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# 11.[29 35]a unique common fixed point theorem under psi varphi contractive condition in partial metric spaces using rational expressions

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### 11.[29 35]a unique common fixed point theorem under psi varphi contractive condition in partial metric spaces using rational expressions

1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.4, 2012 A Unique Common Fixed Point Theorem under psi – varphi Contractive Condition in Partial Metric Spaces Using Rational Expressions K.P.R.Rao Department of Applied Mathematics, AcharyaNagarjunaUnivertsity-Dr.M.R.Appa Row Campus, Nuzvid- 521 201, Krishna District, Andhra Pradesh, India E-mail address: kprrao2004@yahoo.com G.n.V. Kishore Department of Mathematics, Swarnandhra Institute of Engineering and Technology,Seetharampuram, Narspur- 534 280 , West Godavari District, Andhra Pradesh, India E-mail address: kishore.apr2@gmail.comAbstractIn this paper, we obtain a unique common fixed point theorem for two self maps satisfying psi - varphicontractive condition in partial metric spaces by using rational expressions.Keywords: partial metric, weakly compatible maps, complete space.1. IntroductionThe notion of partial metric space was introduced by S.G.Matthews [3] as a part of the study of denotationalsemantics of data flow networks. In fact, it is widely recognized that partial metric spaces play an importantrole in constructing models in the theory of computation ([2, 5, 7 -12], etc).S.G.Matthews [3], Sandra Oltra and Oscar Valero[4] and Salvador Romaguera [6]and I.Altun, Ferhan Sola,HakanSimsek [1] prove fixed point theorems in partial metric spaces for a single map.In this paper, we obtain a unique common fixed point theorem for two self mappings satisfying ageneralized ψ -φ contractive condition in partial metric spaces by using rational expression.First we recall some definitions and lemmas of partial metric spaces.2. Basic Facts and DefinitionsDefinition 2.1. [3].A partial metric on a nonempty set X is a function p :X × X → R+ such that for all x, y, z∈ X:(p1) x = y ⇔ p(x, x) = p(x, y) = p(y, y),(p2) p(x, x) ≤ p(x, y), p(y, y) ≤ p(x, y),(p3) p(x, y) = p(y, x),(p4) p(x, y) ≤ p(x, z) + p(z, y) − p(z, z). (X, p) is called a partial metric space.It is clear that |p(x, y) − p(y, z)| ≤ p(x, z) for all x, y, z ∈ X. 29
2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.4, 2012Also clear that p(x, y) = 0 implies x = y from (p1) and (p2).But if x = y, p(x, y) may not be zero. A basic example of a partial metric space is the pair (R+, p), wherep(x, y) = max{x, y} for all x, y ∈ R+.Each partial metric p on X generates τ0 topology τp on X which has a base the family of open p - balls{Bp(x, ε) / x ∈ X, ε > 0} for all x ∈ X and ε > 0, where Bp(x, ε)= {y ∈ X / p(x, y) < p(x, x) + ε}for all x ∈ X and ε > 0.If p is a partial metric on X, then the function ps : X × X → R+ given by Ps(x, y) = 2p(x, y) − p(x, x) − p(y, y) (2.1)is a metric on X.Definition 2.2. [3]. Let (X, p) be a partial metric space.