1. Introduction

The idea of b-metric space and partial metric space were introduced by S. Czerwik ^[4] and S. G. Matthews ^[12], respectively. S. Shukla ^[15] introduced another generalization which is called a partial b-metric space. Amini Harandi ^[9] introduced a new extension of the concept of partial metric space, called a metric-like space. After that, A. Alghamdi ^[1] introduce the concept of b-metric-like space which generalizes the idea of partial metric space, metric-like space, and b-metric space. They established the existence and uniqueness of fixed points in a b-metric-like space as well as in a partially ordered b-metric-like space.

In 2008, Bashirov et al. ^[3] studied the usefulness of a new calculus, called multiplicative calculus due to Michael Grossman and Robert Katz in the period from 1967 till 1970. By using the concepts of multiplicative absolute values, Bashirov et al. defined a new distance so called multiplicative distance. Also, Ozavsar and Cevikel ^[6] introduced the concept of multiplicative contraction mapings and derive some fixed point results on these mappings on a complete multiplicative metric space.

In this paper, by using the concept of multiplicity we first introduce concept of multiplicative partial metric space, multiplicative metric-like space, multiplicative b-metric space and then we introduce a new generalization of these spaces which is called multiplicative b-metric-like space. Then, we derive some fixed point results. Also, some examples and an application to integral equations are provided for the support of our constructed results.

2. Multiplicative (b)-metric-like Space

We shall begin this section with the introduction to the concept of multiplicative partial metric space.

Definition 2.1. A mapping where is called a multiplicative partial metric on provided that, for all

The pair is a multiplicative partial metric space.

Definition 2.2. A mapping where is called a multiplicative metric-like on provided that, for all

(Ψ1)

(Ψ2)

(Ψ3)

The pair is a multiplicative metric-like space.

S. Czerwik ^[4] give idea of multiplicative b-metric-like space. Now we introduce concept of multiplicative b-metric space.

Definition 2.3. A mapping is called a multiplicative b-metric on provided that, for all and a constant

The pair is a multiplicative b-metric space.

Definition 2.4. A mapping is called a multiplicative b-metric-like on provided that, for all and a constant

The pair is a multiplicative b-metric-like space.

Example 2.5. Suppose and let be defined by where be a fixed real number, is a multiplicative b-metric-like for all with

and so holds. Clearly, and hold.

Similarly.

Example 2.6. Suppose and let be defined by , where be a fixed real number, is a multiplicative b-metric-like space with

Example 2.7. Let The function defined by

where be a fixed real number, is a multiplicative b-metric-like with constant and so is a multiplicative b-metric like space.

As we know, if then

This implies that

Let be a multiplicative b-metric-like space. Then multiplicative open ball with center at and radius is,

Definition 2.8. Let is multiplicative b-metric-like space, and let be a sequence in Then point is the limit of the sequence if and we say that the sequence is convergent to

Definition 2.9. Let is multiplicative b-metric-like space.

A sequence is a Cauchy exists.

A multiplicative b-metric-like space is complete if every cauchy sequence in is convergent. That is,

Proposition 2.10. Let a multiplicative b-metric-like space and let a sequence in such that Then

1. x is unique;

2.

Proof. (1) Suppose there exists such that Then

Hence () gives y = x.

Proof. (2) As we know from ()

and so

Definition 2.11. Let a multiplicative b-metric-like space , let We say is an open subset of if for all there exists such that Also, is closed subset of if is open subset of .

Proposition 2.12. Let be a multiplicative b-metric-like space and let . Then is closed if and only if for any sequence we have ,

Proof. Let is closed and is an open set. Then there exists such that

Since as , then

Hence, there exists such that for all we have

So

which yield to contradiction, since

Conversely, let we have for any sequence in , such that Let Let us prove that for there exist Suppose to the contrary that for , we have Then, for all choose Therefore, for all Hence, as So supposition on implies which is wrong. Then, for all there exists such that That is, is closed.

Lemma 2.13. Suppose be a multiplicative b-metric-like space, and let Then

For Lemma (2.13), we deduce the following result.

Lemma 2.14. Suppose a sequence in multiplicative b-metric-like space such that

for some and each Then

Let be a multiplicative b-metric-like space. Define by

for all

3. Fixed Point of Expansion Mapping in Multiplicative (b)-metric-like Spaces

Many papers have been appeared on the work of expansive mapping see, e.g., (^{[1, 5, 11]}). In this paper we drive fixed point results for expansive mappings in multiplicative b-metric-like space to the corresponding results of A. Alghamdi(see - ^[1]).

