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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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Fixed point and common fixed point theorems in vector metric spaces

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Fixed point and common fixed point theorems in vector metric spaces

  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 Applications247Fixed Point and Common Fixed Point Theorems in Vector MetricSpacesRenu Praveen Pathak* and Ramakant Bhardwaj***Department of Mathematics, Late G.N. Sapkal College of Engineering,Anjaneri Hills, Nasik (M.H.), India** Department of Mathematics,Truba Institute of Engineering & Information Technology, Bhopal (M.P.), Indiapathak_renu@yahoo.co.in, rkbhardwaj100@gmail.comAbstractIn the present paper we prove fixed point and common fixed point theorems in two self mappings satisfy quasitype contraction.Keywords :Fixed point, Common Fixed point ,self mapping, vector Metric space , E-complete.Mathematical Subject Classification: 47B60, 54H25.Introduction and PreliminariesVector metric space, which is introduced in [6] by motivated this paper [7] , is generalisation of metricspace ,where the metric is Riesz space valued .Actually in both of them , the metric map is vector space valued.One of the difference between our metric definition and Huang-Zhang’s metric definition is that there exist acone due to the natural existence of ordering on Riesz space .The other difference is that our definitioneliminates the requirement for the vector space to have a topological structure.A Riesz space (or a vector lattice) is an ordered vector space and alattice. Let E be a Riesz space with thepositive E = x ∈ E ∶ x ≥ 0 . If a is a decreasing sequence in E such that inf a = a, we write a ↓ aDefinition 1. The Riesz space E is said to be Archimedean if a ↓ 0 holds for every a ∈ E .Definition 2. A sequence (bn) is said to order convergent (or o-convergent) to b, if there is a sequence (an) in Esatisfying a ↓ 0 and |b − (b|( ≤ a for all n, and written b*+ b oro-limb = b , where |a| = sup a, −a for any a ∈ E.Definition3. A sequence (bn) is said to order Cauchy (or o-cauchy) if there exists a sequence (an) in E such thata ↓ 0 and |b -b | ≤ a holds for n and p.Definition4. The Riesz space E is said to be o-cauchy complete if every o-cauchy sequence is o-convergent.VECTOR METRIC SPACESDefinition 4.Let X be anon empty set and E be a Riesz space .The functiond ∶ X × X + E is said to be avector metric (or E-metric) if it is satisfying the following properties :(i) d1x, y2 = 0 if and only if x = y.1ii2d1x, y2 ≤ d1x, z2 + d1y, z2For all x, y, z ∈ X. Also the triple (X , d ,E) (briefly X with the default parameters omitted) is said to be vectormetric space. .Main ResultsRecently ,many authors have studied on common fixed point theorems for weakly compatible pairs (see [1], [3],[4], [8], [9]). Let T and S be self maps of a set X. if y = Tx = Sx for