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Keywords = Krasnoselskii–Schaefer type fixed point theorem

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17 pages, 362 KB  
Article
Approximate Controllability of Non-Instantaneous Impulsive Stochastic Evolution Systems Driven by Fractional Brownian Motion with Hurst Parameter H∈(0,12)
by Jiankang Liu, Wei Wei and Wei Xu
Fractal Fract. 2022, 6(8), 440; https://doi.org/10.3390/fractalfract6080440 - 13 Aug 2022
Cited by 9 | Viewed by 2236
Abstract
This paper initiates a study on the existence and approximate controllability for a type of non-instantaneous impulsive stochastic evolution equation (ISEE) excited by fractional Brownian motion (fBm) with Hurst index 0<H<1/2. First, to overcome the irregular [...] Read more.
This paper initiates a study on the existence and approximate controllability for a type of non-instantaneous impulsive stochastic evolution equation (ISEE) excited by fractional Brownian motion (fBm) with Hurst index 0<H<1/2. First, to overcome the irregular or singular properties of fBm with Hurst parameter 0<H<1/2, we define a new type of control function. Then, by virtue of the stochastic analysis theory, inequality technique, the semigroup approach, Krasnoselskii’s fixed-point theorem and Schaefer’s fixed-point theorem, we derive two new sets of sufficient conditions for the existence and approximate controllability of the concerned system. In the end, a concrete example is worked out to demonstrate the applicability of our obtained results. Full article
37 pages, 473 KB  
Article
Pseudo Almost Automorphic Solutions for Stochastic Differential Equations Driven by Lévy Noise and Its Optimal Control
by Chao Tang and Rong Hou
Symmetry 2021, 13(9), 1674; https://doi.org/10.3390/sym13091674 - 11 Sep 2021
Cited by 1 | Viewed by 2252
Abstract
As we know, the periodic functions are symmetric within a cycle time, and it is meaningful to generalize the periodicity into more general cases, such as almost periodicity or almost automorphy. In this work, we introduce the concept of Poisson Sγ2 [...] Read more.
As we know, the periodic functions are symmetric within a cycle time, and it is meaningful to generalize the periodicity into more general cases, such as almost periodicity or almost automorphy. In this work, we introduce the concept of Poisson Sγ2-pseudo almost automorphy (or Poisson generalized Stepanov-like pseudo almost automorphy) for stochastic processes, which are almost-symmetric within a suitable period, and establish some useful properties of such stochastic processes, including the composition theorems. In addition, we apply a Krasnoselskii–Schaefer type fixed point theorem to obtain the existence of pseudo almost automorphic solutions in distribution for some semilinear stochastic differential equations driven by Lévy noise under Sγ2-pseudo almost automorphic coefficients. In addition, then we establish optimal control results on the bounded interval. Finally, an example is provided to illustrate the theoretical results obtained in this paper. Full article
(This article belongs to the Section B: Mathematics)
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