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We study the existence of monotonic and nonnegative solutions of a nonlinear quadratic Volterra-Stieltjes integral equation in the space of real functions being continuous on a bounded interval. The main tools used in our considerations are the technique of measures of noncompactness in connection with the theory of functions of bounded variation and the theory of Riemann-Stieltjes integral. The obtained results can be easily applied to the class of fractional integral equations and Volterra-Chandrasekhar integral equations, among others.

The aim of this paper is to study of monotonic and nonnegative solutions of the nonlinear quadratic Volterra-Stieltjes integral equation having the form

The main result of the paper is contained in Theorem

This paper can be considered as a continuation of [

At the beginning, we provide some basic facts concerning functions of bounded variation and the Riemann-Stieltjes integral. We refer to [

Now, we recall two useful properties of the Riemann-Stieltjes integral, which will be employed in the sequel.

(a) If

(b) Suppose that

In what follows we will use the Riemann-Stieltjes integral of the form

Now, we deal with the discussion of basic facts connected with measures of noncompactness. We refer to [

A mapping

the family

if

An important example of a measure of noncompactness is the Hausdorff measure of noncompactness defined by the formula

The key role in our studies will be played by Darbo’s fixed point theorem.

Let

The considerations in this paper will be placed in the Banach space

Finally, we turn our attention to the

In this section, we will investigate the nonlinear quadratic Volterra-Stieltjes integral equation which has the form

At the beginning, let us consider the following conditions.

The functions

The function

The function

For any

For each

It can be shown (see [

The operator

There exists a positive real number

Observe that if

Now, let us consider the operators

Let conditions (i)–(vii) hold. Then, the operator

The basic idea of the proof of Theorem

Additionally, all solutions of (

We can now formulate our main result about monotonicity and nonnegativity of the solutions of (

The functions

the function

the function

(a) The function

the function

for each

for any function

Or

(b) The function

the function

the function

the function

for each

for any function

For each

The following theorem is a completion of Theorem

Suppose that conditions (i)–(vii) and (i^{'})–(iii^{'}) are fulfilled. Then, (

Let

It is easily seen that

Assume that condition (

Now, assume that condition (

It can be shown (see for instance [

The topic of this section is to present some applications of Theorem

Let us consider the equation

there exists a positive real number

Obviously, when

Now, let us consider the equation

Let us observe that if we put

We finish by providing an example illustrating Theorem

Let us consider the following integral equation:

The author declares that there is no conflict of interests regarding the publication of this paper.