Nonlinear second order differential inclusions with a series of convex functions (Q965016)

The existence of solutions to the following type of systems are established: \[ (\|x'(t)\|^{p-2}x'(t))' \in \partial g(t,x(t)) + F(t,x(t),x'(t))\quad\text{a.e. }t \in [0,b], \] \[ \|x'(0)\|^{p-2}x'(0) \in \partial j_1(x(0)),\quad -\|x'(b)\|^{p-2}x'(b) \in \partial j_2(x(b)); \] where \(j_1, j_2, g(t,\cdot)\) are l.s.c., convex proper functions, and the multivalued map \(F\) has compact convex values, and is measureable in \(t\) and u.s.c. with respect to the last two variables. The proofs rely on the theory of maximal monotone operators, and on a fixed point result due to \textit{R. Bader} [Z. Anal. Anwend. 20, No. 1, 3--15 (2001; Zbl 0985.34053)].