Nonlinear problems with asymmetric principal part (Q2884916)

The authors investigate the following boundary value problem NEWLINE\[NEWLINE \begin{aligned} & x'' = -\lambda f(x^{+}) + \mu f(x^{-}) + h(t, x, x'), \\ & x(0) = x(1) = 0, \end{aligned} NEWLINE\]NEWLINE where \((\lambda, \mu) \in\mathbb{R}^{2}\), \(x^{+} = \max\{x, 0\}\), \(x^{-} = \max\{-x, 0\}\), the function \(f: [0, +\infty)\to[0, +\infty)\) is Lipschitzian, \(h: [0,1] \times \mathbb{R}^{2} \to \mathbb{R}\) is continuous and Lipschitzian in \(x\) and \(x'\). The authors provide sufficient conditions on \((\lambda, \mu)\), \(f\) and \(h\) which guarantee the existence of a solution to the problem. They suppose that \(h\) is bounded, \(f\) is bounded by two linear functions and \((\lambda,\mu)\) is in one of ``good'' regions (with respect to the Fučík spectrum \(\Sigma\)). For \(f(x)=x\), the ``good'' regions coincide with the well known regions of type I (components of \(\mathbb{R}^2 \setminus \Sigma\) which contain at least one point \((\lambda,\lambda)\), \(\lambda \not \in \sum\)).