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1 IESRT INTERNTIONL OURNL OF ENGINEERING SCIENCES & RESERCH TECHNOLOGY HYBRID FIED POINT THEOREM FOR NONLINER DIFFERENTIL EQUTIONS Sidhshwar Sagram Bllal*, Gash Babrwa Dapk * Dparm o Mahmaics, Daaad Scic Collg, Lar,Maharashra, Idia School o Egirig, Maoshri Praishha,Nadd. (M.S.), Idia DOI: /zodo BSTRCT For Hbrid id poi horm or oicrasig mappig i pariall ordrd compl mric spac o prov isc as wll as iiial val problm o oliar irs ordr ordiar dirial qaios. KEYWORDS: Hbrid id poi horm, o-liar dirial qaio. INTRODUCTION Th mid hpohsis o algbra, opolog ad gomr h i is calld as hbrid id poi horm ad hs hbrid id poi horm cosi a w sram o hbrid id poi hor i h sbjc o o-liar cioal aalsis. I is wll kow ha h hbrid id poi horm which ar obaid sig h mid argm rom dir brachs o Mahmaics,pariclarl h hor o o-liar dirial ad igral qaios, (s Hikkila ad Lakhsmikaham [6], Zidlr[9 ]ad Dhag[2,3,5]) This id poi horm combis h mric id poi horm o Baach wih a opological id poi horm o schadr i a Baach spac ad kraoslskiihim sl applid i o som o-liar igral qaios o mid p or provig h isc rsls dr mid Lipschiz ad compacss codiios. Rcl, Ra ad Rcrigs [8] iiiad h sd o hbrid id poi horms i pariall ordrd ss which is rhr coid i Nio ad Rodrigz Lopz [7] ad provd h hbrid id poi horm or h mooo. Diiio 1.1:- pariall ordrd mric spac,, d is calld rglar i is a odcrasig (rspcivl oicrasig) sqc i sch ha as h (rspcivl all N. Lopz ad Nio[7] givs ollowig diiio. Codiio (NL):- pariall ordrd mric spac wih mric d is said o sais codiio (NL) i or vr covrg sqc i o h poi whos cosciv rms ar comparabl h hr iss a sbsqc o sch ha vr rms is comparabl o h limi. k Th ollowig hbrid id poi horm or oicrasig mappig is provd i Nio ad Lopz [7]. Thorm 1.1(Nio ad Rodrigz-Lopz [7]):- L, b a pariall ordrd s ad sppos ha hr is a mric d i sch ha, d is a compl mric spac. L : b a mooo oicrasig mappig sch ha hr iss a cosa,1 sch ha d, k d, (1.1) lm,,. ssm ha ihr is coios or ) or saisis codiio (NL). Frhr i hr is a lm saisig or h has a id poi which is rhr iq i vr pair o lm i has a lowr ad a ppr bod. Iraioal oral o Egirig Scics & Rsarch Tcholog [272]

