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Fixed point result in probabilistic metric space
 

Fixed point result in probabilistic metric space

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    Fixed point result in probabilistic metric space Fixed point result in probabilistic metric space Document Transcript

    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications259Fixed Point Result in Probabilistic Metric SpaceRuchi Singh, Smriti Mehta1and A D Singh21Department of Mathematics, Truba Instt. Of Engg. & I.T. Bhopal2Govt. M.V.M. College BhopalEmail : smriti.mehta@yahoo.comABSTRACTIn this paper we prove common fixed point theorem for four mapping with weak compatibility in probabilisticmetric space.Keywords: Menger space, Weak compatible mapping, Semi-compatible mapping, Weakly commuting mapping,common fixed point.AMS Subject Classification: 47H10, 54H25.1. INTRODUCTION:Fixed point theory in probabilistic metric spaces can be considered as a part of Probabilistic Analysis,which is a very dynamic area of mathematical research. The notion of probabilistic metric space is introduced byMenger in 1942 [9] and the first result about the existence of a fixed point of a mapping which is defined on aMenger space is obtained by Sehgel and Barucha-Reid.Recently, a number of fixed point theorems for single valued and multivalued mappings in mengerprobabilistic metric space have been considered by many authors [1],[2],[3],[4],[5],[6]. In 1998, Jungck [7]introduced the concept weakly compatible maps and proved many theorems in metric space. In this paper weprove common fixed point theorem for four mapping with weak compatibility and rational contraction withoutappeal to continuity in probabilistic metric space. Also we illustrate example in support of our theorem.2. PRELIMINARIES:Now we begin with some definitionDefinition 2.1: Let R denote the set of reals and the non-negative reals. A mapping : → is called adistribution function if it is non decreasing left continuous with inf ( ) 0 sup ( ) 1t R t RF t and F t∈ ∈= =Definition 2.2: A probabilistic metric space is an ordered pair ( , ) where X is a nonempty set, L be set of alldistribution function and : × → . We shall denote the distribution function by ( , ) or , ; , ∈and , ( ) will represents the value of ( , ) at ∈ . The function ( , ) is assumed to satisfy thefollowing conditions:1. , ( ) = 1 > 0 ! =2. , (0) = 0 #$# ! , ∈3. , = , #$# ! , ∈4. , ( ) = 1 ,(!) = 1 (ℎ# ,( + !) = 1 #$# ! , , ∈ .In metric space ( , ) , the metric d induces a mapping : × → such that, ( ) = , = + ( – ( , )) for every , ∈ and ∈ , where H is the distribution function defined as+( ) = -0, if x ≤ 01, if x > 02Definition 2.3: A mapping ∗: [0, 1] [0, 1] → [0, 1] is called t-norm if1. ( ∗ 1) = ∀ ∈ [0,1]2. (0 ∗ 0) = 0, ∀ , 7 ∈ [0,1]3. ( ∗ 7) = (7 ∗ ),4. (8 ∗ ) ≥ ( ∗ 7) 8 ≥ , ≥ 7, and5. ( ( ∗ 7) ∗ 8 ) = ( ∗ (7 ∗ 8 ))Example: (i) ( ∗ 7) = 7, (ii) ( ∗ 7) = : ( , 7)(iii) ( ∗ 7) = : ( + 7 − 1; 0)Definition 2.4: A Menger space is a triplet ( , ,∗) where ( , )a PM-space and ∆ is is a t-norm with thefollowing condition<,=( + !) ≥ <,>( ) ∗ >,=(!)The above inequality is called Menger’s triangle inequality.EXAMPLE: Let = , ( ∗ 7) = : ( , 7) , 7 ∈ (0,1) and
