Abstract

In this paper, we have introduced JS-multiplicative metric space and proved some fixed point theorems in this space. This new metric function is a generalized form of several functions such as multiplicative metric, dislocated multiplicative metric, multiplicative b-metric and multiplicative b-metric-like.

2010 Mathematics Subject Classification: 46S40; 47H10; 54H25.

1. Introduction

Ozaksar and Cevical ¹ investigated multiplicative metric space and proved its topological properties. Mongkolkeha et al. ² described the concept of multiplicative proximal contraction mapping and proved best proximity point theorems for such mappings. Recently, Abbas et al. ³ proved some common fixed point results of quasi weak commutative mappings on a closed ball in the setting of multiplicative metric spaces. They also describe the main conditions for the existence of common solution of multiplicative boundary value problem. For further results on multiplicative metric space, see ^{4, 5, 6, 7}. In 2017, Ali et al. ⁸ introduced the notion of -multiplicative and proved some fixed point result. As an application, they established an existence theorem for the solution of a system of Fredholm multiplicative integral equations. Bakht Zada and Usman Riaz ⁹ introduced the idea of multiplicative b-metric-like space. Jleli and Samet ¹⁰ introduce a new generalization of metric space called generalized metric space (Js-metric space) and proved some fixed point theorems (see ^{11, 12, 13} for further results).

In this paper, we present a new concept of Js multiplicative metric space that covers different spaces including multiplicative metric space, multiplicative b-metric space and multiplicative b-metric-like space. Also we prove Ciric type fixed point theorem and some fixed point theorems with partial order in Js multiplicative metric space.

2. Js Multiplicative Metric Space

Definition 2.1 Let and let be a mapping. For all we define the set as follows:

Definition 2.2 Let and be a given mapping. Then is called Js-multiplicative metric space, if it satisfies the following conditions:

For all we have and

For all we have

there exists such that if for all then

The pair is called a Js multiplicative metric space.

Remark 2.3 Clearly if the set is empty for all then is a Js multiplicative metric space if and only if and are satisfied.

Example 2.4 Let and let be define by

where be a finite fixed real number is a Js multiplicative metric space for all Let then

So except possibly for finite number of terms. Let be the smallest natural number such that then

So hold with Clearly, and hold. Thus is Js multiplicative metric space.

Definition 2.5 A sequence in Js multiplicative metric space is converges to if

Proposition 2.6 Let be Js multiplicative metric space. Suppose is a sequence in and If is convergent to both and then

Definition 2.7 A sequence in Js multiplicative metric space is called Cauchy, if

Definition 2.8 A Js multiplicative metric space is called complete if every Cauchy sequence in it is convergent to some element in

Definition 2.9 Let and a function is called a multiplicative b-metric-like on if satisfies the following conditions for all and a constant

then

The pair is called a multiplicative b-metric-like space. If we take then becomes dislocated multiplicative metric space. If we take then then becomes multiplicative b-metric space. If we take then and then becomes multiplicative metric space.

Proposition 2.10 Every multiplicative b-metric-like space, multiplicative b-metric space, dislocated multiplicative metric space and multiplicative metric space is Js multiplicative metric space.

Let and let be a given mapping. For all let us define the set as below

Definition 2.11 ¹⁰

Let and be a given mapping. Then is called Js-metric space if it satisfies the following conditions for all :

;

;

there exists such that and

Remark 2.12 Every Js metric space generates a Js multiplicative metric space defined as

3. Ciric Type Result in a Js multiplicative Metric Space

This section deals with fixed point for Ciric type mappings in Js multiplicative metric space.

Definition 3.1 Let be a Js multiplicative metric space and be a self mapping. Let then is called Js k-quasicontraction if

where

Proposition 3.2 Let be a Js multiplicative metric space and be a Js k-quasicontraction for some Then any fixed point of satisfies

Theorem 3.3 Let be complete Js multiplicative metric space and let be a Js k-quasicontraction mapping for some such that If there exists an element such that then the sequence converges to and if and then ‘’ is the fixed point of . Moreover for each fixed point of such that and then

Proof We shall prove that is a Cauchy sequence. Let as is a Js k-quasicontraction, for each we have

(1)

As

So, (1) implies

So, we obtain

(2)

Taking (2) into account and by the Definition of for every we have

Using the fact that and we get

This shows that is a Cauchy sequence. By completeness of we must have some such that is convergent to . Now, we suppose that Using the inequality (2)

(3)

For every by the property there exists some constant such that

(4)

Now,

By using (3) and (4), we obtain

Again, using the above inequality, we have

Consequently, we get

for all Therefore, we obtain

Since and Using the property we get

which implies that

Since and Then is the fixed point of By using Proposition 3.2, we have

For uniqueness, suppose that is another fixed point of such that and By using Proposition 3.2, we have

Since is a Js k-quasicontraction, we obtain

This implies that

Corollary 3.4 Let be a complete dislocated multiplicative metric space and let a mapping for which there exists such that

where

if there exists such that

Then has a unique fixed point. Moreover, the sequence converges to fixed point of

Corollary 3.5 Let be a complete multiplicative b-metric space with constant and let be a mapping for which there exists with such that

if there exist we have

Then has a unique fixed point. Moreover, the sequence converges to fixed point of

Example 3.6 Let and where (a finite fixed real number). Clearly is Js multiplicative metric space for all Let us define a function by

Thus, with

Let then it is clear that Now and so on. Clearly is Cauchy sequence and all the properties of Theorem 3.3 are satisfied. So has a unique fixed point.

4. Results in a Js Multiplicative Metric Space with Partial Order

Definition 4.1 Let be a Js multiplicative metric space with partial order and let a mapping. Then is weakly continuous if converges to then there exists a subsequence of such that is convergent to as

Definition 4.2 Let with partial order A mapping is called nondecreasing if

Definition 4.3 The pair is called regular if for every sequence satisfies for each with being convergent to then there exist a subsequence of such that for every

Definition 4.4 A function is called weakly Js k-contraction for some if or we have

Theorem 4.5 Let be a complete Js multiplicative metric space with partial order and let be a function. Assume that the following conditions satisfied:

(i) be a weakly continuous;

(ii) be a weakly Js k-contraction for some

(iii) there exists such that and

(iv) is nondecreasing.

Then, converges to such that is fixed point of Moreover, if then

Proof. Since is nondecreasing and then for all we obtain

By transitivity of for every we have

There, for each and are always comparable. As is weak Js k-contraction for each we have

This implies that

So, we obtain that

(5)

Taking (5) into account and by the definition of for every we obtain

Using the fact that and we get

This shows that is a Cauchy sequence. By completeness of , we must have some such that is convergent to Since is weakly continuous so there is a subsequence of such that is convergent to as By uniqueness of the limit, we get and is a fixed point of Now if then as and is weak js k-contraction, we have

which is possible only if

The weak continuity assumption of in the previous theorem can be replaced by Definition 4.3 to obtain the following result.

Theorem 4.6 Let be a complete Js multiplicative metric space with partial order and let be a function. Assume that the following conditions satisfied:

(i) is regular;

(ii) be a weakly Js k-contraction for some

(iii) there exists such that and

(iv) is nondecreasing.

Then converges to such that is fixed point of Moreover, if then

Proof. As we have proved in previous proof that is convergent to Moreover, we have

Since is regular, there exists a subsequence of such that for each As is weak Js k-contraction, we have

Using the inequality above, we get

This implies that is converges to By uniqueness of the limit, we obtain

As in the previous proof

Competing Interests

The authors declare that they have no competing interests.

References