Successive ApproxiInations and Osgood's Theorenl

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1 Revisa de la Unin Maemaica Argenina Vlumen 40, Niimers 3 y 4, Successive ApprxiInains and Osgd's Therenl Calix P. Caldern Virginia N. Vera de Seri.July 29, 1996 Absrac The Picard's mehd fr slving Y' = f (x, y), y (x) = Y, is cnsidered here fr If (x, YI) - f (x, Y2)1 :::; 'P (IYI - Y21). I is shwn ha fr raher general Osgd's funcins 'P, he difference f w successive apprximains cnverges a expnenially decreasing rae. An applicain parablic parial differenial equains is given as well. Key Wrds: Successive apprximains. Thereical apprxi. main f sluins. Osgd's funcins. 1 Inrducin In a landmark paper W. Osgd (1898) inrduced a cndiin weaker han he well knwn Cauchy-Lipschiz ne ha guaranees he uniqueness f he sluin f he iniial value prblem fr a firs rder equain. Laer, A. Winner (1946) shwed ha Osgd's uniqueness cndiin implies he cnvergence f he successive apprximain n a sufficienly small inerval. Laer n, J. Lasalle (1949) exended Winner's mehd he case f he uniqueness cndiin fr he sluin given by P. Mnel and M. Nagum in Lasalle's mehd is essenially Winner's bu adaped a differen cndiin. One f he imprvemens inrduced by Lasalle is he fac ha he zer apprximain can be chsen be anyne bu sufficienly small in L 1991 Mahemaics Subjec Classificain: Primary 34A45 Secndary 35K05

2 74 nrm. Lasalle and Winner's papers raise quesins cncerning he speed f he cnvergence f he successive apprximains. I is he aim f his paper answer hese quesins fr he cases f raher general Osgd's funcins. In fac, we shw ha in cases such as r.p () = Ilgl,8 0 < f3 ::; 1, he difference f w successive apprximains IYn+! - Yn I cnverges a expnenially decreasing speed. The resul in his paper can be easily exended he case f firs rder sysems very much in he same way as Lasalle (1949). Fr such a generalizain we refer he reader he paper by Lasalle.. We will deal wih he equain Y' = f (x, y) wih iniial cndiin Y (x) = Y. f is a cninuus funcin which verifies ha If (x, yj) - f (x, Y2)1 ::; r.p (IYI - Y2J), where a ::; x ::; band c ::; Y ::; d. r.p saisfies he mdified Osgd's cndiins belw in 2.1. Fr he sake f simpliciy we may assume ha -a ::; x ::; a, -c ::; Y ::; c and X = 0, Y = O. We cnsider he successive apprximains x Yn+! (x) = J and sudy he mdali.y f cnvergence f f (, Yn ()) d fr Ixl ::; {j and {j > 0 suiably small. Wihin ha range we will have fr n ;:::: n, fr sme n, r,o < r < I, and C a psiive cnsan. Cnsider he auxiliary ierain x Zn+! (4) = J r.p (zn ()) d, in general Wn ::; Zn. We sudy he speed f cnvergence zer f Zn (x), which aumaically will imply he resul. In he final secin we briefly indicae sme her applicains parablic parial differenial equains. '(1)

3 75 2 Osgd's funcins A real valued funcin <p is an Osgd's funcin if he fllwing cndiins are me: 1) <p is nn-negaive cninuus and mnne n (0,6), fr sme 6 > O. 2) f; <p(~) = 00, fr any :, 0 < : ~ 6. Remark 1 1) and 2) abve imply immediaely ha limx + <p (x) = 0 and ha i.p is nn-decreasing. We will define <p (0) =. 2.1 Mdified Osgd's cndiin Thrughu his paper we shall be cncerned wih Osgd's funcins f he ype b i.p() = JW~s) ds, where b > 0 is a given cnsan. W is a nn-negaive cninuus nndecreasing funcin n [0, b], such ha Nice ha i.p (0+) = O. Frm nw n we will assume b ~ 1. b J W ~s) ds =00. D Occasinally we use he nain (Y () = f w(s) ds.. 8 Lemma 1 The funcin i.p saisfies he fllwing prperies: (i) There is sme a, a> 0, such ha <p (a) = a. (ii) lim <p' () > l. -+O+ There is sme 6,0 < 6 ~ a, such ha fr,o < ~ 6, i hlds ha (iii)i.p' and (Yare decreasing, (iv) (Y (n) ~ n(y (b,,;;:l ), fr all n = 1,2,... (v) he quien ;,<l) = :Jl) is bunded, and (vi)<p2f) = a2 () is increasing. b

