Semipositone higher-order differential equations

AbstractKrasnoselskii's fixed-point theorem in a cone is used to discuss the existence of positive solutions to semipositone conjugate and (n, p) problems

Positive solutions for nonlinear semipositone nth-order boundary value problems

In this paper, we investigate the existence of positive solutions for a class of nonlinear semipositone $n$th-order boundary value problems. Our approach relies on the Krasnosel'skii fixed point theorem. The result of this paper complement and extend previously known result

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

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