A Unique Common Fixed Point Theorem Using ψ − ϕ Condition in a Partial Metric Space Using an ICS Mapping

(1)

A Unique Common Fixed Point Theorem Using

ψ

−

ϕ

Condition in a Partial Metric Space Using

an ICS Mapping

K.P.R.Rao

1

_,

_{G.N.V.Kishore}

2,∗

_,

_{P.R.Sobhana Babu}

3

1_{Department of Mathematics, Acharya Nagarjuna University,Nagarjuna Nagar,}

Guntur - 522 510, Andhra Pradesh, India

2_{Department of Mathematics, Baba Institute of Technology and Sciences,P.M.Palem,}

Madhurawada Visakhapatnam - 530048, Andhra Pradesh, India

3_{Department of Mathematics, Ramachandra College of Engineering, Vatluru(V),}

Eluru-534007, West Godavari Dist., Andhra Pradesh, India ∗_{Corresponding Author: [email protected]}

Abstract

The ICS mapping was introduced by K.P.Chi [On a fixed point theorem for certain class of maps satisfying a contractive condition depended on an another function, Lobachevskii J. math., 30(4), 2009, 289 - 291.] In this paper, we obtain a unique common fixed point theorem in partial metric spaces by using ICS mapping and also introduced supported example to our main theorem.

Keywords

Partial Metric, ICS Mapping, Complete Space

Mathematics Subject Classification (2000): 54H25, 47H10, 54E50

1 Introduction and Preliminaries

The notion of partial metric space was introduced by Matthews [10] as a part of the study of denotational semantics of data flow networks. In fact, it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation and domain theory in computer science (see e.g. [12, 13, 14, 15, 16, 17, 18, 19, 20]).

Matthews [10, 11], Oltra and Valero [23] and Romaguera [21] and Altun et al. [2] proved some fixed point theorems in partial metric spaces for a single map (see also [1, 6, 7, 8, 9, 3, 4, 22, 17]).

First we recall some basic definitions and lemmas which play crucial role in the theory of partial metric spaces. Definition 1.1 (See [10, 11]) A partial metric on a nonempty setX is a functionp:X×X →R+ _{such that for} allx, 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).

The pair(X, p)is called a partial metric space (PMS).

Clearlyp(x, y) = 0 impliesx=y andx̸=y impliesp(x, y)>0.

Ifpis a partial metric on X, then the functiondp:X×X→R+ given bydp(x, y) = 2p(x, y)−p(x, x)−p(y, y) is

(2)

Example 1.2 (See e.g. [11, 7, 1]) ConsiderX= [0,∞)withp(x, y) = max{x, y}. Then(X, p)is a partial metric space. It is clear that pis not a (usual) metric. Note that in this case dp(x, y) =|x−y|.

Each partial metric pon X generates a T0 topology τp on X which has as a base the family of open p-balls

{Bp(x, ε), x∈X, ε >0}, where Bp(x, ε) ={y∈X :p(x, y)< p(x, x) +ε}for allx∈X andε >0.

We now state some basic topological notions (such as convergence, completeness, continuity) on partial metric spaces (see e.g. [10, 11, 2, 1, 7, 9].)

Definition 1.3

1. A sequence{xn} in the PMS (X, p) converges to the limitxif and only if p(x, x) = lim

n→∞p(x, xn).

2. A sequence{xn} in the PMS (X, p) is called a Cauchy sequence if

lim

n,m_→∞p(xn, xm)exists and is finite.

3. A PMS (X, p)is called complete if every Cauchy sequence{xn} inX

converges with respect to τp, to a point x∈X such that

p(x, x) = lim

n,m_→∞p(xn, xm).

4. A mapping F : X → X is said to be continuous at x0 ∈X if for every ϵ >0, there exists δ > 0 such that

F(Bp(x0, δ))⊆Bp(F x0, ϵ).

We need the following lemmas in PMS([10, 11, 1, 2, 7, 9]). Lemma 1.4

1. A sequence {xn}is a Cauchy sequence in the PMS(X, p)if and only if it is a Cauchy sequence in the metric

space(X, dp).

2. A PMS (X, p)is complete if and only if the metric space(X, dp)is complete. Moreover

lim

n→∞dp(x, xn) = 0⇔p(x, x) = limn→∞p(x, xn) =n,mlim→∞p(xn, xm) (1.1)

Lemma 1.5 Assume xn →z asn→ ∞ in a PMS(X, p) such thatp(z, z) = 0. Then limn→∞p(xn, y) = p(z, y)

for everyy∈X.

