On the nonautonomous difference equation xn+1=An+xn-1pxnq

Applied Mathematics and Computation 217 (2011) 5573–5580

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

p

On the nonautonomous difference equation xnþ1 ¼ An þ

xn1 q

xn

G. Papaschinopoulos ⇑, C.J. Schinas, G. Stefanidou School of Engineering, Democritus University of Thrace, Xanthi 67100, Greece

a r t i c l e

i n f o

a b s t r a c t In this paper we study the asymptotic behavior and the periodicity of the positive solutions of the nonautonomous difference equation:

Keywords: Difference equations Boundedness Persistence Attractivity Periodicity Asymptotic stability

xnþ1 ¼ An þ

xpn1 ; xqn

n ¼ 0; 1; . . . ;

where An is a positive bounded sequence, p, q 2 (0, 1) and x1, x0 are positive numbers. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction In the papers [1–6,9,11,22,24,25,28,30,31] the authors investigated the equation

xn ¼ A þ

xpnk ; xqnm

n ¼ 0; 1; . . . ;

ð1:1Þ

where A, p, q 2 [0, 1) and k, m 2 N, k – m. Moreover there exist many other papers related with Eq. (1.1) and on its extensions (see [13,14,26,27,29,32–35]). In addition in the papers [10,16,23] the authors studied the behavior of the positive solutions of the difference equation

xnþ1 ¼ pn þ

xn1 ; xn

n ¼ 0; 1; . . . ;

ð1:2Þ

where pn is a bounded positive sequence and the initial values x1, x0 are positive numbers. We note that the papers [18–20] are devoted to Eq. (1.2) and on its some extensions. Finally in [21] the authors investigated the behavior of the positive solutions of the difference equation

xnþ1 ¼ An þ

 p xn1 ; xn

n ¼ 0; 1; . . . ;

where An is a bounded positive sequence, p 2 (0, 1) [ (1, 1) and the initial values x1, x0 are positive numbers. Motivated by the above papers we study the attractivity, the periodicity and the stability of the positive solutions of the difference equation

xnþ1 ¼ An þ

xpn1 ; xqn

n ¼ 0; 1; . . . ;

where An is bounded positive sequence, p, q are positive constants and x1, x0 are positive numbers. ⇑ Corresponding author. E-mail addresses: [email protected] (G. Papaschinopoulos), [email protected] (C.J. Schinas), [email protected] (G. Stefanidou). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.12.031

ð1:3Þ

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We note that since difference equations have many applications in applied sciences there are a lot of papers concerning difference equations and their applications (see for example [1–31]). Finally some recent applications of linear as well as nonlinear difference equations are included in the papers [7,8]. 2. Asymptotic behavior of the positive solutions In this section we find conditions so that if  xn is a fixed solution of (1.3) then  xn attracts all solutions of (1.3). Let

yn ¼

xn ; xn

n ¼ 1; 0; 1; . . . :

ð2:1Þ

Then from (1.3) and (2.1) we take that yn satisfies the difference equation

ynþ1 ¼

An þ

p xp y n1 n1 xqn yqn

An þ

xp n1 xqn

ð2:2Þ

:

To prove the first result of this paper we need the following lemmas. Lemma 2.1. Let yn be a particular positive solution of (2.2) . Suppose that there exists an m 2 {0, 1, . . .} such that

y2m1 P 1;

y2m < 1:

ð2:3Þ

Then

yp2n1 > 1;

yq2n < 1;

yq2n1 > 1;

yp2n < 1;

n ¼ m þ 1; m þ 2; . . . :

ð2:4Þ

In addition if

y2m1 < 1;

y2m P 1;

yp2n1 < 1;

yq2n > 1;

ð2:5Þ

then

yq2n1 < 1;

yp2n > 1;

n ¼ m þ 1; m þ 2; . . . :

ð2:6Þ

The proof of Lemma 2.1 follows immediately from (2.2). Lemma 2.2. Consider the function

