Contraction Mapping Principle Approach to Differential Equations

Transcription

1 epl Journl of Science echnology 0 (009) Conrcion pping Principle pproch o Differenil Equions Bishnu P. Dhungn Deprmen of hemics, hendr Rn Cmpus ribhuvn Universiy, Khmu epl bsrc Using n eension of he conrcion mpping principle, new pproch hs been proposed in proving he eisence of unique soluions of some differenil equions. Key words: conrcion, eisence uniqueness, differenil equions. Inroducion Over he ps few decdes here hs been cler emergence of he ide of conrcion mpping in he relm of nonliner funcionl nlysis. Wih germ of his ide, i ws Bnch who ws firs ble o inroduce prove very powerful principle, clled Conrcion pping Principle. Beginning from he ide of conrcion is fied poin (Bily 966, Rubinsein 998, Yosid 978) we se he principle eend i o ieres i.e. resul similr o conrcion mpping principle is obined for mpping (no conrcion) provided h some iere of is conrcion. his pper is minly concerned wih he erordinry pplicbiliy effeciveness of he principle o evolve number of useful resuls in differenil equions. Priculrly he srengh will be given for new pproch in proving he eisence of unique soluions of some differenil equions wih some iniil condiions. Compred o he proofs hrough Picrd ieres (Brun 993) he mehod of our proof bsed on n eension of conrcion mpping principle is new pproch. In fc he mjor echnicl novely is o obin he resul (3.9) proving h is conrcion for some posiive ineger. Finlly, some emples o illusre he resuls re suibly provided. Preliminries ny equions which re of ineres in pplicions cn be pu in he form 0where is mpping of some subse of meric spce ino iself. Such poinquie nurlly is clled fied poin of. n ncien mehod of solving equions of he form is he mehod of ierion i.e. n iniil pproimionis chosen successive pproimionsre genered by he formul If he mppingis coninuous if he sequence converges o w, hen hus for coninuous mpping if he process (.) converges ll, hen i converges o fied poin of. However, o prove he convergence of (.), we will in generl need condiion on which is much sronger hn coninuiy. Specificlly, we will require h be conrcion in he sense h i lwys mps ny wo poins closer ogeher in uniform wy s epressed by he following definiion. Definiion.: Le (.) be meric spce. hen conrcion of is mpping wih he propery h for some rel number, oe h differenible mpping is conrcion if only if here is number wih.. 49

2 epl Journl of Science echnology 0 (009) heorem.: (Conrcion pping Principle, []) Le : X X be conrcion of complee meric spce ( X, d). hen hs unique fied poin i.e. here eiss unique poin w in X such h (. Furhermore, if 0 is ny poin w) = w of X ( n ) is sequence of ieres defined by n = ( n ), n =,, L. hen0 lim n n = w. his shows h w) is noher fied poin of. From uniqueness, hs fied poin w in X moreover i is unique since ny poin which fies clerly remins fied by h. Finlly o see,, L converges o w, we re-lbel, 3 s g noe h Remrk.: Boh condiions of heorem. re necessry sincehe mpping :(0,] (0,] defined by ( ) = / is conrcion mp bu hs no fied poin since ( 0,] is no complee meric spce. he mpping : R R defined by no fied poin lhough R is complee. If is conrcion mpping, hen is no conrcion hs n where n is posiive ineger, is clerly conrcion mpping. However he converse my no be rue s cn be seen from he following emple. Emple.: he funcion : R R defined by is no conrcion, bu h is. hus we see h provided some iere of is conrcion we sill ge fied poin resul similr o he conrcion mpping principle for. he following heorem is n eension of he principle o ieres. heorem.: Le ( X, d) be complee meric spce le : X X hve he propery h for some ineger > 0, he iere is conrcion of X. hen hs unique fied poin i.e. here eiss unique poin w in X such h. Furhermore, if is in X, hen he sequence of ieres defined by n + = ( n ), n converges o w. Proof: Since is conrcion of he complee meric spce ( X, d), i hs unique fied poin w, sy. I hen follows h ) = ( )) = = ( )) ) + = + L = We now rewrie he sequence,, 3, L, L,, ), ), ), L, ), )), )), )), L., his is cully combinion of he sequences, ), )), L 3,, 3, ), ), 3 ), )), L )), L )), L Ech row in he bove rry is obined by sring some poin of X iering wih he conrcion of g. By heorem., ny such sequence converges o unique fied poin of s g =, nmely w. Since ech row in he bove rry is subsequence of he combined sequence converging o w, he sequence mus lso converge o, s desired. in resuls Le F be rel vlued funcion on nonempy subse D of he Eucliden spce R. rel vlued funcion φ on n inervl I is sid o be soluion of he differenil equion d / d = F( (3.) on he inervl I if only if(φ( ), ) for ll,is differenible on I φ ( = F( φ(, I. D 50

