Periodic problems with asymmetric nonlinearities and nonsmooth potentials (Q1883460)

The authors show the existence of a periodic solution to the nonlinear boundary value problem \[ -\big ( | x'(t)| ^{p-2} x'(t) \big )' \in \partial j (t, x(t)),\text{for a.a. }t \in [0, b], \] \[ x(0) = x(b), x'(0) = x'(b), \] \(p > 1\), where \(j(t, \cdot)\) is a nonsmooth locally Lipschitz function and \(\partial j(t, \xi)\) denotes the generalized Clarke subdifferential of \(\varphi\) given by \[ \partial \varphi (x) \equiv \left \{ x^* \in X^*: \langle x^*, h \rangle_X \leq \varphi^0(x; h)\, \forall h \in X \right \} \, \forall x \in X, \] where \(\varphi^0(x; h)\) is the generalized directional derivative of \(\varphi\), \[ \varphi^0(x;h) \equiv \limsup_{x' \to x, t \searrow 0} \frac{\varphi(x'+th) - \varphi(x)}{t}. \]