Teorem Titik Tetap Pada Ruang 2-Metrik

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1 BULLEIN Bll. Malasa Math. So. Seod Seres of the MALAYSIAN MAHEMAICAL SOCIEY eore t eta Pada Rag -Metr MASHADI DAN ABU OSMAN BIN MD AP Jrsa Mateata Uerstas Ra Kas Ba Wda Paa Peabar Ra Idoesa Jabata Mateat Falt Sas Mateat Uerst Kebagsaa Malasa 46 UKM Bag Selagor Darl Ehsa Malasa Abstra. Dala aalah a erlasa beberaa teore tt teta dega etrasforasa etga-tga tt d dala rag -etr. Perlasaa adalah tertal eada sarat etasaaa tertet ag dataa. Bet etasaaa ag dba adalah selar dega dea Pahatte da jga Sgh da Ra. Abstrat. I ths aer we eted soe fed ot theores b trasforg the three ots the -etr sae. he etesos are sbjeted to the ge artlar eqalt odto. he eqaltes ostrted are arallel to the deas of Pahatte ad also that of Sgh ad Ra.. Pedahla Msala sat set ag egadg serag-raga sr. Peetaa : R ag eeh sfat: M. Ut seta asaga a b dega a b wjd sehgga a b ; M. a b ja seraa serag-raga da sr darada a b adalah saa; M. a b b a a b t sea a b ; M4. a b a b d a d d b t sea a b d dsebt -etr ada da asaga ; dsebt rag -etr ada. Beberaa bet teore tt teta ada rag -etr telah daj oleh d ataraa Gahler [] Lal da Sgh [5] Pahatte [8] Ise [4] Patha da Dbe [9] Cho [] Cho da Par []. Kala derhata bet teore tt teta ada rag -etr ag dbat oleh egaj d atas ta lhat d dala sarat etasaaa tersebt haa da tt sahaja dtrasforasa da tt etgaa selal eal tda t dtraforasa. Cotoha dala sarat etasaaa eetaa eget ada rag -etr bag seta α t sat α. Selajta Mashad da Ab Osa [6 7] telah egaj tt teta

2 Mashad da Ab Osa b Md a ag elbata sarat etga tt da dtrasforasa. Cotoha Mashad da Ab Osa [6] telah ebera sarat etasaaa eetaa -eget ada rag -etr sebaga bag seta α t sat α. Dala aalah a aa elajta aja tersebt dega elhat tt teta sea btara bag beberaa eetaa ada rag -etr ag tertal eada sarat etasaaa tertet selar dega dea ag dtaraa oleh Pahatte [8] da jga Sgh da Ra []. Bag edaha erbaga selajta a bera beberaa tarf ag aa dgaa selajta. arf.. Jja { } ada rag -etr ; dsebt jja -Cah ja had. arf.. Rag -etr ; dsebt -lega bla seta jja -Cah d dala adalah e. arf.. Rag -etr ; dsebt terbatas ja wjd obor ata K > sehgga K t sea d.. Hasl Dala bahaga a adaa ag ; eraa rag -etr terbatas -lega. Ka aa erhata asalah tt teta sea btara bag beberaa eetaa ada ; eada dra sedr tertal eada sarat etasaaa tertet. eore.. Msala da tga eetaa ada rag -etr terbatas -lega ; eada dra sedr da eeh sarat: { [ ] as [ ] } t sea ag. Maa da ea tt teta sea ag btara.

3 eore t eta Pada Rag -Metr Bt. Plh sebarag tt da tarfa da. Maa deroleh ; dega { [ ] as [ ]} [ ] { } as Ja [ ] { } as aa. Ja sebala [ ] { } as [ ] aa [ ] sehgga. Jga ; dega [ ] as [ ]} [ ] { }. as

4 Mashad da Ab Osa b Md a 4 Ja [ ] { } as aa ata. Sebala ja [ ] { } as [ ] aa [ ] sehgga ata. Ja roses ta lag sebaa al aa aa deroleh ata. Maa t sebarag da dega > > deroleh L. L Oleh t. M. M

5 eore t eta Pada Rag -Metr 5 ata. M. M Oleh eraa > M adalah sat ealar da eraa had aa } { eraa jja -Cah. Oleh eraa ; eraa rag -etr ag -lega aa wjd sehggaa jja } { e e. Selajta aa dtja bahawa eraa tt teta sea bag da. Plh sebarag. Maa.. Jad ; dega { [ ] as [ ]} { [ ] as [ ]}.. Ja dabl had ada etasaaa. da dgata e dala etasaaa. aa deroleh. Oleh eraa aa. Maaa. Dega ara ag saa boleh dtja bahawa da jga.

6 6 Mashad da Ab Osa b Md a Bert aa dtja bahawa btara. Msala wjd tt teta bag da. Plh sebarag. Maa ; as { [ ] [ ]} { [ ] [ ]} as { }. as Ja { } ag jga as aa. Aabla roses dlaa sebaa al aa aa deroleh t sat. Oleh eraa aa. Maaa. Catata:. Ja sea eraa eetaa -eget aa sarat etasaaa d dala eore.. aa deh. Oleh ag dea eore.. d atas eraa erlasa eada teore ag dbta oleh Mashad da Ab Osa [6].. Ja dala teore d atas aa teore tersebt jga bear ag aa beta ejad seert bert: eore.. Msala sat eetaa ada rag -etr terbatas -lega ; eada dra sedr da eeh etasaaa { [ ] as [ ] } t sea da. Maa ea tt teta btara.

7 eore t eta Pada Rag -Metr 7 eore.. Msala da tga eetaa ada rag -etr terbatas -lega ; eada dra sedr da eeh etasaaa [ ] { as [ ] ] [ ] [ t sea da. Maa da ea tt teta btara sea. Bt. Plh sebarag tt da tarfa da. Maa deroleh ; dega { [ ] as [ ] ] [ ] [ [ ] { }. as Ja [ ] { } as aa.

8 Mashad da Ab Osa b Md a 8 Sebala ja [ ] { } as [ ] aa ] [ sehgga. Begt jga ; [ ] as [ ]} [ ] { }. as Ja [ ] { } as aa ata. Sebala ja [ ] { } as [ ] / aa ] sehgga ata. Ja roses dlag sebaa al aa aa deroleh ata.

9 eore t eta Pada Rag -Metr 9 Maa t sebarag da ag > > deroleh L L aa sehgga.. M M ata sehgga.. M M. Oleh eraa > M adalah sat ealar da eraa had aa } { eraa jja -Cah. Oleh eraa ; eraa rag -etr ag -lega aa wjd sehggaa jja } { e e. Selajta aa dtja bahawa eraa tt teta sea dar da. Plh sebarag. Maa. Jad ; dega [ ] { } as [ ] ] [ ] [

10 Mashad da Ab Osa b Md a [ ] { as [ ] ] [ ] [.4 Ja dabl had ada etasaaa.4 da dgata e dala etasaaa. aa aa deroleh. Oleh eraa aa. Maaa. Dega ara ag saa boleh dtja bahawa da jga. Bert aa dtja bahawa btara. Msala wjd ag jga tt teta sea bag da. Plh sebarag. Maa ; dega [ ] { as [ ] [ ] [ ] as. Ja { as aa da hal stahl. Sebala ja as

11 aa Jad harslah eore t eta Pada Rag -Metr. I berert da hal stahl. as sehgga. Aabla roses dlaa sebaa al aa aa deroleh t sat. Oleh eraa aa harslah. Maaa. Catata. Ut sebarag tarfa da. Maa eore. d atas boleh debaga ejad teore bert ag bta a tggala eraa sera dega ebta eore.. eore.4. Msala L asg-asga eetaa ada rag -etr terbatas -lega : eada dra sedr da eeh sarat etasaaa { as [ ] [ ] [ [ ] ] t sea t sea da. Maa wjd sehggaa L.

12 Mashad da Ab Osa b Md a Rja. Y.J. Cho Fed ot for oatble ag of te A Math. Jao Y.J. Cho da S.C. Par Codee theore -etr sae SEAMS Bll. Math S. Gahler -etrshe rae d hr toologshe strtr Math. Nahr K. Ise Fed ot theore -etr sae Math. Se. Note S.N. Lal ad A.K. Sgh. A aaloge of Baah s otrato rle for -etr saes Bll. Astral. Math. So Mashad da Ab Osa b Md a eore tt teta eetaa -eget ada rag -etr. Aa terbt dala J. Mateata UM. 7. Mashad da Ab Osa b Md a eore tt teta eetaa selajar -orbtal ada rag -etr. Praeta dserah t eerbta J. Sas Malasaa. 8. B.G. Pahatte Fed ot theores for otrato te ag o -etr sae Pro. Nat. Aad. S. Ida 4 II H.K. Patha. da R.P. Dbe Soe fed ot theore -etr sae he Math. Ed S.L. Sgh da B. Ra A ote o the oergee of seqees of ag ad ther oo fed ot -etr sae Math. Se. Note

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