MỘT SỐ BẤT ĐẲNG THỨC TÍCH PHÂN CHO TOÁN TỬ ĐẠO HÀM TRÊN THANG THỜI GIAN VÀ ÁP DỤNG

B GIO DC V O TO TRNG I HC QUY NHN TRN èNH PHNG MT S BT NG THC TCH PHN CHO TON T O HM TRấN THANG THI GIAN V P DNG LUN N TIN S TON HC BèNH NH - NM 2017 B GIO DC V O TO TRNG I HC QUY NHN TRN èNH PHNG MT S BT NG THC TCH PHN CHO TON T O HM TRấN THANG THI GIAN V P DNG Chuyờn ngnh: Toỏn Gii Tớch Mó s: 62.46.01.02 Phn bin 1: Phn bin 2: Phn bin 3: TP TH HNG DN: PGS TS inh Thanh c GS TSKH V Kim Tun BèNH NH - NM 2017 Li cam oan Lun ỏn ny c hon thnh ti Trng i hc Quy Nhn, di s hng dn ca PGS TS inh Thanh c v GS TSKH V Kim Tun Tụi xin cam oan õy l cụng trỡnh nghiờn cu khoa hc ca tụi Cỏc kt qu Lun ỏn l trung thc, c cỏc ng tỏc gi cho phộp s dng v cha tng c cụng b trc ú Tỏc gi Trn ỡnh Phng Li cm n Lun ỏn c thc hin v hon thnh ti Trng i hc Quy Nhn di s hng dn nhit tỡnh v y tn tõm ca PGS TS inh Thanh c v GS TSKH V Kim Tun Trc tiờn, tỏc gi xin by t lũng bit n sõu sc n Thy inh Thanh c, ngi ó hng dn tỏc gi t nhng bc i u tiờn nghiờn cu khoa hc, Thy khụng ch hng dn mt cỏch tn tỡnh, nh hng, giỳp tỏc gi vt qua khú khn quỏ trỡnh nghiờn cu khoa hc m cũn s quan tõm giỳp v mt vt cht ln tinh thn cho tỏc gi sut quỏ trỡnh hc v nghiờn cu ca mỡnh Tỏc gi xin by t lũng bit n chõn thnh n Thy V Kim Tun, ngi ó nhit tõm giỳp tỏc gi quỏ trỡnh nghiờn cu khoa hc v giỳp tỏc gi hc hi thờm c nhiu iu v nghiờn cu khoa hc v cuc sng mc dự thi gian lm vic chung vi tỏc gi khụng nhiu Tỏc gi xin trõn trng gi li cm n chõn thnh n Ban Giỏm hiu Trng i hc Quy Nhn, Phũng o to sau i hc, Khoa Toỏn cựng Quý thy cụ giỏo ging dy lp nghiờn cu sinh Toỏn gii tớch khúa ó tn tỡnh giỳp v to mi iu kin thun li cho tỏc gi sut thi gian hc v nghiờn cu Tỏc gi xin chõn thnh cm n Thy Nguyn D Vi Nhõn Thy ó giỳp tỏc gi tn tỡnh quỏ trỡnh nghiờn cu khoa hc cng nh vic hon thnh Lun ỏn Cui cựng, tỏc gi xin c t lũng bit n chõn thnh n gia ỡnh, bn bố, nhng ngi luụn sỏt cỏnh ng viờn, chia s giỳp tỏc gi hon thnh Lun ỏn Mc lc Danh mc cỏc ký hiu iii M u Chng Mt s kin thc c bn v gii tớch trờn thang thi gian 1.1 Cỏc nh ngha c bn 1.2 Phộp tớnh vi phõn 11 1.3 Phộp tớnh tớch phõn 14 Chng Bt ng thc loi Opial trờn thang thi gian v ỏp dng 22 2.1 Bt ng thc loi Opial cho hm mt bin 24 2.2 Bt ng thc loi Opial cho hm nhiu bin 38 2.3 Mt s ỏp dng 68 Chng Tớnh dao ng ca mt s phng trỡnh ng lc trờn thang thi gian 77 3.1 Bt ng thc loi Lyapunov trờn thang thi gian 79 3.2 Tớnh dao ng ca phng trỡnh thun nht 84 3.3 Tớnh dao ng ca phng trỡnh khụng thun nht 92 Chng ng nht thc loi Picone trờn thang thi gian v ỏp dng 110 i 4.1 Mt s ng nht thc v bt ng thc loi Picone 112 4.2 Bt ng thc loi Wirtinger v loi Hardy trờn thang thi gian 118 4.3 nh lý