Type \(K\) monotone systems with an order-increasing invariant function (Q1814865)

The asymptotic behavior of solutions of the type \(K\) monotone system (1) \(\dot x=F(x)\), \(x\in\mathbb{R}^n_+\) is considered, where \(F_i(x_1,\dots,x_n)\) is nondecreasing in \(x_k\) for all \(k\neq i\). If \(F\) is continuous on \(\mathbb{R}^n_+\) and satisfies the conditions for existence and uniqueness of solutions of the initial value problem, \(F(0)=0\), and (1) possesses an order-increasing invariant function \(H\), then every bounded solution of (1) converges to an equilibrium.