The scope of this note encompasses an extension of common fixed-point results from their initial establishment in complex-valued metric spaces to the realm of complex-valued fuzzy metric spaces. By adopting a nuanced approach, these outcomes are further demonstrated within the context of complex-valued fuzzy metric spaces, focusing on employing more lenient contractive criteria. Building upon the groundwork laid by previous researchers and considering a range of other relevant studies, our current research makes significant strides in enhancing and expanding upon their findings. While these prior works have primarily focused on complex-valued metric spaces, our study takes a significant leap forward by applying their insights to complex-valued fuzzy metric spaces. This extension of previous research deepens our understanding of these intricate spaces and opens up new avenues for exploring the complexities inherent in such settings. To enhance the clarity of our core findings, we offer illustrative examples that lend visual support to our assertions.
In this article, we do a study of common fixed point theorems for six self-maps in FM-Spaces using common limit in range property concerning two pairs of products of two different self-maps. We use the properties (CLRTH) and (CLRSR) along with contractive type implicit relations to prove our results. In support of our result, an example has been provided. Our findings are like those of Kumar and Chouhan [12]. Kumar and Chauhan demonstrated their primary result in [12] by improving and generalizing Aalam, Kumar, and Pants' [1] results. In past, many authors have done study of common fixed point using (E-A) property (like Aalam et. al. [1] proved results using this property), and then these results were improved and generalized by using common (E-A) property as this property is superior to (E-A) property, as the closeness of subspace is required to prove a required result on common fixed point by using these properties, which is a drawback. We improve and generalize all results on these properties using common limit in range property. The goal of this note is to refine and generalize Kumar and Chauhan's [12] results on a common fixed point, as well as some earlier comparable results.
In this research article, we give a new concept of common α-fixed point, α-compatible mappings, weakly α-compatible mappings, α-commuting mappings, weakly α-commuting mappings, and α-continuous mappings and then prove some common α-fixed point theorems for these mappings under new contractive conditions. Further, we generalize the results of Singh and Chouhan [17] for common α-fixed points and give some results for common α-fixed points under this newly introduced concept along with α- contractive conditions. Many examples have also been given and proved in support of our concept and results.
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