DSpace at VNU: Oscilation and Convergence for a Neutral Difference Equation

DSpace at VNU: Oscilation and Convergence for a Neutral Difference Equation tài liệu, giáo án, bài giảng , luận văn, luậ...

\'N H l u ur n a l o f SCICỈKC M a U i c n i a l i c s - IMiysics ( 0 S } P ' V Oscilation and Convergence for a Neutral D i n ' c r c n c c IZqualion Diiih Conti H u on u * Di'fH- OỈ Sỉí il ỉì ( ) u \ - XỈÌOỈI I ' i i i y c r s i t y ỉ " ỉ ) Aii Dỉíoỉ ìí ^ riioỉìíỉ O i i y n ỉ i o ỉ ì ỉiỉììỉìíỉììììì, \ '}ưĩ}i(i})} R cccivcd 24 Apriỉ 200S A ỉ)stra ct i h c o s c i l l a t i o n a n d c o u \ c i ' i zci i cc o f l l ì c s o l u l i o i i s ol ' I i c u l r a ! d i t l c i c i i c c c q u a t u u i r ỉ -) I 'S ì n , (//)/■'(,/',, ì // I) i - - - arc i n v c s t i t i a t c d u h c r c // /; ^ FJ() V / ! /■ a n d /■’ is a l u i i c t i o n Iiìappint: E to s K c y w o r k s: N e u l r a l d i í ì c r c ĩ i c c ctỊualìoĩi oscillaluK i, Iioiioscillalioiụ c (u i\'crg cn cc, In trod uctio n It IS well-known that dillcivncc equation i ,-) I w h e r e it G N- th e (Operator A is d e f in e d as A /‘„ (1) th e t u n c t i o n n ( / / ) IS d e f i n e d on N ÍÌ IS a ciìiìstant- r is a p o s i t i \ c i n t e g e r a n d Ơ is a n o n n c i z a t i\ c in tcu cr w a s first c o n s i d e r e d b \ H U tin a n d w i l l o i m h l n f r o m t he n u m e r i c a l p o i n t I ) f \IC\V (see I 1|) In r c c c n l \ c a r s , t he íisyiiiỊìtotic h c h a v i o r t il's iilu tiiin s o f th is e q u a ti o n h a s b e e n s t u d i e d c x t c n s i \ c l \ ' (see | - | ) , In |4 | , th e o s c i l l a t i o n lit s o l u t u m s o f th e d i l l c r c n c c c q u a tiiin (1) w a s d is c u s s c d , MiUivatcd b\ the work ab(i\c in this paper, wc aim ti) stud\ the tiscillatit)n and convcrgcncc Í s o l u t io n s o f n e u tra l d i l i c r c n c c e q u a ti o n r A (.r„ (I f o r s o m e P u t /1 for n - 1 7/^/2- ■ ■ ■ ' arc defined tin N and the function /'’ is defined ÍÌI1 R r j ÌÌ > ) a € PJ w h e r e r j n Tixcd p ( i s it i\ c iu n c tid iis I i i a x j r i / / | ■ ■■ i / / ; } T h e n h \ a s o lu tio n o f (2) w c m e a n a f u n c ti o n w h i c h is d e f in e d — '1 : i n d s a s t i s t i c s t h e c c Ị u a l Ì D i i ( ) r/,, t o r tỉ G f ' ] ÌỈ C ic a rh ' i f —/I —.'I Ỉ arc Lii\cn then (2) has a unique solution, Lind it can be constnictcd A integers, th e m m triv ia l 112 ỳ ii\ s u c h th a t arc n scillator\- s o lu tio n o th e rw is e , o f (2) is ca llcd osalỉriỉoiỊỊ Ỉ recursively it fo r anv /7] a there exists -1 ÍÍ 0- T h e ciiíĩcrcncc e q u a ti o n (2) is c a llc d o s c i l l a t o n ' i f all its s o l u t io n s it is c a l l c d ’ T d - S ‘^741 I'^-niaiL d c o n u li u o im i/> alnH).coni ncm oscillaton- 0.