Well-ordered and non-well-ordered lower and upper solutions for periodic planar systems
The authors consider the following periodic problem \[ (P) \left\{ \begin{array}{ll} x'=f(t,x,y), & y'=g(t,x,y),\\ \\ x(0)=x(T), & y(0)=y(T), \end{array} \right. \] where \(f: {\mathbb R}^3\to {\mathbb R}\) and \(g: {\mathbb R}^3\to {\mathbb R}\) are continuous functions, \(T\)-periodic in their first variable. The authors introduce a general definition of a lower and an upper solution, that improves previous ones given in the literature, and prove the existence of a solution of problem \((P)\). The new definition allows to the lower and upper solutions to be not differentiable at some points of its domain, moreover some discontinuities at related functions on the definition may be also considered. They deduce the existence of solutions of problem \((P)\) on the classical well order case and in the non well ordered one by assuming, in this last case, some growth conditions on \(f\) and \(g\) in order to avoid resonance. A direct application is given for the \(\phi\)-Laplacian equation \[ (\phi(x'(t))'=h(t,x(t),x'(t)), \] with \(\phi:I \to J\) is an increasing homeomorphism between two intervals \(I\) and \(J\) containing \(0\), and \(\phi(0)=0\).
scientific article; zbMATH DE number 7374861
scientific article