On oscillation of higher-order advanced trinomial differential equations

Abstract

We study the oscillatory property of the higher-order trinomial differential equation with advanced effects x(n)(t)+p(t)x′(t)+q(t)x(σ(t))=0,σ(t)≥t.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ x^{(n)}(t)+p(t)x'(t)+q(t)x \\bigl(\\sigma (t) \\bigr)=0,\\quad \\sigma (t) \\geq t. $$\\end{document} Suppose that all solutions of the corresponding (n-1)th-order two-term differential equation y(n−1)(t)+p(t)y(t)=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ y^{(n-1)}(t)+p(t)y(t)=0 $$\\end{document} are non-oscillatory. In order to supplement the research in the theory of oscillation proposed by (Džurina et al. in Electron. J. Differ. Equ. 2015:70, 2015), two types of clearly confirmable criteria for oscillatory behavior of the investigated equation are obtained. Some examples are offered to describe our main results.