This paper’s main objective is to define Menger space (PQM) and the concept of weakly compatible by using the notion of property (EA) &amp; JSR maps to define new property to prove a common fixed point theorem for 4 self maps in Menger space (PQM). Key Words: Fixed Point, Probabilistic Metric Space, Menger space, JSR mappings, property EA Subject classification: 47H10, 54H2

Dhagat, Vanita Ben

Mehta, Smriti

Singh, A.D.

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications  313 Fixed Point Result in Menger Space with EA Property  Smriti Mehta, A.D.Singh* and Vanita Ben Dhagat Department of Mathematics, Truba Institute of Engineering & I.T. Bhopal *Government M V M College Bhopal Email: smriti.mehta@yahoo.com  ABSTRACT This paper’s main objective is to define Menger space (PQM) and the concept of weakly compatible by using the notion of property (EA) & JSR maps to define new property to prove a common fixed point theorem for 4 self maps in Menger space (PQM). Key Words: Fixed Point, Probabilistic Metric Space, Menger space, JSR mappings, property EA  Subject classification: 47H10, 54H25                       1. INTRODUCTION  The notion of probabilistic metric space is introduced by Menger in 1942 [10] 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.    A number of fixed point theorems for single valued and multivalued mappings in menger probabilistic metric space have been considered by many authors [2],[3],[4],[5],[6],[7]. In 1998, Jungck [8] introduced the concept weakly compatible maps and proved many theorems in metric space. Hybrid fixed point theory for nonlinear single  valued and multivalued maps is a new development in the domain of contraction type multivalued theory ([4], [7], [11], [12], [13], [14] ).Jungck and Rhoades [8] introduced the weak compatibility to the setting of single valued and multivalued maps. Singh and Mishra introduced (IT)-commutativity for hybrid pair of single valued and multivalued maps which need not be weakly compatible. Recently, Aamri and El Moutawakil [1] defined a property (EA) for self maps which contained the class of noncompatible maps. More recently, Kamran [9] extended the property (EA) for a hybrid pair of single valued and multivalued maps and generalized the (IT) commutativity for such pair. The aim of this paper is to define a new property which contains the property (EA) for hybrid pair of single valued and multivalued maps and give some common fixed point theorems under hybrid contractive conditions in probabilistic space.  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 ( ) 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 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 p, q ∈  X and x ∈  R, where H is the distribution function defined as      +() =  20,   ≤ 01,   > 0 4                                                     Definition 2.3: A mapping 5:  [0, 1]  [0, 1]  →  [0, 1] is called t-norm if  1. 5 (, 1) =   ∀  ∈ [0,1] 2.  5 (0, 0)  =  0, 3. 5 (, 9)  =  5 (9, ), Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications  314 4. 5 (:,  )  ≥  5 (, 9)  : ≥  ,   ≥  9, and  5. 5 (5 (, 9), : )  =  5 ( , 5(9, : ))     Example: (i) 5 (, 9)  =  9,                (ii) 5 (, 9)  =  < (, 9)  (iii) 5 (, 9) = < ( + 9 − 1; 0)  Definition 2.4: A Menger space is a triplet (, , 5) where (, )a PM-space and ∆ is is a t-norm with the following condition >,?( + !) ≥ ∆( >,A(), A,?(!) The above inequality is called Menger’s triangle inequality. EXAMPLE: Let  =  , 5 (, 9) =  <(, 9)  , 9 ∈  (0,1) and     >,A() =  2   +()   B ≠ $1        B = $ 4 where   +() =  D 0       ≤ 0      0 ≤  ≤ 11          ≥ 14 Then (, , 5 ) is a  Menger space.  Definition 2.5: Let (, , ∆)  be a Menger space. If B ∈  , E >  0, F ∈  (0, 1),  then an (E, F) neighbourhood of u, denoted by G> (E, F) is defined as         G>(E, F) = H$ ∈ ; >,A(E) > 1 − FI.       If (, , 5) be a Menger space with the continuous t-norm t, then the familyG>(E, F); B ∈ ; E > 0, F ∈(0,1) of neighbourhood induces a hausdorff topology on X and if   JBKLM 5(, ) = 1, it is metrizable.                           