On a class of second-order nonlinear difference equation

Abstract

In this paper, we consider the rule of trajectory structure for a kind of second-order rational difference equation. With the change of the initial values, we find the successive lengths of positive and negative semicycles for oscillatory solutions of this equation, and the positive equilibrium point 1 of this equation is proved to be globally asymptotically stable.

Mathematics Subject Classification (2000)

39A10[PUBLICATION ABSTRACT]

Dongsheng et al. Advances in Difference Equations 2011, 2011:46 http://www.advancesindifferenceequations.com/content/2011/1/46

RESEARCH Open Access

On a class of second-order nonlinear difference equation

Li Dongsheng1*, Zou Shuliang1 and Liao Maoxin2

* Correspondence: mailto:[email protected]

Web End =lds1010@sina. mailto:[email protected]

Web End =com

1School of Economics and Management, University of South China, Hengyang, Hunan 421001, Peoples Republic of ChinaFull list of author information is available at the end of the article

Abstract

In this paper, we consider the rule of trajectory structure for a kind of second-order rational difference equation. With the change of the initial values, we find the successive lengths of positive and negative semicycles for oscillatory solutions of this equation, and the positive equilibrium point 1 of this equation is proved to be globally asymptotically stable.

Mathematics Subject Classification (2000)39A10

Keywords: rational difference equation, trajectory structure rule, semicycle length; periodicity, global asymptotic stability

1 Introduction and preliminaries

Motivated by those work [1-17], especially [10], we consider in this paper the following second-order rational difference equation

xn+1 = 1 + xknxln1 + a

xkn + xln1 + a

, n = 1, 0, 1, . . . , (1:1)

the initial values x-1, x0 (0, +), a (0, +) and k, l (-, +).

Mainly, by analyzing the rule for the length of semicycle to occur successively, we describe clearly out the rule for the trajectory structure of its solutions and further derive the global asymptotic stability of positive equilibrium of Equation (1.1).

It is easy to see that the positive equilibrium x of Equation (1.1) satisfies

x = 1 + xk+l + a

xk + xl + a

.

From this, we see that Equation (1.1) possesses a positive equilibrium x = 1. In this

paper, our work is only limited to positive equilibrium x = 1.

Here, for readers convenience, we give some corresponding definitions.

Definition 1.1. A positive semicycle of a solution {xn}n=1of Equation (1.1) consists of

a string of terms {xr, xr+1, ..., xm}, all greater than or equal to the equilibrium x, with r

-1 and m such that

either r = 1 or r > 1 and xr1 < x

2011 Li et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Dongsheng et al. Advances in Difference Equations 2011, 2011:46 http://www.advancesindifferenceequations.com/content/2011/1/46

and

either m = or m < and xm+1 < x.

A negative semicycle of a solution {xn}n=1of Equation (1.1) consists of a string of

terms {xr, xr+1, ..., xm}, all less than the equilibrium x, with r -1 and m such that either r = 1 or r > 1 and xr1 x

and

either m = or m < and xm+1 x.

The length of a semicycle is the number of the total terms contained in it. Definition 1.2. A solution {xn}n=1of Equation (1.1) is said to be eventually positive if

xn is eventually greater than x = 1. A solution {xn}n=1of Equation (1.1) is said to be

eventually negative if xn is eventually smaller than x = 1.

Definition 1.3. We can divide the solutions of Equation (1.1) into two kinds of types: trivial ones and nontrivial ones. A solution {xn}n=1of Equation (1.1) is said to be even

tually trivial if xn is eventually equal to x = 1; otherwise, the solution is said to be

nontrivial.

If the solution is a nontrivial solution, then we can further divide the solution into two cases: non-oscillatory solution and oscillatory solution. A nontrivial solution {xn}n=1of

Equation (1.1) is regarded as non-oscillatory solution if xn is eventually positive or negative; otherwise, the nontrivial solution is oscillatory.

For the other concepts in this paper, see Refs.[1,2].

2 Trajectory structure rule

The solutions of Equation (1.1) include trivial ones, non-oscillatory ones and oscillatory ones, and their trajectory structure rule of the solutions is as follows.

