Topological properties of solution sets for Sobolev-type fractional stochastic differential inclusions with Poisson jumps

ABSTRACT

In this paper, we investigate the topological properties of solution sets for Sobolev-type fractional stochastic differential inclusions with Poisson jumps in Caputo and Riemann–Liouville fractional derivatives of order $(1,2)$, respectively. We show that the solution set is nonempty, compact and an $R_{\delta }$-set under some suitable conditions, which implies that the solution set may not be a singleton, but in the point of view of algebraic topology, it is equivalent to a point in the sense that it has the same homology group as one-point space.

COMMUNICATED BY:

• Salvatore Leonardi

KEYWORDS:

• Sobolev-type fractional stochastic differential inclusions
• compact $R_{\delta }$-set
• Poisson jumps

MATHEMATICS SUBJECT CLASSIFICATION(2000):

• 34A08
• 34A60
• 93E03

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author's work was partly supported by National Natural Science Foundation of China [11671331]. The second author's work was supported Natural Science Foundation of Shaanxi Province [2017JM1017].