Sufficient conditions on existence of solution for nonlinear fractional iterative integral equation

1036
Downloads

1411
Views
Authors
Faten H. M. Damag
 Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia.
Adem Kilicman
 Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia.
Abstract
In this article, we study nonlinear quadratic iterative integral equations and establish sufficient conditions for the existence
of Volterra solutions for fractional iterative integral equations and solvency in Banach space and \(C_{\ell,\beta}\). In the present work we
use the principle of contraction, Schaefer’s fixed point theorem and the nonexpansive operator method as essential tools. In
this study we consider RiemannLiouville differential operator and prove some related theorems, further provide an example as
an application.
Keywords
 Fractional integral equation
 existence of solution
 Schaefer’s fixed point theorem
 nonexpansive operator.
MSC
References

[1]
A. Atangana, R. T. Alqahtani, Numerical approximation of the spacetime CaputoFabrizio fractional derivative and application to groundwater pollution equation, Adv. Difference Equ., 2016 (2016), 13 pages.

[2]
A. Atangana, I. Koca, Chaos in a simple nonlinear system with AtanganaBaleanu derivatives with fractional order, Chaos Solitons Fractals, 89 (2016), 447–454.

[3]
V. Berinde, Iterative approximation of fixed points, Second edition, Lecture Notes in Mathematics, Springer, Berlin (2007)

[4]
V. Berinde, Existence and approximation of solutions of some first order iterative differential equations, Miskolc Math. Notes, 11 (2010), 13–26.

[5]
S. S. Cheng, J.G. Si, X.P.Wang, An existence theorem for iterative functional differential equations, Acta Math. Hungar., 94 (2002), 1–17.

[6]
F. H. Damag, A. Kılıçman, R. A. A. Abdulghafor, Approximate solutions and existence result for some integral equation with modified argument of fractional order, Adv. Difference Equ., (2016), submitted

[7]
F. H. Damag, A. Kılıçman, R. W. Ibrahim, Approximate solutions for nonlinear iterative fractional differential equations, Innovations Through Mathematical and Statistical Research, Proceedings of the 2nd International Conference on Mathematical Sciences and Statistics (ICMSS2016) , AIP Publishing,1739 ((2016). )

[8]
F. H. Damag, A. Kılıçman, R. W. Ibrahim, Findings of fractional iterative differential equations involving first order derivative, Int. J. Appl. Comput. Math., (2016), 1–10.

[9]
A. ElSayed, H. Hashem, Existence results for nonlinear quadratic integral equations of fractional order in Banach algebra, Fract. Calc. Appl. Anal., 16 (2013), 816–826.

[10]
A. Granas, J. Dugundji, Fixed point theory, Springer Science and Business Media, (2013)

[11]
R.W. Ibrahim, A. Kılıçman, F. H. Damag, Existence and uniqueness for a class of iterative fractional differential equations, Adv. Difference Equ., 2015 (2015 ), 13 pages.

[12]
R. W. Ibrahim, A. Kılıçman, F. H. Damag, Extremal solutions by monotone iterative technique for hybrid fractional differential equations, Turkish J. Anal. Number Theory, 4 (2016), 60–66.

[13]
S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59 (1976), 65–71.

[14]
M. Lauran , Existence results for some integral equation with modified argument, Gen. Math., 19 (2011), 85–92.

[15]
M. Lauran, Existence results for some nonlinear integral equations, Miskolc Math. Notes, 13 (2012), 67–74.

[16]
M. Lauran, Solution of first iterative differential equations, An. Univ. Craiova Ser. Mat. Inform, 40 (2013), 45–51.

[17]
K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A WileyInterscience Publication, John Wiley & Sons, Inc., New York (1993)

[18]
D. O’Regan, Existence results for nonlinear integral equations, J. Math. Anal. Appl., 192 (1995), 705–726.

[19]
I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)

[20]
S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikolskiĭ, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)

[21]
J.R. Wang, M. Fečkan, Y. Zhou, Fractional order iterative functional differential equations with parameter, Appl. Math. Model., 37 (2013), 6055–6067.

[22]
P.P. Zhang, X.B. Gong, Existence of solutions for iterative differential equations, Electron, J. Differential Equations, 2014 (2014 ), 10 pages.