度量空间中Meir-Keeler型压缩的不动点定理
Fixed point theorems for Meir-Keeler type contractions in metric spaces
摘要：
我们建立了一个简单而强大的引理，提供了一个标准的序列在度量空间柯西。利用这个引理，很容易地证明了Picard迭代序列$\{T^nx\}$（其中$T$是Meir-Keeler型的压缩序列或渐近压缩序列）是Cauchy序列.作为应用，对满足Meir-Keeler型压缩条件的完全度量空间的连续渐近正则自映射的不动点存在性的几个已知结果给出了新的简单证明.这些结果包括Proinov在[Petko D. Proinov，度量空间中的不动点定理，非线性分析。\textbf{46}（2006）546- 557]中，Suzuki的渐近压缩的不动点定理[Tomonari Suzuki，A definitive result on asymptomatic contractions，J. Math. Anal. \textbf{335}（2007）707- 715]等。我们还证明了一些新的不动点定理。
英文摘要：
We establish a simple and powerful lemma that provides a criterion for sequences in metric spaces to be Cauchy. Using the lemma, it is then easily verified that the Picard iterates $\{T^nx\}$, where $T$ is a contraction or asymptotic contraction of Meir-Keeler type, are Cauchy sequences. As an application, new and simple proofs for several known results on the existence of a fixed point for continuous and asymptotically regular self-maps of complete metric spaces satisfying a contractive condition of Meir-Keeler type are derived. These results include the remarkable fixed point theorem of Proinov in [Petko D. Proinov, Fixed point theorems in metric spaces, Nonlinear Anal. \textbf{46} (2006) 546--557], the fixed point theorem of Suzuki for asymptotic contractions in [Tomonari Suzuki, A definitive result on asymptotic contractions, J. Math. Anal. Appl. \textbf{335} (2007) 707--715], and others. We also prove some new fixed point theorems.
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