Theorem 3.1. Let ba a complete multiplicative b-metric-like space. Suppose is onto mapping, such that

(3.1)

where Then has a fixed point.

Proof. Let as is onto, so there exists such that Similarly, there exists such that for all In case for some then is a fixed point of Now let for all Then from (3.1) with and we have

which implies

and so

Then by Lemma(2.14) we get exists (and is finite), so is a Cauchy sequence. Since is a complete multiplicative b-metric-like space, the sequence so that

Since is onto, there exists such that From (3.1) we have

By taking limit in the above, we get

which implies From Proposition 2.10(1), we have That is,

If we take I = 0 in theorem (3.1), then we have the following corollary.

Corollary 3.2. Let ba a complete multiplicative b-metric-like space. Suppose is onto mapping and satisfies

where Then has a fixed point.

Example 3.3. Let and let a multiplicative b-metric-like be defined by

where be a fixed real number.

Clearly is a complete multiplicative b-metric-like space. be defined by

Clearly, as J onto, so consider the following cases:

• Let then

• Let then

• Let then

• Let and let then

• Let and let then

• Let and let then

That is, for all where The conditions of Corollary 3.2 are satisfied and has a fixed point

Let be the class of functions such that it satisfy the condition where

Theorem 3.4. Let be a complete multiplicative b-metric-like space. Suppose the mapping is onto and satisfies

(3.2)

where Then J has a fixed point.

Proof. Let as J is onto, so there exists such that Similarly, there exists such that for all In case for some then it is clear that is a fixed point of Now suppose that for all From (3.2) with and we get

(3.3)

Then the sequence is decreasing in and so there exists such that Let us prove that Suppose to the contrary that by (3.3) with and we get

Taking limit in the above, we get Hence

which is contradiction. That is, m = 0. We shall show that

Suppose to the contrary that

By (3.2) we have

That is,

Then by () we get

therefore,

Taking limit as in the above, since and

then we obtain

which implies

and so

which is contradiction. Hence,

Now, since exists (and finite), so is a Cauchy sequence. Since is complete multiplicative b-metric-like space, in converges to so that

As is onto, so there exists such that Let us prove that Suppose to the contrary that Then by (3.2) we have

Taking limit as and by applying proposition 2.10(2), we have

and hence

which is contradiction. Indeed Since for all therefore That is,

Example 3.5. Let and be defined by

Clearly, is a complete multiplicative b-metric-like space. Let be defined by

Also define by Mapping is an onto.

Suppose Now, since

so

equivalently,

and hence

That is

The condition of theorem(3.4) hold and J has a fixed point (x = 0 is a fixed point of J).

NOTE. We can obtain the following corollaries because multiplicative b-metric-like spaces are extension of multiplicative partial metric, multiplicative metric-like and multiplicative b-metric spaces.

Corollary 3.6. Let be a complete multiplicative partial metric space. Suppose is onto and satisfies

where Then J has a fixed point.

Corollary 3.7. Let be a complete multiplicative metric-like space. Suppose is onto and satisfies

where Then J has a fixed point.

Corollary 3.8. Let be a complete multiplicative b-metric space. Suppose is onto and satisfies

(3.4)

where Then J has a fixed point.

4. Partially Ordered Multiplicative (b)-metric-like Spaces and Fixed Point Theorems

A. Alghamdi ^[1] proved fixed point point results which extend results of A. Harandi and someothers (see ^{[2, 9]}). Now, we prove some fixed point results in partially ordered multiplicative b-metric-like space to the corresponding results of A. Alghamdi ^[1].

Let be the class of functions such that it satisfy the condition where

Theorem 4.1. Let be a partially ordered complete multiplicative b-metric-like space, and suppose the mapping is non-decreasing such that

(4.1)

with where is bounded function and

and

Also, assume that the following assertions hold:

(1) for there exists

(2) for an increasing sequence where we have for all then has a fixed point.

