Solutions of superlinear Sturm-Liouville problems in Banach spaces (Q1919586)

Let \(E\) be a Banach space and \(I = [0,1]\). Consider the Sturm-Liouville problem (1) \(L \varphi = f(x, \varphi)\), \(x \in I\), (2) \(\alpha_0 \varphi (0) - \beta_0 \varphi' (0) = \theta\), \(\alpha_1 \varphi (1) + \beta_1 \varphi' (1) = \theta\), where \(L \varphi \equiv - (p(x) \varphi'(x))' + q(x) \varphi (x)\), \(p(x) \in C^1 (I)\), \(p(x) > 0\) \((x \in I)\), \(q(x) \in C(I)\), \(q(x) \geq 0\) \((x \in I)\), \(f \in C [I \times E,E]\), \(\alpha_i\), \(\beta_i\) \((i = 0,1)\) are nonnegative constants, and \(\theta\) is the zero element of \(E\). In this paper, the author uses the topological degree theory and the cone theory to investigate the existence of nontrivial and positive solutions of (1), (2). An example for an infinite system of equations is given.