On the Asymptotic Behavior of Second Order Quasilinear Difference Equations

Transcription

1 Applied Mthemtic, 6, 7, 6-63 Publihed Olie Augut 6 i SciRe. O the Aymptotic Behvior of Secod Order Quilier Differece Equtio Vdivel Sdhivm, Po Sudr, Amli Sthi PG d Reerch Deprtmet of Mthemtic, Thiruvlluvr Govermet Art College, Ripurm, mkkl, Idi Om Murug College of Art d Sciece, Slem, Idi Received 5 Jue 6; ccepted 6 Augut 6; publihed 9 Augut 6 Copyright 6 by uthor d Scietific Reerch Publihig Ic. Thi work i liceed uder the Cretive Commo Attributio Itertiol Licee (CC BY. Abtrct I thi pper, we ivetigte the ymptotic behvior of the followig quilier differece equtio ( y y p y y where = {, +, +, }, =, (E. We clified the olutio ito ix type by me of their ymptotic behvior. We etblih the ecery d/or ufficiet coditio for uch equtio to poe olutio of ech of thee ix type. Keyword Aymptotic Behvior, Poitive Solutio, Homogeeou, Quilier Differece Equtio. Itroductio Recetly, the ymptotic propertie of the olutio of ecod order differetil equtio [] [] differece equtio of the type (E d/or relted equtio hve bee ivetigted by my uthor, for exmple ee, [3]-[9] d the referece cited there i. Followig thi tred, we ivetigte the exitece of thee ix type of olutio of the Equtio (E howig the ecery d/or ufficiet coditio c be obtied for the exitece of thoe olutio. For the geerl bckwrd o differece equtio, the reder i referred to the moogrph []-[4]. I 996, PJY Wg d R.P. Agrwl [5] coidered the quilier equtio ( ( y σ q f ( y + = ( How to cite thi pper: Sdhivm, V., Sudr, P. d Sthi, A. (6 O the Aymptotic Behvior of Secod Order Quilier Differece Equtio. Applied Mthemtic, 7,

2 d obtied ocilltio criteri for the Equtio (. I 996, E. Thdpi, M.M.S. Muel d R.P. Agrwl [6] hve tudied the qui-lier differece equtio ( x x q f ( x + = (. I, Po Sudrm d E. Thdpi [7] coidered the followig qui-lier fuctiol differece equtio ( y y f y( σ +, = (3 d they hve etblihed ecery d ufficiet coditio for the olutio of Equtio (3 to hve vriou type of oocilltory olutio. Further they hve etblihed ome ew ocilltio coditio for the ocilltio of olutio of Equtio (3. I 997, E. Thdpi d R. Arul [8] tudied, the followig qui-lier equtio ( pφ ( y f( y+ +, =. (4 They etblihed ecery d ufficiet coditio for the olutio of (4 to hve vriou type of oocilltory olutio. I 4, E. Thdpi et l. [9] tudied the equtio ( ( k ( + y py + q f y =,, d etblihed coditio for the exitece of o-ocilltory olutio. S.S. Cheg d W.T. Ptul [3] tudied the differece equtio ( y p p k y k k (5 + = (6 where p > d proved exitece theorem for Equtio (6. I, M. Mizukmi et l. [] dicued the ymptotic behvior of the followig equtio y y = p t y y. (7 Dicrete model re more uitble for udertdig the problem i Ecoomic, geetic, popultio dymic etc. I the qulittive theory of differece equtio ymptotic behvior of olutio ply vitl role. Motivted by thi, we coider the dicrete logue of (7 of the form ( y ( y ( p ( y ( y ( where = {, +, +, }, y = y( + y. = (8 d i the forwrd differece opertor defied by We ume the followig coditio o Equtio (8 d re poitive cott { p( } i rel equece uch tht p( > for ll >. For implicity, we ofte employ the ottio γ * γ iterm of which Equtio (8 c be expreed i γ x = x x = x g x, x R, γ >, * * ( y ( = p ( ( y (. By olutio of Equtio (8, we me rel equece y : R, together with y y exit d tifie Equtio (8 for ll. We here cll Equtio (8 uper-homogeeou or ub-homogeeou ccordig < or > If = Equtio (8 i ofte clled hlf-lier. Our ttetio i mily pid to the uper-homogeeou d ub-homo- 63