2.1 Nontrivial solution

Theorem 2.1. A positive solution {xn}n=1of Equation (1.1) is eventually trivial if and

only if

(x1 1)(x0 1) = 0. (2:1) Proof. Sufficiency. Assume that Equation (2.1) holds. Then according to Equation(1.1), we know that the following conclusions are true:(i) If x-1 = 1, then xn = 1 for n 1.(ii) If x0 = 1, then xn = 1 for n 1.

Necessity. Conversely, assume that

(x1 1)(x0 1) = 0. (2:2) Then, we can show xn 1 for any n 1. For the sake of contradiction, assume that for some N 1,

xN = 1 and that xn = 1 for any 1 n N 1. (2:3) Clearly,

1 = xN = 1 + xkN1xlN2 + a

xkN1 + xlN2 + a

.

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From this, we can know that

0 = xN 1 =

(xkN1 1)(xlN2 1)

xkN1 + xlN2 + a

,

which implies xN-1 = 1, or xN-2 = 1. This contradicts with Equation (2.3).

Remark 2.2. Theorem 2.1 actually demonstrates that a positive solution {xn}n=1of

Equation (1.1) is eventually nontrivial if (x-1 - 1)(x0 - 1) 0. So, if a solution is a nontrivial one, then xn 1 for any n -1.

2.2 Non-oscillatory solution

Lemma 2.3. Let {xn}n=1be a positive solution of Equation (1.1) which is not eventually

equal to 1, then the following conclusion is true:

(A) If kl < 0, then (xn+1 - 1)(xn - 1)(xn-1 - 1) < 0, for n 0;(B) If kl > 0, then (xn+1 - 1)(xn - 1)(xn-1 - 1) > 0, for n 0;

Proof. First, we consider (A). According to Equation (1.1), we have that

xn+1 1 =

(xkn 1)(xln1 1)

xkn + xln1 + a

, n = 0, 1, ....

Considering kl < 0,

(xn+1 1)(xn 1)(xn1 1) < 0.

Noting that kl < 0, that is k (-, 0) and l (0, +), or k (0, + -, 0), and l (-, 0), one has (xkn 1)(xn 1) > 0, (xln1 1)(xnl 1) < 0, or

(xln1 1)(xnl 1) > 0, (xln1 1)(xnl 1) > 0. From those, one can get the result

easily.

The proof of (B) is similar to (A).

Theorem 2.4. Let kl < 0, there exist non-oscillatory solutions of Equation (1.1) with x-

1, x0 (0, 1), which must be eventually negative. There do not exist eventually positive non-oscillatory solutions of Equation (1.1).

Proof. Consider a solution of Equation (1.1) with

x1, x0 (0, 1).

We then know from Lemma 2.3 (A) that 0 <xn < 1 for n N, where N 1, 2, 3, ....So, this solution is just a non-oscillatory solution and furthermore eventually negative.

Suppose that there exists eventually positive non-oscillatory of Equation (1.1). Then, there exists a positive integer N such that xn > 1 for n N. Thereout, for n N + 1,

(xn+1 1)(xn 1)(xn1 1) 0.

This contradicts Lemma 2.3. So, there do not exist eventually positive non-oscillatory of Equation (1.1), as desired.

From Lemma 2.3 (B), we can get the result as follows, also.

Theorem 2.5. Let kl > 0, there exist non-oscillatory solutions of Equation (1.1) with x-

1, x0 (1, +), which must be eventually positive. There do not exist eventually negative non-oscillatory solutions of Equation (1.1).

Dongsheng et al. Advances in Difference Equations 2011, 2011:46 http://www.advancesindifferenceequations.com/content/2011/1/46

2.3 Oscillatory solution

Theorem 2.6. Let kl < 0, and {xn}1be a strictly oscillatory of Equation (1.1), then the

rule for the lengths of positive and negative semicycles of this solution to occur succes

sively is ..., 2+, 1-, 2+, 1-, ....

Proof. By Lemma 2.3, one can see that the length of a negative semicycle is at most 3, and a positive semicycle is at most 2. On the basis of the strictly oscillatory character of the solution, we see that, for some integer p 0, one of the following 32 cases must occur:

case 1: xp < 1, xp+1 < 1;case 2: xp > 1, xp+1 < 1;case 3: xp < 1, xp+1 > 1;case 4: xp > 1, xp+1 > 1.case 1 cannot occur. Otherwise, the solution is a non-oscillatory solution of Equation (1.1).

