Solvability of 4-Point Boundary Value Problems at Resonance for Fourth-Order Ordinary Differential Equations

Abstract:

      In this paper, we consider the following fourth order ordinary differential equation $$x^{(4)}(t)=f(t,x(t),x'(t),x''(t),x'''(t)),~~t\in(0,1) \tag E$$ with the four-point boundary value conditions: $$x(0)=x(1)=0,\; \alpha x''(\xi_1)-\beta x'''(\xi_1)=0,\;\gamma x''(\xi_2)+\delta x'''(\xi_2)=0, \tag B$$ where $0<\xi_1<\xi_2< 1.$ At the resonance condition $\alpha \delta+\beta\gamma+\alpha\gamma(\xi_2-\xi_1)=0,$ an existence result is given by using the coincidence degree theory. We also give an example to demonstrate the result.