3 geeou ce, d the hlf-lier i lmot excluded from our coidertio.. The Clifictio of All Solutio of Equtio (8 To clify ll olutio of Equtio (8, we eed the followig lemm. Lemm. Let y( be locl olutio of Equtio (8 er = d [, w, mximl itervl of exitece. The we hve either y er w or y er w. Tht i w, be it right y doe ot chrge trictly it ig ifiitely my time w. The clifictio of ll (locl olutio of Equtio (8 re give o the bi of Lemm. Sice the proof i ey, we leve it to the reder. Propoitio. Ech locl olutio y / of Equtio (8 fll ito exctly oe of the followig ix type. Sigulr olutio of the firt kid: type ( S there exit uch tht d y / for, d y for. Decyig olutio: type (D, y( c be cotiued to, d tifie y y lim y =. < for ll lrge, 3 Aymptoticlly cott olutio: type (AC y( c be cotiued to, d tifie y y < for ll lrge d R { } lim y. 4 Aymptoticlly lier olutio: type (AL y( c be cotiued to d tifie y y ll lrge d y lim R { }. > for 5 Aymptoticlly uper-lier olutio: type (AS y( c be cotiued to d tifie y y > for ll lrge d 6 Sigulr olutio of ecod kid: type uch tht y lim = ±. S y h the fiite ecpe time; tht i, there exit > lim y = ±. 3. Mi Reult for the Super-Homogeeou Equtio Before we lit our mi reult for the ce <. Throughout thi ectio we ume tht <. Theorem. Equtio (8 h o olutio of type ( S. Theorem 3. Equtio (8 h olutio of type (D if d oly if Theorem 4. Equtio (8 h olutio of type (AC if d oly if Theorem 5. Equtio (8 h olutio of type (AL if d oly if p( =. (9 p( <. ( <. p ( 64

4 Theorem 6. Equtio (8 h olutio of type (AS if ( hold. Theorem 7. Equtio (8 doe ot hve olutio of type (AS if there re cott tifyig d ρ σ + σ ρ limif ( > Remrk. The et of ll pir (, (, (, ρ > d σ (, p ( σ + σ ρ σ σ ρ ρτ tifyig iequlitie (3 i ot empty. I fct, the pir + ( ρσ, =, belog to it Theorem 8. Equtio (8 h olutio of type ( S. Remrk. Theorem 7 h the me cocluio tht thee re ot olutio of type (AS. However, Theorem 7 i till vlid for the ce tht p i oegtive. For exmple, it i formed by thi exteded verio of Theorem 7 tht the equtio doe ot hve olutio of type (AS. Exmple Let ( y y = + t y y, <, coider the Equtio (8 with p = ( σ y y = y y, d σ R. (4 For thi equtio, we hve the followig reult: Equtio (4 h olutio of type (D if d oly if σ (Theorem 3. Equtio (4 h olutio of type (AC if d oly if σ < (Theorem 4. 3 Equtio (4 h olutio of type (AL if d oly if σ < (Theorem 5. 4 Equtio (4 h olutio of type (AS if d oly if σ < (Theorem Mi Reult for the Sub-Homogeeou Equtio Below we lit our mi reult for the ce >. Throughout thi ectio we ume tht >. Theorem 9. Equtio (8 h olutio of type ( S. Theorem. Equtio (8 h olutio of type (D if Theorem. Equtio (8 doe ot hve olutio of type (D if (3 p( <. (5 + lim if p. > (6 Theorem. Equtio (8 doe ot hve olutio of type (D if there re cott tifyig d σ σ ρ ρ > d σ (, ρ limif + p > (7 σ + σ + ρ σ σ + ρ. Remrk 3. The et of ll pir ( ρσ, (, (, tifyig iequlitie (8 i ot empty. I fct, the (8 65

5 pir +, =, ( ρσ belog to it. Theorem 3. Equtio (8 h olutio of type (AC if d oly if (5 hold. Theorem 4. Equtio (8 h olutio of type (AL if d oly if p <. Theorem 5. Equtio (8 h olutio of type (AS if d oly if Theorem 6. Equtio (8 h o olutio of type =. p (9 S. Exmple. Let > d coider the Equtio (4 gi. We hve the followig reult: Equtio (4 h olutio of type (D if d oly if σ < (Theorem d. Equtio (4 h olutio of type (AC if d oly if σ < (Theorem 4. 3 Equtio (4 h olutio of type (AL if d oly if σ < (Theorem 5. 4 Equtio (4 h olutio of type (AS if d oly if σ (Theorem Auxillry Lemm I thi ectio, we collect xillry lemm, which re mily cocered with locl olutio of Equtio (8. A comprio lemm of the followig type i ueful, d will be ued i my plce. Lemm. Suppoe tht { pp },{ pq( } re uch tht < p < p for b. Let yi (, i =, d b be olutio of the equtio ( i i i i i y y = p y y, i =, repectively. If y ( y ( d y ( y (, the y < y d y y b. Proof. We hve * * * ( i ( i i( i < for < y = y + p y, b, i =, ( * * = + ( + (,, =,. ( yi yi yi pi r yi r b i By the hypothee we hve y < y i ome right eighborhood of. If y y poit i < b, we c fid c uch tht < c b tifyig y < y for < < c d y ( c y ( c But, thi yield cotrdictio, becue = y c y c Hece we ee tht y y = + + c * * y( y( ( y( p( ( y( * * ( y( + p( ( y( <. < for b b. The proof i complete. for ome =. <. Returig to (, we fid tht y y < for 66

