In this article, a special mapping, the so-called enriched nonexpansive mapping with center zero and its asymptotic version are introduced and corresponding fixed point properties are investigated in the setting of complete normed spaces. Further, using approximate fixed point sequences, the fixed points of such mappings are analysed where Banach spaces have Kadec-Klee Property. Finally, a convexically enriched nonexpansive mapping is launched as a special case of the studied mappings.

In this paper, we propose one step convex combination of proximal point algorithms for countable collection of monotone vector fields in CAT(0) spaces. We establish $ \Delta $-convergence and strong convergence theorems for approximating a common solution of a countable family of monotone vector field inclusion problems. Furthermore, we apply our methods to solve a family of minimization problems, compute Frechét mean and geometric median in CAT(0) spaces, and solve a kinematic problem in robotic motion control. Finally, we give a numerical example to show the efficiency and robustness of the proposed scheme in comparison to a known scheme in the literature.