CHAPTER 3 COMMON FIXED POINT THEOREMS IN GENERALIZED SPACES

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1 CHAPTER 3 COMMON FIXED POINT THEOREMS IN GENERAIZED SPACES 3. INTRODUCTION After the celebrated Baach cotracto prcple (BCP) 9, there have bee umerous results the lterature dealg wth mappgs satsfyg the cotracto codtos of varous types cludg eve olear expressos. Oe of the mportat geeralzato of t s gve by Jugck [83] whch he establshed commo fxed pot theorems for commutg par of maps. Ths cocept of commutatvty of maps s relaxed to weakly commutatvty, compatblty, weakly compatblty, etc., by several authors, (see Sgh ad Mshra [47], Sessa [84], Jugck ad Rhoades [85], Jugck [37]-[39], Mshra [86], Mshra et al [87]). Aamr ad Moutawakl [87] defed a property (E.A) for self maps whch s exteded for hybrd maps by Kamra [88]. Bracar [89] obtaed a fxed pot theorem for map satsfyg a aalogue of Baach cotracto prcple for tegral type equalty. Ths result was further geeralzed by may authors, see, for stace [9]-[97], [9] ad refereces thereof. Further study o fxed pots ad commo fxed pots of the maps are carred out by varyg structures of the space by may authors (see, [6], [87]-[88], [9]-[9], []-[4], [7]-[8], [37]-[39], [47]-[5], [86]-[99]). 69

2 The purpose of ths chapter s to mprove ad geeralze some recet commo fxed pot results geeralzed metrc spaces ad θ fuzzy metrc spaces for the maps satsfyg tegral type cotractve codtos. Frst, we obta some fxed pot results for hybrd par of sgle ad multvalued maps the settgs of b metrc spaces. Thereafter, we defe θ fuzzy metrc space ad obta some fxed pot theorems t. Some recet results are derved as specal cases. We frst prove some commo fxed pot ad cocdece pot theorems for hybrd par of maps b metrc spaces satsfyg commo property (E. A.) ad a tegral equalty. 3. COMMON FIXED POINT THEOREM USING INTEGRA INEQUAITY Theorem 3... et ( X, d) be a complete b metrc space ad f, g : X X ad F, G : X CB( X ) such that () FX gx, GX fx, () The pars ( F, f ) ad ( G, g) satsfy the commo property (E.A), () for all x, y X, where φ : R + Gy R ) M ( x, y) φ (3..) ( dt q φ( dt + s ebesgue tegrable mappg whch s summable, o-egatve ad such that ad ε φ( t ) dt >, for each ε > (3..) M ( x, y) = max{ fx, gy), fx), gy), [ gy) + fx)]/ } qb wth qb <, λ b <, where λ = max{ q, }. qb If fx ad gx are closed subspace of X, the (3..3) () f ad F have a cocdece pot, 7

3 () g ad G have a cocdece pot, (3) f ad F have a commo fxed pot provded that f s F weakly commutg at u ad ffu = fu for u C( f, F), C ( f, F) = {x : x s a cocdece pot of f ad F }. (4) g ad G have a commo fxed pot provded that g s G weakly commutg at v ad ggv = gv for v C( g, G). (5) f, g, F ad G have a commo fxed pot provded (3) ad (4) are true. Proof. et x X. From () we ca costruct a sequece { y } X such that y+ = fx+ Gx + = gx+ Fx+ y for all. It follows from equato (3..) that y+ 3 ) h ( G+, F + ) M ( x +, x + φ ( dt = ( ) ( ) φ t dt q, φ t dt h ( y+, M ( x +, x+ ) = max{ fx+, gx+ ), Fx+, fx+ ), Gx+, gx+ ), [ d ( Fx +, gx+ ) + Gx+, fx+ )]/ } Thus, = max{d ( y +, y+ ), y+, y+ 3 ), y+, y + 3 ) / } h ( y+, y max{ d ( y+, y+ ), d ( y+, y+ 3 ), d ( y+, y+ 3 ) / + 3) φ ( dt φ( dt Smlarly, λ. y+, + y ) φ( dt. h ( y+, y y, + ) φ ( dt λ y + ) φ( dt, qb λ = max{ q, }. qb Thus, we have proved that for all 7

4 y+ ) y, y+ ) y, y φ ( dt λ φ( dt λ φ( dt. y+, ) qb Hece for all m ad otg λ = max{ q, }, a costat, we have qb The ym, y ) φ ( dt φ( dt y) ( ) λ φ t dt λ m y, y+ ) m y, y, y = = λ ) φ( dt. lm m, ym, y ) φ ( dt =,.e., y } s a Cauchy sequece. { Sce { y } s a Cauchy sequece, there exst a z satsfyg lm y = z = lm fx + = lm gx+. Sce fx ad gx are closed, there exst u, v such that fu = z = gv. A smlar argumet proves that lm Fx + = lmgx+ If z lm Fx + = lmgx+. lm Fx + = A ad lm Gx + = B, commo property (E. A) the z A B. Thus ( F, f ) ad ( g, G) satsfy We clam that gv Gv. To prove t, we take y = v (3..), Fx, Gv) φ ( dt q M ( x, v) φ( dt, M ( x, v) max{ fx, gv), Fx, fx ), Gv, gv), = [ d ( Fx, gv) + Gv, fx )]/ } Takg the lmt as, we obta A, Gv) φ ( dt q M ( A, v) φ( dt, M ( A, v) = max{ fu, gv), A, fu), Gv, gv), [ A, gv) + Gv, fu)]/ } = Gv, gv). 7

