Impulsive retarded differential equations in Banach spaces via Bochner-Lebesgue and Henstock integrals
By means of the integral of Henstock, the authors provide conditions for the existence and uniqueness of solutions as well as continuous dependence on the initial conditions of the system \[ x'(t)=f(t,x_{t}), \quad t\not=t_{k}, \quad k=1,\ldots,m,\quad \Delta x(t_{k})=I_{k}(x(t_{k})), \quad k=1,\dots,m,\;x_{0}=\phi, \] where \(\phi\) is a given continuous function defined on \([-r,t_{0}], \;r\geq 0, \;f\) is a continuous map from an open set \(\Omega\subset \mathbb{R}\times C([-r,t_{0}],X),\) \(t_{k}, \;k=1,\dots,m\), are pre-assigned moments of impulses from the interval \([t_{0},t_{0}+a], \;a\geq 0, \;x\to I_{k}(x)\) maps the Banach space \(X\) into itself and \(\Delta x(t_{k}):=x(t_{k}^{+})-x(t_{k}^{-}), \;k=1,\dots, m.\)
scientific article; zbMATH DE number 1788019
scientific article