A new characteristic property of Mittag-Leffler functions and fractional cosine functions
Tom 220 / 2014
Studia Mathematica 220 (2014), 119-140
MSC: Primary 45N05; Secondary 47D09.
DOI: 10.4064/sm220-2-2
Streszczenie
A new characteristic property of the Mittag-Leffler function $E_\alpha (at^\alpha )$ with $1<\alpha <2$ is deduced. Motivated by this property, a new notion, named $\alpha $-order cosine function, is developed. It is proved that an $\alpha $-order cosine function is associated with a solution operator of an $\alpha $-order abstract Cauchy problem. Consequently, an $\alpha $-order abstract Cauchy problem is well-posed if and only if its coefficient operator generates a unique $\alpha $-order cosine function.
