Picone Type Comparison Theorems for Two-Iterval Sturm-Liouville Equations with Transmission Conditions

In this study we consider a Sturm-Liouville equation \(\begin{aligned} (k(x)y')'+ \sigma (x)y=0, \end{aligned}\) $$\begin{array}{c}\hfill {(k(x)y^{\prime })}^{\prime }+\sigma (x)y=0,\end{array}$$ defined on two disjoint intervals [a, c) and (c, b], the left and right parts of the solutions of which are connected with interaction conditions of the form \( y(c+0)=\alpha y(c-0), y'(c+0)=\beta y'(c-0),\) $y(c+0)=\alpha y(c-0),y^{\prime }(c+0)=\beta y^{\prime }(c-0),$ the so-called transmission conditions. We have obtained new Picone type identities to establish some comparison and oscillation theorems. In the special case \(\alpha =\beta =1,\) $\alpha =\beta =1,$ our results are equivalent to the corresponding classical results, so the results obtained extend and generalize the corresponding results from the Sturm’s classical comparison and oscillation theory.