k2 <...<_kq . Then

(1) the unique positive equilibrium point y¯=1 of (1.10) is globally asymptotically stable;

(2) the unique positive equilibrium point y¯=1 of (1.11) is globally asymptotically stable.

2. Auxiliary Results and Notation

In this section, we will introduce some useful notation and lemmas. Consider the following notation(for similar ones see [14]), which play an important role in the paper: [figure omitted; refer to PDF] Employing α and β , define a mapping _Φ1 :^...+q [arrow right]... as follows: [figure omitted; refer to PDF] Then (1.10) can be rewritten as [figure omitted; refer to PDF] or [figure omitted; refer to PDF] with 3≤q∈... being odd, and r∈_...+ .

By the notation defined by (2.1), define the other function _Φ2 :^...+q [arrow right]... such that: [figure omitted; refer to PDF] Then (1.11) can be rewritten as [figure omitted; refer to PDF] or [figure omitted; refer to PDF] with 3≤q∈... being odd, and r∈_...+ .

Lemma 2.1.

If r∈(0,1] , then both (2.4) and (2.7) have the unique positive equilibrium point y¯=1.

Proof.

Suppose that _λ1 >0 is an equilibrium of (2.4), then [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] Obviously _λ1 =1 , due to the different signs of both sides of the last equality for the case _λ1 ≠1. Likewise, let _λ2 >0 be an equilibrium of (2.7); then [figure omitted; refer to PDF] which indicates [figure omitted; refer to PDF] Assume that _λ2 ≠1 . If _λ2 >1 , then by the monotonicity of the map h(x)=(x+1)/(x-1) , x>1 we have that [figure omitted; refer to PDF] which contradicts (2.11).Similarly, if 0<_λ2 <1 , then by the monotonicity of the function l(x)=(1+x)/(1-x) , x∈(0,1), we have that [figure omitted; refer to PDF] which also contradicts (2.11). Thus _λ2 =1 .

The proof is complete.

Lemma 2.2.

(1) Let _Φ1 be defined by (2.2); then _Φ1 is monotonically increasing in _xj if and only if ^{∏i=1,i≠jq} (_xi -1)<0, and monotonically decreasing in _xj if and only if ^{∏i=1,i≠jq} (_xi -1)>0, for j=1,2,...,q.

(2) Let _Φ2 be defined by (2.5), then _Φ2 is monotonically decreasing in _xj if and only if ^{∏i=1,i≠jq} (_xi -1)<0, and monotonically increasing in _xj if and only if ^{∏i=1,i≠jq} (_xi -1)>0, for j=1,2,...,q.

Proof.

The results follow directly from the facts below: [figure omitted; refer to PDF]

Remark 2.3.

The second statement (i.e., (2)) in Lemma 2.2 can also be found in Stevic's paper [34] (see Lemma 1 and Corollary 1 ).

For r∈^...+ ,3≤q∈... odd, define a map Ψ:_...+ [arrow right]... such that [figure omitted; refer to PDF] which has the following simple property: [figure omitted; refer to PDF]

Lemma 2.4.

Suppose that 0<ξ<1, and let Γ=Ψ(ξ),Φ∈{_Φ1 ,_Φ2 }. If _x1 ,_x2 ,...,_xq ∈[ξ,1/ξ], then [figure omitted; refer to PDF]

Proof.

Since Φ(_x1 ,_x2 ,...,_xq ) is symmetric in _x1 ,_x2 ,...,_xq , without loss of generality, we suppose that ξ≤_x1 ≤_x2 ≤...≤_xq ≤1/ξ. If there exists j∈{1,...,q} such that _xj =1 , then by (2.2) and (2.5) we can easily get that Φ(_x1 ,_x2 ,...,_xq )=1 . Thus, assume _xj ≠1 for all j∈{1,...,q}.

Then we have the following q+1 cases to consider: [figure omitted; refer to PDF] By Lemma 2.2, for the above cases, we have that [figure omitted; refer to PDF] Obviously, Γ≤_Φ1 (_x1 ,_x2 ,...,_xq )≤1/Γ follows directly from the above inequalities.