5 Sce gv = fu A, t follows from the deftos of Hausdorff metrc that Gv, gv) A, Gv) Gv, gv), whch mples that gv Gv. O the other had, by codto () aga we have, Fu, Gy ) φ ( dt q M ( u, y ) φ( dt, M ( u, y ) = max{ fu, gy ), Fu, fy ), Gy, gy ), [ Fu, gy ) Gy, fu)]/ } + ad smlarly, we obta fu, Fu) Fu, B) fu, Fu). Hece fu Fu. Thus f ad F have a cocdece pot u ad g ad G have a cocdece pot v. Ths completes the proofs of part () ad (). Furthermore, by vrtue of codto (3), we obta ffu = fu ad ffu Ffu. Thus u = fu Fu. Ths proves (3). A smlar argumet proves (4). The (5) holds mmedately. If we put φ ( = ad b = Theorem 3.., the followg result of u et al [88, Th..8] s obtaed. Corollary 3.. [88]. et ( X, d) be a complete metrc space ad f, g : X X ad F, G : X CB( X ) such that () FX gx, GX fx, () The pars ( F, f ) ad ( G, g) satsfy the commo property (E.A). et q (, ) be a costat, such that for all x y X, h ( Gy) q max{ fx, gy), fx), gy), [ gy) + fx)]/ } If fx ad gx are closed subspace of X, the () f ad F have a cocdece pot, () g ad G have a cocdece pot, 73

6 (3) f ad F have a commo fxed pot provded that f s F weakly commutg at u ad ffu = fu for u C( f, F), (4) g ad G have a commo fxed pot provded that g s G weakly commutg at v ad ggv = gv for v C( g, G), (5) f, g, F ad G have a commo fxed pot provded (3) ad (4) are true. As a applcato of Theorem 3.. we obta the followg result. Theorem 3... et ( X, d) be a complete b metrc space ad f, g : X X ad F, G : X CB( X ) such that () FX gx, GX fx, () The pars ( F, f ) ad ( G, g) satsfy the commo property (E.A.), () for all x, y X, where φ : R + Fx R, Gy) M ( x, y) ( ) ( ), φ t dt q φ t dt + s a ebesgue tegrable mappg whch s summable, o-egatve ad ε such that φ( t ) dt >, for each ε >. M ( x, y) = α fx, gy) + β max{ fx), gy)} + γ max{ d ( gy) + fx), fx) + gy)} (3..4) qb wth α + β + γ <, qb <, λb <, where λ = max{ q, }. qb If fx ad gx are closed subspace of X, the () f ad F have a cocdece pot, () g ad G have a cocdece pot, (3) f ad F have a commo fxed pot provded that f s F weakly commutg at u ad ffu = fu for u C( f, F), (4) g ad G have a commo fxed pot provded that g s G weakly commutg at v ad ggv = gv for v C( g, G), (5) f, g, F ad G have a commo fxed pot provded (3) ad (4) are true. 74

7 Proof. From equato (3..4), we have Gy) α fx, gy) + β max{ fx), gy)} + γ max{[ d ( gy) + fx)]/,[ fx) + gy)]/ }. But, max{ d ( fx), gy)} ( fx) + gy)) /, so, Gy) α fx, gy) + β max{ fx), gy)} + γ max{[ d ( gy) + fx)]/, fx), gy)}. et q = α + β + γ <. Followg (3..4), t s easy to see that h ( Gy) q max{ fx, gy), fx), gy), [ gy) + fx)]/ }. Thus by Theorem 3.., we arrve at the coclusos of Theorem 3... If we put φ ( = ad b = Theorem 3.., we get the followg result of u et al [Corollary.9, 88]. Corollary 3... [88] et ( X, d) be a complete metrc space ad f, g : X X ad F, G : X CB( X ) such that () FX gx, GX fx, () The pars ( F, f ) ad ( G, g) satsfy the commo property (E.A). et λ (, ) be a costat, such that for all x y X, Gy) α fx, gy) + β max{ fx), gy)} + γ max{ d ( gy) + fx), fx) + gy)}. ad α + β + γ <. If fx ad gx are closed subspace of X, the () f ad F have a cocdece pot, () g ad G have a cocdece pot, (3) f ad F have a commo fxed pot provded that f s F weakly commutg at u ad ffu = fu for u C( f, F), (4) g ad G have a commo fxed pot provded that g s G weakly commutg at v ad ggv = gv for v C( g, G), (5) f, g, F ad G have a commo fxed pot provded (3) ad (4) are true. 75

