Complex networks represent various real-world systems. Overlapping community detection is one of the critical tasks in studying these networks and has significance to a wide variety of applications, including the exploration of online social networks because of the natural attitude of persons to participate in multiple communities at the same time. Despite a large number of existing community detection algorithms for detecting disjoint communities, the efficient and fast uncovering of overlapping communities has remained a challenging problem. To provide an efficient solution, on the one hand, the balanced link density label propagation (BLDLP) algorithm, proposed by the authors of the current study, is a fast, stable, and efficient method for disjoint community detection. On the other hand, the fuzzy theory is a worthwhile approach for overlapping community detection since it provides the membership rate of the overlapping nodes as well as the detection of overlapping communities. Hence, in this paper, based on the synergy of the BLDLP algorithm and the fuzzy theory, a novel method, called fuzzy BLDLP, for overlapping community detection is proposed.Fuzzy BLDLP is fast and efficient. The proposed method needs no prior information about the number of network communities to discover them. The experiments on both synthetic and real-world known networks, including Zachary, Dolphins, and COVID-19 Co-authorship, have revealed that the proposed method successfully detects the overlapping nodes and communities and hence is comparable with the state-of-the-art overlapping community detection algorithms in terms of recall, precision, F-score and overlapping normalized mutual information.