Generalized Fixed Point Theorem. in Three Metric Spaces

Transcription

1 It. Joural of Math. Aalysis, Vol. 4, 00, o. 40, Geeralized Fixed Poit Thee i Three Metric Spaces Kristaq Kikia ad Luljeta Kikia Departet of Matheatics ad Coputer Scieces Faculty of Natural Scieces, Uiversity of Gjirokastra, Albaia kristaqkikia@yahoo.co, gjoileta@yahoo.co Abstract We prove a related fixed poit thee f three appigs i three etric spaces usig a iplicit relatio. This result geeralizes ad uifies several of well-kow fixed poit thees i coplete etric spaces. Keywds: Cauchy sequece, coplete etric space, fixed poit, iplicit relatio. Matheatics Subject Classificatio: 47H0, 54H5. Itroductio I [8] ad [4] the followig thees are proved: Thee. ( Nug )[8] Let ( X, d),( Y, ρ ) ad ( Z, σ ) be coplete etric spaces ad suppose T is a cotiuous appig of X ito Y, S is a cotiuous appig of Y ito Z ad R is a cotiuous appig of Z ito X satisfyig the iequalities drstxrsy (, ) cax{ dxrsy (, ), dxrstx (, ), ρ( ytx, ), σ ( SySTx, )} ρ( TRSy, TRz) c ax{ ρ( y, TRz), ρ( y, TRSy), σ( z, Sy), d( Rz, RSy)} σ( STRz, STx) c ax{ σ( z, STx), σ( z, STRz), d( x, Rz), ρ( Tx, TRz)} f all x i X, y i Y ad z i Z, where 0 c <. The RST has a uique fixed

2 996 K. Kikia ad L. Kikia poit u i X, TRS has a uique fixed poit v i Y ad STR has a uique fixed poit w i Z. Further, Tu = v, Sv = w ad Rw= u. Thee. ( Jai et. al.)[7] Let ( X, d),( Y, ρ ) ad ( Z, σ ) be coplete etric spaces ad suppose T is a appig of X ito Y, S is a appig of Y ito Z ad R is a appig of Z ito X satisfyig the iequalities (, ) ax { (, ) (, ), (, ) (, ), d RSy RSTx c d x RSy ρ y Tx ρ y Tx d x RSTx d( x, RSTx) σ( Sy, STx), σ( Sy, STx) d( x, RSy)} ( TRzTRSy, ) cax { ( ytrz, ) ( zsy, ), ( zsy, ) ( ytrsy, ), ρ ρ σ σ ρ ρ( y, TRSy) d( Rz, RSy), d( Rz, RSy) ρ( y, TRz)} σ ( STx, STRz) c ax { σ( z, STxdxRz ) (, ), dxrz (, ) σ ( zstrz, ), σ( z, STRz) ρ( Tx, TRz), ρ( Tx, TRz) σ( z, STx)} f all x i X, y i Y ad z i Z, where 0 c <. If oe of the appigs R, ST, is cotiuous, the RST has a uique fixed poit u i X, TRS has a uique fixed poit v i Y ad STR has a uique fixed poit w i Z. Further, Tu = v, Sv = w ad Rw= u.. Mai results This result geeralizes ad uifies several well-kow fixed poit thees obtaied i [8,4,0,9]. F this, we will use the iplicit relatios. ( ) Let Φ k be the set of cotiuous fuctios with k variables k ϕ :[0, + ) [0, + ) satisfyig the properties: a. ϕ is o decreasig i respect with each variable t, t,..., t k b. ϕ( tt,,..., t) t, N. ( ) ( ) Deote Ik = {,,..., k }.F k < k we have Φk Φ k. F k = 6 we ca give these exaples: Exaple. ϕ ( t, t, t3, t4, t5, t6)=ax{ t, t, t3, t4, t5, t6}, with =. Exaple. ϕ( t, t, t3, t4, t5, t6)=ax{ tt i j: i, j I6}, with =. Exaple.3 ϕ ( t, t, t3, t4, t5, t6)=ax{ t p, t p, t p 3, t p 4, t p 5, t p 6}, with = p.