(i) A sequence {xn} in (X, p) is said to converge to a point x ∈ X if and only if p(x, x) = lim p(x, xn). n→∞(ii) A sequence {xn} in (X, p) is said to be Cauchy sequence if lim p(xm, xn) exists and is finite . m,n→∞(iii) (X, p) is said to be complete if every Cauchy sequence {xn} in X converges, w.r.to τp, to a pointx ∈ X such that p(x, x) = lim m,n→∞ p(xm, xn).Lemma 2.3. [3].Let (X, p) be a partial metric space.(a) {xn} is a Cauchy sequence in (X, p) if and only if it is a Cauchy sequence in the metric space (X, ps).(b) (X, p) is complete if and only if the metric space (X, ps) is complete. Furthermore,lim ps(xn, x) = 0 if and only if p(x, x) = lim p(x, xn) = lim p(xm, xn).n→∞ n→∞ m,n→∞3. Main ResultTheorem 3.1. Let (X, p) be a partial metric space and let T, f : X → X be mappings such that(i) ψ(p (Tx, Ty)) ≤ψ (M (x, y)) − φ (M (x, y)) , ∀ x, y ∈ X,  1 + p(fx, Tx) where M (x, y) = max p(fy, Ty) , p(fx, fy) and ψ: [0, ∞) → [0, ∞) is continuous,  1 + p(fx, fy) non-decreasing with ψ (t) = 0 if and only if t = 0 and φ:[0, ∞) → [0, ∞) is lower semi continuous withφ(t) = 0 if and only if t = 0,(ii) T(X) ⊆ f(X) and f(X) is a complete subspace of X and(iii) the pair (f, T) is weakly compatible.Then T, f have a unique common fixed point of the form α in X.Proof: Let x0 ∈ X. From (ii), there exist sequences {xn} and {yn} in X such thatyn = fxn+1 = Txn, n = 0, 1, 2,3, · · ·Case(a): Suppose yn = yn+1 for some n.Then fz = Sz , where z = xn+1. Denote fz = Tz = α.ψ(p(Tα, α)) = ψ (p(Tα, Tz)) ≤ ψ(M(α, z)) − φ(M(α, z)). 30
3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.4, 2012  1 + p(fα , Tα )  M(α, z) = max p(fz, Tz) , p(fα , fz)  1 + p(fα , fz)   1 + p(Tα , Tα )  = max p(α , α ) , p(Tα ,α )  1 + p(Tα , α )  = p(Tα, α), from (p2).Thereforeψ(p(Tα, α)) ≤ ψ(p(Tα, α)) − φ(p(Tα, α)) < ψ(p(Tα, α)).It is a contradiction.Hence fα = Tα = α.Hence α is a common fixed point of f and T.Let β be another common fixed point of f and T such that α ≠ β. ψ(p(α, β)) = ψ(p(Tα, Tβ))  1 + p(α , α )     − ϕ  max p(β , β ) 1 + p(α , α ) , p(α , β )     ≤ ψ max p( β , β )  , p(α , β )       1 + p(α , β )     1 + p(α , β )   = ψ(p(α, β)) – φ(p(α, β)), from (p2) < ψ(p(α, β)).It is a contradiction. Hence α = β.Thus α is the unique common fixed point of T and f.Case(b): Suppose yn≠ yn+1for all n.ψ(p(yn, yn+1)) = ψ(p(Txn, Txn+1)) ≤ ψ(M(xn, xn+1)) − φ(M(xn, xn+1)) = ψ(max{p(yn, yn+1), p(yn, yn-1)}) − φ(max{p(yn, yn+1), p(yn, yn-1)}).If p(yn, yn+1)is maximum, thenψ(p(yn, yn+1)) ≤ ψ(p(yn, yn+1)) - φ(p(yn, yn+1)) < ψ(p(yn, yn+1)).It is a contradiction.Hence p(yn, yn-1) is maximum.Thereforeψ(p(yn, yn+1)) ≤ ψ(p(yn, yn-1)) - φ(p(yn, yn-1)) (3.1) ≤ ψ(p(yn, yn-1)).Since ψ is non – decreasing, we have p(yn, yn+1) ≤ p(yn, yn-1)Similarly p(yn+2, yn+1) ≤ p(yn, yn+1).Thus p(yn, yn+1) ≤ p(yn, yn-1), n = 1, 2, 3, …Thus { p(yn, yn+1) } is a non - increasing sequence of non-negative real numbers and must converge to a realnumber, say, L ≥ 0.Letting n→ ∞ in (3.1), we getψ(L) ≤ ψ(L) - φ(L) so that φ(L) ≤ 0. Hence L = 0. 31