Definition 2.6: A sequence NOP in (, , ∆) is said to be convergent to a point  ∈   if for every E >  0 and λ >  0, there exists an integer Q =  Q(E, F) such that O ∈ G(E, F) for all  ≥  Q or equivalently   RS,R(T) > 1 − F for all  ≥  Q.                 Definition 2.7: A sequence NOP   in (, , ∆)  is said to be Cauchy sequence if for every E > 0 and  F >  0, there exists an integer Q =  Q(E, F) such that S,U(T) > 1 − F for all  , < ≥  Q.                   Definition 2.8: A Menger space (, , 5) with the continuous t-norm ∆ is said to be complete if every Cauchy sequence in X converges to a point in X.                 Lemma 2.9 [14]: Let NOP  be a sequence in Menger space  (, , 5)  where ∆ is continuous and ∆(, )  ≥   for all  ∈ [0, 1]. If there exists a constant  V ∈  (0, 1) such that  >  0 and  ∈ Q   S,SWX(V) ≥SYX,S(), then NOP is a Cauchy sequence. Definition 2.10: Let J:  →    Z ∶  → \]() be mappings in Menger space (X, F, ∆) then,  (1)    s is said to be T weakly commuting at   ∈   JJ ∈ ZJ.           (2)   s and T are weakly compatible if they commute at their coincidence points,        i.e. if JZ = ZJ ^ℎ##$# J ∈ Z.  (3)     s and T are (IT) commuting at  ∈   JZ ⊂ ZJ ^ℎ##$# J ∈ Z.           Definition 2.11: Let (X, F, ∆) be a Menger space. Maps , `:  →  are said to satisfy the property (EA) if there exists a sequence NOP  in x such that limO →∞ O = limO →∞ `O =  d ∈ . Definition 2.12: -Maps :  →    Z ∶  → \]() are said to satisfy the property (EA) if there exists a sequence NOP   in X, some z in X and A in CB(X) such that         limO →∞ O = d ∈ e = limO →∞ ZO .                            Definition 2.13: Let , `, f, g:  →    be mappings in Menger space. The pair (f, S) and (g, G) are said to satisfy the common property (EA) if there exist two sequences NOP, N!OP in X and some z in X such that limO →∞ g!O = limO →∞ fO = limO →∞ O = limO →∞ `!O =  d .  Definition 2.14:  Let , `:  →    f, g ∶  → \]() be mappings on Menger space. The maps pair (f, S) and (g, G) are said to satisfy the common property (EA) if there exist two sequences NOP, N!OP  in X and some z in X, and A, B in CB(X) such that limO →∞ fO =  e   limO →∞ g!O =  ], limO →∞ O =  limO →∞ `!O =  d ∈ e ∩ ]. Definition 2.15:- Let (X, F, ∆) be a Menger space. Let f and g be two self maps of a Menger space. The Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications  315 pair {f, g} is said to be f-JSR mappings iff   i(`O , `O; ) ≥  i(O , O; ) where µ = lim Sup or lim inf and NOP is a sequence in X such that  limO →∞ O = limO →∞ `O =  d  J<# d ∈      ∆(, ) > . Example Let X = [0, 1] with    and f, g are two self mapping on X defined by Now the sequence  in X is defined as then we have   Clearly we have . Thus pair {f, g} is f-JSR mapping. But this pair is neither compatible nor weakly compatible or other non commuting mapping S. Hence pair of JSR mapping is more general then others. Let :  →  self map of a Menger space (X, F, ∆) and f:  → \]() be multivalued map. The pair {f, S} is said to be hybrid S-JSR mappings for all ∆ (p, p) > p  if and only if  i(fO , O; ) ≥  i(ffO , fO; ) where µ = lim Sup or lim inf and NOP is a sequence in X such that  limO →∞ O =  d ∈ e = limO →∞ fO . Let j:  →  be continuous and satisfying the conditions  (i)  j is nonincreasing on R, (ii) j(() > (,  for each ( ∈ (0, ∞).      3. MAIN RESULTS Theorem 3.1:  Let (, , 5) be a Menger space. Let , `:  →    f, g ∶  → \]() such that  (3.1.1)    (, f)   (`, g)  satisfy the common property (EA),   (3.1.2)   ()   `()  are closed, (3.1.3)   Pair (, f) is f − kf maps and pair (`, g)  is g − kf maps, (3.1.4) lR,mn(V) ≥ j[minHpR,qn  (), pR,lR(), qn,mn (), pR,mn (), lR,qn()I] Then f, g, S and G have a common fixed point in X. Proof:  By (3.1.1) there exist two sequences NOP   N!OP  in X and B ∈ , A, B in CB(X) such that limO →∞ fO =  e   limO →∞ g!O =  ],   limO →∞ O =  limO →∞ `!O = B ∈ e ∩ ]. Since () and `() are closed, we have B = $   B = `  for some $,  ∈ . Now by (3.1.4) we get  lRS,m'(V) ≥ j rmin s pRS,q'  (), pRS ,lRS(),q',m'  (), pRS,m'  (), lRS,q'()tu        On taking limit → ∞, we obtain v,m'(V) ≥ jwminHpA,q'  (), pA,v(), q',m' (), pA,m' (), v,q'()Ix ≥ j q',m'() > q',m' () Since ` = $ ∈ e  and q',m'  () ≥ v,m'  () > q',m' ().      