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In this paper, we establish a Hausdorff metric over the family of nonempty closed subsets of an extended b-metric space. Thereafter, we introduce the concept of multivalued fuzzy contraction mappings and prove related α-fuzzy fixed point theorems in the context of extended b-metric spaces that generalize Nadler’s fixed point theorem as well as many preexisting results in the literature. Further, we establish α-fuzzy fixed point theorems for Ćirić type fuzzy contraction mappings as a generalization of previous results. Moreover, we give some examples to support the obtained results.

In this paper, we introduce the notion of controlled rectangular metric spaces as a generalization of rectangular metric spaces and rectangular b-metric spaces. Further, we establish some related fixed point results. Our main results extend many existing ones in the literature. The obtained results are also illustrated with the help of an example. In the last section, we apply our results to a common real-life problem in a general form by getting a solution for the Fredholm integral equation in the setting of controlled rectangular metric spaces.