Medical, Pharma, Engineering, Science, Technology and Business

**Balaji R Wadkar ^{1}, Ramakant Bhardwaj^{2}, Lakshmi Narayan Mishra^{3*} and Basant Singh^{1}**

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

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

^{3}Department 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.

**Visit for more related articles at** Fluid Mechanics: Open Access

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.

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

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 T_{0}-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, such that for all x, y ∈ X the following inequality holds [3],

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

**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 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 T_{0} quasi metric or di-metric. The latter condition is referred as the T_{0} condition.

Example 4.1: On R×R, we define the real valued map d given by 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 d^{t} or d^{–}. It is easy to verify that the function d^{s} defined by d^{s}=d∨d^{-1}

i.e.

defines a metric on X whenever d is a T_{0} 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, d^{s}) 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 and the matrix n^{s} is complete on (X ,d).

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

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

denotes the closed ∈ − ball at x.

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

We say that has the mixed binary intersection property provided that

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

**Proposition 4.1:** If (X ,d) is an extended Isbell-complete quasipseudo metric space then (X ,d^{s}) 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 T_{0} – condition, then d is called a T_{0}- ultra quasi metric on X. Notice that is an ultra-metric on X whenever d is a T_{0}- ultra quasi metric.

In a literature, T_{0}- ultra quasi metric spaces are also known as non Archimedean T_{0}- 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 be a family of points in X and let be (X ,d) is q- spherical complete provided that each family [8]

Satisfying , whenever i, j∈I is such that

**Proposition 5.1:** Each q- spherically complete T_{0} ultra quasi metric space (X, d) is bi-complete [8].

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 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 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, such that for all x, y ∈ X the following inequality holds.

**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, such that for all x, y ∈ X the following inequality holds.

**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 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 is a generalized ccontraction.

Since for any x, y ∈ X we have

We see that is a generalized C-pseudo contraction therefore

and

Hence , for all x, y ∈ X

and so is a generalized C- contraction.

By assumption (X,d) is a bi complete. Hence (X,d^{s}) 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 T_{0}-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 T_{0} 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 is an generalized S –contraction.

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

We shall first show that is a generalized Ccontraction.

Since for any x, y ∈ X we have

We see that is a pseudo contraction.

Therefore

and

Hence

and so T : (X,d^{s})→(X,d^{s}) is a generalized s-contraction.

By assumption (X,d) is a bi complete. Hence (X,d^{s}) 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 T_{0} ultra quasi metric space has a unique fixed point.

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