Tripled Fixed Points and Tripled Best Proximity Points in Modular Function Spaces

We establish a modular-space framework for the study of tripled fixed points and tripled best proximity points. Under suitable assumptions on the underlying modular (convexity, the Δ2 property, uniform continuity, and uniform convexity-type properties), we prove that Banach theorems guarantee the existence, uniqueness, and convergence of modular iterative schemes. In particular, we develop results for cyclic ρ–Kannan contraction maps and pairs, showing that both tripled fixed points and tripled best proximity points arise uniquely and attract all iterative trajectories. An illustrative example in the space L2[0,1] with integral operators demonstrates the applicability of the theory and the predicted rate of convergence. These results extend classical fixed point methods to a broader modular setting and open the way for applications in nonlinear functional equations.