The proof of the case Φ=_Φ2 is analogous and hence omitted.

Lemma 2.5.

Suppose that 0<r≤1,ξ∈(0,1) is fixed and let Γ=Ψ(ξ). Then we have [figure omitted; refer to PDF]

Proof.

By the monotonicity of the function h(x)=(1+x)/(1-x),x∈(0,1), we have that [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] Therefore, (^{(1+ξr )q} -^{(1-ξr )q} )/(^{(1+ξr )q} +^{(1-ξr )q} )>ξ , that is, Γ>ξ.

The proof is complete.

The following corollary follows directly from Lemma 2.4 and Lemma 2.5.

Corollary 2.6.

Assume that 0<ξ<1. If any positive solution (_yn^{)n=-_kq +∞} to (2.4) or (2.7) has the initial values [figure omitted; refer to PDF] then we have _yn ∈[ξ,1/ξ], for n∈_...0 .

Define two sequences (_ξi^)i=0+∞ and (_ηi^)i=0+∞ as follows: [figure omitted; refer to PDF] with initial values _ξ0 ,_η0 >0 .

Lemma 2.7.

For the sequences (_ξi^)i=0+∞ and (_ηi^)i=0+∞ defined by (2.24), if 0<_ξ0 <1, and _ξ0η0 =1 , then [figure omitted; refer to PDF]

Proof.

Inductively, we can simply obtain that 0<_ξi <1<_ηi <+∞ , i∈_...0 . Through simple calculations, by (2.16), we have that [figure omitted; refer to PDF] Therefore by Lemma 2.4 and Lemma 2.5, we get that [figure omitted; refer to PDF] which implies that the sequences (_ξi^)i=0+∞ and (_ηi^)i=0+∞ converge to some limits (denoted by ^ξ* and ^η* , resp.), that is, [figure omitted; refer to PDF] By taking limits on both sides of the first identity of (2.24), we get [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] Suppose that ^ξ* ≠1 ; then by the monotonicity of the function f(x)=(1+x)/(1-x) , x∈(0,1), we have that [figure omitted; refer to PDF] which contradicts (2.29). Hence, we have that ^ξ* =1 and then obviously it follows by (2.26) and (2.28) that ^ξ* =^η* =1 .

The proof is complete.

3. Stability

In this section, we give a new, concise and clear proof of Stevic's Theorem 1.5, by the lemmas in Section 2.

Proof of Theorem 1.5.

Employing Lemma 2.2, the linearized equations of (2.4) and (2.7) about the equilibrium y¯=1 are both [figure omitted; refer to PDF] Then by the Linearized Stability Theorem, y¯=1 is locally stable.

Thus it suffices to confirm that y¯=1 is also a global attractor for all positive solutions of (2.4) and (2.7).

Let (_yn^{)n=-_kq +∞} be a positive solution to (2.4) or (2.7) with initial values [figure omitted; refer to PDF] We need to prove that _{lim n[arrow right]∞yn} =1.

Apparently, there exists _ξ0 ∈(0,1) such that [figure omitted; refer to PDF] where _η0 =1/_ξ0 . Employing Corollary 2.6, we have [figure omitted; refer to PDF] Let sequences (_ξi^)i=0+∞ and (_ηi^)i=0+∞ be defined by (2.24). Let Φ∈{_Φ1 ,_Φ2 }; then in light of Lemma 2.4, (3.4), and (2.26), we get [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF] In view of (3.6), (2.26) and Lemma 2.4, we have that [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF] Reasoning inductively, we can get [figure omitted; refer to PDF] By Lemma 2.7 and (3.9), we obtain [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF]

The proof is complete.

4. Exponential Convergence

In this section, we will prove that all positive solutions to (2.4) and (2.7) with 0<r≤1 are exponentially convergent, by using an approach from paper [42].

Theorem 4.1.

If r∈(0,1], then every positive solution to (2.4) and (2.7) exponentially converges to 1.

Proof.

