Back
 AJCM  Vol.6 No.2 , June 2016
On Existence of Solutions of q-Perturbed Quadratic Integral Equations
Abstract: 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.
Cite this paper: 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.
References

[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.
http://dx.doi.org/10.1017/S0305004100045060

[3]   Jleli, M., Mursaleen, M. and Samet, B. (2016) Q-Integral Equations of Fractional Orders. Electronic Journal of Differential Equations, 2016, 1-14.
http://dx.doi.org/10.1186/s13662-015-0739-5

[4]   Kac, V. and Cheung, P. (2002) Quantum Calculus. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4613-0071-7

[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.
http://dx.doi.org/10.1016/j.cnsns.2011.01.026

[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.
http://dx.doi.org/10.1016/j.cnsns.2014.12.014

[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.
http://dx.doi.org/10.1006/jmaa.1998.5941

[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.
http://dx.doi.org/10.1016/j.camwa.2003.11.001

[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.
http://dx.doi.org/10.1016/j.jmaa.2004.11.054

[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.
http://dx.doi.org/10.1016/j.jmaa.2008.04.050

[17]   Darwish, M.A. (2005) On Quadratic Integral Equation of Fractional Orders. ournal of Mathematical Analysis and Applications, 311, 112-119.
http://dx.doi.org/10.1016/j.jmaa.2005.02.012

[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.
http://dx.doi.org/10.1016/j.nahs.2009.06.010

[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.
http://dx.doi.org/10.2298/AADM0701311R

[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.

 
 
Top