On Existence of Solutions of q-Perturbed Quadratic Integral Equations

DOI: 10.4236/ajcm.2016.62018   PDF   HTML   XML   1,617 Downloads   2,131 Views  


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.

Share and Cite:

Al-Yami, M. (2016) On Existence of Solutions of q-Perturbed Quadratic Integral Equations. American Journal of Computational Mathematics, 6, 166-176. doi: 10.4236/ajcm.2016.62018.

Received 18 April 2016; accepted 26 June 2016; published 29 June 2016

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


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


Also, the q-analog of the power is given by




Notice that, exists and we will denote it by.

More generally, for we define




Notice that. Therefore, if, then.

The q-gamma function is defined by


where Or, equivalently, and satisfies

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 is defined by


If f is given on the interval and then


The operator is defined by


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




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


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


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


Lemma 1. Let and f be a function defined on. Then the following formulas are verified:


Lemma 2. [21] For, 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 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




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








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

Now, we define the map on as


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.


a5) The function is continuous on and nondecreasing and separately. Moreo-


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



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




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


Notice that, we have used


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




Therefore, if we get from assumption a7) the following


Therefore, maps into itself.

We define the subset of by


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,




As and are continuous operators, such that


Also, such that


Furthermore, such that


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


The last estimate implies


and, consequently,


Since the function k is uniformly continuous on 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 and with. Then, by our assumption, we obtain our suggestions, we have


Now, we will prove that


We find that


But, because is increasing with respect to t, then


and, since is negative for then


Inequalities (50) and (51) imply that

This inequality and (47) gives us


The above estimate implies that




Step 7: is contraction with respect to the measure of noncompactness.

Inequalities (46) and (54) give us that




But, then


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

This finishes our proof.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Jackson, F.H. (1910) On q-Definite Integrals. The Quarterly Journal of Pure and Applied Mathematics, 41, 193-203.
[2] Agarwal, R.P. (1969) Certain Fractional q-Integrals and q-Derivatives. Proceedings of the Cambridge Philosophical Society, 66, 365-370.
[3] Jleli, M., Mursaleen, M. and Samet, B. (2016) Q-Integral Equations of Fractional Orders. Electronic Journal of Differential Equations, 2016, 1-14.
[4] Kac, V. and Cheung, P. (2002) Quantum Calculus. Springer-Verlag, New York.
[5] Annaby, M.H. and Mansour, Z.S. (2012) q-Fractional Calculus and Equations. Lecture Notes in Mathematics, Springer, Heidelberg, 2056.
[6] Abdeljawad, T. and Baleanu, D. (2011) Caputo q-Fractional Initial Value Problems and a q-Analogue Mittag-Leffler Function. Communications in Nonlinear Science and Numerical Simulation, 16, 4682-4688.
[7] Salahshour, S., Ahmadian, A. and Chan, C.S. (2015) Successive Approximation Method for Caputo q-Fractional IVPs. Communications in Nonlinear Science and Numerical Simulation, 24, 153-158.
[8] Banaś, J., Lecko, M. and El-Sayed, W.G. (1998) Existence Theorems of Some Quadratic Integralequations. Journal of Mathematical Analysis and Applications, 222, 276-285.
[9] Banaś, J., Caballero, J., Rocha, J. and Sadarangani, K. (2005) Monotonic Solutions of a Class of Quadratic Integral Equations of Volterra Type. Computers & Mathematics with Applications, 49, 943-952.
[10] Caballero, J., Lopez, B. and Sadarangani, K. (2005) On Monotonic Solutions of an Integral Equation of Volterra Type with Supremum. ournal of Mathematical Analysis and Applications, 305, 304-315.
[11] Darwish, M.A. (2007) On Solvability of Some Quadratic Functional-Integral Equation in Banach Algebra. Communications in Applied Analysis, 11, 441-450.
[12] Darwish, M.A. (2007) On a Singular Quadratic Integral Equation of Volterra Type with Supremum. IC/2007/071, Trieste, Italy, 1-13.
[13] Darwish, M.A. (2008) On Existence and Asympototic Behaviour of Solutions of a Fractional Integral Equation with Linear Modification of the Argument. arXiv: 0805.1422v1.
[14] Darwish, M.A. (2008) On Monotonic Solutions of a Singular Quadratic Integral Equation with Supremum. Dynamic Systems and Applications, 17, 539-550.
[15] Banaś, J. and Rzepka, B. (2007) Monotonic Solutions of a Quadratic Integral Equation of Fractional Order. Journal of Mathematical Analysis and Applications, 332, 1370-1378.
[16] Banaś, J. and O’Regan, D. (2008) On Existence and Local Attractivity of Solutions of a Quadratic Integral Equation of Fractional Order. Journal of Mathematical Analysis and Applications, 345, 573-582.
[17] Darwish, M.A. (2005) On Quadratic Integral Equation of Fractional Orders. ournal of Mathematical Analysis and Applications, 311, 112-119.
[18] Darwish, M.A. and Ntouyas, S.K. (2009) Monotonic Solutions of a Perturbed Quadratic Fractional Integral Equation, Nonlinear Analysis: Theory. Methods and Applications, 71, 5513-5521.
[19] Banaś, J. and Olszowy, L. (2001) Measures of Noncompactness Related to Monotonicity. Commentationes Mathematicae, 41, 13-23.
[20] Bhaskar, T.G., Lakshmikantham, V. and Leela, S. (2009) Fractional Differential Equations with a Krasnoselskii-Krein Type Condition. Nonlinear Analysis: Hybrid Systems, 3, 734-737.
[21] Rajković, P.M., Stanković, S.D. and Miomir, S. (2007) Fractional Integrals and Derivatives in q-Calculus. Applicable Analysis and Discrete Mathematics, 1, 311-323.
[22] Banaś, J. and Goebel, K. (1980) Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, New York, 60.
[23] Dugundji, J. and Granas, A. (1982) Fixed Point Theory. Monografie Mathematyczne, PWN, Warsaw.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.