some ,x ∈ X, then y is said to be a point of coincidence and xis said to be coincidence pointof T and S. If T and S are weakly compatible ,that is they are commuting at their coincidence point on X, thenthe point of coincidence y is the unique common fixed point of these maps [1].Theorem 1 :Let X be a vector space with E is Archimedean. Suppose the mappings S, T ∶ X × X +X satis7ies the following conditions1i2for all x, y ∈ X, d1Tx, Ty2 ≤ ku1x, y2 (1)where k ∈ 90,12is a constant andu1x, y2 ∈ ;d1Sx, Sy2, d1Sx, Tx2, d1Sy, Ty2, d1Sx, Ty2, d1Sy, Tx2,129d1Sx, Tx2 + d1Sy, Tx2=>(ii) T1x2 ⊆ S1x2(iii) S1x2 or T1x2is a E-Complete subspace of x.Then T and S have a unique fixed point of coincidence in X . Moreover, if S and Tare weakly compatible ,thenthey have a unique common fixed point in X.Proof : Let x@, x ∈ X. De7ine the sequence 1x 2 by Sx = Tx = y for n ∈ N.We first show thatd1y , y 2 ≤ k d1y - , y 2 (2)For all n. we have that
  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 Applications248d1y , y 2 = d1Tx , Tx 2 ≤ ku1x , x 2Fir all n. Now we have to consider the following three cases:If u1x , x 2 = d1y - , y 2 then clearly 122holds . If u1x , x 2 = d1y , y 2Then according to Remark 1 d1y , y 2 = 0 and 122is immediate. Finally , suppose thatu1x , x 2 =Dd1y - , y 2. Then,d1y , y 2 ≤k2d1y - , y 2 ≤k2d1y - , y 2 +12d1y , y 2Holds , and we prove (2).We haved1y , y 2 ≤ k d1y@, y 2For all n and p,dEy , y FG ≤ d1y , y 2 + d1y , y D2 + ⋯ + d1y F- , y F2≤ 1k + k + k D+ ⋯ + k F- 2 d1y@, y 2dEy , y FG ≤k1 − kd1y@, y 2Holds .Now since E is Archimedean then (y@) is an E-cauchy sequence. Since the range of S contains the rangeof T and the range of at least one is E-Complete , there exists a z ∈ S1X2such that SxI.JKL z. Hence there exists a sequence 1a 2in E such that a ↓ 0and d1Sx , z2 ≤ a . On the other hand , we can 7ind w ∈ X such that Sw = z.Let us show that Tw = z we haved(Tw, z) ≤ d1Tw, Tx 2 + d1Tx , z2 ≤ ku1x , w2 + afor all n . Sinceu1x , w2 ∈ Pd1Sx , Sw2, d1Sx , Tx 2, d1Sw, Tw2, d1Sx , Tw2, d1Sw, Tx 2,129d1Sx , Tw2 + d1Sw, Tx 2=QAt least one of the following four cases holds for all n.Case 1 :d1Tw, z2 ≤ d1Sx , Sw2 + a ≤ a + a ≤ 2aCase 2 :d1Tw, z2 ≤ d1Sx , Tx 2 + a ≤ d1Sx , z2 + 2a ≤ 3aCase 3:d1Tw, z2 ≤ kd1Sw, Tw2 + a ≤ kd1Tw, z2 + a ,that is d1Tw, z2 ≤11 − ka .Case 4 :d1Tw, z2 ≤ d1Sx , Tw2 + a ≤ d1Sx , z2 + d1Tw, z2 + 3a ≤ 4aCase 5 :d1Tw, z2 ≤ kd1Sw, Tx 2 + a ≤ k d1Tx , z2 + 2a ≤ 3aCase 6 :d1Tw, z2 ≤D9d1Sx , Tw2 + d1Sw, Tx 2= + a≤12d1Sx , Tw2 +32a≤Dd1Sx , z2 +Dd1Tw, z2 +TDa≤12d1Tw, z2 + 2aThat is , d1Tw, z2 ≤ 4aSince the infimum of the sequence on the right side of last inequality are zero, thend1Tw, z2 = 0, i. e. Tw = z . Therefore , z is a point of coincidence of T and S.If z is another point of coincidence then there is w ∈ X with z = Tw = Sw .Now from 112, it follows thatd (z, z ) =d1Tw, Tw 2 ≤ ku1w, w 2.Whereu1w, w 2 ∈ Ud1Sw, Sw 2, d1Sw, Tw2, dESw ,Tw G, dESw ,Tw GdESw ,Tw G,129d1Sw, Tw 2 + d1Sw , Tw2=V= 0, d1z, z 2 .
