Some Common Fixed Point Theorems for a Pair of Non expansive Mappings in Generalized Exponential Convex Metric Space

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1 Mish Kumr Mishr D.B.OhU Ktoch It. J. Comp. Tch. Appl. Vol ( Som Commo Fi Poit Thorms for Pir of No psiv Mppigs i Grliz Epotil Cov Mtric Spc D.B.Oh Mish Kumr Mishr U Ktoch (Rsrch scholr Drvii Uivrsit ohb@hoo.co.i mkm78@riffmil.comu_shiv@hoo.com Dprtmt of Mthmtics R.K.G.I.T Dlhi-Mrut Ro Ghzib U.P./3 INDIA Dprtmt of Mthmtics H.I.E.T. Ghzib U.P.(INDIA. Abstrct: I this ppr w cosir Ishikw itrtio schm with rrors to pproimt th uiqu commo fi poit for Pir of No psiv Mppigs i Grliz Cov Mtric Spc. K wors: Commo fi poit No psiv mppig cov mtric spc.. Itrouctio I th litrtur commo fi poit thor m uthors prov svrl covrgc thorms for commo fi poits of m o psiv mppigs. Also stui th sm problm; (Gu Li 8 th cosir th Ishikw itrtio procss to pproimt th commo fi poit of m o psiv mppigs i uiforml cov Bch spc. (Tkhshi 97 first itrouc otio of cov mtric spc which is mor grl spc ch lir orm spc is spcil mpl of th spc. Lt o (Ciric 3 prov th covrgc of Ishikw tp itrtio procss to pproimt th commo fi poit of pir of mppigs ur coitio B. Vr rctl (Wg Liu 9 giv som sufficic cssr coitios for Ishikw tp itrtio procss with rrors to pproimt commo fi poit of two mppigs i grliz cov mtric spc. Ispir motivt b th bov fcts w will cosir th Ishikw tp itrtio procss with rrors which covrgs to th uiqu commo fi poit of th pir of smptoticll opsiv mppigs i grliz cov mtric spc. Our rsults t improv th corrspoig rsults.. Dfiitios Prlimiris Dfiitio. (s Tkhshi 97. Lt ( b mtric spc I = []. A mppig ω : I is si to b p-cov structur o if for ( α u th followig iqulit hols: ( ω( α u ( u α + ( u α. ( ( If ( is mtric spc with cov structur ω th ( is cll cov mtric spc. Morovr ompt subst α E of is si to b cov if α E I ( for ll (. Dfiitio. (s Y.-. Ti 5. Lt ( b mtric spc I = [] { { b { c rl squcs i [ ] with b = +. A mppig ω : I is si to b cov structur o if for 3 3 ( b c I u th followig iqulit hols: ( ω( b c ( ( u ( u ( z u. 33

2 Mish Kumr Mishr D.B.OhU Ktoch It. J. Comp. Tch. Appl. Vol ( If ( is mtric spc with cov structur ω th ( is cll grliz cov mtric spc. Morovr ompt subst E of is si to b cov if ω ( b c E for ll 3 3 ( b c E I. Rmrk.3. It is s to s tht vr grliz cov mtric spc is cov mtric spc ( lt =. c Dfiitio.4. Lt ( b mtric spc I = [] { { b { c rl squcs i [ ] with =. A mppig ω : 3 I 3 is si to b cov structur o 3 3 if for ( b c I u th followig iqulit hols: ( u ( u ( z u ( ω( b c log (.3 Dfiitio.5. Lt ( b grliz cov mtric spc with cov structur ω : 3 I 3 E ompt clos cov subst of. Lt S T : E E b pir of smptoticll opsiv mppigs { { b { c { ' si squcs i [ ] with ' = =... for giv E s follows: fi squc { = ω( S u b c + = ω T v ' b' c'.4 ( ( Am { i i Si Ti ui vi : i m whr { u { = v r two squcs i E stisfig th followig coitio. If for ogtiv itgrs m < m δ ( A > th m m i m { ( : { u v { S T u v < δ ( A δ i i (A ( A = sup.5 m Am ( th { is cll th Ishikw tp itrtio procss with rrors of pir of smptoticll opsiv mppigs S T. Rmrk.6 (s Wg Liu 9. Lt E b ompt clos cov subst of complt cov mtric spc S T : E E uiforml qusi- Lipschitzi mppigs with L > L ' > F = F( T F( T = : T = ( { Suppos tht { is th Ishikw tp itrtio procss with rrors fi b [.] { u { v stisf (A { { b { c { i [ ] stisfig ' r si squcs ' = = <.6 covrgs to fi poit of S T lim if F = whr F = if{ p : p F. th { if ol if ( ( ( Rmrk.7. Lt F ( T = { : T = /. A mppig T : is cll uiforml qusi- Lipshitzi if thr ists L > such tht ( T L(.7 for ll p F( T. Rso bhi this stu is th lmtr iqulit + b b ( > b > Assrts th covit of th fuctio of which m sil covic oslf b mkig th substitutio m 34

3 Mish Kumr Mishr D.B.OhU Ktoch It. J. Comp. Tch. Appl. Vol ( = b= which givs (B Th stu of th coitios ur which th suprpositios of th fuctios of crti clsss tur out to b cov or cocv is of fiit itrst. Lt b ompt cov subst of R show tht st is cov for. W ssum tht is ompt cov subst of R such tht st is cov for rbitrr. Th rltio + is stisfi. Lt us put t =. Hc for rl umbr α ( w hv ( t + ( ( α α α α A so ( + α + ( α α + ( α.8 B ssumptio th st is cov. W obti α + ( α =. ( t + log ( α + ( α Hols for α ( rbitrr poit. This ms b fiitio tht is cov. 3. Mi Rsults Now w will prov th strog covrgc of th itrtio schm (.4 to th uiqu commo fi poit of pir of smptoticll opsiv mppigs S T i complt grliz cov mtric spcs. Thorm 3.. Lt E b ompt clos cov subst of complt grliz p-cov mtric spc S T : E E pir of smptoticll opsiv mppigs with b F = F( T. / Suppos{ s i (.4 { u { v stisf (B { { b { c { ' r si squcs i [ ] stisfig ' = = < 3. th { covrg to th uiqu commo fi poit of S T if ol if lim if ( F = whr ( F = if{ ( : p F. Proof. Th cssit of coitios is obvious. Thus w will ol prov th sufficic. Lt p F for ll E ( S ( ( S ( p ( ( p S log ( ( [ ( p S ( ( ] [ ( p S log ] 3. implis ( ( S ( b c ( 3.3 which il (usig th fct tht + c b Sp ( ( p K 3.4 whr < K = ( /( b c.. Similrl T ( p ( p w lso hv K B Rmrk.7 w gt tht S T r two uiforml qusi-lipschitzi mppigs ( with L = L' = K >. Thrfor from Thorm.6 w kow tht { covrgs to commo fi poit of S T. Fill w prov th uiquss. Lt p = Sp = Tp p = Sp = Tp th b (* w hv ( [ ] c[ p log( ( p p ( p p ( p p ( p p + ( p log ( p. ( + c = Sic ( + c < w obti p = p. This complts th proof. 35

4 Mish Kumr Mishr D.B.OhU Ktoch It. J. Comp. Tch. Appl. Vol ( Rmrk 3.. (i W cosir sufficit cssr coitio for th Ishikw tp itrtio procss with rrors i complt grliz cov mtric spc; our mppigs r th mor grl mppigs ( pir of smptoticll opsiv mppigs so our rsult t grliz th corrspoig rsults. covrgs to th uiqu fi poit (ii Sic { of S T w hv improv Thorm.6 i (C. Wg L. W. Liu 9. Corollr 3.3. Lt E b ompt clos cov subst of Bch spc S T : pir of smptoticll opsiv mppigs tht is ( S T ( ( S ( T ( T ( S log 3.5 with b F = F( T /. For giv E { is Ishikw tp itrtio procss with rrors fi b = S u + ' T c' v = whr { u { v E r two bou squcs { { b { c { ' r si squcs i [ ] stisfig ' = = < 3.7 Th { covrgs to th uiqu commo fi poit of S T if ol if ( F lim if = whr ( F ( = if { : p F. Proof. From th proof of Thorm 3. w hv Sp ( p ( T ( p p ( K K 3.8 whr K = ( /( b c. Hc S T r two uiforml qusi-lipschitzi mppigs i Bch spc. Sic Thorm.6 lso hols i Bch spcs w c prov tht thr ists p F such tht lim p =. Th proof of uiquss is th sm to tht of Thorm.. Thrfor { covrgs to th uiqu commo fi poit of S T. Corollr 3.4. Lt E b ompt clos cov subst of Bch spc S T : pir of smptoticll opsiv mppigs tht is ( ( ( ( ( log b c S T S T T 3.9 with b F = F( T /. For giv E { is Ishikw tp itrtio procss with rrors fi b + 3. S = log( α + ( α whr { α { β = T = log( β + ( β ( α <. Th { r two squcs i [] stisfig covrgs to th uiqu commo fi poit of S T if ol if ( F lim if = whr ( F = if{ p : p F. Proof. Lt = α ' = β c = c'. Th rsult c b uc immitl from Corollr 3.3. This complts th proof. Coclusio: As w kow vrious pplictios of frctl sts i crptogrph stgogrph igitl sigtur commitmt schm k grmt protocol tht ms ll commuictios btw two commuictors with scurit. Th pplictio of 36

5 Mish Kumr Mishr D.B.OhU Ktoch It. J. Comp. Tch. Appl. Vol ( Ishikw itrtios i frctl sts is wll kow(yshwt S Chuh Rshri R Ashish Ngi. Our pproch provis mor grliz cov structur o th bsis of qutio.8 it ts th rsults of prvious litrtur. Our pproch will l th pltform for tsio of cov structur to iv structur i commo fi poit litrtur for futur vors. Rfrcs S. C. Bos 978 Commo fi poits of mppigs i uiforml cov Bch spc Jourl of th Loo Mthmticl Socit vol. 8 o. pp Z. Gu Y. Li 8 Approimtio mthos for commo fi poits of m opsiv mppig i Bch spcs Fi Poit Thor Applictios vol. Articl ID pgs. W. Tkhshi 97 A covit i mtric spc opsiv mppigs. I Koi Mthmticl Smir Rports vol. pp L. B. Ciric J. S. Um M. S. Kh 3 O th covrgc of th Ishikw itrts to commo fi poit of two mppigs Archivum Mthmticum vol. 39 o. pp C. Wg L. W. Liu 9 Covrgc thorms for fi poits of uiforml qusi- Lipschitzi mppigs i cov mtric spcs Nolir Alsis: Thor Mthos & Applictios vol. 7 o. 5 pp Y.-. Ti 5 Covrgc of Ishikw tp itrtiv schm for smptoticll qusi-opsiv mppigs Computrs & Mthmtics with Applictios vol. 49 o.-pp Yshwt S Chuh Rshri R Ashish Ngi Nw Juli Sts of Ishikw Itrts Itrtiol Jourl of Computr Applictios ( Volum 7 No.3. 37

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