On the number of positive solutions for a four-point boundary value problem with generalized Laplacian
In this paper, the authors study the four-point boundary value problem of a nonlinear ordinary differential equation with generalized Laplacian \[ \begin{cases} -\phi(u')'=\lambda h(t)f(t,u), ~ t\in (0,1),\\ u(0) = \alpha u(\xi), u(1) = \beta u(\eta), \end{cases} \] where \(\phi: \mathbb{R}\to \mathbb{R}\) is an odd increasing homeomorphism, \(\lambda > 0,\) \( 0 \le \alpha, \beta < 1,\) \( 0 <\xi < \eta < 1,\) \(f: [0,1]\times [0,\infty)\to [0,\infty)\) is continuous, \(h\not\equiv 0\) on any subinterval in \((0, 1)\) and \(h\) may have a singularity at \(t = 0\) and/or \(t = 1.\) The relations between the number of positive solutions and two different kinds of asymptotic behaviors of the nonlinearity at \(0\) and \(\infty\) were obtained, by using the fixed point theorem of expansion/compression of a cone and Schauder's fixed point theorem.
scientific article; zbMATH DE number 7381166
scientific article