Fðx; y; zÞ ¼

z þ xy ; zþx

x; y; z > 0:

ð2:7Þ

Then the following statements are true: (i) F is an increasing function in y for any x, z 2 (0, 1); (ii) F is an increasing (resp. decreasing) function in x for any y 2 (1, 1) (resp. y 2 (0, 1)) and z 2 (0, 1); (iii) F is an increasing (resp. decreasing) function in z for any y 2 (0, 1) (resp. y 2 (1, 1)) and x 2 (0, 1). Proof. Statement (i) is obvious. Since

@F zðy  1Þ ¼ ; @x ðx þ zÞ2

@F xð1  yÞ ¼ ; @z ðx þ zÞ2

the proof of statements (ii) and (iii) follows immediately.

h

Using the same argument to prove Proposition 2.1 of [21] and using Theorem 2.6.2 of [15] we take the following lemma. Lemma 2.3. Suppose that An is a bounded sequence such that

0 < m ¼ lim inf An ; n!1

M ¼ lim sup An < 1:

ð2:8Þ

n!1

Suppose also that

0 < p < 1:

ð2:9Þ

Then every positive solution of Eq. (1.3) is bounded and persists. We state now our proposition. Proposition 2.1. Consider Eq. (1.3) where An is a bounded positive sequence such that (2.8) holds. Suppose also that

0 < p þ q < 1;

q > p:

ð2:10Þ

G. Papaschinopoulos et al. / Applied Mathematics and Computation 217 (2011) 5573–5580

5575

Let  xn be a fixed solution of (1.3) and xn be an arbitrary solution of (1.3) . Then

lim yn ¼ 1;

ð2:11Þ

n!1

where yn is defined in (2.1). Proof. Using Lemma 2.3 and relation (2.1) we have

0 < g ¼ lim inf yn ;

h ¼ lim sup yn < 1;

n!1

n!1

0 < k1 ¼ lim inf xn ;

ð2:12Þ

k2 ¼ lim sup xn < 1:

n!1

n!1

We suppose that there exists an m 2 {1, 2, . . .} such that either (2.3) or (2.5) holds. Consider that (2.3) is satisfied. Using (2.2) we obtain that for n P m

A2n þ y2nþ1 ¼

p xp y 2n1 2n1 xq2n yq2n

A2n þ

xp 2n1 xq

xp

;

y2nþ2 ¼

p

y2n

A2nþ1 þ xq 2n

2n

q

y2nþ1

2nþ1

xp

:

ð2:13Þ

A2nþ1 þ xq 2n

2nþ1

Then from Lemmas 2.1 and 2.2, and relations (2.8), (2.12) and (2.13) we have

hgq 6

mgq þ khp ; mþk

p

mhq þ kgp ; mþk

hq g P

k¼

k2 q; k1

and so

mgq hq1 þ khpþq1 P mhq gq1 þ kgpþq1 :

ð2:14Þ

Relations (2.10) and (2.14) imply that

mgq1 hq1 ðh  gÞ 6 kðhpþq1  gpþq1 Þ ¼ kðghÞ1þpþq ðg1pq  h1pq Þ 6 0; and so we have that h = g which implies that there exists limn?1yn. Using (2.4) we have that (2.11) is true. Similarly we can prove that if (2.5) is satisfied then (2.11) holds. Suppose now that neither (2.3) nor (2.5) holds. Then from Lemma 2.1 we get

yn < 1;

or yn P 1;

n P 1:

ð2:15Þ

Without loss of generality we may suppose that

yn < 1;

n P 1:

ð2:16Þ

We claim that

yqnþ1 > ypn ;

n P 1:

ð2:17Þ

Suppose on the contrary that there exists a q

v P 1 such that

p

yv þ1 6 yv :

