Comparison principles for impulsive hyperbolic equations of first order (Q1903647)

The authors prove strong and weak comparison principles for the solutions of the initial value problem for impulsive hyperbolic equations of first order: \[ z_t(t, x)+ f(t, x)\cdot z_x(t, x)= F(t, x, z(t, x)),\quad t\neq t_k,\;x\in [\alpha(t), \beta(t)], \] \[ z(0, x)= \varphi(x),\quad x\in [\alpha(0), \beta(0)], \] \[ z(t_k, x)- z(t^-_k, x)= g_k(t_k, x, z(t^-_k, x)),\quad k= 1,\dots, p,\;x\in [\alpha(t_k), \beta(t_k)]. \] It should be mentioned that the theory of impulsive PDE is of growing interest in the recent years. The first monograph in this new area is the authors' [Approximate solutions of impulsive hyperbolic equations, Academic Publishers (Calcutta 1996)].