Fixed point result in menger space with ea property

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Fixed point result in menger space with ea property

  1. 1. 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 Applications313Fixed Point Result in Menger Space with EA PropertySmriti Mehta, A.D.Singh*and Vanita Ben DhagatDepartment of Mathematics, Truba Institute of Engineering & I.T. Bhopal*Government M V M College BhopalEmail: smriti.mehta@yahoo.comABSTRACTThis paper’s main objective is to define Menger space (PQM) and the concept of weakly compatible by using thenotion of property (EA) & JSR maps to define new property to prove a common fixed point theorem for 4 selfmaps in Menger space (PQM).Key Words: Fixed Point, Probabilistic Metric Space, Menger space, JSR mappings, property EASubject classification: 47H10, 54H251. INTRODUCTIONThe notion of probabilistic metric space is introduced by Menger in 1942 [10] and the first result aboutthe existence of a fixed point of a mapping which is defined on a Menger space is obtained by Sehgel andBarucha-Reid.A number of fixed point theorems for single valued and multivalued mappings in menger probabilisticmetric space have been considered by many authors [2],[3],[4],[5],[6],[7]. In 1998, Jungck [8] introduced theconcept weakly compatible maps and proved many theorems in metric space. Hybrid fixed point theory fornonlinear single valued and multivalued maps is a new development in the domain of contraction typemultivalued theory ([4], [7], [11], [12], [13], [14] ).Jungck and Rhoades [8] introduced the weak compatibility tothe setting of single valued and multivalued maps. Singh and Mishra introduced (IT)-commutativity for hybridpair of single valued and multivalued maps which need not be weakly compatible. Recently, Aamri and ElMoutawakil [1] defined a property (EA) for self maps which contained the class of noncompatible maps. Morerecently, Kamran [9] extended the property (EA) for a hybrid pair of single valued and multivalued maps andgeneralized the (IT) commutativity for such pair.The aim of this paper is to define a new property which contains the property (EA) for hybrid pairof single valued and multivalued maps and give some common fixed point theorems under hybridcontractive conditions in probabilistic space.2. PRELIMINARIESNow we begin with some definitionDefinition 2.1: Let R denote the set of reals and the non-negative reals. A mapping : → iscalled a distribution function if it is non decreasing left continuous withinf ( ) 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 beset 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 satisfythe 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, > 04Definition 2.3: A mapping 5: [0, 1] [0, 1] → [0, 1] is called t-norm if1. 5 ( , 1) = ∀ ∈ [0,1]2. 5 (0, 0) = 0,3. 5 ( , 9) = 5 (9, ),
  2. 2. 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 Applications3144. 5 (:, ) ≥ 5 ( , 9) : ≥ , ≥ 9, and5. 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 withthe 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 = $4where +( ) = D0 ≤ 00 ≤ ≤ 11 ≥ 14Then ( , , 5 ) is a Menger space.Definition 2.5: Let ( , , ∆) be a Menger space. If B ∈ , E > 0, F ∈ (0, 1), then an (E, F)neighbourhood of u, denoted by G> (E, F) is defined asG>(E, F) = H$ ∈ ; >,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 JB KLM 5( , ) = 1, it is metrizable.Definition 2.6: A sequence N OP in ( , , ∆) is said to be convergent to a point ∈ if for everyE > 0 and λ > 0, there exists an integer Q = Q(E, F) such that O ∈ G (E, F) for all ≥ Q or equivalentlyRS,R(T) > 1 − F for all ≥ Q.Definition 2.7: A sequence N OP 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 everyCauchy sequence in X converges to a point in X.Lemma 2.9 [14]: Let N OP 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 N OP 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 thatlimO →∞ O = limO →∞ ` O = d ∈ .Definition 2.12: -Maps : → Z ∶ → ]( ) are said to satisfy the property (EA) if there existsa sequence N OP in X, some z in X and A in CB(X) such thatlimO →∞ 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 saidto satisfy the common property (EA) if there exist two sequences N OP, N!OP in X and some z in X such thatlimO →∞ g!O = limO →∞ f O = limO →∞O = limO →∞ `!O = d .Definition 2.14: Let , `: → f, g ∶ → ]( ) be mappings on Menger space. The mapspair (f, S) and (g, G) are said to satisfy the common property (EA) if there exist two sequences N OP, N!OP in Xand some z in X, and A, B in CB(X) such thatlimO →∞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
  3. 3. 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 Applications315pair {f, g} is said to be f-JSR mappings iffi ( ` O, ` O; ) ≥ i ( O, O; )where µ = lim Sup or lim inf and N OP is a sequence in X such thatlimO →∞O = limO →∞` O = d J <# d ∈ ∆( , ) > .Example Let X = [0, 1] with and f, g are two self mapping on X defined byNow the sequence in X is defined as then wehaveClearly we have .Thus pair {f, g} is f-JSR mapping. But this pair is neither compatible nor weakly compatible or othernon 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 ifi (f O, O; ) ≥ i (ff O, f O; )where µ = lim Sup or lim inf and N OP is a sequence in X such thatlimO →∞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 RESULTSTheorem 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[minH pR,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 thatlimO →∞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 getlRS,m(V ) ≥ j rmin spRS,q ( ), pRS,lRS( ),q,m ( ), pRS,m ( ), lRS,q( )tuOn taking limit → ∞, we obtainv,m(V ) ≥ jwminH pA,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
  4. 4. 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 Applications316SimilarlylA,mnS(V ) ≥ j[minH pA,qnS( ), pA,lA( ), qnS,mnS( ), pA,mnS( ), lA,qnS( )I]lA,y(V ) ≥ j[minH pA,q ( ), pA,lA( ), q,y ( ), pA,y ( ), lA,q( )I]≥ j pA,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 ∈ fand similarly as pair (g, G) is G-JSR maps therefore `B ∈ gpRS,q> ( ) ≥ lRS,m>(V )≥ j[minH pRS,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)tuBy triangular inequality and as B ∈ e ∩ ], we obtain>,q>( ) ≥ >,q>( )⟹ `B = B.Againp>,qRS( ) ≥ l>,mRS(V )≥ j[minH p>,qRS( ), p>,l>( ), qRS,mRS( ), p>,mRS( ), l>,qRS( )I]On taking limit → ∞,we obtainp>,>( ) ≥ jwminH p>,> ( ), p>,l>( ), >,m> ( ), p>,y ( ), l>,>( )Ix≥ j rmin sp>,> ( ), p>,l>( ), >,m> ( ),p>,> ( /2), >,y ( /2), l>,>( )tuBy triangular inequality and as B ∈ e ∩ ], we obtainp>,> ( ) ≥ 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 isthe 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,mƒn(V ) ≥ j[min ~ pR,qn ( ), pR,l‚R( ), qn,mƒn ( ), pR,mƒn ( ), l‚R,qn( )•]
  5. 5. 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 Applications317Then , `, 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, ScientiaSinica 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 ofAmerica 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 touniform spaces, Honam Math. J., 6 (1984), 1-12
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