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**A note on the stability of the integral-differential equation of the hyperbolic type in a Hilbert space.**
*(English)*
Zbl 1146.45001

Summary: The initial-value problem for integral-differential equations of the hyperbolic type in a Hilbert space \(H\) is considered. The unique solvability of this problem is established and the stability estimates for its solution are obtained. The difference scheme approximately solving this problem is presented and the stability estimates for its solution are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented and the stability estimates for its solutions are obtained.

### MSC:

45K05 | Integro-partial differential equations |

26D10 | Inequalities involving derivatives and differential and integral operators |

47G20 | Integro-differential operators |

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\textit{M. Ashyraliyev}, Numer. Funct. Anal. Optim. 29, No. 7--8, 750--769 (2008; Zbl 1146.45001)

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### References:

[1] | Ashirov S., Izv. Vyssh. Uchebn. Zav., Matematika 9 pp 3– (1987) |

[2] | Ashirov S., UMJ 40 pp 510– (1988) |

[3] | Krein S.G., Linear Differential Equations in a Banach Space (1966) |

[4] | Fattorini H.O., Second Order Linear Differential Equations in Banach Spaces (1985) · Zbl 0564.34063 |

[5] | Piskarev S., Taiwanese Journal of Mathematics 1 pp 3585– (1997) |

[6] | M. Ashyraliyev ( 2005 ). Generalizations of Gronwall’s integral inequality and their discrete analogies . CWI Reports of MAS1, MAS-E0520. Available athttp://www.cwi.nl/ftp/CWIreports/MAS/MAS-E0520.pdf . |

[7] | DOI: 10.1155/S1085337501000501 · Zbl 1007.65064 |

[8] | Ashyralyev A., New Difference Schemes for Partial Differential Equations (2004) · Zbl 1060.65055 |

[9] | DOI: 10.1155/DDNS.2005.183 · Zbl 1094.65077 |

[10] | DOI: 10.1080/01630560500431068 · Zbl 1098.65055 |

[11] | Sobolevskii P.E., Difference Methods for the Approximate Solutions of Differential Equations (1975) |

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