An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set.For any integer fc > 1,let h(k) be the smallest integer such that every set of points in the plane,no three collinear,with at least h(k) interior points,has a subset of points with exactly fc or fc + 1 interior points of P.We prove that h(5) = 11.
By the fxed point index theory, the existence of one, two and three positive solutions to(k, n-k) conjugate boundary value problems is obtained, where n > 2, 1 ≤ k ≤ n-1, the nonlinear term may be noncontinuous and singular at any point of [0,1]. Our results extend some of the existent results.
