## Abstract

We study the Volterra integro-differential equation in R^{n}({star operator} )frac(d x, d t) = X (t, x, ∫_{0}^{t} K (t, s) g (x (s) d s)) . We establish a connection between system ({star operator}) with a kernel of the form ({star operator} {star operator})K (t, s) = underover(∑, j = 1, ∞) C_{j} F_{j} (t) G_{j} (s) and a countable system of ordinary differential equations. Such a reduction allows use of results obtained earlier for the countable systems of differential equations in the study of integro-differential equations. In this paper we discuss problems related to the stability of systems ({star operator}) and ({star operator}{star operator}), as well as applications of the method of normal forms to solving some problems in the qualitative theory of integro-differential equations. In particular, it can be employed for the study of critical cases of stability and bifurcation problems in integro-differential equations.

Original language | English |
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Pages (from-to) | 1553-1569 |

Number of pages | 17 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 68 |

Issue number | 6 |

DOIs | |

State | Published - 15 Mar 2008 |

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics