Semilinear Equations in Banach Spaces with Lower Fractional Derivatives

Abstract

In the first part of the work we find conditions of the unique classical solution existence for the Cauchy problem to solved with respect to the highest fractional Caputo derivative semilinear fractional order equation with nonlinear operator, depending on the lower Caputo derivatives. Abstract result is applied to study of an initial-boundary value problem to a modified Oskolkov–Benjamin–Bona–Mahony–Burgers nonlinear equation with time-fractional derivatives. In the second part of the work the unique solvability of the generalized Showalter–Sidorov problem for semilinear fractional order equation with degenerate linear operator at the highest-order Caputo derivative is researched. The nonlinear operator, generally speaking, depends on the lower fractional Caputo derivatives. Here the result on the unique solvability of the Cauchy problem to equation, solved with respect to the highest Caputo derivative, is used also. The abstract result from the second part of the work is demonstrated on an example of an initial-boundary value problem to a nonlinear system of partial differential equations, not solvable with respect to the highest time-fractional derivative.