IJITE Vol.2 Issue-11, (November 2014) ISSN: Impact Factor

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1 IJITE Vol Issue-, (November 4) ISSN: ATTRACTIVITY OF A HIGHER ORDER NONLINEAR DIFFERENCE EQUATION Guagfeg Liu School of Zhagjiagag Jiagsu Uiversit of Sciece ad Techolog, Zhagjiagag, Jiagsu 56,PR Chia Abstract:I this paper, the global asmptotic behavior of the followig ratioal differece equatio r p k,,,,, q k is ivestigated, where the parameters pqr,, (, ), ad the iitial coditios k,, are positive real umbers ad is a positive real umber Kewords: Locall stabilit; Ratioal differece equatio; Global attractivit; Global asmptotical stabilit Itroductio Cosider the followig ratioal differece equatio r p k,,,,, () q k where the parameters pqrare,, positive real umbers ad the iitial coditios,, [, ) ad (, ) k For the equatio r p q id- irjmss@gmailcom Page,,,,, () M R S Kuleovic ad G Ladas i their moograph [, P 8] preseted the followig problem Cojecture 5 Assume that pqr,, (, )The the followig statemets are true for Eq() (a) Local Stabilit of the positive equilibrium implies global stabilit (b) Whe Eq() has o prime period-two solutio, the equilibrium is globall asmptoticall stable

2 IJITE Vol Issue-, (November 4) ISSN: (c) Whe Eq() possesses a period-two solutio, the equilibrium is a saddle poit (d) The period-two solutio of Eq(),whe it exists, is locall asmptoticall stable, but ot globall Motivated b the above cojecture, more geerall, our mai aim i this paper is to ivestigate the local stabilit ad the global attractivit of the equilibrium of Eq() Here, for the sake of coveiece, we ow give some correspodig defiitios, also review some kow lemmas which will be useful i the sequel Cosider the differet equatio x F( x x,, x ),,,, (3), k where is a positive iteger, ad the fuctio F( u, u,, u k ) has cotiuous partial derivatives A poit x is called a equilibrium of Eq(3), if That is x x F( x, x,, x) x for is a solutio of Eq(3) or equivaletl, x is a fixed poit of F The liearized equatio associated with Eq(3) about the equilibrium poit x is Its characteristic equatio is k F x ( x, x,, x) xi,, u i i i k F ( x, x i,, x ),, u I the followig, we preset some Lemmas, which will be used i the sequel Lemma A[] Assume that, {,,, } The i (4) is a sufficiet coditio for the asmptotic stabilit of the differece equatio x x x,,,,, (5) k Suppose i additio that oe of the followig two cases holds: id- irjmss@gmailcom Page

3 IJITE Vol Issue-, (November 4) ISSN: (a) k is odd ad ; (b) k is eve ad ; The (4) is also a ecessar coditio for the asmptotic stabilit of Eq(5) Lemma B[] Let H [[, ),(, )] be a oicreasig fuctio ad let x deote the (uique) fixed poit of H The the followig statemets are equivalet: (a) x is the ol fixed poit of H (, ) ; (b) x is a global attractor of all positive solutio of the equatio with x [, ) Lemma C[] Cosider the differece equatio x H( x),,,,, (6) x x f ( x, x,, x ),,,,, (7), k k r where ki ( i,, r) are positive itegers Deote b k the maximum of k,, kr Also, assume that the fuctio f satisfies the followig hpotheses: r (H) f C[, ) [, ),(, )] ad g r r [[, ),(, )], where g( u, u,, u ) u f ( u, u,, u ) for u (, ) ad u,, [, ) u, ad lim g(, u,, u ) g( u, u,, u ); r u r r r (H) f ( u, u,, u r ) is oicreasig i u,, ur ; (H3) The equatio f ( x, x,, x) has a uique positive solutio x ; (H4) Either the fuctio f ( u, u,, u r ) does ot deped o u or for ever x> ad u, [ f ( x, u, u,, u) f ( x, u, u,, u)]( x x) with [ f ( x, x, x,, x) f ( x, x, x,, x)]( x x) for x x Now, defie a ew fuctio: max xx G( x, ) for x x, Fx ( ) mi xx G( x, ) forx x, id- irjmss@gmailcom Page 3

