Geodesic
Singularly Perturbed Nonlinear Integrodifferential Systems with Fast Varying Kernels
Matematičeskie zametki, Tome 72 (2002) no. 5, pp. 654-664

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We consider nonlinear singularly perturbed integro-differential equations with fast varying kernels. It is assumed that the spectrum of the limiting operator lies in the closed left half-plane $\operatorname {Re}\lambda \le 0$. We derive an algorithm for obtaining regularized (in the sense of Lomov) asymptotic solutions in both the nonresonance and resonance cases. In deriving the algorithm, we essentially use the regularization apparatus for integral operators with fast varying kernels, developed earlier by the authors for linear integral and integro-differential systems. The algorithm is justified and the existence of a solution of the original nonlinear problem is proved by means of the Newton method for operator equations. 
@article{MZM_2002_72_5_a2,
     author = {A. A. Bobodzhanov and V. F. Safonov},
     title = {Singularly {Perturbed} {Nonlinear} {Integrodifferential} {Systems} with {Fast} {Varying} {Kernels}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {654--664},
     publisher = {mathdoc},
     volume = {72},
     number = {5},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_72_5_a2/}
}
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A. A. Bobodzhanov; V. F. Safonov. Singularly Perturbed Nonlinear Integrodifferential Systems with Fast Varying Kernels. Matematičeskie zametki, Tome 72 (2002) no. 5, pp. 654-664. http://geodesic.mathdoc.fr/item/MZM_2002_72_5_a2/