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

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.

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.

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.

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]).

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.

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).

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).

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).

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