^{1}

^{2}

^{1}

^{2}

The paper is devoted mainly to the study of the existence of solutions depending on two variables of a nonlinear integral equation of Volterra-Stieltjes type. The basic tool used in investigations is the technique of measures of noncompactness and Darbo’s fixed point theorem. The results obtained in the paper are applicable, in a particular case, to the nonlinear partial integral equations of fractional orders.

The theory of differential and integral equations of fractional order creates nowadays a large subject of mathematics which found in the last three decades numerous applications in physics, mechanics, engineering, bioengineering, viscoelasticity, electrochemistry, control theory, porous media, and other fields connected with real world problems [

It turns out that a lot of results of the theory of differential and integral equations of fractional order can be considered from a unified point of view with help of the theory of the so-called Volterra-Stieltjes integral equations (cf. [

Such an approach can be also applied to investigations associated with the theory of differential and integral equations of fractional orders in two variables. That subject of nonlinear analysis was recently studied in a few papers [

This section is devoted to provide the notation, definitions, and other auxiliary facts which will be needed in our further study.

At the beginning let us assume that

For the properties of functions of bounded variation we refer to [

If

It is worthwhile mentioning that several conditions ensuring Stieltjes integrability may be found in [

In the sequel we will utilize a few properties of the Stieltjes integral contained in the below quoted lemmas (cf. [

If

Let

In what follows we will also consider Stieltjes integrals having the form

Even more, in our considerations we will use the double Stieltjes integrals of the form

Now, let us assume that

Further on, in order to simplify our investigations, we will always assume that

Now, we recall some facts concerning measures of noncompactness, which will be applied in the sequel. To this end assume that

Next, for a given nonempty bounded subset

Notice that the concept of a measure of noncompactness may be defined in a more general way [

In fact, in our further considerations, we will work in the space

Now we recall a fixed point theorem of Darbo type which will be utilized in our investigations (cf. [

Let

Next we recall a few facts concerning the so-called superposition operator [

The properties of the superposition operator may be found in [

The superposition operator

Let us notice that in our considerations concerning the superposition operator

Finally, we recall some fundamental facts associated with fractional calculus (cf. [

It may be shown that the fractional integral operator

Investigations of this paper are connected mainly with the solvability of the following nonlinear quadratic integral equation of Volterra-Stieltjes type having the form

Let us recall that details concerning the notation used in (

In order to formulate the assumptions under which (

We will study (

the function

for all

the function

the function

for any

for

for all

Before formulating further assumptions concerning (

The function

Let assumptions (iii)–(v) be satisfied. Then, for arbitrarily fixed number

Under assumptions (iii)–(v) the function

As an immediate consequence of the above lemma we derive the following corollary.

There exists a finite positive constant

In what follows let us denote by

Now, we can formulate the last assumption used further on.

There exists a positive solution

such that

Our main result is contained in the following theorem.

Under assumptions (i)–(viii) there exists at least one solution

We start with the following notation:

Moreover, for further purposes let us define the function

Further, for a fixed function

Next, evaluating similarly as above, in view of Lemmas

Observe that taking into account the fact that the function

Further, using the imposed hypotheses, in light of Lemmas

Next, using similar reasonings, we arrive at the following estimate:

Now, we estimate the last term appearing on the right-hand side of inequality (

Now, linking estimates (

On the other hand, keeping in mind Lemma

In the sequel we show that the operator

In what follows let us fix an arbitrary function

Next, let us take a nonempty subset

The proof is complete.

In considerations conducted in the above proof, taking two points

Observe that all possible cases can be always converted to that indicated above. For example, if we assume that

We start with providing some facts concerning assumption (v) imposed in investigations conducted in the preceding section (cf. also [

In order to formulate the announced condition assume (as we have done previously) that

for arbitrary

From results proved in [

Suppose the function

Indeed, in the case when

Further, based on results obtained in [

Let us fix

Similarly as above fix

In what follows we will consider the fractional integral equation with functions involved depending on two variables, which has the form

Let us mention that (

Obviously, (

Now, we show that the functional integral equation of fractional orders (

In fact, take the functions

for

Then, it can be easily seen that (

To formulate such a result let us first calculate the constants

Now, we present the above announced result.

Assume that the function

There exists a positive solution

such that

Then there exists at least one solution

In the above conducted calculations we used the well-known formula

In what follows we consider the functional integral equation of the so-called Volterra-Chandrasekhar type in two variables having the form

Observe that taking the function

Assume that the function

There exists a positive solution

such that

Then there exists at least one solution

Finally, we consider the functional integral equation having the form linking equations (

An existence theorem concerning (

There exists a positive solution

such that

We omit other details.

In this section we focus briefly on possible generalizations of results presented in previous sections.

First of all let us notice that instead of the functional integral equation of Volterra-Stieltjes type in two variables (

Other assumptions concerning (

Let us mention that we can also investigate equations considered precedingly, that is, (

One can expect that in the case of (

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

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (363-001-D1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.