In this paper, we obtain a unique common fixed point theorem for two mappings using ICS mapping in Partial metric spaces. Our result generalizes the recent several known results.

Recently [5] introduced the concept of ICS mapping as follows.

Definition 1.6 [5] Let (X, d) be a metric space. A mapping T : X → X is said to be ICS if T is injective, continuous and has the property : for every sequence{xn}inX,if{T xn}is convergent then{xn}is also convergent.

2 MAIN RESULT

Let Ψ denote the set of all continuous and monotonically increasing functionsψ: [0,∞)→[0,∞).

Let Φ denote the set of all lower semi continuous functionsϕ: [0,∞)→[0,∞) such thatϕ(t)>0 fort >0.

Theorem 2.1 Let (X, p) be a partial metric space and T : X → X be an ICS mapping and F, G : X → X be satisfying

ψ(p(T F x, T Gy)) ≤ψ

(

max

{

p(T x, T y), p(T x, T F x), p(T y, T Gy),

1

2[p(T x, T Gy) +p(T y, T F x)] })

−ϕ

(

max

{

1

2[p(T x, T Gy) +p(T y, T F x)]

})

,

∀x, y∈X, whereψ∈Ψandϕ∈Φ. Then F andGhave a unique common fixed point inX. Letx0∈X. Definex2n+1=F x2n, x2n+2=Gx2n+1, n= 0,1,2,· · · and yn=T xn, n= 0,1,2,· · ·

Case(a): Supposey2n+1=y2n for somen.

ThenT x2n+1 =T x2n.

(3)

SupposeT α̸=T Gα. Consider

ψ(p(T α, T Gα)) = ψ(p(T F α, T Gα))

≤ ψ

(

max

{

p(T α, T α), p(T α, T F α), p(T α, T Gα),

1

2[p(T α, T Gα) +p(T α, T F α)]

})

−ϕ

(

max

{

p(T α, T α), p(T α, T F α), p(T α, T Gα),

1

2[p(T α, T Gα) +p(T α, T F α)]

})

= ψ(p(T α, T Gα))−ϕ(p(T α, T Gα)), from (p2)

< ψ(p(T α, T Gα)),

a contradiction.

HenceT α=T Gα.SinceT is injective, we haveα=Gα.

Thusαis a common fixed point of F andG.

Ifβ is another common fixed point ofF andG,thenT α̸=T β. ψ(p(T α, T β)) = ψ(p(T F α, T Gβ))

≤ ψ

(

max

{

p(T α, T β), p(T α, T F α), p(T β, T Gβ),

1

2[p(T α, T Gβ) +p(T β, T F α)]

})

−ϕ

(

max

{

p(T α, T β), p(T α, T F α), p(T β, T Gβ),

1

2[p(T α, T Gβ) +p(T β, T F α)]

})

= ψ

(

max

{

p(T α, T β), p(T α, T α), p(T β, T β),

1

2[p(T α, T β) +p(T β, T α)]

})

−ϕ

(

max

{

p(T α, T β), p(T α, T α), p(T β, T β),

1

2[p(T α, T β) +p(T β, T α)]

})

= ψ(p(T α, T β))−ϕ(p(T α, T β)), from (p2)

< ψ(p(T α, T Gα)),

a contradiction.

Thusαis the unique common fixed point of F andG. Case(b) : Assume thatyn̸=yn+1 for alln.

Denotepn=p(yn, yn+1).

ψ(p2n) = ψ(p(y2n, y2n+1))

= ψ(p(T x2n+1, T x2n))

= ψ(p(T F x2n, T Gx2n−1))

≤ ψ

({

p(y2n, y2n₋1), p(y2n, y2n+1), p(y2n₋1, y2n),

1

2[p(y2n, y2n) +p(y2n−1, y2n+1)]

})

−ϕ

({

p(y2n, y2n−1), p(y2n, y2n+1), p(y2n−1, y2n),

1

2[p(y2n, y2n) +p(y2n−1, y2n+1)]

})

= ψ(maxp2n−1, p2n)−ϕ(maxp2n−1, p2n), from (p4)

Ifp2n is maximum, thenψ(p2n)≤ψ(p2n)−ϕ(p2n)< ψ(p2n),a contradiction.

Hence

ψ(p2n) ≤ ψ(p2n₋1)−ϕ(p2n₋1) (2.1)

< ψ(p2n−1).

p2n< p2n₋1, sinceψis monotonically non - decreasing.

