Periodic or bounded solutions of Carathéodory systems of ordinary differential equations
This paper is devoted to the extension of the guiding function approach related to the solution of a nonautonomous ordinary differential equation (ODE) under boundary conditions: \(dx/dt=f(x,t)\) a.e. for \(t\in[a,b]\), \(x(a)=x(b)\), where \(x\) is a vector of \(\mathbb R^n\), and \(f\) satisfies the Carathéodory conditions. A solution is an absolutely continuous function satisfying this ODE almost everywhere. The general foundation of the guiding functions method, due to Krasnosel'skji in 1961, consists in obtaining a solution by finding a \(C^1\) real function \(V(x)\) such that the inner product on \(\mathbb R^n\) \(\langle dV/dx,f(x,t)\rangle\) does not change sign on \([a,b]\times G\), in a suitable region \(G\) of \(\mathbb R^n\), and such that the associated gradient system \(dx/dt=-dV/dx\) has a nonempty ``stable'' set of equilibriums in the bounded component of \(\mathbb R^n\setminus G\). This approach appears as a version of Lyapunov's function notion, used for defining stability conditions for ODE solutions with initial conditions, here adapted to a boundary value problem. In this text, the authors generalize the concept of guiding function on \(G\), and give with proof two theorems related to the existence of periodic or bounded solutions for the above ODE, which are also a generalization of recent results obtained by \textit{J. Mawhin} and \textit{J. R. Ward} jun. [Discrete Contin. Dyn. Syst. 8, 39--54 (2002; Zbl 1087.34518)].
scientific article; zbMATH DE number 2057854