8 Now we gve a commo fxed pot result for four self mappgs a b- metrc space. Theorem et ( X, d) be a complete b-metrc space ad A, B, T : X X be such that () AX TX, BX SX, () The pars ( A, S) ad ( B, T ) are weakly compatble, () for all x, y X, where φ : R + R + Ax, By) φ ( dt q φ( dt M ( x, y) s a ebesgue tegrable mappg whch s summable, o-egatve ad such that ad ε φ( t ) dt >, for each ε >, M ( x, y) = max{ Sx, Ty), Sx, Ax), Ty, By), [ Sx, By) + Ty, Ax)]/ } (3..5) qb wth qb <, λ b <, where λ = max{ q, }. qb If TX or SX s closed, the A, B, S ad T have a commo fxed pot. Proof. et x X. From () we ca costruct a sequece { y } X such that y = Tx = Ax ad y + = Sx + = Bx As Theorem 3.., we ca prove that { y } s a Cauchy sequece. Sce X s complete, the sequece { y } coverges to a pot z X. Cosequetly, the subsequeces { Ax},{ Bx+ },{ Sx+ } ad { Tx + } of { y } also coverge to the same lmt z. Now suppose that T (X) s closed. The sce { Tx + } T( X ), there exsts a pot u X such that z = Tu. The by usg (3..5), wth Ax, Bu) M ( x, ) ( ) ( ), u φ t dt q φ t dt = ad y = u we get x x M ( x, u) = max{ Sx, Tu), Sx, Ax ), Tu, Bu), [ Sx, Bu) + Tu, Ax ettg, we get )]/ } 76

9 Bu) M ( u) ( ) ( ), φ t dt q φ t dt M ( u) = max{ z), z), Bu), [ Bu) + z)]/ }. Thus, Bu) M ( Bu) ( ) ( ), φ t dt q φ t dt whch s a cotradcto. Ths mples that z = Bu. Therefore Tu = z = Bu. Hece by the weak compatblty of the par ( B, T ) t mmedately follows that BTu = TBu, that s, Bz = Tz. Next, we shall show that z s a commo fxed pot of B ad T. By settg = ad y = z (3..5) we have x x Ax, Bz) M ( x, ) ( ) ( ), z φ t dt q φ t dt M ( x, z) = max{ Sx, Tz), Sx, Ax ), T Bz), [ Sx, Bz) + T Ax ettg, ad otg that lm Ax = z = lm Sx ad Bz = T we get Bz) M ( z) ( ) ( ), φ t dt q φ t dt M ( z) = max{ Bz), z z), B Bz), [ Bz) + B z)]/ } = Bz) )]/ } Bz) d ( Bz) Thus, ( ) ( ), φ t dt q φ t dt whch s a cotradcto, so z = Bz. Thus we have z = Bz = T.e., z s a commo fxed pot of B ad T. Further, z = Bz mples that z BX SX. Therefore there exsts a pot v X such that = z = Sv. We ow show that Av = Sv. Ideed, by settg x = v ad y x (3..5) ad takg, we get Av = z. Hece Av = z = Sv. The by the weak compatblty of the par ( A, S) we mmedately have SAv = Sz = ASv = Az. Hece Az = Sz. 77

10 = Now, by settg x = z ad y x (3..5) ad followg the earler argumets, t ca easly be verfed that z s a commo fxed pot of A ad S as well. Hece z s a commo fxed pot of A, B, S ad T. The uqueess of z as a commo fxed pot of A, B, S ad T ca easly be verfed. I fact, f z z s aother commo fxed pot of the gve mappgs, the by settg x = z ad y = z (3..5) we get d ( z ) φ ( dt = φ( dt q A Bz ) M ( z ) φ( dt, M ( z ) = max{ S Tz ), S Az), Tz, Bz ), [ S Bz ) + Tz, Az)]/ } Thus we get, d ( z ) max{ d ( z ), z), z, z ), [ z ) + z, z)]/ } = z ) d ( z ) φ ( dt q φ( dt, a cotradcto. Thus z = z ad z s a uque commo fxed pot of A, B, S ad T. Applcato Dyamc Programmg Problem Assume that X ad Y are b-metrc spaces, S X s the state space ad D Y s the decso space. et R = (, ) ad deote by B(S) the set of all bouded real valued fuctos o S. The basc form of the fuctoal equato of dyamc programmg as gve Bellma ad ee [3] s as follows. f ( x) = opt H( x, y, f ( T( x, y))), y where x ad y represet the state ad decso vectors respectvely, T represets the trasformato of the process ad f (x) represets the optmal retur wth tal state x ad opt deotes max or m of the fucto. Here, we study the exstece ad uqueess of a commo soluto of the followg fuctoal equatos arsg dyamc programmg. f ( x) = sup H ( x, y, f ( T ( x, y))), x y D (3..6) g ( x) = sup F ( x, y, g ( T ( x, y))), x y D (3..7) where T : S D S ad H, F : S D R R, =,. 78

11 Suppose the mappgs A ad T ( =, ) are defed by A x) = sup H ( x, y, T ( x, y))), for all x h B( S), =,. y D T k( x) = sup F ( x, y, k( T ( x, y))), for all x k B( S), =,. y D (3..8) Theorem Suppose that the followg codtos are satsfed () H ad F are bouded for =,. H ( x, y, ) H ( x, y, k ( ) M ( h, k ) φ( dt M ( h, k) = max{ T Tk(, T A, Tk( Ak(, [ T Ak( + Tk( A ]} for all ( x, y) S D, h, k B( S) ad t S. A ad T ( =, ) are as defed (3..8). () φ ( dt q () For ay sequece { k } B( S) ad k B(S) wth lm sup ( x) k( x) =, there exsts h B(S) such that k = T h for = or =. (v) For ay h B(S), there exst k k B( ), such that, S A x) = Tk( x), A x) = Tk ( x), x S. k x S (v) For ay h B(S) wth A h = T h ( =, ), we have T A h = AT h. The the system of fuctoal equatos (3..6) ad (3..7) have a uque commo soluto B(S). Proof. It s well kow that B(S) edowed wth the metrc h, k) = sup x) k( x), for ay h, k B( S) x S s a complete metrc space. From codto (), there exsts, M > satsfyg sup{ H ( x, y,, F( x, y, : ( x, y, S D R M. Usg Theorem.34 of Hewtt ad Stromberg [3] ad codto o φ, we coclude that for each ε >, there exsts δ > satsfyg C φ( t ) dt < ε, C [, M ] wth m ( C) δ, (3..9) where m(c) deotes the ebesgue measure of C. 79