Proof. Let If then the result is proved. Now we assume that Define a sequence by for all Since is non-decreasing and then

(4.2)

and hence the sequence is non-decreasing. If for some then the result is satisfied as is a fixed point of In what follows we will suppose that for all From (4.1) and (4.2) we have

where

Then

(4.3)

On the other hand, from () we have

and

Then

and hence

That is

Now by (4.3) we get

If then

which is contradiction. Hence,

(4.4)

and so is decreasing sequence. Then there exists such that By (4.4) we have

Taking in the above inequality, we get

and so Now we want to show that

Suppose to the contrary that

At first,

(4.5)

and

That is,

(4.6)

Now by (4.1) we have

and so from (4.5) and (4.6) we get

(4.7)

By () we have

Taking in the above inequality, we have

Then by (4.7) we deduce

Now, since then

On the other hand, since

hence

This implies that

which is contradiction. Thus, Now, since exists (and finite), so is a cauchy sequence. Since is complete multiplicative b-metric-like space, the sequence in converges to so that

From(2) and(4.1), with and we obtain

(4.8)

On the other hand,

and

Then Again, from Proposition 2.10(2) and 4.8, we have

Now, if then

This implies

which is contradiction. Hence, That is,

Example 4.2. Let and be defined by

Clearly, is a complete multiplicative b-metric-like space, let be defined by

Also define by Let That is for all we have

which implies

equivalently,

and so

So Theorem (4.1) hold and J has a fixed point.

Also we derive the following corollaries.

Corollary 4.3. Let be a partially ordered complete multiplicative partial metric space, and suppose the mapping is non-decreasing such that

(4.9)

with , where is bounded function and

and

Also, assume that the following assertions hold:

(1) for there exists ;

(2) for an increasing sequence where we have for all then has a fixed point.

Corollary 4.4. Let be a partially ordered complete multiplicative b-metric-like space, and suppose the mapping is non-decreasing such that

(4.10)

for all with , where is bounded function and

and

Also, suppose that the following assertions hold:

(1) for there exists ;

(2) for an increasing sequence where we have for all then has a fixed point.

5. Fixed Point Results for Cyclic Contraction

A.Alghamdi ^[1] proved some results which is the generalization of the results proved by Edelstein ^[7], Suzuki ^[16] and Kirk ^[10]. Ozavar and Cevikel ^[6] introduce the concepts of Banach-contraction in multiplicative metric spaces. By using the idea, in this section we derive some results to the setting of multiplicative b-metric-like spaces corresponding to the results proved by Alghamdi ^[1].

Theorem 5.1. Let be a multiplicative b-metric-like space, and suppose the family of non-empty closed subsets of with Let be a map satisfying

(5.1)

Assume that

This implies,

(5.2)

for all and where and Then has a fixed point in

Proof. Let and define a sequence in the following way:

(5.3)

We have for some then, clearly, the fixed point of the map is Hence, we assume that for all Clearly, Now, from (5.2) we have

which implies

(5.4)

From () we have

and

and so

(5.5)

Also,

Then

(5.6)

Hence, by (5.4), (5.5) and (5.6) we get

and then

where

Now since then

which implies

Then by lemma 2.14 we have Now, since exists (and finite), so is a cauchy sequence. Since is complete multiplicative b-metric-like space, the sequence in converges to so that

As Since subsequence the subsequence and similarly the subsequence All the m subsequences are convergent in the closed sets and all have the same limit Assume that there exists such that the following inequalities Satisfied:

Then

which is contradiction. Hence, for every we have

and so by (5.2) we have

(5.7)

or

(5.8)

Assume that (5.7) holds. Then, by taking limit as in (5.7), we get

and hence by using proposition 2.10(2) we obtain

Therefore,

If we take and then That is, Hence, i.e., If (5.8) holds, then by a similar method, we can deduce that

If we take in the above theorem for all m, then we have the following corollary.

Corollary 5.2. Let be a complete multiplicative b-metric-like space, and let be a self-mapping on . Suppose that

This implies

where and Then has a fixed point.

If in theorem (5.1) we take then we deduce the following corollary.

Corollary 5.3. Let be a multiplicative b-metric-like space, and let be a family of non-empty closed subsets of with Let be a map satisfying

Assume that

This implies

for all and where

Then J has a fixed point in

If in Corollary(5.2) we take then we deduce the following corollary.

Corollary 5.4. Let be a complete multiplicative b-metric-like space, and let be a self-mapping on . Assume that

This implies

for all and where

Then J has a fixed point in

Corollary 5.5. Let be a complete multiplicative metric-like space, let be non-empty closed subsets of and Suppose that is an operator such that

(1) is a cyclic representation of with respect to

(2) Suppose there exists such that

where

for where and is Lebesgue-integrable mapping satisfying for Then J has a fixed point.

