Three-point third-order problems with a sign-changing nonlinear term

Abstract

In this article we study a well-known boundary value problem $$\displaylines{ u'''(t) = f(t, u(t)), \quad 0 1/2$. It is well-known that the standard method successfully applied to the semi-positone problem with $\eta > 1/2$ does not work for $\eta =1/2$ in the same setting. We treat the latter as a problem with a sign-changing term rather than a semi-positone problem. We apply Krasnosel'skii's fixed point theorem [4] to obtain positive solutions.

Keywords

• Green's function
• fixed point theorem
• positive solutions
• third-order boundary-value problem