## 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

### Keywords:

fixed point; homotopy method; global convergence; nonconvex set
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### References:

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