Asymptotic Properties of Solutions of Two Dimensional Neutral Difference Systems

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1 Avilble t Appl Appl Mth ISSN: Vol 8, Iue 2 (December 21), pp Applictio d Applied Mthemtic: A Itertiol Jourl (AAM) Aymptotic Propertie of Solutio of Two Dimeiol Neutrl Differece Sytem Thigr Revthi Deprtmet of Mthemtic Quee Mry College Chei-6 4, Tmil Ndu, Idi klyrevthi@yhoocom Received:September 7, 212; Accepted: July 8, 21 Abtrct I thi pper we obti ufficiet coditio for the ymptotic propertie of olutio of two dimeiol eutrl differece ytem Our reult exted ome exitig reult i the literture A exmple i give to illutrte the reult Keyword: Differece ytem, eutrl type, Noocillotory, ymptotic AMS-MSC 21 No: 9A1 1 Itroductio The problem of ocilltio d oocilltio of ecod order olier differece equtio i of prticulr iteret becue they re dicrete logue of ecod order differetil equtio with phyicl pplictio (Kocic et l (199), Pott (1981)) It i iteretig problem to extet ocilltio criteri for ecod order olier differece equtio to the ce of olier two dimeiol differece ytem ice uch ytem iclude, i prticulr, ecod order olier, hlf lier d quilier differece equtio pecil ce I the qulittive theory of differece equtio, ocilltory d oocilltory behvior of olutio ply importt role Further thee type of olutio re ocited with my phyicl d biologicl pheome uch vibrtig mechicl ytem, electricl circuit d 585

2 586 Thigr Revthi popultio dymic Thdpi et l (1996), coidered the followig eutrl differece equtio of the form ( ( x p x )) q ( x )=, N( ) k 1l They obtied the ocilltio criteri of the equtio Further they clified ll oocilltory olutio of the me equtio ito four cle d etblihed coditio for the exitece d oexitece of olutio i thee cle Szfrki et l (199) coidered the two dimeiol differece ytem of the form ( x ) = b y y = f( x ) N( ) They etblihed the coditio for the ocilltio d ymptotic behvior of olutio of the bove ytem Gref et l (1999) coidered the two dimeiol differece ytem of the form ( x) = b g( y) y = f( x ) N( ) They obtied coditio for ll olutio of the ytem to be ocilltory Huo d Li (21) coidered the followig Emde-Fowler differece ytem ( x) = b g( y) y = f( x ) r N( ) They etblihed ome criteri for the ocilltio of the ytem Motivted by the bove, i thi pper we tudy the ymptotic behvior of two dimeiol olier differece ytem of eutrl type Coider the olier two-dimeiol differece ytem of eutrl type x ) = p y (1) ( x ( ) y = q f( x ), N( ) ( ) d ( ) = {, 1, }, o-egtive iteger The followig coditio re umed:

3 AAM: Iter J, Vol 8, Iue 2 (December 21) { } i poitive rel equece { ( )} d { ( )} re equece of iteger d re icreig with lim ( ) = lim ( ) = 2 { p } d { q } re oegtive rel equece with q for ifiitely my vlue of d p = = f : i cotiuou, uf (u) > for u d f ( u) K u where K i poitive cott Let = mx{if > ( ),if > ( )} By olutio of the ytem (1) we me rel equece X ({ x },{ y }), which i defied for ll d tifie the ytem (1) for ll = N( ) Deote by W the et of ll olutio X ({ x },{ y }) of the ytem (1) which exit for N( ) d tify = up{ x y : N} > for y iteger N N A rel equece defied o N ( N) i id to be ocilltory if it i either evetully poitive or evetully egtive d oocilltory otherwie A olutio X W i id to be ocilltory if both compoet re ocilltory d it will be clled oocilltory otherwie Some ocilltio reult for differece ytem (1) whe = for N( N) d ( ) = hve bee preeted i Gref et l (1999), Huo d Li (21) I prticulr whe p > for ll, N( ) the differece ytem reduce to the ecod order olier differece equtio 1 ( x x( ) ) = qf( x( ) ) p (2) Alo if p = 1 for N ) d if f ( u) = u gu, the bove equtio become ( 1 ( x x ( ) ) = x ( ) gx ( ) p () The ocilltory d ymptotic behvior of the equtio of type (2) d () re tudied by Hoker d Ptul (198), Steml et l (1998), Thdpi (1992), Thdpi et l (1995), Zg (199)

