We prove results on the existence and continuous dependence of solutions of a nonlinear quadratic integral Volterra equation on a parameter. This dependence is investigated in terms of Hausdorff distance. The considerations are placed in the Banach space and the Fréchet space.

In this paper we investigate the following nonlinear quadratic integral Volterra equation:

The main aim of the paper is to formulate assumptions that guarantee continuous dependence of solutions of (

Quadratic integral equations appear in theories of radiative transfer and neutron transport and in kinetic theory of gases (cf. [

Existence results for (

In this section we collect some definitions and results which will be needed later. Assume that

A mapping

The family

If

For our purposes we will only need the following fixed point theorem [

Let

In the sequel we will work in the Banach space

Now we recollect the definition of the measure of noncompactness which will be used further on. This measure was introduced in [

In what follows, we will also work in the space

A nonempty subset

Further, let

We accept the following definition of the notion of a sequence of measures of noncompactness [

A sequence of functions

The family

If

We have the following two facts (see [

The family of mappings

Let

Let

Let

A mapping

From (

First we suppose that (

Let us put

Now we assume that (

In this section we give an existence result for the following nonlinear integral Volterra equation:

This theorem will be a starting point of our further investigations on the continuous dependence of solutions on parameter.

Observe that the above equation includes several classes of functional, integral, and functional integral equations considered in the literature [

Equation (

The function

The function

The function

Consider

Then we can formulate our existence result.

Under assumptions

Consider the operator

Now, let us take a nonempty subset

Our next result is concerned with (

Equation (

The function

The function

The function

Consider

Then we can formulate the next existence result.

Under assumptions

In contrast to papers [

Let us define the operator

Applying (

In this section we will investigate (

For fixed

The aim of this paper is to provide the conditions concerning the functions involved in (

First we consider case of the bounded set

Consider

The following example shows that conditions

Let us consider the equation

There is an integrable function

Under assumptions

Theorem

Let us fix arbitrarily

Summarizing, for fixed

Consider the following equation:

If

Finally we will consider (

Consider

There is an integrable function

Under the above assumptions we can formulate a theorem analogous to the previous one.

Under assumptions

The proof of this theorem uses Lemma

Notice that, in hypotheses

Finally, we provide an example of an integral equation of the form (

Consider the following functional integral equation:

Joining all above facts we infer that assumptions

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