Geodesic
Branching equation for a first-order differential equation in a Banach space with quadratic perturbations of a small parameter
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 4, Tome 233 (2024), pp. 99-106

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This paper is devoted to the study of the behavior as $\varepsilon\to0$ of solutions of the Cauchy problem for a first-order differential equation in a Banach space with quadratic operator pencils with the derivative of the unknown function. The branching equation is obtained and analyzed by using the Newton diagram. The conditions of the appearing of a boundary layer near the initial point are identified and the structure of boundary-layer functions is determined.
Keywords: branching equation, first-order differential equation, Fredholm operator, Banach space, small parameter, boundary layer
Mots-clés : quadratic perturbation 
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     author = {V. I. Uskov},
     title = {Branching equation for a first-order differential equation in a {Banach} space with quadratic perturbations of a small parameter},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {99--106},
     publisher = {mathdoc},
     volume = {233},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2024_233_a8/}
}
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V. I. Uskov. Branching equation for a first-order differential equation in a Banach space with quadratic perturbations of a small parameter. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 4, Tome 233 (2024), pp. 99-106. http://geodesic.mathdoc.fr/item/INTO_2024_233_a8/