  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 Applications249Hence d1z, z 2 = 0 that is z = z .If S and T are weakly compatible then it is obvious that z is unique common fixed point of T and S by [1].Theorem 2 : Let X be an vector metric space with E is Archimedean. Suppose the mappings S, T ∶ X +X satis7ies the following conditions1i2 for all x, y ∈ X,d1Tx, Ty2 ≤ ku1x, y2 132Where k ∈ 90,12is a constant andu1x, y2 ∈ Wd1Sx, Sy2,129d1Sx, Tx2 + d1Sy, Ty2=,129d1Sx, Ty2 + d1Sy, Tx2=,129d1Sx, Tx2 + d1Sx, Ty2=,129d1Sy, Ty2 + d1Sy, Tx2=X1ii2 T1X2 ⊑ S1X2,1iii2 S1X2 or T1X2 is E- Complete subspace of X.Then T and S have a unique point of coincidence in X. Moreover, if S and T are weakly compatible, then theyhave a unique common fixed point in X.Proof : Let us define the sequence 1x 2and 1y 2 as in the proof of Theorem 1 , we 7irstShow thatd1y , y 2 ≤ k d1y - , y 2 (4)For all n. Notice thatd1y , y 2 = d1Tx , Tx 2 ≤ ku1x , x 2,For all n.As in Theorem 1, we have to consider three cases: u1x , x 2 = d1y - , y 2,u1x , x 2 =D9d1y - , y 2 + d1y , y 2= and u1x , x 2 =Dd1y - , y 2.First and third have been shown in the proof of Theorem 1. Consider only the second case.If u1x , x 2 =129d1y - , y 2 + d1y , y 2=, then from 132we haved1y , y 2 ≤129d1y - , y 2 + d1y , y 2= ≤k2d1y - , y 2 +12d1y , y 2 .Hence. (4) Holds.In the proof of this Theorem 1 we illustrate that (y ) is an E-Cauchy sequence. Then there exist z ∈ S1X2, w ∈X and 1a 2 in E such that Sw = z, d1Sx , z2 ≤ a and a ↓ 0Now, we have to show that Tw = z. We haved1Tw, z2 ≤ d1Tw, , Tx 2 + d1Tx , z2 ≤ u1x , w2 + aFor all n. sinceu1x , w2 ∈ Wd1Sx , Sw2,129d1Sx , Tx 2 + d1Sw, Tw2=,129d1Sx , Tw2 + d1Sw, Tx 2=,129d1Sx , Tx 2 + d1Sx , Tw2=,129d1Sw, Tx 2 + d1Sw, Tw2=XAt least three of the five holds for all n. Consider only the cases ofu1x , w2 =129d1Sx , Tx 2 + d1Sw, Tw2=,129d1Sx , Tx 2 + d1Sx , Tw2=,129d1Sw, Tx 2 + d1Sw, Tw2=Because the other four cases have shown that the proof of Theorem 1 . It is satisfied thatd1Tw, z2 ≤129d1Sx , Tx 2 + d1Sw, Tw2= +129d1Sx , Tx 2 + d1Sx , Tw2=+129d1Sw, Tx 2 + d1Sw, Tw2= + a≤129d1Sx , z2 + d1Tx , z2 + d1z, Tw2= +129d1Sx , z2 + d1Tx , z2 + d1Sx , z2 + d1z, Tw2=+129d1Tx , z2 + d1z, Tw2= + a≤12d1Sx , z2 +12d1Tx , z2 +12d1z, Tw2 + a≤12a +12d1z, Tw2 +32a
  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 Applications250d1Tw, z2 ≤12d1z, Tw2 + 2aThat is d1z, Tw2 ≤ 4a . Since 4a ↓ 0 then Tw = z . Hence z is a point of coincidenceof T and S. The uniqueness of z as in the proof of Theorem 1. Also, If S and T are weaklycompatible, then it is obvious that z is unique common 7ixed point of Tand S by 91=.Theorem 3 :Let X be an vector space with E is Archimedean. Suppose the mappingsS, T: X + X satis7ies the following conditions(i) for all x , y ∈ Xd1Tx, Ty2 ≤ α d1Sx, Tx2 + β d1Sy, Ty2 + γ d1Sx, Ty2 + δ d1Sy, Tx2 + η d1Sx, Sy2 +12μ d1Sx, Tx2 + d1Sy, Ty2(ii) T1X2 ⊆ S1X2,(iii) S(X) or T(X) is E-complete subspace of XThen T and S have a unique point of coincidence in X .Moreover, If S and T are weakly compatible, then theyhave a unique common fixed point in X .Proof :Let us define the sequence 1x 2and 1y 2 as in the proof of Theorem 1 , we have to show thatd1y , y 2 ≤ k d1y - , y 2 (5)For some k ∈ 90,12 and for all n. Consider Sx = Tx = y for all n. Thend1y , y 2 ≤ 1α + η2d1y - , y 2 + β d1y , y 2 + γd1y - , y 