Existences and uniqueness of solutions for some classes of iterative fractional functional integral and differential equations
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 sti...
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Format:  Thesis 
Language:  English 
Published: 
2017

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Online Access:  http://psasir.upm.edu.my/id/eprint/68642/1/FS%202018%208%20%20IR.pdf 
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Summary:  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, BrowderGhodeKirk Fixed Point Theorem, Schaefer Fixed Point Theo
rem, Burton fixed point theorem, Weakly Picard Operator, Power Series, Non
Expansive Operators and function gNon 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 integrodifferential 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. 
