We use the duality between functional and differential equations to solve several classes of abstract Cauchy problems related to special functions. As a general framework, we investigate operator functions which are multiplicative with respect to convolution of a hypergroup. This setting contains all representations of (hyper)groups, and properties of continuity are shown; examples are provided by translation operator functions on homogeneous Banach spaces and weakly stationary processes indexed by hypergroups. Then we show that the concept of a multiplicative operator function can be used to solve a variety of abstract Cauchy problems, containing discrete, compact, and noncompact problems, including -groups and cosine operator functions, and more generally, Sturm–Liouville operator functions.
"Multiplicative operator functions and abstract Cauchy problems." Banach J. Math. Anal. 12 (2) 347 - 373, April 2018. https://doi.org/10.1215/17358787-2017-0042