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The aim of this paper is to present common fixed point results of quasi-weak commutative mappings on a closed ball in the framework of multiplicative metric spaces. Example is presented to support the result proved herein. We also study sufficient conditions for the existence of a common solution of multiplicative boundary value problem. Our results extend and improve various recent results in the existing literature.

The letters

Consistent with [

The multiplicative absolute value function

Using above definition of multiplicative absolute value function, we deduce the following proposition.

For arbitrary

Bashirov et al. [

Let

Let

Let

Let

Let

For more examples of multiplicative metric spaces, we refer to [

Let

A sequence

A multiplicative metric space

A sequence

For sake of brevity we skip the proof of the following lemma.

Let

Let

fixed point of

coincidence point of the pair

common fixed point of the pair

One of the simplest and most useful results in fixed point theory is the Banach-Caccioppoli contraction mapping principle, a powerful tool in analysis for establishing existence and uniqueness of solution of problems in different fields. Over the years, this principle has been generalized in numerous directions in different spaces. These generalizations have been obtained either by extending the domain of the mapping or by considering a more general contractive condition on the mappings.

Recently, Ozavsar and Cervikel [

Let

Let

Let

Let

commutative if

weakly commutative if

He et al. [

Let

The study of contractive conditions on the entire domain has been at the center of vigorous research activity (see [

If a mapping

The purpose of this paper is to establish the existence and uniqueness of common fixed point of quasi-weak commutative contractive mappings defined on a closed ball in a multiplicative metric space. Our results extend and improve the results of He et al. [

In this section, we obtain several common fixed point results of mappings on multiplicative closed balls in the framework of multiplicative metric spaces.

We start with the following result.

Let

Let

Assume

Similarly,

As

The fact that

Suppose

Because of the fact that

By contractive condition (

Hence

Since

We proceed to show the uniqueness of the common fixed point of the maps

The following result generalizes Theorem

Let

Let

Hence

Let

Put

Let

Consider a system of two multiplicative differential equations

Define

The authors declare that they have no competing interests.