Common Fixed Point Theorem for Expansive Maps in. Menger Spaces through Compatibility

Transcription

1 Iteratioal Mathematical Forum 5 00 o Commo Fixed Poit Theorem for Expasive Maps i Meger Spaces through Compatibility R K Gujetiya V K Gupta M S Chauha 3 ad Omprakash Sikhwal 4 Departmet of Mathematics Govt P G College Neemuch Idia gujetiya7@gmailcom Departmet of Mathematics Govt Madhav Sciece College Ujjai Idia dr_vkg6@yahoocom 3 Departmet of Mathematics Govt Nehru P G College Agra-Malwa Idia drmsc@rediffmailcom 4 Departmet of Mathematics Madsaur Istitute of Techology Madsaur Idia opbhsikhwal@rediffmailcom Abstract I this paper we preset commo fixed poit theorem for four expasive mappigs i Meger spaces through semi ad weak compatibility Mathematics Subject Classificatio: 47H0 54H5 Keywords: Meger space t-orm Semi-compatible Weak compatible Expasive mappigs INTRODUCTION Baach [] proved that a cotractio mappig i a complete metric space possesses a uique fixed poit Due to its applicatio i various disciplies of Mathematics ad

2 348 R K Gujetiya V K Gupta M S Chauha ad O Sikhwal Mathematical Scieces the Baach Cotractio Priciple has bee extesively studied ad geeralized o may settigs ad fixed poit theorems have bee established Bharucha-Reid [] uderlied the eed of the research regardig the study of fixed poits of mappigs o probabilistic metric space (PM-spaces) Sehgal [0] iitiated the study of fixed poit of cotractio mappig o a PM-space He itroduced the otio of cotractio mappig o a PM-space as a geeralizatio of the Baach Cotractio Priciple ad proved some fixed poit theorems for such mappigs These results were first published i 97 by Sehgal ad Bharucha-Reid Subsequetly several geeralizatios of Sehgal's results were obtaied o PM-spaces Jugck [6] geeralized the Baach cotractio priciple by itroducig a cotractio coditio for a pair of commutig self mappigs o a metric space ad poited out of potetial of commutig mappigs for geeralizig fixed poit theorems i metric spaces Jugck s [7] result has bee further geeralized by cosiderig geeral type of cotractive or fuctioal coditios o the pair of mappigs I fact the theory of cotractive priciples got a spirit due to Jugck's result [7] ad its first geeralizatio due to Sigh [] Baach cotractio priciple yields a fixed poit theorem for a diametrically opposite class of maps viz expasive maps Fixed poit theorem for expasive maps i metric spaces is proved by Gillespie ad Williams [4] Taiguchi [3] exteded these theorems I this paper we combie the ideas of ciric [3] ad Sigh ad Kasahara [] ad itroduce the otio of geeralized cotractio quadruplet for four self mappigs o a PM-space We also preset a extesio of the mai result of Vasuki [4] for four expasive mappigs i meger space PRELIMINARIES Defiitio A mappig F: R R + is called a distributio fuctio if it is odecreasig left-cotiuous with if{ Ft ( ) t R} = 0 ad sup{ Ft ( ) t R} = We shall deote by Λ the set of all distributio fuctios while H will always deote the 0 t 0 specific distributio fuctio defied by Ht () = t > 0 Defiitio A Probabilistic metric space (PM-space) is a ordered pair ( X F ) where X is a o empty set ad F is a mappig from X X to Λ the collectio of all distributio fuctios Fuv () x The value of F at () uv X X is represeted by F uv The ordered pair ( X F) is called PM-space if it satisfies the followig coditios:

3 Commo fixed poit theorem for expasive maps 349 ( PM ) F () x = for all x > 0 if u= v; uv ( PM ) F (0) = 0 ; uv ( PM 3) Fuv () x = Fvu () x ; ( PM 4) If F () x = ad F ( y ) = the F ( x+ y) = uv vw Defiitio 3 [0] A mappig t :[0] [0] [0] is called a t-orm if it is associative o-decreasig i each co-ordiate ad () a 0 uw ta = for every [ ] Defiitio 4 [0] A Meger space is a triplet ( X Ft ) ( X F ) is a PM-space ad t is a t-orm such that the iequality where ( PM 5) Fuw ( x+ y) t{ Fuv ( x) Fvw ( y)} for all uvw X ad xy 0 Defiitio 5 [0] A sequece { x } i a Meger space ( X Ft ) is said to be coverget ad coverges to a poit x i X if ad oly if for each ε > 0 ad λ > 0 there is a iteger M( ε λ) such that F () ε > λ for all M( ε λ) x x Further the sequece { x } is said to be Cauchy sequece if for ε > 0 ad λ > 0 there is a iteger M( ε λ) such that F () ε > λ for all m Mε ( λ) x x m A Meger space ( X Ft ) is said to be complete if every Cauchy sequece i X coverges to a poit i X Defiitio 6 Self maps S ad T of a Meger space ( X Ft ) are said to be compatible if F () x ad wheever { x } is a sequece i X such that Sx Tx STx TSx u for some u i X as Defiitio 7 Self maps S ad T of a Meger space ( X Ft ) are said to be semicompatible if F () x for all x > 0 wheever { x } is a sequece i X such that Sx Tx u STx T u for some u i X as It follows that the pair ( ST ) is semi-compatible ad Sy = Ty imply STy = TSy by takig { x } = y ad u= Sy= Ty

4 350 R K Gujetiya V K Gupta M S Chauha ad O Sikhwal Defiitio 8 Self maps S ad T of a Meger space ( X Ft ) are said to be weakly compatible if they commute at their coicidece poits that is if Sp = Tp for some p X the STp = TSp Defiitio 9 Three mappigs P Q T o a PM-space ( X F ) will be called a geeralized cotractio triplet ( P QT ; ) if ad oly if there exists a costat h (0) such that for every uv X FPu Qu ( h x) mi{ FTu Tv ( x ) FPu Tu ( x ) FQv Tv ( x ) FPu Tv ( x ) FQv Tu ( x)} holds for all x > 0 Defiitio 0 Four mappigs P Q S T o a PM-space ( X F ) will be a geeralized cotractio quadruplet ( P QST ; ) if ad oly oly if there exists a costat h (0) such that for every uv X FPu Qv( h x) mi{ FSu Tv ( x ) FPu Su ( x ) FTv Qv( x ) FSu Qv( x ) FPu Tv ( x )} holds for all x > 0 Defiitio A cotractio mappig o a metric space ( X d ) is a fuctio F : X X with the property there is a some real umber k < such that for all x y X d( f( x) f( y)) kd( x y) If k the above mappig is called o-expasive otherwise is called expasive mappig Propositio If S ad T are compatible self maps of a Meger space ( X Ft ) where t is cotiuous ad txx () xfor all x [0] ad Sx Tx u for some u i X The TSx Su provided S is cotiuous Lemma Let { p } be a sequece i a Meger space( X Ft ) with cotiuous t- orm ad txx () x Suppose for all x [0] there exists k (0) such that for all x > 0 ad N F ( kx) F ( x) = P P+ P P The { p } is Cauchy sequece i X Lemma Let ( X Ft ) be a Meger space If there exists k (0) such that for p q X FPq ( kx) FPq ( x) the p = q Propositio If ( ST ) is a semi-compatible pair of self maps i a Meger space ( X Ft ) ad T is cotiuous the ( ST ) is compatible

5 Commo fixed poit theorem for expasive maps 35 Proof Cosider a sequece { x } i X such that{ Sx } u ad { Tx } u As T is cotiuous we get TSx Tu By semi-compatibility of ( ST ) we have FSTx () T ε for all ε > 0 ie for ε > 0 ad λ > 0 there is a iteger M( ε λ ) such u that F ( ε/) λ ad F ( ε /) λ for all M( ε λ) STx T u TSx T Now FSTx TSx () ε > t{ FSTx T ( ε/) FTSx T ( ε/)} u u u > ( λ λ) > λ ie F () ε for all ε > 0 Hece the pair ( ST ) is compatible STx TSx Propositio 3 If self mappigs A ad S of a Meger space( X Ft ) are compatible the they are weakly compatible Proof Suppose Ap= Sp for some p i X Cosider the costat sequece { p } = p Now { Ap } Ap ad { Sp } Sp( = Ap) As A ad S are compatible we have FASp SAp() x = for all x > 0 Thus ASp = SAp ad we get ( A S) is weakly compatible Theorem [4] Let ( X Ft ) be a complete Meger space with tab () = mi() ab for every ab [ 0] If mappigs S T: X (i) Sx () = x; T( x) = x; F ( p) mi{ F ( p/ a) F ( p/ b) F ( p/ c)} (ii) Sx Ty x Sx y Ty x y for all x y X with x y where δ = mi{ abc } > X satisfy the followig coditios: 3 MAIN RESULTS Now we preset exteded result of Vasuki [4] for four self mappigs through cocept of semi-compatibility ad weak compatibility Theorem 3 Let A B S ad T be expasive maps o a complete Meger space( X Ft ) with tab () mi() ab ab 0 satisfyig = for all [ ]

