Fixed point theorems for continuous single-valued and upper semicontinuous set-valued mappings in p-vector and locally p-convex spaces

The goal of this paper is to establish a general fixed point theorem for compact single-valued continuous mappings in Hausdorff p-vector spaces, and a fixed point theorem for upper semicontinuous set-valued mappings in locally p-convex spaces for p∈ (0, 1]. These results not only provide a solution to Schauder conjecture in the affirmative under the setting of p-vector spaces for compact single-valued continuous mappings, but also show the existence of fixed points for upper semicontinuous set-valued mappings defined on s-convex subsets in Hausdorff locally p-convex spaces, which would be fundamental for nonlinear functional analysis, where s, p ∈ (0, 1].