Positive Solutions Of Multi-point Boundary Value Problems For ODE Of Fourth Order

This paper is concerned with the m- point boundary value problems of fourth order ordinary differential equation (ODE)where α,β ∈ R, ξi∈ (0, 1), ai, bi ∈ [0, ∞) , i ∈ {1, 2, ... , m - 2} are constants. By using fixed point index theories in cones, we establish some theorems related to the existence of positive solution to (E1) as well as the uniqueness of positive solution to (E1) for a kind of special case. In addition, some result is given when dealing with the corresponding eigenvalue problemsWe establish the criteria for judging the existence of positive solutions, two positive solutions, and no solution.The main results of this paper are as follows:(i). The existence theorem of positive solutions for (E1) is established . (ii). The uniqueness of positive solution for (E1) is discussed for a special case, (iii). The theorem by judging the existence and multiplicity as well as thenonexistence of positive solutions to (E2) is presented .