Mild Solution for Nonlocal Fractional Functional Differential Equation with not Instantaneous Impulse

In the last few decades fractional differential equations are more interested in developing the numerical methods and theoretical analysis, because it is recently proved that it has memory and hereditary property which is valuable in serval field of engineering and sciences. For some recent development theory and applications of fractional differential equations reader can see the monographs [1–5] and with various conditions such as initial, impulse and nonlocal, one can see the papers [6–12, 14, 21, 22, 26, 28–30] and reference therein. It has been seen that the theory of existence result for fractional differential equations is not yet sufficiently elaborated, compared to that of theory of ordinary differential equations. Due to this fact, it is important and necessary to study the existence result for semi-linear functional fractional differential equations. The spacial type of functional differential equations is delay differential equations in which delay may state-dependent or constant with different type of conditions. See for more details of relevant update theory of state dependent delay in the cited papers [13, 15–20, 23–25, 27]. Zhou et al. [11] study the following Cauchy problem