A new characteristic property of Mittag-Leffler functions and fractional cosine functions

Zhan-Dong Mei¹ ; Ji-Gen Peng¹ ; Jun-Xiong Jia¹

¹School of Mathematics and Statistics Xi'an Jiaotong University Xi'an 710049, China

Studia Mathematica, Tome 220 (2014) no. 2, pp. 119-140

Résumé

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.

Détail

DOI : 10.4064/sm220-2-2

Keywords: characteristic property mittag leffler function alpha alpha alpha deduced motivated property notion named alpha order cosine function developed proved alpha order cosine function associated solution operator alpha order abstract cauchy problem consequently alpha order abstract cauchy problem well posed only its coefficient operator generates unique alpha order cosine function