Recent Developments in Fixed Points and Best Proximity Points Theory and Applications

Abstract

This work explores fixed point and best proximity point theorems in complex-valued metric spaces, providing new observations and extending the existing results. It establishes the existence and uniqueness of fixed points for interpolative rational-type contractive conditions with a complex-valued degree of freedom. The study also refines rational contractive conditions with control functions in bicomplex space combining them with locally contractive conditions. A fast iterative method is introduced to find fixed points in complex-valued Banach spaces, specifically under contraction and weak contraction conditions. Additionally, the research extends best proximity point theorems for complex-valued metric spaces under various contractive conditions and introduces new rational-type contractive conditions involving control functions. The study analyses fractal dynamics, particularly in Julia and Mandelbrot sets, using generalized rational complex polynomials and advanced approximation techniques. Finally, it offers simple proofs for solving several fractional integral equations.