COMMON FIXED POINT THEOREM OF FOUR MAPPINGS IN CONE METRIC SPACE

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1 Joural of Mathematical Scieces: Avaces a Applicatios Volume 5 Number 00 Pages COMMON FIXED POINT THEOREM OF FOUR MAPPINGS IN CONE METRIC SPACE XIAN ZHANG School of Scieces Jimei Uiversit Xiame 60 P. R. Chia zhagxiajmj@sia.m Abstract I this paper we replace max b i tractive itio obtai a ew mmo fixe poit theorem of four mappigs i e metric space. The result uifies a extes ma well-kow fixe poit theorems i metric spaces a e metric spaces.. Itrouctio I 974 Ćirić [] itrouce a stuie quasi-tractio mappig i metric space. The well kow Ćirić s result is that see [ 4 8]: T Theorem. Let ( X ρ) be a mplete metric space. A mappig : X X such that for some stat λ ( 0 ) a for ever x X ρ ( Tx T) λ max{ ρ( x ) ρ( x Tx) ρ( T) ρ( x T) ρ( Tx)}. The T possesses a uique fixe poit. Guag a Xia [5] replace the real umbers b a orere Baach space a efie a e metric space. The prove some fixe poit 00 Mathematics Subject Classificatio: 47H0 54H5 55M0 54E5. Kewors a phrases: e metric space mmo fixe poit tractive mappig. Receive December Scietific Avaces Publishers

2 88 XIAN ZHANG theorems for tractive mappigs o e metric spaces. Ilić a Rako cevic ( [7] geeralize Theorem to e metric space. The prove the followig theorem. Theorem. Let ( X ) be a mplete e metric space with ormal e. A mappig T : X X such that for some stat λ ( 0 ) a for ever x X there exists s C( T x ) such that ( Tx T) λs where C ( T x ) = { ( x ) ( x Tx) ( T) ( x T) ( Tx)}. The T possesses a uique fixe poit. I [] Das a Naik extee Theorem to mmo fixe poit of two maps. The prove the followig theorem. Theorem. Let ( X ρ) be a mplete metric space. Let f be a tiuous self-map o X a g be a self-map o X that mmutes with f. Further let f a g satisf g( X ) f ( X ) a there exists a stat λ ( 0 ) such that for ever x X ρ ( gx g) λ max{ ρ( fx f) ρ( fx gx) ρ( f g) ρ( fx g) ρ( f gx)}. The f a g have a uique mmo fixe poit. I [6] Ilić a Rako cevic ( extee Theorem to e metric space. The prove the followig theorem. Theorem 4. Let ( X ) be a mplete e metric space with ormal e. Let f g : X X be two self-maps o X such that f is tiuous a f a g are mmutig a g( X ) f ( X ). Suppose that there exists a stat λ ( 0 ) such that for ever x X there exists s C( x ) such that ( gx g) λs where C ( x ) = { ( fx f) ( fx gx) ( f g) ( fx g) ( f gx)}. The f a g have a uique mmo fixe poit.

3 COMMON FIXED POINT THEOREM OF FOUR 89 I this paper a ew mmo fixe poit theorem is give. The result geeralizes rrespoig results i metric space a e metric space. Now followig [5] we give some efiitios a auxiliar results. Let E be a real Baach space a P be a e i E we efie a partial orerig with respect to P b x if a ol if x P. We shall write x << if x it P where it P eotes the iterior of P. Defiitio [5]. Let X be a oempt set. Suppose the mappig : X X E satisfies () 0 ( x ) for all x X a ( x ) = 0 if a ol if x = ; space. () ( x ) = ( x) for all x X; () ( x ) ( x z) ( z ) for all x z X. The is calle a e metric o X a ( X ) is calle a e metric Let ( X ) be a e metric space { x } be a sequece i X a x X. If for ever c E with 0 << c there is N such that for all > N ( x x ) << c the { x } is sai to be verget a { x } verges to x a x is the limit of { x }. Let { x } be a sequece i X. If for a c E with 0 << c there is N such that for all m > N ( x xm ) << c the { x } is calle a Cauch sequece i X. If ever Cauch sequece is verget i X the X is calle a mplete e metric space. A mappig T : X X is tiuous if Tx Tx wheever x x X. Let E be a real liear space x E A E a λ be a oegative real umber. With x A a x λa we eote there is a s A such that x s a there is a s A such that x λs respectivel.

4 90 XIAN ZHANG Let E be a real liear space A be a subset of E with A we eote the vex hull of A.. Mai Theorem I this sectio we shall prove our mai theorem. Theorem 5. Let ( X ) be a mplete e metric space. Let S T I J be four mappigs from X to X such that (i) S( X ) I( X ) a T ( X ) J( X ); (ii) I a J are tiuous; (iii) S J a T I are mmutative i.e. for all x X. SJx = JSx a TIx = ITx If S T I J satisf followig tractive itio: ( Sx T) λ{ 0 ( Jx I) ( Jx Sx) ( I T) ( ( Jx T) ( I Sx) )} (.) for all x X. The S T I J have a uique mmo fixe poit i X. Proof. Take a arbitrar x0 X let 0 = Jx 0. Sice S( X ) I( X ) a T ( X ) J( X ) there are x x X such that : = Sx0 = Ix : = Tx = Jx. Geerall we have { x } a { } such that for all N : = Sx = Ix : = Tx = Jx. From the tractive itio (.) ( ) = ( Sx Tx ) 0 λ { 0 ( Jx0 Ix ) ( Jx0 Sx0 ) ( Ix Tx ) ( ( Jx0 Tx ) ( Ix Sx0 ))} = λ { 0 ( 0 ) ( 0 ) ( ) ( ( 0 ) ( ))}

