The Study of Fractional Quadratic Integral Equations Involves General Fractional Integrals

This paper investigates the well-posedness of analytical solutions to fractional quadratic differential equations, which involve generalized fractional integrals with respect to other functions. The nonlinear components f and h depend on spatial variables and the general fractional integral, respectively. We use the operator equation T1ωT2ω+T3ω=ω to investigate the existence of solutions, marking the first study of its kind. Using an auxiliary function and Boyd and Wang’s fixed-point theorem, the uniqueness and continuous dependence of the solution are obtained. In particular, we applied nonlinear functional analysis to investigate Hyers-Ulam and Hyers-Ulam-Rassias stabilities for fractional quadratic integral equations. New results are provided for specific values of the parameter z, and a fundamental inequality is formulated to ensure the existence of maximal and minimal solutions. Some examples are given to illustrate our results.