Abstract
We have established, in the context of metric spaces, that an F-contraction restricted to appropriate neighborhood of its fixed point is an almost contraction and obtain definition of retractions on complete metric spaces. We have illustrated such an inclusion using an example studied in Wardowski (Fixed Point Theory Appl 94:1–6, 2012) and in Minak et al. (Bull Belg Math Soc Simon Stevin 22:411–422, 2015).
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