Iteration method of approximate solution of the Cauchy problem for a singularly perturbed weakly nonlinear differential equation of an arbitrary order.

*(English)*Zbl 1424.34062Summary: We construct an iteration sequence converging (in the uniform norm in the space of continuous functions) to the solution of the Cauchy problem for a singularly perturbed weakly nonlinear differential equation of an arbitrary order (the weak nonlinearity means the presence of a small parameter in the nonlinear term). The sequence thus constructed is also asymptotic in the sense that the difference of its \(n\)th element from the solution of the problem is proportional to the \((n+1)\)th power of the perturbation parameter.

##### MSC:

34A45 | Theoretical approximation of solutions to ordinary differential equations |

34E15 | Singular perturbations, general theory for ordinary differential equations |

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |

47N20 | Applications of operator theory to differential and integral equations |

##### Keywords:

singular perturbations; Banach contraction principle; method of asymptotic iterations; Routh-Hurwitz stability criterion
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\textit{A. R. Alimov} and \textit{E. E. Bukhzhalev}, Turk. J. Math. 42, No. 5, 2841--2853 (2018; Zbl 1424.34062)

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##### References:

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