Cauchy type problem for some quasilinear equations with riemann – liouville derivatives and a sectorial operator

We studies the issues of solvability of the Cauchy type problem for quasi-linear equations solved with respect to the highest fractional Riemann – Liouville derivative, the operator in the linear part at an unknown function in the equation is assumed to be sectorial. In this case, the nonlinear operator depends on low-order fractional derivatives with an arbitrary fractional part. Theorems on the local and global existence of a unique solution are obtained under the condition of local Lipschitz continuity and Lipschitz continuity of a nonlinear operator, respectively, in the case of its continuity in the norm of the graph of the sectorial operator. The Cauchy type problem for a quasi-linear equation is reduced to an integro-differential equation in a specially selected functional space. To prove the existence of a unique solution, Banach Theorem on the fixed point of a compressive map in a complete metric space is used. The abstract result obtained is applied for the study of the existence and uniqueness of a solution of a class of initial boundary value problems for nonlinear partial differential equations with polynomials from a self-adjoint elliptic operator in spatial variables and with fractional derivatives in time.