On Fixed Point Theorem in Fuzzy2- Metric Spaces for Integral type Inequality

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1 On Fixed Poin Theorem in Fuzzy- Meric Spaces for Inegral ype Inequaliy Rasik M. Pael, Ramakan Bhardwaj Research Scholar, CMJ Universiy, Shillong, Meghalaya, India Truba Insiue of Engineering & Informaion Technology, Bhopal, M.P, India Absrac: In he presen paper we are proving a common fixed poin heorem for fuzzy - meric spaces for raional expressions. Mahemaics Subjec Classificaion: Primary 47H, 54H5. Keywords: Fuzzy meric space, Fuzzy -meric space, Common fixed poin.. Inroducion and Preliminaries Impac of fixed poin heory in differen branches of mahemaics and is applicaions is immense. The firs resul on fixed poins for conracive ype mapping was he much celebraed Banach s conracion principle by S. Banach [37] in 9. In he general seing of complee meric space, his heorem runs as he follows, Theorem.(Banach s conracion principle) Le (X, d) be a complee meric space, c (, ) and f: X X be a mapping such ha for each x, y X, d (, ) c d(x, y) Then f has a unique fixed poin a X, such ha for each x X,lim =. Afer he classical resul, R.Kannan [3] gave a subsequenly new conracive mapping o prove he fixed poin heorem. Since hen a number of mahemaicians have been worked on fixed poin heory dealing wih mappings saisfying various ype of conracive condiions. In, A. Branciari [] analyzed he exisence of fixed poin for mapping f defined on a complee meric space (X,d) saisfying a general conracive condiion of inegral ype. Theorem.(Branciari) Le (X, d) be a complee meric space, c (, ) and le f: X X be a mapping such ha (,) (,) for each x, y X, () () Where: [,+ ) [,+ ) is a Lebesgue inegrable mapping which is summable on each compac subse of [,+ ), non negaive, and such ha for each >o, (), hen f has a unique fixed poin a X such ha for each x X, lim = Afer he paper of Branciari, a lo of a research works have been carried ou on generalizing conracive condiion of inegral ype for a differen conracive mapping saisfying various known properies. A fine work has been done by B.E. Rhoades [] exending he resul of Brianciari by replacing he condiion [.] by he following; (,) () (,),(,),(,), (,)(,) ()(.3) The aim of his paper is o generalize some mixed ype of conracive condiions o he mapping and hen a pair of mappings, saisfying a general conracive mapping such as R. Kannan ype [3], S.K. Charerjee ype [36],T. Zamfirescu ype [38],Turkoglu[39] ec. In 965, he concep of fuzzy ses was inroduced by Zadeh [4].Afer ha many auhors have expansively developed he heory of fuzzy ses and applicaions. Especially, Deng [], Erceg [], Kaleva and Seikkala [5, 6], Kramosil and Michalek [7], have inroduced he concep of fuzzy meric spaces in differen ways. Recenly, many auhors [3-5, 3, 9,, 3, 4, 9, 3, 34, and 35] have also sudied he fixed poin heory in he fuzzy meric spaces and [6-9,, 8,] have sudied for fuzzy mappings which opened an avenue for furher developmen of analysis in such spaces and such mappings. Consequenly in due course of ime some meric fixed poin resuls were generalized o fuzzy meric spaces by various auhors. Gähler in a series of papers [5, 6, and 7] invesigaed -meric spaces. Sharma, Sharma and Iseki [33] sudied for he firs ime conracion ype mappings in -meic space. We [4, 4] have also worked on -Meric spaces and - Banach spaces for raional expressions. Definiion.: A binary operaion *: [, ] [, ] is called a coninuous -norm if ([, ], *) is an abelian opological monoid wih uni such ha whenever a a, b b, c c for all,,,,,, are in [, ]. Definiion. : he 3-uple (X, M,*) is called a fuzzy - meric space if X is an arbirary se, * is a coninuous - norm and M is a fuzzy se in [, ) saisfying he following condiions: for all x, y, z, u X and,, >, () M(x, y, z, ) =, () M(x, y, z, ) = for all > (only when he hree simplex x, y, z degenerae) (3) M(x, y, z, ) = M(x, z, y, ) = M (y, z, x, ) =.. (4) M(x, y, z, w,,, ) *(M(x, y, u, )*M(x, u, z, )*M (u, y, z, ) (5) M(x, y, z,): [, ) [, ] is lef coninuous. 59

2 Definiion.3: Le (X, M,*) be a fuzzy- meric space. () A sequence { } in fuzzy - meric space X is said o be convergen o a poin x X (denoed by lim = if for any λ (,) and >, here exiss N such ha for all n and X, M (,,, ) > λ. Tha lim M (,,, ) = for all X and >. () A sequence { } in fuzzy- meric space X is called a Cauchy sequence, if for any λ (,) and >, here exiss N such ha for all m, n and a X, M (,,, ) > λ. (3) A fuzzy- meric space in which every Cauchy sequence is convergen is said o be complee. Definiion.4: Self A funcion M is coninuous in fuzzy -meric space if,, hen lim M (,,, ) = M(,,, ) >. is Definiion.5: Two mappings A and S on fuzzy -meric space X are weakly commuing if M (ASu, SAu,, ) M (Au, Su,, ), >.. Some Basic Resuls Lemma.: [9] for all,, M(, ) is nondecreasing. Lemma.: [9] Le { } be a sequence in a fuzzy meric space (X, M,*) wih condiion (FM-6) If here exiss a number (,) such ha M (,, q) M (,, ) for > and n =,,3.hen { } is a Cauchy sequence in X. Lemma.3: [3] for all, and for a number (,) such ha M (,, q) M (, y, ) for > hen =. Lemma (.,., and.3) are also rue for fuzzy-meric spaces. 3. Main Resul Theorem 3.: Le (X, M, *) be a complee fuzzy -meric space and le S and T be coninuous mappings of X in X. hen S and T have a common fixed poin in X if here exiss coninuous mapping A of X ino S(X) T (X) which commue weakly wih S and T, (3.a) M (Ty, Ay, a, ), M (Sx, Ax, a, ), M (Sx, Ty, a, ) min M (Sx, Ty, a, ),, M (Ax, Ty, a, ) M(Ax, Sx, a, ) M(Ax,Ay, a,q), M (Ax, Ty, a, ), M(Ay, Sx, a, ) For all x, y, a X, > and (,) (3.b) lim M (x, y,, ) = for all,, X. Then F, T and A have a unique common fixed poin in X. Proof: We define a sequence { } in X such ha = = for n =,... We shall prove ha { } is a Cauchy sequence. For his suppose = and = in (3.a), we wrie 53