Hence ` ∈ g Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications  316 Similarly lA,mnS(V) ≥ j[minHpA,qnS  (), pA,lA(), qnS,mnS  (), pA,mnS  (), lA,qnS()I] lA,y(V) ≥ j[minHpA,q'  (), pA,lA(), q',y (), pA,y (), lA,q'()I] ≥ jpA,lA() > pA,lA() Since $ = ` ∈ ]  and  pA,lA() ≥ y,lA() > pA,lA(), We get $ ∈ f$. Now as pair (f, S) is an S-JSR map therefore  ∈ f  and similarly as pair (g, G) is G-JSR maps therefore `B ∈ g                                    pRS ,q> () ≥ lRS,m>(V)                                  ≥ j[minHpRS,q> (), pRS ,lRS(), q>,m> (), pRS,m> (), lRS ,q>()I] On taking limit  → ∞,we obtain  >,q>() ≥ jwminH>,q> (), >,v(), q>,m> (), >,m> (), v,q>()Ix                                        ≥ j rmin s >,q> (), >,v(), q>,m> (),>,m> (), v,>(/2), >,q> (/2)tu                By triangular inequality and as B ∈ e ∩ ], we obtain >,q>() ≥ >,q>() ⟹ `B = B. Again                                    p>,qRS  () ≥ l>,mRS(V)                                  ≥ j[minHp>,qRS  (), p>,l>(), qRS,mRS  (), p>,mRS  (), l>,qRS()I] On taking limit  → ∞,we obtain       p>,>() ≥ jwminHp>,> (), p>,l>(), >,m> (), p>,y (), l>,>()Ix ≥ j rmin s p>,> (), p>,l>(), >,m> (),p>,> (/2), >,y  (/2), l>,>()tu By triangular inequality and as B ∈ e ∩ ], we obtain p>,> () ≥ p>,> () ⟹ B = B. Hence B = B ∈ fB   B = `B ∈ fB.  Example: Let X = [1,∞) with usual metric. Define f:  →  J f = |R}  and Z: \]() →  J Z =[1,2 + ]. Consider the sequence NOP = ~3 + MO. Then all conditions are satisfies of the theorem and hence 3 is the common fixed point.  Theorem 3.2:  Let (, , 5) be a Menger space. Let , `:  →    f , g ∶  → \]() such that  (3.2.1)    (, f)  (`, g) satisfy the common property (EA),   (3.2.2)   ()   `() are closed, (3.2.3)   Pair (, f) is f − kf maps and pair (`, g)  is g − kf maps, (3.2.4)   lR,mn(V) ≥ j[min ~pR,qn  (), pR,lR(), qn,mn (), pR,mn (), lR,qn()] Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications  317 Then , `, f  g have a common fixed point in X. Proof: Same as theorem 3.1 for each sequence f  g.  4. REFERENCES [1]       Amri M. and Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270(2002), no. 1, 181-188.  [2]  A.T.Bharucha Ried, Fixed point theorems in Probabilistic analysis, Bull. Amer.Math. Soc, 82 (1976), 611-617 [3]  Gh.Boscan, On some fixed point theorems in Probabilistic metric spaces,    Math.balkanica, 4 (1974), 67-70 [4] S. Chang, Fixed points theorems of mappings on Probabilistic metric spaces with applications, Scientia Sinica SeriesA, 25 (1983), 114-115 [5]  R. Dedeic and N. Sarapa, Fixed point theorems for sequence of mappings on Menger spaces, Math. Japonica, 34 (4) (1988), 535-539 [6]  O.Hadzic, On the (ε, λ)-topology of LPC-Spaces, Glasnik Mat; 13(33) (1978), 293-297. [7]  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 [8]  G.Jungck and B.E. Rhodes, Fixed point for set valued functions without continuity, Indian J. Pure. Appl. Math., 29(3) (1998), 977-983 [9]       Kamran T., Coincidence and fixed points for hybrid strict contraction, J.              Math.Anal. Appl. 299(2004),no. 1, 235-241 [10]  K. Menger, Statistical Matrices, Procedings of  the National academy of sciences of the United states of America 28 (1942), 535-537 [11]  S. N. Mishra, Common fixed points of compatible mappings in PM-Spaces, Math. Japonica, 36(2) (1991), 283-289 [12]  B.Schweizer and A.Sklar, Statistical metrices spaces, pacific Journal of Mathematics 10(1960),313- 334 [13]  S. Sessa, On weak commutativity conditions of mapping in fixed point consideration, Publ. Inst. Math. Beograd, 32(46) (1982), 149-153  [14]    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    This academic article was published by The International Institute for Science, Technology and Education (IISTE).  The IISTE is a pioneer in the Open Access Publishing service based in the U.S. and Europe.  The aim of the institute is Accelerating Global Knowledge Sharing.  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Fixed Point Result in Menger Space with EA Property

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Fixed Point Result in Menger Space with EA Property

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