ð2:18Þ

Then from (2.2) and (2.18) we get xp

yv þ2 ¼

p

yv

Av þ1 þ xq v

q

v þ1 yv þ1 xp Av þ1 þ xq v v þ1

P 1;

which contradicts to (2.16). So (2.17) is true. Using relations (2.10), (2.16) and (2.17) we get

yqnþ1 > ypn > yqn ;

n P 1;

which implies that

ynþ1 > yn ;

n P 1:

ð2:19Þ

Moreover from (2.2) we have for n P 0

jynþ1  1j ¼

xp n1 xqn

 p   p  yn1  yn1     : <  1  1   yq  xp  yq n1 n n An þ xq n

ð2:20Þ

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Using (2.16), (2.17) and (2.20) we get

1  ynþ1 < 1 

ypn1 ; yqn

n P 0:

ð2:21Þ

In addition from (2.16) and (2.19) we have that

lim yn ¼ k 6 1:

ð2:22Þ

n!1

Hence relations (2.21) and (2.22) imply that

kqpþ1 P 1; and so from (2.10) and (2.22) we have that k = 1. Similarly if the second relation of (2.15) is satisfied we can prove that k = 1. This completes the proof of the proposition. h

3. Periodicity and stability In the next proposition we find sufficient conditions for the existence, the uniqueness of 2-periodic and 3-periodic solutions for Eq. (1.3) and the convergence of the positive solutions of (1.3) to the periodic solutions. Proposition 3.1. Consider Eq. (1.3) . Then the following statements are true: (i) Suppose that An is a positive two-periodic sequence such that

Anþ2 ¼ An ;

n ¼ 0; 1; . . . :

ð3:1Þ

Suppose also that (2.10) are satisfied. Then Eq. (1.3) has a unique two periodic solution and every positive solution of (1.3) tends to the unique 2-periodic solution. (ii) Suppose that An is a positive periodic sequence of period three such that

Anþ3 ¼ An ;

n ¼ 0; 1; . . . :

ð3:2Þ

Suppose also that p, q satisfy (2.10) and there exists a positive number p

ðB þ Þ < ; Cq

pq

þ

C 2ðqþ1pÞ

q < h; C

p C qþ1p

2

þ

q



C qþ2p



  and a h 2 0; 12 such that

< h;

ð3:3Þ

where

B ¼ max fA0 ; A1 ; A2 g;

C ¼ min fA0 ; A1 ; A2 g:

ð3:4Þ

Then Eq. (1.3) has a unique periodic solution of period three and every positive solution of (1.3) tends to the unique 3-periodic solution. Proof. (i) First we prove that (1.3) has a unique 2-periodic solution. Let xn be a solution of (1.3). Using (3.1), xn is periodic of period two if and only if the initial values x1, x0 satisfy

x1 ¼ x1 ¼ A0 þ

xp1 ; xq0

x 0 ¼ x2 ¼ A 1 þ

xp0 : xq1

ð3:5Þ

We set x1 = x, x0 = y then from (3.5) we obtain the system of equations

x ¼ A0 þ

xp ; yq

y ¼ A1 þ

yp : xq

ð3:6Þ

Þ,   > 0. From the first relation of (3.6) we get We prove that (3.6) has a solution ð x; y x > 0; y p

y¼

xq 1

ð3:7Þ

:

ðx  A0 Þq

From (3.7) and the second relation of (3.6) we have p

xq

 A1  1

ðx  A0 Þq

x

p2 q2 q p

ðx  A0 Þq

¼ 0:

ð3:8Þ

We consider the function p

f ðxÞ ¼

xq

1

ðx  A0 Þq

 A1 

x

p2 q2 q p

ðx  A0 Þq

:

ð3:9Þ

G. Papaschinopoulos et al. / Applied Mathematics and Computation 217 (2011) 5573–5580

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From (3.9) we get

f ðxÞ ¼

p

1

xq p

ðx  A0 Þq

ðx  A0 Þ

x 1p

p2 q2 q

!  A1 :

ð3:10Þ

q

From relations (2.10), (3.9) and (3.10) we obtain

lim f ðxÞ ¼ 1;

x!A0 þ

lim f ðxÞ ¼ A1 :

