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In this article, we prove ergodic convergence for a sequence of nonexpansive mappings in Hadamard (completeCAT(0){\rm{CAT}}\left(0)) spaces. Our result extends the nonlinear ergodic theorem by Baillon for nonexpansive mappings from Hilbert spaces to Hadamard spaces. We further establish the standard nonlinear ergodic theorem and apply our results to the problem of finding a common fixed point of a countable family of nonexpansive mappings. Finally, we propose some applications of our results to solve convex optimization problems in Hadamard spaces.

Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications