Impulsive retarded differential equations in Banach spaces via Bochner-Lebesgue and Henstock integrals. (English) Zbl 1011.34070

Federson, M.; Táboas, P.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 50, No. 3, 389-407 (2002).

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 \(\Delta x(t_{k}):=x(t_{k}^{+})-x(t_{k}^{-}), \;k=1,\dots, m.\)

Reviewer: Mouffak Benchohra (Sidi Bel Abbes)

MSC:

34K45	Functional-differential equations with impulses
34K30	Functional-differential equations in abstract spaces

Keywords:

retarded equations; impulses; Henstock integrals; existence theorems

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