ð3:11Þ

x!1

So Eq. (3.9) has a solution  x > A0 . Then if p

¼ y

xq

1

ðx  A0 Þq

;

 is a periodic solution of period two. Finally using Propwe have that the solution  xn of (1.3) with initial values x1 ¼  x; x0 ¼ y osition 2.1 it is obvious that xn is the unique periodic solution of period two and every positive solution of (1.3) tends to the unique periodic solution of period two. (ii) Using (3.2) xn is a 3-periodic solution of (1.3) if the initial values x1, x0 satisfy

x2 ¼ x1 ¼ A1 þ

xp0 ; xq1

x3 ¼ x0 ¼ A 2 þ

xp1 : xq1

ð3:12Þ

We set x1 = x, x0 = y in (3.12) and we take the system of nonlinear equations

x ¼ A1 þ

yp ; ðhðx; yÞÞq

y ¼ A2 þ

ðhðx; yÞÞp ; xq

hðx; yÞ ¼ A0 þ

xp : yq

ð3:13Þ

We consider the function

H : ½A1 ; A1 þ   ½A2 ; A2 þ  ! IR; such that

Hðx; yÞ ¼ ðf ðx; yÞ; gðx; yÞÞ;

f ðx; yÞ ¼ A1 þ

yp ; ðhðx; yÞÞq

gðx; yÞ ¼ A2 þ

ðhðx; yÞÞp : xq

ð3:14Þ

First we prove that H is in [A1, A1 + ]  [A2, A2 + ]. Obviously

A1 < f ðx; yÞ;

A2 < gðx; yÞ;

ðx; yÞ 2 ½A1 ; A1 þ   ½A2 ; A2 þ :

ð3:15Þ

Moreover from (3.3), (3.4) and (3.13) we get for (x, y) 2 [A1, A1 + ]  [A2, A2 + ]

f ðx; yÞ ¼ A1 þ

yp ðA2 þ Þp ðA2 þ Þp ðB þ Þp 6 A1 þ < A1 þ  ; q 6 A1 þ q 6 A1 þ  q p Cq ðhðx; yÞÞ A0 A þx 0

ð3:16Þ

yq

  p p B þ ðBþC qÞ ðhðx; yÞÞp ðB þ Þp gðx; yÞ ¼ A2 þ 6 A þ < A þ < A2 þ  : 2 2 q xq C Cq

ð3:17Þ

Therefore from (3.14)–(3.17) we have that H is in [A1, A1 + ]  [A2, A2 + ]. We prove that H is a contraction in [A1, A1 + ]  [A2, A2 + ]. We prove that

  @f    < h; @x

   @f    < h; @y

  @g    < h;  @x

  @g    < h: @y

ð3:18Þ

From (3.14) we get

@f pq ¼ ; qp 1p @x y x ðhðx; yÞÞqþ1

@f p q 2 xp ¼ 1p ; q þ @y y ðhðx; yÞÞ yqþ1p ðhðx; yÞÞqþ1

@g qðhðx; yÞÞp p2 þ ; ¼ qþ1 q qþ1p @x x y x ðhðx; yÞÞ1p

@g pq : ¼ @y xqp y1þq ðhðx; yÞÞ1p

ð3:19Þ

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From (2.10), (3.3), (3.4), (3.13) and (3.19) we get for all (x, y) 2 [A1, A1 + ]  [A2, A2 + ]

  @f  pq pq  < < 2ðqpþ1Þ < h; @x q2pþ1 qþ1 C C A0   2  @f  p q ðB þ Þp p q2  < þ qpþ1 qþ1 < qpþ1 þ qþ2p < h; @y 1p q C C A0 C C A0  p   p p p   q A0 þ ðA1 þqÞ 2 q B þ ðBþC qÞ A2 @g  p p2 qðB þ Þp p2 q pq  < þ 2qpþ1 1p < þ 2ðqpþ1Þ < þ 2ðqpþ1Þ < < h; þ  @x  qþ1 qþ1 qþ1 C C 2ðqpþ1Þ C C C C C C A0   @g  pq pq  < < 2ðqpþ1Þ < h: @y 2qpþ1 1p A0 C C

