Stability and convergence of common fixed point algorithms for a countable infinite family of enriched nonexpansive mappings

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.