UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY

UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY YIFEI PAN, MEI WANG, AND YU YAN ABSTRACT We estblish soe uniqueness results ner 0 for ordinry differentil equtions of the type u n t = fut which stisfies u0 = u 0 = = u 1 0 = 0 nd u 0 0 with n We lso show tht lthough f is not Lipschitz ner 0, it is lwys Hölder continuous 1 INTRODUCTION A stndrd theore in ODE theory sttes tht in the eqution u t = F t, xt, if the function F t, x is Lipschitz continuous with respect to vrible x in n intervl contining the initil point, then the initil vlue proble hs unique solution loclly Generlly speking, with equtions of rbitrry order, to obtin the uniqueness of solution we need to ssue Lipschitz condition on F In this pper, we study utonoous ODEs of the type u n t = f u t where u C 0, 1] 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 For this type of equtions, 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 For exple, Exple 1 Let ut = t 3, then u0 = u 0 = u 0 = 0, u 3 0 = 6, nd u stisfies u t = 6u 1 3 The function fu = 6u 1 3 is not Lipschitz ner 0 Interestingly, lthough f is not Lipschitz ner 0, we cn prove tht it is lwys Hölder continuous, s surized in Theore 15 below Due to the lck of Lipschitz continuity, the stndrd uniqueness theory no longer pplies to this type of equtions We will show tht if u hs vnishing derivtives t 0 up to certin order, then such solutions re unique ner 0 Specificlly, we hve the following result Theore 11 Let ut C 0, 1 be solution of the differentil eqution 1 u n t = f u t, where n 1 nd f is function Assue tht u stisfies 2 u0 = u 0 = = u 1 0 = 0 nd u 0 = 0 with n Then such solution ut is unique for t ner 0 Typiclly, for n n-th order ODE we need only n initil conditions This theore shows tht in soe sense, the lck of Lipschitz continuity cn be copensted by ssuing dditionl derivtive infortion t the initil point 1

2 YIFEI PAN, MEI WANG, AND YU YAN On the other hnd, if u vnishes to infinite order t 0, the result will not hold either Exple 2 The function is in C 0, 1] nd ut = { e 1 t 0 < t 1 0 t = 0 3 u k 0 = 0 for ll k N Let 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 3 Furtherore, Theore 11 cn be strengthened to the following: Theore 12 Suppose u 1 nd u 2 re two solutions of Eqution 1 nd they stisfy 4 u 1 0 = u 10 = = u 1 1 0 = 0, u 1 0 = 0 nd 5 u 2 0 = u 20 = = u l 1 2 0 = 0, u l 2 0 = b 0 where, l n Then = l, = b nd u 1 u 2 for sll t The proof of Theore 11 is crried out in two steps First, we show the following result concerning the derivtives of u t 0 Le 13 Let ut be solution tht stisfies Equtions 1 nd 2 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, we ke use of unique continution theore in 2] Theore 14 2] Let gx C, b], 0, b], nd g k x 6 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]

UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 3 The proof of Le 13 will be given in Section 2, nd the proof of Theores 11 nd 12 will be given in Section 3 If function ut stisfies Condition 2, then s shown in Section 2, 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 2, then Eqution 1 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] This theore is proved fter Theore 12 in Section 3 A sury of Theores 11 nd 15 is tht ny sooth function of finite order vnishing t 0 is unique locl solution of differentil eqution in the for of 1, where f is differentible in the interior nd uniforly Hölder continuous up to the boundry We note tht in 1] 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 stisfies u v, then u v Theore 11 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 13 Without loss of generlity, we cn ssue tht > 0 Define First, we show tht, the -th derivtive of u t 0, only depends on, n, nd the function f 7 x = Then This iplies tht u 1 x = t + O t +1 1 = t 1 + Ot 8 x t 1 s t 0 We cn lso write u = x Tking the derivtive with respect to t, we get

4 YIFEI PAN, MEI WANG, AND YU YAN 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 8, 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+1 1 + 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 + 1 + O = n n 1 n + 1u + O u n+1 u n+1 by Eqution 7 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 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

UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 5 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 + = 0 = 0 2 d2 t d x 2 = 0 We would like to evlute Eqution 14 t t = 0 Fro x = = = t = t we know tht u 1 t + +1 t+1 + O t +2 1 1 + +1 t + Ot2 1 + 1 ] 1 +1 t + Ot2 = t + +1 t2 + Ot 3, + 1 2 1 1 +1 2 1 t + Ot2 + Ot ] 3 15 Fro 13 we know tht d 2 x 2 t=0 = 2 +1 d x t=0 = 1 nd Thus if we evlute Eqution 14 t t = 0, we get d 2 t d x 2 t=0 = 2b 2 therefore 2 +1 1 + 1 2b 2 = 0, 16 b 2 = +1 Now, fro Eqution 13 we hve

