Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.3, No.6, 2013-Selected fro...
This academic article was published by The International Institute for Science,Technology and Education (IISTE). The IISTE...
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A common fixed point theorem for six mappings in g banach space with weak-compatibility

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A common fixed point theorem for six mappings in g banach space with weak-compatibility

  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 Applications206A Common Fixed Point Theorem for Six Mappings in G-BanachSpace with Weak-CompatibilityRanjeeta JainInfinity Management and Engineering college,Pathariya jat Road Sagar (M.P.)470003ABSTRACTThe aim of this paper is to introduce the concept of G-Banach Space and prove a common fixed point theoremfor six mappings in G-Banach spaces with weak–compatibility.Keywords: Fixed point , common fixed point , G-Banach Space , Continuous mappings , weak compatiblemappings.Introduction : This is well know that the fundamental contraction principle for proving fixed points results isthe Banach Contraction principle. There have been a number of generalization of metric space and Banach space.One such generalization is G-Banach space.The concept of G-Banach space is introduce by ShrivastavaR,Animesh,Yadav R.N.[4 ] which is a probable modification of the ordinary Banach Space.Recently in 2012,R.K. Bharadwaj [2 ] introduced fixed point theorems in G-BanachSpace through weak compatibility and gave the following fixed point theorems for four mappings-Theorems[A]: Let X be a G-Banach Space , such that ∇ Satisfy property with α - α ≤ 1. If A , B , S and T bemapping from X into itself satisfying the following condition:I. A(X) ⊆ T(X) , B(X) ⊆ S(X) and T(X) or S(X) is a closed subset of X.II. The Pair (A,S) and (B,T) are weakly compatible,III. For all x,y∈Xg||By-Ax||≤∇≤||Ty-Sx||||By-Ty||||Ax-Tx||||Ty-Sx||||By-Sx||||Ax-Sx||gggggg1k∇+gggggg2||Ty-Sx||||Ax-Ty||||By-Sx||||Ty-Sx||||By-Ty||||Ax-Sx||maxk( )ggggg3 ||Ty-Sx||||Ax-Ty||||By-Sx||||By-Ty||||Ax-Sx|| ∇∇∇∇+ kWhere k1 , k2 , k3 > 0 and 0 < k1 + k2 + k3 < 1. Then A , B , S and T have a unique common fixedpoint in X.In 1980, Singh and Singh[5] gave the following theorem on metric space for self maps which is use toour main result-Theorems[B]: Let P, Q and T be self maps of a metric space (X, d) such that(i) PT = TP and QT = TQ, (ii) P(X) ∪ Q(X) ⊆ T(X), (iii) T is continuous,(iv) d(Px , Qy ) ≤ cλ (x, y),where λ (x, y) = max{d(Tx, Ty), d(Px, Tx), d(Qy, Ty),21[d(Px, Ty)+d(Qy, Ty)]}for all x, y ∈ X and 0 ≤ < 1. Further if(v) X is complete then P, Q, T have a unique common fixed point in X.Just we recall the some definition of G-Banach space for the sake of completeness which as follows-N be the set of natural numbers and R+be the set of all positive numbers let binary operation ∇ : R+× R+→R+satisfies the following conditions:i. ∇ is associative and commutative,ii. ∇ is continuous.Five typical example are as follows:i. a∇ b = max (a,b)ii. a∇ b = a+biii. a∇ b =a.biv. a∇ b = a.b+a+b
