Fixed Point Results Related To Soft Sets

Transcription

1 Autralia Joural of Baic ad Applied Sciece (6) Noveber 6 Page: 8-37 AUSTRALIAN JOURNAL OF BASIC AND APPLIED SCIENCES ISSN: EISSN: Joural hoe page: Fied Poit Reult Related To Soft Set Balaji Raghuath Wadkar Raakat Bhardwaj 3 Vihu Naraya Mihra Baat Sigh Departet of Matheatic AISECT Uiverity Bhopal-Chikload Road Bhopal (M.P.) Idia Departet of Matheatic TIT Group of Ititute Bhopal (MP) Idia. 3 Applied Matheatic & Huaitie Dept. Sardar Vallabhbhai Natioal Ititute of Techology Surat Gujarat Idia. Addre For Correpodece: Applied Matheatic & Huaitie Dept. Sardar Vallabhbhai Natioal Ititute of Techology Surat Gujarat Idia. A R T I C L E I N F O Article hitory: Received Septeber 6 Accepted Noveber 6 Publihed 8 Noveber 6 A B S T R A C T Thi paper i divided ito two part. I firt part we prove oe fied poit theore of oft cotractive appig i oft etric pace ad i the ecod part; we prove oe fied poit theore for weak cotractive appig i the ae pace. Keyword: Soft poit oft etric pace oft cotractive appig fied poit theore. INTRODUCTION Molodtov (999) iitiated a cocept of oft et theory a a ew atheatical tool for dealig with ucertaitie. A oft et i a collectio of approiate decriptio of a object. Thi theory ha rich potetial applicatio ad provide a very geeral fraework with the ivolveet of paraeter. May tructure o oft et theory cotributed by ay reearcher like Rhoade (977) Ali et al. (5) Maji et al. () Maji et al. (3) Da ad Saata (3). Shabir ad Naz () Rhoade (977) Huai ad Ahad () Bayraov ad Guduz (3) were tudied about oft topological pace. Maji ad et al. () proved how the oft et theory i applicable i a deciio akig proble. Da ad Saata () dicued about Soft real et oft real uber ad proved oe propertie of oft real uber. Fied poit theory i a iportat brach of atheatic which i ued to olve boudary valued proble. May author like Deepala ad Pathak (3) Mihra et al. (6) Deepala (4) Razai et al. () Aghagi ad Roha () Rahii ad Khezerloo () Faroj (5) Parvaeh et al. () Bagheri et al. () Geetharaai (5) proved fied poit theore by ivetigatig differet cocept. I thee tudie the cocept of oft poit i epreed by differet approache. Recetly Da ad Saata (3) itroduced a differet otio of oft etric pace by uig a differet cocept of oft poit ad ivetigated oe baic propertie of thee pace. Yazar ad et al. (3) proved fied poit for oft cotractive appig. I the preet paper we itroduce oft cotractive appig o oft etric pace ad prove oe fied poit theore of oft cotractive appig. Thi reult i otivated by Yazar ad et al. (3). Before tartig to prove ai reult oe baic defiitio are required. Defiitio.: Ope Acce Joural Publihed BY AENSI Publicatio 6 AENSI Publiher All right reerved Thi work i liceed uder the Creative Coo Attributio Iteratioal Licee (CC BY). To Cite Thi Article: Balaji Raghuath Wadkar Raakat Bhardwaj Vihu Naraya Mihra Baat Sigh Fied Poit Reult Related to Soft Set. Aut. J. Baic & Appl. Sci. (6): 8-37