12 Moreover, by codto (), A ad T are self mappgs of B(S) ad by codto (v) t s clear that A B( S)) T ( B( )) ad A B( S)) T ( B( )). ( S ( S Also, by codto (v), the pars A, T ) are weakly compatble for =,. ( Moreover, by (3..8) ad () we have that for ay η > there exst y y D such that A h ( x) < H ( x, y, h ( x)) + η, (3..) x = T( x, y ), =,. Also, A h x) > H ( x, y, h ( )), (3..) ( x A h x) > H ( x, y, h ( )), (3..) ( x, the from (3..), (3..) ad (), we have, A h x) A h ( ) < H x, y, h ( x )) H ( x, y, h ( )) + η (3..3) ( x ( x H x, y, h ( x )) H ( x, y, h ( )) +η ( x max{ T h ( x ) T h ( x ), T h ( x ) A h ( x ), T h [ T h ( x ) A h ( x ) + T h ( x ) A h ( x ) ]} + η ( x ) A h ( x ), max{ d ( Th, Th ), Th, A h ), Th, Ah ), [ Th, Ah ) + Th, A h )} +η From (3..), (3..) ad (), we have A h x) A h ( ) (3..4) ( x max{ d ( Th, Th ), Th, A h ), Th, Ah ), [ Th, Ah ) + Th, A h )} η Ufcato of (3..3) ad (3..4) yelds A h ( x) Ah ( x) max{ d ( Th, Th ), Th, A h ), Th, Ah ), [ Th, Ah ) + Th, A h )} +η (3..5) whch meas that A h ( x) Ah ( x) M ( h, h ) + η φ ( dt q φ( dt, 8

13 M ( h, h ) = max{ Th, Th ), Th, A h ), Th, Ah ), [ Th, Ah ) + Th, A h )} Smlarly, we fer that (3..5) holds also for opt = y D sup y D. A h ( x) Ah ( x) M ( h, h ) + η φ ( dt q φ( dt M( h, h) = max{ dth (, Th) + η, dth (, Ah) + η, Th, Ah) + η, [ dth (, Ah ) dth (, ) ]} Ah + η + + η.e., f max{ dth (, Th) + η, dth (, Ah) + η, dth (, Ah) + η, [ dth (, Ah ) dth (, Ah ) ]} (, ). dth Th + η + + η = + η The we have, A h ( x) Ah ( x) T + h, T h ) d ( T h, T h ) η φ( dt q φ( dt + φ( dt d ( T h, T h ). Thus from (3..9) we have, A h ( x) Ah ( x) M ( h +, h ) η φ( dt q φ( dt + ε. ettg ε the above equalty, we deduce that A h ( x) Ah ( x) M ( h +, h ) η φ ( dt q φ( dt. Therefore codto (3..5) s satsfed by mappgs A, A, T ad T ad hece by Theorem * 3..3, they have a uque commo fxed pot h B( S),.e., h * ( x ) s a uque commo soluto of the fuctoal equatos (3..6) ad (3..7). I the ext secto, we establsh some geeral results θ fuzzy metrc spaces. 8

14 3.3 COMMON FIXED POINT THEOREM IN θ FUZZY METRIC SPACE I 965, Zadeh [98] troduced fuzzy set, whch s further geeralzed to tutostc fuzzy set by Ataassov [6] ad fuzzy set by Gogue [5]. Thereafter several authors defed fuzzy metrc spaces dfferet ways. Cosequetly several fuzzy fxed pot theorems are also establshed ther ew settgs, see for stace, [99]-[], [7]-[8], [4]-[8] ad refereces thereof. We frst recall the prelmares requred for subsequet results. Defto 3.3. [9]. et θ :[, ) [, ) [, ) be a cotuous map wth respect to each varable. Theθ s called a B acto f ad oly f t satsfes the followg codtos () θ (, ) = ad θ ( t, s) = θ ( s, for all t, s, () θ ( s, < θ ( u, v) f s < u ad t v or s u ad t < v, () for each r Im( θ ) {} ad for each s (, r], there exsts t (, r] such that θ ( t, s) = r, where Im( θ ) = { θ ( s, : s, t }, (v) θ ( s, ) s, for all s >. Defto 3.3. [9]. et X be a oempty set. A mappg d θ : X X [, ) s called a θ metrc o X wth respect to B acto, f d θ satsfes the followg () d ( x, y) = ff x = y, θ () dθ ( x, y) = dθ ( y, x), for all x, y X, () d ( x, y) θ( d ( x, z), d ( y)), for all x, y, z X. θ The X, d ) s called a θ metrc space. ( θ θ θ Remark If θ ( s, = s + t, the the θ metrc space becomes a metrc space. Remark If θ ( s, = b( s +, b, the the θ metrc space s the b-metrc space. Now we defe θ fuzzy metrc space followg the defto gve by George ad Veerama []. 8

