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Convergences preserving the fixed point property. (English) Zbl 0985.54035
Summary: Coincidence theorems for very general (non-Hausdorff) topological spaces $$X$$ and $$Y$$ are proved, e.g., if $$\{f_{\gamma }\}_{\gamma \in \Gamma }$$, $$\{g_{\gamma }\}_{\gamma \in \Gamma }$$ are two nets of functions from $$X$$ to $$Y$$ satisfying $$(\forall \gamma \in \Gamma)(\exists x_{\gamma }\in X) (f_{\gamma }(x_{\gamma })=g_{\gamma }(x_{\gamma }))$$, $$\{f_{\gamma }\}_{\gamma \in \Gamma }$$ converges strongly to $$f$$ and $$\{g_{\gamma }\}_{\gamma \in \Gamma }$$ converges strongly to $$g$$, then (under certain conditions posed on $$f$$ and $$g$$) the equation $$f(x) = g(x)$$ has a solution. The paper shows that strong convergence and some other convergences preserve the fixed point property.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects)
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##### References:
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