Completely monotone functions and some classes of fractional evolution equations

Abstract

The abstract Cauchy problem for the distributed order fractional evolution equation in the Caputo and in the Riemann–Liouville sense is studied for operators generating a strongly continuous one-parameter semigroup on a Banach space. Continuous as well as discrete distribution of fractional time-derivatives of order less than one is considered. The problem is reformulated as an abstract Volterra integral equation. It is proven that its kernel satisfies certain complete monotonicity properties. Based on these properties, the well-posedness of the problem is established and a series expansion of the solution is obtained. In case of ordered Banach space this representation implies positivity of the solution operator. In addition, a subordination formula is obtained.

Keywords:

• completely monotone function
• Bernstein function
• fractional evolution equation
• distributed order fractional derivative
• Mittag-Leffler function

AMS Subject Classification:

• 26A33
• 33E12
• 35R11
• 47D06

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was partially supported by the Bulgarian National Science Fund [grant no. DFNI-I02/9/12.12.2014] and the Bilateral Research Project SASA-BAS (2014–2016): ‘Mathematical modelling via integral-transform methods, partial differential equations, special and generalized functions, numerical analysis’.