4 IJITE Vol Issue-, (November 4) ISSN: where k G( x, ) f (, x,, x) f ( x, x,, x, )[ f ( x, x,, x)] (8) The (a) FC[(, ),(, )] ad F is oicreasig i (, ) (b) Assume that F has o periodic poits of prime period two The x is a global attractor of all positive solutios of Eq(7)\ Lemma D[] Let F, H C[(, ),(, )] be oicreasig fuctios i F, H (, ) be such that ad Assume that x is the ol fixed poit of F i (, ) Lemma E[] Cosider the differece equatio F(x)=H(x)=x, [ H ( x) F( x)]( x x) for x H i (, ) The x is also the ol fixed poit of f (, k ),,,, (9) where k {,, }, let I=[a,b] be some iterval of real umbers ad assume that f :[ a, b] [ a, b] [ a, b] is a cotiuous fuctio satisfig the followig properties: (a) f(x,) is oicreasig i each of its argumets; (b) If ( m, M) [ a, b] [ a, b] is a solutio of the sstem the m=m f ( m, m) M ad f ( M, M) m, The Eq(9) has a uique positive equilibrium ad ever solutio of Eq(9) coverges to id- irjmss@gmailcom Page 4

5 IJITE Vol Issue-, (November 4) ISSN: Lemma F[] Let I [ a, b] be a ivariat iterval uder a cotiuous fuctio G(x, ) which is oicreasig i x for each I ad odecreasig i for each x I Assume that I is a uique equilibrium poit of the equatio If the sstem Has exactl oe solutio i Boudess ad Persistece G(, k ),,, (*) x G(, x) ad G( x, ) () I, the is a global attractor with basi I this sectio, we will cosider the boudedess ad persistece of Eq() We have the followig result Lemma Assume that pqr,, (, ) with p q, the ever positive solutio of Eq() is bouded ad persists I Proof Let{ } Eq() that be a arbitrar positive solutio of Eq() The, it follows from ad for, p p p k r k k r p k q q p, q q q q This complete the proof 3 Local stabilit ad Global stabilit k k k r p k p k r rq q p k q k ( q ) p( q ) q I this sectio, we will cosider the global asmptotic stabilit of Eq() where pqr,, (, ), the iitial coditios Eq() has the uique positive equilibrium give b,, [, ) ad (, ) (3) k id- irjmss@gmailcom Page 5

6 IJITE Vol Issue-, (November 4) ISSN: p p r q ( ) 4 ( ) ( q ) The liberalized equatio associated with equatio Eq() about is Z ad its characteristic equatio is ( p q) qr ( p q) r Z,,,,, Z k ( q ) ( q ) k ( p q) qr k ( q ) ( p q) r ( q ) From this ad b lemma A, we have the followig result Theorem 3 Assume that pqr,, (, ) ad the iitial coditios Hold The the followig results are true k,, [, ) ad (, ) (3) () If ( q p) qr,( p q) r ad r ( q ), the the uique positive equilibrium of Eq() is locall asmptoticall stable () If ( q p) r ad ( q p) ( q ) r ( q ), the the uique positive equilibrium of Eq() is locall asmptoticall stable (3) If ( q p) qr ad ( q p) ( q ) r ( q ), the the uique positive equilibrium of Eq() is locall asmptoticall stable (4) If ( q p) r ad ( q p) ( q ) r ( q ), the the uique positive equilibrium of Eq() is ustable Theorem 3 Assume that pqr,, (, ) The the uique positive equilibrium of Eq() is a global attractor provided that oe of the followig coditios is satisfied: i) p >q; ii) p q Proof First, we cosider the case i) p > q The Eq() ca be rewritte as follows: id- irjmss@gmailcom Page 6

7 IJITE Vol Issue-, (November 4) ISSN: r p k k q k Set r uk p u f ( u, u ) ad g( u, u ) u f ( u, u ) k k k qu uk It is eas to verif that the fuctio f satisf the hpotheses of (H)-(H4) of Lemma C Furthermore the fuctio G, which is defied b (6) takes the form r pz r+z+p r++p G(, z) [ ] qz q+z q + k- Next, we will costruct the fuctio F defied b(6) Sice d r pz r z p ( p q) qr( z ) r( z) ( q p) z ( ) dz qz q z ( qz ) ( q z) Case() ( p q) qr( z ) r( z ( q p z ) ) The the fuctio pz r qz is odecreasig i z ad so Also Hece the fuctio F is give b r pz r z p r p r p max z qz q z q q r pz r z p r p r p mi z qz q z q q r p r p N F( ) [ ], q q I order to appl Lemma D to show that is the ol fixed poit of F i (, ) Let id- irjmss@gmailcom Page 7

8 IJITE Vol Issue-, (November 4) ISSN: r p r p N H( ) [ ], q q where N=+k, is a sufficietl large positive iteger From this we kow that the fuctio H( ) C[(, ),(, )] is strictl decreasig i (, ), ad H( ) F( ) Let L=H(M) ad M> is the fixed poit of H ( ), that is to sa, H ( M ) M The r p r p H( L) [ ] N q q L M ad r p r p H( M ) [ ] N q q M L hece N L[ ] M[ ] q L q M N Let N R( x) x[ ] for x (, ) q x The ' N N R( x) [ ] ( ) q x q Sice N is a sufficietl large positive iteger, the for a x (, ), we ca obtai a sufficietl large positive iteger N, such that ( N ) q Thus the fuctio R(x) is strictl decreasig for x (, ) ad M=L So is the uique fixed poit of H ( ) i (, ) For we kow that id- irjmss@gmailcom Page 8

