Existences and uniqueness of solutions for some classes of iterative fractional functional integral and differential equations - Damag, Faten Hasan Mohammed (2017)
The iterative functional equations are important classes and deal with fractional differential and integral equations which involve the composition of unknown. The related theory for some classes of iterative fractional integral and differential equations are not established yet and there are still many open problems in this field. In this thesis, we focus on some classes of initial value problems in the iterative fractional integral and differential equations. The existence and uniqueness of the solutions for such equations, including their qualitative behavior are investi- gated. Fixed Point Theorems, Schauder Fixed Point Theorem, Banach Fixed Point The- orem, Browder-Ghode-Kirk Fixed Point Theorem, Schaefer Fixed Point Theo- rem, Burton fixed point theorem, Weakly Picard Operator, Power Series, Non Expansive Operators and function g-Non Expansive Operators are used to prove the existences and uniqueness of solutions for particular classes of iterative frac- tional integral and differential equations. We establish the sufficient conditions for the existence of solutions for three types of iterative fractional equations i.e iterative fractional differential equations, it- erative fractional integro-differential equations and iterative fractional integral equations. Further, the uniqueness and existence of solutions are proved. The convergence of solutions for special type of iterative fractional differential equa- tions is also studied. The existences and uniqueness of solutions for generalized iterative fractional differential equations, iterative fractional differential equations with state depen- dent, and system of iterative fractional differential equations, and the existences and uniqueness of solutions for generalized classes of iterative fractional integral equations and system of iterative fractional integral equations are proved. We also examine the existences and uniqueness of solutions for nonlinear itera- tive fractional differential equations by testing the convergence of solutions. The sufficient conditions for nonlinear quadratic iterative integral equations are es- tablished. A new type of stability based on the Burton fixed point theorem for the general iterative fractional differential equations is investigated. In order to verify the theorems, several examples are provided.