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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 $$\cases u'(t)=Au(t)+f(t,u(t)),\quad 0\le t\le K,\ t\ne t_i,\\ u(0)+g(u)=u_0,\\ \Delta u(t_i)=I_i(u(t_i)),\quad i=1,2,\dots,p,\ 0<t_1<t_2<\cdots < t_p<K.\endcases$$ 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.

34K30Functional-differential equations in abstract spaces
34K45Functional-differential equations with impulses
Full Text: DOI
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