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications260<,>( ) = -+( ) ? ≠ $1 ? = $2where +( ) = A0 ≤ 00 ≤ ≤ 11 ≥ 12Then ( , ,∗ ) is a Menger space.Definition 2.5: Let ( , ,∗) be a Menger space. If ? ∈ , B > 0, C ∈ (0, 1), then an (B, C) neighbourhood of u,denoted by D< (B, C) is defined asD<(B, C) = E$ ∈ ; <,>(B) > 1 − CF.If ( , ,∗) be a Menger space with the continuous t-norm t, then the familyD<(B, C); ? ∈ ; B > 0, C ∈ (0,1) ofneighbourhood induces a hausdorff topology on X and if supJKL(a ∗ a) = 1, it is metrizable.Definition 2.6: A sequence N OP in ( , ,∗) is said to be convergent to a point ∈ if for every B > 0 andC > 0, there exists an integer Q = Q(B, C) such that O ∈ D (B, C) for all ≥ Q or equivalently RS,R(T) >1 − C for all ≥ Q.Definition 2.7: A sequence N OP in ( , ,∗) is said to be Cauchy sequence if for every B > 0 and C > 0, thereexists an integer Q = Q(B, C) such that S, U(T) > 1 − C for all , : ≥ Q.Definition 2.8: A Menger space ( , ,∗) with the continuous t-norm ∆ is said to be complete if every Cauchysequence in X converges to a point in X.Definition 2.9: A coincidence point (or simply coincidence) of two mappings is a point in their domain havingthe same image point under both mappings.Formally, given two mappings , V ∶ → X we say that a point x in X is a coincidence point of f and g if( ) = V( ).Definition 2.10: Let ( , ,∗) be a Menger space. Two mappings , V ∶ → are said to be weakly compatibleif they commute at the coincidence point, i.e., the pair N , VP is weakly compatible pair if and only if = Vimplies that V = V .Example: Define the pair Y, Z: [0, 3] → [0, 3] byY( ) = -, ∈ [0, 1)3, ∈ [1, 3]2 , Z( ) = -3 − , ∈ [0, 1)3, ∈ [1, 3].2Then for any ∈ [1, 3], YZ = ZY , showing that A, S are weakly compatible maps on [0, 3].Definition 2.11: Let ( , ,∗) be a Menger space. Two mappings Y, Z ∶ → are said to be semi compatible if[RS,R(() → 1for all ( > 0 whenever N OP is a sequence in such that Y O, Z O → for some p in as → ∞.It follows that (Y, Z) is semi compatible and Y! = Z! imply YZ! = ZY! by taking N OP = ! = Y! = Z!.Lemma 2.12[15]: Let N OP be a sequence in Menger space ( , ,∗) where ∗ is continuous and ( ∗ ) ≥ forall ∈ [0, 1]. If there exists a constant ] ∈ (0, 1) such that > 0 and ∈ Q S, S^_(] ) ≥ S`_, S( ), thenN OP is a Cauchy sequence.Lemma 2.13[13]: If ( , ) is a metric space, then the metric d induces a mapping : × → , defined by( , ) = + ( – ( , )) , , ∈ ∈ . Further more if ∗: [0,1] × [0,1] → [0,1] is defined by( ∗ 7) = : ( , 7), then ( , ,∗) is a Menger space. It is complete if ( , ) is complete. The space ( , ,∗)so obtained is called the induced Menger space.Lemma 2.14[10]: Let ( , ,∗) be a Menger space. If there exists a constant ] ∈ (0, 1) such that R,a(]() ≥R,a((), for all , ! ∈ and ( > 0 then = ! .3. MAIN RESULT:Theorem 3.1: Let ( , ,∗) be a complete Menger space where ∗ is continuous and (( ∗ () ≥ ( for all ( ∈ [0,1].Let A, B, T and S be mappings from X into itself such that3.1.1. Y( ) ⊂ Z( ) c( ) ⊂ d( )3.1.2. Z d are continuous3.1.3. The pair (Z, Y) and (d, c) are Semi compatible3.1.4. There exists a number ] ∈ (0,1) such that[R,ea(]() ≥ R,fa(() ∗ R,[R(() ∗ [R,fa(() ∗ fa,ea(() ∗ R,eag(2 − h)(i, ! ∈ , h ∈ (0,2) ( > 0.Then, Y, c, Z and d have a unique common fixed point in X.Proof: Since Y( ) ⊂ Z( ) for any j ∈ there exists a point L ∈ such thatY j = Z L. Since c( ) ⊂d( ) for this point L we can choose a point k ∈ such that d L = c k.Inductively we can find a sequence N!OP as follows!kO = Y kO = Z kO L!kO L = c kO L = d kO k
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications261For = 0, 1, 2, 3 … … …. by (3.1.4.), for all ( > 0 h = 1 − with ∈ (0,1), we haveamS,amS^_(]() = [RmS ,eRmS^_(]()≥ RmS,fRmS^_(() ∗ RmS^_,[RmS^_(() ∗ [RmS,fRmS^_(() ∗ fRmS^_,eRmS^_(() ∗ RmS,eRmS^_g(1 + )(i= amS`_,amS(() ∗ amS,amS^_(() ∗ amS,amS^_(() ∗amS,amS^_(() ∗ amS`_,amS^_g(1 + )(i≥ amS`_,amS(() ∗ amS,amS^_(() ∗ amS`_,amS(() ∗ amS,amS^_( ()= amS`_,amS(() ∗ amS,amS^_(() ∗ amS,amS^_( ()Since t-norm is continuous, letting → 1, we haveamS,amS^_(]() ≥ amS`_,amS(() ∗ amS,amS^_(()SimilarlyamS^_,amS^m(]() ≥ amS,amS^_(() ∗ amS^_,amS^m(()SimilarlyamS^m,amS^n(]() ≥ amS^_,amS^m(() ∗ amS^m,amS^n(()ThereforeaS,aS^_(]() ≥ aS`_,aS(() ∗ aS,aS^_(() ∈ oConsequentlyaS,aS^_(() ≥ aS`_,aS(]pL() ∗, aS,aS^_(]pL() ∈ oRepeated application of this inequality will imply thataS,aS^_(() ≥ aS`_,aS(]pL() ∗ aS,aS^_(]pL() ≥ ⋯ … … … . ≥ aS`_,aS(]pL() ∗ aS,aS^_(]pr(), ∈ oSince aS,aS^_(]pr() → 1 s → ∞, it follows thataS,aS^_(() ≥ aS`_,aS(]pL() ∈ oConsequentlyaS,aS^_(]() ≥ aS`_,aS(() for all ∈ oTherefore by Lemma [2.12], N!OP is a Cauchy sequence in X. Since X is complete,N!OPconverges to a point t ∈. Since NY kOP, Nc kO L P, NZ kO L P Nd kO k P are subsequences of N!OP , they also converge to the pointz,. #. as → ∞, Y kO, c kO L , Z kO L d kO k → t.Case I: Since S is continuous. In this case we haveZY O → Zt, ZZ O → ZtAlso (Y, Z)is semi-compatible, we have YZ O → ZtStep I: Let = Z O, ! = O u (ℎ h = 1 in (3.1.4) we get[RS,eRS(]() ≥ RS,fRS(() ∗ RS,[RS(() ∗ [RS,fRS(() ∗ fRS,eRS(() ∗ RS,eRS(()v,v(]() ≥ v,v(() ∗ v,v(() ∗ v,v(() ∗ v,v(() ∗ v,v(()v,v(]() ≥ v,v(()So we get Zt = t.Step II: By putting = t, ! = O u (ℎ h = 1 in (3.1.4) we get[v,eRS(]() ≥ v,fRS(() ∗ v,[v(() ∗ [v,fRS(() ∗ fRS,eRS(() ∗ v,eRS(()[v,v(]() ≥ v,v(() ∗ v,[v(() ∗ [v,v(() ∗ v,v(() ∗ v,v(()[v,v(]() ≥ [v,v(()So we get Yt = t.Case II: Since T is continuous. In this case we havedc O → dt, dd O → dtAlso (c, d)is semi-compatible, we have cd O → dtStep I: Let = O, ! = d O u (ℎ h = 1 in (3.1.4) we get[RS,efRS(]() ≥ RS,ffRS(() ∗ RS,[RS(() ∗ [RS,ffRS(() ∗ ffRS,efRS(() ∗ RS,efRS(()v,fv(]() ≥ v,fv(() ∗ v,v(() ∗ v,fv(() ∗ fv,fv(() ∗ v,fv(()v,fv(]() ≥ v,fv(()So we get dt = t.Step II: By putting = O, ! = t u (ℎ h = 1 in (3.1.4) we get[RS,ev(]() ≥ RS,fv(() ∗ RS,[RS(() ∗ [RS,fv(() ∗ fv,ev(() ∗ RS,ev(()v,ev(]() ≥ v,v(() ∗ v,v(() ∗ v,v(() ∗ v,ev(() ∗ v,ev(()ev,v(]() ≥ ev,v(()So we get ct = t.Thus, we have Yt = Zt = dt = ct = t.
    • Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications262That is z is a common fixed point of Z, d, Y and c.For uniqueness, let u (u ≠ t) be another common fixed point of Z, d, Y and c .Then Yu = Zu == cu =du = u .Put = t, ! = u and w = 1, in (3.1.4.), we get[v,e=(]() ≥ v,f=(() ∗ v,[=(() ∗ [v,f=(() ∗ f=,e=(() ∗ v,e=(()v,=(]() ≥ v,=(() ∗ v,=(() ∗ v,=(() ∗ =,=(() ∗ v,=(()v,=(]() ≥ v,=(() ∗ v,=(() ∗ v,=(() ∗ 1 ∗ v,=(()v,=(]() ≥ v,=(()Thus we havet = u. Therefore z is a unique fixed point of A,S, B and T.This completes the proof of the theorem.COROLLARY 3.2: Let ( , ,∗) be a complete Menger space where ∗ is continuous and (( ∗ () ≥ ( forall ( ∈ [0,1]. Let A, and S be mappings from X into itself such that3.2.1. Y( ) ⊂ Z( )3.2.2. Z is continuous3.2.3. The pair (Z, Y) is semi compatible3.2.4. There exists a number ] ∈ (0,1) such that[R,a(]() ≥ R,a(() ∗ R,[R(() ∗ [R,a(() ∗ a,[a(() ∗ R,[ag(2 − h)(i, ! ∈ , h ∈ (0,2) ( > 0.Then, Y, and Z have a unique common fixed point in X.4. BIBLIOGRAPHY:[1] A.T.Bharucha Ried, Fixed point theorems in Probabilistic analysis, Bull. Amer.Math. Soc, 82 (1976),611-617[2] Gh.Boscan, On some fixed pont theorems in Probabilistic metric spaces, Math.balkanica, 4 (1974), 67-70[3] S. Chang, Fixed points theorems of mappings on Probabilistic metric spaces with applications, ScientiaSinica SeriesA, 25 (1983), 114-115[4] R. Dedeic and N. Sarapa, Fixed point theorems for sequence of mappings on Menger spaces, Math.Japonica, 34 (4) (1988), 535-539[5] O.Hadzic, On the (ε, λ)-topology of LPC-Spaces, Glasnik Mat; 13(33) (1978), 293-297.[6] O.Hadzic, Some theorems on the fixed points in probabilistic metric and random normed spaces, Boll.Un. Mat. Ital; 13(5) 18 (1981), 1-11[7] G.Jungck and B.E. Rhodes, Fixed point for set valued functions without continuity, Indian J. Pure.Appl. Math., 29(3) (1998), 977-983[8] G.Jungck, Compatible mappings and common fixed points, Internat J. Math. and Math. Sci. 9 (1986),771-779[9] K. Menger, Statistical Matrices, Procedings of the National academy of sciences of the United states ofAmerica 28 (1942), 535-537[10] S. N. Mishra, Common fixed points of compatible mappings in PM-Spaces, Math. Japonica, 36(2)(1991), 283-289[11] B.Schweizer and A.Sklar, Probabilistic Metric spaces, Elsevier, North-Holland, New York, 1983.[12] B.Schweizer and A.Sklar, Statistical metrices spaces, pacific Journal of Mathematics 10(1960), 313-334[13] V.M. Sehgal, A.T. Bharucha-Reid, Fixed points of contraction mappings in PM spaces, Math. SystemTheory 6 (1972) 97-102.[14] S. Sessa, On weak commutativity conditions of mapping in fixed point consideration, Publ. Inst. Math.Beograd, 32(46) (1982), 149-153[15] S.L.Singh and B.D. Pant, Common fixed point theorems in Probabilistic metric spaces and extention touniform spaces, Honam Math. J., 6 (1984), 1-12[16] D.Xieping, A common fixed point theorem of commuting mappings in probabilistic metric spaces,Kexeue Tongbao, 29 (1984), 147-150
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