Existence and Uniqueness of Solutions for Higher-Order Three-Point Boundary Value Problems

Abstract

We are concerned with the higher-order nonlinear three-point boundary value problems: x(n)=f(t,x,x′,…,x(n−1)),n≥3, with the three point boundary conditions g(x(a),x′(a),…,x(n−1)(a))=0; x(i)(b)=μi,i=0,1,…,n−3;h(x(c),x′(c),…,x(n−1)(c))=0, where a<b<c,f:[a,c]×ℝn→ℝ=(−∞,+∞) is continuous, g,h:ℝn→ℝ are continuous, and μi∈ℝ,i=0,1,…,n−3 are arbitrary given constants. The existence and uniqueness results are obtained by using the method of upper and lower solutions together with Leray-Schauder degree theory. We give two examples to demonstrate our result.