4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.4, 2012Thus lim p(yn, yn+1) = 0. (3.2) n→∞Hence from (p2), lim p(yn, yn) = 0. (3.3) n→∞By definition of ps, we have ps(yn, yn+1) ≤ 2p(yn, yn+1).From (3.2), we have lim ps(yn, yn+1) = 0. (3.4) n→∞Now we prove that {yn} is Cauchy sequence in metric space (X, ps). On contrary suppose that {yn} is notCauchy. Then there exists an ε> 0 for which we can find two subsequences {ym(k)}, {yn(k)} of {yn} such thatn(k) is the smallest index for which n(k) > m(k) > k, ps(ym(k), yn(k)) ≥ ε (3.5)and ps(ym(k), yn(k)-1) < ε. (3.6)From (3.5) and (3.6),ε ≤ ps(ym(k), yn(k)) ≤ ps(ym(k), yn(k)-1) + ps(yn(k)-1, yn(k)) < ε + ps(yn(k)-1, yn(k))Letting k → ∞ and using (3.2), we have lim ps(ym(k), yn(k)) = ε. (3.7) k →∞From (p2), we get lim p(ym(k), yn(k)) = ε . (3.8) k →∞ 2Letting k → ∞ and using (3.2) and (3.7) in the inequality | ps(ym(k )-1, yn(k)) – ps(ym(k), yn(k)) | ≤ ps(ym(k), ym(k)-1)we get lim ps(ym(k )-1, yn(k)) = ε. (3.9) k →∞From (p2), we get lim p(ym(k )-1, yn(k)) = ε . (3.10) k →∞ 2Letting k → ∞ and using (3.2) and (3.7) in the inequality | ps(ym(k ), yn(k)+1) – ps(ym(k), yn(k)) | ≤ ps(yn(k), yn(k)+1)we get lim ps(ym(k ), yn(k)+1) = ε. (3.11) k →∞ 32
5. 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.4, 2012From (p2), we get lim p(ym(k ), yn(k)+1) = ε . (3.12) k →∞ 2ψ(p(ym(k ), yn(k)+1)) = ψ(p(Txm(k ), Txn(k)+1)) ≤ ψ(M(xm(k ), xn(k)+1)) – φ(M(xm(k ), xn(k)+1))    1 + p(y m(k) -1, y m(k) )    = ψ max p(y m(k) , y m(k) +1 ) , p(y m(k) -1 , y n(k) )    1 + p(y m(k) -1 , y n(k) )          1 + p(ym(k)-1, y m(k) )    - ϕ  max p(ym(k) , ym(k) +1 ) , p(ym(k)-1 , yn(k) )  .   1 + p(ym(k) -1, y n(k) )      Letting k → ∞ and using (3.12), (3.2), (3.10) and (3.8), we get ε ε ε ε ψ  ≤ ψ  − ϕ   < ψ  2 2 2 2It is a contradiction.Hence {yn} is Cauchy sequence in (X, ps).Thus lim ps(yn, ym) = 0. m,n→∞By definition of ps and from (3.2), we get lim ps(yn, ym) = 0. (3.13) m,n→∞Suppose f(X) is complete.Since {yn} ⊆ f(X) is a Cauchy sequence in the complete metric space (f(X), ps), it follows that {yn}converges in (f(X), ps).Thus lim ps(yn, α) = 0 for some α n →∞ ∈ f(X).There exists w ∈ X such that α = fw.Since {yn+1} is Cauchy in X and {yn} → α, it follows that {yn+1} → α.From Lemma 2.3(b) and (3.13), we havep(α, α) = lim p(yn, α) = lim p(yn+1, α) = lim p(yn, ym) = 0. (3.14) n→∞ n→∞ m,n→∞p(Tw, α) ≤ p(Tw, Txn+1) + p(Txn+1, α) – p(Txn+1, Txn+1) ≤ p(Tw, Txn+1) + p(Txn+1, α).Letting n→∞, we have p(Tw, α) ≤ lim p(Tw, Txn+1). n→∞Since ψ is is continuous and non - decreasing, we haveψ(p(Tw, α)) ≤ lim ψ(p(Tw, Txn+1)) n→∞ 33
