A common fixed point theorem and its application in dynamic programming

Abstract

This master’s thesis establishes a significant common fixed point theorem for four self-mappings

defined on complete metric spaces, under the assumptions of weak compatibility and a gener- alized Φ-contraction condition. This contribution extends and generalizes previous results in

fixed point theory, demonstrating its applicability by solving functional equations that arise in dynamic programming. These equations typically represent recursive formulations of optimal value functions in multistage decision processes. The authors construct a comprehensive framework using metric fixed-point tools, ensuring the existence and uniqueness of solutions to these functional systems. An illustrative example to confirm the theoretical assumptions. This contribution effectively bridges abstract fixed-point theory with applied optimization and control problems, enriching the mathematical foundation of dynamic programming models and opening avenues for further research in optimization.