2 HYBRID FIED POINT THEORY W cosidr h ollowig diiio i wha ollows. Codiio (D):- pariall ordrd mric spac wih mric d is said o sais codiio (D) i vr sqc i whos cosciv rms ar comparabl has a mooo i..odcrasig or oicrasig sbsqc. Thr do is sqc i wih codiio (D). For ampl, i w cosidr = Ɍ, h h sqc 1 1 i Ɍ did b 1 has wo sbsqc, o is odcrasig aohr is oicrasig. gai, 1 1 h sqc 1,, 3,... saisis h codiio (D) b o codiio (NL). 2 4 Diiio 2.1:- L b a mric spac ad L b a mappig. Giv a lm, w di a orbi O ;,d o a b 2 O :,,, Th is calld T T or ach O,T -orbiall coios o * sch ha a :.Th mric spac is calld as i or a sqc O ; T orbiall coii ad complss implis T :, w hav ha orbiall compl i Cach sqc i Covrgs o i. No:-Th coii implis ha o a mric spac, b covrs ma o b r. Diiio 2.2(Dhag [5]):- mappig is calld pariall coios a a poi i or ϵ hr iss a δ < ϵ whvr is comparabl o a ad <δ. coios o, h i is coios o vr chai C coaid i W rql d a damal rsl cocrig Cach sqc i wha ollows. For, w d h ollowig diiio. Diiio 2.3 (Dhag [4]):- mappig : R R is calld a Domiaig cio or, i shor, cio i i. a orbiall complss a E Calld pariall is a ppr smi-coios ad moooic odcrasig cio saisig ( ). Lmma 2.1:- L : R R b D -cio saisig r r or r. h lim. or R ad vic vrsa. Now w ar rad o sa a k rsl i rms o D -cio characrizig h Cach sqcs i a mric spac. Lmma 2.2:- I is a sqc i a mric spac, d saisig d d (2.1), 1 1 or all N, whr is a cio sch ha r r, r, is Cach. Thorm 2.1:- L,, d b a pariall ordrd mric spac. L : b a mooo oicrasig mappig sch ha hr is a D -cio sch ha 2 d, T d, (2.2) D h lm comparabl o T, whr r r, r. Sppos ha ihr is -orbiall compl ad is orbiall coios or is pariall orbiall coios ad is rglar ad saisis codiio (D). Frhr i hr is a lm saisig or T, h T has a id poi * ad h sqc T o iraios covrgs o *. Thorm 2.2:- L,, d b a pariall ordrd mric spac. L : b a mooo oicrasig mappig sch ha hr is a D -cio sch ha d, d, (2.3) comparabl lms,, whr r r, r. sppos ha ihr is orbiall compl ad is -orbiall coios or is pariall -orbiall coios ad is rglar ad saisis codiio (D). Frhr i hr is a lm saisig T or T, h Has a id poi ad h Iraioal oral o Egirig Scics & Rsarch Tcholog [273]

3 sqc o iraios covrgs o which is rhr iq i vr pair o lms i has a lowr ad a ppr bod". Thorm 2.3:- L b a pariall ordrd mric spac ad L b mooo mappig T,,d T : (mooo icrasig or mooo dcrasig) saisig (2.2). Sppos ha ihr is -orbiall coios or is pariall T-orbiall coios ad is rglar ad saisis codiio (D). I hr is a wih or, h T has a id poi ad h sqc o iraios covrgs T T T o. Thorm 2.4:- L b a pariall ordrd compl mric spac. L b a mooo mappig (mooo oicrasig or mooo dcrasig) saisig (2.3). Sppos ha ihr is T -orbiall compl ad is rglar ad saisis codiio (D). I hr iss a wih or, Th has a id poi ad h sqc o iraios o a covrgs o which is rhr iq vr pair o lms i has a lowr ad a ppr bod". T,, d T : T PPLICTIONS TO HYBRID DIFFERENTIL EQUTIONS Giv a closd ad bodd irval o h ral li or som, a R wih a,cosidr h iiial val problm ( i.. IVP ) o irs ordr ordiar oliar Hbrid dirial qaios ( I shor HDE ). Whr, a,. g,, R, g : R R is coios cio. C, is h spac o coios ral vald cios did o. Th HDE (3.1) is wll-kow i h lirar ad discssd a lgh or isc as wll as ohr aspcs o C, R o coios did o. W di B a solio o h HDE (3.1). W ma a cio C, R ha saisis qaio (1.1), whr R h solios. Th HDE (3.1) is cosidrd i h cio spac a orm. ad h ordr rlaio i C, R b R (3.1) d sp (3.2) (3.3). Clarl R C, i a Baach spac wih rspc o abov sprmm orm ad also pariall ordrd w.r.o h abov pariall ordr rlaio rglar as wll as laic. C, R is said o b a lowr solio o h HDE (1.1) i i saisis Diiio 3.1:- cio, g, W cosidr h ollowig s o assmpios i wha ollows. Thr is cosas ad, wih sch ha 1 1 ad, R,. Th HDE (1.1) has a lowr solio C, R 2 Cosidr h IVP o h HDE C, is i i. I is kow ha h pariall ordrd Baach spac R, g,, g,. Iraioal oral o Egirig Scics & Rsarch Tcholog [274]