6 YIFEI PAN, MEI WANG, AND YU YAN 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 12 nd the bove expressions for the powers of t, we hve u n = 1 n + 1t n + +1 + 1 n + 2t n+1 + O t n+2 = 1 n + 1 x n + nb 2 x n+1 + O x n+2] + +1 + 1 n + 2 x n+1 + n + 1b 2 x n+2 + O +O x n+2 = 1 n + 1 x n + 1 nb 2 + +1 + 1 n + 2] x n+1 + O Thus by fu = u n nd 7, we hve x n+3] x n+2 fu = 1 n + 1 x n + 1 nb 2 + +1 + 1 n + 2 ] x n+1 + O u n = 1 n + 1 + 1 nb 2 + +1 + 1 n + 2 u n+2 +O = n n 1 n + 1u + +O This ens tht n 1 ] u n+1 ] 1 nb2 + n 1 +1 + 1 n + 2 u n+2 x n+2 u n+1 n 1 1 nb2 + n 1 +1 + 1 n + 2 fu n 1 n + 1u n = li u 0 u n+1

UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 7 By 16, this cn be written s n 1 1 n +1 + n 1 +1 + 1 n + 2 fu n 1 n + 1u n = li u 0 u n+1 After collecting siilr ters, we get Consequently, 17 +1 n 1 + 1 n + 2 1 2 n] fu n 1 n + 1u n = li u 0 u n+1 +1 = n+1 li u 0 fu n 1 n + 1u n u n+1 + 1 n + 2 1 2 n Since we hve proved tht only depends on, n, nd f, Eqution 17 shows tht +1 is lso copleted deterined by, n, nd the behvior of f ner 0 By 16, 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 +1 + + +k t +k + +k+1 t +k+1 + O t +k+2, 19 t = x + b 2 x 2 + + 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 14, 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 + 2 3 d3 t d x 3 2 d x d 3 t d x d x 3 Tking the derivtive of both sides of 20, we get

8 YIFEI PAN, MEI WANG, AND YU YAN 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 19 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

UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 9 By definition 7 nd Eqution 18, t + +1 t +1 + +2 t +2 + + +k t +k + +k+1 t +k+1 + O t +k+2 x = = t 1 + +1 { = t 1 + 1 t + +2 +1 1 1 t + +2 +1 t2 + + +k t2 + + +k + 1 2 1 t + +2 + + 1 k + 1! 1 1 1 +1 1 k tk + +k+1 t k+1 + O t k+2 1 tk + +k+1 t k+1 + O t k+2] t2 + + +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 t2 + + +k tk 1 24 x = t + λ 2 t 2 + λ 3 t 3 + + 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 24 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 22 t t = 0 nd ke use of 23 nd 25:

10 YIFEI PAN, MEI WANG, AND YU YAN +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 19 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] 2 + 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 2 + + 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+2 + + 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+3 + + 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,

UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 11 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+4 + + 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+2 + + 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+3 + + c k, n+k+1 x n+2k+1 31 + 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 18 we obtin u n = 1 n + 1t n + +1 + 1 n + 2t n+1 + + +k + k + k 1 n + k + 1t n+k + +k+1 + k + 1 + k n + k + 2t n+k+1 + O t n+k+2

12 YIFEI PAN, MEI WANG, AND YU YAN Then by 27 to 32 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 { + +1 + 1 n + 2 x n+1 + c 1, n+1 x n+2 + c 2, n+1 x n+3 + + 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+2 + + 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 + 1 + 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+3 + + 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 + 1 + 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+1 + 33 p n+2 x n+2 + + p n+k x n+k + 1 nb k+2 + +k+1 + k + 1 + k + k n + 2 ] +p n+k+1 x n+k+1 + O x n+k+2

UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 13 Now becuse of u n = fu nd definition 7, we hve u n u n+1 fu = 1 n + 1 + p n+1 + n+2 n+k u u p n+2 + + p n+k + 1 nb k+2 + +k+1 + k + 1 + k + k n + 2 +p n+k+1 ] u n+k+1 + O Due to 26, we cn rewrite the bove eqution s u n+k+2 u n u n+1 fu = 1 n + 1 + p n+1 + n+2 n+k u u p n+2 + + p n+k + ] +{ + k + 1 + k + k n + 2 1 2 n +k+1 34 + 1 nq + p n+k+1 } u n+k+1 + O u n+k+2 Fro 34 we get ] + k + 1 + k + k n + 2 1 2 n +k+1 + 1 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 + 1 + k + k n + 2 1 2 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 26 Therefore, by theticl induction, ll derivtives of f t 0 re deterined copletely by, n, nd f This copletes the proof of Le 13 Theore 11 3 THE PROOFS OF THEOREMS 11, 12 AND 15 Proof: Suppose there re two solutions ut nd vt, both stisfy Equtions 1 nd 2 By Le 13, 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

