Existence of global solutions for second order impulsive abstract partial differential equations
The authors consider second order impulsive differential equations of the form: \[ x'(t)=Ax(t)+f(t, x(t),x(\rho(t))),\;t\in J, \] \[ \begin{aligned} \Delta x(t_{k})&=I_{k}(x(t_{k}^{-})), \quad k\in \mathbb{F}\subset\mathbb N,\\ \Delta x'(t_{k})&=\bar I_{k}(x(t_{k}^{-})),\quad k\in \mathbb{F}\subset\mathbb N,\end{aligned} \] \[ x(0)=x_0\in E, \] where \(J:=[0,b]\) or\ \(J=[0,\infty)\), \(A\) is the infinitesimal generator of a family of cosine operators. \(f: J\times E\times E\to E\) is a Carathédory function, \(\rho: J\to J\) is a continuous function. The functions \(I_k, \bar I_k\) characterize the jump of the solutions at impulse points \(t_k\) (\(k\in\mathbb{F}\)). Existence results are proved, when the right-hand side is a Lipschitz or Carathédory function. The proofs are based on a nonlinear alternative of Leray-Schauder type. One example is presented.
scientific article; zbMATH DE number 5959724
scientific article