The main purpose of this work can be divided in to three aspects.First, a study of the existence and uniqueness of the solution for special types of linear operator equations, namely the Lyapunov equation.
Second, a discussion of the range for the quaii-Jordan*-derivation.Third, some special types of Lyapunov equation, namely stein equation.
The aim of this thesis is to prove the existence and uniqueness of the mild solutions of semilinear initial value control problems in a suitable Banach spaces as well as their controllability. Some theorems regarding controllability, local and global existence as well as uniqueness of the mild solution in infinite dimensional spaces have been developed in suitable Banach space using the Schauder fixed point theorem and the semigroup theory (compact semigroup).
The objective of this thesis studying and deriving with some modification as a new approach of Runge-Kutta method including explicit, semi-explicit and implicit methods as well as studying stability of convergence of these methods.Also, one of most important themes of the thesis is to introduce variable step size and variable order methods using an extrapolation method which has the utility of controlling the local truncation error to be less than a prespecified tolerance error.
