Xiao-Bao Shu | Searcharxiv

arXiv2021

Existence of solutions for first-order Hamiltonian stochastic impulsive differential equations with Dirichlet boundary conditions

In this paper, we study the sufficient conditions for the existence of solutions of first-order Hamiltonian stochastic impulsive differential equations under Dirichlet boundary value conditions. By using the variational method, we first obtain the corresponding energy functional. And by using Legendre transformation, we obtain the conjugation of the functional. Then the existence of critical point is obtained by mountain pass lemma. Finally, we assert that the critical point of the energy functional is the mild solution of the first order Hamiltonian stochastic impulsive differential equation.Finally, an example are presented to illustrate the feasibility and effectiveness of our results.

math.DS↗

arXiv2019

A new study on the mild solution for impulsive fractional evolution equations

In this article, we consider mild solutions to a class of impulsive fractional evolution equations of order $0<α<1$. After analyzing analytic results reported in the literature using Mittag-Leffer function, $α$-resolvent operator theory, we propose a more appropriate new definition of mild solutions for impulsive fractional evolution equations by replacing the impulse term operator $S_α(t-t_i)$ with $S_α(t)S_α^{-1}(t_i)$, where $S_α^{-1}(t_i)$ denotes the inverse of the fractional solution operator $S_α(t)$ at $t=t_i, (i=1,2,\cdots m)$.

math.CA↗