EXISTENCE OF EQUILIBRIUM PROBLEM AND VARIATIONAL INEQUALITY ON HADAMARD MANIFOLD VIA GENRALIZATION OF SOME NOTIONS OF NONLINEAR ANALYSIS

In this work, we first introduce a new version of KKM map for nonself maps on Hadamard manifolds. Then, by using this KKM map, the finite intersection property is stated in the setting of Hadamard manifolds. Afterwards, this property is utilized to extend the Fan-KKM theorem for nonself maps on Hadamard manifolds. Consequently, inspired by this KKM theorem, we present a new existence result of solutions for equilibrium problem for nonself maps on Hadamard manifolds. As an application, at first, we provide a new generalization of variational inequality problem in the setting of Hadamard manifolds. Moreover, we obtain a new existence theorem for generalized variational inequality problem on Hadamard manifolds. Finally, some examples are exposed throughout the paper and our results are compared with the results of some recent papers.