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JOURNALS // Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory // Archive

Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 2022 Volume 211, Pages 83–95 (Mi into1026)

On one integro-differential equation with fractional Hilfer operator and nonlinear maximums

T. K. Yuldasheva, B. J. Kadirkulovb

a National University of Uzbekistan named after M. Ulugbek, Tashkent
b Tashkent State Institute of Oriental Studies

Abstract: In this paper, we discuss the unique solvability of the initial-value problem for a nonlinear fractional integro-differential equation of the Hilfer type with a degenerate kernel and nonlinear maximums. USing a simple integral transformation based on the Dirichlet formula, we reduce the initial-value problem to a nonlinear, fractional integral equation of the Volterra type with nonlinear maximums. The theorem of existence and uniqueness of a solution of the initial-value problem considered is proved. The stability of solutions with respect to the parameter and the initial data is also proved. Illustrative examples are given.

Keywords: ordinary integro-differential equation, equation with nonlinear maximums, Hilfer operator, unique solvability, degenerate kernel.

UDC: 517.911

MSC: 34K29, 45D05, 41A30

DOI: 10.36535/0233-6723-2022-211-83-95



© Steklov Math. Inst. of RAS, 2025