INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010

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1 Fixed Poits theorem i Fuzzy Metric Space for weakly Compatible Maps satisfyig Itegral type Iequality Maish Kumar Mishra 1, Priyaka Sharma 2, Ojha D.B 3 1 Research Scholar, Departmet of Mathematics, Sighaia Uiversity 2 Research Scholar, Departmet of Mathematics, Mewar Uiversity 3 Departmet of mathematics, R.K.G.I.T. U.P.T.U Delhi Meerut road Ghaziabad mkm2781@rediffmail.com ABSTRACT I this paper, we prove some fixed poit theorems for weakly compatible maps i fuzzy metric space satisfyig itegral type iequality but without assumig the completeess of the space or cotiuity of the mappigs ivolved.we exted this cocept to fuzzy metric space ad establish the existece of commo fixed poits for a pair of self mappigs. The result obtaied i the fuzzy metric space by usig the otio of ocomapatible maps or the property (E.A) are very iterestig. we prove commo fixed poit theorems for weakly compatible maps i fuzzy metric space by usig the cocept of (E.A) property, however, without assumig either the completeess of the space or cotiuity of the mappigs ivolved. We also fid a affirmative aswer i fuzzy metric space to the problem of Rhoadese. Symbols Used Not equal to, epsilo, α alpha φ phi propersubset ε varepsilo > grater tha < less tha Mathematics Subject Classificatio : 47H1, 54A4, 54E99 Keywords: Fuzzy metric space, o compatible maps, weakly compatible maps, commo fixed poit, E.A property. 1. Itroductio Fuzzy set has bee defied by Zadeh(1965). Kramosil ad Michalek (1975),itroduced the cocept of fuzzy metric space, may authors exteded their views as some George ad veera mai (1994), Grabiec (1988), Subramaya (1995),Vasuki(1999). Pat ad Jha (24) obtaied some aalogous results proved by Balasubramaiam et al. subsequetly, it was developed extesively by may authors ad used i various fields. I 1986 Jugck (1986) itroduced the otio of compatible maps for a pair of self maps. Several papers have come up ivolvig compatible maps i provig the existece of commo fixed poits both i the classical ad fuzzy metric space. However, the study of commo 315

2 fixed poits of ocomapatible mappigs is also iterestig. Pat ( ) iitiated work alog these lies by employig the otio of poit wise R weak commutativity. I the study of commo fixed poits of compatible mappigs we ofte require assumptio o completeess of the space or cotiuity of mappigs ivolved besides some cotractive coditio but the study of fixed poits of ocomapatible mappigs ca be exted to the class of o expasive or Lipschitz type mappig pairs eve without assumig the cotiuity of the mappigs ivolved or completeess of the space. Aamri ad El Moutawakil (24) geeralized the cocepts of ocomapatibility by defiig the otio of (E.A) property ad proved commo fixed poit theorems uder strict cotractive coditio. Recetly Chouha ad Badshah(21) established fixed poits theorem i fuzzy metric spaces for weakly compatible maps. The result obtaied i the fuzzy metric space by usig the otio of ocomapatible maps or the property (E.A) are very iterestig. Questio arises whether, by usig the cocept of ocomapatibility or its geeralized otio, that is, the property (E.A), ca we fid equally iterestig results i fuzzy metric space also? we aswer i affirmative. I the preset paper, we prove commo fixed poit theorems for weakly compatible maps i fuzzy metric space by usig the cocept of (E.A) property, however, without assumig either the completeess of the space or cotiuity of the mappigs ivolved. We begi with defiitios ad prelimiary cocepts. 2. Materials ad Methods ( is called a cotiuous t orm if [ ] ) 2.1 Defiitio A biary operatio :[,1] [,1] [,1],1,, is a abelia topological mooid with the uit 1 such that a b c d ad wheever a c ad b d for all a, b, c, d [,1]. 2.2 Defiitio The triplet ( X, M, ) is said to be fuzzy metric space if X is a arbitrary set,* is a X X cotiuous t orm ad M is a Fuzzy set o [, ] [,1 ] satisfyig the followig coditios : for all x, y, z X ad s, t >. M ( x, y,) =, 1. (FM1) 2. (FM2) M ( x, y, t ) = 1 for all t > if ad oly if x y, 3. (FM3) M ( x, y, t) = M ( y, x, t ); 4. (FM4) M ( x, y, t) M ( y, z, s) M ( x, z, t + s ); 5. (FM5) M ( x, y,.) : (,1 ) [,1] is cotiuous. 2.3 Example 316