6 , y d = provided tht y + y. The uiquee of the trivil olutio c be cocluded for the ce. Lemm 3. Let d. If y( i locl olutio of Equtio ( tifyig y( = y( = the y for <. Proof. Aume the cotrry. We my uppoe tht y / for <. The, we c fid, uch tht < tifyig y + y = d y + y > for <. Summig (8, we obti The uiquee of locl olutio with o-zero iitil dt c be eily proved. Tht i, for give ( y, Equtio (8 h uique locl olutio y( tifyig y( = y, y( y We therefore hve y = p( ( y( *, y = p( r ( y( r *,. y p( { y( + y( } ( { },. (3 y p r y r + y r Put w = mx y( ξ + y( ξ. We ee tht w( w ξ ig. From ( d (3, we c get y ( w p( =, > for < d w i odecre-, y ( w p( r,. Let τ. The from thi obervtio we ee tht where Coequetly, we hve If =, from (4, we hve G (4 we hve ( w G ( τ τ τ ( τ (, y + y w G w G v v G v p p r = + ( w w G,. (4,, <. Thi i cotrdictio becue. G =. If <, from <. Thi i lo cotrctio becue G( w( + = + =. The proof i complete. Lemm 4. Let. The ll locl olutio of Equtio (8 c be cotiued to d, tht i, ll olutio of Equtio (8 exit o the whole itervl [,. Proof. Let y( be locl olutio of Equtio (8 i eighborhood of. Suppoe the cotrry tht the right mximl itervl of exitece of y( i of the form [, w, w <. The, it i eily ee tht y w = ±. Summig (8 twice, we hve 67

7 where c = y( d c y( Put z = mx y( ξ ξ y c c p r y r * = ( * + + =. Accordigly, y c + c + p( r y( r, < w.. The, Put moreover u mx c, z y c + c + z( p( r, < w. { } =. The, i the proof of Lemm 3, we hve, ( where H = + p(. Sice ( c for w z Let ext let >. The (6 implie tht z c + H u < w (5 y w = ±, there i uch tht < < w uch tht <. Therefore it follow from (5 tht ( u c + H u, < w. (6 =. The, uig dicrete Growll iequlity, we ee tht u( w ( u c + u H, < w. <, we hve u( w <. Thi i cotrdictio too. Hece <, which i cotrdictio. Sice y c be cotiued to. The cotiubility to the left ed poit i verified i imilr wy. The proof i complete. The followig lemm etblihe more th i tted i Theorem 8. Accordigly the proof of Theorem 8 will be omitted. Lemm 5. Let < d d c > be give. The there exit M= M( c, > uch tht the right mximl itervl of exitece of ech olutio y( of Equtio ( tifyig y( c d y( M i fiite itervl,, = y <, d lim y =. Proof. Let > be fixed, d put mi p ( = m >. There i M > tifyig ( (( + + c M + m v c <. = doe ot z ex- We firt clim tht the olutio of Equtio (8 with the iitil coditio z = c, z M exit o [, ; tht i z( blow up t ome (, ]. To ee thi uppoe the cotrry tht it t let [,. By the defiitio of m, we hve Summig the iequlity form to yield d hece ( z p ( z m( z =,. ( ( (, z M m z + C 68

8 ( z M + m z C ( ( z M + m z C ( z M + m z C,. Filly, ummig the bove iequlity both ide from to, we obti z (( M + m w C, V. Sdhivm et l. o the commo itervl of exi- which i cotrdictio to the choice of M. Hece z( mut blow up t ome (, ], lim z If y( c d y( M, the Lemm implie tht y z tece of y d z d therefore. The proof i complete. y blow up t ome poit before 6. oegtive oicreig Solutio The mi objective of thi ectio i to prove the followig theorem. Theorem 7. For ech y >, the problem ( y y p y y y = y = h exctly oe olutio y uch tht y i defied for d tifie the Furthermore, if y i olutio for of Equtio ( tifyig y( =. y, y for. (7 y > y rep y < y, lim y = rep lim y =. = y d Remrk 4. I the ce, employig Lemm 3, we c tregthe (7 to the property tht y >, y < for. (8 I the ce, ll locl olutio of Equtio (8 c be cotiued to the whole itervl [, Hece i thi ce property (6. lwy hold for ll olutio y( with y( = y d y( > y( [rep y( < y( ]. The property of oegtive oicreig olutio y decribed i Theorem 7 will ply importt role through the pper. Thi ectio i etirely derided to provig Theorem 7. To thi ed we prepre everl lemm. Lemm 6. Let AB, R d t be bouded fuctio o [ b, ] R. The, the two poit boudry vlue problem h olutio. Proof. Let K > be cott uch tht ( (,, y y = f, y, b, y = A y b = B (9 69