ð3:20Þ

Therefore from (3.20) relations (3.18) are true. Moreover there exist ni 2 [A1, A1 + ], gi 2 [A2, A2 + ], i = 1, 2 such that for all x1, x2 2 [A1, A1 + ] and y1, y2 2 [A2, A2 + ]

@f ðx1 ; g1 Þ ðy1  y2 Þ; @y @f ðn1 ; y2 Þ f ðx1 ; y2 Þ  f ðx2 ; y2 Þ ¼ ðx1  x2 Þ; @x @gðx1 ; g2 Þ ðy1  y2 Þ; gðx1 ; y1 Þ  gðx1 ; y2 Þ ¼ @y @gðn2 ; y2 Þ gðx1 ; y2 Þ  gðx2 ; y2 Þ ¼ ðx1  x2 Þ: @x

f ðx1 ; y1 Þ  f ðx1 ; y2 Þ ¼

ð3:21Þ

Relations (3.18) and (3.21) imply that

jf ðx1 ; y1 Þ  f ðx2 ; y2 Þj 6 jf ðx1 ; y1 Þ  f ðx1 ; y2 Þj þ jf ðx1 ; y2 Þ  f ðx2 ; y2 Þj 6 2h max fjx1  x2 j; jy1  y2 jg; jgðx1 ; y1 Þ  gðx2 ; y2 Þj 6 jgðx1 ; y1 Þ  gðx1 ; y2 Þj þ jgðx1 ; y2 Þ  gðx2 ; y2 Þj 6 2h max fjx1  x2 j; jy1  y2 jg; and so

max fjf ðx1 ; y1 Þ  f ðx2 ; y2 Þj; jgðx1 ; y1 Þ  gðx2 ; y2 Þjg 6 2h max fjx1  x2 j; jy1  y2 jg:

ð3:22Þ

  So from (3.22) and since h 2 0; 12 the function H is a contraction in [A1, A1 + ]  [A2, A2 + ]. Hence by Theorem 1.7.1 (Banach Þ 2 ½A1 ; A1 þ   ½A2 ; A2 þ  such that Contraction Principle) (see [15]) there exists a unique ð x; y

x ¼ f ðx; y Þ;

 ¼ gðx; y Þ: y

 is a periodic solution of period three. Using Proposition 2.1 it is Therefore the solution xn with initial values x1 ¼  x; x0 ¼ y obvious that xn is the unique solution of period three and every positive solution of (1.3) tends to the unique 3-periodic solution of (1.3) as n ? 1. This completes the proof of the proposition. h In the last proposition we study the stability of the unique periodic solution of Eq. (1.3). Proposition 3.2. Consider Eq. (1.3) . Then the following statements are true: (i) Suppose that relations (2.10) and (3.1) are satisfied. Suppose also that

p Aq1 A1p 0

þ

p2 þ q2 ðA1 A0 Þqþ1p

þ

p Aq0 A1p 1

< 1:

ð3:23Þ

Then the unique 2-periodic solution of (1.3) is globally asymptotically stable. (ii) Suppose that (2.10), (3.2) and (3.3) hold. Suppose also that

3pq C

2ðpþq1Þ

þ

p3 þ q3 C 3ðpþq1Þ

< 1:

ð3:24Þ

Then the unique 3-periodic solution of (1.3) is globally asymptotically stable. Proof. (i) From Proposition 3.1 there exists a unique periodic solution  xn of Eq. (1.3) of period two. Let

x2n1 ¼ x;

; x2n ¼ y

n ¼ 0; 1; . . . :