Similarly, we can show that p2n+1 < p2n. Thus {pn} is monotonically decreasing sequence of non negative real

numbers and hence{pn} converges to somer≥0.Supposer >0.

Lettingn→ ∞in (2.1), we get

ψ(r)≤ψ(r)−ϕ(r)< ψ(r), sinceϕ(t)>0 for t >0.

It is a contradiction . Hencer= 0.

Thus

lim

n→∞p(yn, yn+1) = 0. (2.2)

Hence from (p2), we get

lim

(4)

From the definition ofdp, using (2.2) and(2.3), we get

lim

n→∞dp(yn, yn+1) = 0. (2.4)

Now we prove that{yn} is a Cauchy sequence in (X, dp).

On contrary suppose that{y2n}is not Cauchy.

Then there exist an ϵ > 0 and monotone increasing sequences of natural numbers {2mk} and {2nk} such that

nk> mk,

dp(y2mk, y2nk)≥ϵ (2.5)

and

dp(y2mk, y2nk−2)< ϵ. (2.6)

From (2.5) and (2.6), we obtain

ϵ ≤dp(y2mk, y2nk)

≤dp(y2mk, y2nk−2) +dp(y2nk−2, y2nk−1) +dp(y2nk−1, y2nk)

< ϵ+dp(y2nk−2, y2nk−1) +dp(y2nk−1, y2nk). Lettingk→ ∞and then using (2.4), we get

lim

k→∞dp(y2mk, y2nk) =ϵ. (2.7)

Hence from definition ofdp and (2.3), we have

lim

k_→∞p(y2mk, y2nk) =

ϵ

2. (2.8)

Lettingk→ ∞and then using (2.7) and (2.4) in

|dp(y2nk+1, y2mk)−dp(y2nk, y2mk)| ≤dp(y2nk+1, y2nk) we obtain

lim

k_→∞dp(y2nk+1, y2mk) =ϵ. (2.9)

Hence, we have

lim

k_→∞p(y2nk+1, y2mk) =

ϵ

2. (2.10)

|dp(y2nk, y2mk−1)−dp(y2nk, y2mk)| ≤dp(y2mk−1, y2mk), we get

lim

k_→∞dp(y2nk, y2mk−1) =ϵ. (2.11)

Hence, we have

lim

k→∞p(y2nk, y2mk−1) =

ϵ

2. (2.12)

|dp(y2mk−1, y2nk+1)−dp(y2mk−1, y2nk)| ≤dp(y2nk+1, y2nk) we obtain

lim

k→∞dp(y2mk−1, y2nk+1) =ϵ. (2.13)

Hence, we get

lim

k→∞p(y2mk−1, y2nk+1) =

ϵ

2. (2.14)

Now,

ψ(p(y2nk+1, y2mk)) = ψ(p(T F x2nk, T Gx2mk−1))

≤ ψ



max

  

p(y2nk, y2mk−1), p(y2nk, y2nk+1),

p(y2mk−1, y2mk),

1

2[p(y2nk, y2mk) +p(y2mk−1, y2nk+1)]

  

 

−ϕ



max

  

p(y2nk, y2mk−1), p(y2nk, y2nk+1),

p(y2mk−1, y2mk),

1

2[p(y2nk, y2mk) +p(y2mk−1, y2nk+1)]

  



(5)

ψ

(_ϵ

2

)

≤ ψ

(

max

{

ϵ

2,0,0, 1 2

[_ϵ

2 +

ϵ

2

]})

−ϕ

(

max

{

ϵ

2,0,0, 1 2

[_ϵ

2 +

ϵ

2

]})

= ψ

(_ϵ

2

)

−ϕ

(_ϵ

2

)

< ψ

(_ϵ

2

)

,

a contradiction. Hence{y2n}is Cauchy.

Lettingn, m→ ∞in

|dp(y2n+1, y2m+1)−dp(y2n, y2m)| ≤dp(y2n+1, y2n) +dp(y2m, y2m+1)

we get lim

n,m_→∞dp(y2n+1, y2m+1) = 0.

Hence{y2n+1} is Cauchy.

Thus{yn}is a Cauchy sequence in (X, dp).

Hence, we have lim

n, m_→∞dp(yn, ym) = 0.

Now, from the definition ofdpand from (2.3), we obtain

lim

n, m_→∞p(yn, ym) = 0. (2.15)

SinceX is complete and{yn}is a Cauchy sequence in complete metric space (X, dp).

Thus

lim

n_→∞dp(yn, T z) = 0 for someT z∈X

Also T is an ICS mapping and {yn} ={T xn} is convergent, it follows that {xn} is convergent to some z ∈ X.

i.e. lim

n_→∞p(xn, z) =p(z, z).

