CHATTERJEA CONTRACTION MAPPING THEOREM IN CONE HEPTAGONAL METRIC SPACE

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1 Fameal Joal of Mahemaic a Mahemaical Sciece Vol. 7 Ie 07 Page 5- Thi pape i aailable olie a hp://.fi.com/ Pblihe olie Jaa 0 07 CHATTERJEA CONTRACTION MAPPING THEOREM IN CONE HEPTAGONAL METRIC SPACE Caolo Roa Boo MA 0-60 USA ampa@homail.com Abac We ioce a cocep of Coe Hepagoal Meic Space a obai he Chaejea Fie Poi Theoem (Chaejea []) i hi eig.. Iocio Hag a Zhag [] ioce he cocep of a coe meic pace. The eplace he e of eal mbe b a oee Baach pace a poe ome fie poi heoem fo coacie pe coiio i coe meic pace. Lae o ma aho hae poe ome fie poi heoem fo iffee coacie pe coiio i coe meic pace; fo eample ee Commo fie poi el fo o commig mappig iho coii i coe meic pace (Abba a Jgc []); Commo fie poi fo map o coe meic pace (Ilic a Raoceic []); Some oe o he pape coe meic pace a fie poi heoem of coacie mappig (Reapo a Hamlbaai [5]). Keo a phae: Chaejea coacio coe meic pace hepagoal pope fie poi heoem. 00 Mahemaic Sbjec Claificaio: 7H0 5H5. Receie Decembe 06; Accepe Decembe Fameal Reeach a Deelopme Ieaioal

2 6 Gag [6] ioce he oio of coe heagoal meic pace a poe Baach coacio mappig piciple i a omal coe heagoal meic pace eig. Ve ecel Aal a Hical [7] poe he Kaa coacio mappig piciple i coe heagoal meic pace. I hi pape ipie b he o of Gag [6] a Aal a Hical [7] e ioce a cocep of coe hepagoal meic pace a poe he Chaejea coacio mappig piciple i hi eig. Thi pape i ogaie a follo. Secio coai ome pelimia iea ha ol be efl i he eqel. Eample.8 ho ha he oio of coe hepagoal meic pace i a pope eeio of coe heagoal meic pace. Secio coai he mai el i paicla he Chaejea coacio mappig piciple i coe hepagoal meic pace i gie b Theoem. a Eample. i gie o illae he Chaejea coacio piciple i coe hepagoal meic pace.. Pelimiaie Noaio.. E ill eoe a eal Baach pace. Defiiio.. P E ill be calle a coe iff (a) P i cloe oemp a P { 0} (b) a b R a b 0 a P implie a + b P (c) P a P implie 0. Noaio.. ill eoe a paial oeig ih epec o P a ill be efie b iff P. We hall ie < o iicae ha b hile << ill a fo i( P) hee i ( P) eoe he ieio of P. Defiiio.. A coe P i calle omal if hee i a mbe > 0 ch ha fo all E he ieqali 0 implie ha. The lea poiie mbe aifig i calle he omal coa of P.

3 CHATTERJEA CONTRACTION MAPPING THEOREM 7 Rema.5. I hi pape e ala ame ha E i a eal Baach pace a P i a coe i E ih ( ) Φ P i a i a paial oeig ih epec o P. Defiiio.6. Le be a oemp e. Sppoe he mappig E : aifie (a) ( ) 0 < fo all a ( ) 0 iff (b) ( ) ( ) fo all (c) ( ) ( ) ( ) + fo all. The i calle a coe meic o a ( ) i calle a coe meic pace. Rema.7. If e eplace (c) of he peio efiiio ih he folloig hich e call he hepagoal pope ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + fo all a fo all iic poi { } he e a i a coe hepagoal meic o a e call ( ) a coe hepagoal meic pace. Eample.8. Le { } R E a {( ) } 0 : P be a coe i E. Defie E : b ( ) 0 fo all ( ) ( ) ( ) 6 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). 0 5

4 8 The i i ea o ee ha ( ) i a coe hepagoal meic pace b i i o a coe heagoal meic pace ice i lac he heagoal pope of Aal a Hical [7] ice ( 6 ) ( ) > ( ) + ( ) + ( ) + ( ) + ( ) ( ) + ( ) + ( ) + ( ) + ( ) ( 5 0) a ( 6 ) ( 5 0) ( ) P. Defiiio.9. Le ( ) be a coe hepagoal meic pace. Le { } be a eqece i a. If fo ee c E ih 0 << c hee ei N ch ha fo all > 0 ( } << c he { } i ai o be coege. Defiiio.0. Le ( ) be a coe hepagoal meic pace. Le { } be a eqece i. If fo ee c E ih 0 << c hee ei 0 N ch ha fo all m > 0 ( m ) << c he { } i calle a Cach eqece i. Defiiio.. Le ( ) be a coe hepagoal meic pace. If ee Cach eqece i coege i he ill be calle a complee coe hepagoal meic pace. Taig ipiaio fom Gag a Agaal [8] e hae he folloig Lemma.. Le ( ) be a coe hepagoal meic pace a P be a omal coe ih omal coa. Le { } be a eqece i he { } coege o iff ( ) 0 a. Taig ipiaio fom Gag a Agaal [8] e hae he folloig Lemma.. Le ( ) be a coe hepagoal meic pace a P be a omal coe ih omal coa. Le { } be a eqece i he { } i a Cach eqece iff ( +m ) 0 a m. Taig ipiaio fom Jleli a Same [9 Lemma.0] e hae he folloig Lemma.. Le ( ) be a complee coe hepagoal meic pace P be a omal coe ih omal coa. Le { } be a Cach eqece i a ppoe hee i a aal mbe N ch ha 0

