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Fixed point theory for generalized contractive maps of Meir--Keeler type. (English) Zbl 1086.47016
Let $(X,d)$ be a complete metric space, $x_{0} \in X$, $r > 0$ and let $B$ denote the closed ball centered at $x_{0}$ and with radius $r$. Assume that $F: B \to X$ and $$M(x,y):= \max\{d(x,y), d(x,F(x)), d(y,F(y)), \tfrac{1}{2}(d(x,F(y)) + d(y,F(x)))\}.$$ The authors prove the following results: (1) If $F$ is continuous, $d(x_{0},F^{n}(x_{0})) < r$ for $n = 1, 2,\dots$, and for every $\varepsilon > 0$ there exists $\delta > 0$ such that for $x,y \in B$ we have $$M(x,y) < \varepsilon + \delta \Rightarrow d(F(x), F(y)) < \varepsilon,$$ then there exists a unique fixed point of $F$ in $B$. (2) If $x_{n} = F(x_{n-1})$ for $n = 1, 2,\dots$ and $x_{n} \to x$ implies that $x = F(x)$ and either (a) $\Phi: X \to [0,\infty)$ is a function such that $$d(x,F(x)) \leq \Phi(x) - \Phi(F(x))$$ and $$\Phi(x_{0}) < r,$$ or (b) for $x \in B$ we have $$d(F(x), F^{2}(x)) \leq kd(x,F(x))$$ and $d(x_{0}, F(x_{0})) < (1-k)r$ for some $k \in (0,1)$, then there exists a fixed point of $F$ in $B$. There are some global versions of these results. Some of them are extended to gauge spaces. Domain invariance, coincidence points and Krasnosel’skii type fixed point results are also given. Finally, random fixed point theorems are presented.

##### MSC:
 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties 54H25 Fixed-point and coincidence theorems in topological spaces
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