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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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