Mild solution for a class of nonlinear impulsive evolution inclusions on Banach space. (English) Zbl 1121.34063

The authors consider the following class of nonlinear impulsive differential inclusions
\[ \dot{x}(t) -Ax(t)\in G(t,x(t)),\quad t \in [0,T]\backslash D, x(0)=x_0, \]
\[ \triangle x(t_i)=J_i(x(t_i)), \quad i=1,...,n, \]
where \(D=\{t_1,\dots,t_n\}\subset (0,T)\), \(A\) is the infinitesimal generator of a \(C_0\)-semigroup in a Banach space \(X\), \(G\) is a multifunction, \(J_i\) (\(i=1,\dots,n\)) are bounded maps from \(X\) to \(X\) and \(\triangle x(t_i)=x(t_i+0)-x(t_i-0)=x(t_i+0)-x(t_i)\). The authors introduce the definition of mild solution for such a problem and they give a sufficient condition for its existence.


34G25 Evolution inclusions
34K45 Functional-differential equations with impulses