Abstract
This paper deals with the abstract generalized vector quasi-equilibrium problem in noncompact Hadamard manifolds. We prove the existence of solutions to the abstract generalized vector quasi-equilibrium problem under suitable conditions and provide applications to an abstract vector quasi-equilibrium problem, a generalized scalar equilibrium problem, a scalar equilibrium problem, and a perturbed saddle point problem. Finally, as an application of the existence of solutions to the generalized scalar equilibrium problem, we obtain a weakly mixed variational inequality and two mixed variational inequalities. The results presented in this paper unify and generalize many known results in the literature.
Highlights
Let K be a nonempty subset of a nonempty set X and f : X × X → R be a bifunction
It is well known that SEP contains a broad class of problems arising in pure and applied mathematics, such as fixed point, minimax and variational inequality, Nash equilibrium, complementarity, and convex optimization problems
Motivated by the recent work mentioned above, the main purpose of this paper is to introduce and study the abstract generalized vector quasi-equilibrium problem in noncompact Hadamard manifolds
Summary
Let K be a nonempty subset of a nonempty set X and f : X × X → R be a bifunction. The scalar equilibrium problem (for short, SEP) is to find x ∈ K such that f (x, y) ≥ for every y ∈ K.
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