( D c H u ong / VN U Jo u rn a l o f Science, M aihem atics - Physics 24 (200H) Ỉ 3 - Ỉ4 134 1, Main results 2,1, The Osciỉỉatỉon Consider neutral difTerence equation V ■ ^ ■ (3) i=l for n e N , n ^ a for some a e N, where r, , m r are fixed positive inteucrs and the functions a j ( n ) are defined on N It is clear that equation (3) is a particular case o f (2) We shall establish some sufficient criterias for the oscillation o f solutions o f the difference equation (3) First o f all vve have Theorem Assume that r (m + E lim inf a ị í n ) > 1, 71—»00 1) where (5 — 0, a i ( n ) ^ 0, n G N, ^ i ^ r andỈ m n i — mill m-i ĩ ĩ i ị Then, (3) is oscillaíoỉy l for all n ^ n i , n i E N Setting Vn — and dividing this inequality by x „, \vc obtain ((i) where n ^ n i + m , m — m a x ĩiĩi Clearly, { x ^ } is nonincreasing with 71 ^ n-[ + ?TI, and so V n ^ for all n ^ r i \ iand ( ) we see that {Un} is a above bounded sequence Putting lim inf tVi = /3, we get *00 nrt.— —*■00 1 r m, ^ —l i m i n f y ^ a^(n) lim su p — Vn p Vji-e^ fJl ^ or fj ^ ^ - ^ lim inf a , ( n ) • 71~»00 2= Since ^ V2 = 177 , we have lim inf a i ( n ) / 3^^ ^ lim inf a^ (n)/3’^, n —+00 ’ n —»00 Vi ~ l , r and - ^ ^ 2= l i m i n f Q iin )/? " * ' < 71—^00 ^ ^ l i m i n f Q i(n )/3 " n —*oc rn From (4) D c ỉỉuoní^ / VNU Jo u rn a l o f Science, hiathem atics - Physics 24 (2008) Ỉ3 - Ỉ4 35 I'rom (7) \vc have 7' í3 - lini inf > ^ 1( 7/,) < ■ M— oc 1=1 But so r lim inf a^(?z) ^ , VI 1=1 77—»cc^ which contradicts condition (4) Hencc, (5) has no eventually positive solution Similarly, we can prove that the inequality r A:r,, + ^ Q i ( n ) X n - m , ^ , t=i n N has no cvenlLially negative solution So, the proof is complete Corollary Assuỉìỉe ỉhaí n where Ố — 0, (\t{n) ^ ?1 E in ĨÌI > 1=] N, ^ i ^ r and m ■= - ( 8) Then, (3) is oscillaiory Proof, We will prove that the inequality (5) has no eventually positive solution Assume, for the sake of contradiction, that (5) has a solution {x„} with Xn > for all n ^ n i , n i G N Using arithmetic and geometric mean inequality, we obtain ( r ^ 11-^00 t=l \ T lim in fa ,:(n ) • ị r ^ ^ T n y r lim inf , 71—*c0 \l= l / which is the same as r ^ ' Z=1 / lim inf ai{ĩi) • /5"^^ ^ ^ n —*oc \ r r TT n —>00 Q;i(n) \z=l This yields 1— \ r / r r liiii inf cxAn) • (3^^^ ^ — 7’ n -^00 t=i By using the inequality (7) we have ^ \i=: lim inf a i ( 77.) n^oo /3^ r I (lim in f Qi(n)) n —*oo 1= (m + ) m+I which contradicts condition ( ) Hence, (5) has no eventually positive solution Next, we consider the equation (3) in case Ô We have the following Lemma Lemma = 2'ii + Lei a i { n ) > f o r all 7Ĩ e N and let {Xn} be an eventually positive solution o f (3) Piii vve have D C HuoniỊ / V N U Jo urn al o f Scicncc, M uthcm atics - Physics 24 (200H) I 3 - N 136 • fa) / / ' - < Ỏ' < 0, then Zn > a iu l Az„ < eventually • (b) I f Ỗ < - and ~n < u?ul A z n ^ eventually Proof, (a) Since O j(n) ^ 0, we have r eventually, so 2:„ cannot be eventually identically zero, if 2n ^ < 0, < eventually, then V/i ^ N e N Since - < Ò' < 0, we get àx^i — r > 7i —T ỉ which implies that X-,1 < Zji + X f i —r ^ Z y -}■ X j i —T' Therefore, x ^ ĩ ^ r n < -.V + X.Y + r n - r = -.V + -i\v + r { n - l ) < Taking n ^ oc in the above inequality, wc have x'-v-t rn < (b) We have ■■• < + ‘/'.V- which is a contradiction to > (J r Azn = - for n suiTicicnt large We shall prove that Zn < 0, t=-:l < 0, eventually Assume, for the sake of a contradiction, that ~ ■^71 "1“—T ^ Oi 11 ^ N , I.e which implies that On letting j ^ oo in the above inequality, we get Xn ^ oc as n oc But r r (9) Azn = - ^ a z (n ).rn -7 i,, ^ - M 1=\ 1=1 for 71 sufficient large, where M > Summing (9) from N to n, we obtain n r ^ —AỈ '^ri4 'y ^ (^) , e=N 1-1 which implies that T h e o re m “ OO as n ^ CO This contradicts the hypothesis that ^,1 ^ n > A' Suppose that 10) 1+ Ỏ ^ i- \ D c\ ĩỉuoní^ / VNU Jou rn a l o f Scicnce \ía ĩh c m tic s - Physics 24 (200H) Ỉ 33-143 w iicrc —1 < () < 0, - n i i n iii ÌÌ Ì Ị ^ and > c\i{v — r), fo r \ ‘A ÌI S ỉ í ị i ì c i c ỉ ỉ i l a f x c , < ; ./■,)- r and Ii'n i T- Then b> the case (a) of Lemma \, Zj, > < and U'jj > i) We liave Au'n A c ,, -f /• 7' T)x„^r~m, - Ỏ^ o , ( n Í - Í - r r “ Ị ^ i-1 g , -7n,' I-I r A ỉ / ’;, -m ^ ^ ^ n ~ r —ini') Ì Í-1 /• Alt',, ^ ^ n ,(//)c„_,„, ^ Ỉ^I /1 \vc have J > and IHittiim liin C;, /J ' X liin i('„ = /i + ỗíi = (1 “I ỗ) p ^ 'riicrcforc f/’„ > (} for ÌÌ sufTicicnt large On the other hand, whicli iniplit's lliat Ở llencc, \vc obtain A ii-„ ÍÍ - '' ^ Í -1 Ĩ-1 or Aỉí',, + ( 11 ) ^ Bv Theorem and in view of condition (10), the inequality (11) has no eventually positive solulion which is a contradiction Lem m a Assume ihal - \ < Ỏ < and value o f Ị ( ị i ) = soluiion o f the equation T > rh - 1, where fa = mill -f ổ / r ) on [ l,o c ) is f{ị3*), in which p* G (1, rì^ Then, the m axim um is a unique real + ỏ / r + ựì - 1) Ỗrị3^ - (rn + 1)(1 + ổ ^ ) ] - Proof Tlie equation f'{3 ) = is equivalent to -4- ỏíV 4- [3 ~ l ) \ Sr : r - (rJi + 1)(1 + ố/:r)] = (12) D c H u o n g / V N U Jo urnal o f Science, M athem atics - Physics 24 (200^) Ỉ 3 - N 138 Put i p i P ) = Ỉ + ÔP^ + { P - \ ) [ Ỗ T ị ì ^ - [m + )(1 + ỏ ír) It is easy to check that a,{Ti - r) fo r n sufficient large, I ^ i ^ r, m = m i l l rrii and \< i< r r ^-1 13) lini inf Q:i(n) > ^»rn + n —^oo 1=1 where /3* G [1, oo) is defined as in Lemma Then, (Ỉ) is oscillatory Proof Suppose to the contrary, and let {x„} be an eventually positive solution o f (3) By the ease (a) o f Lemma 1, we get Zn > 0, Axn < eventually On the other hand, r (M Ali)„ = ^ { Z n + ỖZn-r) ^ - X ] ai{n)Zn-rn, < i= l Putting „ = we have „ ^ for n sufficient large Dividing (14) by r ^n~T ế + n " SỈ ^ /o r n sujjicient large; < = -1 , ( 16 ) 1, m ax Till, r > nu + and = CO Then, (3) is oscillalory Proof Assume the contrary Without loss of generality, let {x„} be an eventually positive solution of (3) By the case (b) of Lemma 1, we have Zn < and A^n ^ Putting W n = Zn + Z n - T , we have Wn ~ Zfi àZi-i—T ^ (1 "1“ ^')^n—Ti which is the same as Zn-T ^ Ồ+ Therefore, it follows that A ĩ L ’„ = A „ Ổ A z„ -r + r - r - r)x n-T ~m ,^ - ^a,(77.)xn-77t, i=\ i-j r r ~ ỖỴ^ai{n)Xn~r-rrH, ^ z-1 r ÙlWỵi i= l ^ ^ Qj ('^) {^n—rnj “ỉ“ ^ r —mi)i r ^W ỵi ^ ^ ^ Q'i(^)-^n—mi ^ Oị 1=1 SO w e g e t r 0^ ^ ^ Qị [ù) ^ ^'Wji + ^ r ^ ^^Oii{ĩl)Wn-rn^ + r ■ 1=1 Setting n = tOn we obtain 7n ^^ ^ ~ ^ T ~I Ĩ" ^ ^ -r i ^ ' 2=1 T— rriị n -m i+ r-£ * ^=1 D c H u o n g / V N U J o u r n a l o f Science, M a th e m a tic s - P h ysics (2008) Ỉ 3 - Ỉ 140 Putting p = l i m i n f r^, w e have / ^ Taking lower limit on both sides of (17), vve obiain n —•■oo y /3 ^ - ^ lim inf -ị- ^ ^ i= l (n ) • /Ỉ— •, n —»oo or /? - ^ ^ ^ (18) liminf a ^ { n ) ■ -f- n-^cc 2= Since Vi = l , r , ^ Ò -|+ 1i n -^ —*oo oo ' l i m i n f Qi(n)/?' ' Ớ -|- Ò + Vi = l , r n -* —*oo oo From (18) we get -ĨT Ĩ i= \ But /3 -1 ^ {t - m - p r-m , SO - — 7— ^— - } Ố + (• ( r — m , — 1)^ ” ** ^ l i m inf Q i ( n ) ^ 1, which contradicts condition (16) Hence, (3) has no eventually positive solution Theorem Su p pos e that \ Ổ (r - where Ô < - , m* = m ax n ii, T _r > ^ m* - > m* + a nd li m in f a i ( n ) > 1, (19) n -^o o [1.1=1 oscillalory Proof Suppose to the contrary, and let {x„} be an eventually positive solution o f (3) Put Zn = Xn + ỗXn-r- By the case (b) o f Lem m a 1, we obtain z„ < and A z „ ^ On the other hand, we have Zn > S x n -T or X n -T > which implies that Xn-rrii > ^Zn+T-m^- Hence, A z „ ^ - 'S ^ Q i(7 l)z „ + r-m iSetting Vn = and dividing (20) by Zn, we obtain ^r , i= l or r —T T ii-1 '2^n+r—mi —£—1 (2^^) D c H u o n g / V N U J o u r n a l o f Science, M a th e m a tic s - Physics 24 (2008) Ỉ 3 - Ỉ Taking lower limit on both sides o f (21) and putting [3 = n —»rv'i n— »CXD 11 vvc have (3 ^ and ^ ^ - V lim in fa frO ‘ Ó n—co 2=1 Wc can prove \T 7n^ ^ ( t - - ???,) ill ^ j ^ ^ ^ — — r / l i m i n f a in( n )) ^ ^ 1)' '5 - r n , — ,,-v T Ổ (t similarly as the proof o f Theorem 4, which contradicts condition (19) Hence, (3) has no eventually positive solution 2.2 The Convergence We give conditions implying that every nonoscillatory solution is convergent To begin with, we have Lemma Let {x„} be a n o n osciilatoiy solution o f (2) P ut z-n = T„ + Sxn-T- • (a) I f {Xn} is eveutuaUy positive (negative), then {z n} ứ eventually iwmncreasing (noudecreosingj • (b) I f {Xn} is eveniuaily p o sitive (negative) and there exists a constant such that -K ^ (22) theĩi eventually Zn > {Zn < 0) Proof Let {x„} be an eventually positive solution o f (2) The case {x'n} is an eventually negative solution of (2) can be considered similarly (a) We have ~ a i { n ) F { x n - m i ) ^ for all large n Thus, {Zn} is eventually 1=1 nonincreasing (b) Suppose the conclusion does not hold, then since by (a) {Zn} is nonincreasing, it follows that eventually either Zn = or < N ow Zn - implies that '^ i { n ) F { x n - m , ) = 0, but this contradicts the fact that a , ( 72) ^ for infinitely many n If < 0, then Xn < -Ỏ X n-T so Ỗ < From (22) it follows that —1 < < and Xn < —^X n-T - Thus, by induction, we obtain Xn-^JT ^ { - i V x n for all p o s i t i v e i n t e g e r s J H e n c e , Xri ^ as n to zero as n —^ cx: This contradicts the fact that Zn < Theorem A ssum e that 00 0 It i m p l i e s t h a t {Zn} d e c r e a s e s r ' ^ Y ^ a i ự ) = co, t=\ i = l (23) - < 7y < (5 ^ (24) and there exists a constant T] such that Suppose fu rth e r that, i f \ x \ ^ c then | F ( x ) | ^ C l where c a n d noĩioscillatory solution o f (2) tends to as n —* oo Cl are p o sitiv e constants Then, every D C H u o n g / VNU Jo urn al o f Science M a th e m a tic s - P hysics 24 (2008) 133-143 142 Proof Let {x„} be an eventually positive solution o f (2), say Xn > 0, x , r - r > and > for n > n o G N Put Zn = Xn + Sxn-T- We first prove that Zn ^ as n oo Note that ( ) implies (2 ) with replace by rj By Lemma vve have {z„} is eventually positive and nonincrcasing Therefore, there exists lim Zn- Put lim = p Now, suppose that (3 > By (24), vvc have Zn ÍỈ Thus n -^ o o n —♦OO there exists an integer Til ^ no e N such that /3 ^ -^n—Trij ^ ^n —mị ^ * I^ Hence, A zn = - Y ^ a ^ { n ) F { x n - m , ) ^ - K l ' ^ a r i n ) , 1= ' i n ^ n\ Ĩ—1 for some positive constant M Summing the last inequality, we obtain n -l Zn ^ - M r ^ f= n i i= l which as n ^ oo, in view o f (23), implies that Zn - o o This is a contradiction Since lim Zn = 0, there exists a positive constant A such that < ^ /1 and so, by (24) we n— n —^O ^co O have x „ -riX n -T ^ + (2 ) Assume that [Xn] is not bounded Then, there exists a subsequence { uk } o f N, so that and Xn, = m ax Xj, \im oc , , - From (25), for k sufficiently large, we get Xrik ^ + Ả and so (1 + lj)Xn^ ^ A , w h ic h as /c ^ CO leads to a contradiction N ow suppose that lim su p X n = a > Then, there exists a subsequence {rik} o f N, with ni large enough so that x„ > for n > n \ — T and Xn^ Cl' as k —> oo Then, from (24), we have Zn^ > and so ^ X rik -T ^ As k ~ •^rifc)- lim Xriị,-T ^ ■ oo, we obtain a ^ Since - r j e (0, 1), it follov/s that Q = 0, i.e x„ ^ as n oo The arguments when {x„} is an eventually negative solution o f (2) is similar Theorem Suppose there exists positive constants M , a i , i = 1,2, ■• • , r such that a,(n)^Q *, i = l,2, -,r, F (x )l^ A /|x |, Ò' ^ Vx G Vn E, GN, (2G) (27) ( 28 ) i:í D C lỉỉions^ / \ ’N l ’ J o u r n a l o f Science M athem atics - Physics 24 (200H} Ỉ 3 - Ỉ TỈÌCỈĨ every ììonosciỉhỉtorv solution o f (2) iends to as ỈI Proof Ixn IỈ ^ Pul DC { / ■„} b e a n e v e n t u a l l y p o s i t i v e s o l u t i o n o f (2), s a v X n > J',, ^^(l G N B \ l e m m a 3, - > and J 'n - n i is e v e n t u a l l y p o s itiv e a n d n o iiin c rc a s iim so t h e r e e x is ts > for liiii c„ n—-X liiii Zjt i S u m m i n i i t h e e q u a t i o n (2 ) f r o m 71 t o o c for it ^ //(), w e o b t a i n n —oc oc 'n =■ r f _ỵ ỵ _2 a ^ ự ) F { X ( - J n J ^ ((= = n u i-=\ n ^ '^^0- N o w b y ( ) a n d ( 27 ) w e g et oc r oc i~n i=l / —71 1= w h i c h i m p l i e s t h a t r,, ^ as n ^ r DC T h e p r o o f is s i m i l a r w h e n { x, j } is e v c n t u a l l v n e c a t i v e Rcfcrences |1| R K Hrayton and R A, Wi l l o uc hby O n the numerical inlaeration o f a svmctric SNSlcm o f diffcrcncc dilĩercntiíìl equations Ỉ'neutral t \ p c J M ath A nal A p p l Vol 18 {1967) | 2] L H iliianti and J s Ju As \ ' mpt o l i c be havi or o f sol uli ons for a class o f dií ĩercnce c q u a l i o n / Síaỉh Anal A p p l Vol 204 (!996) 13] (j, 1- Kordoni s and c G I’hilos, Os cil lati on ol' ncLilral din'crcncc cqualion with periodic ctìcíììcicnts C o m p u ters Math AppUc |4| VoL 33 ( 9 ) IÌ S I.alli and ii (} / h a n t ỉ and J / Li, On the oscillation o f solutions and cxistcncc o l ' p o s i t i v e solutions o f neutral delay dií ĩcrcncc equation, y Sicith A nal A ppl |5| Vol 158 (1991) [Ị S l.alli and B G, Zh a n g , O n c x i s i c nc c o f posi tive solutions bounded oscillations for neutral dcla)' diíĩcrcnce equation J M ath Anal Appl Vol 166 ( 9 ) [ | li S Lalli and B (Ỉ Z h a n c , Os ci l l ati on and compari son theorems for ccrtain neutral delay diHcronce equation, J Ans.iral \ i a t h S()C\ [7] Vol { 9 ) IỈ S Lalli, Oscil iai ion i h co r e ms for certain neutral delay diflcrcncc equation, C o m p u ters \f(iih A ppl Vol 28 ( 994) ... | li S Lalli and B (Ỉ Z h a n c , Os ci l l ati on and compari son theorems for ccrtain neutral delay diHcronce equation, J Ans.iral i a t h S()C [7] Vol { 9 ) IỈ S Lalli, Oscil iai ion i... Sicith A nal A ppl |5| Vol 158 (1991) [Ị S l.alli and B G, Zh a n g , O n c x i s i c nc c o f posi tive solutions bounded oscillations for neutral dcla)' diíĩcrcnce equation J M ath Anal Appl... r, , m r are fixed positive inteucrs and the functions a j ( n ) are defined on N It is clear that equation (3) is a particular case o f (2) We shall establish some sufficient criterias for the