^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

We unify the concepts of

The concept of a

Let

In this case, the pair

The concept of a generalized metric space, or a

Let

Then, the function

A metric-like on a nonempty set

The pair

Below, we give some examples of metric-like spaces.

Let

Let

Let

A

In a

Let

Then,

The following propositions help us to construct some more examples of

Let

From the above proposition and Examples

Let

Let

Each

Now, we introduce the concept of generalized

Let

Then,

The following proposition will be useful in constructing examples of a generalized

Let

It is clear that

According to the above proposition, we provide some examples of generalized

Let

Let

By some straight forward calculations, we can establish the following.

Let

Let

The family of all

Let

A point

Using the above definitions, one can easily prove the following proposition.

Let

Let

Let

We need the following simple lemma about the

Let

Using the rectangle inequality, we obtain

If

Samet et al. [

Let

Denote with

Let

there exists

either

Then,

For more details on

Let

Motivated by [

Let

From now on, let

Let

Also, suppose that the following assertions hold:

there exists

Then,

Let

Since

By Proposition

Taking limit as

Using the rectangle inequality, we get

We replace condition (ii) in Theorem

Under the same hypotheses of Theorem

Repeating the proof of Theorem

A mapping

A function

there exists

Later, Berinde [

Let

there exist

Let

If

the series

the function

It is easy to see that if

In the next example, we present a class of

Any function of the form

From the part

For example, for

Let

Also, suppose that the following assertions hold:

there exists

assume that whenever

Then,

Let

If there exists

Using Proposition

Hence,

By induction, since

Let

Let the part (a) of (ii) holds.

Using the rectangle inequality, we get

Now, let part (b) of (ii) holds.

As

Now, we show that

Let

Let

Also, suppose that the following assertions hold:

there exists

Then,

Define

Similarly, using Theorem

Let

there exists

Then,

We conclude this section by presenting some examples that illustrate our results.

Let

For all elements

Let

Let

For all elements

In this section, we present an application of our results to establish the existence of a solution to a periodic boundary value problem (see [

Let

Consider the first-order periodic boundary value problem

A lower solution for (

Assume that there exists

Problem (

Using variation of parameters formula, we get

Let

Hence, the hypotheses of Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU, for financial support.