If Case 2 occurs, it follows from Lemma 2.3 that xp+2 > 1, xp+3 > 1, xp+4 < 1, xp+5 > 1, xp+6 > 1, xp+7 < 1, xp+8 > 1, xp+9 > 1, xp+10 < 1, ....

This means that rule for the lengths of positive and negative semicycles of the solution of Equation (1.1) to occur successively is ..., 2+, 1-, 2+, 1-, .... The proof for other cases, except Case 1, is completely similar to that of Case 2. So, the proof for this theorem is complete.

Theorem 2.7. Let kl > 0, and {xn}1be a strictly oscillatory of Equation (1.1), then the

rule for the lengths of positive and negative semicycles of this solution to occur succes

sively is ..., 1+, 2-, 1+, 2-, ....

The proof of theorem (2.7) is similar to that of theorem (2.6).3 Local asymptotic stability and global asymptotic stability

Before stating the oscillation and non-oscillation of solutions, we need the following key lemmas. For any integer a, denote Na = {a, a + 1, ...,}.

3.1 Four Lemmas

Lemma 3.1. Let k (0, 1], and {xn}n=1be a positive solution of Equation (1.1) which is

not eventually equal to 1, then the following conclusions are valid:

(a) (xn+1 - xn)(xn - 1) < 0, for n 0;(b) (xn+1 - xn-1)(xn-1 - 1) < 0, for n 0.

Proof. First, we consider (a). From Equation (1.1), we obtain

xn+1 xn =

From k (0, 1] and {xn}n=1 not eventually equal to 1, one can see that

(1 xk+1n)(1 xn) > 0, (1 x1kn)(1 xn) 0, xkn + xln1 > 0.

This teaches us that (xn+1 - xn)(1 - xn) > 0, n = 0, 1, .... That is to say, (xn+1 - xn)(xn -1) < 0, n = 0, 1, .... So, the proof of (a) is complete.

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1 xk+1n + xln1xn(xk1n 1) + a(1 xn) xkn + xln1 + a

,

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Second, one investigates (b). From Equation (1.1), one has

xn+1 xn1 =

1 xknxn1 + xln1(xkn xn1) + a(1 xn) xkn + xln1 + a

, (3:1)

From Equation (1.1), one gets

1 xnx

1kn1 =

xkn1 [parenleftBigg]

1 x

1 k2

n1

[parenrightBigg]

(3:2)

xkn1 + xln2 + a

,

According to k (0, 1] and {xn}n=1 not eventually equal to 1, one arrives at

1 x1 k2 n1

(1 xn1) 0. (3:3)

From Equations (3.2) and (3.3), we know 1 xnx1k n1

(1 xn1) > 0. So, we can

get immediately

1 xknxn1[parenrightBig](1 xn1) > 0. (3:4)

From Equation (1.1), one can have

xn x

1kn1 =

xk+ln1 [parenleftBigg]

1 x

1 k2

n1

[parenrightBigg]

(3:5)

xkn1 + xln2 + a

,

According to k (0, 1] and {xn}n=1 not eventually equal to 1, one arrives at

1 x1 k2 n1

(1 xn1) 0. (3:6)

From Equations (3.5), (3.6), we can obtain that

xn x1k n1

(1 xn1) > 0, i.e.,

xkn xn1[parenrightBig](1 xn1) > 0. (3:7)

By virtue of Equations (3.1), (3.4), (3.7), we see that (b) is true.

The proof for Lemma (3.1) is complete.

Lemma 3.2. Let {xn}n=1be a positive solution of Equation (1) which is not eventually

equal to 1, then (xn+1 - xn-2)(xn-2 - 1) < 0, for n 1.

Proof. By virtue of Equation (1.1), one gets

xn+1 xn2 =

(1 xknxn2) + (xkn xn2)xln1 + a(1 xn2)

xkn + xln1 + a

, n = 0, 1, .... (3:8)

By virtue of Equation (1.1), one obtains that

xn1 x

[parenrightBigg]

+ a

1 k2

n2 =

1 x k3+1 k2 n2

1 x1k2n2[parenrightBigg]+ xln3xkn2 [parenleftBigg]1 x1k3 n2 [parenrightBigg]

xkn2 + xln3 + a

.

(3:9)

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According to k (0, 1] and {xn}n=1 not eventually equal to 1, we get

1 x k3+1 k2 n2

(1xn2) > 0,

1 x1 k2 n2

(1xn2) > 0,

1 x1 k3 n2

(1xn2) > 0.

So,

xn1 x1 k2 n2

(1 xn2) > 0. (3:10)

That is

xkn1 x1k n2

(1 xn2) > 0. (3:11)

By virtue of Equation (1.1), we can know

1 xnx

1kn2 =

xkn1 x1k n2

[parenrightBigg]

+ xln2 [parenleftBigg]

1 x

k+ 1

k n1

[parenrightBigg]

+ a

1 x1k n2 [parenrightBigg]

xkn1 + xln2 + a

(3:12)

.

Utilizing (3.11),(3.12), adding

1 xk+ 1 k n1

(1 xn2) > 0,

1 x1k n2

(1 xn2) > 0

when k (0, 1], we know the following is true

1 xnx1k n2

(1 xn2) > 0.

So,

1 xknxn2[parenrightBig](1 xn2) > 0. (3:13)

Similar to (3.13), we know this is true

xn x1k n2

(1 xn2) > 0.

So,

xkn xn2[parenrightBig](1 xn2) > 0. (3:14)

From (3.8),(3.13)and (3.14), one obtains that the following is true

(xn xn2)(1 xn2) > 0.

This shows Lemma (3.2) is true.

Lemma 3.3. Let x-1, x0 (0, 1), then the following conclusions are true:(a) If l > 0 and -1 <k < 0 or l < 0 and 0 <k <1, then (xn+1 - xn) < 0, for n 0;(b) If k > 0 and -1 <l < 0 or k < 0 and 0 <l < 1, then (xn+1 - xn-1) < 0, for n 0.

The proof of lemma (3.3) can be completed by Equation (1.1), theorem 2.4 and properties of power function easily.

Lemma 3.4. Let x-1, x0 (1, ), then the following conclusions are true:

Dongsheng et al. Advances in Difference Equations 2011, 2011:46 http://www.advancesindifferenceequations.com/content/2011/1/46

(a) If l > 0 and 0 <k < 1 or l < 0 and -1 <k < 0, then (xn+1 - xn) < 0, for n 0;(b) If k > 0 and 0 <l < 1 or k < 0 and -1 <l < 0, then (xn+1 - xn-1) < 0, for n 0.

The proof of lemma (3.4) can be completed by Equation (1.1), theorem 2.5 and properties of power function easily.

First, we consider the local asymptotic stability for unique positive equilibrium point

x of Equation (1.1). We have the following results.

3.2 Local asymptotic stability

Theorem 3.5. The positive equilibrium point of Equation (1.1) is locally asymptotically stable.

Proof. The linearized equation of Equation (1.1) about the positive equilibrium point

x is

yn+1 = 0 yn + 0 yn1, n = 0, 1, . . . ,and so it is clear from the paper [[2], Remark 1.3.7] that the positive equilibrium point x of Equation (1.1) is locally asymptotically stable. The proof is complete.

We are now in a position to study the global asymptotically stability of positive equilibrium point x.

3.3 Global asymptotic stability of oscillatory solution

Theorem 3.6. The positive equilibrium point of Equation (1.1) is globally asymptotically stable when k (0, 1] and l (0, +).

Proof We must prove that the positive equilibrium point x of Equation (1.1) is both

locally asymptotically stable and globally attractive. Theorem 3.5 has shown the local asymptotic stability of x. Hence, it remains to verify that every positive solution {xn}n=1 of Equation (1.1) converges to x as n . Namely, we want to prove lim

n

xn = x = 1. (3:15)

Consider now {xn} to be non-oscillatory about the positive equilibrium point x of

Equation (1.1). By virtue of Lemma 3.1(a), it follows that the solution is monotonic and bounded. So, limn xn exists and is finite. Taking limits on both sides of Equation (1.1), one can easily see that (3.15) holds.

Now let {xn} be strictly oscillatory about the positive equilibrium point of Equation(1.1). By virtue of Theorem 2.6, one understands that the rule for the lengths of positive and negative semicycles occurring successively is ..., 2+, 1-, 2+, 1-, 2+, 1-, .... For simplicity, for some nonnegative integer p, we denote by {xp, xp+1}+ the terms of a positive semicycle of length two, followed by {xp+2}-, a negative semicycle with semi-cycle length one, then a positive semicycle of length two and a negative semicycle of length one, and so on. Namely, the rule for the lengths of positive and negative semi-cycles to occur successively can be periodically expressed as follows:

{xp+3n, xp+3n+1}+, {xp+3n+2}, {xp+3n+3, xp+3n+4}+, {xp+3n+5}, n = 0, 1, 2, ....

Lemma (3.1) (a), (b) and Lemma (3.2) teaches us that the following results are true:

(A) xp+3n >xp+3n+1 >xp+3n+3 >xp+3n+4, n = 0, 1, 2, ....

(B) xp+3n+2 <xp+3n+5 <xp+3n+8, n = 0, 1, 2, ....

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So, from (A) one can see that {xp+3n}n=0 is decreasing with lower bound 1. So, the

limit S = limn xp+3n exists and is finite.Furthermore, From (A) one can further obtain

S = lim

n

xp+3n+1

Similarly, by (B) one can see that {xp+3n+2}n=0 is increasing with upper bound 1. So,

the limit T = limnxp+3n+2 exists and is finite.Now, it suffices to prove S = T = 1.

Noting that

xp+3n+2 =

1 + xkp+3n+1xlp+3n + a

xkp+3n+1 + xlp+3n + a

, (3:16)

, (3:17)

Taking limits on both sides of the Equations (3.16) and (3.17), respectively, we get

T = sk+l + 1 + a

sk + sl + a ,

(3:18)

xp+3n+3 =

1 + xkp+3n+2xlp+3n+1 + a

xkp+3n+2 + xlp+3n+1 + a

S = sk + Tl + 1 + a

skTl + a ,

(3:19)

From this one can see S = 1. Again, by Equation (3.18), we have T = 1, too. These show that (3.15) is true. The proof for Theorem 3.6 is complete.

Theorem 3.7. The positive equilibrium point of Equation (1.1) is globally asymptotically stable when k (0, 1] and l (-, 0).

The proof of theorem 3.7 is similar to that of theorem 3.6 by virtue of theorem 3.5, theorem 2.7, Lemma (3.1), Lemma (3.2) and Equation (1.1).

3.4 Global asymptotic stability of non-oscillatory solution

Theorem 3.8. The positive equilibrium point of Equation (1.1) is globally asymptotically stable when x-1, x0 (0, 1) and one of the following conditions is satisfied:

(a) -1 <k < 0 and l > 0;(b) 0 <k < 1 and l < 0;(c) k > 0 and -1 <l < 0;(d) k < 0 and 0 <l < 1.

The proof of theorem 3.8 is similar to that of theorem 3.6 by virtue of theorem 2.4, theorem 3.5, Lemma (3.3) and Equation (1.1).

Theorem 3.9. The positive equilibrium point of Equation (1.1) is globally asymptotically stable when x-1, x0 (1, +) and one of the following conditions is satisfied:

(a) -1 <k < 0 and l < 0;(b) 0 <k < 1 and l > 0;(c) k < 0 and -1 <l < 0;

Dongsheng et al. Advances in Difference Equations 2011, 2011:46 http://www.advancesindifferenceequations.com/content/2011/1/46

(d) k > 0 and 0 <l < 1.

The proof of theorem 3.9 is similar to that of theorem 3.6 by virtue of theorem 2.5, theorem 3.5, Lemma (3.4) and Equation (1.1).

Acknowledgements

The authors would like to thank the referees for giving useful suggestions and comments for the improvement of this paper. This research is supported by Social Science Foundation of Hunan Province of China (Grant no. 2010YBB287), Science and Research Program of Science and Technology Department of Hunan Province (Grant no.2010FJ3163, 2011ZK3066).

Author details

1School of Economics and Management, University of South China, Hengyang, Hunan 421001, Peoples Republic of China 2School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, Peoples Republic of China

Authors contributions

All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 31 January 2011 Accepted: 26 October 2011 Published: 26 October 2011

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doi:10.1186/1687-1847-2011-46Cite this article as: Dongsheng et al.: On a class of second-order nonlinear difference equation. Advances in Difference Equations 2011 2011:46.

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