Since T is continuous, from above we have lim

n→∞p(T xn, T z) =p(T z, T z).

By Lemma 1.4(2), we have that

p(T z, T z) = lim

n→∞p(T xn, T z) = limn→∞p(yn, T z) =n,mlim→∞p(yn, ym). (2.16)

From (2.15) and (2.16), we have

p(T z, T z) = 0. (2.17)

SupposeT z̸=T F z.

Consider

ψ(p(T F z, y2n+2)) =ψ(p(T F z, T x2n+2))

=ψ(p(T F z, T Gx2n+1))

≤ψmax

({

p(T z, T x2n+1), p(T z, T F z), p(T x2n+1, T x2n+2), 1

2[p(T z, T x2n+2) +p(T x2n+1, T F z)]

})

−ϕmax

({

p(T z, T x2n+1), p(T z, T F z), p(T x2n+1, T x2n+2), 1

2[p(T z, T x2n+2) +p(T x2n+1, T F z)]

})

≤ψmax

({

p(T z, y2n+1), p(T z, T F z), p(y2n+1, y2n+2), 1

2[p(T z, y2n+2) +p(y2n+1, T F z]

})

−ϕmax

({

p(T z, y2n+1), p(T z, T F z), p(y2n+1, y2n+2), 1

2[p(T z, y2n+2) +p(y2n+1, T F z]

})

Lettingn→ ∞and using Lemma 1.5 and (2.17), we get

ψ(p(T F z, T z)) ≤ψmax({ p(T z, T z), p(T z, T F z),0,1₂[p(T z, T z) +p(T z, T F z] })

−ϕmax({ p(T z, T z), p(T z, T F z),0,1₂[p(T z, T z) +p(T z, T F z] }) =ψ(p(T z, T F z))−ϕ(p(T z, T F z))

< ψ(p(T z, T F z)), sinceϕ(t)>0 fort >0.

It is a contradiction. HenceT F z=T z. SinceT is injective, we haveF z=z.

As in case(a),z is the common fixed point ofF andG.

Example 2.2 Let X = [0,1] andp(x, y) = max{x, y} for all x, y∈X. Then(X, p) is a complete partial metric space. LetT :X →X andF, G :X →X defined byT(x) = x₂,F(x) = ₂_xx₊₄ and G(x) = ₄_xx₊₂2 . Then it is clear that T is an ICS mapping. Defineψ∈Ψ, ϕ∈Φby ψ(t) =t andϕ(t) = t

(6)

Also

ψ(p(T F x, T Gy)) = max{F x₂ ,Gy₂ }

= 1₂max{₂_xx₊₄,₄_yy₊₂2 }

= 1₄max{_xx₊₂,_yy₊21 2}

≤ 1

4max{x, y}

= 1₂max{x₂,y₂}

= 1₂p(T x, T y)

≤ 1 2max

{

1

2[p(T x, T Gy) +p(T y, T F x)] }

=ψ

(

max

{

1

2[p(T x, T Gy) +p(T y, T F x)] })

−ϕ

(

max

{

1

2[p(T x, T Gy) +p(T y, T F x)]

})

Clearly0 is unique common fixed point ofF andG.

Corollary 2.3 Let(X, p)be complete partial metric space andT :X →X be an ICS mapping and F, G:X→X

be satisfying

p(T F x, T Gy) ≤φ

(

max

{

1

2[p(T x, T Gy) +p(T y, T F x)]

})

,

∀x, y∈X, whereφ: [0,∞)→[0,∞)is continous function withφ(t)< t fort >0. ThenF andGhave a unique common fixed point in X.

It follows from Theorem 2.1 if we putψ(t) =tandϕ(t) =t−φ(t) in Theorem 2.1. If we take F=Gin Corollary 2.3, we get

Corollary 2.4 Let(X, p)be complete partial metric space andT :X →X be an ICS mapping and F, G:X→X

be satisfying

p(T F x, T F y) ≤φ

(

max

{

p(T x, T y), p(T x, T F x), p(T y, T F y),

1

2[p(T x, T F y) +p(T y, T F x)]

})

,

∀x, y∈X, whereφ: [0,∞)→[0,∞)is continous function withφ(t)< t fort >0. ThenF has a unique fixed point in X.

Example 2.5 Let X = [0,1] andp(x, y) = max{x, y} for all x, y∈X. Then(X, p)is a complete partial metric space. LetT :X →X ddefined byT(x) =x, it is clearly T is an ICS mapping and F :X →X byF(x) = ₂_xx₊₃ andφ(t) =₂t.