5 CHATTERJEA CONTRACTION MAPPING THEOREM 9 (a) (b) (c) m fo all m > N ae iic poi i fo all > N ae iic poi i fo all > N () The. a a.. Mai Rel Theoem.. Le ( ) be a complee coe hepagoal meic pace P be a omal coe ih omal coa. Sppoe he mappig f : aifie he coacie coiio: ( f f) α[ ( f) + ( f) ] fo all a α 0. The (a) f ha a iqe fie poi i (b) fo a he ieaie eqece { f } coege o he fie poi. Poof. Le. Fom he coacie coiio e ece ha ( f f ) α( f ) α[ ( f) + ( f f )] α fom hich i follo ha ( f f ) ( f). Similal e hae α α ( f f ) ( f f ) α ( f). Coiig e ece fo α α + each poiie iege ha ( f f ) α ( f) ( f) α α hee 0 : <. No e iie he poof io o cae. α Fi Cae. Le f m f fo ome m N ih m. Le m >.

6 0 m The f ( f ) f ha i f p hee p m a f. No p p+ p ice p > e hae ( f) ( f f ) ( f). Sice [ 0 ) i follo ha ( f) P a ( f) P hich implie ha ( f) 0 ha i f. Seco Cae. Ame ha f m f fo all m N ih m. Sice + 0 he i follo ha h i i clea ha ( f f ) ( f) ( f) No ( f f ) ( f f ) + ( f f ) ( f) + + ( f) ( + ) ( f) a ice ( f) + e ece ha ( f f ) ( f). ( + ) ( f) + ( + + ) Alo e hae he folloig ( f f ) ( f) a ( + + ) ice ( f) ( f) e ece ha ( + f f ) ( f). Theefoe i i clea ha e alo hae he folloig + ( f f ) ( f) + 5 a ( f f ) ( f). No if m > 5 i o he iig m 5 + l l a ig he fac ha p f f fo p N p e ee ha + m ( f f ) l ( ) ( f)

7 CHATTERJEA CONTRACTION MAPPING THEOREM a ice ( f) l ( ) ( f) + m e ece ha ( f f ) ( f). f p No if m > 5 i ee he iig m + l l a ig he fac ha f fo p N p e ee ha + m ( f f ) l ( ) ( f) a ice ( f) l ( ) ( f) + m e ece ha ( f f ) ( f). + m Th combiig all he cae e hae ( f f ) ( f) all m N. fo No if e ae om o ieqali i he epeio immeiael aboe e + m ece ha ( f f ) ( f). Sice ( f) 0 a i follo ha he eqece { f } i Cach a b he compleee of hee i * ch ha f * a. No e ho eiece of he fie poi. Noice ha ( * f* ) ( * f ) + ( f f ) + ( f f ) + ( f f ) ( f f ) + ( f f* )

8 ( * f ) + ( f f ) + ( f f ) + ( f f ) ( f f ) + α[ ( f f* ) + ( * f )] ( * f ) + ( f f ) + ( f f ) + ( f f ) ( f f ) + α( * f ) + α( f f* ). Taig limi i he aboe ieqali e ge ( * f* ) α( * f* ) a ice α 0 i follo α < h α > 0. Hece fom he ieqali ( * f* ) α( * f* ) e ece ha ( * f* ) 0 ha i * f *. No e ho iqee of he fie poi. If * i aohe fie poi of f he i follo ha ( * * ) ( f* f* ) α[ ( * f * ) + ( * f* )] α[ ( * * )] a ice α 0 i follo ha α > 0. Hece fom he ieqali ( * * ) α( * * ) e ece ha ( * * ) 0 ha i * * a iqee follo. No e illae he mai el ih he folloig Eample.. Le omal coe i E. Le a E C a P {( ) : + i R 0} be a : be gie b Eample.8. The a Eample.8 ho ( ) i a coe hepagoal meic pace b i i o a coe heagoal meic pace ice i lac he heagoal pope of Aal a Hical [7] ice 6 + i ( 6 ) ( ) > ( ) + ( ) + ( ) + ( ) + ( ) 5 + 0i ( 5 0) a ( 6 + i) ( 5 + 0i) ( 6 ) ( 5 0) ( ) + i P. No efie a mappig f : a follo f 6 if ; f if. Noe ha f i o a coacie mappig ih epec o he aa meic ice f f Hoee f aifie ( f f) α[ ( f) + ( f)] fo all { } : { 5 6 7} ih

9 CHATTERJEA CONTRACTION MAPPING THEOREM α. Applig he peio heoem e obai ha f ami he iqe fie 6 poi * 6. Refeece [] S. K. Chaejea Fie poi heoem C. R. Aca. Blgae Sci. 5 (97) [] L. Hag a. Zhag Coe meic pace a fie poi heoem of coacie mappig J. Mah. Aal. Appl. () (007) [] M. Abba a G. Jgc Commo fie poi el fo o commig mappig iho coii i coe meic pace J. Mah. Aal. Appl. () (008) 6-0. [] D. Ilic a V. Raoceic Commo fie poi fo map o coe meic pace J. Mah. Aal. Appl. () (008) [5] S. Reapo a R. Hamlbaai Some oe o he pape coe meic pace a fie poi heoem of coacie mappig J. Mah. Aal. Appl. 5() (008) [6] M. Gag Baach coacio piciple o coe heagoal meic pace Ula Sciei 6() (0) [7] A. Aal a E. Hical The Kaa fie poi heoem i a coe heagoal meic pace A. Re. 7() (06) -9. [8] M. Gag a S. Agaal Baach coacio piciple o coe peagoal meic pace J. A. S. Topol. (0) -8. [9] M. Jleli a B. Same The Kaa fie poi heoem i a coe ecagla meic pace J. Noliea Sci. Appl. (009) 6-67.

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