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The aim of this paper is to show the use of the coupled quasisolutions method as a useful technique when dealing with ordinary differential equations with functional arguments of bounded variation. We will do this by looking for solutions for a first-order ordinary differential equation with an advanced argument of bounded variation. The main trick is to use the Jordan decomposition of this argument in a nondecreasing part and a nonincreasing one. As a necessary step, we will also talk about coupled fixed points of multivalued operators.

In the paper [

To show the application of this technique, we will study throughout this paper the existence of solutions for the following first-order problem:

This paper is organized as follows. In Section

In this section, we introduce some preliminaries that we will use throughout this work. First, we remember some concepts about functions of bounded variation. The reader can see more about this in the monographs [

Given a function

One says that

Functions of bounded variation satisfy the following well-known result, which becomes essential now for our purposes.

A function

The proof of Proposition

The set

To obtain our main result, we will use a generalized monotone method in presence of lower and upper solutions. This is a very well-known tool which is extensively used in the literature of ordinary differential equations. The classical version of this technique uses a pair of monotone sequences which will converge to the extremal solutions of the problem. The generalized version of this technique was developed in [

A metric space

Let

Let

If for all

Now, we develop our generalized monotone method applied to problem (

There exists a closed interval

Assumption

Now, we define what we mean by lower and upper solutions for problem (

One says that

Notice that, under the previous definition, the lower and the upper solutions appear “coupled.” On the other hand, it is assumed that

On the other hand, the fact that

As we said in the Introduction, an essential tool in our work is the use of coupled quasisolutions. So, we introduce now this concept.

One says that two functions

We need the following maximum principle related to problems with advance, as an auxiliar tool, for proving our main result. Compare it with [

Let

If

Let

The main result on this paper concerns the existence of extremal quasisolutions and solutions for problem (

Assume

has the extremal solutions in

whenever

Moreover,

where

In these conditions, problem (

We consider the space

To see that

By application of Theorem

To see that

Now, define the function

Now, we point out some remarks related to Theorem

Condition

As we said in Section

a countable number of discontinuities are allowed to exist outside this interval. Moreover, notice that this interval can be improved if we find another function

For almost all

for any choice of

Theorem

Assume that there exist

has the extremal solutions in

In these conditions problem (

The results obtained in the previous section can be easily reformulated in order to deal with problems with delay. We concrete this idea in the following lines.

Consider the following problem:

As we said, we will show now that we can use our technique to obtain a new result on the existence of solutions for problem (

One says that

One says that two functions

Before introducing our main result for problem (

Let

If

Now, we state our main result in this Section.

Assume

has the extremal solutions in

whenever

Moreover,

where

In these conditions, problem (

The proof is analogous to that done in Theorem

The rest of the proof is analogous, with obvious changes.

Now, Theorem

Assume that there exist

has the extremal solutions in

In these conditions, problem (

We finish this work with two applications of our main results.

Consider the following problem with advance:

Defined that way,

We will show now that the functions

If

If

Then,

Now, we check condition

Finally, notice that for a.a.

By application of Theorem

In the following example, we consider a practical application of Corollary

Consider a bacterial culture governed by a logistic-type equation of the form

To counteract the effects of saturation term, we introduce an electronic mechanism which acts as follows. It counts the number of individuals and then it provides some food that makes the population grow. The amount of food supplied by the machine is proportional to the number of individuals. Moreover, the machine can distinguish only thousands of individuals and it supplies the food with a delay

Finally, we consider an initial population of one thousand individuals. Therefore, we deal with the following initial value problem with delay:

We will show now that problem (

First, notice that

Then,

To check conditions

Finally, as for

Therefore, we can apply Corollary

The author wants to thank the reviewers for their pertinent comments, which contribute to improve the quality of this paper. The paper is partially supported by FEDER and Ministerio de Educación y Ciencia, Spain, Project MTM2010-15314.