Positive Solutions for Third-Order Boundary Value Problems with Indefinite Weight

In this paper, we initially employ the disconjugacy theory to establish some sufficient conditions for the disconjugacy of y′′′+βy′′+αy′=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y'''+\beta y''+\alpha y'=0$$\end{document}. We then utilize Elias’s spectrum theory to demonstrate the spectrum structure of the linear operator y′′′+βy′′+αy′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y'''+\beta y''+\alpha y'$$\end{document} coupled with the boundary conditions y(0)=y(1)=y′(1)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(0)=y(1)=y'(1)=0$$\end{document}. Ultimately, by utilizing the acquired results, we ascertain the existence of positive solutions for the corresponding nonlinear third-order problem with an indefinite weight, based on the principles of bifurcation theory and the Leray–Schauder fixed point theorem.