9 ( ( [ ] f y, K, for y, b, R. We firt clim tht with ech y(, we c ocite uique cott c( y tifyig Further thi c( y tifie ( b c y + f r, y r = B A. (3 * * B A B A K( b + c( y K( b + b b To ee thi let y( be fixed, d coider the fuctio B A < + b If λ K( b * b ( I λ = λ + f r, y r, λ R. λ <. If λ K( b, the I B A B A > + b *. (3, the I( λ > B A. Sice I i trictly icreig cotiuou fuctio, there i uique cott c( y tifyig ( mely (3. The (3 i clerly tified. I c y = B A, By (3, we ee tht there i cott M= M( babk,,,, > tifyig c( y M for ll y(. Chooe L > o lrge tht A L d M + K b b L. Coider the Bch pce B of ll rel equece y = { y } with the uperum orm y up y ow we defie the et d y B d the mppig F : Y B by Y = y B : for y L b ( ( F y = A+ c y + f r, y r, b =. repectively. The the boudry vlue problem (9 i equivlet to fidig fixed elemet of. We how tht F h fixed elemet i Y (vi the Schvder fixed poit theorem Hece F mp Y ito itelf. F y A c y f r y r ( + + (, A + M + K A + M b b L+ L = L, b. ext, to ee the cotiuity of F, ume tht yk ( be equece covergig to y Y uiformly i [, ] We mut prove tht F( yk ( coverge to Fyk ( uiformly i [, ] lim c( yk = c( y. Aume tht thi i ot the ce. The becue of the boudede of { ( k } there i ubequece c y ( tifyig cy c c ( y for ome fiite vlue c. otig the reltio { ( k i } b ( k k k i * c y + f r, y r = B A. i i b. b. A firt tep, we how tht c y, 6

10 We hve ( k k ( b B A= lim c y f r, y r K i i i + b b = c + f r, y r. Thi cotrdict the uiquee of the umber c( y. Hece lim c( y c( y tht lim F( yk = F( y uiformly o [ b, ]. It will be eily ee tht the et re uiformly bouded o [, ] { } { } ( FY = Fy : y Y d Λ F y : y Y =. The we fid imilrly b. The FY i compct. From the bove obervtio we ee tht F h fixed elemet i Y. The thi fixed elemet i olutio of boudry vlue problem (9 i eily proved. The proof i ow complete. Lemm 7. Let > d >. The the two poit boudry vlue problem (, y y = p y y for y = y y = h olutio y( uch tht y d y Proof. Defie the bouded fuctio f o [, ] R by By Lemm 6, the boudry vlue problem h olutio y. for. p y for, y y f( y, = p y for, y y for, y. ( (, y y = f, y, y = y y = We how tht y tifie y for [ τ, τ] [, ] uch tht y( < o [ τ, τ ] d y( τ y( τ tifie the equtio ( y y = o τ τ. Hece y( i lier fuctio o [, ] Obviouly tht thi i cotrdictio. We ee therefore tht y o [, ]. Sice y( d y y o [, ], by the defiitio of t, we fid tht y o [, ]. Hece y y (3. If thi i ot the ce, we c fid itervl = =. The defiitio of f implie tht y τ τ., which implie tht y i deired olutio of problem (3. The proof i complete. Proof of Theorem 7. The uiquee of y tifyig the propertie metored here i eily etblihed i the proof Lemm. Therefore we prove oly the exitece of uch y. By Lemm 7, for ech k, we hve olutio y = y of the boudry vlue problem tifyig yk d yk ( y y p y y,, for = y = y y + k = + k for + k let u exted ech k y over the itervl [, 6

11 by defiig yk for k deired olutio of (8. A firt tep, we prove tht +. Below we how tht { k } k. y coti ubequece covergig to y y y (33 I fct, if thi i ot ce, the yi( > yi+ ( for ome i. Sice yi( = yi+ (. Lemm implie tht yi > yi+ for + i. Puttig = + i, we hve = yi( + i > yi+ ( + i cotrdictio. Accordigly (33 hold, d o lim yk ( = l (, ] exit, ice y k y o [, + k] for y k, { yk ( } i uiformly bouded o ech compct ubitervl of [,. otig tht yk i odecreig d opoitive o [, + k], we hve y( yk ( yk o [, + k], k. Hece { yk ( } i equicotiuou o ech compct ubitervl of [,. From thee coidertio we fid tht there i ubequece { yk } { y } i k d fuctio y tifyig lim ( yk i = y uiki formly o ech compct ubitervl of [,. Filly we hll how tht y i deired olutio of Equtio (8. Let [, k we hve lettig be fixed rbitrrily. For ll ufficietly lrge i ki, we obti * = + ( + yk y y i k p r y i k r i. * y = y + l + p r y r Tkig differece i thi bove equlity, we re tht y olve Equtio (8 o [ (7 i evidet. The proof of Theorem 7 i complete. 7. Proof of Mi Reult for the Super-Homogeeou Equtio. Tht y tifie Throughout thi ectio, we ume tht <. Proof of Theorem. The theorem i immedite coequece of the uiquee of the trivil olutio (Lemm 3. Proof of Theorem 4. eceity Prt: Let y( be poitive olutio of Equtio (8 for of type (AC. It i ey to ee tht y d y y( >. Hece ummig (8 twice, we hve from which we fid tht y( + y( = p( r y( r ( y p( r <. Thi i equivlet to (. Sufficiecy Prt: Let ( hold. Fix l > d chooe o tht p r We itroduce the Bch pce l of ll bouded, rel equece { y( } with orm y up y Defie the et Y = { y l : l y, l } d the mppig F : Y l by ( l., =. 6

12 Fy = l + p( r y( r,. We below how vi the Schuder-Tychooff fixed poit theorem tht F h t let oe fixed elemet i Y. y Y. The Firtly, let Thu Fy Y, d hece FY Y coverig to y Y d l Fy l ( l + p ( r l + l =, l.. Secodly, to ee the cotiuity of F, let { k } uiformly o ech compct ubitervl of [ ice p( r yk ( r ( l p( r <. The Lebegue domited covergece theorem implie tht Fyk Fy ubitervl of [ ice for y ( Y,, { } ( Fy p( y( ( l p(,. The et ( Fy ( : y Y i uiformly bouded o [,, y be equece i Y p i bouded for uiformly o ech compct. Thi implie tht Y F i compct. From there obervtio we fid tht F h proved elemet y i Y uch tht Fy = y. Tht thi y i olutio of Equtio ( of type (AC i eily proved. The proof i complete. Proof of Theorem 3. Sufficiecy Prt: Let y( be olutio of Equtio (8 tifyig y( >, y < for. The exitece of uch olutio i eured by Theorem 7. Obviouly, y( i either of type (D or type (AC. Theorem 4 how tht uder umptio (9, Equtio (8 doe ot poe olutio of type (AC. Hece y( mut be of type (D. eceity Prt: Let y( be poitive olutio of Equtio (8 for of type (D. Clerly y( tifie y = p( r y( r,. To verify (9, uppoe the cotrry tht (9 fil to hold. The, othig tht we hve Accordigly, y y p( r,. y p( r,. y i decreig for, The left hd ide ted to becue of <, where the right hd ide ted to. Thi cotrdictio verifie (9. The proof i complete. Proof of Theorem 5. eceity Prt: Let y( be poitive olutio of Equtio (8 er of type (AL. There i cott c > d tifyig Summtio of Equtio (8 from to yield y c,. (34 * * ( y y = p y,. 63

13 Sice lim y y (, =, thi i equlity implie tht p y <. (35 Combiig (35 with (34, we fid tht ( hold. Sufficiecy Prt: We fix l > rbitrrily, d chooe lrge eough o tht p l. Let l be the Bch pce i the proof Theorem 4. Defie the et Y l follow The mppig F : Y l defied by { } Y = y l : l y l,. Fy = ( l p( r y( r,. A i the proof of the ufficiecy prt of Theorem 4, we c how tht F h fixed elemet y the Schvder-Tyehooff fixed poit Theorem y = ( l p( r y( r,. Tkig twice for thi formul we ee tht y lim = l. Thu L Hopitl rule how tht i complete. Y by y i poitive olutio of Equtio (8 for. y i olutio of Equtio (8 of type (AL. The proof Lemm 8. Let y ( >. If ( hold, the there i poitive olutio of Equtio (8 for (AL tifyig y( = y(. Proof. By Theorem 5, there i (AL-type poitive olutio z borhood of : < lim = lim z <. Let y( be poitive olutio of Equtio (8 for tifyig y( = y( d y( >, y < for. Tke > uch tht y z y < z for. By Lemm if λ > y( i ufficietly elver to y( y( of Equtio (8 with y( = y( d y( = λ exit t let o [, ] d tifie y < y < z, y < y < z. The Lemm gi implie tht y < y < z log exit. Sice y( d z(, thi me tht d tifie y y z exit for hve y exit for y y z < <,. otig tht y( i the uique olutio of (8 tifyig of type z of Equtio (8 defied i ome eigh- < d, the the olutio < <,. The we y lim = d pig through the poit (, y ( we hve y lim lim (,. Therefore Proof of Theorem 6. For λ >, we deote by yλ coditio y( = y d y( = λ. The mximl itervl of exitece of y, the uique olutio of Equtio (8 with i iitil my be fiite or ifiite. Defie the et S (, by y i of type (AL. The proof i complete. k 64

