{'fraction_non_alphanumeric': 0.029832935560859187, 'fraction_numerical': 0.0059665871121718375, 'mean_word_length': 4.668918918918919, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Common fixed point theorems for hybrid pairs of mappings with some weaker conditions

In this paper, we prove a common fixed point theorem for hybrid pairs of set and single valued mappings without assuming compatibility and continuity of any mapping on noncomplete metric spaces. To prove the theorem, we use a noncompatible condition, that is, weak commutativity of type (KB). We show that completeness of the whole space is not necessary for the existence of common fixed point. Our result improves, extends and generalizes the results of Fisher [5], Sastry and Naidu [18]. We give an example to validate our result. We also prove a common fixed point theorem on compact metric spaces. At the end, we improve our theorem by omitting the assumption of compactness. We also improve and generalize the results of Ahmed [2] and Fisher [5].