3 Let ( X, d ) a b = mi be a metric space. Defie { a, b } t >.The (,, ) ad all 2.4 Defiitio ad M ( x, y, t ) t = t + d x y (, ) for all x, y X X M is a fuzzy metric space. It is called the Fuzzy metric space iduced by d. Let U ad V be two self maps of a fuzzy metric space ( X, M, ) U M ( UVx, VUx, t ) 1 if as, wheever { x } is a sequece i X such that, for some z X. ad V are said to be compatible Ux, Vx z as 2.5 Defiitio Two self maps U ad V of Fuzzy metric space ( X, M, ) are said to be weakly compatible if they commute at their coicidece poit, i.e. UVu = VUu wheever Uu = Vu u X. The cocept of weak compatibility is most geeral amog all the commutativity cocepts, clearly each pair of compatible self maps is weakly compatible but the coverse is ot true always. 2.6 Defiitio Let U ad V be two self maps of a fuzzy metric space ( X, M, ) property, if there exists a sequece { x } i.e., ( Ux ) ( ) x t Vx x t i X such that Ux, Vx x,, =,, 1 as for some t X. we say that U ad V satisfy E.A as, for some x X, 2.7 Weakly commutig Let f ad g be two self maps of a metric space (X,d) ad f ad g to be weakly commutig if for all x X d ( fgx, gfx ) d ( gx, fx ). It ca be see that commutig maps ( fgx = gfx x X ) are weakly compatible, but coverse is false. Let (X, d ) be a complete metric space, α [,1], f : X X a mappig such that for each x, y X, d ( fx, fy ) ϕ ( t ) α d ( x, y ) ϕ ( t ), Where ϕ ; + R R is a lebesgue itegrable mappig which is summable, ε ε >, > oegative ad such that, for each. The f has a uique commo fixed z X such that for each lim f x = z. x X, Rhoades[18], exteded this result by replacig the above coditio by the followig 317

4 d ( fx, fy ) 1 max{ d ( x, y ), d ( x, fx ), d ( y, fy ), [ d ( x, fy ) + d ( y, fx )] 2 α Ojha et al.(21) Let ( X, d ) be a metric space ad let f : X X, F : X CB ( X ) be a sigle ad a multi valued map respectively, suppose that f ad F are occasioally weakly commutative (OWC) ad satisfy the iequality P J ( Fx, Fy ) ad fx fy d fx Fx ad fx fy d fy Fy φ P 1 P 1 (, ) (, ), (, ) (, ), max P 1 P 1 ad ( fx, Fx ) d ( fy, Fy ), cd ( fx, Fy ) d ( fy, Fx ) ( t) φ ( ) for all x, y i X,where p 2 is a iteger a ad < c < 1 the f ad F have uique commo fixed poit i X. t 3. Results ad Discussios 3.1 Theorem Let f ad g be two weak compatible self maps of a fuzzy metric space ( X, M, ), satisfyig the property (E.A) ad (i) fx gx, (ii) (iii) M ( fx, fy, kt ) M ( gx, gy, t ) φ( t), k 2 2 { } φ( t) > M ( fx, ffx, t) mi M ( gx, gfx, t ), M ( fx, gx, t ), M ( f x, gfx, t), M ( fx, gfx, t ), M ( gx, f x, t ) wheever fx 2 f x. if the rage of f or g is a complete subspace of X, the f ad g have a commo fixed poit. Proof. Sice f ad g are satisfy the property (E.A), there exists a sequece { x } fx, gx z as, for some z X. i X such that Sice z fx ad fx gx, there exists some poit u i X such that z = gu,where gx z as. If fu gu the Takig limit, we get M ( fx, fu, kt) M ( gx, gu, t ) φ( t) 318