4 76 Prf. (i) hlds because 'P (0) = 0, 'P (b) = 0 and 'P is cncave. (ii) and (iii) fllw frm he represenain 'P' () = a () - w () = lb W (s ) ds - w (). s (iv) We shw he case b = 1, he hers fllw by cnsidering he funcin it (T) = a (bt) and he change f variable = bt. Le a, (3, 0 < a, (3 ~ 1, hen J w (as) ds + J w (s) ds < J w (s) ds + J w (s) ds s s - s s (3 0: (3 0: a (a) + a (,8). ( ) ~.. u() 1 0+ V <p'() = u()-w() ---+,as ---+ (vi) i (a ()2) = a ()(a() - 2w ()) >. 2.2 Ms cmmn mdified Osgd's funcins I is sraighfrward shw ha he fllwing funcins saisfy he mdified Osgd's cndiins: 'Pk () = (lg (1/ ) )(31 (lg lg (l/))f32... (lg lg lg (1 /) )(3" fr sme 0 ~,8i < 1, i = 1,..., k - 1,0 <,8k ~ 1, k = 1,2,... These are he ms cmmn mdified Osgd's funcins. Observe ha fr all f hem, he cnsan b saisfies ha b ~ 1. 3 Auxiliary lemmaa Frm nw n we will assume b ~ 1. The fllwing lemmaa are he echnical ls fr prving ha Ilwnll ~ Crn fr sme r, 0 < r < 1, C a psiive cnsan. Assume ha 0 < ~ I), where I) > 0 is as in Lemma 1.

5 77 L (( )n) A <p(x+1 emma 2 'P >''P x ~ nb x ' fr any >. ~ 1,n = 1,2,... Prf. Frm 'P () = fj () and Lemma 1 ( iii), (iv) i fllws ha 'P (>''P (x) - >''P (x fj (>''P (x) - >''P (x fj (>.xnfj (x)n) < >''P (x fj (xn) < n~'p (x (J (x) >. 'P (x+1 - n- b x L x ()n 1 <p(:i:),,+l <p(x)n+1 _ emma 3 10 'P d ~ <p'(x) n+1 ~ n+1,n - 1,2,... Prf. Lemma 1 (iii) and (ii) imply ha r 'P ( d < r 'P ( <p' () d = _1_'P (x+1 < 'P (x+1 J - J 'P' (x) 'P' (x) n n + 1 Lemma 4 There exiss a psiive cnsan C such ha I: ~ d ~ n~l 'P (x) n,,n = 1,2,... 'Prf. Lemma 1 (v)-{vi) and Lemma 3 give r 'P(r d J _ r 'P ()2 'P (-2 d J < 'P (X)2 r 'P (r-2 d x J 'P(X)2 1 'P(xr- 1 < '-x-'p'(x) n-1 C (-n < n_ 1'P x ). Lemma 5 Le Z be a nn-negaive funcin defined fr Ixl ~ fj, Z (x) ~ 1], x fj and 1] small enugh. Suppse Zn+ 1 (x) = I 'P (Zn ()) d, hen,

6 78 ~ Zl (x) ~X, ~ Zn(X) ~ cn-2cp(x)", n = 2,3,..., C > 1 a suiable cnsan. Prf. cp is a cninuus funcin wih cp (0) = 0, s fr 11 > 0 sufficienly small, IX ~ Zl (x) ~ I cp (1IZID d ~ x.., Pu C = ~', where 0' is as in Lemma 4 and such ha 0' > b. Fr n = 2, we use Lemma 3 ge x x 1 ~ Z2 (x) ~ I cp (Zl (» d ~ I cp () d ~ "2CP (X)2 ~ cp (X)2.. 0 Nw, by inducin n n, fr n Z 2, Z < zn+1 (x) == I cp{zn (» d ~ I 'P (cn-2cp() d 0 cn-2 IX cp ()"+l < n-- d b < cn-2 C' ()n+1 n---cp x b n _ n-lcp (x)"+1, IX because f Lemmas 2 and 4. 4 Rae f cnvergence f he successive apprximains In rder analyze he rae f cnvergence f Wn, bserve ha IX W'HI (x) ~ I If (, Yn+l (» - f (, Yn (»1 d ~ I cp (wn (» d. 0 Z

