New nonlinear Henry–Gronwall type inequality with time-varying delay and applications to fractional differential equations

In this paper, four types of new nonlinear Henry–Gronwall type integral inequalities have been established. As for the first type, by employing inequality techniques, we overcome the limitation of traditional methods in which dividing the range of parameter β ∈ ( 0 , 1 ) into two parts is needed. For the second type, we derive a bound that is more precise than previous studies by comparative analysis. Regarding the third type and the fourth type, they are new models that are studied in our work. Specifically, the third type extends our proposed inequality to case when β ≥ 1 , and the fourth type constitutes a new variant of the nonlinear Bihari-type inequality with time-varying delay that offers greater generality. As applications of the derived results, the existence of solutions to the fractional differential equations has been discussed by fixed point theorems and two examples are provided to illustrate the validity of the theorems.