6 35 R K Gujetiya V K Gupta M S Chauha ad O Sikhwal (i) TX ( ) AX ( ) ad SX ( ) BX ( ); (ii) Either A or S is cotiuous; (iii) ( A S ) is semi-compatible ad ( BT ) is weak compatible; (iv) There exists k (0) such that F ( k) mi{ F ( k/ a) F ( k/ b) F ( k/ c)} Ap Bq Sp Aq Tq Bq Sp Tq for all p q X with p q where δ = mi{ abc } > The A B S ad T have a uique commo fixed poit i X Proof Let x0 X from coditio (i) there exist x x such that T( x0) = A( x) = y0 ; S( x) = B( x) = y X Iductively we ca costruct sequeces { x } ad { y } i X such that Tx ( ) = Ax ( ) = y ; Sx ( + ) = Bx ( + ) = y + ; Now + F k F k 0 () = () Sx Ax Tx Bx Sx Tx y y Ax Bx mi{ F ( k/ a) F ( k/ b) F ( k/ c)} mi{ F ( k/ a) F ( k/ b) F ( k/ c)} y y y y y y 0 mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} as δ = mi{ abc } y y0 y y y y mi{ F ( k/ δ ) F ( k/ δ )} y y0 y y mi{ F ( k/ δ )} y y = F ( k/ δ ) y y ie F () k = F ( k/ δ) Similarly y0 y y y F () k F ( k/ δ) y y y y3 Now agai F () k = F () k y y+ Ax+ Bx + mi{ F ( k/ a) F ( k/ b) F ( k/ c)} Sx+ Ax+ Tx+ Bx+ Sx+ Ax+ mi{ F ( k/ a) F ( k/ b) F ( k/ c)} y+ y y+ y+ y+ y+ mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} y+ y y+ y+ y+ y+ as δ = mi{ abc }

7 Commo fixed poit theorem for expasive maps 353 The = mi{ F ( k/ δ ) F ( k/ δ )} y+ y y+ y+ F ( k/ δ ) mi{ F ( k/ δ) F ( k/ δ)} y y+ y+ y y+ y+ Hece F () k F ( k/ δ ) as δ > y y+ y+ y+ Now agai F () k = F () k = F y+ y+ Bx+ Ax + 3 Ax Bx () k F ( k) mi{ F ( k/ a) F ( k/ b) F ( k/ c)} y+ y+ Sx+ 3 Ax+ 3 Tx+ Bx+ Sx+ 3 Ax+ mi{ F ( k/ a) F ( k/ b) F ( k/ c)} y+ 3 y+ y+ y+ y+ 3 y+ mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} y+ 3 y+ y+ y+ y+ 3 y+ = mi{ F ( k/ δ ) F ( k/ δ )} y+ 3 y+ y+ y+ as δ = mi{ abc } Hece F () k F ( k/ δ ) as δ > y+ y+ y+ 3 y+ Hece for all we have F () k F ( k/ δ ) as δ > y y+ y+ y+ Now F () k F ( ck) as c = / δ δ > y y+ y+ y+ Or F ( ck) F ( k) y+ y+ y y+ It follows by lemma () that { y } is a Cauchy sequece i X Thus there exist some poit z X to which { y } coverges Now its subsequeces { Ax+ } z{ Bx+ } z { Sx } z{ Tx } z + + Case I S is cotiuous I this case we have SAx + Sz ad S x + Sz The semi-compatibility of ( A S ) gives SAx + Sz Step I By puttig p = Sx + q = x + i (iv) we have