Corollary 5.6. Let be a complete multiplicative metric-like space, and let such that for any there exists such that

where

and is Lebesgue-integrable mapping satisfying for Then J has a fixed point.

6. Application to Integral Equations

Integral equation method is very useful for solving problems in applied fields. Many papers consist on the problem of existence of solutions of nonlinear integral equations and the results are established by using different fixed point techniques, see e.g., ^{[1, 8, 13, 14]}. Inspired by the work. Consider the following integral equation

(6.1)

where and

are continuous functions. Let be a set of real continuous functions on We endow with the complete multiplicative b-metric-like

Clearly, is a complete multiplicative b-metric-like space.

Let be such that

(6.2)

Assume that for all , we have

(6.3)

and

(6.4)

Let, for all be a decreasing function, that is,

(6.5)

Assume that

(6.6)

Also suppose that for all for all with ( and ) or ( and ).

(6.7)

where and

Theorem 6.1. Under assumptions (6.2)-(6.7), integral equation (6.1) has a solution in

Proof. Define the closed subsets of and by

and

Also define the mapping by

Let us prove that

(6.8)

Suppose that that is,

Applying condition (6.5), since for all we obtain that

The above inequality with condition (6.3) imply that

for all Then we have

Similarly, let that is,

Using condition (6.5), since for all we obtain that

This implies the above inequality with condition (6.4)

for all Then we have Also, we deduce that (6.8) holds.

Now, let that is, for all

From condition(6.2) that for all

Now, by condition (6.6) and (6.7), we have, for all

which implies

Similarly, the above inequality hold if

So, Conditions of Theorem (5.1) satisfied and has a fixed point in

That is, is the solution to,

References

[1]	M. A. Alghamdi, N. Hussain, and P. Salimi, Fixed point and coupled fixed point theorems on b-metric-like spaces. Journal of Inequalities and Applications, article 402, (2013).
	In article		View Article
[2]	Agarwal, El-Gebeily, ORegan, Generalized contractions in partially ordered metric spaces. Appl. Anal., (2008).
	In article		View Article
[3]	A. E. Bashirov, E. M. Kurpinar, A. Ozyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (2008), 36-48.
	In article		View Article
[4]	S. Czerwik, Contraction mappings in b-metric spaces. Acta Mathematica et Informatica Universitatis Ostraviensis, vol. 1, (1993).
	In article
[5]	Daffer, Kaneko, On expansive mappings. Math. Jpn. (1992).
	In article
[6]	Ozavsar, Cevikel, Fixed points of multiplicative contraction mappings on multiplicative metric spaces. arXiv:1205.5131v1 [math.GM] (2012).
	In article
[7]	Edelstein, On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 37, (1962).
	In article
[8]	S. Gaulyaz, E. Karapinar and V. Rakocevic and P. Salimi, Existence of a solution of integral equations via fixed point theorem. Journal of Inequalities and Appl, (2013).
	In article		View Article
[9]	Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points. Fixed Point Theory and Applications, (2012).
	In article
[10]	Kirk,WA, Srinavasan, PS, Veeramani, Fixed points for mapping satisfying cyclical contractive conditions. Fixed Point Theory 4, (2003).
	In article
[11]	Kumar, Garg, Expansion mapping theorems in metric spaces. Int. J. Contemp. Math. Sci. (2009).
	In article
[12]	S. G. Matthews, Partial metric topology. Annals of the New York Academy of Sciences. General Topology and Applications, vol. 728, (1994).
	In article
[13]	H. K. Pathak, M. S. Khan and R. Tiwari, A common fixed point theorem and its application to nonlinear integral equations. Computers and Mathematics with Applications, (2007).
	In article		View Article
[14]	H. K. Pathak, S. N. Mishra and A. K. Kalinde, Common xed point theorems with applications to nonlinear integral equations. Demonstratio Math., XXXII (1999).
	In article
[15]	S. Shukla, Partial b-metric spaces and fixed point theorems. Mediterranean Journal of Mathematics, (2013).
	In article
[16]	Suzuki, A new type of fixed point theorem in metric spaces. Nonlinear Anal. 71(11), (2009).
	In article