Geodesic
On one integro-differential equation with fractional Hilfer operator and nonlinear maximums
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 211 (2022), pp. 83-95

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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. 
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     author = {T. K. Yuldashev and B. J. Kadirkulov},
     title = {On one integro-differential equation with fractional {Hilfer} operator and nonlinear maximums},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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     publisher = {mathdoc},
     volume = {211},
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T. K. Yuldashev; B. J. Kadirkulov. On one integro-differential equation with fractional Hilfer operator and nonlinear maximums. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 211 (2022), pp. 83-95. http://geodesic.mathdoc.fr/item/INTO_2022_211_a5/