^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

We prove a result on the existence and uniform attractivity of solutions of an Urysohn integral equation. Our considerations are conducted in the Banach space consisting of real functions which are bounded and continuous on the nonnegative real half axis. The main tool used in investigations is the technique associated with the measures of noncompactness and a fixed point theorem of Darbo type. An example showing the utility of the obtained results is also included.

The theory of nonlinear functional integral equations creates an important branch of the modern nonlinear analysis. The large part of that theory describes a lot of classical nonlinear integral equations such as nonlinear Volterra integral equations, Hammerstein integral equations, and Urysohn integral equations with solutions defined on a bounded interval (cf. [

Nevertheless, more important and simultaneously, a more difficult part of that theory is connected with the study of solutions of the mentioned integral equations defined on an unbounded domain. Obviously, there are some known results concerning the existence of solutions of those integral equations in such a setting but, in general, they are mostly obtained under rather restrictive assumptions [

On the other hand, the use of some tools of nonlinear analysis enables us to obtain several valuable results under less restrictive assumptions (cf. [

In the paper, we will use the above described approach associated with the technique of measures of noncompactness in order to obtain a result on the existence of solutions of a quadratic Urysohn integral equation. Applying the mentioned technique in conjunction with a fixed point theorem of Darbo type, we show that the equation in question has solutions defined, continuous, and bounded on the nonnegative real half axis

The results obtained in this paper generalize several results obtained earlier in numerous papers treating nonlinear functional integral equations, which were quoted above. Particularly, we generalize the results concerning the Urysohn or Hammerstein integral equations obtained in the papers [

In this section, we establish some notations, and we collect auxiliary facts which will be used in the sequel.

By the symbol

Moreover, if

In what follows, we will accept the following definition of the concept of a measure of noncompactness [

A mapping

The family

If

The family

Observe that the set ^{°} is a member of the family

Now, we formulate a fixed point theorem of Darbo type which will be used further on [

Let

Denote by

In what follows, we will work in the Banach space

Now, we recall the construction of a measure of noncompactness in the space

For further purposes, we introduce now the concept of attractivity (stability) of solutions of operator equations in the space

We say that solutions of (

Notice that the previous definition comes from [

We will consider the existence and asymptotic behaviour of solutions of the quadratic Urysohn integral equation having the form

In our study, we will impose the following assumptions.

The function

For each

The following equalities hold:

Let us observe that in view of assumptions (ii) and (v) we can define the following finite constants:

It is worthwhile mentioning that in the theory of improper Riemann integral with a parameter there has been considered the concept of

We say that the integral (

Equivalently (cf. [

Let us observe that if integrals appearing in assumption (v), that is, the integrals

It may be also shown that the converse implications are, in general, not valid [

It can be also shown [

Now, we formulate our last assumption.

The inequality

has a positive solution

Assume that

Then, we obtain

Now, we are prepared to formulate our main result.

Under assumptions (i)-(vii), (

Consider the operator

Further, let us notice that in view of assumptions (ii) and (iv), the function

In what follows, we show that the function

Further, observe that from (

Now, let us take a nonempty subset

Fix numbers

In what follows assume, as previously, that

In the last step of our proof, we show that the operator

Finally, using the above established facts and (

The proof is complete.

It is worthwhile mentioning that the above result generalizes those obtained in [

Now, we are going to illustrate the result contained in Theorem

Let us consider the following quadratic Urysohn integral equation:

Observe that this equation is a special case of (

Further, let us note that the function

Moreover, we get

Now, fix arbitrarily a number

Next, observe that taking into account (

Now, we are coming to the last assumption of Theorem

For example, it is easy to check that if we take

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (470/363/1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.

The authors declare that there is no conflict of interests in the submitted paper.