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 CHUYấN NGNH: TON GII TCH M S: 62.46.01.02 TểM TT LUN N TIN S TON HC BèNH NH - NM 2017 Cụng trỡnh c hon thnh ti: Trng i hc Quy Nhn Tp th hng dn: PGS TS inh Thanh c GS TSKH V Kim Tun Phn bin 1: Phn bin 2: Phn bin 3: Lun ỏn s c bo v trc Hi ng ỏnh giỏ lun ỏn ti Trng i hc Quy Nhn vo lỳc gi ngy thỏng nm 2017 Cú th tỡm hiu lun ỏn ti: -Th vin Quc gia Vit Nam -Trung tõm thụng tin t liu Trng i hc Quy Nhn 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 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 luụn 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 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 toỏn vi phõn 1.3 Phộp toỏn tớch phõn Chng Bt ng thc loi Opial trờn thang thi gian v ỏp dng 2.1 Bt ng thc loi Opial cho hm mt bin 2.2 Bt ng thc loi Opial cho hm nhiu bin 10 2.3 Mt s ỏp dng 13 Chng Tớnh dao ng ca mt s phng trỡnh ng lc trờn thang thi gian 14 3.1 Bt ng thc loi Lyapunov trờn thang thi gian 14 3.2 Tớnh dao ng ca phng trỡnh thun nht 15 3.3 Tớnh dao ng ca phng trỡnh khụng thun nht 17 Chng ng nht thc loi Picone trờn thang thi gian v ỏp dng 19 4.1 Mt s ng nht thc v bt ng thc loi Picone 19 4.2 Bt ng thc loi Wirtinger v loi Hardy trờn thang thi gian 21 4.3 nh lý Ried cho mt lp h ng lc cp mt 22 Kt lun 23 Danh mc cỏc cụng trỡnh ca tỏc gi 25 Ti liu tham kho 29 i 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 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 [34] 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 [8] 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 phỏt trin v m rng cỏc bt ng thc Opial (0.1) v (0.2) theo nhiu hng khỏc Nm 1968, Willett [57] ln u tiờn a bt ng thc loi Opial m rng Bt ng thc (0.2) theo hng nõng bc o hm Sau ú, Boyd [13], Das [17], Pachpatte [35] ó tip tc phỏt trin kt qu theo hng m rng ny nghiờn cu cỏc tớnh cht nh tớnh v nh lng ca cỏc phng trỡnh o hm riờng, Agarwal [1], Cheung [16], Yang [60], Pachpatte [37] ó m rng cỏc Bt ng thc (0.1) v (0.2) cho hm s nhiu bin Mt hng m rng khụng tm thng khỏc ú l xột cỏc trng hp khỏc i vi cỏc s m ca hm s v o hm ca nú Theo hng nghiờn cu ny, cú mt s cụng trỡnh tiờn phong ca Hua [20] v Yang [59] Nm 1972, Godunova v Levin [19] ó a cỏc dng m rng cho cỏc Bt ng thc (0.1) v (0.2) liờn quan n hm li Cỏc kt qu ny ó c Pecaric [39], Pachpatte [37], v Andric cựng cỏc cng s [7] m rng cho hm s nhiu bin Bt ng thc Opial dng ri rc ó c Lasota [26] xut vo nm 1968 C th, Lasota ó a cỏc dng ri rc tng ng vi cỏc Bt ng thc (0.1) v (0.2) nh sau: Cho {xi }N i=0 l mt dóy s thc Nu x0 = xN = 0, thỡ N N 1 N +1 |xi |2 , (0.3) |xi xi | 2 i=0 i=0 ú l toỏn t sai phõn tin v [ã] l hm phn nguyờn Nu x0 = thỡ N |xi xi | i=0 N N |xi |2 i=0 (0.4) Sau ú, cỏc Bt ng thc (0.3) v (0.4) ó c m rng bi Lee [27] v Pachpatte [36] Cỏc bt ng thc Opial cựng vi cỏc dng m rng ca chỳng ó c chng minh l mang tớnh ng dng rng rói nhiu lnh vc ca toỏn hc khụng ch k tha ý tng t bt ng thc Poincarộ m cũn chớnh bn thõn cỏc bin th C th, cỏc bt ng thc loi Opial l cụng c hu ớch vic chng minh s tn ti v tớnh nht nghim, xột tớnh b chn, tớnh n nh v tớnh dao ng ca nghim, nghiờn cu cỏc bi toỏn v giỏ tr riờng v nhiu khỏc Trong [6], Agarwal v Pang ó tng hp rt nhiu dng m rng khỏc ca bt ng thc Opial v cỏc ỏp dng ca chỳng Gn õy, Nhõn, c, v Tun [30] (2013) ó a mt bt ng thc loi Opial tng quỏt cú dng m b Gj fj n (x) a s (n) s (x)dx m b N j=1 Gj a j=1 (n) |fj (x)|p n (x)j (x)dx p (0.5) Dng ri rc v m rng cho hm nhiu bin ca Bt ng thc (0.5) ln lt c Nhõn, c, Tun, v V [32], c, Nhõn v Xuõn [18] a sau ú Nm 1988, nh toỏn hc Hilger ngi c lun ỏn tin s ca mỡnh ó a lý thuyt gii tớch trờn thang thi gian nhm mc ớch thng nht gii tớch liờn tc v gii tớch ri rc S i ca gii tớch trờn thang thi gian l mt ũi hi mang tớnh tt yu bi rt nhiu mụ hỡnh toỏn hc thc t luụn ũi hi d liu va cú tớnh liờn tc va cú th ri rc Gii tớch trờn thang thi gian cho phộp chỳng ta cú th nghiờn cu cỏc phng trỡnh trờn mt úng tựy ý ca R, m ta gi l thang thi gian, thng c kớ hiu l T Cỏc vớ d in hỡnh nht ca thang thi gian T ú l R, Z v qZ = {q t , t Z} {0} ú q > Do ú vic thit lp cỏc bt ng thc 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 thi gian Vic nghiờn cu cỏc bt ng thc loi Opial trờn thang thi gian c Bohner v Kaymakácalan [9], Agarwal, Bohner v Peterson [2] xng vo nm 2001: Nu hm s f : [0, b]T R l hm kh vi cho f (0) = 0, thỡ b b |[f (x) + f (x)]f (x)|x b |f (x)|2 x (0.6) Ngay sau ú, Bt ng thc (0.6) ó thu hỳt c s quan tõm ca nhiu nhúm nghiờn cu khỏc Cú hai dng m rng t nhiờn nht cho Bt ng thc (0.6) ú l cỏc bt ng thc loi Opial b + p b |f (x)| |f (x)| (x)x S1 a p |f (x)| (x)x (0.7) a v b b |f (x) + f (x)| |f (x)| (x)x S2 a p |f (x)| (x)x + p , (0.8) a f (a) = hoc/v f (b) = 0, ú p > 1, > 0, > 0, S1 , S2 l cỏc hng s Hai dng bt ng thc ny c nghiờn cu cỏc cụng trỡnh [10, 24, 25, 29, 45, 46, 47, 48, 52, 54, 55, 61, 62] Sau ú, cỏc dng (0.7) v (0.8) ó c phỏt trin theo mt s hng nh: nõng bc o hm (xem Wong v cỏc cng s [58], Sirvastava v cỏc cng s [54], Saker, Agarwal, v Oregan [50], Saker [48]), cho nhiu ă hm s (xem Karpuz v Ozkan [25]), cho hm s nhiu bin (xem Agarwal, Oregan v Saker [5]) T nhng bt ng thc gn õy ca Nhõn, c, v Tun [30, 32] v nhiu nh toỏn hc khỏc trờn th gii cho trng hp liờn tc, ri rc cựng vi nhng ng dng khỏc ca chỳng lnh vc phng trỡnh vi phõn, sai phõn, ta cú th thy rng s i ca chỳng cú nht nh vic phỏt trin lý thuyt phng trỡnh vi phõn v sai phõn S phỏt trin mnh m ca nhiu bt ng thc trờn thang thi gian thi gian gn õy ó gúp phn vo vic phỏt trin lý thuyt phng trỡnh ng lc trờn thang thi gian Vỡ l ú, vic nghiờn cu nhm ci tin v xut cỏc bt ng thc mi trờn thang thi gian luụn l quan trng v cú tớnh cp thit lnh vc gii tớch núi chung m c bit l vic nghiờn cu lý thuyt phng trỡnh ng lc núi riờng Tip ni nhng kt qu gn õy, chỳng tụi tip tc phỏt trin v m rng cỏc Bt ng thc loi Opial (0.7) v (0.8) cho hm mt bin v hm nhiu bin trờn c s s dng phng phỏp v ý tng c xut bi Nhõn, c v Tun [30, 32] cựng vi lý thuyt gii tớch trờn thang thi gian Hn na, chỳng tụi cũn ỏp dng cỏc kt qu mi thit lp mt s bt ng thc loi Lyapunov mi, hu ớch vic nghiờn cu s phõn b cỏc khụng im tng quỏt ca nghim ca mt s lp phng trỡnh ng lc Tuy nhiờn, vic xõy dng nhng kt qu mi ny l khụng h d dng m chỳng ũi hi nhng k thut cao, tớnh toỏn phc nhm khc phc nhng im khỏc bit gia liờn tc v ri rc Cựng vi bt ng thc Opial, mt cụng c quan trng khỏc nghiờn cu lý thuyt phng trỡnh vi phõn, tớch phõn, v o hm riờng ú l ng nht thc Picone, c chớnh nh toỏn hc ngi í Picone xut nm 1910 [40] nh sau: d u (P0 (t)u v P1 (t)uv ) = (P0 (t) P1 (t))(u )2 + (Q1 (t) Q0 (t))u2 dt v u u + P1 (t) v + (v [u] uL2 [v]) v v (0.9) T ú, Picone ó s dng nú chng minh cỏc nh lý so sỏnh Sturm cho cỏc toỏn t vi phõn [u] L2 [v] = (P1 (t)v ) + Q1 (t)v = (P0 (t)u ) + Q0 (t)u, ng nht thc Picone khụng ch l mt cụng c rt mnh nghiờn cu tớnh dao ng nghim ca cỏc phng trỡnh vi phõn, nghiờn cu cỏc bi toỏn giỏ tr riờng phng trỡnh vi phõn m cũn c dựng thit lp cỏc bt ng thc tớch phõn liờn quan ti cỏc hm s v o hm ca nú chng hn nh cỏc bt ng thc loi Wirtinger v loi Hardy K t i, ng nht thc Picone ó nhn c nhiu s quan tõm nghiờn cu, m rng theo nhiu hng khỏc Nm 1999, Jaros v Kusano [23] ó m rng ng nht thc Picone (0.9) cho cỏc toỏn t vi phõn na tuyn tớnh cp hai [u] = (P0 G+1 (u )) + Q0 G+1 (u) L [v] = (P1 G+1 (u )) + Q1 G+1 (u), ú > v Gp (t) = |t|p1 sign(t) vi p > v ỏp dng nú nghiờn cu lý thuyt Sturm cho cỏc phng trỡnh thun nht [u] = v khụng thun nht [u] = f Cỏc phiờn bn ri rc v thang thi gian ca ng nht thc Picone ln lt c Rehỏk [43], Agarwal, Bohner v Rehỏk [3] a sau ú Nm 2011, Jaros [21] ó a ng nht thc loi Picone cho toỏn t L [v] t ú thit lp mt bt ng thc loi Wirtinger Gn õy, Jaros [22] (2013) v Tiryaki [56] (2015) ó thu c cỏc ng nht thc v bt ng thc loi Picone cho h phng trỡnh vi phõn phi tuyn u = A(t)u + B(t)G (v) +1 (0.10) v = C(t)G (u) D(t)v +1 T ú, ỏp dng chỳng nhn c mt s bt ng thc loi Wirtinger Tuy nhiờn cỏc kt qu ny cũn mt s hn ch cn phi khc phc Mt cõu hi t nhiờn c t l: Chỳng ta cú th xõy dng c cỏc ng nht thc loi Picone cho cỏc hm s xỏc nh trờn thang thi gian t ú thit lp cỏc bt ng thc loi Wirtinger, m cỏc kt qu ny thng nht cỏc kt qu dng liờn tc v dng ri rc ó c cp trờn hay khụng? N lc u tiờn tr li cõu hi ny thuc v Agarwal cựng cỏc cng s [4] H ó thit lp bt ng thc loi Wirtinger bng cỏch nghiờn cu mt lp bt phng trỡnh vi phõn tuyn tớnh cp hai Gn õy nht, vo nm 2015, Saker, Mahmoud v Peterson [51] ó c gng m rng ng nht thc loi Picone v bt ng thc loi Wirtinger c thit lp bi Jaros [21] cho cỏc hm s xỏc nh trờn thang thi gian Tuy nhiờn, Cụng trỡnh [51] ca h cú mt vi thiu sút, cỏc kt qu [51, Theorem 2.1] khụng chớnh xỏc Do ú, chỳng tụi nhn thy rng vic xõy dng cỏc ng nht thc v bt ng thc loi Picone mi trờn thang thi gian l thc s cn thit, cú tớnh thi s v cú ý ngha khoa hc Trong lun ỏn ny, chỳng tụi va hiu chnh cỏc kt qu Cụng trỡnh [51] va m rng chỳng bng cỏch thit lp mt s ng nht thc v bt ng thc loi Picone mi cho h ng lc phi tuyn u = Au + BG (v) +1 v = CG (u ) Dv +1 T ú, chỳng tụi s dng chỳng thu c cỏc c lng tiờn nghim cho mt s phng trỡnh v h phng trỡnh ng lc trờn thang thi gian Theo hng nghiờn cu ny, chỳng tụi thit lp mt s bt ng thc loi Wirtinger v loi Hardy mi trờn thang thi gian, hu ớch vic nghiờn cu cỏc tớnh cht nh tớnh cho nghim ca h ng lc m chỳng tụi ang xột C th, chỳng tụi thu c cỏc kt qu nh nh lý hoỏn v vũng quanh Reid, nh lý so sỏnh Sturm, nh lý tỏch Sturm v mt nguyờn lý bin phõn lý thuyt dao ng Mc ớch chớnh ca Lun ỏn l xõy dng mt s bt ng thc tớch phõn cho toỏn t o hm trờn thang thi gian nh bt ng thc loi Opial, bt ng thc loi Wirtinger, bt ng thc loi Hardy v cỏc ỏp dng ca chỳng Lun ỏn, ngoi phn M u, Kt lun v Ti liu tham kho, gm cú chng: Chng 1: Mt s kin thc c bn v gii tớch trờn thang thi gian Chng 2: Bt ng thc Opial trờn thang thi gian v ỏp dng Chng 3: Tớnh dao ng ca mt s phng trỡnh ng lc trờn thang thi gian Chng 4: ng nht thc Picone trờn thang thi gian v ỏp dng Bỡnh nh, thỏng 05 nm 2017 Tỏc gi Trn ỡnh Phng Chng Mt s kin thc c bn v gii tớch trờn thang thi gian Trong chng ny, chỳng tụi trỡnh by mt s kin thc c bn v thang thi gian, phộp tớch vi phõn v tớch phõn trờn thang thi gian Hu ht cỏc kt qu chng ny c chỳng tụi trớch dn t cỏc cụng trỡnh [11, 12] ca Bohner v Peterson, [14, 15] ca Cabada v Vivero 1.1 Cỏc nh ngha c bn nh ngha 1.1 ([11]) Thang thi gian (time scale) T l mt khụng rng, úng ca R Nh vy, cỏc R, Z, N, qZ := {q k : k Z} {0} vi q > 1, hZ := {hk : k Z} vi h > 0, Cantor l cỏc thang thi gian Cỏc Q, R \ Q, 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 ([11]) Cho T l mt thang thi gian tựy ý Toỏn t nhy tin (forward jump operator) : T T c nh ngha nh sau: (t) := inf{s T : s > t} vi mi t T Toỏn t nhy lựi (backward jump operator) : T T c nh ngha nh sau: (t) := sup{s T : s < t} vi mi t T Hm ht (graininess function) : T R+ c nh ngha nh sau: à(t) := (t) t vi mi t T Trong nh ngha ny ta t inf := sup T (tc l (t) = t nu t l im ln nht ca T) v sup := inf T (tc l (t) = t nu t l im nh nht ca T), ú l rng nh ngha 1.3 ([11]) Cho T l mt thang thi gian, t T Nu (t) > t thỡ t c gi l im cụ lp phi (right-scattered) Nu (t) < t thỡ t c gi l im cụ lp trỏi (left-scattered) Nu t va l im cụ lp phi va l im cụ lp trỏi thỡ t c gi l im cụ lp (isolated) Tng t, nu f l mt nghim khụng tm thng ca Phng trỡnh (3.4) cho f Lpb ([c, (b)]T , ) v f (c)f (c) 0, thỡ T (c, 1) sup |(x)| + p1 (c) c (x)| | sup x[c,b]T > (3.14) x[c,(b)]T nh lý 3.3 Phng trỡnh (3.4) khụng tiờu im trờn [a, (b)]T nu (a, (b), a ) max T0 ((b), ) + W0 (a, (b), (b) ), T0 (a, ) + W (3.15) nh lý 3.4 Nu Phng trỡnh (3.4) cú mt nghim khụng tm thng f vi hai khụng im tng quỏt liờn tip a v b, thỡ tn ti c, (d) [a, b]T , c (d), cho T (c, ) + W (a, c, c ) > (3.16) (d, (b), d ) > T (d, ) + W (3.14) v H qu 3.5 Gi s vi mi c, (d) [a, b]T , c (d), ta cú (d, (b), d ) max T0 (c, ) + W0 (a, c, c ), T0 (d, ) + W (3.17) Khi ú, Phng trỡnh (3.4) khụng dao ng trờn [a, b]T 3.2 Tớnh dao ng ca phng trỡnh thun nht Xột phng trỡnh ( Gp (f )) + Gp (f ) = (3.19) nh lý 3.6 Nu f Lpa ([a, c]T , ) l mt nghim khụng tm thng ca Phng trỡnh (3.19) v f (c)f (c) 0, thỡ p1 (c) c sup (t)t > (3.20) x x[a,c]T Hn na, f khụng cú cc tr (a, c)T , thỡ p1 (c) c sup (t)t x[a,c]T > (3.21) x Nu f l mt nghim khụng tm thng ca Phng trỡnh (3.19) tha f Lpb ([c, (b)]T , ) v f (c)f (c) 0, thỡ x p1 (c) sup (t)t x[c,(b)]T > (3.22) c Hn na, nu f khụng cú cc tr (c, b]T , thỡ x p1 (c) sup (t)t x[c,(b)]T 15 c > (3.23) H qu 3.7 Phng trỡnh (3.19) khụng tiờu im trờn [a, (b)]T nu (b) max (t)t , sup x[a,b]T x x sup (t)t a x[a,(b)]T 1p (b) q p (x)x (3.24) a nh lý 3.8 Cho a v b l hai khụng im tng quỏt ca mt nghim khụng tm thng f ca Phng trỡnh (3.19) Khi ú, tn ti hai on ri I1 v I2 ca [a, b]T , tha 1p (b) (x)x > p (x) (x) I1 I2 q p (x)x , (3.25) a (x)x (3.26) [a,(b)]T \(I1 I2 ) H qu 3.9 Phng trỡnh (3.19) l khụng dao ng trờn [a, b]T nu vi mi on I1 v I2 ca [a, (b)]T , cho 1p (b) (x)x p I1 I2 q p (x)x (3.31) a H qu 3.10 T h qu trờn ta nhn c phiờn bn trờn thang thi gian ca bt ng thc Lyapunov nh lý 3.11 Gi s tn ti mt nghim khụng tm thng ca Phng trỡnh (3.19) cú (n + 1) khụng im tng quỏt trờn [a, b]T Khi ú tn ti 2n on ri Ij1 v Ij2 , j = 1, , n ca [a, b]T cho (b) n< q p p p (x)x (x)x a (3.34) I (x)x (3.35) [a,(b)]T \n j=1 (Ij1 Ij2 ) nh lý 3.12 Gi n l giỏ tr riờng th n ca phng trỡnh ( Gp (f )) (x) = (x)Gp (f (x)), x (a, b)T , (3.39) ú a v b l cỏc khụng im tng quỏt ca f Khi ú tn ti 2n on ri Ij1 v Ij2 , j = 1, , n ca [a, b]T , cho 1p (b) p p n > n q p (x)x (x)x a (3.40) I (x)x (3.41) [a,(b)]T \n j=1 (Ij1 Ij2 ) nh lý 3.13 Nu f l mt nghim dao ng ca Phng trỡnh (3.19), K > trờn [a, )T , v lim sup x p1 K (t)t < 2p (3.42) (I1 I2 )(x,) vi mi > v vi bt k hai on ri I1 v I2 ca [x, x + ]T , thỡ khong cỏch gia hai khụng im liờn tip ca f khụng b chn x nh lý 3.14 Cho f l mt nghim dao ng ca Phng trỡnh (3.19) Nu K > trờn [a, )T v tn ti > cho vi mi on ri I1 v I2 ca [x, x + ]T cho lim x (t)t = 0, (I1 I2 )(x,0 ) thỡ khong cỏch gia hai khụng im tng quỏt liờn tip ca f phi dn n vụ cựng x 16 (3.43) 3.3 Tớnh dao ng ca phng trỡnh khụng thun nht Xột cỏc phng trỡnh sau ( Gp (f )) + Gp (f ) + Gp (f ) = h, < p (3.47) < k < 2p (3.48) n k Gk (f ) = h, ( Gp (f )) + k=1 ( Gp (f )) + G (f ) = h, < < 2p (3.49) nh lý 3.15 Nu f l mt nghim khụng tm thng ca Phng trỡnh (3.47) thuc vo lp hm Lpa ([a, c]T , ) v f (c)f (c) 0, thỡ 1 T (c, ) + 2(1)(p1) W (a, c, c )V (a, c) + ú c (x) = c x (t)t, Hc (x) = T (c, ) = x c h(t)t, p W (a, c, c ) c ( (x)) x | (x)| c a = (x)p1 (x)|(x)|q x c , , q c (a) (t) (t)t (3.52) (1)(p1) p p c V (a, c) = p L (a, c, Hc ) > , L (a, c, Hc ) = q p , q q (x)|Hc (x)| (x)x a a Tng t, nu f l mt nghim khụng tm thng ca Phng trỡnh (3.47) thuc vo lp hm v f (c)f (c) 0, thỡ Lpb ([c, (b)]T , ) 1 c )V (c, (b)) + T (c, ) + 2(1)(p1) W (c, (b), c (x) = ú x c (t)t, c (x) = H T (c, ) = p (t) (t)t (b) |c (x)| ( (x)) x c b (x) p1 (x)|(x)|q x (3.53) (1)(p1) W (c, (b), c ) = p , q , c p p (b) V (c, (b)) = c x h(t)t, (c, (b), H c ) > , L , (b) (c, (b), H c) = L c q p c (x)|q (x)x (x)|H q c nh lý 3.17 Phng trỡnh (3.47) khụng tiờu im trờn [a, (b)]T nu 1 T0 ((b), ) + 2(1)(p1) W0 (a, (b), (b) )V0 (a, (b)) + L0 (a, (b), H(b) ) , (3.61) (1)(p1) v 1 a )V0 (a, (b)) + T0 (a, ) + 2(1)(p1) W (a, (b), 17 (1)(p1) (a, (b), H a ) L (3.62) nh lý 3.18 Nu Phng trỡnh (3.47) cú mt nghim khụng tm thng f vi hai khụng im tng quỏt liờn tip a v b thỡ tn ti c, (d) [a, b]T , c (d) cho 1 T (c, ) + 2(1)(p1) W (a, c, c )V (a, c) + (3.63) (1)(p1) 1 d )V (c, (b)) + T (d, ) + 2(1)(p1) W (d, (b), L (a, c, Hc ) > , (d, (b), H d ) > L (3.64) (1)(p1) H qu 3.19 Gi s vi mi c, (d) [a, b]T , c (d), ta cú 1 T0 (c, ) + 2(1)(p1) W0 (a, c, c )V (a, c) + v L0 (a, c, Hc ) , (1)(p1) 1 T0 (d, ) + 2(1)(p1) W (d, (b), d )V (c, (b)) + (d, (b), H c ) L (1)(p1) Khi ú, Phng trỡnh (3.47) khụng dao ng trờn [a, b]T nh lý 3.20 Gi s f l mt nghim khụng tm thng ca Phng trỡnh (3.48) tha f Lpa ([a, c]T , ) v f (c)f (c) Nu f khụng cú cc tr (a, c)T , thỡ tn ti n + on Ik , k = 1, , n + ca [a, c]T tha n n k (t)t k=1 Ik (k 1)(p1) k (t)t + Ik k=1 h(t)t > In+1 c k (t)t 0, k = 1, , n, q p 2(1p) (t) (t)t , (3.65) a h(t)t (3.66) [a,c]T \In+1 [a,c]T \Ik Tng t, gi s f l mt nghim khụng tm thng ca (3.48) cho f Lpb ([c, (b)]T , ) v f (c)f (c) 0, hn na nu f khụng cú cc tr (c, (b))T , thỡ tn ti n+1 on Jk , k = 1, , n+1 ca [c, (b)]T cho n n k (t)t k=1 Jk (k 1)(p1) k=1 k (t)t + Jk h(t)t > Jn+1 (b) q p 2(1p) , (t) (t)x c (3.67) (t)t 0, k = 1, , n, [c,(b)]T \Jk h(t)t (3.68) [c,(b)]T \Jn+1 nh lý 3.21 Gi s a v b l hai khụng im tng quỏt liờn tip ca mt nghim khụng tm thng ca Phng trỡnh (3.49) Khi ú, tn ti bn on ca [a, (b)]T l I1 , I2 , J1 , v J2 tha I1 J1 = I2 J2 = v (b) 2(1)(p1) (x)x + I1 J1 I2 J2 h(x)x > 2p (1)(p1) 18 a q p 1p (x) (x) (x)x (3.74) Chng ng nht thc loi Picone trờn thang thi gian v ỏp dng Chỳng tụi dnh chng ny 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 Cỏc kt qu mi chng ny c trớch t cụng trỡnh [31] Xột h ng lc cú dng u = Au + BG (v) +1 v = CG (u ) Dv (4.1) +1 4.1 Mt s ng nht thc v bt ng thc loi Picone nh lý 4.1 ([11]) Cho x R v t0 T c nh Khi ú bi toỏn y = x(t)y, y(t0 ) = (4.2) cú mt nghim nht trờn T B 4.1 (Bt ng thc Gronwall [11, Corollary 6.7]) Cho y Crd (T, R), u R+ , u v R Nu cú bt ng thc t y(t) + y()u() vi mi t T t0 thỡ ta cng cú bt ng thc y(t) eu (t, t0 ) vi mi t T nh lý 4.2 Cho x AC(I0 ), (u, v) P(I0 ), E R+ v t 1 [1 sA(t)] +1 [1 sE(t)] +1 EA (t) := lim , sà(t) s sT 19 t I0 (4.3) Khi ú, tn ti mt hm khụng õm (x, u, v, EA ) cho ng nht thc |x|+1 v G+1 (u) |x |+1 v |x EA x |+1 +1 C|x | + (E D) (x, u, v, EA ) B G+1 (u ) = (4.4) xy hu khp ni trờn I0 c bit, nu E = D, thỡ |x|+1 v G+1 (u) |x DA x |+1 C|x |+1 , B hu khp ni trờn I0 (4.5) ng thc (4.5) xy nu v ch nu x(t) = Ku(t)e(A) (DA ) (t, t0 ), hu khp ni trờn I0 , (4.6) vi K l hng s v t0 I0 B 4.2 Nu sj vi j = 1, , l cỏc s thc cho s3 s4 > 0, thỡ |s1 |+1 |s2 |+1 G+1 (s3 ) G+1 (s4 ) G+1 (s3 s4 ) |s1 s2 |+1 (4.7) ng thc xy v ch s1 s4 = s2 s3 H qu 4.3 Cho x AC(I0 ) v u l mt nghim dng ca bt phng trỡnh (P G+1 (u )) + QG+1 (u ) + RG+1 (u ) 0, (4.11) ú > 0, l mt s thc v R Crd (I0 ) Khi ú, |x|+1 P G+1 (u ) G+1 (u) P |u | |x |+1 |x |+1 (Q|u | + R), (4.12) ng thc xy v ch x v u t l vi Xột h ng lc cú dng x = A1 x + B1 G (y), +1 y = C G (x ) D y, +1 (4.14) H qu 4.4 Cho (x, y) l nghim ca h (4.14), (u, v) P(I0 ) v t (1 sA(t)) (1 sA1 (t))+1 , sà(t) s(1 sA(t)) E(t) := lim t I0 (4.15) sT Khi ú, tn ti mt hm khụng õm (x, u, v, A1 ) cho |x|+1 v xy G+1 (u) = (x, u, v, A1 ) + (A1 D1 )x y + (C C1 )|x |+1 + 1 B1 B +1 |x A1 x | |x |+1 v (E D) G+1 (u ) (4.16) xy trờn I Do ú, xy |x|+1 v G+1 (u) + |A1 D1 | |x |+1 +1 1 |A1 D1 | |x A1 x |+1 B1 B + B1+1 C C1 20 (4.17) 4.2 Bt ng thc loi Wirtinger v loi Hardy trờn thang thi gian nh ngha cỏc hm s (x; a, b), a (x; u, v), v b (x; u, v) nh sau: ab (x) nu a < (a) v nu a < (a) v limdb ad (x) dT (x; a, b) := limca+ cb (x) nu a = (a) v cT (a) v limca+ ,db cd (x) nu a = (b) < b, (b) = b, (b) < b, (4.19) (b) = b, c,dT ú d cd (x) := c a (x; u, v) := |x DA x |+1 C|x |+1 t, B limta+ |x|+1 v G+1 (u) tT a(a) (x) v b (x; u, v) := limtb tT |x|+1 v G+1 (u) (t) |x|+1 v G+1 (u) |x|+1 v G+1 (u) (t) nu a = (a), (4.20) ((a)) nu a < (a), nu (b) = b, (4.21) ((b)) + (b)b (x) nu (b) < b nh lý 4.5 Cho (u, v) P(I0 ) v [1 sA(t)] +1 [1 sD(t)] +1 DA (t) := lim , sà(t) s t I0 (4.22) sT Nu x AC(I0 ) cho (4.19), (4.20), v (4.21) tn ti v hu hn, thỡ (x; a, b) a (x; u, v) + b (x; u, v) (4.23) ng thc (4.23) xy nu v ch nu x tha x(t) = Ku(t)e(A) (DA ) (t, t0 ), hu khp ni trờn I0 , (4.24) vi K l s thc v t0 I0 B 4.3 Nu A, B, C, D Crd (I), A R+ , v u(a) = v(a) = 0, thỡ (4.1) ch cú nghim tm thng trờn I nh lý 4.6 Cho I b chn, (u, v) P(I0 ) v x AC(I) cho x , x L+1 (I ) v x(t) = {a, b} t l im trự mt v u(t) = Khi