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The method of upper and lower solutions for Carathéodory $$n$$-th order differential inclusions. (English) Zbl 1046.34023
The authors consider the following $$n$$th-order differential inclusion $x^{(n)}(t)\in F(t,x(t))\text{ a.e. on } J=[0,a],\quad x^{(i)}(0)=x_i\in \mathbb{R}, \tag{1}$ where $$F$$ is a multifunction defined on $$J\times\mathbb R$$ with values in the class of nonempty subsets of $$\mathbb R$$.
By assuming that $$F$$ (i) has compact and convex values, (ii) is $$L^1$$ Carathéodory and the existence of an upper and a lower solution for (1), the authors prove – via a fixed-point theorem due to M. Martelli [Boll. Unione Mat. Ital., IV. Ser. 11, Suppl. Fasc. 3, 70–76 (1975; Zbl 0314.47035)] – that the above inclusion has at least one solution. Moreover, by assuming that $$F$$ satisfies a monotonicity condition, they also establish the existence of extremal solutions.
##### MSC:
 34A60 Ordinary differential inclusions
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