8 354 R K Gujetiya V K Gupta M S Chauha ad O Sikhwal F ( k) mi{ F ( k/ a) F ( k/ b) F ( k/ c)} ASx+ Bx+ SSx+ ASx+ Tx+ Bx+ SSx+ Tx+ Lettig we have mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} Sz z Sz Sz z z Sz z SSx+ ASx + Tx+ Bx+ SSx+ Tx+ F ( k) mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} Thus FSz z () k mi{ FSz z( k/ δ )} FSz z ( k) FSz z ( k / δ ) Sice δ > therefore (0) δ Usig lemma () Sz = z Step II By puttig p = z q= x + i (iv) we have F ( k) mi{ F ( k/ a) F ( k/ b) F ( k/ c)} Az Bx+ Sz Az Tx+ Bx+ Sz Tx+ mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} Lettig we have Sz Az Tx+ Bx+ Sz Tx+ F ( k) mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} Az z Sz Az z z Sz z Hece Az mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} zaz zz zz as δ = mi{ abc } as δ = mi{ abc } mi{ FzAz ( k/ δ )} FzAz ( k/ δ ) ie FAz z () k Fz Az ( k/ δ ) as δ > = z Thus Sz = z = Az As T( X) A( X) the there exists w X such that Tw = Az for some z X Therefore z = Az = Sz = Tw Step III By puttig p = x + q= w i (iv) we have F ( k) mi{ F ( k/ a) F ( k/ b) F ( k/ c)} Ax+ Bw Sx+ Ax+ Tw Bw Sx+ Tw mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} Sx+ Ax+ Tw Bw Sx+ Tw

9 Commo fixed poit theorem for expasive maps 355 Lettig we have F ( k) mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} zbw zz TwBw ztw mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} zz zbw zz mi{ F ( k/ δ )} zbw ie FzBw () k FzBw ( k/ δ ) as δ > Thus z = Bw Hece Bw = Tw = z As ( B T ) is weak compatible we have TBw = BTw Bz = Tz Step IV By puttig p = z q= z i (iv) we have F ( k) mi{ F ( k/ a) F ( k/ b) F ( k/ c)} Az Bz Sz Az Tz Bz Sz Tz Az Bz z Bz mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} Sz Az Tz Bz Sz Tz mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} zz zbz zbz mi{ F ( k/ δ )} zbz F () k F ( k/ δ ) Thus z = Bz Therefore z = Bz = Tz Hece z = Az = Sz = Bz = Tz Therefore z is a commo fixed poit of A B S ad T Case II A is cotiuous I this case we have ASx + Az ad A x + Az ad the semi-compatibility of ( A S) gives ASx + Sz By uiqueess of limit i Meger space we get Az Step V By puttig p = z q= x + i (iv) we have Az Bx+ Sz Az Tx+ Bx+ Sz Tx+ = Sz F ( k) mi{ F ( k/ a) F ( k/ b) F ( k/ c)} mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} as δ = mi{ abc } Lettig we have Sz Az Tx+ Bx+ Sz Tx+

10 356 R K Gujetiya V K Gupta M S Chauha ad O Sikhwal F ( k) mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} Az z Sz Az z z Sz z mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} Az Az z z Az z mi{ F ( k/ δ )} Az z ie FAz z() k FAz z( k/ δ ) Thus z = Az Hece z = Az = Sz = Bz = Tz that is z is a commo fixed poit of A B S ad T Uiqueess Let u be aother commo fixed poit of A B S ad T the u = Au = Su = Bu = Tu So by puttig p = u ad q = z i (iv) we have F ( k) mi{ F ( k/ a) F ( k/ b) F ( k/ c)} Au Bz Su Au Tz Bz Su Tz mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} Su Au Tz Bz Su Tz F ( k) mi{ F ( k/ δ ) F ( k/ δ) F ( k/ δ)} mi{ F ( k/ δ )} uz uu zz uz uz Hece Fuz () k Fuz ( k/ δ )} k > 0 δ > u = z Therefore z is the uique commo fixed poit of A B S ad T CONCLUSION: I this paper we have preseted commo fixed poit theorems for four expasive mappigs i meger spaces through cocept of semi ad weak compatibility ACKNOWLEDEGMENT: The Authors are thakful to the aoymous referees for his valuable suggestios for the improvemet of this paper REFERENCES [] Baach S Surles operatios das les esembles abstraits et leur applicatios aux equatios itegrals FudMath3 (9) 33-8 [] Bharucha Reid AT ad Sehgal V M fixed poits of cotradictio maps o probabilistic metric space MathSystemTheory 6 (97) 97-0

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12 358 R K Gujetiya V K Gupta M S Chauha ad O Sikhwal [] Sigh SL ad Kasahara S O some recet result o commo fixed poits Idia J Pure Appl Math 3 (98) [3] Taiguchi T Commo fixed poits theorems of expasive type maps o complete metric spacees Maths Japa 34C (989) [4] Vasuki R Fixed poit ad Commo fixed poit theorems for expasive maps i meger spaces BullCal Math Soc 83 (99) Received: Jue 00