On refined rational convex contractions with applications to matrix and implicit functional integral equations.

This research presents an advancement in fixed point theory by extending the renowned Banach contractive condition and introducing the new convex contractive condition. Central to this work is exploring and developing new convex contraction mappings characterized by rational expressions. Our investigation not only proves fixed point results for these contractions but also highlights their uniqueness compared to well-established contractions like Banach, Kannan, Chatterjea, Jaggi, and other known contraction mappings. A significant aspect of this research is the presentation of new rational-type convex contraction mappings, demonstrating their independence from traditional contractions. Through illustrative examples, we showcase the comprehensive generalization and inherent advantages of our novel results. Furthermore, the practical applications of our findings are discussed, particularly in the context of solving matrix and functional integral equations. This is achieved by applying our theoretical results to establish the existence and uniqueness of solutions for these equations, thereby underscoring our research's practical significance and potential impact in the broader field of mathematical studies. [ABSTRACT FROM AUTHOR]