Behavior of directions of solutions of differential equations
In this interesting and difficult paper the author investigates the following problem described below. Assume that \(K\) is a normal cone with nonempty interior in a Banach space \(X\). Let \(f:R_ +\times K\to X\) be a continuous function, locally Lipschitzian and positively homogeneous of degree 1. Denote by \(x(t,\xi)\) a solution of the Cauchy problem \(x'=f(t,x)\), \(x(0)=\xi\). Then the vector \(x(t,\xi)/\| x(t,\xi)\|\) creates the direction of the solution \(x(t,\xi)\) at the time \(t\). The main result of the paper establishes some conditions guaranteeing that the directions \(x(t,\xi_ 1)/\| x(t,\xi_ 1)\|\) and \(x(t,\xi_ 2)/\| x(t,\xi_ 2)\|\) behave in a similar way for any two points \(\xi_ 1\), \(\xi_ 2\in\text{int }K\). Some related problems are also discussed.
scientific article; zbMATH DE number 90490