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Existence results for semilinear differential equations with nonlocal and impulsive conditions. (English) Zbl 1193.35099
The authors consider the following impulsive differential equation with nonlocal conditions: \[ \begin{aligned} u'(t)= Au(t)+ f(t,u(t)),\qquad & 0\leq t\leq b,\;t\neq t_i,\\ u(0)= g(u),\\ \Delta u(t_i)= I_i(u(t_i)),\qquad & i= 1,2,\dots,p,\;0< t_i<\cdots < t_p< b,\end{aligned} \] where \(A\) is the infinitesimal generator of a \(C_0\)-semigroup on a real Banach space \(X\) and \(f\), \(g\), \(I_i\) are appropriate continuous functions.
Existence results are obtained for mild solutions without the compactness or Lipschitz continuity assumptions on impulsive functions. Two examples are given to illustrate the results.

MSC:
35K58 Semilinear parabolic equations
47D06 One-parameter semigroups and linear evolution equations
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