  We investigate the variable-exponent Abel integral equations and
corresponding fractional Cauchy problems. The main contributions of the work
are enumerated as follows: (i) We develop an approximate inversion technique
for variable-exponent Abel integral operators, based on which we analyze the
corresponding integral and differential equations; (ii) We prove that the
sensitive dependence of the well-posedness of classical Riemann-Liouville
fractional differential equations on the initial value could be resolved by
adjusting the initial value of the variable exponent; (iii) We prove that the
singularity of the solutions to the Riemann-Liouville fractional differential
equations could also be eliminated by adjusting the variable exponent and its
derivatives at the initial time, which, together with (ii), demonstrates the
advantages of introducing the variable exponent. The proposed approximate
inversion technique provides a potential means to convert the intricate
variable-exponent integral and differential problems to feasible forms, and the
above findings suggest that the variable-exponent fractional problems may serve
as a connection between integer-order and fractional models by adjusting the
variable exponent at the initial time.
