W-convergence of the proximal point algorithm in complete CAT(0) metric spaces

‎In this paper‎, ‎we generalize the proximal point algorithm to complete CAT(0) spaces and show‎ ‎that the sequence generated by the proximal point algorithm‎ $w$-converges to a zero of the maximal‎ ‎monotone operator‎. ‎Also‎, ‎we prove that if $f‎: ‎Xrightarrow‎ ‎]-infty‎, +‎infty]$ is a proper‎, ‎convex and lower semicontinuous‎ ‎function on the complete CAT(0) space $X$‎, ‎then the proximal point algorithm $w$-converges to a zero of the subdifferential of $f$‎, ‎i.e., a minimizer of $f$‎. ‎Some strong convergence results (convergence in metric) are also presented with additional assumptions on the monotone operator and‎ ‎the convex function $f$.