14 YIFEI PAN, MEI WANG, AND YU YAN In order to pply Theore 14 we need to show tht w stisfies Condition 6 w n t = u n t v n t = fut fvt Without loss of generlity we ssue > 0 By Eqution 10, 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 37 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 1, we know tht f is differentible with respect to t, since u n t is differentible with respect to t Condition 2 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 35, since f is differentible on sll intervl 0, δ, α is lso differentible on sll intervl 0, δ By the Men Vlue Theore

UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 15 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 36, 38, nd 39, 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 6 By Theore 14, w 0, which ens u v This copletes the proof of Theore 11 Theore 12 Proof: Without loss of generlity we ssue > 0 We pply the se nlysis s in the proof of Le 13 to u 1 nd u 2, respectively Siilr to 11 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 + 1 1 n + 1 li s l n l s 0 n ll 1 l n + 1 1 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 11 we know tht u 1 u 2 for sll t

16 YIFEI PAN, MEI WANG, AND YU YAN Theore 15: Proof: : The proof of Le 13 shows tht ner 0, t is function of u, therefore u n t cn be expressed s function f of u Thus Eqution 1 holds when t > 0 is sll Fro Condition 2 we define f0 = 0 By the first two equtions in 10 nd the discussions in Appendix A bout differentiting Tylor expnsion, we know tht there is function h C 1 0, ɛ] for soe ɛ > 0, such tht fu = 1 n + 1 x n + h x x n By definition 7 we hve u n u 1 u 40 fu = 1 n + 1 + h Since 0 n < 1 nd 0 < 1 on the closed intervl 0, 1] with Hölder coefficients n the first ter in 40 is Hölder continuous on 0, 1] 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 40 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 in detil the differentition of the Tylor expnsion of function tht is frequently used in the proof of Le 13 In generl, suppose gx C k+1, b], by the Tylor Theore we cn write 41 gx = g + g x + g x 2 + + gk x k + hxx k 2! k! where li hx = 0 An explicit expression for hx is x 42 where < ξ < x Usully we denote 41 s 43 gx = g + g x + g 2! hx = gk+1 ξ x k + 1! Tking the derivtive on both sides of Eqution 41, we get x 2 + + gk x k + O x k+1 k! 44 g x = g +g x + + gk k 1! x k 1 +h xx k +khxx k 1

UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY 17 Fro 41 we know tht hx is C 1 on, b] Next we show tht it is ctully C 1 up to the boundry, on, b] Define h = 0, so h is continuous on, b] Denote then P x = g + g x + g 2! By the definition of liits, 45 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!, hx = where we hve used the fct tht P k+1 = 0 When x >, h x = d dx gx P x x k x 2 + + gk x k, k! gx P x x k by pplying the L Hospitl Rule k + 1 ties = 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 46 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! li x h x = gk+1 kgk+1 k! k + 1! = gk+1 k + 1! = gk+1 k! = kgk+1 k + 1!

18 YIFEI PAN, MEI WANG, AND YU YAN Equtions 45 nd 46 show tht h x 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 42 we know tht hx C 2 x for soe constnt C 2, thus khxx k 1 kc2 x k Therefore, 44 cn be denoted s 47 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, Eqution 47 is the Tylor expnsion of g x t to order k 1 This shows tht we cn forlly differentite 43 to get 47 Acknowledgeent: The reserch of Yu Yn ws prtilly supported by the Scholr in Residence progr t Indin University-Purdue University Fort Wyne REFERENCES 1] Ynyn Li nd Louis Nirenberg A geoetric proble nd the hopf le ii Chinese Annls of Mthetics, Series B, 27B2:193 218, 2006 2] Yifei Pn nd Mei Wng When is function not flt? Journl of Mtheticl Anlysis nd Applictions, 3401:536 542, 2008 DEPARTMENT OF MATHEMATICAL SCIENCES, INDIANA UNIVERSITY-PURDUE UNIVERSITY FORT WAYNE, FORT WAYNE, INDIANA 46805 E-il ddress: pn@ipfwedu DEPARTMENT OF STATISTICS, UNIVERSITY OF CHICAGO, CHICAGO, ILLINOIS 60637 E-il ddress: eiwng@gltonuchicgoedu DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, HUNTINGTON UNIVERSITY, HUNTINGTON, INDIANA 46750 E-il ddress: yyn@huntingtonedu