Solutions of semi-linear stochastic evolution integro-differential inclusions with Poisson jumps and non-local initial conditions

Abstract

In this paper, we mainly investigate properties on solution sets of a semi-linear stochastic evolution integro-differential inclusion with Poisson jumps and non-local initial conditions. We firstly establish the existence of solutions to the addressed equation via weak topology technique. Through establishing the estimate on noncompactness measure of integrals for Poisson jumps, we further show that the solution set is a compact $R_{\delta }$-set when the resolvent operator is compact or non-compact, respectively.

Keywords:

• Stochastic evolution integro-differential inclusions
• Poisson jumps
• Compact $R_{\delta }$-set

Mathematics Subject Classification (2010):

• 34A60
• 34K30
• 60H15

Acknowledgements

Authors would like to thank the anonymous referee and the associate editor for carefully reading this manuscript and giving valuable comments to improve the previous version of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by NSF of Shaanxi Province of China [2020JM-183].