Cosine and sine families on time scales and abstract second order dynamic equations on time scales

Abstract

Abstract cosine and sine functions defined on a Banach space are useful tools in the study of wide classes of abstract evolution equations. In this paper, we introduce a definition of cosine and sine functions on time scales, which unify the continuous, discrete and the cases “in between.” Our definition includes several types of time scales such as real numbers set, integers numbers set, quantum scales, among others. We investigate the relationship between the cosine function on time scales and its infinitesimal generator, proving several properties concerning it. Also, we investigate the sine functions on time scales, presenting their main properties. Finally, we apply our theory to study the homogeneous and inhomogeneous abstract Cauchy problem on time scales in Banach spaces.