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**Fixed points and stability of an integral equation: nonuniqueness.**
*(English)*
Zbl 1066.45002

Summary: We consider a paper of J. Banaś and B. Rzepka [ibid. 16, No. 1, 1–6 (2003; Zbl 1015.47034)] which deals with existence and asymptotic stability of an integral equation by means of fixed-point theory and measures of noncompactness. By choosing a different fixed-point theorem, we show that the measures of noncompactness can be avoided and the existence and stability can be proved under weaker conditions. Moreover, we show that this is actually a problem about a bound on the behavior of a nonunique solution. In fact, without nonuniqueness, the theorems of stability are vacuous.

Reviewer: Ulrich Kosel (Freiberg)

### MSC:

45G10 | Other nonlinear integral equations |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |

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

45M05 | Asymptotics of solutions to integral equations |

45M10 | Stability theory for integral equations |

### Keywords:

Fixed points; asymptotic stability; Integral equations; Nonuniqueness; measures of noncompactness### Citations:

Zbl 1015.47034
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\textit{T. A. Burton} and \textit{B. Zhang}, Appl. Math. Lett. 17, No. 7, 839--846 (2004; Zbl 1066.45002)

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### References:

[1] | Banaś, J.; Rzepka, R., An application of a measure of noncompactness in the study of asymptotic stability, Appl. math. lett., 16, 1, 1-6, (2003) · Zbl 1015.47034 |

[2] | Smart, D.R., Fixed point theorems, (1980), Cambridge University Press New York · Zbl 0427.47036 |

[3] | Burton, T.A., A fixed-point theorem of Krasnoselskii, Appl. math. lett., 11, 1, 85-88, (1998) · Zbl 1127.47318 |

[4] | Burton, T.A.; Furumochi, T., A note on stability by Schauder’s theorem, Funkcialaj ekvacioj, 44, 73-82, (2001) · Zbl 1158.34329 |

[5] | Burton, T.A., Differential inequalities for integral and delay equations, (), 43 |

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