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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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