14 { λ : λ exit for d i of type } S = > y AL V. Sdhivm et l. We kow by Lemm 8 tht S Q d by Lemm 5 tht λ S for ll ufficietly lrge λ >. Hece up S = λ (, exit. For λ there re three poibilitie: λ S λ S d y ( i of type (AS 3 λ S d λ ( λ i of type ( S. To prove the theorem, we below how tht ce (b occur. For implicity, we write y for y λ below. lim y y l, y < l,. By coditio Suppoe tht the ce ( occur. The = ( = ( d ( we c fid > tifyig p y + l < l l Chooe λ λ y, yλ < l. The, for uch λ, yλ c be exteded to, d tifie yλ <, l. I fct, if thi i ot the ce, there i > tifyig yλ < l for < d yλ = l. It follow therefore tht yλ y ( + lt for. Summig the Equtio (8 (with y = y λ for yield > cloe eough to λ o tht λ exit t let o [ ] d ( l = ( yλ = ( yλ ( + p( yλ( Thi cotrdictio implie tht y ( ( l + p y + l l + p y + l < l. λ exit for d tifie yλ <, l. Thee obervtio how tht S λ > λ, which cotrdict the defiitio of λ. Hece ce ( doe ot occur. ext, uppoe tht ce (c occur. Let > be the poit uch tht y( = y( =. By Lemm 5, there i M > uch tht olutio y( of Equtio (8 tifyig y(, y( M mut = y, : y = y =. For ufficietly mll >, we hve blow up t ome fiite y( > y( > M. The if λ < λ i ufficietly cloe to λ, the y let to, d tifie yλ ( >, yλ ( > M yλ blow up t ome fiite poit by the defiitio of M. Thi fct how tht uch λ( λ λ c be cotiued t. The, eve through y λ c be cotiued to, < doe ot belog to S, cotrdictig the defiitio of λ, gi. Coequetly ce (b occur, d hece the proof of Theorem 6 i complete. Proof of Theorem 7. The proof i doe by cotrdictio. Let y( be olutio of Equtio ( of type (AS. We uppoe tht y( exit for d tifie Put z y ( y ( = >. The y C, y C, for ome C > (36 + ( ( + = y+ ( y + ( y p z = y y + y y = + ( y y+ p ( y ( y p y ( y + ( y ( y( + = + p y y y y z + y. y 65

15 ow, we employ the Youg iequlity of the form ( σ i the lt iequlity. It follow therefore tht where C C ( σ σ σ σ X+ Y σ σ X Y for XY, > d < σ < (37 ( + ( ( σ σ σ σ σ z C z y y p, =,, > i cott. We rewrite i iequlity ( ( ( ( σ σ ρ σ σ ρ σ + ρ z C y y p z,. otig (7.3 d coditio (3, we obti + ( σ ( σ ρ + ρ 3 z C p z,, where C3 = C3( σ,,, ρ, C, C > i cott. Dividig both ide by ( z, we hve ρ becue lim z =. Coequetly, we hve ρ σ + σ ρ ( z C p, 3 σ +ρ d ummig from to ρ ρ ρ + σ ρ σ ( y C 3 ( p(,. ρ y( Lettig, we get cotrdictio to umptio (. Thi complete the proof. A w metioed i Sectio 5, the proof of Theorem 8 i omitted. I fct, more geerl reult i proved i Lemm Proof of Mi Reult for the Sub-Homogeeou Equtio Throughout thi ectio, we ume tht >. Proof of Theorem 9. Let, be fixed o tht d put d + m= mi p > d ρ = >. The there re cott L > d c > tifyig Coider the Bch pce the ubet Y of B by L p( r L, C m ( ρ + c + ρ + ρ c L for. B of ll rel equece y { y } { [, ] : ( e for } Y = y C c y L = with up orm y = up y. Defie 66

16 d Fy = p r y r,, =, other wie. We how tht the hypothei of the Schvder fixed poit theorem i tified for Y d F. Let y Y. The, Fy L for. Moreover obviouly Fy m c r ( m ( ρ + + ρ + c ρ + ρ + ρ c, for. Hece FY Y FY : y Y c be eily etblihed. Accordigly there i y Y tifyig FY = y. By tkig differece twice, we fid tht y( i olutio of Equtio ( for d tht y( > for d y( = y( =. ow, we put. The cotiuity of F d the boudede of the et FY d { } It i ey to ee tht y y =. y i olutio of equtio (8 for S. The proof i complete. Theorem 4 d 5 c be proved eily i the proof of Theorem 4 d 5 repectively. We therefore omit the proof. Proof of Theorem. By our umptio we c fid poitive olutio yk, k of Equtio (8 t- ifyig y k ( =. Sice >, we ee by Lemm 4 tht ech yk ( exit for. We how tht the K equece { yk ( } h the limit fuctio y(, d it give rie to poitive olutio of Equtio (8 of type (D. We firt clim tht d i of type y > y > > yk > yk+ > >,. If thi i ot true, the yi = yi+ for ome i d oegtive oicreig olutio of Equtio (8 pig through the poit (, i Theorem 7. We therefore hve (38 d o y = y exit oberve tht y (38 lim k k yk = + p( r yi( r,. K Lettig k, we obti vi the domited covergece theorem y = p( r y( r,.. Thi me however tht there re two y. Thi cotrdictio to tifie We ee tht y( i oegtive olutio of Equtio (8 tifyig y ( = y( > for Fix > rbitrrily. The proof of Theorem implie tht there i olutio y ( > < d. We clim tht for y = for I fct, if thi fil to hold, the k. It remi to prove tht y > y for for ll k. (39 k 67

