The paper deals with the existence of multiple positive solutions for the boundary value problem (ϕ(p(t)u (n−1))(t))′ + a(t)f(t, u(t), u′ (t), . . . , u(n−2)(t)) = 0, 0 < t < 1, u (i) (0) = 0, i = 0, 1, . . . , n − 3, u (n−2)(0) = mP−2 i=1 αiu (n−2)(ξi), u(n−1)(1) = 0, where ϕ: R → R is an increasing homeomorphism and a positive homomorphism with ϕ(0) = 0. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.
In this paper, using a fixed point theorem on a convex cone, we consider the existence of positive solutions to the multipoint one-dimensional $p$-Laplacian boundary value problem with impulsive effects, and obtain multiplicity results for positive solutions.
This paper is a continuation of Y. Liu, Anti-periodic solutions of nonlinear first order impulsive functional differential equations, Math. Slovaca 62 (2012), 695–720. By using Schaefer's fixed point theorem, new existence results on anti-periodic solutions of a class of nonlinear impulsive functional differential equations are established. The techniques to get the priori estimates of the possible solutions of the mentioned equations are different from those used in known papers. An example is given to illustrate the main theorems obtained. One sees easily that Example 3.1 can not be solved by Theorems 2.1–2.3 obtained in Liu's paper since (G2) in Theorem 2.1, (G4) in Theorem 2.2 and (G6) in Theorem 2.3 are not satisfied.