Resolvent Positive Operators and Positive Fractional Resolvent Families. (6th October 2021) - Li, Nan-Ding; Liu, Ru; Li, Miao
Abstract : 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.