ú, (x; a, b) a (x; u, v) + b (x; u, v) (4.31) Hn na, nu u(a)u (a) v x(a) = x(b) = 0, thỡ (x; a, b) (4.32) ng thc (4.32) xy v ch tn ti K R v t0 I cho x(t) = Ku(t)e(A) (DA ) (t, t0 ), 21 hu khp ni trờn I (4.33) nh lý 4.7 Gi s a T hu hn v H phng trỡnh (4.1) cú mt nghim (u, v) tha uv > trờn b I0 Khi ú, vi hm s f cho e (DA ) (t, )f (.) L1 (I) vi mi t I v a B |f |+1 t < , ta cú b +1 (t) C(t) a e a b t (DA ) (t, s)f (s)s a |f (t)|+1 t B (t) (4.36) nh lý 4.8 Cho a, b T hu hn v gi s H phng trỡnh (4.1) cú mt nghim (u, v) P(I0 ) Cho b b f tha e (DA ) (t, )f (.) L1 (I) vi t I, a e (DA ) (b, t)f (t)t = 0, v a B |f |+1 t < Khi ú, b C(t) a +1 (t) e a (DA ) (t, s)f (s)s b t a |f (t)|+1 t B (t) (4.37) 4.3 nh lý Ried cho mt lp h ng lc cp mt nh lý 4.9 (nh lý hoỏn v vũng quanh Reid) Cỏc khng nh sau õy tng ng: (i) H phng trỡnh (4.1) khụng dao ng trờn I (ii) H phng trỡnh (4.1) cú mt nghim (u, v) cho uu > trờn I , tc l u khụng cú khụng im tng quỏt trờn I (iii) (x; a, b) khụng õm trờn U(a, b) H qu 4.10 (nh lý tỏch Sturm) Cho (u1 , v1 ) v (u2 , v2 ) l hai nghim khụng tm thng ca H phng trỡnh (4.1) cho u1 (t) v u2 (t)e(A) (DA ) (t, a) khụng t l trờn I Khi ú, gia hai khụng im tng quỏt bt kỡ ca uj trờn I tn ti ớt nht mt khụng im tng quỏt ca ui vi i, j = 1, 2, i = j H qu 4.11 (nh lý so sỏnh Sturm) Gi s A1 = DA , C1 C, v B1 B trờn I Nu H phng trỡnh (4.14) khụng dao ng trờn I, thỡ H phng trỡnh (4.1) cng vy H qu 4.12 (Nguyờn lý bin phõn) H phng trỡnh (4.1) khụng dao ng nu v ch nu tn ti a T cho (x; a, ) > vi mi nghim khụng tm thng x U(a), ú U(a) = {x AC(Ia ) : b > a cho x(t) = nu t / (a, b)T } Mt khỏc, H phng trỡnh (4.1) dao ng nu v ch nu vi bt kỡ a T tn ti mt hm th c khụng tm thng x U(a) cho (x; a, ) 22 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 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 [49] 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 [51] T ú, chỳng tụi nhn c bt ng thc loi Wirtinger trờn thang 23 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 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, nghiờn cu tớnh dao ng nghim ca cỏc lp h phng trỡnh ny Thit lp mt s bt ng thc hm quan trng nh Poincarộ, Poincarộ-Wirtinger, Hardy-Littlewood, trờn thang thi gian v ng dng ca chỳng Thit lp cỏc bt ng thc ma trn trờn thang thi gian, nghiờn cu cỏc phng trỡnh ma trn trờn thang thi gian Thit lp cỏc tiờu chun dao ng loi Leighton-Wintner v loi Hinton-Lewis cho mt s lp h phng trỡnh ng lc phi tuyn 24 Danh mc cụng trỡnh ca tỏc gi (1) N D V Nhan, D T Duc V K Tuan, and T D Phung (2016), Dynamic Picones identity and its applications, J Math Anal Appl., (ó gi ng) (2) N D V Nhan, T D Phung, D T Duc, and V K Tuan (2016), Opial and Lyapunov inequalities on time scales and their applications to dynamic equations, Kodai Math J., 40, 254-277 (3) T D Phung (2016), Some inequalities for partial derivatives on time scales, Acta Math Vietnam., 42, 369-394 (4) T D Phung and D T Duc (2017), New Opial-type inequalities on time scales, Tp khoa hc-Trng H Quy Nhn, Tp 11, S 1, 47-59 (5) T D Phung, N D V Nhan, D T Duc, and V K Tuan (2016), Disfocality and disconjugacy for second order forced dynamic equations, (tin n phm) 25 Ti liu tham kho [1] R P Agarwal (1983), An integro-differential inequality, General Inequalities III, eds E F Beckenbach and W Walter, Birkhă auser, Basel, 503, 501503 [2] R Agarwal, M Bohner, and A Peterson (2001), Inequalities on time scales: A survey, Math Inequal Appl., 4, 535557 [3] R P Agarwal, M Bohner, and P Rehỏk (2003), Half-linear dynamic equations, Nonlinear Analysis and Applications: To V Lakshmikantham on his 80th Birthday, Vol 1,2, 157 (Kluwer Academic Publishers, Dordrecht, 2003) [4] R P Agarwal, V O Espinar, K Perera, and D R Vivero (2008), Wirtingers inequalities on time scales, Canad Math Bull., 51, 161171 [5] R Agarwal, D Oregan, and S H Saker (2014), Dynamic Inequalities on Time Scales, Springer International Publishing, Switzerland [6] R P Agarwal and P Y H Pang (1995), Opial Inequalities with Applications in Differential and Difference Equations, Kluwer Academic Publishers, Dordrecht [7] M Andric, A Barbir, and J Pecaric (2014), Generalizations of Opial-type inequalities in several independent variables, Demonstr Math., XLVII, 839847 [8] P R Beesack (1962), On an integral inequality of Z Opial, Trans A mer Math Soc., 104, 470475 [9] M Bohner and B Kaymakácalan (2001), Opial inequalities on time scales, Ann Polon Math., 77, 1120 [10] M Bohner, R R Mahmoud, and S H Saker (2015), Improvements of dynamic Opial-type inequalities and applications, Dynamic Syst Appl., 24, 229242 [11] M Bohner and A Peterson (2001), Dynamic Equations on Time Scales: An Introduction with Applications, Birkhă auser, Boston [12] M Bohner and A Peterson (Eds.) (2003), Advances in Dynamic Equations on Time Scales, Birkhăauser, Boston [13] D W Boyd (1969), Best constants in inequalities related to Opials inequality, J Math Anal Appl., 25, 378387 [14] A Cabada and D R Vivero (2005), Criterions for absolute continuity on time scales, J Difference Equ Appl., 11, 10131028 26 [15] A Cabada and D R Vivero (2006), Expression of the Lebesgue -integral on time scales as a usual Lebesgue integral: application to the calculus of -antiderivatives, Math Comput Modelling., 43, 194207 [16] W S Cheung (1989), On Opial-type inequalities in two variables, Aequationes Math., 38, 236244 [17] K M Das (1969), An inequality similar to Opials inequality, Math Soc., 22, 258261 [18] D T Duc, N D V Nhan, and N T Xuan (2015), Inequalities for partial derivatives and their applications, Canad Math Bull., 58, 486496 [19] E K Godunova and V I Levin (1967), On an inequality of Maroni, Mat Zametki., 2, 221224 [20] L K Hua (1965), On an inequality of Opial, Scientia Sinica, 14, 789790 [21] J Jaros (2011), On an integral inequality of the Wirtinger type, Appl Math Lett., 24, 13891392 [22] J Jaros (2013), Wirtinger inequality and nonlinear differential systems, Arch Math., 49, 3541 [23] J Jaros and T Kusano (1999), A Picone type identity for second order half-linear differential equations, Acta Math Univ Comenian., 68, 137151 ă Ocalan ă [24] B Karpuz, B Kaymakácalan, and O (2010), A generalization of Opials inequality and applications to second-oder dynamic equations, Differ Equa Dyn Sys., 18, 1118 ă [25] B Karpuz and U M Ozkan (2011), Some generalizations for Opials inequality involving several function and their derivatives of arbitrary order on arbitrary time scales, Math Inequal Appl., 14, 7992 [26] A Lasota (1968), A discrete boundary value problem, Ann Polon Math., 20, 183190 [27] C M Lee (1968), On a discrete analogue of inequalities of Opial and Yang, Canad Math Bull., 11, 7377 [28] C F Lee, C C Yeh, C H Hong, and R P Agarwal (2004), Lyapunov and Wirtinger inequalities, Appl Math Lett., 17, 847853 [29] L Li and M Han (2014), Some new dynamic Opial-type inequalities and applications for second order integro-differential dynamic equations on time scales, Appl Math Comput., 232, 542547 [30] N D V Nhan, D T Duc, and V K Tuan (2013), Weighted norm inequalities for derivatives and their applications, Kodai Math J., 36, 228245 [31] N D V Nhan, D T Duc V K Tuan, and T D Phung (2016), Dynamic Picones identity and its applications, J Math Anal Appl., (submitted) [32] N D V Nhan, D T Duc, V K Tuan, and N D H Vu (2014), Inequalities for differences and their applications, J Difference Equa Appl., 20, 13191343 [33] N D V Nhan, T D Phung, D T Duc and V K Tuan (2017), Opial and Lyapunov inequalities on time scales and their applications to dynamic equations, Kodai Math J., 40, 254-277 27 [34] Z Opial (1960), Sur une inộgalitộ, Ann Polon Math., 8, 2932 [35] B G Pachpatte (1986), On Opial-type integral inequalities, J Math Anal Appl., 120, 547556 [36] B G Pachpatte (1987), On some new discrete inequalities involving partial sums, Indian