Monotone method for differential inequalities for the boundary value problems with deviating argument and discontinuous nonlinearity (Q1911822)

The author studies the nonlinear boundary value problem of second-order with deviating argument \(- x''(t)+ qx'(t)+ rx(t)= f(t, x(t), x(\varphi(t))\) for a.e. \(t\in I\), with general condition of Dirichlet/Neumann type \(B_1 x(t)= \alpha_0 x(t)- \alpha_1 x'(t)= \alpha(t)\), \(t\in I_1\), \(B_2 x(t)= \beta_0 x(t)+ \beta_1 x'(t)= \beta(t)\), \(t\in I_2\), where \(f\) is assumed to be continuous at the end points of the interval \(I\) and bounded on its domain.