Let (_yn^{)n=-_kq ∞} be a positive solution to (2.4) or (2.7); then by Theorem 1.5, there exists a sufficiently large natural number N such that for arbitrary fixed [varepsilon]>0 we have |_yn -1|<[varepsilon] for all n≥N .

Denote _Tn =|_yn -1|,n≥-_kq ; then _Tn <[varepsilon] for all n≥N .

(1). For (2.4).

Let 0<[varepsilon]≤1-1/qr; then by (2.4), we have [figure omitted; refer to PDF]

(2). For (2.7).

Let 0<[varepsilon]<1 be fixed; then by (2.7), we get [figure omitted; refer to PDF]

From this inequality and Lemma 1 in [43] (see also Corollary 1 therein), the result directly follows.

5. Other Simple Results

In this section, we will present some elementary results of (2.3) and (2.6) with r>1 .

Proposition 5.1.

If r>1 , then there is no positive solution (_yn^{)n=-_kq +∞} to (2.3) such that _{lim n[arrow right]∞ yn} =+∞ .

Proof.

Suppose (_yn^{)n=-_kq +∞} is a positive solution to (2.3) such that [figure omitted; refer to PDF] Then for some fixed M>1 , there exists N∈... such that [figure omitted; refer to PDF] Employing (2.3) and (5.2), we can simply get that [figure omitted; refer to PDF] which contradicts (5.2). The proof is complete.

Proposition 5.2.

We have the following simple statements:

(1) if r>1 , then (2.6) has nonoscillatory positive solutions with all initial values _yi ≥1 , -_kq ≤i≤-1, or 0<_yi <1,-_kq ≤i≤-1;

(2) let H={_yi |"_yi ≠1,-_kq ≤i≤-1} and denote by ||H|| the cardinality of the set H . If ||H||<q , then for any positive solution (_yn^{)n=-_kq +∞} to (2.3) or (2.6), we get _yn ≡1, for all n≥0 .

6. Conclusions

In the following, let ^a* =max {a,1/a} for any a∈_...+ as defined in [20] and firstly we present [20, Theorem 1 ].

Theorem 6.1 1 (see [20]).

Let f,g satisfy the following two conditions:

(H1) ^{[f(_u1 ,_u2 ,...,_uk )]*} =f(^u1* ,^u2* ,...,^uk* ) and ^{[g(_u1 ,...,_ul )]*} =g(^u1* ,...,^ul* );

(H2) f(^u1* ,^u2* ,...,^uk* )≤^u1* .

Then x¯=1 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: [figure omitted; refer to PDF] where f∈C(^R+k ,_R+ ) and g∈C(^R+l ,_R+ ) with k,l∈{1,2,...},0≤_r1 <...<_rk , and 0≤_m1 <...<_ml , and the initial values are positive real numbers.

Remark 6.2.

Equation (1.10) is a special case of equation (1) in [20].

Proof.

Let k≥3,_f1 (u)=^ur (u>0) , r∈_...+ , and define a recursive equation [figure omitted; refer to PDF] for all 2≤j≤q. Then the following difference equation: [figure omitted; refer to PDF] where 1≤_r1 <_r2 <...<_rq and the initial values _y-rq ,_{y-rq +1} ,...,_y-1 ∈(0,+∞) , is the very Equation (1.10) in this paper.

Remark 6.3.

Let f(u)=^ur , r∈(0,1] , and g(_u1 ,_u2 ,...,_uk )=_fk (_u1 ,_u2 ,...,_uk ) . Then through simple calculations, we have

(H1 ): ^[f(u)]* =f(^u* ) and ^{[g(_u1 ,_u2 ,...,_uk )]*} =g(^u1* ,^u2* ,...,^uk* ) ;

(H2 ): (_H2 )f(^u* )≤^u* .

Thus the conditions (H1) and (H2) of [20, Theorem 1 ] hold. By [20, Theorem 1 ], we know that the unique positive equilibrium y¯=1 of (1.10) (also (6.3)) is globally asymptotically stable.

Remark 6.4.

Although the stability of (1.10) can be also obtained as a corollary from Theorem 1 of the paper by Sun and Xi [20], the method of proof of Theorem 1.5 in 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 China (no. 10771227).

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