88285bb6-59a9-4045-8abd-2c37040cd96920210218104355485wseamdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS1109-276910.37394/23206http://wseas.org/wseas/cms.action?id=40512720202720201910.37394/23206.2020.19http://wseas.org/wseas/cms.action?id=23185Nonlinear Singularly Perturbed Integro-Differential Equations and Regularization MethodAbdukhafizBobodzhanovNational Research University, Moscow Power Engineering Institute, Moscow, RUSSIABurkhanKalimbetovAhmet Yassawi University, Natural Sciences Institute, Turkestan, KAZAKHSTANValerySafonovNational Research University, Moscow Power Engineering Institute, Moscow, RUSSIAThe paper considers a nonlinear integro-differential system of singularly perturbed equations. We discuss the question of the spectrum of its operator, which does not coincide with the spectrum of its limit operator and includes an additionally identically zero point. In the case of linear systems, this difference does not play a special role, since the regularization and construction of the space of solutions of the corresponding iterative problems are realized at nonzero points of the spectrum. In the case of nonlinear problems, the identically zero point of the spectrum plays an essential role in the construction of the solution space in the resonance and nonresonance cases (see below); therefore, in most works using the regularization method in nonlinear problems, only the nonresonance case is usually considered. In the paper, for the classical integrodifferential system, regularization (according to Lomov) is carried out and the corresponding algorithm for constructing asymptotic solutions taking into account the zero point of the spectrum is developed.692020692020301311https://www.wseas.org/multimedia/journals/mathematics/2020/a605106-1227.pdf10.37394/23206.2020.19.30https://www.wseas.org/multimedia/journals/mathematics/2020/a605106-1227.pdfLomov, S.A., Introduction to General Theory of Singular Perturbations, vol. 112 of Translations of Math.Monographs, American Math. Society, Providence, USA, 1992.Safonov, V.F., Bobodzhanov, A.A., Course of higher mathematics. Singularly perturbed equations and the regularization method: textbook,Moscow, Publishing House of MPEI, 2012. (in Russian)Lomov, S.A., Lomov, I.S., Foundations of mathemathical theory of boundary layer, Moscow, Publishing House of MSU, 2011. (in Russian)10.1070/sm2001v192n08abeh000586Bobodzhanov, A.A., Safonov, V.F., Volterra integral equations with rapidly changing kernels and their asymptotic integration, Math. collection, Vol. 192. No 8, 2011, pp. 53-78.Bryuno, A.D., The analytical form of differential equations, Proceed. Moscow Math. Society, Vol. 25, 1971, pp. 119-262.(In Russian)