4 , g,, whr, g: R R ad g,, g,, Rmark: - No ha h codiio opraor F (3.4) (3.5), g is coios o R ad so h associad sprposiio Nmski is igrabl o is a solio o h HDE (3.4) i i is a posiio o h HDE (1.1) o. Lmma 3.1:- cio C, R is a solio o h HDE (3.4) i i is a solio o h oliar igral qaio c s. gai, a cio C, R s s, sds gs, s ds (3.6) whr C is a ral mbr did b Thorm 3.1:- ssm ha hpohsis Did o 1 c ad h sqc s Whr covrgs o Proo: S E C(, R) 1 ad C 2. hold.h h HDE (1.1) has a iq solio o sccssiv approimaios did b s s, sds gs, s ds (3.7). s c ad di wo opraors o E b s s, sds gs, s ds, (3.8) From h coii o h igral, i ollows ha dis h map : E E. Now b lmma (3.1) Th HDE (3.1) is qival o h opraor qaio., W shall show ha h opraor saisis all h codiio o horm (2.1) Firs w show ha is mooo oicrasig o E, l, E b sch c c s s, s gs, s, s gs, s ds ds or all. This shows ha is oicrasig opraor o E io E. N, L, E b sch ha.h (3.9) Iraioal oral o Egirig Scics & Rsarch Tcholog [275]

5 s s s, s g s, s ds s, s gs, s B ssmpio 1 s s d ds 1 1 s s s s s s s s s ds ds 1 s B hc ds 1.Takig sprmm ovr, w obai j r, E wih, whr is a D -cio did b r r, r. Hc 1 r Saisis h coracio codiio (2.3) o E which rhr implis ha is a pariall coios ad cosql pariall T -orbiall coios o E. N, w show ha saisis h opraor, h HDE (1.1) has a lowr solio.th w hav. B hpohsis 2 ds, g, 3.1, ddig o boh sids o h irs iqali i (3.1), w obai, g,, 3.11 gai, mliplig h abov iqali (3.11) b, g,, g, (B ssmpio 2 ), g, (3.12) Takig igraio o h boh sid rom s o s, s g s, sds c, w g w.r.o s Iraioal oral o Egirig Scics & Rsarch Tcholog [276]

6 Hc qaio o. s s, s gs, s ds c (3.13) rom diiio o h opraor or all.. Ths saisis h codiio o horm (2.2) ad w appl i o cocld ha h opraor i ollows ha has a solio. Cosql h igral qaio ad h HDE (1.1) has a solio. Frhrmor, h sqc o sccssiv approimaio did b (3.7) covrgs o *. Hc h proo. * did REFERENCES [1] G.Birkho, Laic hor, mr.mah. Soc.Pbl [2] B.C. Dhag, O sio o Tarski s id poi horm ad applicaio, pr ppl. Mah. Sci. 25 (1987), [3] B.C. Dhag, Som id poi horms or i ordrd Baach Spacs ad applicaios, Mahmaics sd 61(1992), [4] B.C.Dhag, Fid poi horm i ordrd Baach algbras ad applicaios, Pa mr. Mah..9 (4) (1999), [5] B.C.Dhag, Hbrid Fid poi hor i pariall ordrd ormd liar spacs ad applicaiosto racioal igral qaios, Dir. Eq ppl.5 (213), [6] S. Hikkila ad V.Lakshmikaham, Mooo Iraiv Tchiqs or Discoios Noliar Dirial Eqaios, Marcl Dkkr ic., Nw York [7]..Nio ad R.Rodrigz-Lopz, Eisc ad Uiqss o id poi i pariall ordrd ss d applicaios o ordiar dirial qaios, a Mah. Siica (Eglish Sris) 23 (27), [8].C.M. Ra, M.C.R. Rrigs. id poi horm i pariall ordrd ss ad som pplicaios o mari qaios, proc. mr. Mah. Soc.132 (23), [9] E.Zidlr.Noliar Fcioal alsis ad is pplicaios: par I, Sprigr Vrlag Iraioal oral o Egirig Scics & Rsarch Tcholog [277]