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Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. (English) Zbl 1173.34048
Summary: We study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive differential equation with nonlocal conditions
\[ \begin{cases} u'(t)=Au(t)+f(t,u(t)),\quad 0\leq t\leq K,\;t\neq t_i,\\ u(0)+g(u)=u_0,\\ \Delta u(t_i)=I_i(u(t_i)),\quad i=1,2,\dots,p,\;0<t_1<t_2<\cdots < t_p<K.\end{cases} \]
by combining and extending some earlier work on equations with nonlocal conditions and equations with impulsive conditions. Here, \(A\) is the generator of a strongly continuous semigroup in a Banach space, \(g\) constitutes a nonlocal condition, and \(\Delta u(t^+_i)-u(t^-_i)\) constitutes an impulsive condition. New results are obtained.

34K30 Functional-differential equations in abstract spaces
34K45 Functional-differential equations with impulses
Full Text: DOI
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