Also

p(T F x, T F y) = max{₂_xx₊₃,₂_yy₊₃}

=1₂max{_x₊x3 2

,_y₊y3 2}

≤1

2max{x, y}

=1₂p(T x, T y)

≤1 2max

{

1

2[p(T x, T F y) +p(T y, T F x)] }

=φ

(

max

{

1

2[p(T x, T F y) +p(T y, T F x)]

})

Clearly0 is unique fixed point ofF.

REFERENCES

[1] T. Abdeljawad, E. Karapınar, K. Tas, Existence and uniqueness of a common fixed point on partial metric spaces, Appl. Math. Lett. 24 (11),1894–1899(2011) (2011).

[2] I. Altun, F. Sola and H. Simsek, Generalized contractions on partial metric spaces, Topology and its Applica-tions. 157 (18) (2010) 2778-2785.

(7)

Advanced Mathematical and Studies, Volume 4, Number 2, (2011).

[5] K.P.chi, On a fixed point theorem for certain class of maps satisfying a contractive condition depended on an another function, Lobachevskii J. math., 30(4), 2009, 289 - 291.

[6] D. Ilić, V. Pavlović, V. Rakoˇcević, Some new extensions of Banach’s contraction principle to partial metric spaces, Appl. Math. Letters, doi:10.1016/j.aml.2011.02.025.

[7] E. Karapınar, I. M. Erhan, Fixed point theorems for operators on partial metric spaces, Applied Mathematics Letters 24 (11),1900-1904 (2011), 10.1016/j.aml.2011.05.013.

[8] Karapınar, E.: Weak ϕ-contraction on partial contraction and existence of fixed points in partially ordered sets, Mathematica Aeterna, 1(4)237-244(2011).

[9] Karapınar, E.: Generalizations of Caristi Kirk’s Theorem on Partial metric Spaces, Fixed Point Theory and. Appl. 2011:4, doi:10.1186/1687-1812-2011-4

[10] S.G. Matthews. Partial metric topology. Research Report 212. Dept. of Computer Science. University of Warwick, 1992.

[11] S.G. Matthews, Partial metric topology, in Proceedings of the 8th Summer Conference on General Topology and Applications, vol. 728, pp. 183-197, Annals of the New York Academy of Sciences, 1994.

[12] M.Schellekens, The Smtth comletion: a common foundation for denotational semantics and complexity anal-ysis, Electronic Notes in Theoretical Computer Science, vol 1, 1995, 535 - 556.

[13] P.Waszkiewicz, Quantitative continuous domains, Applied Categorical Structures, vol 11, no. 1, 2003, 41 - 67. [14] P.Waszkiewicz, Partial metrizebility of continuous posets, Mathematical Structures in Computer Sciences, vol

16, no. 2, 2006, 359 - 372.

[15] R. Heckmann, Approximation of metric spaces by partial metric spaces, Appl. Categ. Structures,no.1-2, 7, 1999, 71-83.

[16] R. Kopperman, S.G. Matthews, and H. Pajoohesh: What do partial metrics represent?, Spatial represen-tation: discrete vs. continuous computational models, Dagstuhl Seminar Proceedings, No. 04351, Interna-tionales Begegnungs- und Forschungszentrum fr Informatik (IBFI), Schloss Dagstuhl, Germany, (2005). MR 2005j:54007

[17] S.J. ONeill: Two topologies are better than one, Tech. report, University of Warwick, Coventry, UK, http://www.dcs.warwick.ac.uk/reports/283.html, (1995).

[18] H.P.A. K¨unzi, H. Pajoohesh, and M.P. Schellekens: Partial quasi-metrics, Theoret. Comput. Sci. 365 no.3 (2006) 237-246. MR 2007f:54048

[19] S. Romaguera and M. Schellekens: Weightable quasi-metric semigroup and semilattices, Electronic Notes of Theoretical computer science, Proceedings of MFCSIT, 40, Elsevier, (2003).

[20] M.P. Schellekens: A characterization of partial metrizability: domains are quantifiable, Topology in computer science (Schlo Dagstuhl, 2000), Theoretical Computer Science 305 no. 1-3 (2003) 409-432. MR 2004i:54037 [21] S. Romaguera, A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory

and Applications, Volume 2010, Article ID 493298, 6 pages, 2010.

[22] O. Valero, On Banach fixed point theorems for partial metric spaces, Applied General Topology. 6 (2) (2005) 229-240.