Approximation of mild solution of the delay fractional integro-differential equations on the Banach spaces

This work deals with the approximation of a mild solution to a fractional integro-differential equation involving delay and an initial history condition. The approximation observes the mild solution \(\{\widetilde{\xi }_\varrho \}_{\varrho >0}\) ${\{{\tilde{\xi }}_{\varrho }\}}_{\varrho >0}$ of a corresponding family of fractional differential equations with piecewise constant arguments that varies the semilinear term with a parameter \(\varrho\) $\varrho$ . The main result is to obtain of the solution in terms of a difference equation on a Banach space, and necessary conditions ensuring uniform convergence of approximate solution \(\widetilde{\xi }_\varrho\) ${\tilde{\xi }}_{\varrho }$ to the mild solution of fractional integro-differential equation as \(\varrho\) $\varrho$ approaches to zero on both compact and unbounded intervals. The stability of the approximation is discussed using the stability of the solution operator and Halanay’s inequality. An example is presented illustrating the obtained abstract result.