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

Abstract

In this study we consider a Sturm-Liouville equation (k(x)y ')'+sigma(x)y=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (k(x)y')'+ \sigma (x)y=0, \end{aligned}$$\end{document}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),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ y(c+0)=\alpha y(c-0), y'(c+0)=\beta y'(c-0),$$\end{document} 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,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\beta =1,$$\end{document} 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.