5 COMMON FIXED POINT THEOREM OF FOUR 9 ( ) λ{ 0 ( ) ( ) ( ) ( ) ( )} ( ) λ{ 0 ( )}. 0 ( ) = ( Sx Tx ) λ { 0 ( Jx Ix ) ( Jx Sx ) ( Ix Tx ) ( ( Jx Tx ) ( Ix Sx ))} = λ { 0 ( ) ( ) ( ) ( ( ) ( ))} ( ) λ{ 0 ( ) ( ) ( ) ( ) ( )} ( ) λ{ 0 ( )} ( ) λ { 0 ( )}. 0 Suppose that ( ) λ { 0 ( )} 0 a ( ) λ { 0 ( )}. 0 The ( ) ( Sx Tx ) = λ { ( Jx Ix ) ( Jx Sx ) ( Ix Tx ) 0 ( ( Jx Tx ) ( Ix Sx ))} λ { ( ) ( ) ( ) = 0 ( ( ) ( ))}

6 XIAN ZHANG 9 ( ) { ( ) ( ) 0 λ ( ) ( ) ( )} ( ) { ( )} 0 λ ( ) { ( )}. 0 0 λ ( ) ( ) = Tx Sx { ( ) ( ) ( ) 0 λ Tx Ix Sx Jx Ix Jx ( ( ) ( ))} Sx Ix Tx Jx { ( ) ( ) ( ) 0 λ = ( ( ) ( ))} ( ) { ( ) ( ) 0 λ ( ) ( ) ( )} ( ) { ( )} 0 λ ( ) { ( )}. 0 0 λ Hece it is prove b iuctio that for all ( ) { ( )}. 0 0 λ

7 COMMON FIXED POINT THEOREM OF FOUR 9 There are 0 a such that For 0 < m < ( ) λ a ( ). 0 ( ) ( ) ( ) L ( ) m m m m m m ( λ a L λ a ) ( ) m 0 m λ ( ). 0 λ λ For c E 0 << c there is N such that ( 0 ) << c λ m N. So for m N a all we have m for all ( m ) << c. Hece { } is a Cauch sequece. Sice X is mplete let some X. B the itios (ii) a (iii) J = lim J = lim JSx = lim SJx = lim S for I = lim I = lim ITx = lim TIx = lim T. B the tractive itio (.) for all ( S T ) λ { ( J I ) ( J S ) ( I T ) 0 ( ( J T ) ( I S ))}. There are real a b c with a b c 0 a b c such that

8 94 XIAN ZHANG ( S T ) λ( a ( J I ) b ( J S ) c ( I T ) ( ( J T ) ( I S ))). (.) There are subsequeces { a } { b } { c } { } of { a } { b } { c } m m m m { } such that a a b b c c as m with m m a b c 0 a b c. Replace the subscript b i (.) a let m we get m m ( J I ) λ( a ) ( J I ). (.) J Sice λ( a ) < ( I ) = 0. Hece I = J. Sice ( S T ) m λ { ( J I ) ( J S ) ( I T ) 0 ( ( J ) ( ))} T I S there are real a b c 0 a b c such that ( S T ) λ( a ( J I ) b ( J S ) c ( I T ) ( ( J T ) ( I S ))). (.4) As the proof of (.) there are a b c 0 a b c such that ( S I )

9 COMMON FIXED POINT THEOREM OF FOUR 95 λ( a ( J I ) b ( J S ) c ( I I ) ( ( J I ) ( I S ))) = λ( b ) ( S J ). Sice λ( b ) < ( S J ) = 0. Hece S = J. Similarl T = I. From ( S ) ( S Tx ) = λ { ( J Ix ) ( J S ) ( Ix Tx ) 0 ( ( J Tx ) ( Ix S ))} there are real a b c 0 a b c such that ( S ) λ( a ( J Ix ) b ( J S ) c ( Ix Tx ) ( ( J Tx ) ( Ix S ))). (.5) As the proof of (.) there are a b c 0 a b c such that ( J ) λ( a ) ( I ). Sice λ( a ) < ( J ) = 0. So J =. Therefore is a mmo fixe poit of S T I J. Now if x is also a mmo fixe poit of S T I J the ( x ) = ( Sx T )

10 96 XIAN ZHANG λ { 0 ( Jx I ) ( Jx Sx ) ( I T ) ( ( Jx T ) ( I Sx ))} = λ { 0 ( x )}. This implies that x =. Hece the mmo fixe poit of S T I J is uique. Remark. I Theorem 5 let I = J = i (the ietit operator o X) or S = T or X be a metric space we ca get ma versios of fixe poit a mmo fixe poit theorems. Refereces [] M. Abbas a G. Jugck Commo fixe poit results for o-mmutig mappigs without tiuit i e metric spaces J. Math. Aal. Appl. 4 (008) [] Lj. B. Ćirić A geeralizatio of Baachs tractio priciple Proc. Amer. Math. Soc. 45 (974) [] K. M. Das a K. V. Naik Commo fixe poit theorems for mmutig maps o a metric space Proc. Amer. Math. Soc. 77() (979) [4] Lj. Gajić a V. Rako cevic Pair of o-self-mappigs a mmo fixe poits Appl. Math. Comput. 87 (007) [5] H. L. Guag a Z. Xia Coe metric spaces a fixe poit theorems of tractive mappigs J. Math. Aal. Appl. (007) [6] Deja Ilić a Vlaimir Rako cevic Commo fixe poits for maps o e metric space J. Math. Aal. Appl. 4 (008) [7] Deja Ilić a Vlaimir Rako cevic Quasi-tractio o a e metric space Appl. Math. Lett. i press. [8] V. Rako cevic Fuctioal Aalsis Nau c a kjiga Beogra 994. g