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Fixed-point theorem for asymptotic contractions of Meir–Keeler type in complete metric spaces. (English) Zbl 1101.54047
Motivated essentially by the work of W. A. Kirk [J. Math. Anal. Appl. 277, No. 2, 645–650 (2003; Zbl 1022.47036)] and T.-C. Lim [Nonlinear Anal., Theory Methods Appl. 46, No. 1(A), 113–120 (2001; Zbl 1009.54044)], the author introduces the concept of asymptotic contraction of Meir-Keeler ($ACMK$) type on a metric space and obtains fixed point theorems fo such maps. The main result states that if $T$ is an $ACMK$ on a complete metric space $X$ and if ${T}^{l}$ is continuous for some natural number $l$, then $T$ has a unique fixed point.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces