Iteration for a Third-Order Three-Point BVP with Sign-Changing Green's Function

Abstract

We are concerned with the following third-order three-point boundary value problem: $u‴(t)=f(t\mathrm{,u}(t)),\mathrm{}\mathrm{t}\in [\mathrm{0,1}],u\prime (\mathrm{0})=u(\mathrm{1})=\mathrm{0,}\mathrm{}\mathrm{u}″(\eta )+\mathrm{\alpha u}(\mathrm{0})=\mathrm{0}$, where $\alpha \in [\mathrm{0,2})$ and $\eta \in [\mathrm{2}/\mathrm{3,}\mathrm{1})$. Although corresponding Green's function is sign-changing, we still obtain the existence of monotone positive solution under some suitable conditions on $f$ by applying iterative method. An example is also included to illustrate the main results obtained.