Impulsive problems for semilinear differential equations with nonlocal conditions
This paper deals with the impulsive differential equation with nonlocal conditions \[ \begin{aligned} u'(t) &= Au(t) + f(t, u(t)), 0\leq t\leq b, t\neq t_i,\\ u(0) &= u_0-g(u),\\ \Delta u(t_i) &= I_i(u(t_i)), i = 1, 2, \dots, p, 0 < t_1 < t_2 < \dots < t_p < b, \end{aligned} \] where \(A : D(A) \subseteq X \to X\) is the infinitesimal generator of a strongly continuous semigroup \(T (t), t \geq 0,\) \(X\) is a real Banach space, \(\Delta u(t_i) =u(t_i^+)-u(t_i^-),\) \( u(t_i^+), u(t_i^-)\) denotes the right and the left limit of \(u\) at \(t_i,\) respectively and \(f, g, I_i \) are appropriate continuous functions. By using the Hausdorff measure of noncompactness and fixed point techniques, the author proves the existence of a mild solution without the Lipschitz continuity of the mapping \(f,\) in the cases when (i) \(g\) and \(I_i\) are Lipschitz and the semigroup \(T (t), t > 0\), generated by the linear operator \(A\) is compact, and (ii) \(g\) is not Lipschitz and not compact.
scientific article; zbMATH DE number 5655116
scientific article