From (1.3) we get

x2nþ1 ¼ A0 þ

xp2n1 ; xq2n

x2nþ2 ¼ A1 þ

xp2n : xq2nþ1

ð3:25Þ

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5579

If we set x2n1 = zn, x2n = wn in (3.25) we get

znþ1 ¼ A0 þ

zpn ; wqn

wnþ1 ¼ A1 þ

wpn : zqnþ1

ð3:26Þ

Þ is the positive equilibrium of (3.26) and the linearized system of (3.26) about ð Þ is the system: Then ð x; y x; y

v nþ1 ¼ Bv n ; where:

0 B¼@

1

qxp qþ1 y

p x1p y q p 1p xq y

pq qp xqþ2p y

A;

q2

þ ðxyÞqþ1p

vn ¼



 zn : wn

The characteristic equation of B is the following

k2  k

p q2 þ þ q xq y 1p ðxy x1p y Þqþ1p p

! þ

p2 Þqþ1p ðxy

¼ 0:

ð3:27Þ

 satisfy (3.6) we have   > A1 and so relation (3.23) implies that Since  x; y x > A0 ; y

p q x1p y

þ

p 1p xq y

þ

p2 þ q2 p p2 þ q2 p < q 1p þ þ < 1: qþ1p Þ A1 A0 ðxy ðA1 A0 Þqþ1p Aq0 A1p 1

Þ is locally asymptotically stable. So from Remark 1.3.1 of [15] all the roots of (3.27) are of modulus less than 1. Hence ð x; y Finally using Proposition 3.1 we have that the unique 2-periodic solution of (1.3) is globally asymptotically stable. This completes the proof of the statement (i). (ii) From Proposition 2.1 there exists a unique 3-periodic solution  xn of (1.3). Let

x3n1 ¼ x;

; x3n ¼ y

x3nþ1 ¼ A0 þ

xp ¼ z; q y

n ¼ 0; 1; . . . :

From (1.3) we get

x3nþ1 ¼ A0 þ

xp3n1 ; xq3n

x3nþ2 ¼ A1 þ

xp3n ; xq3nþ1

x3nþ3 ¼ A2 þ

xp3nþ1 ; xq3nþ2

n ¼ 0; 1; . . . :

ð3:28Þ

If we set x3n2 = un, x3n1 = vn, x3n = wn in (3.28) we get

unþ1 ¼ A0 þ

v pn

; wqn

v nþ1 ¼ A1 þ

wpn ; uqnþ1

wnþ1 ¼ A2 þ

upnþ1

v qnþ1

;

n ¼ 0; 1; . . . :

ð3:29Þ

Þ is the positive equilibrium of (3.29) and the linearized system of (3.29) about the equilibrium ðz;  Þ is the Then ðz;  x; y x; y following

0

znþ1 ¼ Tzn ;

0 r1 B T ¼ @ 0 r2 0 r3

s1

1

C s2 A; s3

0

1 un B C zn ¼ @ v n A; wn

where

p qxp pq ; s1 ¼  qþ1 ; r 2 ¼  1p qp qþ1 ; x1p y x y q   z y p q2 xp p2 pq2 s2 ¼ 1p q þ qþ1p qþ1 ; r3 ¼ qþ1p q 1p þ 2þqp qp qþ1p ; z  x  z x  z  z y y y y pq pq q3 : s3 ¼  qp qþ1 1p  1þq 1p qp  x y x y  z  z zÞqþ1p ðxy

r1 ¼

The characteristic equation of the matrix T is

k k2 þ k

pq pq pq q3 þ þ þ qþ1 z1p x1p y qpzqþ1 x1þq y 1p zqp ðxy xqp y zÞqþ1p

! 

p3 zÞqþ1p ðxy

! ¼ 0:

ð3:30Þ

Using (2.10) and (3.24) and Remark 1.3.1 of [15] all the roots of (3.30) are of modulus less than 1. Hence the unique 3-periodic solution of (1.3) is locally asymptotically stable. Finally from Proposition 3.1 the unique 3-periodic solution is globally asymptotically stable. This completes the proof of the proposition. h

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