This work intoduced a modified Halpern iterate for a countably infinite family of enriched nonexpansive mappings within convex metric spaces. Under the Aoyama-Kimura-Takahashi-Toyoda (AKTT) condition, we established that the generated sequence serves as an approximating common fixed point sequence of enriched nonexpansive mappings. Furthermore, strong convergence theorems were presented, ensuring that the iterative sequence converges to a common fixed point of the countably infinite family of enriched nonexpansive mappings in convex metric spaces, provided that the AKTT and the Song-Zheng (SZ) conditions are satisfied. Additionally, the concept of a $\mathtt{\text{W}}$-mapping was extended from Banach spaces to the convex metric spaces, thereby broadening and refining existing results in the literature. We also gave a numerical illustration in the framework of convex metric spaces to show the efficiency of our proposed algorithm.

This paper proposes a novel modified one-step iterative scheme for approximating common fixed points of three nonexpansive mappings in uniformly convex Banach spaces. The proposed scheme extends the classical one-step iteration associated with two mappings, which can be recovered as a particular case by an appropriate choice of the third mapping. Weak convergence of the generated sequence is established under both the Kadec-Klee property (KKP) and the Opial condition, while strong convergence is obtained by assuming a modified condition (B). Moreover, we introduce a new Condition (P) tailored for three mappings, which reduces to the well-known Condition (S) in the case of two mappings. The results presented here broaden and unify several existing contributions in the theory of common fixed point approximation.