Theoretical fixed point results involving double averaged mappings with new enriching techniques and applications

This dissertation delves into the intricate development of the classical fixed point theorem by delving deep into the fixed point theory in metric spaces. The main focus of this research is on the enriching technique, which was first introduced by Berinde and P\u{a}curar [Approximating fixed points of enriched contraction mappings in Banach spaces. J. Fixed Point Theory Appl. 22, 38 (2020)] published in 2020. In this study, we aim to explore the limitations of classical fixed point theorems and propose new ways to enhance its efficacy. To this end, we are the first to introduce a novel class of mappings called double averaged mappings and conduct an in-depth investigation of their properties. Our findings on double averaged mappings provide a fresh perspective on the development of classical fixed point theory. By utilizing a variety of the enriching technique, which we refer to as the new enriching technique, we are able to demonstrate how our results on double averaged mappings can be utilized to refine classical fixed point theorems with fewer strictly necessary conditions. Lastly, we apply our newly-developed fixed point results to the existence theory of nonlinear fractional integral equations, demonstrating the practical relevance of our research. Overall, this dissertation represents a significant contribution to the field of fixed point theory and lays the foundation for future research in this area.