Positive solutions for a multi-parameter system of second-order ordinary differential equations
The authors investigate the following multi-parameter boundary value problem \[ \begin{aligned} u''(t) + \lambda h_{1}(t) f_{1}(u(t), v(t)) = 0, & \quad t\in(0,1),\\ v''(t) + \kappa h_{2}(t) f_{2}(u(t), v(t)) = 0, & \quad t\in(0,1),\\ u(0) = u(1) = v(0) = v(1) = 0, & \end{aligned} \] where \(\lambda,\kappa\in \mathbb{R}^{+}\), \(h_{i}\in C([0,1], \mathbb{R}^{+})\), \(f_{i}\in C(\mathbb{R}^{+}\times\mathbb{R}^{+}, \mathbb{R}^{+}_{0})\), \(i=1,2\), \(\mathbb{R}^{+} = [0,+\infty)\) and \(\mathbb{R}^{+}_{0} = (0,+\infty)\). Existence and multiplicity of positive solutions to this boundary value problem are investigated involving a new class of superlinear nonlinearities in which \(f_{1}\) is superlinear with respect to \(u\) at infinity and \(f_{2}\) is superlinear with respect to \(v\) at infinity. The authors establish the product formula for the fixed point index on a product cone. The main result is proved by using this product formula and the fundamental properties of the fixed point index.
scientific article; zbMATH DE number 5908404