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Almost-periodic and bounded solutions of Carathéodory differential inclusions. (English) Zbl 1017.34011
The author first proves a fixed-point theorem (theorem 1) for a class of $$\mathcal J$$-maps in locally convex topological vector spaces. By this theorem, he gives sufficient conditions for existence results to a large family of multivalued boundary value problems on noncompact intervals in theorem 3. This theorem represents a slight generalization of corollary 2.37 in J. Andres, G. Gabor and L. Górniewicz [Trans. Am. Math. Soc. 351, No. 12, 4861-4903 (1999; Zbl 0936.34023)] and improves in the single-valued case its analogies in M. Cecchi, M. Furi and M. Marini [Nonlinear Anal., Theory Methods Appl. 9, 171-180 (1985; Zbl 0563.34018) and Boll. Unione Mat. Ital., VI. Ser., C. Anal. Funz. Appl. 4, 329-345 (1985; Zbl 0587.34013)]. Applying theorem 3 to Carathéodory quasi-linear differential inclusions and investigating the topological structure of solution sets to the associated linearized system, a new effective criterion for the existence of an entirely bounded solution on $$\mathbb{R}$$ to the considered quasi-linear differential inclusions is given in theorem 4. Next, the almost-periodicity in the sense of H. Weyl for multifunctions is introduced. Theorem 5 presents conditions for the existence of an almost-periodic solution to quasi-linear differential inclusions with Lipschitz continuous right-hand sides. In concluding remarks, some open problems are posed.

##### MSC:
 34A60 Ordinary differential inclusions 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 47H04 Set-valued operators 54C60 Set-valued maps in general topology 34B15 Nonlinear boundary value problems for ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations