It is known that the class of p-vector spaces (0 < p ≤ 1) is an important generalization of usual norm spaces with rich topological and geometrical structure, but the most tools and general principles with nature in nonlinearity have not been developed yet, the goal of this paper is to develop some useful tools in nonlinear analysisby applying the best approximation approach for the classes of 1-set contractive set-valued mappings in p-vectorspaces. In particular, we first develop the general fixed point theorems of condensing mappings which provideanswer to Schauder conjecture in 1930’s in the affirmative way under the setting of p-vector spaces. Then one bestapproximation result for upper semi-continuous and 1-set contractive set-valued is established, which is used as auseful tool to establish fixed points of non-self set-valued mappings with either inward or outward set conditions,and related various boundary conditions.