Fixed Point Theorems in Complete Metric Spaces: Banach, Kannan, and Common Fixed-Point Results with Applications to Volterra Integral Equations

Fixed point theory occupies a central position in modern mathematics, providing existence and uniqueness results that underlie fundamental theorems in functional analysis, differential equations, integral equations, topology, and operator theory. This paper presents a self-contained, rigorous treatment of fixed-point theorems in complete metric spaces, beginning with the foundational concepts of metric spaces and Cauchy completeness, progressing through the Banach Contraction Principle and its complete proof, generalizing to the Kannan fixed point theorem (which does not require continuity of the mapping), and culminating in common fixed point results for pairs of mappings. The independence of Banach and Kannan conditions is demonstrated through explicit counterexamples. The theoretical results are applied to establish existence and uniqueness of solutions to Volterra integral equations of the second kind, demonstrating the practical power of fixed-point methods. Convergence rates of Banach iteration are compared for different contraction constants with numerical illustration. All theorems are proved in full detail with concrete worked examples throughout.