Effects of varying nonlinearity and their singular perturbation flavour (Q1580480)

The authors consider the second-order differential equation \[ y''+ f(y, p)= 0, \] where \(p\) is a parameter and \(f(\cdot, p)\) is odd, strictly increasing and continuous on \((-\infty, \infty)\). For \(p\to\infty\) the family of functions approaches \(f_0(y)\) which is continuous on \((-a,a)\), the convergence being uniform on \((\varepsilon- a,a-\varepsilon)\). At \(\pm a\) discontinuities are allowed. The convergence of the periodic solutions to those of the ``limit'' equation \[ y''= f_0(y) \] is considered. Further, a study is made for the nonperiodic solutions to \[ y''- f(y, p)= 0. \]