17 y = y for ome i d <. i By thi me, before, tht there re two oegtive oicreig olutio fo Equtio (8 pig through the poit (, y (. Thi cotrdictio how tht (39 hold. Hece by lettig i i (39 we hve y y > for <. Sice > i rbitrry, we ee tht y( > for. The proof i complete. Proof of Theorem. The proof i doe by cotrdictio. Let y( be poitive olutio of equtio (8 for of type (D. Uig (6. We obti from Equtio (8 ( ( y C y,. (4 where C i poitive cott. We fix rbitrry d coider iequlity (4 oly o the itervl, for momet. A ummtio of (4 from to, give [ ] From which, we hve ( y( + ( y C ( y( ( y C ( y( ( y y C ( y( ( y, y,. ( y( ( y y( y ( du ( ( u y We c fid cott C > tifyig Therefore (43 implie tht from which we hve C C, ( (4 y y( y v d v C,. x + v dv C x, x. ( y C( y( C, y ( ( C y y C,. Lettig, we hve cotrdictio. The proof i complete. Proof of Theorem. The proof i doe by cotrdictio. Let y( be olutio of Equtio (8 of type (D. We otice firt tht I fct, ice y >, we c compute follow lim y =. (4 68

18 ( ( y = y y Therefore (4 hold. We my uppoe tht for ome C > d But z y( y ( = >. The y p = y for lrge. < y C, < y C,. (43 ( ( + z = y y y y + ( y y+ p ( y ( y y( y p ( y y = + proceedig i the proof of Theorem 8, we obti y y = + ( y ( y y( y + p y ( y ( ( ( σ + σ ρ σ σ + ρσ ρ z C y y p z, where C i cott. We obti from (43 d umptio (8 where 3 tht i, + σ ( σ σ ρ ρ 3, z C p z C > i cott. Dividig both ide by ( z ρ ρ σ + σ ρ σ ( z C ( p(, ρ ρ 3 ρ σρ σρ σ + σ ρ ( ( d ummig from to, we hve y y C p,. 3 Lettig, we get cotrdictio to umptio (7 by (4. The proof i complete. Proof of Theorem 6. Sufficiecy Prt: By Theorem 7 d ( of Remrk 6., there i poitive olutio y =. Thi y( i either of type (AL or of type (AS. But by Theorem y mut be of type (AS. eceity Prt: Let y( be poitive olutio of Equtio (8 for of type (AS. To prove (9, we y of equtio (8 tifyig 5, we ee tht uppoe the cotrry tht p where c = y( d c = y( let z <. A i the proof of Lemm 5.3, we hve y = ρ + C + p( r y( r, ( ξ y = mx. It follow tht ξ ξ σ 69

19 where y c i cott. put w mx c, z Sice y( i of type (AS, C + c + z( r p( r, c + c + z r p r { } =. We the hve z c + w + r f ( r, m. y i ubouded for, tifyig w z for. Thu w c + ( w + r p( r d o i c + ( w + r p( r,. z. Accordigly, there i Sice <, thi implie the boudede of w, which i cotrctio. Hece, we mut hve (9. The proof i complete. Theorem 6 i cler becue of ll olutio of equtio (8 with < exit for [ee Lemm 5]. Referece [] Mizukmi, M., ito, M. d Umi, H. ( Aymtotic Behvior of Solutio of Cl of Secod Order Quilier Ordiry Differetil Equtio. Hirohim Mthemticl Jourl, 3, [] Tiryki, A. ( Some Criteri for the Aymptotic Behvior of Certi Secod Order olier Perturbed Differetil Equtio. Advce i Pure Mthemtic,, [3] Drozdowic, A. d Poped, J. (987 Aymptotic Behvior of the Solutio of the Secod Order Differece Equtio. Proceedig of the Americ Mthemticl Society, 99, [4] Drozdowic, A. d Poped, J. (993 Aymptotic Behvior of Solutio of Differece Equtio of Secod Order. Jourl of Computtiol d Applied Mthemtic, 47, [5] Elbert, A. d Kwo, T. (99 Ocilltio d oocilltio Theorem for Cl of Secod Order Quilier Differetil Equtio. Act Mthemtic Hugric, 56, [6] He, X.Z. (993 Ocilltory d Aymptotic Behvior of Secod Order olier Differece Equtio. Jourl of Mthemticl Alyi d Applictio, 75, [7] Kigurdze, I.T. d Chturi, T.A. (99 Aymptotic Propertie of Solutio of o Autoomou Ordiry Differetil Equtio. uk, Mocow. [8] Kubiczyk, I. d Sekr, S.H. ( Ocilltio Theorem for Secod Order Sublier Dely Differece Equtio. Mthemtic Slovc, 5, [9] Kuo, T., ito, K. d Ogt, A. (994 Strog Ocilltio d oocilltio of Quilier Differetil Equtio of Secod Order. Differetil Equtio d Dymicl Sytem,, -. [] Luo, J. ( Ocilltio Criteri for Secod Order Quilier eutrl Differece Equtio. Computer & Mthemtic with Applictio, 43, [] Migd, M. d Migd, J. (998 Aymptotic Behvior of the Solutio of the Secod Order Differece Equtio. Archivum Mthemticum, 34, [] Sdhivm, V., Sudr, P. d Sthi, A. (6 Ocilltio d Aymptotic Behvior of Solutio of Secod Order Homogeeou eutrl Differece Equtio with Poitive d egtive Coefficiet. IOSR Jourl of Mthemtic,,

