Impulsive hyperbolic differential inclusions with variable times (Q1428723)

The authors deal with the existence of solutions for the second-order impulsive hyperbolic differential inclusions with variable times \[ \begin{aligned} {\partial^2u\over\partial t\,\partial x}\in F(t,x,u(t,x))\quad\text{a.e. }(t,x)\in J_a\times J_b,&\quad t\neq \tau_k(u(t,x)),\\ u(t^+, x)= I_k(u(t,x)),&\quad t= \tau_k(u(t,x)),\\ u(t,0)= \psi(t),&\quad t\in J_a,\\ u(0,x)= \varphi(x),&\quad x\in J_b,\end{aligned} \] where \(F: J_a\times J_b\times \mathbb{R}^d\to P(\mathbb{R}^d)\) is a multivalued map with compact values, \(J= J_a\times J_b=\) \([0,a]\times [0,b]\), \(I_k\in C^1(\mathbb{R}^d,\mathbb{R}^d)\), \(\varphi\in C(J_a, \mathbb{R}^d)\), \(u(t^+,y):= \lim_{(h,x)\to (0^+,y)} u(t+ h,x)\), \(u(t^-,y):= \lim_{(h,x)\to (0^-,y)} u(t- h,x)\). To this end the authors use the nonlinear alternative of Leray-Schauder theory.