Representation and stability of distributed order resolvent families

Chen-Yu Li; Chen-Yu Li

doi:10.3934/math.2022650

AIMS Mathematics

2022, Volume 7, Issue 7: 11663-11686. doi: 10.3934/math.2022650

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Research article

Representation and stability of distributed order resolvent families

Chen-Yu Li ^,

College of computer science, Chengdu University, Chengdu, China

Received: 22 February 2022 Revised: 08 April 2022 Accepted: 11 April 2022 Published: 15 April 2022
MSC : 26A33, 45K05, 47A10, 47A60, 45D05

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.
- integral representation,
- stability,
- distributed order calculus,
- functional calculus,
- resolvent families
Citation: Chen-Yu Li. Representation and stability of distributed order resolvent families[J]. AIMS Mathematics, 2022, 7(7): 11663-11686. doi: 10.3934/math.2022650

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Abstract

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.

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