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ISSN: 2476-2296
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Common Fixed Point Theorem in T0 Quasi Metric Space

Balaji R Wadkar1, Ramakant Bhardwaj2, Lakshmi Narayan Mishra3* and Basant Singh1

1Department of Mathematics, AISECT University, Bhopal-Chiklod Road, Bhopal, Madhya Pradesh, India

2Department of Mathematics, TIT Group of Institutes, Anand Nagar, Bhopal, Madhya Pradesh, India

3Department of Mathematics, Mody University of Science and Technology, Lakshmangarh, Sikar Road, Sikar, Rajasthan, India

*Corresponding Author:
Mishra LN
Department of Mathematics
Mody University of Science and Technology
Lakshmangarh, Sikar Road, Sikar
Rajasthan 332 311, India
Tel: +919913387604
E-mail: [email protected]

Received Date: January 02, 2017; Accepted Date: January 30, 2017; Published Date: February 08, 2017

Citation: Wadkar BR, Bhardwaj R, Mishra LN, Singh B (2017) Common Fixed Point Theorem in T0 Quasi Metric Space. Fluid Mech Open Acc 4: 143. doi: 10.4172/2476-2296.1000143

Copyright: © 2017 Wadkar BR, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this paper, we prove fixed point theorems for generalized C-contractive and generalized S-contractive mappings in a bi-complete di-metric space. The relationship between q- spherically complete T0 Ultra-quasi-metric space and bi-complete diametric space is pointed out in proposition 5.1. This work is motivated by Petals and Fvidalis in a T0-ultraquasi- metric space 2000 AMS Subject Classification: 47H17, 74H05, 47H09.

Keywords

Fixed point; Generalized C-contraction; Generalized S-contraction; Spherically complete; Bi-complete di-metric

Introduction

In Agyingi [1] proved that every generalized contractive mapping defined in a q- spherically complete T0-ultra-quasi metric space has a unique fixed point. In Petals and Fvidalis [2] proved that every contractive mapping on a spherically complete non Archimedean normed space has a unique fixed point. Agyingi and Gega proved fixed point theorems in a T0-ultra-quasi-metric space [3-6]. Later many authors published number of papers in this space [7-10].

In this paper we shall prove a fixed point theorem for generalized c- contractive and generalized s-contractive mappings in a bi-complete di-metric space.

If we delete, in the used definition of the pseudo metric d on the set X. the symmetry condition, d(x, y)=d(y, x), whenever x, y ∈ X we are led to the concept of quasi-pseudo metric.

Definition 1.1: Let (X ,m) be a metric space. Let T: X→ X map is called a C-contraction if there exist, equation such that for all x, y ∈ X the following inequality holds [3],

equation

Definition 1.2: Let (X,m) be a metric space. A map T: X→ X is called a S-contraction if there exist equation such that for all x, y ∈ X the following inequality holds [3],

equation

Preliminaries

Now we recall some elementary definitions and terminology from the asymmetric topology which are necessary for a good understanding of the work below.

Definition 4.1: Let X be a non empty set. A function equation is called quasi pseudo metric on X if

d(x, x) = 0,∀x∈ X

d(x, z) ≤ d(x, y) + d( y, z),∀x, y, z∈ X

Moreover if d(x, y) = 0 = d( y, x)⇒ x = y then d is said to be a T0 quasi metric or di-metric. The latter condition is referred as the T0 condition.

Example 4.1: On R×R, we define the real valued map d given by equation then (R,d) is a di metric space.

Remark 4.1

Let d be quasi-pseudo metric on X, then the map d-1 defined by d-1 (x, y)=d(y, x) whenever x, y X is also a quasi-pseudo metric on X, called the conjugate of d.

It is also denoted by dt or d. It is easy to verify that the function ds defined by ds=d∨d-1

i.e. equation

defines a metric on X whenever d is a T0 quasi pseudo metric.

In some cases, we need to replace [0, ∞) by [0, ∞) (where for a d attaining the value ∞, the triangle inequality is interpreted in the obvious way). In such case we speak of extended quasi- pseudo metric.

Definition 4.3: The di metric space (X, d) is said to be bi complete if the metric space (R, ds) is complete.

Example 4.2: Let X=[0, ∞) define for each x, y∈ X , n(x, y) = x if x > y and n(x, y) = 0 if x < y . It is not difficult to check that (X,n) is a T0 quasi pseudo metric space [9]. Notice that, for x, y∈[0,∞) , we have equation and equation the matrix ns is complete on (X ,d).

Definition 4.3: Let (X ,d) be quasi pseudo metric space, for x,∈ X&∈> 0

equation

denotes the open ∈ − ball at x. The collection of such balls is a base for a topology τ(d) induced by d on X. Similarly for x,∈ X&∈≥ 0

equation

denotes the closed ∈ − ball at x.

Definition 4.4: Let (X ,d) be quasi pseudo metric space Let (xi),i∈I be a family of points in X and let equation be a family of non negative real numbers.