2+12μ9d1y - , y 2 + d1y , y 2=Andd1y , y 2 ≤ α d1y , y 2 + 1β + η2d1y - , y 2 + δd1y - , y 2For all n. Hence,d1y , y 2 ≤α + β + γ + δ + 2η + μ2 − α + β + γ + δd1y - , y 2If we choose k =c d e f Dg μD-c d e f, then k ∈ 90,12and 152is hold.In the proof of Theorem 1 we illustrate that 1y 2 is an E-Cauchy sequence .then there exist z ∈ s1X2, w ∈X and 1a 2 in E such that Sw = z , d1Sx , z2 ≤ a and a ↓ 0 .Let us show that Tw = z .we haved1Tw, z2 ≤ d1Tw, Tx 2 + d1Tx , z2≤ 1α + δ + μ2d1Tw, z2 + 1β + δ + η2d1Sx , z2 + 1β + γ + 12d1Tx , z2≤ 1α + δ + μ2 d1Tw, z2 + 1β + δ + η2a + 1β + γ + 12 + a≤ 1α + δ + μ2 d1Tw, z2 + 12β + γ + δ + η + 12aThat is d1Tw, z2 ≤1Dd e f g 2-1c f μ2a for all n.Then d1Tw, z2 = 0 , i. e. Tw = z . Hence ,Z is a point of coincidence of T and S. The uniqueness of z is easily seen. Also ,If S and T are weaklycompatible , then it is obvious that z is unique common fixed point of T and S by [1].Corrollary 1: Let X be an vector space with E is Archimedean. Suppose the mappingsS, T: X + X satis7ies the following conditions(i) For all x , y ∈ X ,d1Tx, Ty2 ≤ k d1Sx, Sy2 (6)Where k< 1(ii) T1X2 ⊑ S1X2,(iii) S1X2 or T1X2 is E- Complete subspace of X.Then T and S have a unique point of coincidence in X .Moreover, If S and T are weakly compatible, then theyhave a unique common fixed point in X .Now we give an illustrative exampleExample :Let E = ℝDwith coordinatwise ordering 1since ℝDis not Archmedean withlexicogra7ical ordering ,then we can not use this ordering ), X= ℝD, d1x, y2 = 1|x − y |, α|x − y |2, α > 0, Tx =2xD+ 1 and Sx = 4xD. Then , for all x, y ∈ X we haved1Tx, Ty2 =12d1Sx, Sy2 ≤ k d1Sx, Sy2For k ∈ lD, 1m,T1X2 = 91, ∞2 ⊆ 90, ∞2 = S1X2
  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 Applications251And T(X) is E-complete subspace of X. Therefore all conditions of corollary 1 are satisfied. Thus T and S have aunique point of coincidence in X .References[1] M. Abbas, G. Jungck, “Common fixed point results for no commuting mappings without continuity in conemetric spaces”, J. Math. Anal.Appl, 341(2008), 416-420.[2] C.D Apliprantis, K. C. Border, “Infinite Dimensional Analysis”, Springer-Verlag, Berlin, 1999.[3] I. Altun,D. Turkoglu,“Some fixed point thermos for weakly compactible mappings satisfying an implicitrelation”, Taiwanese J. Math,13(4)(2009),1291-1304.[4] I Altun,D.Turkoglu, B.E> Rhoades, “Fixed points of weakly compactible maps satisfying a generalcontractive condition of integral type”, Fixed Point Theory Appl.,2007,Art.ID 17301,9 pp.[5] IshakAltun and CiineytCevi’k, “some common fixed point theorem in vector metric space”,Filomat 25:1(2011) 105,113 DOI: 10.2298/FIL1101105A.[6] C. Cevik, I. Altun, “Vector metric spaces and some properties”, Topo.Met.Nonlin. Anal, 34(2) (2009) .375-382.[7] L.G .Huang, X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings”, J.Math.Anal.Appl. ,332(2007), 1468-1476.[8] G. Jungck, S. Radenovic, V. Rakocevic, “Common fixed point Theorems on weakly compatible pairs oncone metric spaces”, Fixed Point Theory Appl. ,2009, Art. ID 643840, 13 pp.[9] Z. Kadelburg, S. Radenovic, V. Rakocevic, “Topological vector space-valued cone metric spaces and fixedpoint theorems”, Fixed Point Theory Appl. , 2010, Art. ID 170253, 17 pp.
  6. 6. 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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