  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 Applications207v. a∇ b =)1,,max( baabDefinition 1: The binary operation ∇ is said to satisfy α-property if there exists a positive real number α ,such that a ∇ b ≤ α max (a,b) for every a,b Є R+Example:If we define a ∇ b = a + b for each a,b Є R+then for α ≥ 2, we havea ∇ b ≤ α max (a,b)if we define a ∇ b =)1,,max( baabfor each a,b Є R+then for α ≥ 1, we havea ∇ b ≤ α max (a,b)Definition 2: Let x be a nonempty set ,A Generalized Normed Space on X is a function ║ ║g : x × x → R+thatsatisfies the following conditions for each x,y,z, Є X1. ║x-y║g > 02. ║x-y║g = 0 if and only if x = y3. ║x-y║g = ║y-x║g4. ║ α x║g = │ α │║x║g for any scalar α5. ║x-y║g ≤ ║x-z║g ∇ ║z-x║gThe pair (x, ║ ║g) is called generalized Normed Space or simply G-Normed Space.Definition 3: A Sequence in X is said converges to x if ║xn-x║g→0 , as n → ∞. That is for each є > 0 thereexists n0 Є N such that for every n ≥ n0 implies that ║xn-x║g < єDefinition 4: A sequence {xn} is said to be Cauchy sequence if for every є > 0 there exists n0 є N such that║xm-xn║g < є for each m,n ≥ n0. G-Normed Space is said to be G-Banach Space if for every Cauchy sequence isconverges in it.Definition 5: Let (x, ║ ║g) be a G-Normed Space for r>0 we defineBg (x,r) = {y є X : ║x-y║g < r }Let X be a G-normed Space and A be a subset of X, then for every x є A, there exists r > 0 such that Bg (x,r) ⊆A , then the subset A is called open subset of X. A subset A of X is said to be closed if the complement of A isopen in X.Definition 6: Let A and S be mappings from a G-Banach space X into itselt. Then the mappings are said tobe weakly compatible if they are commute at there coincidence point that is Ax= Sx implies that ASx = SAxHere we generalized and extend the results of R.K.Bhardwaj[2] (theorem A) for six mappingsopposed to four mappings in G-Banach space using the concept of weak-compatibility.Main Result :THEOREM(1) : Let X be a G-Banach Space , such that ∇ Satisfy property with α - α ≤ 1. If P , Q , A , B , Sand T be mapping from X into itself satisfying the following condition:I. A(X) ⊆ Q(X) ∪ T(X) , B(X) ⊆ P(X) ∪ S(X) and T(X) or S(X) is a closed subset of X.II. The Pair (A,S) and (P,S) , (B,T) and (Q ,T) are weakly compatible,III. For all x,y∈Xg||By-Ax|| ≤ +∇+∇∇≤ggggggggggggg||By-Sx||||By-Px||||Ty-Bx||||Qx-Px||||Ax-Px||||Ax-Qy||||By-Qy||||Ty-Sx||||Py-Sx||||By-Qy||||Ax-Px||||Ax-Ty||||By-Sx||maxα∇∇∇∇∇∇+ggggggggggg||Qy-Px||||Sx-Qy||||Ty-Sx||||Qy-Px||||By-Qy||||Ax-Px||||Sx-Qy||||Ty-Px||||By-Qy||||Sx-Px||||Ty-Sx||maxβWhere α , β > 0 and 0 < α + β < 1. Then P , Q , A , B , S and T have a unique commonfixed point in X.Proof: Let x0 be an arbitrary point in X then by (i) we choose a point x1 in X such that y0 = Ax0 =Tx1 = Qx1 and y1 = Bx1 = Sx2 = Px2 .
  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 Applications208In general there exists a sequence {yn} such thaty2n = Ax2n = Tx2n+1 = Qx2n+1 andy2n+1 = Bx2n+1 = Sx2n+2 = Px2n+2 , for n = 1 , 2 , 3 , --------------we claim that the sequence {yn} is a Cauchy sequence.By (iii) we haveg12n2n ||y-y|| + = g12n2n ||Bx-Ax|| + +∇+∇∇≤++++++++++++g12n2ng12n2ng12n2ng2n2ng2n2ng2n12ng12n12ng12n2ng12n2ng12n12ng2n2ng2n12ng12n2n||Bx-Sx||||Bx-Px||||Tx-Bx||||Qx-Px||||Ax-Px||||Ax-Qx||||Bx-Qx||||Tx-Sx||||Px-Sx||||Bx-Qx||||Ax-Px||||Ax-Tx||||Bx-Sx||maxα∇∇∇∇∇∇++++++++++++g12n2ng2n12ng12n2ng12n2ng12n12ng2n2ng2n12ng12n2ng12n12ng2n2ng12n2n||Qx-Px||||Sx-Qx||||Tx-Sx||||Qx-Px||||Bx-Qx||||Ax-Px||||Sx-Qx||||Tx-Px||||Bx-Qx||||Sx-Px||||Tx-Sx||maxβ +∇+∇∇≤+++−g2n1-2ng2n1-2ng12n2ng1-2n1-2ng2n1-2ng2n2ng12n2ng2n1-2ng2n1-2ng12n2ng2n1-2ng2n2ng12n1-2n||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||maxα∇∇∇∇∇∇+ ++g2n1-2ng1-2n2ng2n1-2ng2n1-2ng12n2ng2n1-2ng1-2n2ng2n1-2ng12n2ng1-2n1-2ng2n1-2n||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||||y-y||maxβg12n2n ||y-y|| + ≤ (α β+ ) g2n1-2n ||y-y||That is by induction we can show thatg12n2n ||y-y|| + ≤ (α β+ )ng10 ||y-y|| g12n2n ||y-y|| +As n → ∞ g12n2n ||y-y|| + → 0 , for any integer m ≥ nIt follows that the sequence {yn} is a Cauchy sequence which converges to y∈X .This implies that lim n → ∞ yn = lim n → ∞ Ax2n = lim n → ∞ Tx2n+1 = lim n → ∞ Qx2n+1 = lim n → ∞Bx2n+1 = lim n → ∞ Sx2n+2 = lim n → ∞ Px2n+2 = yNow let us assume that T(X) is closed subset of X, then there exists v∈X such thatTv = Qv = yWe now prove that Bv = y ,By (iii) we get +∇+∇∇≤g2ng2ng2ng2n2ng2n2ng2ngg2ng2ngg2n2ng2ng2n||Bv-Sx||||Bv-Px||||Tv-Bx||||Qx-Px||||Ax-Px||||Ax-Qv||||Bv-Qv||||Tv-Sx||||Pv-Sx||||Bv-Qv||||Ax-Px||||Ax-Tv||||Bv-Sx||maxα∇∇∇∇∇∇+g2ng2ng2ng2ngg2n2ng2ng2ngg2n2ng2n||Qv-Px||||Sx-Qv||||Tv-Sx||||Qv-Px||||Bv-Qv||||Ax-Px||||Sx-Qv||||Tv-Px||||Bv-Qv||||Sx-Px||||Tv-Sx||maxβg||Bv-y|| ≤ (α β+ ) g||Bv-y||g2n ||Bv-Ax||
  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 Applications209Which is contradiction , it follows that Bv = y = Tv = Qv . Since (B,T) and (Q,T) are weakly compatiblemappings , then we haveBTv = TBv and QTv = TQvBy = Ty and Qy = TyWhich implies By = Qy.Now we prove that By = y for this by using (iii) we getg2n ||By-Ax|| +∇+∇∇≤g2ng2ng2ng2n2ng2n2ng2ngg2ng2ngg2n2ng2ng2n||By-Sx||||By-Px||||Ty-Bx||||Qx-Px||||Ax-Px||||Ax-Qy||||By-Qy||||Ty-Sx||||Py-Sx||||By-Qy||||Ax-Px||||Ax-Ty||||By-Sx||maxα∇∇∇∇∇∇+g2ng2ng2ng2ngg2n2ng2ng2ngg2n2ng2n||Qy-Px||||Sx-Qy||||Ty-Sx||||Qy-Px||||By-Qy||||Ax-Px||||Sx-Qy||||Ty-Px||||By-Qy||||Sx-Px||||Ty-Sx||maxβg2n ||By-Ax|| ≤ (α β+ ) g||By-y||Which is contradiction.Thus By = y = Ty = Qy --------------------(A)SinceB(X) ⊆ P(X) ∪ S(X) , there exists w∈ X. such that sw = y = Pw . we show that Aw = yFrom (iii) we haveg||By-Aw|| +∇+∇∇≤ggggggggggggg||By-Sw||||By-Pw||||Ty-Bw||||Qw-Pw||||Aw-Pw||||Aw-Qy||||By-Qy||||Ty-Sw||||Py-Sw||||By-Qy||||Aw-Pw||||Aw-Ty||||By-Sw||maxα∇∇∇∇∇∇+ggggggggggg||Qy-Pw||||Sw-Qy||||Ty-Sw||||Qy-Pw||||By-Qy||||Aw-Pw||||Sw-Qy||||Ty-Pw||||By-Qy||||Sw-Pw||||Ty-Sw||maxβg||y-Aw|| ≤ (α β+ ) g||y-Aw||Which is contradiction , so that Aw = y = Sw = Pw .Since (A , S) and (P , S) are weakly compatible , thenASw = Saw and PSw = SPwAy = Sy Py = SyTherefore Ay = PyNow we Show that Ay = y,From (iii) we haveg||By-Ay|| +∇+∇∇≤ggggggggggggg||By-Sy||||By-Py||||Ty-By||||Qy-Py||||Ay-Py||||Ay-Qy||||By-Qy||||Ty-Sy||||Py-Sy||||By-Qy||||Ay-Py||||Ay-Ty||||By-Sy||maxα
  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 Applications210∇∇∇∇∇∇+ggggggggggg||Qy-Py||||Sy-Qy||||Ty-Sy||||Qy-Py||||By-Qy||||Ay-Py||||Sy-Qy||||Ty-Py||||By-Qy||||Sy-Py||||Ty-Sy||maxβg||y-Ay|| ≤ (α β+ ) g||y-Ay||Which is contradiction thus Ay = y and thereforeAy = Sy = Py = y ---------------------------------------------(B)Now from equation (A) and (B) we getAy = By = Py = Qy = Sy = Ty = y .Hence y is a common fixed point of A ,B , S , T , P and Q .Uniqueness:Let us assume that x is another fixed point of A ,B , S , T , P and Q different from y in X.Then from (iii) we haveg||By-Ax|| ≤ +∇+∇∇≤ggggggggggggg||By-Sx||||By-Px||||Ty-Bx||||Qx-Px||||Ax-Px||||Ax-Qy||||By-Qy||||Ty-Sx||||Py-Sx||||By-Qy||||Ax-Px||||Ax-Ty||||By-Sx||maxα∇∇∇∇∇∇+ggggggggggg||Qy-Px||||Sx-Qy||||Ty-Sx||||Qy-Px||||By-Qy||||Ax-Px||||Sx-Qy||||Ty-Px||||By-Qy||||Sx-Px||||Ty-Sx||maxβg||y-x|| ≤ (α β+ ) g||y-x||Which is contradiction . Thus x = y . This completes the proof of the theorem.COROLLARY:Let X be a G-Banach Space such that ∇ Satisfy α- property with α ≤ 1. If T be a mapping from X into itself,satisfying the following condition:gsr||yT-xT|| +∇+∇∇≤grgsgsgrgrgrgsgggsgrgrgs||xT-x||||yT-x||||yT-y||||xT-x||||xT-x||||xT-y||||yT-y||||y-x||||y-x||||yT-y||||xT-x||||xT-y||||yT-x||maxα∇∇∇∇∇∇+ggrgrggsgrgrgsgsgrg||y-x||||xT-y||||xT-x||||y-x||||yT-y||||xT-x||||xT-y||||yT-x||||yT-y||||xT-x||||y-x||maxβFor non-negative α , β suvh that 0 < α + β < 1 and r , s ∈ N(set of natural number). Then Thas a unique common fixed point in X.
  6. 6. 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 Applications211REFERENCES :1. Ahmad A. and Shakil M. “Some fixed point theorems in Banach spaces ”, Nonlinear func. Anal.Appl.11(2006) 343-349.2. Bhardwaj R.K. “ Some contraction on G-Banach Space”,Global journal of Science frontier researchMathematics and Decision Sciences vol.12 issue 1 version 1.o, jaanuary (2o12).pp.81-893. Ciric Lj.B and J.S Ume “some fixed point theorems for weakly compatible mappings”J.Math.Anal.Appl.314(2) (2006) 488-499.4. Shrivastava R ,Animesh,yadav R.N. “some Mapping on G-Banach Space” international journal ofMathematical Science and Engineering Application, Vol.5, No. VI , (2011),pp.245-260.5. Singh,S.L. and Singh,s.p. “A fixed point theorm” Indian journals pure and Applied. Math.,volno.11(1980), 1584-1586.6. Saini R.K. and Chauhan S.S “Fixed point theorem for eight mappings on metric space”journal ofmathematics and system science,vol.1 no.2 (2005).7. V.Popa,” A general fixed point theorems for weakly compatible mapping satisfying an Implictrelation”,Filomat,19(2005) 45-51.
  7. 7. This academic article was published by The International Institute for Science,Technology and Education (IISTE). The IISTE is a pioneer in the Open AccessPublishing service based in the U.S. and Europe. The aim of the institute isAccelerating Global Knowledge Sharing.More information about the publisher can be found in the IISTE’s homepage:http://www.iiste.orgCALL FOR PAPERSThe IISTE is currently hosting more than 30 peer-reviewed academic journals andcollaborating with academic institutions around the world. There’s no deadline forsubmission. Prospective authors of IISTE journals can find the submissioninstruction on the following page: http://www.iiste.org/Journals/The IISTE editorial team promises to the review and publish all the qualifiedsubmissions in a fast manner. All the journals articles are available online to thereaders all over the world without financial, legal, or technical barriers other thanthose inseparable from gaining access to the internet itself. Printed version of thejournals is also available upon request of readers and authors.IISTE Knowledge Sharing PartnersEBSCO, Index Copernicus, Ulrichs Periodicals Directory, JournalTOCS, PKP OpenArchives Harvester, Bielefeld Academic Search Engine, ElektronischeZeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe DigtialLibrary , NewJour, Google Scholar

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