On the solution method of operator equations using power series expansion of abstract functions depending on numerical variables

In this thesis, the basic properties of abstract functions and operator-functions depending on numerical variables are examined in detail. For these functions, the concepts of limit, continuity, differential and Riemann integral as well as their properties are given. Power series expansion of abstract functions is shown, and then Abel's theorem and its proof are presented. Analytical abstract functions are given and their expansion into Taylor series is shown. Then, using the power series expansion of abstract functions and operator-functions, the solution methods of operator equations in the form of Ax-λCx=y and A(λ)x=y(λ) are presented. Finally, an example of the solution of the parameter dependent operator equation is given to demonstrate the effectiveness of these proposed solution methods.