7 79 We will cnsider a differen ierain. If we begin wih small iniial values, namely: W (x) $.1], fr Ixl $. 6, we sudy he auxiliary ierain z Zn+l (x) = J I' (Zn ()) d, where Z = 1]. I is clear ha WI $. Zl and, in general Wn $. Zn, which is a cnsequence f he mnniciy f <p Then i will be enugh prve he esimae (1) fr Zn. I will be wrh nice ha because f he cncaviy f <p, plus <p (0) =. The nex sep will be sudy he speed f cnvergence zer f Zn (x), Ixl ::; 6 and shw ha fr Zn (x) he esimae (1) is valid. Indeed, frm Lemma 5, here exiss a cnsan C > 1 such ha 0 $. Zn (x) $. C n - 2<p (x. By he cninuiy f <p and he fac ha <p(0) = 0, fr any r, 0 < r < 1, here is sme 6',0 < 6' $. 6, fr which C<p (x) $. r, fr Ixl $. 6'. Thus, we have prven he fllwing herem: Therem 6 Le f be a cninuus funcins n he recangle R = [-c, c] x [-d,d]. Suppse ha fr Ixl$. c,iyil $. d, i = 1,2, where <p verifies he mdifiea Osgd's cndiins saed in secin 2.1. Then, he successive apprximains z YnH (x) = J IYl $. 1],1]> 0 sufficienly small, saisfy f (, Yn ()) d, 6. fr sme r,o < r < 1, sme C large enugh and Ixl $. 6, fr sme psiive

8 80 5 An applicain parial differenial equains Cnsider he fllwing parablic parial differenial equain U=~u+F(u,x,), U(x,O) = O. Assume ha F is a cninuus funcin ha saisfies where IIIL!; is an ad-hc nrm in he space variables and i.p is a mdified Osgd's funcin as in secin 2.l. Le K (x, ) be he usual L1 fundamenal sluin! and implemen he ierains Un+1 (x, ) = J fa"" K(x - y, - T) F (un (y, T), y, T) dydt, U is chsen be C in R m wih small L nrm. Then I\Un+1 -'- unll x, which is a funcin f, saisfies ha IIUn+l - unllx S; J llfa", K (x - y, - T) (F (un, y, T) - F (un_i, y, T)) dy IL dt < jllf (un,., T) - F (un-i," T)llx dt < J'P (liun - un-illx) dt. Wih a similar prcedure as he ne emplyed befre, by chsing U suiably small in he apprpriae ad-hc nrm, i fllws ha fr 0 S; < {;, sme {; > 0 and sme r, 0 < r < 1. 1 As i is well knwn, B. Frank Jnffi has cnsruced fundamenal sluins ha are n f he usual ype.

9 81 References [1] Iyanaga, S., Uber die Uniasbedingungen der Lasung der DifIerenialgleichung, Jap. J. Mah. 5 (1928), [2] LaSalle, J. Uniqueness herems and successive apprximains, Annals f Mahemaics 50 (1949), [3] Mnel, P., Sur 1 'inegrale superieure e 1 'inegrale inferieure d 'une equain difierenielle, Bull. Sci. Mah. (2) 50 (1926), [4] Muller, M., Uber das Fundamenalherem in der Therie der gewahnlichen DifIerenialgleichungen, Mah. Zei. 26 (1927), [5] Nagum, M., Eine hinreichende Bedingung fur die Uni a der Lasung vn DifIerenialgleichungen erser Ordnung, Jap. J. Mah. 3 (1926), [6] Osgd, W., Beweise der Exisenz einer Lasung der DifIerenialgleichungen dy/dx = f(x,y) hne Hinzunahme der Cauchy-Lipschizschen Bedingung, Mnashefe fur Mah. und Physik 9 (1898), [7] Perrn, 0., Eine hinreichende Bed.ingung fur die Unia der Lasung vn DifIerenialgleichungen erser Ordnung, Mah. Zei. 28 (1928), [8] Winner, A., On he cnvergence f successive apprximains, Amer. J. Mah. 68 (1946), Insiu de Ciencias Basicas, Universidad Nacinal de Cuy Deparmen f Mahemaics, Saisics and Cmpuer Sciences, Universiy f illinis a Chicag Faculad de Ciencias Ecnmicas, Universidad Nacinal de Cuy Cenr U niversiari 5500 Mendza Argenina address:vvera@raiz.uncu.edu.ar Recibid en Ags de 1996