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From the Schauder fixed-point theorem to the applied multivalued Nielsen theory. (English) Zbl 0958.34015

The starting point of the paper is the fixed-point theorem of Schauder and its applications to boundary value problems. There are described Lefschetz- and Nielsen-type theorems as well as their applications to the boundary value problems \[ x'(t)\in F(t,x(t)),\quad x\in S\text{ or }x'(t)+A(t)x(t)\in F(t,x(t)),\quad Lx=0. \] Here, \(F:J\times \mathbb{R}^n\to \mathbb{R}^n\) is a Carathéodory multifunction and \(S\subseteq AC_{loc}(J,\mathbb{R}^n)\). In addition, \(A:J\to \mathbb{R}^{n\times n}\) is a continuous matrix function and \(L:C(J,\mathbb{R}^n)\to \mathbb{R}^n\) is a linear function. A nontrivial example shows the power of the results.
Reviewer: C.Ursescu (Iaşi)

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
34A60 Ordinary differential inclusions
47H04 Set-valued operators
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