Study of equilibrium problem in Hadamard manifolds by extending some concepts of nonlinear analysis

Abstract

In this paper, we attempt to define a new KKM map for nonself maps in the setting of Hadamard manifolds, which is then utilized to state the finite intersection property in this framework. Subsequently, this property is used to develop the Fan-KKM theorem for nonself maps on Hadamard manifolds. Moreover, a new definition of upper semicontinuouty and a generalization of the closedness of a set are also proposed. Inspired by this extension, we establish a new existence theorem of a solution to the equilibrium problem for nonself maps on Hadamard Manifolds. As an application of our KKM theorem, we obtain an existence result of maximal elements for nonself set valued mappings in Hadamard manifold frameworks. Finally, for the sake of clarity, a number of examples are presented throughout the paper and our results are compared with the results of some other papers.