3 Geeralized fixed poit thee 997 Exaple.4 ϕ ( t, t, t3, t4, t5, t6)=ax{ t p, t p, t p 3, t p 4}, with = p, etc. t+ t + t3 t+ t Exaple.5 ϕ( t, t, t3, t4, t5, t6) = with =, etc. 3 Thee.6. Let ( X, d),( Y, ρ ) ad ( Z, σ ) be coplete etric spaces ad suppose T is a appig of X ito Y, S is a appig of Y ito Z ad R is a appig of Z ito X, such that at least oe of the is a cotiuous appig. Let ( ) ϕi Φ 6 f i =,,3. If there exists q [0,) ad the followig iequalities hold () d ( RSy, RSTx) qϕ ( d( x, RSy), d( x, RSTx), ρ( y, Tx), ρ( y, TRSy), ρ( Tx, TRSy, σ( Sy, STx)) () ρ ( TRzTRSy, ) qϕ ( ρ( ytrz, ), ρ( ytrsy, ), σ( zsy, ), σ( z, STRz), σ( Sy, STRz), d( Rz, RSy)) (3) σ ( STx, STRz) qϕ3 ( σ( z, STx), σ( z, STRz), d( x, Rz), d( x, RSTx), d( Rz, RSTx), ρ( Tx, TRz)) f all x X, y Y ad z Z, the RST has a uique fixed poit α X, TRS has a uique fixed poit β Y ad STR has a uique fixed poit γ Z. Further, Tα = β, Sβ = γ ad Rγ = α. Proof. Let x0 X be a arbitrary poit. We defie the sequeces ( x),( y ) ad ( z ) i X, Y ad Z respectively as follows: x =( RST) x0, y = Tx, z = Sy, =,,... Deote d = d( x, x+ ), ρ = ρ( y, y+ ), σ = σ( z, z+ ), =,,... We will assue that x x+, y y+ ad z z+ f all, otherwise if x = x + f soe, the y+ = y+, z+ = z+ we ca take α = x+, β = y+, γ = z+. By the iequality (), f y = y ad z = z ρ ( y, y+ ) qϕ( ρ( y, y), ρ( y, y+ ), σ( z, z), σ( z, z), σ( z, z), d( x, x)) ρ qϕ(0, ρ, σ, σ,0, d ) ( 4) F the codiates of the poit (0, ρ, σ, σ, 0, d ) we have: ρ ax{ d, σ } = λ, N (5) because, i case that ρ >ax{ d, σ } f soe, if we replace the codiates with ρ ad apply the property (b) of ϕ (,,,,, ) ρ qϕ ρ ρ ρ ρ ρ ρ qρ

4 998 K. Kikia ad L. Kikia This is ipossible sice 0 q <. By the iequalities (4), (5) ad properties of ϕ ρ qϕ( λλλλλλ,,,,, } qλ = q ax { d, σ }. Thus ρ qλ = qax { d, σ } (6) By the iequality (3), f x = x ad z = z σ ( z, z+ ) qϕ3( σ( z, z), σ( z, z+ ), d( x, x), d( x, x ), d( x, x, ρ( y, y )) + σ qϕ (0, σ, d, d,0, ρ ) (7) 3 I siilar way, σ qax{ d, ρ }, N. By this iequality ad (6) σ qax{ d, σ }, N (8) By () f x = x ad y = y d ( x, x ) qϕ ( d( x, x ), d( x, x ), ρ( y, y ), ρ( y, y, ρ( y, y, σ( z, z )) d qϕ (0, d, ρ, ρ,0, σ) (9) I siilar way, d qax{ ρ, σ}, N. By this iequality ad the iequalities (6), (8) d qax{ ρ, σ } q( qax{ d, σ }) = = q( q) ax{ d, σ } qax{ d, σ } d qax{ d, σ } (0) By the iequalities (6), (8) ad (0), usig the atheatical iductio, d( x, x ) r ax{ d( x, x ), σ ( z, z )} + ρ( y, y ) r ax{ d( x, x ), σ( z, z )} + σ( z, z ) r ax{ d( x, x ), σ( z, z )} +

5 Geeralized fixed poit thee 999 where q = r <. Thus the sequeces ( x),( y ) ad ( z ) are Cauchy sequeces. Sice the etric spaces ( X, d),( Y, ρ ) ad ( Z, σ ) are coplete etric spaces, we have: li x = α X, li y = β Y, li z = γ Z. it follows Assue that T is a cotiuous appig. The by li Tx = li y. Tα = β. () By (), f y = Tα ad x = x d ( RSTα, x+ ) qϕ( d( x, RSTα), d( x, x+ ), ρ( Tα, y+ ), ρ( Tα, TRSTα), ρ( y+, TRSTα), σ( STα, z+ )) By this iequality ad (), lettig ted to ifiity, d ( RSTα, α) qϕ ( d( α, RSTα),0,0, ρ( β, TRSβ), ρ( β, TRSβ), σ( Sβ, γ)) () q[ax { ρβ (, TRSβ), σ( Sβγ, )}] By (), f z = z ad y = β ρ ( y+, TRSβ) qϕ( ρβ (, y+ ), ρβ (, TRSβ), σ( z, Sβ), σ ( z, z+ ), σ( Sβ, z+ ), d( x, RSβ)) By this iequality lettig ted to ifiity ad usig (), ρ ( β, TRSβ) qϕ (0, ρ( β, TRSβ), σ( γ, Sβ),0, σ( Sβ, γ), d( α, RSTα)). (3) q[ax { d( α, RSTα), σ( γ, Sβ) }] By (3) f x = α, z = z σ ( STα, z+ ) qϕ3( σ( z, STα), σ( z, z+ ), d( α, x), d( α, RSTα), d( x, RSTα), ρ( Tα, y+ )) By this iequality lettig ted to ifiity ad usig (), we have: σ ( Sβ, γ) qϕ3 ( σ( γ, Sβ),0,0, d( α, RSTα), d( α, RSTα),0,) (4) qd ( α, RSTα) By the iequalities (),(3) ad (4) d ( RSTα, α) q d ( RSTα, α) Thus d( RSTα, α) = 0 RSTα = α (5) By (4) ad (3) we obtai

6 000 K. Kikia ad L. Kikia Sβ = γ TRSβ = β STRγ = STR( Sβ) = S( TRSβ) = Sβ = γ Thus, we proved that the poits α, β ad γ are fixed poits of RST, TRS ad STR respectively. I the sae coclusio we would arrive if oe of the appigs R T would be cotiuous. We ow prove the uiqueess of the fixed poits α, β ad γ. Let us prove fα. Assue that there is α a fixed poit of RST differet fro α. By () f x = α ad y = Tα d ( αα, ) = d ( RSTα, RSTα ) qϕ ( d( α, RSTα), d( α, RSTα ), ρ( Tα, Tα ), ρ( Tα, TRSTα), ρ( Tα, TRSTα, σ( STα, STα )) = = qϕ ( d( α, α),0, ρ( Tα, Tα ),0, ρ( Tα, Tα), σ( STα, STα ) q[ax{ d( α, α), ρ( Tα, Tα), σ( STα, STα )}] d ( αα, )= q(ax A) (4) where A= { d( α, α); ρ( Tα, Tα ); σ( STα, STα )}. We distiguish the followig three cases: Case I: If ax A= d( α, α), the the iequality (4) iplies d ( α, α ) qd ( α, α) α = α. Case II: If ax A= ρ( Tα, Tα ), the the iequality (4) iplies d ( α, α ) qρ ( Tα, Tα ) (5) Cotiuig our arguetatio f the Case, by () f z = STα ad y = Tα we have: ρ ( Tα, Tα ) = ρ ( TRSTα, TRSTα ) qϕ ( ρ( Tα, TRSTα), ρ( Tα, TRSTα ), σ( STα, STα ), σ( STα, STRSTα), σ( STα, STRSTα), d( RSTα, RSTα)) = = qϕ ( ρ( Tα, Tα),0, σ( STα, STα ),0, σ( STα, STα), d( α, α )) = q(ax A) (6) Sice i Case II, ax A= ρ( Tα, Tα ), by (6) it follows ρ ( Tα, Tα ) qρ ( Tα, Tα ) ρ( Tα, Tα )=0.

7 Geeralized fixed poit thee 00 By (5), it follows d( α, α ) = 0. Case III: If ax A= σ ( STα, STα ), the by (4) it follows d ( α, α ) qσ ( STα, STα ) (7) By the iequality (3), f x = RSTα, z = STα, i siilar way we obtai: σ ( STα, STα ) q(ax A) = qσ ( STα, STα ) It follows ad by (7) it follows σ ( STα, STα )=0 d( α, α )=0. Thus, we have agai α = α. I the sae way, it is proved the uiqueess of β adγ. Exaple.7. Let X = [0,], Y = [, ], Z = [,] ad d = ρ = σ is the usual etric f the real ubers. Defie: ad if 0 x < if z < 4 3 Tx = Sy = 4 4 Rz = 3 if x 5 if z 4 The S is cotiuous but T ad R are ot cotiuous. We have 3 3 STx =, RSy =, TRz = 3 3 RSTx =, TRSy =, STRz = RST=, TRS 3 = 3, STR 3 = 3 ad T= 3, S 3 = 3, R 3 = These iequalities (), () ad (3) are satisfies sice the value of left had side of each iequality is 0. Hece all the coditios of Thee.6 are satisfies We ephasize the fact that it is ecessary the cotiuity of at least oe of the appigs T, S ad R. The followig exaple shows this. Exaple.8. Let X = Y = Z = [0,]; d = ρ = σ such that d( x, y)= x y, x, y [0,]. We cosider the appigs T, S, R:[0,] [0,] such