3 Bishnu P. Dhungn/Conrcion pping Principle... Definiion 3.: Le F be coninuous rel vlued funcion on nonempy subse D of he Eucliden spce R. rel vlued funcionφ on n inervl I conining c is sid o be soluion of he inegrl equion [, b] on hence hese re he soluions o he differenil equion (3.3). o prove he heorem i suffices o show h hs unique fied poin. We firs show h some iere of is conrcion of X ino iself. o ech y C[, b],, we hve if only if on I for ll (φ(, D for ll I I he inegrl c F ( s), s) ds (3.), is coninuous (φ is defined for ech I since he funcion is coninuous on I. I is esy o see h he differenil ( c) = is equion (3.) wih he iniil condiion 0 equivlen o he inegrl equion (3.). heorem 3.: Le F : R wo vribles such h R be funcion of is defined for ll R. ssume h F is coninuous h here eiss rel number L wih F( F( y, L y for ll R. hen he differenil equion (3.3) subjec o n iniil condiion of he ype hs unique soluion. Proof: Le. X =C [, b] hen X is complee meric spce wih he meric. Define : X X by ( ( ))( = β + F( ( s), s) ds. hen he fied poins of re he soluions of he inegrl equion ( ( ))() ( ( )() = ( )))() ( ))() L ( ))( s) ( )( s) dsl ( s ) ds L ( ) =! By inducion, i is esy o see (3.4) Bu F(( ))( s), s) F(( ( s), s) ds L ( ) h ( ( ))( ( ( ))(.! I hen follows h ( ), ( ) sup ( ( ))( ( )( L = b ( b )! L sup! b ( ) = (3.5) b )]! Clim: 0 s. Proof of he clim: Le be he smlles posiive ineger such h L( b ). hen [ L ( b )! ]! (3.6) 5

4 epl Journl of Science echnology 0 (009) Bu for >, =! + + L since he produc L <, we hve + +! +! < (3.7) Le ε > 0 be rbirry. Choose. hen +! hus + <! = ε < ε ε.. ε (3.8) From (3.6), (3.7) (3.8), b )] we hve <.! b )]! ε hus 0 s. (3.9) Clerly here eiss posiive ineger such h b )]! < hence from (3.5), we hve b )] ( ), ( ).! I proves h is conrcion of X ino iself. By heorem., hs unique fied poin, sy φ( which is he unique soluion o he differenil equion (3.3). In prcice, finding wheher here is such n L s in heorem 3. is mjor quesion. his cn be seen from he following heorem some emples. heorem 3.: Le F : R [, b] R be funcion of wo vribles such h F ( is defined for ll R. ssume h F is coninuous, h F is prilly differenible wih respec o h is bounded hroughou R differenil equion. hen he subjec o n iniil condiion of he ype Proof: ssume h hs unique soluion. by ( ) F( men vlue heorem forin R F / L for ll R. Fi define funciong of lone G =. Clerlyis differenible by for some z (. Hence he proof is complee by heorem 3.. Emple 3.: Le F( = /( + ) for R condiion. hen he differenil equion wih some iniil hs unique soluion on [ 0,0]. For, le P be ny posiive rel number ny rel number. hen P + [ /( P + )] = ( P + ) /[ P( P + which yield /( P + ) P so h )] 0 3 /( P + ) /( P + ) P P. (3.0) Using (3.0) F( / = /( + ), we hve Since he sme conclusion cn be drwn for ny inervl [ n, n] he given differenil equion hs unique soluion defined for R. 5

5 Bishnu P. Dhungn/Conrcion pping Principle... Emple 3.: Le [, b] be n inervl conined in ( 0, ). Le for R. hen he differenil equion wih soluion on [, b]. For, F / = e /( + e ) /( + e hs unique ). References Rubinsein, I. L. Rubinsein 998. Pril differenil equions in clssicl mhemicl physics. Cmbridge Universiy Press, Unied Kingdom. 675 pp. Biley, D.F Some heorems on conrcion mppings. Journl of London hemicl Sociey 4: Yosid, K Funcionl nlysis. Springer, Berlin. Brun, Differenil equions heir pplicions, Springer-Verlg, ew York. 578 pp. 53

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