Theorem on the existence of an invariant section over \(\mathbb R^m\) for the indefinite monotone system in \(\mathbb R^m\times\mathbb R^n\) (Q384036)

The paper deals with the system \[ \dot \varphi = a(\varphi, x), \dot x = b(\varphi, x), \tag{1} \] whose phase space is the direct product \(\mathbb{R}^m \times \mathbb{R}^n\), \(a(.,.): \mathbb{R}^m\times \mathbb{R}^n \to \mathbb{R}^m\) and \(b(.,.): \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n\) are locally Lipschitz mappings. A bounded invariant section of system (1) is called the graph of a continuous bounded mapping \(u(.): \mathbb{R}^m \to \mathbb{R}^n\) invariant under the (local) flow of system (1). By some properties of indefinite coercivity and indefinite monotonicity, the authors establish sufficient conditions for the existence of bounded Lipschitz invariant sections over \(\mathbb{R}^m.\) The proposed approach is based on the combination of certain modification of the topological Ważewski principle with the Schauder-Tikhonov fixed point theorem.