2 9 Balaji Raghuath Wadkar et al 6 Autralia Joural of Baic ad Applied Sciece (6) Noveber 6 Page: 8-37 Let X ad E be repectively a iitial ivere et ad a et of paraeter. A pair (Y E) i called a oft et over X if oly if Y i a appig fro E ito the et of all ubet of the et X i.e. Y : E P( X ) where P() i the power et of X. Defiitio.: Let A deoted by C Y ad Z B be two oft et over X. The iterectio of two oft et Y Aad B I ad i give by Y A Z B I C Z i a oft et where C A B ad C I( ) Y( ) Z( ). Defiitio.3: The uio of two oft et Y A ad Z B over X i the oft et C Y( ) if A B I( ) Z( ) if B A Y ( ) Z( ) A B. Y A Z B I C Thi relatiohip i deoted by. I where C A B ad C () Defiitio.4: A oft et (Y A) over X i aid to be a ull oft et deoted by if Y ( ) A. Defiitio.5: The oft et Y A over X i called a a abolute oft et if A Y( ) X. Defiitio.6: Let E et E F ad G E be two oft et over X the the differece of two oft et F E ad E H over X deoted by F E \ G E ad i defied a H() = F() \ G() E. Defiitio.7: Let A c c Y A Y A Y be a oft et. The copleet of oft et Y A i deoted by A c where Y c : A P( X ) i a appig give by ( ) X Y( ) for all. Y c G i oft Y ad i defied a Defiitio.8: Let R be the et of real uber ad B(R) be the collectio of all o epty bouded ubet of R ad E take a a et of paraeter. The a appig Y : E B(R) i called a oft real et. It i deoted by (Y E). If pecifically (Y E) i a igleto oft et the idetifyig (Y E) with the correpodig oft eleet it will be called a oft real uber ad deoted a r t etc. are the oft real uber where ( e ) ( e) for all e E repectively. Defiitio.9: For two oft real uber i. u v if u ( ) v ( ) for all E ; ii. u v if u ( ) v ( ) for all E ; iii. u v if u ( ) v ( ) for all E ; iv. u v if u ( ) v ( ) for all E. Defiitio.: A oft et (P E) over X i aid to be a oft poit if there i eactly oe X ad P( ') epty ' E /{ }. It will be deoted by. E uch that P() = {} for oe Defiitio.: Two oft poit are aid to be equal if ad P( ) P( ) i.e. = y. Hece y or. Defiitio.:

3 3 Balaji Raghuath Wadkar et al 6 Autralia Joural of Baic ad Applied Sciece (6) Noveber 6 Page: 8-37 A appig : SP( X ) SP( X ) R(E) * i aid to be a oft etric o the oft et X if atifie the followig coditio: SM. for all y X y SM. y if ad oly if y SM3. for all y X y y SM4. z 3 y y z for all y z X 3. 3 The oft et X with a oft etric o X i called a oft etric pace ad deoted by X E. Defiitio.3: Let u coider a oft etric X E ad be a o egative oft real uber. The oft ope ball with ceter at ad radiu i give by B y X : ( y ) SP( X ) ' ad the oft cloed ball with ceter at ad radiu i give by e B X : ( y ) SP( X ). ' ' ' ' Defiitio.4: A equece of oft poit i oft etric pace X E i aid to be coverget i X E if there i a oft poit y X uch that y a. That i for every choe arbitrary there i a atural uber N N( ) uch that ( y ) wheever N. Defiitio.5: Let X E be a oft etric pace. The the equece of oft poit i X E i aid to be a Cauchy equece i X if correpodig to every N uch that i j y i j i.e. i j y a i j. i j i j Defiitio.6: The oft etric pace X E i called i coplete if every Cauchy equece i X coverge to oe poit of X. Defiitio.7: Let X E be a oft etric pace. A fuctio ( f ) : X E X E i called a oft cotractio appig if there eit a oft real uber R uch that for every poit y SP( X) we have f f y y ( )( )( )( ). (). Mai Reult:. Soe fied poit theore of cotractive Mappig i oft Metric Space: Theore..: Let ( X E) be a oft coplete etric pace. Let appig ( f ) : ( X E) ( X E) followig oft cotractive coditio: ( f )( )( f )( y ) a y b ( f )( ) ( f )( y ) y. Where a b the there eit a uique oft poit SP( X ) uch that ( f )( ). atifie the (3) Proof: Let be ay oft poit i SP(X).

