Fixed Point Result in Probabilistic Metric Space

Transcription

1 Fixed Point Result in Probabilistic Metric Space Ruchi Singh, Smriti Mehta 1 and A D Singh 2 1 Department of Mathematics, Truba Instt. Of Engg. & I.T. Bhopal 2 Govt. M.V.M. College Bhopal smriti.mehta@yahoo.com ABSTRACT In this paper we prove common fixed point theorem for four mapping with weak compatibility in probabilistic metric space. Keywords: Menger space, Weak compatible mapping, Semi-compatible mapping, Weakly commuting mapping, common fixed point. AMS Subject Classification: 47H10, 54H 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 by Menger in 1942 [9] and the first result about the existence of a fixed point of a mapping which is defined on a Menger space is obtained by Sehgel and Barucha-Reid. Recently, a number of fixed point theorems for single valued and multivalued mappings in menger probabilistic 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 we prove common fixed point theorem for four mapping with weak compatibility and rational contraction without appeal to continuity in probabilistic metric space. Also we illustrate example in support of our theorem. 2. PRELIMINARIES: Now we begin with some definition Definition 2.1: Let R denote the set of reals and the non-negative reals. A mapping : is called a distribution function if it is non decreasing left continuous with inf F( t) = 0 and sup F( t) = 1 Definition 2.2: A probabilistic metric space is an ordered pair (,) where X is a nonempty set, L be set of all distribution function and :. We shall denote the distribution function by (,) or, ;, and, () will represents the value of (,) at. The function (,) is assumed to satisfy the following 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 0 +()= - 1,if x>0 2 t R Definition 2.3: A mapping : [0,1] [0,1] [0,1] is called t-norm if 1. ( 1)= [0,1] 2. (0 0)= 0,,7 [0,1] 3. ( 7) = (7 ), 4. (8 ) ( 7) 8, 7, and 5. (( 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 the following condition <,= (+!) <,> () >,= (!) The above inequality is called Menger s triangle inequality. EXAMPLE: Let =, ( 7)= :(,7),7 (0,1) and t R 259

2 2 Mathematical Theory and Modeling +()? $ <,> ()= - 1? =$ where +()= A Then (,, ) 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 as D < (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) of neighbourhood induces a hausdorff topology on X and if sup JKL (a a)=1, it is metrizable. Definition 2.6: A sequence N O P in (,, ) is said to be convergent to a point if for every B > 0 and C > 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 O P in (,, ) is said to be Cauchy sequence if for every B > 0 and C > 0, there exists 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 Cauchy sequence 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 having the 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 compatible if they commute at the coincidence point, i.e., the pair N,VP is weakly compatible pair if and only if = V implies that V = V. Example: Define the pair Y,Z:[0,3] [0,3] by Y()=-, [0,1) 3, [1,3] 2 3, [0,1), Z()=- 3, [1,3]. 2 Then 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 O P 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 O P=! =Y! =Z!. Lemma 2.12[15]: Let N O P be a sequence in Menger space (,, ) where is continuous and ( ) for all [0,1]. If there exists a constant ] (0,1) such that > 0 and Q S, S^_ (]) S`_, S (), then N O P 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 that Y() Z() c() d() Z d are continuous The pair (Z,Y) and (d,c) are Semi compatible There exists a number ] (0,1) such that [R,ea (]() \R,fa \R,[R [R,fa fa,ea \R,ea g(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! O P as follows! ko =Y ko =Z kol! kol =c kol =d kok 260

3 For = 0,1,2,3. by (3.1.4.), for all ( >0 h =1 with (0,1), we have (]() = [RmS,eR ms^_ (]() \RmS,fR ms^_ \RmS^_,[R ms^_ [RmS,fR ms^_ frms^_,er ms^_ \RmS,eR ms^_ g(1+)(i = ams`_,a ms ams`_,a ms^_ g(1+)(i ams`_,a ms ams`_,a ms = ams`_,a ms Since t-norm is continuous, letting 1, we have (]() ams`_,a ms Similarly ams^_,a ms^m (]() ams^_,a ms^m Similarly ams^m,a ms^n (]() ams^_,a ms^m ams^m,a ms^n Therefore (]() as`_,a S o Consequently as`_,a S (] pl (), (] pl () o Repeated application of this inequality will imply that as`_,a S (] pl () (] pl (). as`_,a S (] pl () (] pr (), o Since (] pr () 1 s, it follows that as`_,a S (] pl () o Consequently (]() as`_,a S for all o Therefore by Lemma [2.12], N! O P is a Cauchy sequence in X. Since X is complete,n! O Pconverges to a point t. Since NY ko P,Nc kol P,NZ kol P Nd kok P are subsequences of N! O P, they also converge to the point z,.#. as, Y ko,c kol,z kol d kok t. Case I: Since S is continuous. In this case we have ZY O Zt, ZZ O Zt Also (Y,Z)is semi-compatible, we have YZ O Zt Step I: Let =Z O,! = O u(h h =1 in (3.1.4) we get [\RS,eR S (]() \\RS,fR S \\RS,[\R S [\RS,fR S frs,er S \\RS,eR S \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,er S \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 have dc O dt, dd O dt Also (c,d)is semi-compatible, we have cd O dt Step I: Let = O,! =d O u(h h =1 in (3.1.4) we get [RS,efR S (]() \RS,ffR S \RS,[R S [RS,ffR S ffrs,efr S \RS,efR S 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,[R S [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. 261

4 That 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 (( () ( for all ( [0,1]. Let A, and S be mappings from X into itself such that Y() Z() Z is continuous The pair (Z,Y) is semi compatible There exists a number ] (0,1) such that [R,\a (]() \R,\a \R,[R [R,\a \a,[a \R,[a g(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), [2] Gh.Boscan, On some fixed pont theorems in Probabilistic metric spaces, Math.balkanica, 4 (1974), [3] S. Chang, Fixed points theorems of mappings on Probabilistic metric spaces with applications, Scientia Sinica SeriesA, 25 (1983), [4] R. Dedeic and N. Sarapa, Fixed point theorems for sequence of mappings on Menger spaces, Math. Japonica, 34 (4) (1988), [5] O.Hadzic, On the (ε, λ)-topology of LPC-Spaces, Glasnik Mat; 13(33) (1978), [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), [8] G.Jungck, Compatible mappings and common fixed points, Internat J. Math. and Math. Sci. 9 (1986), [9] K. Menger, Statistical Matrices, Procedings of the National academy of sciences of the United states of America 28 (1942), [10] S. N. Mishra, Common fixed points of compatible mappings in PM-Spaces, Math. Japonica, 36(2) (1991), [11] B.Schweizer and A.Sklar, Probabilistic Metric spaces, Elsevier, North-Holland, New York, [12] B.Schweizer and A.Sklar, Statistical metrices spaces, pacific Journal of Mathematics 10(1960), [13] V.M. Sehgal, A.T. Bharucha-Reid, Fixed points of contraction mappings in PM spaces, Math. System Theory 6 (1972) [14] S. Sessa, On weak commutativity conditions of mapping in fixed point consideration, Publ. Inst. Math. Beograd, 32(46) (1982), [15] S.L.Singh and B.D. Pant, Common fixed point theorems in Probabilistic metric spaces and extention to uniform 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),

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