J Pure Appl Math., 9, 640647 [37] B G Pachpatte (1987), On two independent variable Opial-type integral inequalities, J Math Anal Appl., 125, 4757 [38] B G Pachpatte (1993), Some inequalities similar to Opial inequality, Demonstr Math., 26, 643 647 [39] J Pecaric (1993), An integral inequality, Hadronic Press, Palm Harbor, FL., 471478 [40] M Picone (1910), Sui valori eccezionali di un parametro da cui dipende unequazione differenziale lineare ordinaria del secondordine, Ann Scuola Norm Sup Pisa., 11, 1141 [41] T D Phung (2017), Some inequalities for partial derivatives on time scales, Acta Math Vietnam., 42, 369-394 [42] T D Phung and D T Duc (2017), New Opial-type inequalities on time scales, Tp khoa hc-Trng H Quy Nhn, Tp 11, S 1, 47-59 [43] P Rehỏk (2001), Oscillatory properties of second order half-linear difference equations, Czechoslovak Math J., 51(126), 303321 [44] T D Phung, N D V Nhan, D T Duc, and V K Tuan (2016), Disfocality and disconjugacy for second order forced dynamic equations, (manuscript) [45] S H Saker (2011), Some Opial-type inequalities on time scales, Abs Appl Anal., Vol 2011, Article ID 265316, 19 pages [46] S H Saker (2012), Opials type inequalities on time scales and some applications, Ann Polon Math., 104, 243260 [47] S H Saker (2012), Some new inequalities of Opials type on time scales, Abs Appl Anal., Vol 2012, Article ID 683136, 14 pages [48] S H Saker (2012), New inequalities of Opials type on time scales and some of their applications, Discrete Dyn Nature Soc., Vol 2012, Article ID 362526, 23 pages [49] S H Saker (2013), Applications of Opial inequalities on time scales on dynamic equations with damping terms, Math Comput Modelling, 58, 17771790 [50] S H Saker, R P Agarwal, and D Oregan (2014), Higher order dynamic inequalities on time scales, Math Ineq Appl., 17, 461472 [51] S H Saker, R R Mahmoud, and A Peterson (2015), A new Picones dynamic inequality on time scales with applications, Appl Math Lett., 48, 162169 28 [52] S H Saker, M M Osman, D Oregan, and R Agarwal (2015), Some new Opial dynamic inequalities with weight functions on time scales, Math Inequal Appl., 18, 11711187 [53] O Shisha (1973), On the discrete version of Wirtingers inequality, Amer Math Monthly, 80, 755760 [54] H M Srivastava, K L Tseng, S J Tseng, and J C Lo (2010), Some weighted Opial-type inequalities on time scales, Taiwanese J Math., 14, 107122 [55] H M Srivastava, K L Tseng, S J Tseng, and J C Lo (2011), Some generalizations of Maronis inequality on time scales, Math Inequal Appl., 14, 469480 [56] A Tiryaki (2005), Sturm-Picone type theorems for nonlinear differential systems, Electron J Differential Equations, Vol 2005, No 154, pp 19 [57] D Willett (1968), The existence-uniqueness theorem for an nth order linear ordinary differential equation, Amer Math Monthly, 75, 174178 [58] F H Wong, W C Lin, S L Yu, and C C Yeh (2008), Some generalization of Opials inequalities on time scales, Taiwanese J Math., 12, 463471 [59] G S Yang (1966), On a certain result of Z Opial, Proc Japan Acad., 42, 7883 [60] G S Yang (1982), Inequality of Opial-type in two variables, Tamkang J Math., 13, 255259 [61] L Yin and C Zhao (2013), Some new generalizations of Maroni inequality on time scales, Demonstr Math., XLVI, 645654 [62] Z Zhao, Y Xu, and Y Li (2005), Dynamic inequalities on time scales, Inter J Pure Appl Math., 22, 4956 29 ... tớch trờn thang thi gian Chng 2: Bt ng thc Opial trờn thang thi gian v ỏp dng Chng 3: Tớnh dao ng ca mt s phng trỡnh ng lc trờn thang thi gian Chng 4: ng nht thc Picone trờn thang thi gian 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 thi gian Vic nghiờn cu cỏc bt ng thc loi Opial trờn thang thi gian c Bohner... thc trờn thang thi gian thi gian gn õy ó gúp phn vo vic phỏt trin lý thuyt phng trỡnh ng lc trờn thang thi gian Vỡ l ú, vic nghiờn cu nhm ci tin v xut cỏc bt ng thc mi trờn thang thi gian luụn