Abstract
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 Reλ≤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.
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REFERENCES
S. A. Lomov, An Introduction to the General Theory of Singular Perturbations [in Russian], Nauka, Moscow, 1981.
M. I. Imanaliev, Methods of Solution of Nonlinear Inverse Problems and Their Applications [in Russian], Frunze, 1977.
A. B. Vasil′eva and V. F. Butuzov, Asymptotic Expansions of Solutions of Singularly Perturbed Equations [in Russian], Nauka, Moscow, 1973.
V. F. Safonov, “The regularization method for singularly perturbed systems of nonlinear differential equations,” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.], 43 (1979), no. 3, 628-653.
V. F. Safonov and O. D. Tuichiev, “Regularization of singularly perturbed integral equations with fast varying kernels and their asymptotics,” Differentsial′nye Uravneniya [Differential Equations], 33 (1997), no. 9, 1199-1211.
L. S. Srubshchik and V. I. Yudovich, “On the application of the Newton method in problems of the asymptotic integration of nonlinear equations,” in: Abstracts of Papers Presented at the All-Union Intercollege Conference on the Application of Functional Analysis Methods to the Solution of Nonlinear Problems [in Russian] Baku, 1975, p. 103. Moscow Institute of Power Engineering
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Bobodzhanov, A.A., Safonov, V.F. Singularly Perturbed Nonlinear Integrodifferential Systems with Fast Varying Kernels. Mathematical Notes 72, 605–614 (2002). https://doi.org/10.1023/A:1021444603184
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DOI: https://doi.org/10.1023/A:1021444603184