Bounded solutions of Carathéodory differential inclusions: a bound sets approach
Here, the authors prove the existence of bounded solutions to the following differential inclusion: \(x'\in F(t,x)\), \(t\in\mathbb{R},\) where \(F:\mathbb{R}\times \mathbb{R}^N\rightarrow\mathbb{R}^N\) is a Carathéodory multivalued map, with nonempty, compact and closed values. In order to do so, they first solve an appropriate Floquet boundary value problem using a bound sets technique, which relies on the construction of bounding functions which are either continuous or locally Lipschitz and also a modification of a continuation principle due to the first author, \textit{G. Gabor} and \textit{L. Górniewicz} [Trans. Am. Math. Soc. 351, 4861--4903 (1999; Zbl 0936.34023)]. Then, they apply a sequential approach to get a solution to the original problem. The paper is very clear, discusses in detail previous results in the area, contains two examples and concludes with suggestions for further research: generalization into \(L^2\) spaces and problems related to retarded functional-differential inclusions.
scientific article; zbMATH DE number 1929998