Upper and lower solutions method for differential inclusions with integral boundary conditions (Q871319)

For a second-order differential inclusion, the authors consider a boundary value problem with nonlinear integral boundary conditions \[ y''(t)\in F(t, y(t)),\quad t\in [0,1], \] \[ y(0)- k_1 y'(0)= \int^1_0 h_1(y(s))\,ds, \] \[ y(1)+ k_2y'(1)= \int^1_0 h_2(y(s))\,ds. \] Here, \(F\) is a multivalued map with convex values that satisfies the Carathéodory conditions. The functions \(h_1\), \(h_2\) are continuous and nondecreasing, \(k_1\), \(k_2\) are nonnegative numbers. In terms of upper and lower functions, sufficient conditions are formulated for the existence of a solution. The proof is based on a nonlinear alternative of Leray-Schauder type.