On a Generalized Sturm-Liouville Problem

Basic results of our paper are devoted to a generalized Sturm-Liouville problem for an equation of the form $-(p(t)y^{\prime }(t){)}^{\prime }+q(t)y(t)=F(t,y(\cdot ))$ with conditions $$\{\begin{array}{l}{\alpha }_{1}y(a)+{\alpha }_2{y}^{{}^{\prime }}(a)=0\\ {\beta }_{1}y(b)+{\beta }_2{y}^{{}^{\prime }}(b)=0\end{array}$$ where ${\alpha }_1^2+{\alpha }_2^2,{\beta }_1^2+{\beta }_2^2>0$, $p(t)\ne 0$ for $t\in [a,b]$, $q\in C([a,b])$ and $F$ is a continuous transformation from $[a,b]\times C([a,b])$ to $C([a,b])$. It is required that the Green’s function associated with this problem be nonnegative.