High order nonlinear differential inequalities with distributed deviating arguments and applications (Q1294323)

The authors establish results on the asymptotic behaviour of solutions to the following nonlinear differential inequalities with distributed deviating arguments: \[ {d^m\over dt^m}(y(t)+ \lambda(t)y(t-\tau))+ \int^\beta_\alpha Q(t,\xi) F(y(h(t, \xi))) d\sigma(\xi)\lesseqqgtr 0.\tag{\(*\)} \] The results are used to obtain oscillation criteria for solutions to equation \((*)\) and for the following partial differential equation: \[ {\partial^m\over\partial t^m} (u+\lambda(t) u(x,t-\tau))+ \int^\beta_\alpha q(x,t,\xi) F(u(x,h(t, \xi))) d\sigma(\xi)= \] \[ =a(t) \Delta u+ \sum^k_{i=1} a_i(t) \Delta u(x,\tau_i(t)). \]