Geodesic
The Cauchy problem for a singularly perturbed Volterra integro-differential equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 5, pp. 805-812

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The Cauchy problem for a singularly perturbed Volterra integro-differential equation is examined. Two cases are considered: (1) the reduced equation has an isolated solution, and (2) the reduced equation has intersecting solutions (the so-called case of exchange of stabilities). An asymptotic expansion of the solution to the Cauchy problem is constructed by the method of boundary functions. The results are justified by using the asymptotic method of differential inequalities, which is extended to a new class of problems. 
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     author = {N. N. Nefedov and A. G. Nikitin and T. A. Urazgil'dina},
     title = {The {Cauchy} problem for a~singularly perturbed {Volterra} integro-differential equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {805--812},
     publisher = {mathdoc},
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     number = {5},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_5_a3/}
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N. N. Nefedov; A. G. Nikitin; T. A. Urazgil'dina. The Cauchy problem for a singularly perturbed Volterra integro-differential equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 5, pp. 805-812. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_5_a3/