Analysis of Fixed Points in Controlled Metric Type Spaces with Application

This article, presents a study on the existence and uniqueness of fixed points using Ćirić type and Reich type contractions in the context of controlled type metric spaces. By incorporating graph theory into these contraction mappings, we aim to provide a more comprehensive understanding of fixed point results. The utilization of graphs helps visualize and analyze the behavior of these mappings, leading to improved insights into the structure and properties of fixed points. Through examples, we demonstrate the practical applications of these results in various situations. Additionally, we apply these findings to solve a second order differential equation using a second kind Fredholm integral equation. By utilizing the power of contraction mappings, we are able to establish important theoretical results and solve practical problems. Our work highlights the usefulness of contraction mappings in mathematics and their wide range of applications in nonlinear differential equations. This study contributes to the growing body of research on the applications of contraction mappings in mathematical analysis.