Boundary Value Problems For Second-order Differential Equations And The High-order Equation

In this paper, we study the existence of single or multiple positive solutions of three-point boundary value problems, under the assumption of resonance or non-resonance, and we also study the existence of nontrivial periodic solutions for high-order equation.In the first chapter we study the p-Laplacian equation — (g{t)φ(x'))' = λF(t, x) , subject to three-point boundary conditions using the Krasnoselskii's fixed point theorem, where g(t) is an increasing function. We derive the explicit intervals for A such that the existence of positive solutions are guaranteed.In the second chapter we consider the equation - (g(t)x')' = λF(t,x) in the resonance case, where g(t) is a differential function. We prove the existence of solutions on this problem by the coincidence degree theory.In chapter three, we study the existence of nontrivial periodic solutions for the high-order equation in X = Hⁿ⁺¹(0, L) ∩ H₀ⁿ(0, L)The tool we use is the variational method.