Ried cho mt lp h ng lc cp mt 124 Kt lun 130 Danh mc cỏc cụng trỡnh ca tỏc gi 133 Ti liu tham kho 134 Ch mc 144 ii Danh mc cỏc kớ hiu T : Thang thi gian R : Tp cỏc s thc Z : Tp cỏc s nguyờn N : Tp cỏc s t nhiờn T \ ((sup T), sup T] nu sup T < : T nu sup T = T [a, b]T : [a, b] T : Toỏn t nhy tin : Toỏn t nhy lựi : Hm ht : Toỏn t o hm trờn thang thi gian f : f [a, b] T [a, b) T : (a, b] T (a, b) T I nu a < (a) v (b) < b, nu a < (a) v (b) = b, nu a = (a) v (b) < b, nu a = (a) v (b) = b I0 : (a, b) T Ia : [a, ) T n : Thang thi gian n chiu x : (x1 , , xn ) n xy : xj yj vi mi j [1, n]N : a ch s = (1 , , n ) : (1, , 1) (b) : (1 1 (b1 ), , nn (bn )) (hay [a, b]) : {x n : a x b} x : {x n : a x (b)} : {t n : a t x} iii x : {t n : x t (b)} : [a2 , b2 ]T2 ì ã ã ã ì [an , bn ]Tn f (x)x : f (x) x : AC(I) bn b1 ã ã ã an f (x1 , , xn )x1 a1 || f (x) x1 ãããn xnn ã ã ã xn : Tp tt c cỏc hm s nhn giỏ tr thc v liờn tc tuyt i trờn mi on úng ca I Crd (I) : Tp tt c cỏc hm s nhn giỏ tr thc v rd-liờn tc trờn I C1rd (I) : Tp tt c cỏc hm s nhn giỏ tr thc, xỏc nh trờn I cho cỏc o hm ca chỳng thuc lp Crd (I) Lp (I), p : Tp tt c cỏc hm s o c f xỏc nh trờn I cho I |f (x)|p x < Lp ([a, b]T , ), p : Tp tt c cỏc hm s f o c, xỏc nh trờn [a, b]T cho b |f (x)|p (x)x a < , ú W([a, b]T ) Lpa ([a, b]T , ), p : Tp tt c cỏc hm s f AC([a, b]T ) cho f Lp ([a, b]T , ) v f cú mt khụng im tng quỏt l a f g g : : f g 1+àg g 1+àg ef (ã, x0 ) : Nghim nht ca bi toỏn y = f (x)y, Gp (t), p > : |t|p1 sign(t) P(I0 ) : Tp tt c cỏc nghim (u, v) ca h ng lc phi tuyn u = Au + BG f (x0 ) = +1 (v) v = CG+1 (u ) Dv ú A, B, C v D thuc lp hm Crd (I0 ) vi B > v A, D R+ , cho u khụng cú khụng im tng quỏt I0 iv R : Tp tt c cỏc hm hi quy R+ : Tp tt c cỏc hm hi quy f tha + à(x)f (x) > vi mi x T U(a, b) : Tp tt c cỏc hm th Cn rd () : Tp tt c cỏc hm s f : R cú cỏc o hm riờng k1 +ããã+kj f (x) k k x1 ãããj xj j vi kj [1, j ]N , j [1, n]N l cỏc hm rd-liờn tc W() Lpa (, , ), p : Tp tt c cỏc hm trng trờn n () : Tp tt c cỏc hm s f : R thuc lp Crd cho v | kj f (x) = vi kj [0, j 1]N , j [1, n]N , k j xj j xj =aj f (x) p | (x)x x < , ú W() Lpa ([a, b], ), p : Tp tt c cỏc hm s f : [a, b] R thuc lp Cnrd1 ([a, b]) cho f cú khụng im tng quỏt l a v b f (x) p | x1 | (x)x a v < , ú W([a, b]) M u Bt ng thc khụng ch xut hin v úng mt vai trũ quan trng hu ht cỏc lnh vc ca toỏn hc thun tỳy, toỏn ng dng m cũn cú nhiu ng dng nhiu lnh vc khỏc ca cuc sng, chng hn nh khoa hc t nhiờn, khoa hc k thut v kinh t Cỏc bt ng thc hm l mt nhng c s quan trng xõy dng gii tớch núi chung v lnh vc phng trỡnh vi phõn, o hm riờng v tớch phõn núi riờng Trong lnh vc phng trỡnh vi phõn, tớch phõn v o hm riờng, cỏc bt ng thc tớch phõn cho toỏn t o hm l nhng cụng c vụ cựng hu hiu vic nghiờn cu cỏc tớnh cht nh tớnh v nh lng cho nghim ca cỏc lp phng trỡnh ny Mt s i din quan trng ca lp cỏc bt ng thc tớch phõn cho toỏn t o hm l cỏc bt ng thc Opial, Wirtinger v Hardy Di gúc gii tớch thun tỳy, cú th thy rng bt ng thc Opial l dng ni suy ca bt ng thc Poincarộ mt chiu vi mt s iu kin biờn no ú, bt ng thc Wirtinger l dng ca bt ng thc Poincarộ mt chiu i vi cỏc hm tun hon Nm 1960, Opial [63] nh toỏn hc ngi Ba Lan ó a bt ng thc b |f (x)f (x)|dx b b |f (x)|2 dx, (0.1) ú f l hm liờn tc tuyt i v xỏc nh trờn [0, b], nhn giỏ tr phc cho f (0) = f (b) = Trong Bt ng thc (0.1), 4b l hng s tt nht cú th Nm 1962, Beesack [17] ó chng minh rng: Nu f l hm liờn tc tuyt i v xỏc nh trờn [0, b], nhn giỏ tr phc cho f (0) = 0, thỡ b |f (x)f (x)|dx b b |f (x)|2 dx, (0.2) ú 2b l hng s tt nht cú th, ng thc xy v ch f (x) = cx, vi c l hng s Ngay sau ú, nhiu nh toỏn hc trờn th gii ó quan tõm nghiờn cu, phỏt trin, m rng v tng quỏt húa cỏc bt ng thc Opial (0.1) v (0.2) theo nhiu hng khỏc ng thi cng ó a cỏc dng ri rc tng ng Nm 1968, Willett [98] ln u tiờn a mt m rng cho Bt ng thc Kt lun Mc ớch chớnh ca Lun ỏn l thit lp mt s bt ng thc tớch phõn cho toỏn t o hm trờn thang thi gian nh bt ng thc loi Opial, loi Wirtinger, loi Hardy v cỏc ỏp dng ca chỳng lnh vc phng trỡnh v h phng trỡnh ng lc trờn thang thi gian Lun ỏn c chia lm chng khụng bao gm Mc lc, Danh mc cỏc kớ hiu, Ti liu tham kho v Danh mc cỏc cụng trỡnh ca tỏc gi Chng trỡnh by tt c s toỏn hc gii tớch trờn thang thi gian Cỏc Chng 2, v l cỏc chng trng tõm ca Lun ỏn Chng bao gm nhiu dng khỏc ca bt ng thc Opial trờn thang thi gian trng hp mt hoc nhiu bin cui chng, chỳng tụi a mt s ỏp dng ca bt ng thc loi Opial trờn thang thi gian Trong Chng 3, chỳng tụi trung nghiờn cu tớnh dao ng ca mt s phng trỡnh ng lc trờn thang thi gian bng cỏch xột bi toỏn v s phõn b ca cỏc khụng im tng quỏt ca nghim Chỳng tụi dnh Chng thit lp ng nht thc loi Picone trờn thang thi gian cho nghim ca mt lp h phng trỡnh ng lc phi tuyn cp v s dng nú c lng tiờn nghim cho mt s phng trỡnh v h phng trỡnh ng lc trờn thang thi gian Theo ú, chỳng tụi thu c mt s bt ng thc loi Wirtinger v loi Hardy mi trờn thang thi gian v ỏp dng chỳng nghiờn cu tớnh cht nh tớnh cho nghim ca h ng lc m chỳng tụi ang xột C th hn, Lun ỏn ó t c cỏc kt qu chớnh sau õy: Thit lp mt s bt ng thc tớch phõn cú trng cho toỏn t o hm tỏc ng lờn tớch cỏc hm s trờn thang thi gian, mt s bt ng thc tớch phõn cú trng cho toỏn t o hm tỏc ng lờn hp ca cỏc hm s trờn thang thi gian T ú, ỏp dng thit lp mt s bt ng thc loi Opial tng quỏt trờn thang thi gian Bờn cnh ú, chỳng tụi cũn ỏp dng cỏc kt qu mi