9 IJITE Vol Issue-, (November 4) ISSN: r p r p k r p [ H ( ) F( )]( ) [ ] [( ) ]( ) q q q r p r p ( ) [ ] ( r p) ( r p) q q k ( r p) ( q ) ( q ) B Lemma D, is also the ol fixed poit of global attractor of all positive solutios of Eq() F i (, ) B Lemma C, The is a Case() ( p q) qr( z ) r( z) ( q p) z So the fuctio pz r qz is decreasig ad r pz r z p r p r p max z qz q z q q Also Hece the fuctio F is give b Let r pz r z p r p r p mi z qz q z q q q r p r p k F( ) [ ] q q q r p r p k H( ) [ ] q q where N k, is a sufficietl large positive iteger From this we kow that the fuctio H( ) C[(, ),(, )] is strictl decreasig i (, ), ad H( ) f ( ) B a similar wa to above, we ca obtai that i this case is a global attractor of all positive solutios of Eq() We the stud the case ii)<p < q From Theorem we see that ever solutio { } Eq() is bouded ad persists whe p<q Hece of id- irjmss@gmailcom Page 9

10 IJITE Vol Issue-, (November 4) ISSN: limif ad = lim sup, exist ad are fiite So, for a positive umber, there exists a positive iteger N such that for N Ad moreover, at this time, the fuctio r p z q z is decreasig i for z, (, ) Accordig to Eq(), oe has, for N, There out, oe ca derive r p( ) r p k r p( ) q( ) q ( ) k r p( ) r p( ) q( ) ( ) I view of the arbitrar ature of, oe has This idicates r p r p q q r p ( q ) r p So, ( p) ( p ) Noticig <p, we have Agai, Therefore, The proof is over For the global attractivit of the positive equilibrium poit of Eq(), we also have the followig results Theorem 33 Assume that pqr,, (, ) The the uique positive equilibrium of Eq() is a global attractor provided that oe of the followig coditios is satisfied: r p i) p < q ad ; q p q r qr ii) p < q ad ; q p p( q ) r p Proof We first cosider the case i) p<q ad ; From Theorem, we q p q id- irjmss@gmailcom Page 3

11 IJITE Vol Issue-, (November 4) ISSN: p qr kow I (, ) q p( q ) is a ivariat iterval At this time, we set (, ) r pu v f u v qu v Oe ca see the fuctio f is oicreasig i x ad odecreasig i Eq()ca be writte ito The the sstem ad (, ) r p k,,, f k q k (, ) r p x x f x p x (, ) r px f x px has ol oe solutio =x So, b Lemma F, we see that the uique positive equilibrium of Eq(), the is a global attractor r qr p qr We ow cosider Case ii) p < q ad We kow I (, ) q p p( q ) q p( q ) a ivariat iterval Set is (, ) r pu v f u v qu v Oe ca see the fuctio f is decreasig i both variables From f ( m, m) M ad f ( M, M ) m, ie, ad r pm m M qm m r pm M m, qm M id- irjmss@gmailcom Page 3

12 IJITE Vol Issue-, (November 4) ISSN: we have m=m So, all of the coditios i Lemma E are satisfied Accordigl, i view of Lemma E, Eq() has a uique positive equilibrium, ad ever solutio of Eq() coverges to The proof is complete for case ii) Refereces [] V L Kocic, GLadas, Global behavior of oliear differece equatios of higher order with applicatios, Kluwer Academic Publishers, Dordrecht, 993 [] M R S Kuleovic, G Ladas, Damics of Secod Order Ratioal Differece Equatios, with Ope Problems ad Cojectures, Chapma ad Hall/CRC, [3] Guagfeg Liu, Xiai Li, Global attractivit of a oliear differece equatio, Idia Joural of Mathematics, 5(3), (8), [4] Youhui Su, Watogh Li, Global attractivit of a higher order oliear differece equatio, J Differece Equ Appl (), (5), [5] Xiai Li, Demig Zhu, Global asmptotic stabilit i a ratioal equatio, J Differece Equal Appl, 9 (9), (3), [6] Xiai Li, Demig Zhu, Global asmptotic stabilit of a oliear recursive sequece, Appl Math Letters, 7(7), (4), [7] Xiai Li, Global behavior for a fourth-order oliear recursive sequece, J Math Aal Appl, 5, 3(), id- irjmss@gmailcom Page 3