Some Fixed Point Results in S-Metric Spaces

K. Prudhvi

Journal of Mathematical Sciences and Applications

Some Fixed Point Results in S-Metric Spaces

K. Prudhvi

Department of Mathematics, University College of Science, Saifabad, Osmania University, Hyderabad, Telangana State, India

Abstract

In this paper, we prove some fixed point results on complete S-metric spaces. Our results extend and improve some recent results in the references.

Cite this article:

  • K. Prudhvi. Some Fixed Point Results in S-Metric Spaces. Journal of Mathematical Sciences and Applications. Vol. 4, No. 1, 2016, pp 1-3. http://pubs.sciepub.com/jmsa/4/1/1
  • Prudhvi, K.. "Some Fixed Point Results in S-Metric Spaces." Journal of Mathematical Sciences and Applications 4.1 (2016): 1-3.
  • Prudhvi, K. (2016). Some Fixed Point Results in S-Metric Spaces. Journal of Mathematical Sciences and Applications, 4(1), 1-3.
  • Prudhvi, K.. "Some Fixed Point Results in S-Metric Spaces." Journal of Mathematical Sciences and Applications 4, no. 1 (2016): 1-3.

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1. Introduction

In 2006, Z. Mustafa and B. I. Sims [6] introduced the concept of G-metric space which is a generalization of metric space, and proved some fixed point theorems in G-metric space. Subsequently, many authors were proved fixed point theorems in G- metric space (see, eg. [3, 7, 11]). And B. C. Dhage [4] introduced the notion of D-metric space. In 2007, S. Sedghi, N. Shobe and H. Zhou [10] introduced D*- metric space which is a modification of D-metric space of [4] and proved some fixed point theorems in D*- metric space and later on many authors were proved fixed point theorems in D*- metric space (see, e.g. [1, 5]). In 2012, S. Sedghi et al. [9] introduced the notion of S-metric space which is a generalization of G-metric space of [4] and D*- metric space of [10] and proved some fixed point theorems on S-metric space. Recently, S. Sedghi, N.V. Dung [8] proved generalized fixed point theorems in S-metric spaces which is a generalization of [9]. In this paper, we proved some fixed point results on complete S-metric spaces. Our results extended and improved the results of [8].

2. Preliminaries

2.1. [2] Definition

Let X be a nonempty set. A metric on X is a function d: if there exists a real number such that the following conditions holds for all .

(i) if and only if .

(ii) .

(ii) .

The pair is called a B-metric space.

2.2. [9] Definition

Let X be a nonempty set. An S-metric on X is a function that satisfies the following conditions holds for all .

(i) if and only if .

(ii) .

The pair is called an S-metric space.

2.3. [9] Definition

Let be an S-metric space. For and , we define the open ball and the closed ball with centre x and radius r as follows

The topology induced by the S-metric is the topology generated by the base of all open balls in X.

2.4. [9] Definition

Let be an S-metric space. A sequence converges to if as . That is, for each , there exists such that for all we have . We write for .

3. Main Results

In this section, we have proved some fixed point theorems on complete S- metric spaces.

S. Sedghi, N.V. Dung [8] introduced an implicit relation to investigate some fixed point theorems on S-metric spaces.

Let be the family of all continuous functions of five variables

M: for some . We consider the following conditions.

(C1 ) For all , if with , then .

(C2 ) For all , if , then .

(C3) If for all , , then

Moreover, for all , .

The following theorem was proved in [8] (Theorem 2.6 of [8]).

3.1. [9] Theorem

Let T be a self-map on a complete S-metric space (X, S) and

for all and some Then we have

(i) If M satisfies the condition (C1), then T has a fixed point. Moreover, for any and the fixed point x, we have .

(ii) If M satisfies the condition (C2) and T has a fixed point, then the fixed point is unique.

(iii) If M satisfies the condition (C3) and T has a fixed point, then T is continuous at x.

3.2. Theorem

Let T be a self-map on a complete S-metric space (X, S) and

for some such that and for all . Then T has a unique fixed point in X. Moreover, if , then T is continuous at the fixed point.

Proof: The following ascertain is by using the Theorem 3.1 with

and for all . Indeed, M is continuous. First, we have,

So, if y ≤ M(x, x, 0, z, y) with z ≤ 2x + y, then

Therefore, T satisfies the condition (C1).

Next, if

then y = 0. Since .

Therefore, T satisfies the condition (C2).

Finally, if for , then

More over, , where .

Therefore, T satisfies the condition (C3).

3.3. Theorem

Let T be a self-map on a complete S-metric space and

for some such that and for all . Then T has a unique fixed point in X. Moreover, if , then T is continuous at the fixed point.

Proof: The following ascertain is by using the Theorem 3.1 with for some , and for all . Indeed, M is continuous. First, we have

So, if with , then

Therefore, T satisfies the condition (C1).

Next, if

then . Since .

Therefore, T satisfies the condition (C2).

Finally, if for , then

More over, , where .

Therefore, T satisfies the condition (C3).

3.4. Theorem

Let T be a self-map on a complete S-metric space and

for some such that and for all . Then T has a unique fixed point in X. Moreover, if , then T is continuous at the fixed point.

Proof: The following ascertain is by using the Theorem 2.1 with

for some , .

Indeed, M is continuous. First, we have

So, if with , then

Therefore, T satisfies the condition (C1).

Next, if

Then, . Since .

Therefore, T satisfies the condition (C2).

Finally, if for , then

More over,

Therefore, T satisfies the condition (C3).

References

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