Derivative (finance)

Combinatorics

Mathematics

Financial economics

Fractional calculus

Mathematical analysis

Economics

Initial value problem

Interval (graph theory)

Cauchy distribution

Applied mathematics

Mittag-Leffler function

Fractional Calculus and Applied Analysis

In this paper, we consider a Cauchy problem for fractional evolution equations with Hilfer’s fractional derivative on semi-infinite interval. An elementary fact shows that semi-infinite interval is not compact, the classical Ascoli-Arzelà theorem is not valid. In order to establish the global existence criteria, we first generalize Ascoli-Arzelà theorem into the semi-infinite interval. Next, we introduce a new concept of mild solutions based on cosine/sine family and probability density function and obtain several existence results of mild solutions on semi-infinite interval.

A Cauchy problem for fractional evolution equations with Hilfer’s fractional derivative on semi-infinite interval