20 [3] Thdpi, E. (99 Aymptotic d Ocilltory Behvior of Solutio of Secod Order olier eutrl Dely Differece Equtio. Rivit di Mtemtic dell Uiverità di Prm,, [4] Thdpi, E. d Arul, R. (997 Ocilltio Theory for Cl of Secod Order Quilier Differece Equtio. Tmkg Jourl of Mthemtic, 8, [5] Thdpi, E., Gref, J.R. d Spike, P.W. (996 O the Ocilltio of Solutio of Secod Order Quilier Differece Equtio. olier World, 3, [6] Wog, P.J.Y. d Agrwl, R.P. (996 Ocilltio Theorem for Certi Secod Order olier Differece Equtio. Jourl of Mthemticl Alyi d Applictio, 4, [7] Wog, P.J.W. d Agrwl, R.P. (996 Ocilltio d Mootoe Solutio of Secod Order Quilier Differece Equtio. Fukcilj Ekvcioj, 39, [8] Y, J. d Liu, B. (995 Aymptotic Behvior of olier Dely Differece Equtio. Applied Mthemtic Letter, 8, [9] Rth, R.., Sehdev, P. d Brik, B.L.S. (8 Ocilltory d Aymptotic Behviour of Homogeeou eutrl Dely Differece Equtio of Secod Order. Bulleti of the Ititute of Mthemtic Acdemi Siic (ew Serie, 3, [] Agrwl, R.P. (99 Differece Equtio d Iequlitie Theory, Method d Applictio. Mrcel Dekker, ew York. [] Elydi, S.. (996 A Itroductio to Differece Equtio. Spriger Verlg, ew York. [] Gyori, I. d Ld, G. (99 Ocilltio Theory of Dely Differetil Equtio with Applictio. Clredo Pre, Oxford. [3] Kelly, W.G. d Petero, A.C. (99 Differece Equtio: A Itroductio with Applictio. Acdemic Pre, ew York. [4] Lkhmikth, V. d Trigite, O. (988 Theory of Differece Equtio: umericl Method d Applictio. Acdemic Pre, ew York. [5] Wog, P.J.Y. d Agrwl, R.P. (996 O the Ocilltio d Aymptoticlly Mootoe Solutio of Secod Order Quilier Differece Equtio. Applied Mthemtic d Computtio, 79, [6] Thdpi, E., Muel, M.M.S. d Agrwl, R.P. (996 Ocilltio d o Ocilltio Theorem for Secod Order Quilier Differece Equtio. Fct Uiveritti, Serie: Mthemtic d Iformtic,, [7] Sudr, P. d Thdpi, E. ( Ocilltio d o-ocilltio Theorem for Secod Order Quilier Fuctiol Differece Equtio. Idi Jourl of Pure d Applied Mthemtic, 3, [8] Thdpi, E. d Arul, R. (997 Ocilltio d oocilltio Theorem for Cl of Secod Order Quilier Differece Equtio. Zeitchrift Für Alyi Ud Ihre Aweduge, 6, [9] Thdpi, E., Liu, Z., Arul, R. d Rj, P.S. (4 Ocilltio d Aymptotic Behvior of Secod Order Differece Equtio with olier eutrl Term. Applied Mthemtic E-ote, 4, [3] Cheg, S.S. d Ptul, W.T. (993 A Exitece Theorem for olier Differece Equtio. olier Alyi, Theory Method d Applictio,,

21 Submit or recommed ext mucript to SCIRP d we will provide bet ervice for you: Acceptig pre-ubmiio iquirie through Emil, Fcebook, LikedI, Twitter, etc. A wide electio of jourl (icluive of 9 ubject, more th jourl Providig 4-hour high-qulity ervice Uer-friedly olie ubmiio ytem Fir d wift peer-review ytem Efficiet typeettig d proofredig procedure Diply of the reult of dowlod d viit, well the umber of cited rticle Mximum diemitio of your reerch work Submit your mucript t: