UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY

Transcription

1 UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY YIFEI PAN, MEI WANG, AND YU YAN ABSTRACT We study ordinry differentil equtions of the type u n t = fut with initil conditions u0 = u 0 = = u 1 0 = 0 nd u 0 0 where n, no dditionl ssuption is de on f We estblish soe uniqueness results nd show tht f is lwys Hölder continuous 1 INTRODUCTION The question of finding criteri for the uniqueness of solutions hs been constnt thee in the study of ordinry differentil equtions for very long tie, nd welth of results hve been estblished The ost quoted one in textbooks is perhps the Lipschitz uniqueness theore, which sttes tht in the eqution y n x = fx, y, y,, y n 1, if the function fx, z 1, z 2, z n is Lipschitz continuous with respect to z 1, z 2, z n, then the initil vlue proble hs unique locl solution Generlly speking, to ensure the uniqueness of solutions to n ODE, we need to ssue soe condition on the function f besides continuity, the Lipschitz condition being one exple Most of the reserch in this topic hs been devoted to finding the pproprite condition, nd there re ny nice results such s the clssicl theores by Peno, Osgood, Montel-Tonelli, nd Nguo The onogrph 1] provides n extensive nd systetic tretent of the vilble results In this pper, we pproch the uniqueness proble fro different perspective nd relte it to the unique continution proble We study utonoous ODEs of the type u n t = f u t where u C 0, 1] nd no dditionl ssuption is de on the function f If we ssue the initil conditions u0 = u 0 = = u n 1 0 = 0, the solution is not unique The following is trivil exple Exple 1 ut = t 3 stisfies u t = 6u 1 3 nd u0 = u 0 = 0 Another solution to this initil vlue proble is u 0 It is no surprise tht uniqueness fils in this exple becuse the function fu = 6u 1 3 is not Lipschitz ner 0 Fro nother perspective, this exple shows tht if solution nd its derivtives up to order n 1 ll vnish t 0, it is not gurnteed to be the zero function On the other hnd, however, even if ll its derivtive vnish t 0, the solution still y not be identiclly 0 Exple 2 The function is in C 0, 1] nd ut = { e 1 t 0 < t 1 0 t = 0 1 u k 0 = 0 for ll k N Let 1

2 2 YIFEI PAN, MEI WANG, AND YU YAN Then ut stisfies the eqution fs = { sln s 2 s > 0 0 s = 0 u = fu However, this eqution hs nother solution u 0, which lso stisfies?? This function ut is lso clssicl exple in the study of the unique continution proble, tht is, when cn we conclude tht function is loclly identiclly zero if its derivtives ll vnish t point One result long this line is given in 3] Theore 11 3] Let gx C, b], 0, b], nd g k x 2 g n x C n 1 k=1 for soe constnt C nd soe n 1 Then iplies, x, b] x n k g k 0 = 0 k 0 g 0 on, b] The order of singulrity of x t 0 in??, ie n k, is shrp, s one exple in 3] shows This theore is crucil to the proof of our in theore below The previous two exples suggest tht to gurntee uniqueness ner 0, the solution needs to vnish to sufficiently high order, but not to the infinite order So we ssue tht it stisfies the initil conditions u0 = u 0 = = u 1 0 = 0 nd u 0 = 0, where n Tht is, the order of the lowest non-vnishing derivtive of u t 0 is no less thn the order of the eqution Fro the eqution it is not difficult to see tht f is differentible wy fro 0, however, it is not differentible t 0 In fct, f is not Lipschitz ner 0, s shown by Exple 1 with = 3 Due to the lck of infortion bout the regulrity of f, the vilble uniqueness theory no longer pplies to this type of equtions We will show tht becuse u hs sufficiently high vnishing order t 0, such solutions re unique ner 0 Specificlly, we hve the following result Theore 12 Let ut C 0, 1 be solution of the differentil eqution 3 u n t = f u t, where n 1 nd f is function Assue tht u stisfies 4 u0 = u 0 = = u 1 0 = 0 nd u 0 = 0 with n Then such solution ut is unique for t ner 0 The proof of Theore?? is crried out in two steps First, we show the following result concerning the derivtives of u t 0

