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MATHEMATICS
A class of quasilinear equations with Hilfer derivatives
V. E. Fedorov, A. S. Scorynin Chelyabinsk State University
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.
Keywords:
hilfer derivative, cauchy type problem, mittag – leffler function, quasilinear equation, contraction mapping theorem, local solvability, global solvability.
Received: 30.12.2023 Accepted: 30.12.2023
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