P. K. Palamides
abstract:
Consider the second order nonlinear scalar differential equations
\begin{equation}
x''\pm f(t,x)=0,\;\;\;0\leq t\leq1,
\end{equation}
where $f\in C([0,1]\times[0,\infty),[0,\infty)),$ associated
to the boundary conditions
\begin{equation}
\begin{cases}
\alpha x(0)\pm\beta x'(0)=0, \\
\gamma x(1)\mp\delta x'(1)=0,
\end{cases}
\end{equation}
with $\alpha,\beta,\gamma,\delta\geq0,$ or the more general nonlinear one
\begin{equation}
g(x(0),x'(0))=0=h(x(1),x'(1)).
\end{equation}
Existence of positive solutions of above BVPs are given, under superlinear
and/or sublinear growth in $f$. The approach is based on an analysis of the
coresponding vector field on the $(x,x')$ phase plane and Kneser's
property of solutions funnel.
Mathematics Subject Classification: Primary 34B05, 34B10; Secondary 34B15, 34B18.
Key words and phrases: Sturm-Liouville boundary value problems, positive solution, Kneser's property, vector field, sublinear, superlinear, growth rate.