We say that equation has the mixed binary intersection property provided that

equation

Definition 4.5: Let (X ,d) be quasi pseudo metric space we say that (X ,d) is Isbell complete provided that each family equation that has the mixed binary intersection property is such that

equation

Proposition 4.1: If (X ,d) is an extended Isbell-complete quasipseudo metric space then (X ,ds) is hyper complete. An interesting class of quasi pseudo metric space, for which, we investing a type of completeness are the ultra quasi pseudo metric.

Definition 4.6: Let X be a set & d : X × X →[0,∞) be function a function mapping into the set [0,∞) of non negative real’s then d is ultra quasi pseudo metric on X if

d(x, x) = 0 for all x in X &

d(x, z) ≤ max{d(x, y),d( y, z)} whenever x, y, z∈ X

The conjugate d–1 of d where d–1(x, y) = d( y, x) whenever x, y∈ X is also an ultra quasi pseudo metric on X.

If d also satisfies the T0 – condition, then d is called a T0- ultra quasi metric on X. Notice that equation is an ultra-metric on X whenever d is a T0- ultra quasi metric.

In a literature, T0- ultra quasi metric spaces are also known as non Archimedean T0- quasi metric.

q-spherically Completeness

In this section we shall recall some results about q- spherical completeness belonging mainly to [8].

Definition 5.1: Let (X ,d) be an ultra–quasi pseudo metric space Let equation be a family of points in X and let equation be (X ,d) is q- spherical complete provided that each family [8]

equation

Satisfying equation, whenever i, j∈I is such that

equation

Proposition 5.1: Each q- spherically complete T0 ultra quasi metric space (X, d) is bi-complete [8].

Main Results

We recall the following interesting results respectively due to Chatterji [3] and to Shukla [4]

Theorem 6.a A C- contraction on a complete metric space has a unique fixed point [3].

Theorem 6.b A S- contraction on a complete metric space has a unique fixed point [4]

Following results generalizes the above theorem to setting of a bicomplete di-metric space.

Definition 6.1: Let (X ,d) be a quasi pseudo metric space. A map T : X → X is called a c-pseudo contraction if there exist equation such that for all x, y ∈ X the following inequality holds.

d(Tx,Ty) ≤ k [d(Tx, x) + d( y,Ty)]

Definition 6.2: Let (X ,d) be a quasi-pseudo metric space. A map T : X → X is called a S-pseudo contraction if there exist equation such that for all x, y ∈ X the following inequality holds.

d(Tx,Ty) ≤ k [d(Tx, x) + d( y,Ty) + d(x, y)]

Now we define following definitions

Definition 6.3: Let (X ,d) be a quasi-pseudo metric space. A map T : X → X is called a generalized c-pseudo contraction if there exist k, equation such that for all x, y ∈ X the following inequality holds.

equation

Definition 6.4: Let (X ,d) be a quasi-pseudo metric space. A map T : X → X is called a generalized S-pseudo contraction if there exist k, equation such that for all x, y ∈ X the following inequality holds.

equation

Theorem 6.1:

Let (X ,d) be a bi complete di metric space and let T : X → X be a generalized c- pseudo contraction then T has a unique fixed point.

Proof: Since T : X → X is a generalized c-pseudo contraction then there exist equation such that for all x, y∈ X the following inequality holds:

d(Tx,T) ≤ k{d(,Tx, x) + d( y,Ty) + d(Tx, y) + d(x,Ty)}

We shall first show that equation is a generalized ccontraction.

Since for any x, y ∈ X we have

equation

We see that equation is a generalized C-pseudo contraction therefore

equation

and

equation

equation

Hence equation, for all x, y ∈ X

and so equation is a generalized C- contraction.

By assumption (X,d) is a bi complete. Hence (X,ds) is complete. There fore by theorem (4a) T has a unique fixed point. This completes the proof.

Corollary 6.1: Let (X,d) be a T0-Isbell-Complete quasi pseudo metric spaces and T : X → X be a generalized c-pseudo contraction then T has a unique fixed point.

The proof follows from the proposition 2.1

Corollary 6.2: Any generalized c- pseudo contraction on a q-spherically complete T0 ultra quasi metric space has a unique fixed point.

The proof follows from the proposition 3.1

Theorem 6.2:

Let (X,d) be a bi complete di metric space and let T : X → X be an generalized S pseudo contraction then T has a unique fixed point

Proof: As in the previous proof it is enough to prove that equation is an generalized S –contraction.

Since T : X → X be a S –pseudo contraction then there exist k, equation such that for all x, y ∈ X the following inequality holds:

equation

We shall first show that equation is a generalized Ccontraction.

Since for any x, y ∈ X we have

equation

We see that equation is a pseudo contraction.

Therefore

equation

and

equation

Hence

equation and so T : (X,ds)→(X,ds) is a generalized s-contraction.

By assumption (X,d) is a bi complete. Hence (X,ds) is complete. There fore by theorem (4a) T has a unique fixed point. This completes the proof.

Corollary 6.3: Let ( X ,d ) be a T0-Isbell-Complete quasi pseudo metric spaces and T : X → X be a pseudo contraction then T has a unique fixed point.

The proof follows from the proposition 2.1

Corollary 6.4: Any s-pseudo contraction on a q-spherically complete T0 ultra quasi metric space has a unique fixed point.

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