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Using Hausdorff measure of noncompactness and a fixed-point argument we prove the existence of mild solutions for the semilinear integrodifferential equation subject to nonlocal initial conditions

The concept of

The earliest works related with problems submitted to nonlocal initial conditions were made by Byszewski [

Henceforth, (

On the other hand, the study of abstract integrodifferential equations has been an active topic of research in recent years because it has many applications in different areas. In consequence, there exists an extensive literature about integrodifferential equations with nonlocal initial conditions, (cf., e.g., [

The classical initial value version of (

Most of the notations used throughout this paper are standard. So,

In this work

As we have already mentioned

In the same manner, for

We denote by

We next include some preliminaries concerning the theory of resolvent operator

Let

In what follows we assume that there exists a resolvent operator

Note that property

The existence of solutions of the

A continuous function

The main results of this paper are based on the concept of measure of noncompactness. For general information the reader can see [

Let

Let

If

We next collect some specific properties of the Hausdorff measure of noncompactness which are needed to establish our results. Henceforth, when we need to compare the measures of noncompactness in

Let

For the rest of the paper we will use the following notation. Let

Let

A set of functions

The next property has been studied by several authors; the reader can see [

If

The next result is crucial for our work, the reader can see its proof in [

Let

The following lemma is essential for the proof of Theorem

For all

Clearly, a manner for proving the existence of mild solutions for (

Let

In this section we will present our main results. Henceforth, we assume that the following assertions hold.

Assuming that the function

The following theorem is the main result of this paper.

If the hypotheses

Define

We begin showing that

Let

Therefore

Define

Let

By the hypothesis

Using the inequality (

Since

By an inductive process, for all

In addition, for all

Since

Consequently,

Our next result is related with a particular case of (

Since (

Let

We need the following definitions for proving the existence of a resolvent operator for (

Let

Convolutions of

Let

Let

Finally, we recall that a one-parameter family

The next proposition guarantees the existence of a resolvent operator for (

Suppose that

Integrating in time (

Suppose that

It follows from Proposition

In this section we apply the abstract results which we have obtained in the preceding section to study the existence of solutions for a partial differential equation submitted to nonlocal initial conditions. This type of equations arises in the study of heat conduction in materials with memory (see [

Identifying

We will prove that there exists

With this purpose, we begin noting that

Further, the function

Define

On the other hand, it follows from [

Let

A direct computation shows that for each

Therefore the expression

Since there exists

From Corollary

C. Lizama and J. C. Pozo are partially supported by FONDECYT Grant no. 1110090 and Ring Project ACT-1112. J. C. Pozo is also partially financed by MECESUP PUC 0711.