3 UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 3 Le 13 Let ut be solution tht stisfies Equtions?? nd?? The derivtive of u t 0 of ny order equl to or higher thn, tht is, u k 0 for ny k, depends only on, n, nd the behvior of the function f ner 0 In the second step, suppose there re two solutions u nd v, both stisfying?? nd??, then by the previous le the function ut vt nd ll its derivtives vnish t 0 Mking use of Theore??, we cn show tht u v 0 Typiclly, for n n-th order ODE we need only n initil conditions This theore shows tht in soe sense, the lck of infortion bout f cn be copensted by ssuing dditionl derivtive infortion t the initil point Interestingly, it turns out tht the solution is unique s long s the vnishing order is no less thn the order of the eqution, but the ctul vnishing order nd the vlue of the lowest non-zero derivtive re not essentil Theore 14 Suppose u 1 nd u 2 re two solutions of Eqution?? nd they stisfy 5 u 1 0 = u 10 = = u = 0, u 1 0 = 0 nd 6 u 2 0 = u 20 = = u l = 0, u l 2 0 = b 0 where, l n Then = l, = b nd u 1 u 2 for sll t The proof of Le?? will be given in Section??, nd the proof of Theores?? nd?? will be given in Section?? Nturlly, we would like to sk if the result still holds if one of the solutions in Theore?? hs vnishing order lower thn n, the order of the eqution We propose the following conjecture Conjecture 1 Suppose Eqution?? hs solution ut which stisfies?? with n Then it cnnot possess nother solution vt which stisfies initil conditions where l < n v0 = v 0 = = v l 1 0 = 0 nd v l 0 = b 0, We cn show tht this conjecture is true if = n + 1 or l nd re reltively prie However, there re soe difficulties in the generl cse nd we hve not been ble to prove the full conjecture Although in Theores?? nd?? we do not need to ke ny ssuptions bout the function f, we cn ctully obtin interesting infortion bout it Suppose there is function ut which stisfies Condition??, then s shown in Section??, loclly t cn be expressed s function of u, therefore we cn express u n t loclly s function f of u, so u n t = fu The next theore shows tht the function f is Hölder continuous in n intervl 0, δ] for sll δ > 0 Theore 15 Suppose function ut stisfies Condition??, then Eqution?? holds for soe function f where n, nd there is constnt δ > 0 such tht f is uniforly Hölder continuous in the intervl 0, δ]

4 4 YIFEI PAN, MEI WANG, AND YU YAN This theore is proved fter Theore?? in Section?? A sury of Theores?? nd?? is tht ny sooth function of finite order vnishing t 0 is unique locl solution of differentil eqution in the for of??, where f is differentible in the interior nd uniforly Hölder continuous up to the boundry In Theores?? nd??, the high order vnishing condition?? llows us to obtin uniqueness results without ny extr ssuption on f This phenoenon is only found in utonoous equtions like?? For generl ODEs of the for dn u = f t, u, du,, dn 1 u, such results n n 1 cnnot be expected becuse there is ore thn one expression for f For exple, Exple 3 Let ut = t 4 It stisfies the initil conditions Its derivtives re u0 = u 0 = u 0 = u 3 0 = 0 nd u 4 0 = 24 u t = 4t 3, nd u t = 12t 2 We cn express u s function f of u nd u in different wys such s or or or u t = 192u2 u 2, u t = 3u 2 4u, 2 1 u 7 t = 12 4 uu, u t = 12u u In the first two equtions f is not continuous t the origin, while in the lst two equtions f is Hölder continuous t the origin This siple exple shows tht the function f cn be expressed in vrious wys nd tht in order to study the uniqueness we need to ipose very specific ssuptions on f Our work ws otivted by 2] in which Li nd Nirenberg studied siilr second order PDE: u = fu, where u = ut, x C R k+1 hs non-vnishing prtil derivtive t 0 which cn be expressed in the for ut, x = t + Ot +1, 0, t R nd x R k They showed tht if two solutions u nd v stisfy u v, then u v Theore?? cn be viewed s n iproveent of their result in the one-diensionl cse to rbitrry order nd without the coprison condition u v 2 THE PROOF OF LEMMA?? Without loss of generlity, we cn ssue tht > 0 First, we show tht, the -th derivtive of u t 0, only depends on, n, nd the function f