5 Hece fu = gu. Sice f ad g are weakly compatible. So fgu if ffu fu the by iequality (iii) = gfu ad therefore fgu = ffu = gfu = ggu 2 2 { } φ( t) > M ( fu, ffu, t ) mi M ( gu, gfu, t ), M ( fu, gu, t ), M ( f u, gfu, t), M ( fu, gfu, t ), M ( gu, f u, t ) = = = Which is a cotradictio ad so fu { M fu ffu t M fu fu t M f 2 u ffu t M fu ffu t M fu f 2 u t } mi (,, ), (,, ), (,, ), (,, ), (,, ) { M fu ffu t } mi (,, ) M ( fu, ffu, t ) Hece fu is a commo fixed poit of f ad g. = ffu ad fu = ffu = fgu = gfu = ggu. The case whe fx is a complete subspace X is similar to the above sice fx This completes the proof of the theorem. To illustrate the theorem we give a example. 3.2 Example M ( gu, fu, kt ) M ( gu, gu, t ) φ( t) gx, Let X = [,1] edowed with the usal metric ( ) d x, y = x y, x, y X, f, g : X X ad defied by f ( x ) 5x + 1, if x < 1 = 2x 1, if x = 1, g ( x ) x + 1, if x 1 = x, if x = 1 f ( x) = 5x + 1, g ( x) = x + 1 ad f ( x + 1) = 5( x + 1) + 1 = 5x + 6, ( ) g 5x + 1 = 5x = 5x + 2, t The, M ( fgx, gfx, t ) = t + 4 t t ad M ( fgx, gfx, ) = R t + 4 xr to test R weakly commutig, we observe that t M ( fgx, gfx, t) M ( fx, gx, ) R which gives 1 R, but there exists o R for x =, [,1[ x 319

6 Hece f ad g are ot R weakly commutig. However for x = 1, we have fx = gx = 1 ad fgx = gfx = 1. Hece f ad g are weakly compatible at x = 1, clearly fx coditios of the above theorem. Also the above theorem ca be proved for k = 1. gx. The f ad g satisfy all the Theorem1, has bee proved by usig the cocepts of (E.A) property which has bee itroduced i a recet work by Aamri ad Moutawakil [7].They have show that the (E.A) property is more geeral tha the otio of ocompatibility. It may, however be observed that by usig the otio of ocompatible maps i place of (E.A) property. I ext theorem we will show that if we take ocompatible maps i place of (E.A) property we ca show i additio that the mappigs are discotiuous at the commo fixed poit ad thus fid out a aswer i fuzzy metric space to the problem of Rhoades[14]. 3.3 Theorem Let f ad g be two o compatible weakly compatible self mappig of a fuzzy metric space ( X, M, ), (i) fx gx, (ii) (iii) M ( fx, fy, kt ) M ( gx, gy, t ) wheever φ( t), k 2 2 { } φ( t) > M ( fx, ffx, t) mi M ( gx, gfx, t ), M ( fx, gx, t ), M ( f x, gfx, t), M ( fx, gfx, t ), M ( gx, f x, t ) fx 2 f x. if the rage of f or g is a complete subspace of X, the f ad g have a commo fixed poit ad the fixed poit is the poit of discotiuity. Proof. Sice f ad g are o compatible maps, there exists a sequece { } x i X such that lim fx = lim gx = z..(1) for some z i X, but either ( fx gx t ) lim,, 1 or the limit does ot exist. Sice z fx ad fx gx, there exists some poit u i X such that z = gu,where z = lim gx.we 32

7 claim that If fu = gu.suppose that fu gu the Takig limit, we get M ( fx, fu, kt) M ( gx, gu, t ) φ( t) Hece fu = gu. Sice f ad g are weakly compatible. So fgu Suppose that ffu fu the by iequality (iii) = gfu ad therefore ffu = fgu = gfu = ggu 2 2 { } φ( t) > M ( fu, ffu, t) mi M ( gu, gfu, t ), M ( fu, gu, t), M ( f u, gfu, t ), M ( fu, gfu, t), M ( gu, f u, t ) = = = { M fu ffu t M fu fu t M f 2 u ffu t M fu ffu t M fu f 2 u t } mi (,, ), (,, ), (,, ), (,, ), (,, ) { M fu ffu t } mi (,, ) M ( fu, ffu, t ) Which is a cotradictio ad so fu Hece fu is a commo fixed poit of f ad g. = ffu ad fu = ffu = fgu = gfu = ggu. The case whe fx is a complete subspace X is similar to the above sice fx gx, We ow show that f ad g are discotiuous at the commo fixed poit z = fu = gu. If possible, suppose f is cotiuous, the cosiderig the sequece { } Sice f ad g are weakly compatible so ffu ffx = gfu so fz = gz. gfx = Takig limit, we get lim This, i tur yields, ( fgx gfx t ) lim,, = 1 This cotradicts the fact that ( fgx gfx t ) x of (1) we get lim ffx = fz = z. fz = gfx or z = lim gfx. lim,, 1 or ot exist. M ( gu, fu, kt ) M ( gu, gu, t ) φ( t) 321

8 Hece f is discotiuous at the fixed poit. Similarly we ca prove g is discotiuous at the fixed poit. This completes the proof of the theorem. 4. Coclusio We prove some fixed poit theorems for weakly compatible maps i fuzzy metric space satisfyig itegral type iequality but without assumig the completeess of the space or cotiuity of the mappigs ivolved. We also fid a affirmative aswer i fuzzy metric space to the problem of Rhoades (1988). 5. Refereces 1. Kramosil ad J. Michalek (1975), Fuzzy metric ad statistical metric space, Kyberetika,11, pp George ad P. Veeramai (1994), O some results i fuzzy metric spaces, Fuzzy sets ad Systems, (64), pp M. Grabiec (1988), Fixed poits i fuzzy metric spaces, Fuzzy sets ad Systems, 27, pp P.V. Subrahmayam (1995), Commo fixed poit theorem i fuzzy metric spaces, Iformatio Scieces,83, pp R. Vasuki (1999), Commo Fixed poits for R weakly computig maps i fuzzy metric spaces, Idia J. Pure Appl. Math., (3,4), pp R.P. Pat, K. Jha (24), A remark o commo fixed poits of four mappigs i a fuzzy metric space, J. Fuzzy Math. 12(2), pp M. Aamri ad D. El Moutawakil (22), Some ew commo fixed poit theorem uder strict cotractive coditios, J. Math. Aal. Appl.,27, pp G. Jugck (1986), Compatible mappigs ad commo fixed poits, It. J.Math. & Math. Sci. 9, pp S.N. Mishra (1994), N. Sharma ad S.L. Sigh, commo fixed poits of maps o fuzzy metric space, It. J. Math. & Math. Sci. 17, pp R. P. Pat (1994), Commo fixed poits of ocommutig mappigs, J.Math. Aal. Appl., 188, pp R. P. Pat (1998), Commo fixed poits of cotractive maps, J. Math. Aal. Appl., 226, pp R. P. Pat (1999), Discotiuity ad fixed poits, J. Math. Aal. Appl.,24, pp

9 13. B.E. Rhoades (1988), Cotractive defiitios cotiuity, Cotemp. Math.,72, pp B.E. Rhoades ad G. Jugck (1998), fixed poits for set valued fuctio without cotiuity, Idia J. Pure Appl. Math. 29(3), pp Schweizer ad A. Sklar (196), Statistical metric spaces, Pacific Joural of Mathematics., 1 pp Deo Brat Ojha, Maish Kumar Mishra ad Udayaa Katoch (21),A Commo Fixed Poit Theorem Satisfyig Itegral Type for Occasioally Weakly Compatible Maps, Research Joural of Applied Scieces, Egieerig ad Techology 2(3): pp B.E Rhoades (23).,Two fixed poit theorem for mappig satisfyig a geeral cotractiv coditio of itegral type. It. J. Math. Sci., 3: pp Chouha ad Badshah (21), geerate fixed poits theorem i fuzzy metric spaces for weakly compatible maps, It. J. Cotemp. Math. Scieces, Vol. 5, o. 3, pp