4 588 Thigr Revthi 2 Some Ueful Lemm Deote A (, ) = 1 p t, > For y x we defie z by z = x x (4) ( ) We begi with the followig lemm Lemm 1 Let (C1) - (C) hold d let X = ({ x},{ y}) W be olutio of the ytem (1) with { x } either evetully poitive or evetully egtive for ll N 1 N( ) The ({ x },{ y}) i oocilltory d { z }, { y } re mootoe for N( N1) Proof: Let X = ({ x},{ y}) W d let { x } be evetully poitive The from the ecod equtio of the ytem (1) we hve y for ll N 1 N( ) d y d y re ot ideticlly zero for ifiitely lrge vlue of Hece, { y } i either evetully poitive or evetully egtive N 2 The, ({ x },{ y}) i oocilltory Further from the firt equtio of the ytem (1) we hve z > or z < evetully Hece, { z } i mootoe for ll N N2 The proof i imilr whe { x } i evetully egtive Lemm 2 I dditio to coditio (C1) - (C) ume tht oocilltory olutio of the iequlity 1 for ll N( ) Let { x } be x ( x () x ) > (5) for ufficietly lrge Proof: 1 If ( ) = k for N( ) where k i poitive iteger the { x } i bouded Moreover if 1 <, N( ) for ome cott the lim x = 2 If ( ) = k for, the there exit poitive cott C uch tht x C for ll lrge

5 AAM: Iter J, Vol 8, Iue 2 (December 21) 589 Let { x } be oocilltory olutio of the ytem (1) Without lo of geerlity we my ume tht { x } be evetully poitive olutio of the iequlity (5), the proof for the ce { x } evetully egtive i imilr Aume tht 1 1 Let x > for ll N( ) I view of 1 d ( ) = k we hve 1 xk < x x for ll lrge, which implie { x } i bouded If 1 <, 1 1 N( ) hold for ome poitive cott, the we hve x k < x < x The, 1 x k < lim x which implie tht lim x = 2 Let { x } be oocilltory olutio of the ytem (1) Let x > for ll 1 N( ) I view of 1 d ( ) = k we hve x xk > Thi implie x k > x, which implie tht there exit cott C > uch tht x C for ll lrge Aymptotic Behvior I thi ectio we preet ufficiet coditio for the ymptotic behvior of olutio of the ytem (1) Theorem Aume tht 1 <, ( icott) (6) ( ) = k d ( ) = l with k < l, (7) 1 limupk A(, 1) q / kl>1, (8) = kl d q = (9) p = = kl The, for every oocilltory olutio X = ({ x },{ y }) W, lim x = lim y = hold

6 59 Thigr Revthi Proof: Let ({ x },{ y }) W be o ocilltory olutio of the ytem (1) Without lo of geerlity we my uppoe tht x > for ll N ) By the ecod equtio of the ytem (1) d the hypothee, we hve y for ll N ) 1 ( 2 ( 1 I view of Lemm 1, we hve two ce for ufficietly lrge N ) ; (I) y >, for ; (II) y <, for Ce (I) We coider two poibilitie (A): Let z < for 4 where N ) i ufficietly lrge 4 ( We prove lim z = Sice z } i o-decreig { ( 2 limz = L, L> i cott (1) d z L for 4 Sice x >, by (4), we hve zkl xl <, 4, kl d L z x, (11) kl kl l 4 By the hypothee d the ecod equtio of (1) KLq k l Kq x l q f ( xl ) = y for 4 (12) Summig (12) from to * d the tkig * we obti KL = q k l y, 4

7 AAM: Iter J, Vol 8, Iue 2 (December 21) 591 Multiplyig the lt iequlity by p d uig the firt equtio of the ytem (1), we hve KLp q = z, 4 k l Summig the lt iequlity from N to 1 d lettig, we obti KL q p N L z N < = = k l z N Thi cotrdict (9) coequetly lim z = Sice { z } i bouded, there i cott B > uch tht z B for 4 d by (4), oe h x = 4 xk z xk B xk B for (1) We clim { x } i bouded Let { x } be ubouded the { x k} i ubouded d there i equece { } uch tht d lim x k = d x k = mx k x By (1), 4 x k x k B, x k B, 1 = 1,2, Thi i cotrdictio to lim x =, d, hece, x } i bouded k { Next we clim tht lim x = Let lim x k = c > The, lim up x = c Let { }, = 1,2, be ubequece uch tht lim = d lim up x = c The, lim c By (4), we hve z k x x z x x k, 1,2, d x k, = 1,2, By the lt iequlity, we hve

8 592 Thigr Revthi c = limup x k limup x c, which hold whe 1, cotrdictio to (6) Thi me tht lim x k = d lo limup x = Further, x > hold for N( ), o lim if x = d thi led to lim x = Next we prove lim y = Let lim = D > The, y y D for (14), Summig the firt equtio of ytem (1) from to 1 d uig (4) d (14), we hve 1 = z z D p (15) By (15) d the hypothei, lim y = lim z =, which cotrdict the fct z < for 4 Hece, (B): Let z > for 4 where 4 N( ) i ufficietly lrge By Lemm 2, lim x = hold We prove imilrly i the bove proof tht lim y = The reltio (15) implie lim z = Therefore, by (4) we hve z < x for d tht cotrdict lim x = Hece, lim y = Ce (II) I thi ce, y L for, Li poitive cott (16) Summig the firt equtio of the ytem (1) from to 1, d uig the iequlity (16), we hve 1 = z z L p (17) By (17) d the hypothei, it follow tht lim z = d z < for 4, where 4 N( ) i ufficietly lrge By (4) we hve z > x k, 4 The,

