Babenko's Method for q‐Fractional Integral Equations: A Systematic Framework

ABSTRACT In this paper, we first introduce a generalized form of the ‐Mittag‐Leffler function and use it as the integral kernel to construct a new type of ‐fractional integral operators. We study the boundedness of these operators on the space and characterize their composition structures via discrete convolution for sequences. Subsequently, we apply Babenko's method (also known as the inverse operator method) to investigate the existence and uniqueness of solutions as well as the Hyers‐Ulam stability for a class of ‐fractional integral equations involving such operators. Some illustrative examples and useful connections to other fields (especially fractional calculus based on Sonine kernels) are also discussed.