The Matrix Mittag–Leffler Function and Nonlinear Fractional Integro–Differential Equations

Fractional differential equations with initial conditions or boundary value problems have recently attracted many researchers and emerged as an important field of research due to their applications in various areas of science and engineering such as control theory, wave propagation, mechanics, and biology. There is very little known in the current literature on boundary value problems of fractional integro-differential equations with integral boundary conditions and variable coefficients. Clearly, one important topic in the area is to study uniqueness and existence of solutions to nonlinear fractional differential or integro-differential equations, including partial differential equations. This chapter is comprised of seven independent and self-contained sections which deal with nonlinear equations with various boundary conditions or initial value problems, based on the matrix Mittag-Leffler function given in Section 4.8, fixed point theory, as well as Babenko’s approach.