_{1}

We investigate a <i>q</i>-fractional integral equation with supremum and prove an existence theorem for it. We will prove that our <i>q</i>-integral equation has a solution in <i>C</i> [0, 1] which is monotonic on [0, 1]. The monotonicity measures of noncompactness due to Banaś and Olszowy and Darbo’s theorem are the main tools used in the proof of our main result.

Jackson in [

In several papers among them [

In this paper, we will study the q-perturbed quadratic integral equation with supremum

where

By using Darbo fixed point theorem and the monotonicity measure of noncompactness due to Banaś and Olszowy [

First, we collect basic definitions and results of the q-fractional integrals and q-derivatives, for more details, see [

First, for a real parameter

and a q-analog of the Pochhammer symbol (q-shifted factorial) is defined by

Also, the q-analog of the power

Moreover,

Notice that,

More generally, for

and

Notice that

The q-gamma function is defined by

where

Next, the q-derivative of a function f is given by

and the q-derivative of higher order of a function f is defined by

The q-integral of a function f defined on the interval

If f is given on the interval

The operator

The fundamental theorem of calculus satisfies for

The following four formulas will be used later in this paper

and

where

Notice that, if

Definition 1. [

Notice that, for

Definition 2. [

where

In q-calculus, the derivative rule for the product of two functions and integration by parts formulas are

Lemma 1. Let

Lemma 2. [

or

Second, we recall the basic concepts which we need throughout the paper about measure of noncompactness.

We assume that

Now, let

Moreover, the families

Definition 3. [

1)

2)

3)

4)

5) if

then

Here,

Our result will establish in C(I) the Banach space of all defined, continuous and real functions on

Next, we defined the measure of noncompactness related to monotonicity in

We fix a bounded subset

Moreover, we let

and

Define

and

Notice that, all functions in Y are nondecreasing on I if and only if

Now, we define the map

Clearly, μ verifies all conditions in Definition 3 and, therefore it is a measure of noncompactness in

Definition 4.Let

If

Theorem 1. [

Let us consider the following suggestions:

a_{1})

Moreover,

a_{2}) The superposition operator F generated by the function f satisfies for any nonnegative function y the condition_{1}).

a_{3})

a_{4})

a_{5}) The function

ver,

a_{6})

a_{7})

and

Before, we state and prove our main theorem, we define the two operators

and

respectively. Finding a fixed point of the operator

Theorem 2. Assume the suggestions (a_{1})-(a_{7}) be verified, then Equation (1) has at least one solution

Proof. We divide the proof into seven steps for better readability.

Step 1: We will show that the operator

For this, it is sufficient to show that

Notice that, we have used

Notice that, since the function k is uniformly continuous on

Thus

Step 2:

Now,

Hence

Therefore, if _{7}) the following

Therefore,

We define the subset

It is clear that

Step 3:

By this facts and suggestions a_{1}), a_{4}) and a_{6}), we obtain

Step 4: The operator

To prove this, we fix

Thus, we have

Consequently,

As

Also,

Furthermore,

Now, take

This shows that

Step 5: In recognition of

Now, we take

The last estimate implies

and, consequently,

Since the function k is uniformly continuous on

Step 6: In recognition of

Here, we fix an arbitrary

Now, we will prove that

We find that

But,

and, since

Inequalities (50) and (51) imply that

This inequality and (47) gives us

The above estimate implies that

Therefore,

Step 7:

Inequalities (46) and (54) give us that

or

But

Inequality (57) enables us to use Theorem 1, then there are solutions to Equation (1) in

This finishes our proof.

Maryam Al-Yami, (2016) On Existence of Solutions of q-Perturbed Quadratic Integral Equations. American Journal of Computational Mathematics,06,166-176. doi: 10.4236/ajcm.2016.62018