Half-linear Dynamic Equations: A Survey
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Nonlinear Analysis and Applications
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- dynamic equations; time scales; half-linear equations; Sturmian theory; oscillation; Reid roundabout theorem; Picone Identity; dynamic equations; time scales; half-linear equations; Sturmian theory; oscillation; Reid roundabout theorem; Picone Identity; Mathematics; Statistics and Probability
We survey half-linear dynamic equations on time scales. These contain the well-known half-linear di erential and half-linear di erence equations as special cases, but also other kinds of half-linear equations. Special cases of half-linear equations are the well-studied linear equations of second order. We discuss existence and uniqueness of solutions of corresponding initial value problems and, using a Picone identity, derive a Reid roundabout theorem that gives conditions equivalent to disconjugacy of half-linear dynamic equations, among them solvability of an associated Riccati equation and positive de niteness of an associated functional. We also develop a corresponding Sturmian theory and discuss methods of oscillation theory, which we use to present oscillation as well as nonoscillation criteria for half-linear dynamic equations.