15 Defto The 3-trplet ( X, M, T ) s sad to be a θ fuzzy metrc space, f X s a arbtrary (o-empty) set, T s a cotuous t orm o ad M s a -fuzzy set o X (, ) wth respect to B acto, θ every x, y, z X ad t, s (, ). M satsfyg the followg codtos for () M ( x, y, t ) >, () ( x, y, = for all t >, ff x = y, M () M( x, y, = M ( y, x, (v) T ( M( x, y,, M( y, s)) M ( x, θ ( t, s)), (v) M ( x, y,.):(, ) s cotuous ad lm ( x, y, =. M t I ths case M s called a θ fuzzy metrc space. If M = M, s a tutostc fuzzy set, the the 3-tuple ( X, M,, T ) s sad to be a θ tutostc fuzzy metrc space. M N M N Example 3.3. [5]. et X = [, ] edowed wth the usual metrc x, y) = x y, x, y X ad θ ( t, s) = t + s. et T ( a, b) = ( ab, m( a+ b, )) for all * * a = ( a, a), b= ( b, b), where (, * ) s a complete lattce ad let M, M N be the tutostc fuzzy set o X (, ) defed as follows M M, N t x, y) ( x, y, =, t+ x, y) t+ x, y) The ( X, M M, N, T ) s a tutostc fuzzy metrc space. Defto [9]. A fuzzy metrc space ( X, M, T ) s sad to have the property (C), f t satsfes the followg codto: M ( x, y, = C, for all t > mples C =. emma []. et φ ( :[, ) [, ) satsfes the followg codto = (φ ) It s odecreasg ad φ () t < for all >, terate of φ (, the φ ( t ) < t for all t >. t where ( φ deotes the th 83

16 emma et ( X, M, T ) be a θ fuzzy metrc space. The M ( x, y, s odecreasg wth respect to t, for all x, y X. Proof. Suppose M( x, y, > M( x, y, s) for some < t < s. The T ( M( x, y,, M( y, y, s ) M( x, y, θ ( s t, ). Now from the defto, we have, for each r Im( θ ) {} ad for each s (, r], there exsts t (, r] such that θ ( t, s) = r, where Im( θ ) = { θ ( s, : s, t }. So, θ ( t, s = r, sce s t >. The, T ( M( x, y,, M( y, y, s ) M( x, y, r) > M( x, y,, sce t (, r]. Also, we have ( y, y, s = for all s>. M Thus we have M( x, y, > M( x, y,, a cotradcto. emma et ( X, M, T ) be a θ fuzzy metrc space. Defe + Eλ, M : X R {} by Eλ, M ( x, y) = f{ t > : M( x, y, > N ( λ)} for each λ \{, } ad x, y X. The we have () For ay µ {, } there exsts λ \{, } such that \ E ( x, x ) E ( x, x ) + E ( x, x ) E ( x, x ) µ, M λ, M λ, M 3 λ, M for ay x,..., x. X () The sequece { x } s coverget to x wth respect to θ fuzzy metrc M f ad N oly f t s Cauchy wth E λ,. Proof. For the frst part, we ca fd λ \{, } for every µ \{, } T ( N( λ),..., N( λ )) N ( µ ). Usg the defto of θ fuzzy metrc space, we have M ( x, x, θ( E ( x, x ) E ( x, x ), δ)) λ, M λ, M such that T ( M x x θ Eλ, M x x (,, ( (, ), δ ), M ( x, x, θ ( E ( x, x ), δ ), 3 λ, M 3 M x x θ Eλ, M x x (,, ( (, ), δ )) 84

17 T ( ( N( λ),..., N( λ )) N ( µ ) for every δ >, whch mples that E ( x, x ) θ ( E ( x, x ) + E ( x, x ) E ( x, x ), δ ). µ, M λ, M λ, M 3 λ, M Sce δ > s arbtrary, we have E ( x, x ) θ ( E ( x, x ) + E ( x, x ) E ( x, x )) µ, M λ, M λ, M 3 λ, M E ( x, x ) + E ( x, x ) E ( x, x ). λ, M λ, M 3 λ, M Ths proves (). For (), ote that scem s cotuous ts thrd place, E, ( x, y) s ot a elemet of the set { t > : M( x, y, > N( λ)} as soo as x y. Hece we have, ( x, x, η) > ( λ) Eλ, ( x, x) < η for every η >. Ths completes the proof. λ M emma et A, B, I ad J be mappgs from θ fuzzy metrc space ( X, M, T ) to tself satsfyg () AI( X ) T ( X ), BJ ( X ) S( X ) () M ( AIx, BJy, ϕ ( ) M ( x, y, φ() s ds r φ() s ds M ( x, y, = max{ M ( Sx, Ty,, M ( AIx, Sx,, M ( BJy, Ty,,[ M ( AIx, Ty, + M ( BJy, Sx)]/}, where φ ( satsfes codto (φ ) ad r : s a cotuous fucto such that r( > t for each t = ( t, t) \{, } ad for all, y X. x The the sequece { } y defed by y = Tx + = AIx, y Sx BJx, =,,,... s a Cauchy sequece X. + = + = + Proof. For t >, we have, M ( y, y, ϕ ( ) = M ( AIx, BJx, ϕ ( ) rmax{ M ( Sx, Tx, ϕ ( ), M ( AIx, Sx, ϕ ( ), M( BJx+, Tx+, ϕ ( ), M AIx Tx+ ϕ t + M BJx+ Sx ϕ t [ (,, ()) (,, ())]/} = r( M ( y, y, ϕ ( ) ( y, y, ( )... ( y, y, M ( y, y, ϕ ( ) M ϕ M for =,,... whch mples that, for every λ \{, }, we have 85

