Two-Dimensional Differential Systems with Asymmetric Principal Part

Abstract

We consider the Sturm–Liouville nonlinear boundary value problem

$$\displaystyle\begin{array}{rcl} \left \{\begin{array}{l} x^{\prime} = f(t,y) + u(t,x,y),\\ y^{\prime} = -g(t, x) + v(t, x, y), \end{array} \right.& & {}\\ \begin{array}{l} x(0)\cos \alpha - y(0)\sin \alpha = 0,\\ x(1)\cos \beta - y(1)\sin \beta = 0, \end{array} & & {}\\ \end{array}$$

assuming that the limits $\lim _{y\rightarrow \pm \infty }\frac{f(t,y)} {y} = f_{\pm }$, $\lim _{x\rightarrow \pm \infty }\frac{g(t,x)} {x} = g_{\pm }$ exist. Nonlinearities u and v are bounded. The system includes various cases of asymmetric equations (such as the Fučík one). Two classes of multiplicity results are discussed. The first one is that of A. Perov–M. Krasnosel’skii; the second one has originated from the works by L. Jackson–K. Schrader and H. Knobloch.