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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.

54H25 Fixed-point and coincidence theorems (topological aspects)
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