18 Eλ M ( y, y ) = f{ ϕ ( > : M(( y, y, ϕ ( ) > N ( λ)}, + + = f{ ϕ ( > : M(( y, y, ) > N ( λ)} = ϕ (){f{ t t > : M(( y, y, ) > N ( λ)} = ϕ () teλ, M ( y, y). From emma 3.3.3, for every µ \{, } there exsts λ \{, } such that E ( y, y ) E ( y, y ) + E ( y, y ) E ( y, y ) µ, M m λ, M + λ, M + + λ, M m m m j ϕ ( E, ( y, y)) λ M as m,. j= Thus { y } s a Cauchy sequece X. Now we prove the followg commo fxed pot theorem. Theorem et A, B, I ad J be mappgs from a complete θ fuzzy metrc space ( X, M, T ) to tself satsfyg (), () of emma ad property (C). Suppose that oe of A, B, I ad J s complete ad pars ( AI, S) ad ( BJ, T ) are weakly compatble, the A, B, I ad J have a uque commo fxed pot. Proof. By emma 3.3.4, { } y s a Cauchy sequece ad sce X s complete, therefore,{ y } coverges to some pot z X. Cosequetly, the subsequeces { AIx }, Sx + }, BJx + } ad Tx + } of y } also coverges to z. { { { { Assume that S(X ) s complete, so there exsts a pot u X such that S u = z. If z AIu, from (), we have M ( ux,, φ() s ds r φ() s ds M ( AIu, BJx, ϕ ( t )) M (, u x, = max{ M ( Su, Tx,), t M ( AIu, Su,), t M ( BJx, Tx,),[ t M ( AIu, Tx,) t + M ( BJx, Su)]/} O the other had, by emma 3.3., M( AIu, BJx, ϕ ( ) M( AIu, BJx,. 86

19 Takg the lmt, we get, M (, u = max{ M(,,), z z t M( AIu,,), z t M(,,),[ z z t M( AIu,,) z t + M(,,)]/} z z t M ( AIu, M ( AIu, φ() s ds r φ() s ds whch s a cotradcto. Thus we have, AIu = S u = z. Sce AI( X ) T ( X ), there exsts v X, such that Tv = z. If z BJv, we have M ( BJv, ϕ ( ) M( AIu, BJv, ϕ( ) M( u, v, φ() s ds = φ() s ds r φ() s ds M( u, v, = max{ M ( Su, Tv,, M ( AIu, Su,, M ( BJv, Tv,, [ M ( AIu, Tv, + M ( BJv, Su, ]/ } But by emma 3.3., M ( AIu, BJv, ϕ ( ) M( AIu, BJv,. Thus we have, M (, z BJv, M (, z BJv,) t φ() s ds r φ() s ds Hece, Tv = BJv = AIu = Su = z. Sce the pars ( AI, S) ad ( BJ, T) are weakly compatble, we have AISu = SAIu, ad BJTv = TBJv;.e., AIz = Sz ad BJz = Tz. If AIz ad M( AIz, BJv, = max{ M ( S Tv,, M ( AI S, M ( BJv, Tv,, [ M ( AI Tv, + M ( BJv, S ] / } or, M ( AIz, BJv, = max{ M ( AI, M ( AI AI, M (, [ M ( AI + M ( AI ] / } We get, φ() s ds= φ() sds r φ() sds > φ() sds M( AI ϕ ( ) M( AI BJv, ϕ ( ) M( AI M( AI t ) By emma 3.3., M( AI ϕ ( ) M( AI, so we have 87

20 M ( AI M ( AI t ) φ() s ds φ() s ds, whch s a cotradcto. Hece AIz = z = Sz. Smlarly, we ca prove that BJz = z = Tz. Therefore, z s a commo fxed pot of A, B, I ad J. For uqueess, let f possble The there exsts t > such that M ( z, ϕ ( ) < ad z z be aother commo fxed pot of A, B, I ad J. M ( z, ϕ ( ) M ( AI BJz, ϕ ( ) φ() s ds = φ() s ds But, M ( AI BJz, = max{ M ( S Tz,, M ( AI S, M ( BJz, Tz,, [ M ( AI Tz, + M ( BJz, Sz)]/} = M ( z, Thus, M ` ( AI BJz, ϕ ( t )) M ( z, ϕ ( ) φ( s) ds φ( s) ds... M ( z, M ( z, r φ() s ds φ() s ds By emma 3.3., M( z, ϕ ( ) M ( z,. Hece, M ( z, = C for all t >. Sce, M has the property (C), t follows that C =, therefore z = z,.e., z s a uque commo fxed pot of A, B, I ad J. Ths completes the proof. If we put B = S = I = J = d,.e., the detty map ad φ ( = kt, < k <, ϕ( = ad θ ( s, = s + t, Theorem 3.3., the the followg result of Maro et al [93] s deduced. Corollary [93]. et A ad T be mappgs from a complete fuzzy metrc space ( X, M, T ) to tself satsfyg A( X ) T ( X ) ad M( Ax, Ay, k M( Tx, Ty,. Suppose that ether A (X ) or T (X ) s complete ad A ad T are weakly compatble o X. The Aad T have a uque commo fxed pot. 88

