Study on Existence of Solution for Some Fractional Integro Differential Equations Via the Iterative Process

Study on Existence of Solution for Some Fractional Integro Differential Equations Via the Iterative Process

Somayyeh Dadsetadi Kazem Nouri

Department of Mathematics‎, ‎Faculty of Mathematics‎, ‎Statistics and‎ Computer Sciences‎‎, ‎Semnan University, P.O. Box 35195-363, Semnan, Iran

Corresponding Author Email:
22 April 2018
8 June 2018
30 June 2018
| Citation



We study the existence and uniqueness of solution of nonlinear fractional integro-differential equations of the Hammerstein type, using the iterative method under some suitable conditions in the Banach space. At the end, an example is given to illustrate the theory.


fractional Hammerstein integro-differential equations, Caputo fractional derivative, iterative method

1. Introduction
2. Preliminaries
3. Explanation of the Problem
4. Existence and Uniqueness
5. Application

[1] Aghajani A, Jalilian Y, Trujillo JJ. (2012). On the existence of solutions of fractional integro-differential equations. Fractional Calculus and Applied Analysis 15: 44-69.

[2] Alfifi HY, Ben Saad I, Turki S, El-Abidine ZZ. (2017). Existence and asymptotic behavior of positive solutions for a coupled system of semilinear fractional differential equations. Results in Mathematics 71: 705-730.

[3] Anguraj A, Karthikeyan P, Trujillo JJ. (2011). Existence of solutions to fractional mixed integro-differential equations with nonlocal initial condition. Advances in Difference Equations 690653.

[4] Baghani O, Gachpazan M, Baghani H. (2012). Existence, uniqueness and stability of solutions for a class of nonlinear integral equations under generalized Lipschitz condition. Indian Journal of Pure and Applied Mathematics 43: 309-321.

[5] Balachandran K, Kiruthika S, Trujillo JJ. (2011). Existence results for fractional impulsive integro-differential equations in Banach spaces. Communications in Nonlinear Science and Numerical Simulation 16: 1970-1977.

[6] Benyettou L. (2016). Performance evaluation of a multi-sensor system using fixed point DSP for water leak detection. Advances in Modelling and Analysis D 21: 78-87.

[7] Das S. (2008). Functional fractional calculus for system identification and controls. Springer-Verlag, Berlin, Heidelberg.

[8] Gautam GR, Dabas J. (2014). Existence result of fractional functional integro-differential equation with not instantaneous impulse. International Journal of Advances in Applied Mathematics and Mechanics 1: 11-21. 

[9] Henderson J, Luca R. (2017). Existence of nonnegative solutions for a fractional integro-differential equation. Results in Mathematics 72: 747-763. 

[10] Kilbas AA, Srivastava HM, Trujillo JJ. (2006). Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.

[11] Klafter J, Lim SC, Metzler R. (2011). Fractional Dynamics in Physics: Recent Advances. World Scientific, Singapore. 

[12] Li G. (2017). A quintic spline collocation method for the fractional sub-diffusion equation with variable coefficients. Advances in Modelling and Analysis A 54: 40-49.

[13] Merdan M, Gökdoğan A, Yildirim A. (2013). On numerical solution to fractional non-linear oscillatory equations. Meccanica 48: 1201-1213.

[14] Nouri K, Baleanu D, Torkzadeh L. (2018). Study on application of hybrid functions to fractional differential equations. Iranian Journal of Science and Technology, Transactions A, Science 42: 1343–1350.

[15] Nouri K, Elahi-Mehr S, Torkzadeh L. (2016). Investigation of the behavior of the fractional Bagley-Torvik and Basset equations via numerical inverse Laplace transform. Romanian Reports in Physics 68: 503-514. 

[16] Nouri K, Nazari M, Keramati B. (2017). Existence results for a coupled system of fractional integro-differential equations with time-dependent delay. Journal of Fixed Point Theory and Applications 19: 2927-2943.

[17] Wang F. (2012). Existence and uniqueness of solutions for a nonlinear fractional differential equation. Journal of Applied Mathematics and Computing 39: 53-67.