Mild and strong solutions to few types of fractional order nonlinear equations with periodic boundary conditions

In this paper, the two fractional periodic boundary value problems <Equation ID={''}Equ1{''}> <MediaObject> </MediaObject> </Equation> and <Equation ID={''}Equ2{''}> <MediaObject> </MediaObject> </Equation> will be studied where (0) (C) D (t) (alpha) is the ordinary Caputo fractional derivative and lambda a a{''}e -\{0\}. Under some suitable assumptions on the function f, the existence of at least one mild solution will be proved. Under some conditions, the uniqueness of this mild solution will be proved to both problems. Finally, these mild solutions will be strong solutions under certain conditions.