Resolvent Positive Operators and Positive Fractional Resolvent Families

This paper is concerned with positive α -times resolvent families on an ordered Banach space E (with normal and generating cone), where 0 < α ≤ 2 . We show that a closed and densely defined operator A on E generates a positive exponentially bounded α -times resolvent family for some 0 < α < 1 if and only if, for some ω ∈ ℝ , when λ > ω , λ ∈ ρ A , R λ , A ≥ 0 and sup λ R λ , A : λ ≥ ω < ∞ . Moreover, we obtain that when 0 < α < 1 , a positive exponentially bounded α -times resolvent family is always analytic. While A generates a positive α -times resolvent family for some 1 < α ≤ 2 if and only if the operator λ α − 1 λ α − A − 1 is completely monotonic. By using such characterizations of positivity, we investigate the positivity-preserving of positive fractional resolvent family under positive perturbations. Some examples of positive solutions to fractional differential equations are presented to illustrate our results.

Abstract degenerate fractional differential inclusions in Banach spaces

In the paper under review, we investigate a class of abstract degenerate fractional differential inclusions with Caputo derivatives. We consider subordinated fractional resolvent families generated by multivalued linear operators, which do have removable singularities at the origin. Semi-linear degenerate fractional Cauchy problems are also considered in this context.