6. 6. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.4, 2012   = lim ψ max p(yn , yn +1 ) 1 + p(α , Tw) , p(α , yn )    n→∞  1 + p(α , y n )      1 + p(α , Tw)  - lim ϕ  max p(yn , y n +1 ) , p(α , y n )  n→∞  1 + p(α , yn )     = ψ(0) – φ(0) = 0.It follows that Tw = α. Thus Tw = α = fw.Since the pair (f, T) is weakly compatible, we have fα = Tα.As in Case(a), it follows that α is the unique common fixed point of T and f.Example 3.2. Let X = [0, 1] and p(x, y) = max{x, y} for all x, y ∈X . Let T, f : X → X, f(x) = x/3and T(x) = x2/6, ψ : [0,∞) → [0,∞) by ψ(t) = t and φ: [0,∞) → [0,∞) by φ(t) = t/2 . Then the conditions(ii) and (iii) are satisfied andψ(p(Tx, Ty)) = p(Tx, Ty)  x 2 y2  = max  ,  6 6 x y ≤ max  ,  6 6 1 1  1 + p(fx, Tx)  = p(fx, fy) ≤ max p(fy, Ty) , p(fx, fy) 2 2  1 + p(fx, fy)  = ψ(M(x, y)) – φ(M(x, y)).  1 + p(fx, Tx) Where M(x, y) = max p(fy, Ty) , p(fx, fy)  1 + p(fx, fy) Thus (i) is holds.Hence all conditions of Theorem 3.1 are satisfied and 0 is the unique common fixed point of T and f.ReferencesAltun,I Sola.F and Simsek H, (2010), “Generalized contractions on partial metric spaces”, Topology and itsApplications.157, 6, 2778 - 2785.Heckmann. R, (1999), “Approximation of metric spaces by partial metric spaces”, Appl. Categ.Structures.No.1-2, 7, 71 - 83.Matthews.S.G, (1994), Partial metric topology, in: Proc. 8th Summer Conference on General Topology andApplications, in: Ann. New York Acad. Sci. Vol. 728, 1994, pp. 183 - 197.Oltra. S, Valero. O,(2004) “Banach’s fixed point theorem for partial metric spaces”, Rend.Istit. Mat. Univ.Trieste. Vol XXXVI, 17 - 26.O‘Neill.S.J, (1996)“Partial metrics, valuations and domain theory, Proc. 11th Summer Conference onGeneral Topology and Applications”, Annals of the New York Academy of Sciences, Vol806, pp. 304 - 315. 34
7. 7. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.4, 2012Romaguera. S, (2010) “A Kirk type characterization of completeness for partial metric spaces”, Fixed PointTheory. Vol.2010, Article ID 493298, 6 pages, doi:10.1155/2010/493298.Romaguera.S, Schellekens.M, (2005), “Partial metric monoids and semi valuation spaces”, Topology andApplications. 153, no.5-6, 948 - 962.Romaguera.S, Valero.O, (2009), ‘A quantitative computational modal for complete partial metric space viaformal balls” , Mathematical Structures in Computer Sciences. Vol.19, no.3, 541 - 563.Schellekens. M,(1995), “The Smyth completion: a common foundation for denotational semantics andcomplexity analysis “, Electronic Notes in Theoretical Computer Science. Vol.1, 535 - 556.Schellekens.M, (2003), “A characterization of partial metrizebility: domains are quantifiable”, TheoreticalComputer Sciences. Vol 305,no.1-3, 409 - 432.Waszkiewicz.P,(2003), “Quantitative continuous domains”, Applied Categorical Structures. Vol 11, no. 1,41 - 67.Waszkiewicz.P, (2006), Partial metrizebility of continuous posets, Mathematical Structures in ComputerSciences. Vol 16, no. 2, 2006, 359 - 372. 35