Lower and upper functions and generalized solutions of a boundary value problem for a differential-operator equation (Q2352654)

The authors prove the existence of solutions of the second order functional differential equation \[ x''(t)=F(t,x,x(t),x'(t)),\quad t \in I, \] coupled with the nonlinear functional boundary conditions \(H_i\,x=h_i \in \mathbb{R}\), \(i=1,2\). Here, \(F: I \times C(I) \times \mathbb{R}^2 \to L(I)\) and \(H_i \in C(C(I),\mathbb{R})\) satisfy suitable regularity assumptions. The authors propose a more general definition of lower and upper solutions under a weaker version of the Schrader growth condition on the function \(F\).