| Abstract: |
| In the presentation, I`ll present the existence of nonzero nonnegative or strictly positive solutions of
the boundary value problems (BVPs) of nonlinear Sturm-Liouville (S-L) differential equation:
\begin{equation*}
\label{sec1intro}
-(p(x)u`(x))`+q(x)u(x)=f(x,u(x)) \quad\mbox{for almost every $x\in [0,1]$}
\end{equation*}
subject to the separated boundary condition (BC):
\begin{equation*}
\label{bcintro}
\alpha u(0)-\beta u`(0)=0,\quad \gamma u(1)+\delta u`(1)=0,
\end{equation*}
where $\alpha,\beta,\gamma,\delta\in \mathbb R_{+}$ satisfy $(\alpha+\beta)(\gamma+\delta)>0$ and
$f:[0,1]\times\mathbb R_{+}\to \mathbb R$ is allowed to take negative values and may have no lower bounds.
A sufficient condition for the linear S-L homogeneous equations subject to the above BC
to have only zero solution is provided for the first time.
The sufficient conditions are a key toward obtaining both the Green`s functions to such BVPs
and uniqueness of solutions for the linear S-L nonhomogeneous BVPs including the one-dimensional elliptic BVPs. |
|