Positive solutions for equations and systems with \(p\)-Laplace-like operators (Q1028492)

The authors consider the existence of positive solutions of \[ \big( \phi(u') \big)' + f(t, u) = 0, \quad t \in (0, 1), \] \[ \theta(u(0)) = \beta \theta (u'(0)), \] \[ \theta(u(1)) = -\delta \theta (u'(1)), \] where \(\beta, \delta \geq 0\), as well as related boundary value problems. Here, \(\phi\) and \(\theta\) are odd increasing homeomorphisms and \(f:[0, 1] \times [0, \infty) \to [0, \infty)\) is continuous. They prove three theorems on existence using a maximum principle and the Leray-Schauder degree. In each theorem, they assume that the nonlinear term \(f\) is either super-linear or sub-linear. Additionally, they need to assume that limits involving ratios of the form \(\frac{\theta(\tau s)}{\theta(s)}\) are positive and tend to either \(0\) or \(\infty\).