Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard Manifolds

In this paper, we consider a class of nonsmooth weak vector variational-hemivariational inequality problems (abbreviated as, WVVHVIP) in the framework of Hadamard manifolds. By employing an analogous to the KKM lemma, we establish the existence of the solutions for WVVHVIP without utilizing any monotonicity assumptions. Moreover, a uniqueness result for the solutions of WVVHVIP is established by using generalized geodesic strong monotonicity assumptions. We formulate Auslender, regularized, and Moreau-Yosida regularized type gap functions for WVVHVIP to establish necessary and sufficient conditions for the existence of the solutions to WVVHVIP. In addition to this, by employing the Auslender, regularized, and Moreau-Yosida regularized type gap functions, we derive the global error bounds for the solution of WVVHVIP under the generalized geodesic strong monotonicity assumptions. Several non-trivial examples are furnished in the Hadamard manifold setting to illustrate the significance of the established results. To the best of our knowledge, this is the first time that the existence results, gap functions, and global error bounds for WVVHVIP have been investigated in the framework of Hadamard manifolds via Clarke subdifferentials.