Received 18 April 2016; accepted 26 June 2016; published 29 June 2016
![](//html.scirp.org/file/12-1100523x7.png)
1. Introduction
In several papers among them [8] - [11] , integral equations with nonsigular kernels have been studied. In [12] - [14] Darwish et al. introduced and studied the quadratic Volterra equations with supremum. Also, Banaś et al. and Darwish [13] [15] - [17] studied quadratic integral equations of arbitrary orders with singular kernels. In [18] , Darwish generalized and extended Banaś et al. [15] results to the perturbed quadratic integral equations of arbitrary orders with singular kernels.
In this paper, we will study the q-perturbed quadratic integral equation with supremum
(1)
where
,
, and
.
By using Darbo fixed point theorem and the monotonicity measure of noncompactness due to Banaś and Olszowy [19] , we prove the existence of monotonic solution to Equation (1) in
.
2. q-Calculus and Measure of Noncompactness
First, for a real parameter
, we define a q-real number
by
, (2)
and a q-analog of the Pochhammer symbol (q-shifted factorial) is defined by
(3)
Also, the q-analog of the power
is given by
(4)
Moreover,
(5)
Notice that,
exists and we will denote it by
.
More generally, for
we define
(6)
and
(7)
Notice that
. Therefore, if
, then
.
The q-gamma function is defined by
(8)
where
Or, equivalently,
and satisfies ![]()
Next, the q-derivative of a function f is given by
(9)
and the q-derivative of higher order of a function f is defined by
(10)
The q-integral of a function f defined on the interval
is defined by
(11)
If f is given on the interval
and
then
(12)
The operator
is defined by
(13)
The fundamental theorem of calculus satisfies for
and
, i.e.,
, and if f is continuous at
, then
.
The following four formulas will be used later in this paper
(14)
and
(15)
where
denotes the q-derivative with respect to variable t.
Notice that, if
and
, then
.
Definition 1. [2] Let f be a function defined on
. The fractional q-integral of the Riemann-Liouville type of order
is given by
(16)
Notice that, for
, the above q-integral reduces to (11).
Definition 2. [2] The fractional q-derivative of the Riemann-Liouville type of order
is given by
(17)
where
denotes the smallest integer greater than or equal to
.
In q-calculus, the derivative rule for the product of two functions and integration by parts formulas are
(18)
Lemma 1. Let
and f be a function defined on
. Then the following formulas are verified:
(19)
Lemma 2. [21] For
, using q-integration by parts, we have
(20)
or
(21)
Second, we recall the basic concepts which we need throughout the paper about measure of noncompactness.
We assume that
is a real Banach space with zero element
and we denote by
the closed ball with radius r and centered x, where
.
Now, let
and denote by
and Conv X the closure and convex closure of X, respectively. Also, the symbols
and
stands for the usual algebraic operators on sets.
Moreover, the families
and
are defined by
and
respectively.
Definition 3. [22] Let
If the following conditions
1)
.
2) ![]()
3) ![]()
4)
and
5) if
is a sequence of closed subsets of
with
and ![]()
then
hold. Then, the mapping
is said to be a measure of noncompactness in E.
Here,
is the kernel of the measure of noncompactness
.
Our result will establish in C(I) the Banach space of all defined, continuous and real functions on
with
.
Next, we defined the measure of noncompactness related to monotonicity in
, see [19] [22] .
We fix a bounded subset
of
. For
and
denotes the modulus of continuity of the function y given by
. (22)
Moreover, we let
(23)
and
(24)
Define
(25)
and
(26)
Notice that, all functions in Y are nondecreasing on I if and only if
.
Now, we define the map
on
as
(27)
Clearly, μ verifies all conditions in Definition 3 and, therefore it is a measure of noncompactness in
[19] .
Definition 4.Let
Let
be a continuous operator. Suppose that
maps bounded sets onto bounded ones. If there exists a bounded
with
, then
is said to be satisfies the Darbo condition with respect to a measure of noncompactness
.
If
, then
is called a contraction operator with respect to
.
Theorem 1. [23] Let
be a bounded, convex and closed subset of E. If
is a Contraction operator with respect to
. Then
has at least one fixed point belongs to Q.
3. Existence Theorem
Let us consider the following suggestions:
a1)
is continuous and
![]()
Moreover,
and ![]()
a2) The superposition operator F generated by the function f satisfies for any nonnegative function y the condition
, where c is the same constant as in a1).
a3)
is a continuous operator which satisfies the Darbo condition for the measure of noncompactness
with a constant
. Also,
if
.
a4)
.
a5) The function
is continuous on
and nondecreasing
and separately. Moreo-
ver, ![]()
a6)
is a continuous operator and there is a nondecreasing function
such that
for any
. Moreover, for every function
which is nonnegative on I, the function
is nonnegative and nondecreasing on I.
a7)
such that
(28)
and
.
Before, we state and prove our main theorem, we define the two operators
and
on
as follows
(29)
and
(30)
respectively. Finding a fixed point of the operator
defined on the space
is equivalent to solving Equation (1).
Theorem 2. Assume the suggestions (a1)-(a7) be verified, then Equation (1) has at least one solution
which is nondecreasing on I.
Proof. We divide the proof into seven steps for better readability.
Step 1: We will show that the operator
maps
into itself.
For this, it is sufficient to show that
if
. Fix
and let
and
with
. We have
(31)
Notice that, we have used
(32)
Notice that, since the function k is uniformly continuous on
, then when
we have that
.
Thus
, and therefore, ![]()
Step 2:
applies
into itself.
Now,
, we have
(33)
Hence
(34)
Therefore, if
we get from assumption a7) the following
(35)
Therefore,
maps
into itself.
We define the subset
of
by
(36)
It is clear that
is closed, convex and bounded.
Step 3:
applies the set
into itself.
By this facts and suggestions a1), a4) and a6), we obtain
transforms
into itself.
Step 4: The operator
is continuous on
.
To prove this, we fix
to be a sequence in
with
. We will show that
.
Thus, we have
,
(37)
Consequently,
(38)
As
and
are continuous operators,
such that
(39)
Also,
such that
(40)
Furthermore,
such that
(41)
Now, take
, then (38) gives us that
. (42)
This shows that
is continuous in
.
Step 5: In recognition of
with respect to the quantity
.
Now, we take
Let us fix an arbitrarily number
and choose
and
with
. We will be supposed that
because no generality will be loss. Then, by using our suggestions and inequality (31), we get
(43)
The last estimate implies
(44)
and, consequently,
(45)
Since the function k is uniformly continuous on
and the function f is continuous on
, then the last inequality gives us that
(46)
Step 6: In recognition of
with respect to the quantity d.
Here, we fix an arbitrary
and
with
. Then, by our assumption, we obtain our suggestions, we have
(47)
Now, we will prove that
(48)
We find that
(49)
But,
because
is increasing with respect to t, then
(50)
and, since
is negative for
then
(51)
Inequalities (50) and (51) imply that
![]()
This inequality and (47) gives us
(52)
The above estimate implies that
(53)
Therefore,
(54)
Step 7:
is contraction with respect to the measure of noncompactness
.
Inequalities (46) and (54) give us that
(55)
or
(56)
But
, then
(57)
Inequality (57) enables us to use Theorem 1, then there are solutions to Equation (1) in
.
This finishes our proof.