Time-Scale Calculus and Oscillation Theory of Linear Equations

Abstract

Time-scale calculus unifies continuous and discrete analysis. From this viewpoint, dynamic equations on time scales can treat differential and difference equations simultaneously. Using the exponential function on time scales and, in the double-root case, the delta integral, we obtain general solutions of second-order linear equations with constant coefficients and of Cauchy–Euler equations. We then focus on the oscillation constants, which are the critical parameter values at which the solutions of Cauchy–Euler equations change from oscillatory to nonoscillatory. In this critical case, the logarithm and iterated logarithm functions appear naturally. Finally, we briefly discuss how oscillation constants for p-Laplacian-type equations are related to the best constants in Hardy-type inequalities.