We study Fan's minimax inequality miny ∈ c supx ∈ c f(x, y) ≤ supx ∈ c f(x, x), under different assumptions on f and C in locally convex topological vector spaces. The main generalization is the weaking of "convex" to "acyclic." The aim of this paper is to develop as consequence of several fixed point theorems and coincidence theorems a variety of existence results relevant to minimax inequalities. As an application, some variational inequalities are deduced without monotonicity nor coercivity.
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