21 I the followg Theorem we relax the codto of completess o the space. Theorem et A, B, I ad J be mappgs from a θ fuzzy metrc space ( X, M, T ) to tself satsfyg (), () of emma ad property (C). Suppose that oe of A, B, I ad J s complete ad pars ( AI, S) ad ( BJ, T ) are weakly compatble, the A, B, I ad J have a uque commo fxed pot. Proof. From the proof of Theorem 3.3., we coclude that { y } s a Cauchy sequece X. Assume that S(X ) s complete subspace of X. The the subsequece of y } must get a lmt S(X ). et t be u ad Sv = u. As y } { { s a Cauchy sequece cotag a coverget subsequece, therefore the sequece { y } also coverges mplyg thereby the covergece of subsequece of the coverget sequece. Now we have, M ( AIv, BJx, ϕ ( t )) M( v, x, φ() s ds r φ() s ds M ( v, x, = max{ M ( Sv, Tx,, M ( AIv, Sv,, M ( BJx, Tx,,[ M ( AIv, Tx, + M ( BJx, Sv)]/} Now by emma 3.3., M( AIv, BJx, ϕ ( ) M ( AIv, BJx,. Takg the lmt, we get, M ( v, u, = max{ M( u, u,, M( AIv, Sv,, M( u, u,,[ M( AIv, u, + M( u, u)]/} M ( AIv, u, M ( AIv, u, φ() s ds r φ() s ds, whch s a cotradcto. Thus we have, AIv = S v = u, whch shows that par ( AI, S) has a pot of cocdece. Sce AI( X ) T ( X ), there exsts p X, such that Tp = u. If u BJp, we have M ( u, BJp, ϕ ( ) M ( AIv, BJp, ϕ ( ) φ() s ds = φ() s ds 89

22 We have, M ( AIv, BJp, = max{ M ( Sv, Tp,, M ( AIv, Sv,, M ( BJp, Tp,,[ M ( AIv, Tp, + M ( BJp, Sv, ] / } = max{ M ( u, u,, M ( u, u,, M ( BJp, u,,[ M ( u, u, + M ( BJp, u, ] / } By emma 3.3., M( AIv,BJp, ϕ ( ) M ( AIv,BJp,. Thus we have, φ() s ds r φ() s ds M ( u, BJp, M ( u, BJp, a cotradcto. Hece, Tp = BJp = AIv = Sv = u. Sce the pars ( AI, S) ad ( BJ, T ) are weakly compatble, we have AISv = SAIv, BJTp = TBJp,.e., AIu = Su ad BJu = Tu. ad If AIu u, ad M ( AIu, BJp, t ) = max{ M ( Su, Tp, t ), M ( AIu, Su, t ), M ( BJp, Tp, t ),[ M ( AIu, Tp, t ) + M ( BJp, Su, t )] / } We get = max{ M ( AIu, u,, M ( AIu, AIu,, M ( Tp, Tp,, [ M ( AIu, u, + M ( u, AIu, ] / } M ( AIu, u, ϕ ( t )) M ( AIu, BJp, ϕ ( t )) M( AIu, BJp, ϕ ( ) M( AIu, u, φ() s ds = φ() s ds r φ() s ds > φ() s ds By emma 3.3., M( AIu, u, ϕ ( ) M( AIu, u,, so we have M( AIu, u, M( AIu, u, φ() s ds φ() s ds, whch s a cotradcto. Hece AIu = u = Su. Smlarly, we ca prove that BJu = u = Tu. Therefore u s a commo fxed pot of A, B, I ad J. For uqueess, let f possble u u be aother commo fxed pot of A, B, I ad J. The there exsts t > such that M ( u, u, ϕ ( ) ad M( u, u, ϕ ( ) M( AIu, BJu, ϕ ( ) φ() s ds φ() s ds. 9

23 But, M ( AIu, BJu, = max{ M( Su, Tu,, M( AIu, Su,, M( BJu, Tu,, [ M ( AIu, Tu, + M ( BJu, Su, ]/} = M ( u, u, Thus, M ( AIu, BJ u, ϕ ( ) M( u, u, ϕ ( ) φ() s ds φ() s ds... M( u, u, M( u, u, r φ() s ds φ() s ds. By emma 3.3., M( u, u, ϕ ( ) M ( u, u,. Hece, M ( u, u, = C for all t >. Sce, M has the property (C), t follows that C =, therefore u = u,.e., u s a uque commo fxed pot of A, B, I ad J. Ths completes the proof. If we put B = S = I = J = detty map ad φ ( = kt, < k <, ϕ( = ad θ ( s, = s + t, Theorem 3.3., the the followg result of Maro et al [93] s obtaed. Corollary [93]. et A ad T be mappgs from a fuzzy metrc space ( X, M, T ) to tself satsfyg A( X ) T ( X ) ad M( Ax, Ay, k M ( Tx, Ty,. Suppose that ether A (X ) or T (X ) s complete ad A ad T are weakly compatble o X. The A ad T have a uque commo fxed pot. Theorem All the statemets of the Theorem 3.3. remas true f a weakly compatble property s replaced by ay oe of the followg: () R-weakly commutg property, () R-weakly commutg property of type ( A f ), () R-weakly commutg property of type ( A g ), (v) weakly commutg property. 9

