On the Recursive Sequence

doi:10.1155/2009/327649

Research Article

On the Recursive Sequence

x

_n

1

A

x

p

_n

₋

₁

/x

n

q

C. J. Schinas, G. Papaschinopoulos, and G. Stefanidou

School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece

Correspondence should be addressed to G. Papaschinopoulos,gpapas@env.duth.gr

Received 11 June 2009; Revised 10 September 2009; Accepted 21 September 2009

Recommended by A ˘gacik Zafer

In this paper we study the boundedness, the persistence, the attractivity and the stability of the positive solutions of the nonlinear difference equationxn1 α xp_n−1/xqn,n 0,1, . . . ,where

α, p, q∈0,∞andx−1, x0∈0,∞. Moreover we investigate the existence of a prime two periodic

solution of the above equation and we find solutions which converge to this periodic solution.

Copyrightq2009 C. J. Schinas 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.

1. Introduction

Difference equations have been applied in several mathematical models in biology, economics, genetics, population dynamics, and so forth. For this reason, there exists an increasing interest in studying difference equations see 1–28 and the references cited therein.

The investigation of positive solutions of the following equation

xnA x p n−k

xq_n−m, n0,1, . . . , 1.1

whereA, p, q∈0,∞andk, m∈N,k /m, was proposed by Stevi´c at numerous conferences. For some results in the area see, for example,3–5,8,11,12,19,22,24,25,28.

In 22 the author studied the boundedness, the global attractivity, the oscillatory behavior, and the periodicity of the positive solutions of the equation

xn1ax p n−1

xpn , n

wherea, pare positive constants, and the initial conditionsx−1, x0are positive numberssee also5for more results on this equation.

In 11 the authors obtained boundedness, persistence, global attractivity, and periodicity results for the positive solutions of the difference equation

xn1axn−1

xpn

, n0,1, . . . , _1.3

wherea, pare positive constants and the initial conditionsx−1, x0are positive numbers. Motivating by the above papers, we study now the boundedness, the persistence, the existence of unbounded solutions, the attractivity, the stability of the positive solutions, and the two-period solutions of the difference equation

xn1Ax p n−1

xnq , n

0,1, . . . , 1.4

where A, p,andq are positive constants and the initial values x−1, x0 are positive real numbers.

Finally equations, closely related to1.4, are considered in1–11,14,16–23,26,27, and the references cited therein.

2. Boundedness and Persistence

The following result is essentially proved in22. Hence, we omit its proof.

Proposition 2.1. If

0< p <1, 2.1

then every positive solution of 1.4is bounded and persists.

In the next proposition we obtain sufficient conditions for the existence of unbounded solutions of1.4.

Proposition 2.2. If

p >1 2.2

then there exist unbounded solutions of 1.4.

Proof. Letxnbe a solution of1.4with initial valuesx−1, x0such that

x−1>max

Then from1.4,2.2, and2.3we have

Soxnis unbounded. This completes the proof of the proposition.

3. Attractivity and Stability

In the following proposition we prove the existence of a positive equilibrium.

Proposition 3.1. If either

or

0< p < q 3.2

holds, then1.4has a unique positive equilibriumx.

Proof. A pointx ∈ Rwill be an equilibrium of1.4if and only if it satisfies the following equation

Fx xp−q_{−xA0. 3.3}

Suppose that3.1is satisfied. Since3.1holds and

F_{x p−q x}p−q−1_{−1, 3.4}

we have thatFis increasing in0,p−q1/−pq1andFis decreasing inp−q1/−pq1,∞. MoreoverF0 A >0 and

lim

x→ ∞Fx −∞. 3.5

So if3.1holds, we get that1.4has a unique equilibriumxin0,∞.

Suppose now that3.2holds. We observe thatF1 A >0 and since from3.2and

3.4Fx< 0, we have thatF is decreasing in0,∞. Thus from3.5we obtain that1.4

has a unique equilibriumxin0,∞. The proof is complete.

In the sequel, we study the global asymptotic stability of the positive solutions of1.4.

Proposition 3.2. Consider1.4. Suppose that either

0< p <1< q, A >pq−1 1/q−p1 3.6

or3.1and

0< pq≤1. 3.7

hold. Then the unique positive equilibrium of 1.4is globally asymptotically stable.

Proof. First we prove that every positive solution of 1.4 tends to the unique positive equilibriumxof1.4.

