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

T. K. Yuldashev ; B. J. Kadirkulov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 211 (2022), pp. 83-95.

Résumé

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.

Détail

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

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