Periodic solutions of second order differential equations with discontinuous nonlinearities (Q854718)

The main result of this paper consists of two theorems on the existence of periodic solutions to \[ -\ddot x= f(t, x)\tag{1} \] with \(f\) being periodic in \(t\) and discontinuous in \(x\). Theorem 1 formulates conditions, under which (1) has two \(2\pi\)-periodic solutions \(x^i_k\) with \(2k\) zeros in \([0,2\pi)\) and \(\| x^i_k\|_\infty\to\infty\) as \(k\to\infty\), \(i= 1,2\). Theorem 2 formulates conditions under which (1) has two \(2\pi\)-peridic solutions \(x^i_k\) with \(2k\) zeros in \([0,2\pi)\) and \(\| x^i_k\|_\infty\to 0\) as \(k\to\infty\), \(i= 1,2\). The nature of the proofs is variational as in [\textit{M.-Y. Jiang}, Periodic solutions of second-order differential equations with an obstacle, Nonlinearity 19, No. 19, 1165--1183 (2006; Zbl 1121.34047)].