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In this paper, we discuss about monotone vector fields, which is a typicalextension to the theory of convex functions, by exploiting the tangent spacestructure. This new approach to monotonicity in CAT(0) spaces stands in opposedto the monotonicity defined earlier in CAT(0) spaces by Khatibzadeh and Ranjbar[14] and Chaipunya and Kumam [8]. In particular, this new concept extends thetheory from both Hilbert spaces and Hadamard manifolds, while the known conceptbarely has any obvious relationship to the theory in Hadamard manifolds. Wealso study the corresponding resolvents and Yosida approximations of a givenmonotone vector field and derive many of their important properties. Finally,we prove a generation theorem by showing convergence of an exponential formulaapplied to resolvents of a monotone vector field. Our findings improve severalknown results in the literature including generation theorems of Jost [13,Theorem 1.3.13], Mayer [19, Theorem 1.13], Stojkovic [22, Theorem 2.18], andBac\'ak [4, Theorem 1.5] for proper, convex, lower semicontinuous functions inthe context of complete CAT(0) spaces, and also by Iwamiya and Okochi [11,Theorem 4.1] for monotone vector fields in the context of Hadamard manifolds.

Monotone vector fields and generation of nonexpansive semigroups in complete CAT(0) spaces