Extremal mild solutions to fractional delay integro-differential equations with non-instantaneous impulses

We study the existence of extremal mild solutions to fractional delay integro-differential equations with non-instantaneous impulses by employing the monotone iterative technique coupled with the method of lower and upper solutions. Our existence results are established by appealing to two different theories. In our first result, the semigroup generated by the linear operator associated with the problem is assumed to be compact. On the other hand, in our second result, the semigroup need not be compact and our conclusions are obtained using the theory of measures of noncompactness. In order to illustrate our results, we also discuss a concrete problem regarding heat conduction in a metal bar with predefined impulsive points.