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

P. K. Palamides, Naval Academy of Greece, Piraeus, Greece
A. P. Palamides, University of Peloponesse, Tripolis, Greece.

E. J. Qualitative Theory of Diff. Equ., No. 14. (2008), pp. 1-15.

Abstract: Consider the three-point boundary value problem for the 3$^{rd}$ order differential equation:
$x^{^{\prime \prime \prime }}(t)=\alpha \left( t\right) f(t,x(t),x^{\prime}\left( t\right) ,x^{\prime \prime }\left( t\right) ),\;\;\;0<t<1,$ \\
$x\left( 0\right) =x^{\prime }\left( \eta \right) =x^{\prime \prime }\left(1\right) =0,$
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 \emph{a very simple applications on a new cone of the plane} of the well-known Krasnosel'ski\u{\i}'s fixed point theorem. The main feature of this aproach 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.