Monotone inequalities, dynamical systems, and paths in the positive orthant of Euclidean \(n\)-space. (English) Zbl 1211.47091
Let \(\mathbb{R}^n_+\) denote the positive orthant in the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\). A continuous mapping \(T:\mathbb{R}^n_+ \rightarrow \mathbb{R}^n_+\) such that \(x \leq y \Rightarrow Tx \leq Ty\) is called a monotone operator (here, \(x \leq y\) signifies the fact that \(y-x \in \mathbb{R}^n_+\)). Given a monotone operator on \(\mathbb{R}^n_+\), the author explores the relation between inequalities involving the operator and an induced monotone dynamical system given by \(x^{k+1}=T(x^k)\). Attractivity of the origin (for the given dynamical system) implies stability for this system, as well as the inequality \(T(x) - x \ngeq 0\) (meaning that the vector \(T(x) - x\) has at least one negative component), which is referred to as the no-joint-increase condition. A certain converse is established. A “positive continuous selection problem” is also solved.
Reviewer: K. C. Sivakumar (Chennai)
MSC:
| 47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 47N20 | Applications of operator theory to differential and integral equations |
| 37C65 | Monotone flows as dynamical systems |
| 93B27 | Geometric methods |
Keywords:
monotone operators and inequalities; positive systems; comparison functions; decay sets; omega paths; monotone selectionsReferences:
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