8 00 K. Kikia ad L. Kikia that f x =0 Tx = Rx = Sx = x f x (0,] We have f x =0 STx = RSx = TRx = x f x (0,] 4 ad f x =0 4 RSTx = TRSx = STRx = x f x (0,] 8 We observe that the iequalities (), () ad (3) are satisfied f () ϕ = ϕ = ϕ3 = ϕ Φ 6 withϕ( t, t, t3, t4, t5, t6)=ax{ t, t, t3, t4, t5, t6} ad q =. It ca be see that oe of the appigs RST, TRS, STR has a fixed poit. This is because oe of the appigs T, R, S is a cotiuous appig. 3. Collaries Collary 3. Let ( X, d),( Y, ρ ) ad ( Z, σ ) be coplete etric spaces ad suppose T is a appig of X ito Y, S is a appig of Y ito Z ad R is a appig of Z ito X, such that at least oe of the is a cotiuous appig. If there exists q [0,) ad N such that the followig iequalities hold d ( RSy, RSTx) q ax{( d ( x, RSy), d ( x, RSTx), ρ ( y, Tx), ρ ( y, TRSy), ρ ( Tx, TRSy, σ ( Sy, STx)} ρ ( TRz, TRSy) q ax( ρ ( y, TRz), ρ ( y, TRSy), σ ( z, Sy), σ ( z, STRz), σ ( Sy, STRz), d ( Rz, RSy)} σ ( STx, STRz) q ax( σ ( z, STx), σ ( z, STRz), d ( x, Rz), d ( x, RSTx), d ( Rz, RSTx), ρ ( Tx, TRz)} f all x X, y Y ad z Z, the RST has a uique fixed poit α X, TRS has a uique fixed poit β Y ad STR has a uique fixed poit γ Z. Further, Tα = β, Sβ = γ ad Rγ = α.

9 Geeralized fixed poit thee 003 ( ) Proof. The proof follows by Thee.6 i the case ϕ = ϕ = ϕ3 = ϕ Φ 6 such that ϕ ( t, t, t3, t4, t5, t6) = ax{ t, t, t 3, t 4, t 5, t 6 }. Collary 3. Thee. (Nug [8]) is take by thee.6 f =ad ϕ = ϕ = ϕ3 = ϕ such that ϕ ( t, t, t3, t4, t5, t6) = ax{ t, t, t3, t6} Collary 3.3 Thee. (Jai et. al. [4]) is take by Thee.6 i case () ϕ = ϕ = ϕ3 = ϕ Φ 6 such that ϕ ( t, t, t3, t4, t5, t6)= ax{ tt 3, tt 3, tt 6, tt 6 }. Collary 3.4 Thee Telci (Thee [0]).Let ( X, d),( Y, ρ ) be coplete etric spaces ad T is a appig of X ito Y, S is a appig of Y ito X. ϕi Φ 3 f i =,. If there exists q [0,) such that the followig iequalities hold () dsystx (, ) qϕ ( dxsy (, ), dxstx (, ), ρ( ytx, )). () ρ( Tx, TSy) qϕ ( ρ( y, Tx), ρ( y, TSy), d( x, Sy). f all x X, y Y, the ST has a uique fixed poit α X ad TS has a uique fixed poit β Y. Further, Tα = β, Sβ = γ. Proof. The proof follows by Thee.6 i the case Z = X, σ = d, = ad the appig R as the idetity appig i X. The the iequality () takes the f ('), the iequality () takes the f (') ad the iequality (3) is always satisfied sice his left side is σ ( STx, STx)=0. Thus, the satisfyig of the coditios (), () ad (3) is reduced i satisfyig of the coditios('), ('). The appigs T ad S ay be ot cotiuous, while fro the appigs TS, ad R f which we applied Thee.6, the idetity appig R is cotiuous. This copletes the proof. We have the followig collary. Collary 3.5 Thee Popa (Thee, [9]) is take by Collary 3.4 f ϕ = ϕ = ϕ such that ϕ ( t, t, t3)= ax{ tt, tt 3, tt3} with =. We also ephasize here that the costats c, c ca be replaced by q =ax{ c, c }. Coclusios. I this paper it has bee proved a related fixed poit thee f three appigs i three etric spaces, oe of appigs is cotiuous. This thee geeralizes ad uifies several of well-kow fixed poit thees f cotractive-type appigs o etric spaces, f exaple the thees of Nug [8], Jai et.al [4], Popa [9], Telci [0] ad the thee of Fisher []. As collaries of ai result we ca obtai other propositios deteried by the f of iplicit relatios. Refereces [] B. Fisher, Fixed poit i two etric spaces, Glasik Mate. 6(36) (98),

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