In the paper, stochastic differential equations with random impulses is first brought forward, where the so-called random impulses mean that impulse ranges are driven by some stochastic sequence and impulse times are a sequence of random variables, so these equations extend stochastic differential equations with jumps. Then existence and uniqueness of solutions to such equations are discussed by employing the Gronwall inequality, Lipschtiz condition, and some techniques in stochastic analysis.
Let ${\mathcal T}_{b}$ be the commutator generated by generalized Hardy operator and CMO function. The author considers the boundedness of ${\mathcal T}_{b}$ on the weighted homogeneous Morrey--Herz space. Furthermore, the author obtains the CMO estimates for the commutators and multilinear singular integrals with rough kernels on the homogeneous Morrey--Herz spaces.
Beylkin-Coifman-Rokhlin (B-C-R) algorithm says that an operator can be analyzed by 2n-dimensional wavelets, but here we provide a new method based on n-dimensional wavelets to consider convolution-type Calder\'{o}n-Zygmund (C-Z) operators. We apply this idea to the approximation; our approximation algorithm is much simpler than B-C-R algorithm and our approximation speed is much faster. By the way, we prove that H\"{o}rmander condition can ensure the continuity on Besov spaces $\dot{B}_p^{0,q}\ (1\leq p,\,q \leq\infty)$ and on Triebel-Lizorkin spaces $\dot{F}_p^{0,q}(1
We discuss a kinetic model of the Boltzmann equation: a generalized version of the Tjon--Wu equation. We show the asymptotic stability of the stationary solution of the equation in the $L_{1,1}$ norm.
Firstly, the concept of closure operator on a set is introduced; some properties of closure operator were obtained. Secondly, the quantale completion of ordered semigroup is given; it was proved that the quantale completions of ordered semigroup up to isomorphism were completely determined by topological closure operators on ordered semigroup. Finally, an application of the quantale completion of ordered semigroup is given.
