Resolvent Positive Operators and Positive Fractional Resolvent Families

Abstract

This paper is concerned with positive $\alpha$ -times resolvent families on an ordered Banach space $E$ (with normal and generating cone), where $0<\alpha \le 2$ . We show that a closed and densely defined operator $A$ on $E$ generates a positive exponentially bounded $\alpha$ -times resolvent family for some $0<\alpha <1$ if and only if, for some $\omega \in ℝ$ , when $\lambda >\omega ,\lambda \in \rho \left(A\right)$ , $R\left(\lambda ,A\right)\ge 0$ and $\sup \left\{‖\lambda R\left(\lambda ,A\right)‖\text{:}\lambda \ge \omega \right\}<\infty$ . Moreover, we obtain that when $0<\alpha <1$ , a positive exponentially bounded $\alpha$ -times resolvent family is always analytic. While $A$ generates a positive $\alpha$ -times resolvent family for some $1<\alpha \le 2$ if and only if the operator ${\lambda }^{\alpha -1}{\left({\lambda }^{\alpha }-A\right)}^{-1}$ 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.