Fractional analytic solutions and fixed point results with some applications

Abstract

This study explores a collection of theorems that provide valuable insights into the existence and properties of fixed points in mathematical and real-life problems. The first theorem establishes the existence of fixed points for contractive mappings, guaranteeing the convergence of iterative sequences. Building upon this result, the second theorem extends the concept to complete metric spaces, enabling the convergence analysis of sequences generated by repeated application of the mapping. To demonstrate the practical relevance of these theorems, a real-life example is presented in the context of population dynamics. By formulating the dynamics as a system of equations, the theorems are applied to determine equilibrium points and analyze the long-term behavior of populations. Numerical solutions and graphical representations shed light on the stability and coexistence of species, showcasing the applicability of the theorems in ecological, economic, and engineering contexts. Moreover, the introduction of fractional calculus in the third theorem enriches the analysis by considering fractional derivatives in self-mappings. This theorem establishes a connection between sequence convergence and the existence of fixed points, providing a powerful tool for studying complex systems with fractional dynamics.