A third-order 3-point BVP. Applying Krasnosel'skii's theorem on the plane without a Green's function

Abstract

Consider the three-point boundary value problem for the 3$^{rd}$ order differential equation: \begin{equation*}\left\{ \begin{aligned} & x^{^{\prime \prime \prime }}(t)=\alpha \left( t\right) f(t,x(t),x^{\prime}\left( t\right) ,x^{\prime \prime }\left( t\right) ),\;\;\;0<t<1, \\ & x\left( 0\right) =x^{\prime }\left( \eta \right) =x^{\prime \prime }\left(1\right) =0, \end{aligned}\right.\end{equation*} under positivity of the nonlinearity. Existence results for a positive and concave solution $x\left( t\right) ,\ 0\leq t\leq 1$ are given, for any $1/2<\eta <1.\ $ In addition, without any monotonicity assumption on the nonlinearity, we prove the existence of a sequence of such solutions with \begin{equation*} \lim_{n\rightarrow \infty }||x_{n}||=0. \end{equation*} Our principal tool is a very simple applications on a new cone of the plane of the well-known Krasnosel’skiĭ’s fixed point theorem. The main feature of this approach is that, we do not use at all the associated Green's function, the necessary positivity of which yields the restriction $\eta \in \left( 1/2,1\right) $. Our method still guarantees that the solution we obtain is positive.