4 3 Balaji Raghuath Wadkar et al 6 Autralia Joural of Baic ad Applied Sciece (6) Noveber 6 Page: 8-37 Set ( f )( ) f ( ) ( ) ( f )( ) ( ) f ( ) ( f )( ) f ( ) We have ( ) ( f )( )( f )( ) a ( )( ) ( )( ) b f f a b a b b. a b So for > We get b h h... h where a b h b h h h d h. h. h a h Thi iplie that copletee of X there i Sice. Hece X uch that *. i a oft Cauchy equece. By the ( )( * ) * * * ( )( )( )( ) ( )( f ) f f f * * * * a ( )( ) ( )( ) b f f * * * * a ) b. * * Thi iplie that ( f )( ). So the poit aother fied poit of ( f ) the we obtai * i a fied oft poit of the appig ( ) * * * * y ( f )( )( f )( y * * * * * * a y b ( f )( ) ( f )( y ) y * * * * * * a y b y y * * a y. We kow that a b. So ( f ) i uique. a ad hece * * * f. Now if y i * * y. Therefore the fied oft poit of y

5 3 Balaji Raghuath Wadkar et al 6 Autralia Joural of Baic ad Applied Sciece (6) Noveber 6 Page: 8-37 Theore..: Let ( X E) be a oft etric pace. If the appig ( f ) : ( X E) ( X E) cotractio coditio: ( f )( )( f )( y ) a ( f )( ) ( f )( y ) y atifie the oft b ( f )( ) y ( f )( y ) (4) for all y X with a b / where a b are oft cotat the ( f ) ha a uique oft poit i X. Proof: Let be ay oft poit i SP(X). Set We have ( ) ( f )( ) f ( ) ( f )( ) f ( ) ( ) ( f )( ) f ( ). ( ) ( )( )( )( ) f f d ( f )( ) ( )( ) f b ( f )( ) ( f )( ) a b a b. a b a a h. where h h.... h.. a b a b So for > we get... h h h Thu we obtai h h.... h. h Thi iplie that a copletee of X there i * X uch that. Sice. Hece i a oft Cauchy equece. By the

6 33 Balaji Raghuath Wadkar et al 6 Autralia Joural of Baic ad Applied Sciece (6) Noveber 6 Page: 8-37 ( )( * ) * * * ( )( )( )( ) ( )( f ) f f f * * a ( f )( ) ( )( ) f * * * b ( f )( ) ( f )( ) * * a * * * b ). * * Thi iplie that ( f )( ). So the poit i a fied oft poit of the appig ( f ). Now if aother fied poit of ( f ) the * * * * * y ( f )( )( f )( y * * * * * * * * a ( f )( ) ( f )( y ) y b ( f )( ) y ( f )( y ) a * * * * * * * * y y b y d y * * b y. Sice a b / iplie that ( f ) i uique. Theore..3: b. Hece * * y * * y * y i.therefore the fied oft poit of Let ( X E) be a oft etric pace. If the appig ( f ) : ( X E) ( X E) atifie the oft cotractio coditio: ( f )( )( f )( y ) a y b ( f )( ) ( f )( y ) y c ( f )( ) y ( f )( y ) ; (5) for all y X with a b c where a b c are oft cotat the ( f ) ha a uique oft poit i X. Proof: The Proof follow fro theore.... Fied poit theore for weakly cotractive appig i oft Metric: We ow prove oe fied poit theore for weak cotractive appig i the oft etric pace. Theore..: Let ( X E) be a oft etric pace & F : X X be a elf oft appig atifyig F Fy ) r y r y (6) for all y X where :[ ) [ ) i a lower ei cotiuou fuctio ( t) with () ad r y a ( ) ( ) ( ) ( ) F Fy y F y Fy (7) The there eit a uique oft poit P X uch that FP. P Proof:

7 34 Balaji Raghuath Wadkar et al 6 Autralia Joural of Baic ad Applied Sciece (6) Noveber 6 Page: 8-37 We firt how that r y if ad oly if y a coo fied poit of F. Now if y F Fy the clearly r y. Coverely let r y the y r y F r y F y y r y F y y r y We have y F Fy. Now for the covergece Let X be a arbitrary poit ad chooe a equece i X uch that F for all >. Fro (6) ad (7) ad property of we have F F r r F F F F F a F F F F F a a a a. a Now if the we have Which i a cotradictio thu fro (8) we get r r r. Therefore for all ;. r Thu i ootoic o icreaig fuctio ad bouded below. Therefore there eit r uch that li li r r. (9) The by lower ei cotiuity of we have ( r ) liif r. We clai that r i fact takig upper liit a o either ide o followig iequality r r We get li li liif r r By (9) we get r r liif r ( ) r r that i ( r ). (8)

8 35 Balaji Raghuath Wadkar et al 6 Autralia Joural of Baic ad Applied Sciece (6) Noveber 6 Page: 8-37 Alo () by defiitio ad we get li. () Now we how that the equece i X i a Cauchy equece. Let thi be ot true the there eit uch that for a iteger k N there eiti( k uch that (). i( For every iteger k let i( be the leat poitive iteger eceedig atifyig () ad uch that () Now. i(. i( i( i( i(. The by () & () we get li (3) k e j e ( i( Net let be the leat poitive iteger eceedig i( By (6) we get j i i( j ( i( r a Lettig k ad by () & (4) i( atifyig () & () we have (4) j ( j ( i( i( i( i( j ( li r ( ) ( ) j k k N i k the li r j. ( i( k By lower ei cotiuity of we have By (6) we get ( ) liif r k j ( i ( k j ( ( ) r i k j ( i ( r i ( Lettig k we get. liif r ( ). j ( k i ( Thi i a cotradictio. The the equece i a Cauchy equece. It follow fro the copletee of X that there eit P X uch that P a. Now we how that FP P. We uppoe that FP P FP r P r P. Lettig the we get P FP FP P FP P

9 36 Balaji Raghuath Wadkar et al 6 Autralia Joural of Baic ad Applied Sciece (6) Noveber 6 Page: 8-37 a P FP P P FP P FP P P FP a. Thi i a cotradictio therefore FP P. Now we prove the uiquee of poit P a follow: Suppoe that q P & Fq q the by (6) we get p q Fp Fq r p q r p q Fp p Fq q Fp q Fq p r p q a p q p q p q. Thi i a cotradictio. Therefore p q. Theore..: Let ( X d E) be a oft etric pace & F : X X be a elf oft appig atifyig F Fy ) r y r y (5) for all y X. Here :[ ) [ ) i a lower ei cotiuou fuctio ( t) with () ad r y. F y Fy (6) The there eit a uique oft poit P X uch that FP. P The proof follow fro theore Recoedatio: For future tudie ugget that: ) Thi work ca be eteded to fied poit theory of o epaive ulti-valued appig. ) Reforulatio of our reult replacig etric or b-etric tructure could be coidered a a valuable additio to preet fied poit reult i.e. thee reult ca be eteded to ay directio. 3) Thi theory ha wide applicatio i fuzzy oft etric pace i oft b-etric pace. 4) Thee proved reult lead to differet directio ad apect of oft etric fied poit theory. Cocluio: I thi paper the ivetigatio cocerig the eitece ad uiquee of oft fied poit theore i oft etric pace are etablihed. REFERENCES Razai A. H. Hoei Zadeh A. Jabbari. Coupled fied poit theore i partially ordered etric pace which edow with vector valued etric Autrilia Joural of Baic ad Applied Sciece 6(): 4-9. Aghagi A. J.R. Roha. Coo fied poit theore uder cotractive coditio i ordered etric pace Autrilia Joural of Baic ad Applied Sciece 5(9): Bagheri Vakilabad A. S. Maour Vaezpour. Quai Cotractio ad Coo Fied poit Theore Cocerig Geeralized Ditace Autralia Joural of Baic ad Applied Sciece 5(): Rhoade B.E A copario of variou defiitio of cotractive appig Tra. Aer. Math. Soc. 66: Guduz (Ara) C. A. Soez H. Çakall 3. O Soft Mappig arxiv: v [ath.gm]. D. Che 5. The paraeterizatio reductio of oft et ad it applicatio Coput. Math. Appl. 49: Molodtov D Soft et-theory-firt reult Coput. Math. Appl. 37: 9-3. Deepala ad H.K. Pathak 3. A tudy o oe proble o eitece of olutio for oliear fuctioal-itegral equatio Acta Matheatica Scietia 33(5):

10 37 Balaji Raghuath Wadkar et al 6 Autralia Joural of Baic ad Applied Sciece (6) Noveber 6 Page: 8-37 Deepala 4. A tudy o Fied poit theore for oliear cotractio ad it applicatio Ph.D. Thei Pt. Ravihakar Shukla Uiverity Raipur 49 Chhatigarh Idia. Faroj A Abduabi 5. Fied poit i NG-Group Autrilia Joural of Baic ad Applied Sciece. 9(3): Geetharaai G. C. Sharila Devi ad J. Aru Padia 5. Fuzzy Paraeterized Geeralied Fuzzy Soft Epert Set Autralia Joural of Baic ad Applied Sciece 9(): Pathak H.K. ad Deepala 3. Coo fied poit theore for PD-operator pair uder Relaed coditio with applicatio Joural of Coputatioal ad Applied Matheatic 39: 3-3. Rahii H. S. Khezerloo M. Khezerloo. Uiquee ad eitece of olutio of the periodic firt order differetial equatio with boudary value uig fied poit theore Autrilia Joural of Baic ad Applied Sciece 6(6): Mihra L.N. R.P. Agarwal M. Se 6. Solvability ad ayptotic behavior for oe oliear quadratic itegral equatio ivolvig Erd$\acute{\bo{e}}$lyi-Kober fractioal itegral o the ubouded iterval Progre i Fractioal Differetiatio ad Applicatio (3): Mihra L.N. H.M. Srivatava M. Se 6. O eitece reult for oe oliear fuctioal-itegral equatio i Baach algebra with applicatio It. J. Aal. Appl. (): -. Mihra L.N. M. Se R.N. Mohapatra 6. O eitece theore for oe geeralized oliear fuctioal-itegral equatio with applicatio Filoat accepted. Mihra L.N. S.K. Tiwari V.N. Mihra 5. I.A. Kha; Uique Fied Poit Theore for Geeralized Cotractive Mappig i Partial Metric Space Joural of Fuctio Space Article ID p: 8. Mihra L.N. S.K. Tiwari V.N. Mihra 5. Fied poit theore for geeralized weakly S-cotractive appig i partial etric pace Joural of Applied Aalyi ad Coputatio 5(4): 6-6. DOI:.948/547. Mihra L.N. M. Se 6 O the cocept of eitece ad local attractivity of olutio for oe quadratic volterra itegral equatio of fractioal order Applied atheatic ad coputatio 85: DOI:.6/j.ac Ali M.I. F. Feg X. Liu W.K. Mi ad M. Shabir 5. O oe ew operatio i oft et theory. Coput. Math. Appl. 49: Yazar M.I. C. Guduz (Ara) S. Bayraov 3. fied poit for oft cotractive appig. Corell Uiverity Library. Shabir M. ad M. Naz. O oft topological pace. Coput. Math. Appl. 6: Maji P.K. A.R. Roy R. Biwa. A applicatio of oft et i a deciio akig proble. Coput. Math. Appl. 44: Maji P.K. R. Biwa A.R. Roy 3. Soft et theory. Coput. Math. Appl. 45: Majudar P. ad S.K. Saata. O oft appig. Coput. Math. Appl. 6: Da S. ad S.K. Saata. Soft real et oft real uber ad their propertie J. Fuzzy Math. (3): Da S. ad S.K. Saata 3. O oft etric pace J. Fuzzy Math (3): 7-3. Huai S. ad B. Ahad. Soe propertie of oft topological pace Coputer ad Math. with Applicatio 6: Bayraov S. C. Guduz Ara 3. Soft locally copact ad oft para copact pace Joural of Matheatic ad Syte Sciece 3: -3. Parvaeh V. H. Hoei Zadeh. Soe coo fied poit reult for Geeralized week C- cotractio i ordered etric pace Autrilia Joural of Baic ad Applied Sciece (9):