Representation and stability of distributed order resolvent families

We consider the resolvent family of the following abstract Cauchy problem (1.1) with distributed order Caputo derivative, where $A$ is a closed operator with dense domain and satisfies some further conditions. We first prove some stability results of distributed order resolvent family through the subordination principle. Next, we investigate the analyticity and decay estimate of the solution to (1.1) with operator $A=\lambda >0$, then we show that the resolvent family of Eq (1.1) can be written as a contour integral. If $A$ is self-adjoint, then the resolvent family can also be represented by resolution of identity of $A$. And we give some examples as an application of our result.