9 AAM: Iter J, Vol 8, Iue 2 (December 21) 59 zkl xk >, 4 (18) kl Sice f ( u) k u, for u = x k, we hve f( x ) x, K l (19) l 4 Multiply (19) by Kq d uig (18) we hve zkl Kq Kqxl q f ( xl), 4 (2) kl Uig the ecod equtio of the ytem (1) d (2), we hve A(, t 1) yt = A(, t 1) qt f ( xt l ) KA( ; t 1) q t z tk l tk l, for > 4 (21) Uig ummtio by prt formul we hve 1 A( t, 1) yt = z z A (, ) (22) Combiig (21) d (22), we hve 1 1 tkl A(, t1) yt = z z A(, ) y K A(, t1) qt > 4 (2) tkl z Sice z <, y <, d A (, t), > 4, we hve z z K A t q 1 tkl (, 1) t, > 4 (24) tkl Let = k l d uig the fct tht z k l < d o-icreig, by (24), we obti 1 qt 1 K A(, t 1) qt, 5 k l tk l Thi cotrdict (8) The ecod prt of the proof for the ce x < evetully i imilr to the previou oe d hece the detil re omitted Hece, the proof i complete

10 594 Thigr Revthi Exmple 4 Coider the differece ytem x ( x 1) = 2( 1) y y = x 1 (25) Here =, ( ) = 1, p = 2( 1), q =, ( ) =, f ( u) = u, K = 1, A (, ) = ( 1) ( 1) The ll the coditio of Theorem re tified Hece, every oocilltory olutio ({ x },{ y}) of (25) tifie lim x = lim y = 4 Cocluio The ufficiet coditio for the ymptotic behvior of the two dimeiol eutrl differece ytem hve bee dicued The exmple coidered i thi work upport the reult of the theorem REFERENCES Agrwl, RP (2) Differece Equtio d Iequlitie, Mrcel Dekker, New York Gref, JR d Thdpi, E (1999) Ocilltio of two dimeiol differece ytem, Comput Mth Appl, 8, pp Hooker, JW d Ptul, WT (198) A ecod order olier differece equtio, Ocilltio d ymptotic behviour, J Mth Al Appl, 91, pp 9 29 Hou, HF d Li, WT (21) Ocilltio of the Emde - Fowler differece ytem, J Mth Al Appl, 256, pp Kocic, Vd Ld, G (199) Globl Behviour of Nolier Differece equtio of Higher Order with Applictio, Kluwer Publiher, Dordrecht Li, WT (21) Clifictio cheme for oocilltory olutio of two dimeiol o lier differece ytem, Comput Mth Applic, 42, pp Ocl, O (26) Ocilltio criteri for ytem of differece equtio with vrible, coefficiet, Appl Mth E-ote, 6, pp Pott, R (1981) Exct olutio of differece pproximtio to Duffig equtio, J Autrl Mth Soc (Serie B) 2, pp Steml, A, Szfrki, Z d Zmd, B (1998) Ocilltory d ymptotic behviour of ome differece equtio, Publ It Mth, 6, pp Szfrki, Z d Szmd, B (199) Ocilltory propertie of ome differece ytem, Rd Mt, 6, pp Thdpi, E (1992) Aymptotic d ocilltory behvior of olutio of ecod order olier eutrl dely differece equtio, Riv Mt Uiv prm 1, pp 15 11

11 AAM: Iter J, Vol 8, Iue 2 (December 21) 595 Thdpi, E, Sudrm, P, Gref, JR d Spike, PW (1995) Aymptotic propertie of olutio of olier ecod order eutrl dely differece equtio, Dym Sytem Appl, 4, pp Thdpi, E d Mohkumr, P (27) Ocilltio of differece ytem of eutrl type, Comput Mth Appl, 54, pp Thdpi, E d Pomml, B (2) Aymptotic behvior of olutio of two dimeiol differece ytem, Bull Clcutt Mth Soc, 95, pp Thdpi, E d Arul, R (1997) Ocilltio d oocilltio theorem for cl of ecod order olier differece equtio, Z Al Aweduge, 16, pp 1 11 Thdpi, E, Muel, MMS d Agrwl, RP (1996) Ocilltory d oocilltory Behviour of ecod-order eutrl dely differece equtio, Mthl Comput Modellig, 24, pp 5 11 Zg, BG (199) Ocilltio d ymptotic behviour of ecod order differece equtio, J Mth Al Appl, 17, pp 58 68