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We establish some coincidence point results for self-mappings satisfying rational type contractions in a generalized metric space. Presented coincidence point theorems weaken and extend numerous existing theorems in the literature besides furnishing some illustrative examples for our results. Finally, our results apply, in particular, to the study of solvability of functional equations arising in dynamic programming.

Banach contraction principle is one of the most important aspects of fixed point theory as a source of the existence and uniqueness of solutions of many problems in various branches inside and outside mathematics (see, [

Recently, in 2015, Almeida et al. [

Another offshoot of generalizations of Banach’s theorem is based on extending the axioms of metric spaces. It is worth mentioning that the use of triangle inequality in a metric space

In this paper, we introduce coincidence point theorems for two contraction self-mappings of rational type in generalized metric spaces. Our results improve the results of Almeida et al. [

In this section, we present some preliminaries and notations related to rational type contraction and GMS.

Suppose that

Then we called

The following example shows that GMS are more general than metric spaces.

Suppose that

Suppose that

Definition

A GMS is Hausdorff.

There is a unique limit of a convergence sequence.

Any convergent sequence is a Cauchy sequence.

In 2009, Samet [

Suppose that

Then

there is no

Any Cauchy sequence in GMS converges to a unique point.

Let

Let

Suppose that

In this section we introduce some coincidence point results for two rational contraction self-mappings on GMS.

Let

Then

Define the sequence

Now, by (

If

If

If

In any case, we proved that (

Now, if possible, let

Now, we prove that

Next, we prove that

Suppose

Then

Replace condition (

Then

Put

It is spatial case of Theorem

Let

Consider also that the next conditions hold:

Either

Then

Suppose that

If

If

If

In any case, we proved that (

Now, if possible, let

Now, we prove that

As in the conclusion in the last paragraph of the proof of Theorem

Put

It is spatial case of Theorem

The aim of this section is to use Theorem

We denote by

We note that the space

Let

Suppose that

It is enough to prove that

Consider the assumptions of Proposition

Then (

First, we prove that the mappings in system (

The authors declare that they have no competing interests.

The two authors contributed equally to this work. Both authors read and approved the final paper.