Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives (Q1812327)

The author studies the existence of periodic solutions to \[ x''+ax^+-bx^-+g(x')=p(t) \quad \text{and} \quad x''+ax^+-bx^-+f(x)+g(x')=p(t), \] where \(x^+=\max\{x,0\}, x^-=\max\{-x,0\}; a ,b\) are positive constants lying on one of the Fu\`cik spectrum curves; \(f,g: \mathbb{R}\to \mathbb{R}\) are locally Lipschitzian continuous and \(p: \mathbb{R}\to \mathbb{R}\) is continuous and \(2\pi\)-periodic. Sufficient conditions are derived for the existence of \(2\pi\)-periodic solutions in terms of the limit values of \(f(x)\) and \(g(x)\) as \(x\) tends to \(+\infty\).