2025, Vol. 10, Issue 3, Part B

This paper explores the applications of Fixed-point theory in metric spaces, focusing on the role of real numbers in matrix spaces and iterative methods. Using Banach's Contraction Principle, the study provides mathematical proofs to establish the existence and uniqueness of fixed points, while also analyzing the convergence and stability of iterative methods. In a complete metric space (?, ?), a contraction mapping ?: ? ? ? has a unique fixed point ?? such that ? (??) = ?? and ? (? (?), ? (?)) ? ? ? (?, ?), where 0 ? ? < 1. 

The Intermediate Value Theorem ensures that for a continuous function ?on [?, ?] with ?(?) ? ? ? ?(?), there is a point ? ? [?, ?] such that ?(?) = ?. The Mean Value Theorem states that for a differentiable function ? on (?,), there exists a point ? ? (?, ?) such that ??(?)=?(?) – ?(?) / ???. The Cauchy-Schwarz inequality for vectors ?and ? gives ???, ??? ? ??? ???. For the function (?) =  1/(2 )  (?+3), the fixed point is ?? = 3. In matrix space, the operator (?) =  1/( 2)  ? has a fixed point ? = 0. For ? = cos (?), the successive approximations method converges to ?? ? 0.7391, and the Newton-Raphson method achieves faster convergence to the same fixed point.

This research highlights the practical applications of fixed-point theory, supported by real numbers, as a foundation for accurate mathematical models. It enhances outcome prediction in optimization, stability analysis, and system modeling, emphasizing the critical role of real numbers in both theoretical and applied mathematics.