A Class of Quasilinear Equations with Hilfer Derivatives

Abstract

We investigate the solvability issues of the Cauchy type problem for linear and quasilinear equations with Hilfer fractional derivatives resolved with respect to the higher-order derivative. The linear operator at the unknown function in the equation is assumed to be bounded. The unique solvability of the Cauchy type problem for a linear inhomogeneous equation is proved. Using the resulting solution formula, we reduce the Cauchy type problem for the quasilinear differential equation to an integro-differential equation of the form y = G(y). Under the local Lipschitz condition of the nonlinear operator in the equation, the contraction of the operator G in a suitably chosen metric space of functions on a sufficiently small segment is proved. Thus, we prove the theorem on the existence of a unique local solution to a Cauchy type problem for the quasilinear equation. The result on the unique global solvability of this problem is obtained by proving the contraction of a sufficiently large degree of the operator G in a special space of functions on an initially given segment when the Lipschitz condition on a nonlinear operator in the equation is fulfilled. We use the general results to study Cauchy type problems for a quasilinear system of ordinary differential equations and for a quasilinear system of integro-differential equations.

Acknowledgements
The work was funded by the grant of the President of the Russian Federation for state support of leading scientific schools, project number NSH-2708.2022.1.1.