5 Define UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 5 7 x = Then This iplies tht u 1 x = t + O t +1 1 = t 1 + Ot x 8 1 s t 0 t We cn lso write u = x Tking the derivtive with respect to t, we get du 1 d x = x t 1 + Ot = x t 1 t x + O = d x 1 x 1 1 d x In the second eqution bove nd in the nlysis tht follows, we forlly differentite the Tylor expnsion with the big-o nottion A detiled discussion of this differentition is provided in the Appendix In light of??, it follows tht d x 9 t=0 = 1 By the Inverse Function Theore, t cn be expressed s function of x: t = x + O x 2 Then Siilrly, t n+1 t n = x + O x 2 n = x n 1 + O x = x n O x Thus 10 fu = u n = 1 n + 1t n + O t n+1 x n+1 = 1 n + 1 x n + O u n = 1 n O = n n 1 n + 1u + O u n+1 u n+1 by Eqution?? Therefore, 11 n = li u 0 fu 1 n + 1u n This shows tht is copletely deterined by, n, nd the behvior of the function f ner 0

6 6 YIFEI PAN, MEI WANG, AND YU YAN Next, we show tht the + 1-th derivtive of u t 0 lso only depends on, n, nd f Write u s 12 ut = t + +1 t +1 + Ot +2 We will show tht +1 only depends on, n, nd the behvior of f t 0 Express t s 13 t = x + b 2 x 2 + O x 3 We would like to obtin n expression for b 2 in ters of the derivtives of u t 0 To do this, we tke the derivtive with respect to t on both sides of d x = 1 By the Product Rule nd Chin d x Rule, we hve 14 d d x d x + d x d d x d 2 t d x d x 2 d x d 2 x 2 d x + d x d 2 x 2 d x + We would like to evlute Eqution?? t t = 0 Fro u 1 x = = = t = t t + +1 t+1 + O t t + Ot ] 1 +1 t + Ot2 = t + +1 t2 + Ot 3, we know tht 15 Fro?? we know tht = 0 = 0 2 d2 t d x 2 = t + Ot2 + Ot ] 3 d 2 x 2 t=0 = 2 +1 d x d 2 t t=0 = 1 nd d x 2 t=0 = 2b 2 Thus if we evlute Eqution?? t t = 0, we get b 2 = 0, therefore

7 UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 7 16 b 2 = +1 Now, fro Eqution?? we hve t n = x + b 2 x 2 + O x 3 n = x n 1 + b 2 x + O x 2] n = x n 1 + n b 2 x + O x 2 + O x 2] = x n + nb 2 x n+1 + O x n+2 Siilrly t n+1 = x n+1 + n + 1b 2 x n+2 + O x n+3 t n+2 = O x n+2 Then fro Eqution?? nd the bove expressions for the powers of t, we hve u n = 1 n + 1t n n + 2t n+1 + O t n+2 = 1 n + 1 x n + nb 2 x n+1 + O x n+2] n + 2 x n+1 + n + 1b 2 x n+2 + O +O x n+2 = 1 n + 1 x n + 1 nb n + 2] x n+1 + O Thus by fu = u n nd??, we hve x n+3] x n+2 fu = 1 n + 1 x n + 1 nb n + 2 ] x n+1 + O u n = 1 n nb n + 2 u n+2 +O = n n 1 n + 1u + +O n 1 This ens tht ] u n+1 ] 1 nb2 + n n + 2 u n+2 x n+2 u n+1

8 8 YIFEI PAN, MEI WANG, AND YU YAN n 1 1 nb2 + n n + 2 fu n 1 n + 1u n = li u 0 u n+1 By??, this cn be written s n 1 1 n +1 fu n 1 n + 1u n = li u 0 u n+1 After collecting siilr ters, we get Consequently, n 1 + n n n n] fu n 1 n + 1u n = li u 0 +1 = n+1 li u 0 u n+1 fu n 1 n + 1u n u n n n Since we hve proved tht only depends on, n, nd f, Eqution?? shows tht +1 is lso copleted deterined by, n, nd the behvior of f ner 0 By??, this lso shows tht b 2 depends only on, n, nd f Then, we will use theticl induction to show tht ll the derivtives of u t 0 of order higher thn re copletely deterined by, n, nd f Express u nd t s 18 ut = t + +1 t k t +k + +k+1 t +k+1 + O t +k+2, 19 t = x + b 2 x b k+1 x k+1 + b k+2 x k+2 + O x k+3 Suppose tht for k 1,, +1,, +k, b 2,, b k+1 re ll deterined only by by, n, nd f, we will show tht +k+1 nd b k+2 lso re deterined only by by, n, nd f We strt by obtining n expression for b k+2 in ters of +1,, +k nd +k+1 Tking the derivtive with respect to t on both sides of??, we obtin 20 0 = d3 x 3 d x + d2 x d 2 t 2 d x d x + 2 d x 2 0 = d3 x 3 d x + 3 d x d2 x 2 d2 t d x 2 + d x d2 x d2 t 2 d x d3 t d x 3 2 d x d 3 t d x d x 3

9 UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 9 Tking the derivtive of both sides of??, we get 21 0 = d4 x 4 d x + d3 x d2 t 3 d x d x 2 + d x d2 x d3 t 2 d x d x 3 0 = d4 x 4 +6 d x + 3 d 2 x + 3 d2 x 2 d2 t 2 d x + d x 2 d3 x d2 t 3 d x 2 2 d x d2 x 3 d3 t d x 2 d x + d4 t 3 d x d x 4 d x + 4 d3 x d x 3 d2 t d x + 3 d2 x 2 d2 x 2 d2 t 2 d x 2 2 d2 x 4 d3 t d x 2 d x + d4 t 3 d x 4 If we keep tking the derivtive with respect to t for k ties nd collect the siilr ters fter ech differentition s shown bove, eventully we will rrive t n expression of the for 22 0 = dk+2 x k+2 d x + ters involving dk+1 x dk x d x,,, k+1 k, d x, d2 t d x,, dk+1 t 2 d x k+1 k+2 d x + dk+2 t d x k+2 Fro?? we know tht 23 d x t=0 = 1 d 2 t d x 2 t=0 = 2b 2 = d k+1 t d x k+1 t=0 = k + 1!b k+1 d k+2 t d x k+2 t=0 = k + 2!b k+2 Then we look t d x t=0,, dk+1 x k+1 t=0, nd dk+2 x k+2 t=0

10 10 YIFEI PAN, MEI WANG, AND YU YAN By definition?? nd Eqution??, t + +1 t t k t +k + +k+1 t +k+1 + O t +k+2 x = = t { = t t t t k t k t k + 1! k tk + +k+1 t k+1 + O t k+2 1 tk + +k+1 t k+1 + O t k+2] t k + +k+1 t k+1 + O t k+2 ] k+1 } + O t k+2 After collecting siilr ters we cn write t + +2 tk + +k+1 t k+1 + O t k+2] 2 t k tk 1 24 x = t + λ 2 t 2 + λ 3 t λ k+1 t k+1 + where +k+1 + λ k+2 t k+2 + O t k+3, λ 2 is constnt involving,, nd +1 λ 3 is constnt involving,, +1 nd +2 λ k+1 is constnt involving,, +1,, +k 1, +k λ k+2 is constnt involving,, +1,, +k 1, +k By the inductive hypothesis, λ 2, λ 3, λ k+1, λ k+2 re ll constnts tht only depend on, n, nd the function f Fro Eqution?? we obtin 25 d x t=0 = 1 d 2 x 2 t=0 = 2λ 2 d k+1 x = k+1 t=0 = k + 1!λ k+1 d k+2 x k+2 t=0 = k + 2! +k+1 + λ k+2 Now we evlute?? t t = 0 nd ke use of?? nd??:

11 UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 11 +k+1 0 = k+2! Thus we obtin + λ k+2 1+ ters involving b 2,, b k+1, λ 2,, λ k+1 +1 k+2!b k+2 26 b k+2 = +k+1 + Q, where Q is constnt depending on b 2,, b k+1, λ 2,, λ k+1, λ k+2, hence Q is copletely deterined by, n, nd f Next we will nlyze +k+1 Fro?? we hve t n = x n 1 + b 2 x + + b k+1 x k + b k+2 x k+1 + O x k+2] n { = x n 1 + n b 2 x + + b k+1 x k + b k+2 x k+1 + O x k+2] n n 1 + b2 x + + b k+1 x k + b k+2 x k+1 + O x k+2] n n 1 n k + b2 x + + b k+1 x k + b k+2 x k+1 + O x k+2] k+1 k + 1! } +O x k+2 After collecting siilr ters we cn express t n s t n = x n { 1 + c 1, n x + c 2, n x c k, n x k + nb k+2 + c k+1, n ] x k+1 + O x k+2 }, where c 1, n is constnt depending on nd b 2 ; c 2, n is constnt depending on, b 2 nd b 3 ; ; c k, n is constnt depending on, b 2,, b k+1 ; c k+1, n is constnt depending on, b 2,, b k+1 By the inductive hypothesis, c 1, n, c 2, n,, c k, n nd c k+1, n re ll deterined only by, n, nd f Thus we hve 27 t n = x n + c 1, n x n+1 + c 2, n x n c k, n x n+k + nb k+2 + c k+1, n ] x n+k+1 + O x n+k+2, where c 1, n, c 2, n,, c k, n nd c k+1, n re constnts depending on, n, nd f By the se type of nlysis we obtin siilr expressions for the other powers of t: 28 t n+1 = x n+1 + c 1, n+1 x n+2 + c 2, n+1 x n c k, n+1 x n+k+1 + n + 1b k+2 + c k+1, n+1 ] x n+k+2 + O x n+k+3,

12 12 YIFEI PAN, MEI WANG, AND YU YAN where c 1, n+1, c 2, n+1,, c k, n+1 nd c k+1, n+1 re constnts depending on, n, nd f 29 t n+2 = x n+2 + c 1, n+2 x n+3 + c 2, n+2 x n c k, n+2 x n+k+2 + n + 2b k+2 + c k+1, n+2 ] x n+k+3 + O x n+k+4, where c 1, n+2, c 2, n+2,, c k, n+2 nd c k+1, n+2 re constnts depending on, n, nd f t n+k = x n+k + c 1, n+k x n+k+1 + c 2, n+k x n+k c k, n+k x n+2k 30 + n + kb k+2 + c k+1, n+k ] x n+2k+1 + O x n+2k+2, where c 1, n+k, c 2, n+k,, c k, n+k nd c k+1, n+k re constnts depending on, n, nd f t n+k+1 = x n+k+1 + c 1, n+k+1 x n+k+2 +c 2, n+k+1 x n+k c k, n+k+1 x n+2k n + k + 1b k+2 + c k+1, n+k+1 ] x n+2k+2 + O x n+2k+3, where c 1, n+k+1, c 2, n+k+1,, c k, n+k+1 nd c k+1, n+k+1 re constnts depending on, n, nd f 32 t n+k+2 = O x n+k+2 Fro?? we obtin u n = 1 n + 1t n n + 2t n k + k + k 1 n + k + 1t n+k + +k+1 + k k n + k + 2t n+k+1 + O t n+k+2

13 UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 13 Then by?? to?? we cn write { u n = 1 n + 1 x n + c 1, n x n+1 + c 2, n x n+2 + } +c k, n x n+k + nb k+2 + c k+1, n ] x n+k+1 + O x n+k+2 { n + 2 x n+1 + c 1, n+1 x n+2 + c 2, n+1 x n c k, n+1 x n+k+1 + n + 1b k+2 + c k+1, n+1 ] x n+k+2 +O x n+k+3 } + { + +k + k + k 1 + k n + 1 x n+k + c 1, n+k x n+k+1 +c 2, n+k x n+k c k, n+k x n+2k + } n + kb k+2 + c k+1, n+k ] x n+2k+1 + O x n+2k+2 { + +k+1 + k k + k n + 2 x n+k+1 + c 1, n+k+1 x n+k+2 + c 2, n+k+1 x n+k c k, n+k+1 x n+2k+1 } + n + k + 1b k+2 + c k+1, n+k+1 ] x n+2k+2 + O x n+2k+3 +O x n+k+2 = 1 n + 1 x n + C,, +1, c 1, n x n+1 +C,, +1, +2, c 1, n+1, c 2, n x n+2 + +C,, +1,, +k, c k, n, c k 1, n+1,, c 1, n+k 1 x n+k + 1 nb k+2 + +k+1 + k k + k n + 2 ] +C,, +1,, +k, c k+1, n, c k, n+1,, c 1, n+k x n+k+1 +O x n+k+2 Here C,, +1, c 1, n is constnt depending on,, +1 nd c 1, n ; we denote it s p n+1 to siplify nottions Since, +1 nd c 1, n only depend on, n, nd f, we know tht p n+1 only depends on, n, nd f Siilrly, the other constnts C,, +1, +2, c 1, n+1, c 2, n,, nd C,, +1,, +k, c k+1, n, c k, n+1,, c 1, n+k ll depend only on, n, nd f, nd cn be denoted siply s p n+2,, p n+k, nd p n+k+1 Thus we cn rewrite the bove eqution s u n = 1 n + 1 x n + p n+1 x n p n+2 x n p n+k x n+k + 1 nb k+2 + +k+1 + k k + k n + 2 ] +p n+k+1 x n+k+1 + O x n+k+2

14 14 YIFEI PAN, MEI WANG, AND YU YAN Now becuse of u n = fu nd definition??, we hve u n u n+1 fu = 1 n p n+1 + n+2 n+k u u p n p n+k + 1 nb k+2 + +k+1 + k k + k n + 2 +p n+k+1 ] u n+k+1 + O Due to??, we cn rewrite the bove eqution s u n+k+2 u n u n+1 fu = 1 n p n+1 + n+2 n+k u u p n p n+k + ] +{ + k k + k n n +k nq + p n+k+1 } u n+k+1 + O u n+k+2 Fro?? we get ] + k k + k n n +k nq + p n+k+1 n u = li u 0 fu n + 1 p u n+1 n+k+1 u n+1 p n+k u n+k Note tht + k k + k n n 0, then since the constnts Q,, p n+1,, p n+k+1 ll depend only on, n, nd f, we know tht +k+1 only depends on, n, nd f Consequently, b k+2 lso only depends on, n, nd f becuse of?? Therefore, by theticl induction, ll derivtives of f t 0 re deterined copletely by, n, nd f This copletes the proof of Le?? Theore?? 3 THE PROOFS OF THEOREMS??,?? AND?? Proof: Suppose there re two solutions ut nd vt, both stisfy Equtions?? nd?? By Le??, t t = 0, u nd v hve the se derivtive of ny order Let w = u v, then w k 0 = 0 for ny integer k 0

