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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.

35K58 Semilinear parabolic equations
47D06 One-parameter semigroups and linear evolution equations
Full Text: DOI
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