3 M ( Axn, Axn+, a, q ) M(T xn+, A xn+, a, ), M(S xn, A xn, a, ), M (Sxn, T xn+, a, ), min M(S x, T x, a, ) M(A x,, S x, a, ) n n+ n n, M(A x M(A xn, T xn+, a, ) M(A xn+,, T x, a, ), S x, a, ) n n+ n M ( Ax n, Ax n+, a, q ) Therefore M(Ax n,ax n+, a, ), M(Ax n+, Ax n, a, ), M (Ax n+, Ax n, a, ), min M(A x, A x, a, ) M(A x,, Ax, a, ) n+ n n n+, M(A x M(A x n, A x n, a, ) M(A x n+,,a x, a, ) Ax, a, ) n n n + M(Ax, min n,ax n+, a, ), M(Ax n+, Ax n, a, ), M (Ax n+ Ax n, a, ), M(Ax +, Ax, a, ), M(Ax,Ax +, a, ), = n n n n min M(Ax n,ax n, a, ), M(Ax q n, Ax n, a, ) q M ( Axn, Axn+, a, q ) min M(Ax n,ax n, a, ) q ζ d By inducion () ζ() d M ( Ax, Ax, a, q k m ) min M(Ax m,ax k, a, q ) + ζ d for every k and m in N, furher if m+>k, hen () ζ() d 53

4 M ( Ax, Ax, a, q ) M(Ax,Ax, a, q ) k m+ k m ζ d () ζ() d... M(Ax k ( m ),Ax, a, m q ) + + ζ ( ) d... (3. d) By simple inducion wih (3.c) and (3.d) we have M ( Axn, Axn+ p, a, q) M(Ax, Ax p, a, ) n ζ d q For n = k, p = m+ or n = k+, p = m+ () M ( Ax, Ax, a, q) M(Ax, Ax, a, ) M(Ax, Ax q n n+ p n p If n = k, p =m or n =k+, p = m, For every posiive ineger p and n in N, by nohing ha, a, ) n q... (3. e) Thus { } is a Cauchy sequence. Since he space X is complee here exiss, such ha lim = lim = lim =. I follows ha == and Therefore M(TAz, AAz, a, ), M(Sz, Az, a, ), M (Sz, TAz, a, ) min M (Sz, TAz, a, ),, M(Az, TAz, a, ) M (Az, Sz, a, ) M(Az,AAz, a,q), M (Az, TAz, a, ), M(AAz, Sz, a, ) 53

5 {, } ζ ( ) d ζ ( ) d M(Az,A z, a,q) M (Sz TAz, a, ) Since, lim M (, z, a, q )= = Thus z is common fixed poin of A, S and T. { M (S z, A T z, a, ) } { M(Az,A z, a, ) } ζ ( ) d ζ ( ) d M(Az,A z, a, ) n q ζ ( ) d. For uniqueness, le ( ) be anoher common fixed poin of S, T and A, By (3.a) we wrie M (Tv, Av, a, ), M (Sz, Az, a, ), M (Sz, Tv, a, ) min M (Sz, Tv, a, ),, M (Az, Tv, a, ) M (Az, Sz, a, ) M(Az,Av, a,q), M(Az, Tv, a, ), M(Av, Sz, a, ) { } ζ ( ) d ζ ( ) d M (Az,Av, a,q) M (z,v, a, ) This implies ha { } ζ ( ) d ζ ( ) d M (z,v, a,q ) M (z,v, a, ) 4. Acknowledgemen Dr. Ramakan Bhardwaj, Associae Professor and PI of MPCOST Bhopal for he projec (Applicaion of Fixed Poin Theory), Deparmen of Mahemaics, Truba Insiue of Engineering & Informaion Technology, Bhopal, M.P, India References [4] A.Branciari, A fixed poin heorem for mappings saisfying a general conracive condiion of inegral ype, In.J.Mah.Sci. 9(), no.9, [5] B.E. Rhoades, Two fixed poin heorems for mappings saisfying a general conracive condiion of inegral ype. Inernaional Journal of Mahemaics and Mahemaical Sciences, 63, (3), [6] Badard, R. Fixed poin heorems for fuzzy numbers Fuzzy ses and sysems 3 (984) 9-3 [7] Bose, B.K. and Sahani, D. Fuzzy mappings and fixed poin heorems Fuzzy ses and Sysems (984) [8] Bunariu, D. Fixed poin for fuzzy mappings Fuzzy ses and Sysems 7(98) 9-7 [9] Chang, S.S. Fixed poin heorems for fuzzy mappings Fuzzy Ses and Sysems 7(985) [] Change, S.S., Cho, Y.J., Lee, B.S. and Lee, G.M. Fixed degree and fixed poin heorems for fuzzy 533

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