A Class of Quasilinear Equations with Hilfer Derivatives

Abstract

We study the solvability of a Cauchy type problem for linear and quasilinear equations with Hilfer fractional derivatives solved for the higher-order derivative. The linear operator acting on 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 for the nonlinear operator in the equation, the contraction property of the operator \(G\) in a suitably chosen metric function space on a sufficiently small interval is proved. Thus, we prove a theorem on the existence of a unique local solution of the Cauchy type problem for the quasilinear equation. The result on the unique global solvability of this problem is obtained by proving the contraction property of a sufficiently high power of the operator \(G\) in a special function space on the original interval provided that the Lipschitz condition is satisfied for the nonlinear operator in the equation. 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.