On Existence of Solutions of q-Perturbed Quadratic Integral Equations

We investigate a q-fractional integral equation with supremum and prove an existence theorem for it. We will prove that our q-integral equation has a solution in [ ] C 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.


Introduction
Jackson in [1] introduced the concept of quantum calculus (q-calculus).This area of research has rich history and several applications, see [2]- [4] and references therein.There are several developments and applications of the q-calculus in mathematical physics, especially concerning quantum mechanics, the theory of relativity and special functions [5] [4].Recently, several researchers attracted their attention by the concept of q-calculus, and we could find several new results in [6] [7] and the references therein.
In this paper, we will study the q-perturbed quadratic integral equation with supremum where ( ) 0 , 0,1 , : , and : k I I × →  .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 [ ] 0,1 C .

q-Calculus and Measure of Noncompactness
First, we collect basic definitions and results of the q-fractional integrals and q-derivatives, for more details, see [5] [6] [20] [21] and references therein.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 ( ) Notice that, ( ) lim ; n n a q →∞ exists and we will denote it by ( ) More generally, for ( ) Notice that ( ) ( ) ( ) 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 [ ] 0,b is defined by The operator n q I is defined by , , The fundamental theorem of calculus satisfies for q D and q I , i.e., ( )( ) ( ) , and if f is continuous at 0 t = , then ( )( ) ( ) ( ) . The following four formulas will be used later in this paper where t q D denotes the q-derivative with respect to variable t.Notice that, if 0 β > and a b t Let f be a function defined on [ ] 0,1 .The fractional q-integral of the Riemann-Liouville type of order 0 β ≥ is given by Notice that, for 1 β = , the above q-integral reduces to (11).Definition 2. [2] The fractional q-derivative of the Riemann-Liouville type of order 0 β ≥ is given by where [ ] β denotes the smallest integer greater than or equal to β .
In q-calculus, the derivative rule for the product of two functions and integration by parts formulas are Lemma 1.Let , 0 γ β ≥ and f be a function defined on [ ] 0,1 .Then the following formulas are verified: .
Lemma 2. [21] For 0 β > , using q-integration by parts, we have [ ] Second, we recall the basic concepts which we need throughout the paper about measure of noncompactness.
We assume that ( ) , E ⋅ is a real Banach space with zero element θ and we denote by ( ) , B x r the closed ball with radius r and centered x, where ( ) ⊂ and denote by X and Conv X the closure and convex closure of X, respectively.Also, the symbols X Y + and X λ stands for the usual algebraic operators on sets.
Moreover, the families

{ }
: is relatively compact , 2) Then, the mapping µ is said to be a measure of noncompactness in E.
Here, kerµ is the kernel of the measure of noncompactness µ .
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 ( ) C I , see [19] [22].
We fix a bounded subset Y ≠ ∅ of ( ) ∈ denotes the modulus of continuity of the function y given by Moreover, we let and , then  is said to be satisfies the Darbo condition with respect to a measure of noncompactness µ .If 1 γ < , then  is called a contraction operator with respect to µ .Theorem 1. [23] Let Q ≠ ∅ be a bounded, convex and closed subset of operator with respect to µ .Then  has at least one fixed point belongs to Q.

Existence Theorem
Let us consider the following suggestions: ( ) ( ) 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 a 1 ).a 3 ) ( ) ( ) is a continuous operator which satisfies the Darbo condition for the measure of noncompactness µ with a constant η .Also, The function : is a continuous operator and there is a nondecreasing function :  ( ) Before, we state and prove our main theorem, we define the two operators  and  on ( ) and respectively.Finding a fixed point of the operator  defined on the space ( ) C I is equivalent to solving Equation (1).
Theorem 2. Assume the suggestions (a 1 )-(a 7 ) be verified, then Equation (1) has at least one solution ( ) y C I ∈ 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 ( ) For this, it is sufficient to show that ( ) ( ) q q t t q q q t t q q q t t q q y t y Notice that, since the function k is uniformly continuous on I I × , then when 0 ε → we have that ( ) , and therefore, ( ).
. Γ 1 t q q t q q t q q q y t y t To prove this, we fix ( )

f t y t k t s t qs y s s y t f t y t f t f t k t s t qs y s s a b y k y c y f t qs s a b y k y c y f
to be a sequence in 0 r B + with n y y → .We will show that

y t c y t y t k t s t qs y s s y t k t s t qs y s y s s
As  and  are continuous operators, The last estimate implies Since the function k is uniformly continuous on I I × and the function f is continuous on , then the last inequality gives us that Step 6: In recognition of  with respect to the quantity d.
Here, we fix an arbitrary y Y ∈ and 1 2 , t t I ∈ with 2 1 t t > .Then, by our assumption, we obtain our sugges- tions, we have , t q q t q q t q q t q q y t y t t q q q q t t q q q q t t q q q y t f Now, we will prove that We find that , t t q q t t q q t t q q t t q q k t s   Inequalities ( 50) and (51) imply that ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) The above estimate implies that Step 7:  is contraction with respect to the measure of noncompactness µ .
Inequalities (46) and (54) give us that ∅ is closed, convex and bounded.Step 3:  applies the set 0 r B + into itself.By this facts and suggestions a 1 ), a 4 ) and a 6 ), we obtain  transforms 0 r B + into itself.Step 4: The operator  is continuous on 0 r B + .
Suppose that  maps bounded sets onto bounded ones.If there exists a bounded Notice that, all functions in Y are nondecreasing on I if and only if ( ) 0 d Y = .Now, we define the map µ on ( )