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Bounded solutions of Carathéodory differential inclusions: a bound sets approach. (English) Zbl 1036.34011
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, G. Gabor and 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.

MSC:
34A60 Ordinary differential inclusions
34B15 Nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
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