In this paper, we study the existence of nonnegative solutions of singular (n - 1,1) conjugate boundary value problemwith boundary value conditions u(k)(0) = 0(0≤k≤n - 2) and u(I) = 0. Where a(t) may have finitely many singularities. Our results essentially generalized paper [1] and other related literature (for example [4]).
This paper is concerned with chaos in discrete dynamical systems governed by continuously Frechet differentiable maps in Banach spaces. A criterion of chaos induced by a regular nondegenerate homoclinic orbit is established. Chaos of discrete dynamical systems in the n-dimensional real space is also discussed, with two criteria derived for chaos induced by nondegenerate snap-back repellers, one of which is a modified version of Marotto's theorem. In particular, a necessary and sufficient condition is obtained for an expanding fixed point of a differentiable map in a general Banach space and in an n-dimensional real space, respectively. It completely solves a long-standing puzzle about the relationship between the expansion of a continuously differentiable map near a fixed point in an n-dimensional real space and the eigenvalues of the Jacobi matrix of the map at the fixed point.