ny thit lp mt s bt ng thc loi Lyapunov trờn thang thi gian, hu ớch v cn thit vic nghiờn cu cỏc liờn quan n nghim ca s mt lp phng trỡnh 130 ng lc na tuyn tớnh nh: tớnh khụng tiờu im, tớnh dao ng, cỏc chn di ca cỏc giỏ tr riờng, khong cỏch gia cỏc khụng im tng quỏt ca nghim, m s lng khụng im tng quỏt c bit, chỳng tụi ó gii mt bi toỏn m Saker t [84] Nhn c mt s kt qu v s phõn b cỏc khụng im tng quỏt ca nghim ca mt s lp phng trỡnh ng lc khụng thun nht trờn thang thi gian C th, nhn c mt s iu kin cho tớnh khụng tiờu im, tớnh khụng dao ng ca cỏc lp phng trỡnh ny Bờn cnh ú, thit lp mt s bt ng thc loi Lyapunov, loi Hartman trờn thang thi gian phc v cho vic nghiờn cu tớnh dao ng ca mt lp phng trỡnh ng lc khụng thun nht trờn thang thi gian Thit lp mt s bt ng thc loi Opial cho hm nhiu bin Thit lp mt s ng nht thc v bt ng thc loi Picone cho mt lp h phng trỡnh ng lc cp mt trờn thang thi gian c bit, kt qu ca chỳng tụi hiu chnh v tng quỏt húa kt qu ca Saker, Mahmoud v Peterson [87] T ú, chỳng tụi nhn c bt ng thc loi Wirtinger trờn thang thi gian v bt ng thc loi Hardy trờn thang thi gian Bờn cnh ú, chỳng tụi ỏp dng cỏc kt qu mi nghiờn cu nh tớnh nghim ca mt lp h phng trỡnh ng lc cp mt trờn thang thi gian C th, nhn c nh lý hoỏn v vũng quanh Ried, nh lý tỏch Sturm, nh lý so sỏnh Sturm v mt nguyờn lý bin phõn lý thuyt dao ng Cỏc kt qu trờn l mi v l nhng úng gúp thc s vo hng nghiờn cu v cỏc bt ng thc tớch phõn cho toỏn t o hm trờn thang thi gian v ỏp dng Chỳng cú ý ngha khoa hc, mang tớnh thi s v c s quan tõm ca nhiu tỏc gi lnh vc nghiờn cu ca Lun ỏn Trờn c s nhng kt qu thu c Lun ỏn, chỳng tụi d kin tng lai s nghiờn cu cỏc sau: Thit lp cỏc bt ng thc tớch phõn cho toỏn t o hm bc khụng nguyờn nh bt ng thc loi Opial, loi Wirtinger v loi Hardy trờn thang thi 131 gian v ỏp dng ca chỳng Thit lp ng nht thc v bt ng thc loi Picone cho cỏc h phng trỡnh vi phõn bc khụng nguyờn trờn thang thi gian Nghiờn cu tớnh dao ng ca cỏc phng trỡnh, h phng trỡnh vi phõn bc khụng nguyờn trờn thang thi gian Xõy dng mt s dng bt ng thc Lyapunov trờn thang thi gian cho mt s lp h phng trỡnh ng lc trờn thang thi gian T ú ỏp dng, 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ca mt s phng trỡnh ng lc trờn thang thi gian Chng 4: ng nht thc Picone trờn thang thi gian v... trờn thang thi gian, l cn thit cho vic nghiờn cu cỏc tớnh cht nh tớnh v nh lng cho nghim ca phng trỡnh ng lc trờn thang thi gian T ú nhiu nh toỏn hc ó quan tõm n lý thuyt bt ng thc trờn thang. .. C, (0, 1) khụng phi l cỏc thang thi gian Tụpụ trờn thang thi gian T l tụpụ cm sinh t tụpụ chun trờn cỏc s thc R nh ngha 1.2 ([22, Definition 1.1]) Cho T l mt thang thi gian tựy ý Toỏn t nhy tin