Geodesic
Asymptotic expansions of solutions of singularly perturbed equations
Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 1 (2022), pp. 18-32

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We consider a first-order equation in a Banach space with a small parameter at the derivative and a second-order perturbation of smallness on the right-hand side. A solution to the Cauchy problem is constructed in the form of an asymptotic expansion in powers of a small parameter by the Vasilieva-Vishik-Lyusternik method. The operator A on the right-hand side is degenerate: we consider the case of possessing the property of having a number 0 by a normal eigenvalue and a two-dimensional kernel; core elements have no attached. Formulas for calculating the components of the regular and boundary layer parts of the expansion are determined. A condition for the regularity of degeneration is obtained. The expansion is shown to be asymptotic. An illustrative example is given.
Keywords: first-order equation in a Banach space, small parameter at the highest derivative, perturbation square on the right-hand side, closed operator, 0-normal eigenvalue, asymptotics, Vasil'eva-Vishik-Lyusternik method. 
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     author = {V. I. Uskov},
     title = {Asymptotic expansions of solutions of singularly perturbed equations},
     journal = {Vestnik Tverskogo gosudarstvennogo universiteta. Seri\^a Prikladna\^a matematika},
     pages = {18--32},
     publisher = {mathdoc},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTPMK_2022_1_a1/}
}
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V. I. Uskov. Asymptotic expansions of solutions of singularly perturbed equations. Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 1 (2022), pp. 18-32. http://geodesic.mathdoc.fr/item/VTPMK_2022_1_a1/