We introduce a large class of contractive mappings, called enriched contractions, a class which includes, amongst many other contractive type mappings, the Picard–Banach contractions and some nonexpansive mappings. We show that any enriched contraction has a unique fixed point and that this fixed point can be approximated by means of an appropriate Krasnoselskij iterative scheme. Several important results in fixed point theory are shown to be corollaries or consequences of the main results of this paper. We also study the fixed points of local enriched contractions, asymptotic enriched contractions and Maia-type enriched contractions. Examples to illustrate the generality of our new concepts and the corresponding fixed point theorems are also given.

中文翻译：

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Banach空间中压缩收缩的逼近定点

我们介绍了一大类收缩映射，称为“富集收缩”，该类在众多其他收缩类型映射中包括Picard–Banach收缩和一些非扩展映射。我们表明，任何富集收缩都有一个唯一的固定点，并且可以通过适当的Krasnoselskij迭代方案来近似该固定点。定点理论中的几个重要结果被证明是本文主要结果的推论或结果。我们还研究了局部富集收缩，渐近富集收缩和Maia型富集收缩的不动点。还给出了一些例子来说明我们新概念的一般性以及相应的不动点定理。