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Convergence of approximating fixed point sets for multivalued mappings. (English) Zbl 0929.47027

Let \(X\) be a normed space, \(C\subset X\), and \(T:C\to X\) a multivalued map. Moreover, let \(T_{\lambda, y}:C\to X\) be defined for \(0<\lambda< 1\) and \(y\in C\) by \(T_{\lambda, y}(x)= \lambda T(x)+(1- \lambda)y\). The authors study conditions under which the set \[ F_\lambda= \bigcup_{y\in C}\{x\in C:x\in T_{\lambda, y}(x)\} \] converges (in a certain sense) to the fixed point set of \(T\), as \(\lambda\to 1\).

MSC:

47H04 Set-valued operators
47H10 Fixed-point theorems
54C60 Set-valued maps in general topology
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