Quasilinear equations with Riemann — Liouville derivatives in Hölder type spaces

The issues of the unique solvability of a Cauchy type problem for a quasilinear equation in a Banach space with several minor fractional derivatives in the nonlinear part and with a linear operator generating an analytical resolving family of operators of a linear homogeneous equation are investigated. Using the Banach contraction mapping theorem, the existence and uniqueness of local and global solutions in specially constructed Hölder type spaces is proved. Abstract results are used for the study of an initial boundary value problem for a modified time-fractional order system of the phase field equations.