24 Proof. () Sce all the codtos of Theorem 3.3. are satsfed, the exstece of cocdece pots for both the pars ( AI, S) ad ( BJ, T ) are esured. et x be a arbtrary pot of cocdece for the par ( AI, S) ad ( BJ, T ). The usg R-weakly commutatvty, oe gets ad M( AISx, SAIx, ϕ ( ) M( AIx, Sx, t / R) = M ( Sx, Sx, t / R) = M( BJTx, TBJx, ϕ ( ) M( BJx, Tx, t / R) = M ( Tx, Tx, t / R) = From emma 3.4., M( AISx, SAIx, ϕ ( ) M ( AISx, SAIx, ad M( BJTx, TBJx, ϕ ( ) M ( BJTx, TBJx, So we have AISx = SAIx ad BJTx = TBJx. Thus the pars ( AI, S) ad ( BJ, T) are weakly compatble. Now applyg Theorem 3.3., we coclude that A, B, I ad J have a uque commo fxed pot. () I case par ( AI, S) satsfyg property R-weakly commutatvty of type A ), we have M( SSx, AISx, ϕ ( ) M( Sx, AIx, t / R) = M ( AIx, AIx, t / R) = ( f ad we have AISx = SSx. From emma 3.3., M( AISx, SAIx, ϕ ( ) M( AISx, SAIx,. So, M( AISx, SAIx, T ( M( AISx, SSx, t /), M( SSx, SAIx, t /)) Ths mples that AISx = SAIx. = T ( M( AISx, AISx, t /), M ( x, x, t /)) T (, ) =. Smlarly, f par ( BJ, T ) satsfes property R-weakly commutatvty of type A ), we get BJTx = TBJx. () I case par ( AI, S) satsfyg property R-weakly commutatvty of type A ), we have M( AIAIx, SAIx, ϕ ( ) M( AIx, Sx, t / R) = M ( AIx, AIx, t / R) = ad we have AIAIx = SAIx. From emma 3.3., M( AISx, SAIx, ϕ ( ) M ( AISx, SAIx,. So, M( AISx, SAIx, T ( M( AISx, AIAIx, t /), M( AIAIx, SAIx, t /)) ( g ( f = T ( M( x, x, t /), M ( AIAIx, AIAIx, t /)) T (, ) =. 9

25 Ths mples that AISx = SAIx. Smlarly, f par ( BJ, T ) satsfes property R-weak commutatvty of type A ), we get BJTx = TBJx. Smlarly, we ca show that, f pars ( AI, S) ad ( BJ, T) are weakly commutg the ( AI, S) ad ( BJ, T ) also commutes at ther pot of cocdece. Now, vew of Theorem 3.3., all fve cases, A, B, I ad J have a uque commo fxed pot. Ths completes the proof. ( f Now we gve a example support of our result metoed Theorem Example et X = [, ] wth usual metrc d, θ ( s, = s + t, ad for each t \{, }, defe M ( x, t y, = x y, N( x, y, = t + x y t + x y. Clearly ( X, M M, N, T ) s a complete tutostc fuzzy metrc space. et A, B, I ad J be defed by f x = f x = Ax = x x X, Ix = Tx = f < x 5, 3 f x > x f x > 5 f x = or x > 5 f x =, x > 5 f x = Bx =, Jx =, Sx = 6 f < x 5 3x f < x 5 6 f x > The, AI ( X ) = {, 3} {, } (.5, ] = T ( X ) ad BJ ( X ) = {, 6} {, 6} = S( X ) ad A s cotuous. Defe r : by r ( = t for t \{, } ad r( = for t =. The r( > t for t \{, }. Also φ() s ds r φ() s ds x, y X. M( AIx, BJy, ϕ ( ) M( Sx, Ty, ϕ ( ) For x =, AIx =, Sx =, AISx = ad SAIx =. 93

26 Thus M ( AISx, SAIx, = = M ( AIx, Sx, t / R) ad N ( AISx, SAIx, = = N ( AIx, Sx, t / R). For x >, AIx = 3, Sx = 6, AISx = 3 ad SAIx = 6. Thus M ( AISx, SAIx, M ( AIx, Sx, t / R) ad N ( AISx, SAIx, N ( AIx, Sx, t / R) ad the par ( AI, S) s R-weakly commutg o X. Smlarly, for x =, BJx =, Tx =, BJTx = ad TBJx =. Thus M ( BJTx, TBJx, = = M ( BJx, Tx, t / R) ad N ( BJTx, TBJx, = = N ( BJx, Tx, t / R). For < x 5, BJx =, Tx =, BJTx = ad TBJx =. t Thus M ( BJTx, TBJx, = = M ( BJx, Tx, t / R) t + R R ad N ( BJTx, TBJx, = = N( BJx, Tx, t / R). t + R Aga for x > 5, BJx =, Tx = x/, BJTx = ad TBJx =. Thus t M ( BJTx, TBJx, = = M ( BJx, Tx, t / R) (4 x) t + R ad 4 x R N ( BJTx, TBJx, = = N( BJx, Tx, t / R). 4 x t + R ad the par ( BJ, T ) s R-weakly commutg o X. Thus, all the codtos of Theorem are satsfed ad s the commo fxed pot of A, B, I ad J. 94