Applications of fixed point theory to equations of curvature type.

This thesis investigates applications of Fixed Point Theory to singular ordinary differential equations arising from curvature conditions on rotational surfaces. The primary objective is to establish existence and uniqueness results for solutions of curvature equations derived from Weingarten surface relations in Euclidean space. The study focuses on rotational surfaces satisfying a prescribed functional relationship between mean curvature and Gauss curvature, particularly under the condition of orthogonal intersection with the axis of rotation. The research formulates the governing curvature equations as singular initial value problems and transforms them into equivalent integral equations suitable for fixed point analysis. The Banach Contraction Principle and Picard's Theorem are employed to construct nonlinear operators on appropriate function spaces and to demonstrate the existence and uniqueness of local solutions. This analytical framework allows singular behavior at the rotation axis to be handled rigorously without relying on classical regularity assumptions. The results provide a systematic method for proving the solvability of singular curvature equations and yield classification criteria for rotational Weingarten surfaces satisfying the given curvature relations. The findings contribute to the mathematical understanding of curvature-driven surface theory and illustrate the effectiveness of fixed-point methods in geometric analysis.