Exploring new fuzzy fractional integral operators with applications over fuzzy number convex and harmonic convex mappings

Modeling of real-world phenomena frequently contains uncertainties mainly due to impreciseness and ambiguity. In this context, fuzzy sets can be an important tool for mitigate them. In turn, fractional calculus has shown be relevant to improve physical models and to provide more accurate solutions for problems in science and engineering. Fractional integrals combined with the fuzzy theory are an important tool to explore the aforementioned uncertainties whenever we desire to model and simulate distinct phenomena. In this paper, the generalized fuzzy fractional integral operator with exponential kernels is introduced as an intriguing generalization of the fuzzy fractional integral operator with exponential kernels, and its significant properties are examined. Several Hermite–Hadamard- (ℋℋ-) and Pachpatte-type integral inclusion relations are established via fuzzy harmonic convexity using these newly proposed operators in fuzzy fractional calculus. Specifically, enhanced ℋℋ-generalized fractional integral inclusions for the fuzzy convexity are suggested. Based on variations in the parameter γ and differentiable function ξ , theoretical representations of the results are provided to determine the accuracy of the obtained inclusion relations in the study. Additionally, interval-valued fractional inclusions are exceptional cases of this article's new fuzzy fractional operators where interval-valued mappings and fuzzy number mappings are integrable, respectively. The results contributes to the existing literature and may be useful in better understanding of automatic control models, artificial intelligence, computer science, physics, biosystems, mathematical economics, or econophysics, among other domains.