Assume first that3.6is satisfied. Letxnbe a positive solution of1.4. From3.6and

Proposition 2.1we have

0< llim inf

Then from1.4and3.8we get,

L≤AL_lqp, l≥A_Llp_q, 3.9

and so

Llq_≤Alq_Lp_{, lL}q_≥ALq_lp_{. 3.10}

Thus,

ALq_lq−1_lp_lq−1 _≤Alq_Lq−1_Lp_Lq−1_{. 3.11}

This implies that

ALq−1_lq−1_L−l≤Lpq−1_−lpq−1_{. 3.12}

Suppose for a while thatpq−2≥0. We shall prove thatlL. Suppose on the contrary that

l < L. If we consider the functionxpq−1_{, then there exists ac∈l, Lsuch that}

Lpq−1_−lpq−1

L−l

pq−1 cpq−2 ≤pq−1 Lpq−2. 3.13

Then from3.12and3.13we obtain

ALq−1_lq−1_{≤pq−1 L}pq−2 _3.14

or

AL1−p_lq−1_{≤pq−1. 3.15}

Moreover, since from1.4,

L≥A, l≥A, 3.16

from3.6and3.15we get

AA1−p_Aq−1_Aq−p1_{≤pq−1 3.17}

Suppose thatpq−2<0. Then from3.12and arguing as above we get

ALq−1_lq−1_{≤pq−1 l}pq−2_{. 3.18}

Then arguing as above we can prove thatxntends to the unique positive equilibriumx. Assume now that3.7holds. From3.7and3.12we obtain

ALq−1_lq−1_L−l≤ 1

L1−p−q − 1

l1−p−q

l1−p−q_−L1−p−q

L1−p−q_l1−p−q ≤0, 3.19

which implies thatL l. So every positive solutionxnof1.4tends to the unique positive equilibriumxof1.4.

It remains to prove now that the unique positive equilibrium of 1.4 is locally asymptotically stable. The linearized equation about the positive equilibrium x is the following:

yn2qxp−q−1yn1−pxp−q−1yn 0. 3.20

Using13, Theorem 1.3.4the linear3.20is asymptotically stable if and only if

qxp−q−1_<−pxp−q−1_{1<2. 3.21}

First assume that3.6holds. Since3.6holds, then we obtain that

A >pq p−q/q1−pqp−1 . 3.22

From3.6and3.22we can easily prove that

Fc>0, wherecpq 1/q1−p. 3.23

Therefore

x >pq 1/q1−p_{, 3.24}

which implies that3.21is true. So in this case the unique positive equilibriumxof1.4is locally asymptotically stable.

Finally suppose that3.1 and 3.7 are satisfied. Then we can prove that 3.23 is satisfied, and so the unique positive equilibriumxof 1.4satisfies3.24. Therefore 3.21

4. Study of 2-Periodic Solutions

Motivated by 5, Lemma 1, in this section we show that there is a prime two periodic solution. Moreover we find solutions of 1.4 which converge to a prime two periodic solution.

Proposition 4.1. Consider1.4where

0< p <1< q. 4.1

Assume that there exists a sufficient small positive real number1, such that

1

A1q−p

> 1, 4.2

A1p/q₁−1/q< A−₁p/qA1p

2_−q2_/q

. 4.3

Then1.4has a periodic solution of prime period two.

Proof. Letxnbe a positive solution of1.4. It is obvious that if

x−1 A

xp₋₁

xq₀ , x0A xp₀

x₋q₁, 4.4

thenxnis periodic of period two. Consider the system

xAxp

yq, yA

yp

xq, 4.5

Then system4.5is equivalent to

y−A− y_xp_q 0, y xp/q

x−A1/q, 4.6

and so we get the equation

Gx xp/q

x−A1/q −A−

xp2_−q2_/q

x−Ap/q 0. 4.7

We obtain

Gx 1

x−A1/q

xp/q_−xp2_−q2_/q

and so from4.1

lim

x→AGx ∞. 4.9

Moreover from4.3we can show that

GA1<0. 4.10

Therefore the equationGx 0 has a solutionxA0, where 0< 0 < 1, in the interval

A, A1. We have

y xp/q

x−A1/q. 4.11

We consider the function

H Ap−q−. 4.12

Since from4.1H p−qAp−q−1−1<0 and we have

H0> H1. 4.13

From4.2we haveH1>0, so from4.13

H0 A0p−q−0>0, 4.14

which implies that

xA0< A0 p/q

1/q 0

y. 4.15

Hence, ifx−1x,x0y, then the solutionxnwith initial valuesx−1,x0is a prime 2-periodic solution.

In the sequel, we shall need the following lemmas.

Lemma 4.2. Let{x_n}be a solution of 1.4. Then the sequences{x2n}and{x2n1}are eventually

monotone.

Proof. We define the sequence{un}and the functionhxas follows:

Then from1.4forn≥3 we get

Working inductively we can easily prove relations4.20. Similarly if4.19is satisfied, we can prove that4.21holds.

Proposition 4.4. Consider1.4where4.1,4.2, and4.3hold. Suppose also that

A1<1. 4.23

Then every solution xn of 1.4 with initial values x−1, x0 which satisfy either 4.18 or 4.19,

Proof. Letxnbe a solution with initial valuesx−1, x0which satisfy either4.18or4.19. Using Proposition 2.1andLemma 4.2we have that there exist

lim

n→ ∞x2n1 L, nlim→ ∞x2nl. 4.24

In addition from Lemma 4.3we have that either Lor lbelongs to the interval A, A1. Furthermore fromProposition 3.1we have that 1.4has a unique equilibriumxsuch that 1 < x <∞. Therefore from4.23we have thatL /l. Soxnconverges to a prime two-period solution. This completes the proof of the proposition.

Acknowledgment

The authors would like to thank the referees for their helpful suggestions.

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