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In this paper, we introduce the concept of a rectangular metric-like space, along with its topology, and we prove some fixed point theorems for different contraction types. We also introduce the concept of modified metric-like spaces and we prove some topological and convergence properties under the symmetric convergence. Some examples are given to illustrate the new introduced metric type spaces.

The generalization of Banach contraction principle, which has many applications in different branches of science and engineering, depends on either generalizing the metric type space or the contractive type mapping (see [

Let

In this case, the pair

Let

In this case, the pair

In [

Let

In this case, the pair

It is clear that every rectangular metric space is a rectangular partial metric space, but the converse is not true.

Let

For convergence, completeness, and examples of RM, PM, and RPM spaces, we refer to [

Let

In this case, the pair

Every metric-like space is a topological space whose topology is generated by the base consisting of the open

Metric-like spaces lose some topological and convergence properties as known for metric spaces. We state the following. For example, limits are not unique in

Upon Remark

Let

The pair

It is clear that every partial metric space is an

Let

As in the case of metric-like spaces, the open

We shall say that a sequence

We shall denote

Let

If

If

If

If

Assume

and

Let

Let

Since

and

Therefore,

If we take

Assume that

and

From what is mentioned, it follows that

Let

Hence

Assume that

and

Now, for each

and

Finally, letting

and thus

Now, we introduce the concepts of rectangular metric-like and rectangular modified metric-like spaces.

Let

then the pair

Let

then the pair

Let

Let

Let

A sequence

(resp.

A sequence

(resp.,

A rectangular metric-like space

(resp.,

The convergence defined in Definition

Let

Our second main result concerns an existence and uniqueness theorem on rectangular metric-like spaces.

Let

Let

We rewrite (

We deduce that

Next, we present the following example.

Let

Our third main result is as follows.

Let

Fix

We rewrite (

Now, let

No data were used to support this study.

The authors declare thast they have no conflicts of interest regarding the publication of this paper.

The authors would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RG-DES-2017-01-17.