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In this paper, we point out that a recent characterization of geodesic convex hull on Hadamard manifolds is not rigorous and explain why the characterization does not hold like it in linear spaces. Therefore, a definition of geodesic pseudo-convex combination is proposed to show that the Knaster–Kuratowski–Mazurkiewicz theorem still holds under some mild conditions on Hadamard manifolds.

In this paper, some new results, concerned with the geodesic convex hull and geodesic convex combination, are given on Hadamard manifolds. An S-KKM theorem on a Hadamard manifold is also given in order to generalize the KKM theorem. As applications, a Fan---Browder-type fixed point theorem and a fixed point theorem for the a new mapping class are proved on Hadamard manifolds.

In this paper, a class of hemivariational inequality problems are introduced and studied on Hadamard manifolds. Using the properties of Clarke’s generalized directional derivative and Fan-KKM lemma, an existence theorem of solution in connection with the hemivariational inequality problem is obtained when the constraint set is bounded. By employing some coercivity conditions and the properties of Clarke’s generalized directional derivative, an existence result and the boundedness of the set of solutions for the underlying problem are investigated when the constraint set is unbounded. Moreover, a sufficient and necessary condition for ensuring the nonemptiness of the set of solutions concerned with the hemivariational inequality problem is also given.

In this paper, a generalized mixed variational inequality on Hadamard manifolds is introduced and studied. Some gap functions for the generalized mixed variational inequality on Hadamard manifolds are obtained under suitable conditions. By using these gap functions, global error bounds for the generalized mixed variational inequality are derived on Hadamard manifolds. The main results presented in this paper generalize and improve corresponding known results.

In this paper, a relationship between a vector variational inequality and a vector optimization problem is given on a Hadamard manifold. An existence of a weak minimum for a constrained vector optimization problem is established by an analogous to KKM lemma on a Hadamard manifold.