15 UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 15 In order to pply Theore?? we need to show tht w stisfies Condition?? w n t = u n t v n t = fut fvt Without loss of generlity we ssue > 0 By Eqution??, we cn write ] 35 fu = n n 1 n + 1u + αu nd fv = where α is function with the order n 1 n + 1v n + αv ], αs = O s n+1 So we cn write 36 w n t = n 1 n + 1 u n n v + αu αv If = n, then n 1 n + 1 u n n v = 0 If > n, by the Men Vlue Theore, 37 u n n v n n ζ u v where ζt is between ut nd vt Since ut = t + O t +1 nd vt = t + O t +1, we know tht ζt = t + O t +1, which iplies ζ n = n t n 1 + Ot Ct n for soe constnt C > 0 when t is sufficiently sll Thus ζ n u v C 1 u v t n nd by eqution?? we know tht 38 n u n 1 n + 1 for nother constnt C > 0 Next, we estite αu αv n v C u v t n Fro Eqution??, we know tht f is differentible with respect to t, since u n t is differentible with respect to t Condition?? shows tht du t 0 when t > 0 is sufficiently sll Then by the Inverse Function Theore, t is differentible with respect to u Thus, when u is sll nd positive, f is differentible with respect to u nd df du = df du Then by Eqution??, since f is differentible on sll intervl 0, δ, α is lso differentible on sll intervl 0, δ By the Men Vlue Theore

16 16 YIFEI PAN, MEI WANG, AND YU YAN αu αv = α ηu v where ηt is between ut nd vt Becuse ut = t +O t +1 nd vt = t +O t +1, we know tht ηt = t + O t +1 Ct when t is sll, thus Fro αs = O s n+1 Thus for soe C > 0 39 η n 1 = O t n 1 we get α s = O s n 1 Therefore α η = O η n 1 = O t n 1 αu αv Ct n 1 u v Cobining Equtions??,??, nd??, we conclude Ct n u v since 0 < t < 1 w n ut vt t C t n = C wt t n Finlly, extend the doin of wt to 1, 1] by defining wt = w t when 1 t < 0 Then w C 1, 1] nd it stisfies Condition?? By Theore??, w 0, which ens u v This copletes the proof of Theore?? Theore?? Proof: Without loss of generlity we ssue > 0 We pply the se nlysis s in the proof of Le?? to u 1 nd u 2, respectively Siilr to?? we hve nd n = li u1 0 b n l = li u2 0 fu 1 1 n + 1u n 1 fu 2 ll 1 l n + 1u l n l 2 = li s 0 = li s 0 fs 1 n + 1s n fs ll 1 l n + 1s l n l Suppose l, without loss of generlity we ssue < l Dividing the two equtions, we get n b n l = = ll 1 l n n + 1 li s l n l s 0 n ll 1 l n n + 1 li s 0 s n n l Since < l, li s 0 s n n l = 0 However, n b n l 0 This is contrdiction Therefore = l, consequently = b Then by Theore?? we know tht u 1 u 2 for sll t

17 UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 17 Theore??: Proof: : The proof of Le?? shows tht ner 0, t is function of u, therefore u n t cn be expressed s function f of u Thus Eqution?? holds when t > 0 is sll Fro Condition?? we define f0 = 0 By the first two equtions in?? nd the discussions in Appendix??, we know tht there is function h which is C 1 on the closed intervl 0, ɛ] for soe ɛ > 0, such tht By definition?? we hve fu = 1 n + 1 x n + h x x n 40 fu = 1 n + 1 u n Since 0 n < 1 nd 0 < 1 on the closed intervl 0, 1] with Hölder coefficients n the first ter in?? is Hölder continuous on 0, 1] u + h 1 u n < 1, it is well known tht u n nd u 1 re Hölder continuous nd 1, respectively This iplies tht Since h is C 1 on 0, ɛ], it is lso Hölder continuous on 0, ɛ] Then since the coposition of two Hölder continuous functions is Hölder continuous, we know tht h u 1 is Hölder continuous with respect to u on closed intervl 0, δ] with δ > 0 Next, becuse the product of two Hölder continuous functions is lso Hölder continuous, we know tht h u 1 n u is Hölder continuous Thus the second ter in?? is Hölder continuous on 0, δ] Therefore, f is Hölder continuous on 0, δ] nd the theore is proved APPENDIX A DIFFERENTIATION OF THE TAYLOR EXPANSION We will discuss the regulrity of the reinder ter in the Tylor expnsion of function tht is used in the proof of Theore?? nd the differentition of the Tylor expnsion tht is frequently used in the proof of Le?? In generl, suppose function gx C k+1, b], by the Tylor Theore we cn write 41 gx = g + g x + g 2! where li x hx = 0 An explicit expression for hx is 42 where < ξ < x x gk x k + hxx k k! hx = gk+1 ξ x k + 1! Fro?? we know tht hx is C 1 on, b] Next we show tht it is ctully C 1 up to the boundry, on, b] Tking the derivtive on both sides of Eqution??, we get

18 18 YIFEI PAN, MEI WANG, AND YU YAN 43 g x = g +g x + + gk k 1! x k 1 +h xx k +khxx k 1 Define h = 0, so h is continuous on, b] Denote then P x = g + g x + g 2! By the definition of liits, hx = x gk x k, k! gx P x x k 44 h = hx h li x x = gx P x li x x k+1 = gk+1 P k+1 k + 1! = gk+1 k + 1!, by pplying the L Hospitl Rule k + 1 ties where we hve used the fct tht P k+1 = 0 When x >, h x = d dx gx P x x k = g x P x x k gx P xkx k 1 x 2k = g x P x x k k gx P x x k+1 By repetedly pplying the L Hospitl Rule, we know tht nd Therefore g x P x li = gk+1 P k+1 x x k k! k gx P x li = k g k+1 P k+1 x x k+1 k + 1! = gk+1 k! = kgk+1 k + 1!

19 UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY li x h x = gk+1 kgk+1 k! k + 1! = gk+1 k + 1! Equtions?? nd?? show tht hx is C 1 on the closed intervl, b] Furtherore, we know tht for ny x, b], h x C for soe constnt C 1, thus h xx k C1 x k Since gx C k+1, b], fro?? we know tht hx C 2 x for soe constnt C 2, thus khxx k 1 kc2 x k Therefore,?? cn be denoted s 46 g x = g + g x + + gk k 1! x k 1 + O x k Since the first, second,, nd k 1-th derivtives of g x t re g, g 3,, nd g k, respectively, Eqution?? is the Tylor expnsion of g x t to order k 1 Usully we denote?? s 47 gx = g + g x + g 2! This shows tht we cn forlly differentite?? to get?? x gk x k + O x k+1 k! Acknowledgent: The reserch of Yu Yn ws prtilly supported by the Scholr in Residence progr t Indin University-Purdue University Fort Wyne REFERENCES 1] R P Agrwl nd V Lkshiknth Uniqueness nd Nonuniqueness Criteri for Ordinry Differentil Equtions World Scientific, ] YnYn Li nd Louis Nirenberg A geoetric proble nd the hopf le ii Chinese Annls of Mthetics, Series B, 27B2: , ] Yifei Pn nd Mei Wng When is function not flt? Journl of Mtheticl Anlysis nd Applictions, 3401: , 2008

20 20 YIFEI PAN, MEI WANG, AND YU YAN DEPARTMENT OF MATHEMATICAL SCIENCES, INDIANA UNIVERSITY-PURDUE UNIVERSITY FORT WAYNE, FORT WAYNE, INDIANA E-il ddress: DEPARTMENT OF STATISTICS, UNIVERSITY OF CHICAGO, CHICAGO, ILLINOIS E-il ddress: DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, HUNTINGTON UNIVERSITY, HUNTINGTON, INDIANA E-il ddress: