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Generalizations of fixed point theorems and computation. (English) Zbl 1167.47044

Let \(C \subset \mathbb R^n\) be a convex and compact set with non empty interior. Let \(\Lambda\) be a nonempty and connected subset of \(C\) satisfying that, for each point \(x \in A\), \(\text{int}_{R^n} (N(x, \delta, \Lambda)) \neq \phi\) for any \(\delta > 0\), where \( (N(x, \delta, \Lambda)) = \{y \in A : \|y-x\| < \delta\}\). Let \(D\) denote the closure of \(C \backslash A\) and \(\overline{A}\) the closure of \(A\). Obviously, \(C = D \cup A\). Also, \(A\) satisfies certain properties. The authors then present two fixed point results. First, they show that if \(f\) is a continuous mapping from \(D\) to itself, then \(f\) has a fixed point. In the second result, they show that if \(F\) is a point to set mapping from \(D\) to the set of nonempty convex subsets of \(D\) and if \(F\) is upper semicontinuous on \(D\), then there exists a point \(x^{*} \in D\) such that \(x^{*} \in F(x^*)\). In the remaining part of the paper, they develop a globally convergent homotopy method for computing fixed points on this class of nonconvex sets.

MSC:

47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
47J25 Iterative procedures involving nonlinear operators
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