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**Attractivity results for a nonlinear functional integral equation.**
*(English)*
Zbl 1214.45003

Nonlinear functional integral equations have been studied in recent years for various characterizations of solutions on unbounded intervals of the real axis. In a recent paper, M. A. Darwish [Electron. J. Qual. Theory Differ. Equ. 2007, Paper No. 21 (2007; Zbl 1178.45005)] obtained an existence theorem concerning the attractivity of solutions, however, the main result is not correct. It is a motivation of this paper to correct and generalize the main existence result of [loc. cit.] under weaker conditions. In this paper, the authors consider a reasonably more general nonlinear functional integral equation and they prove two existence results concerning the global attractivity and global asymptotic attractivity for a certain functional nonlinear integral equation. The existence results include several existence and attractivity results obtained earlier by Darwish and Hu-Yan as special cases under weaker conditions. A fixed point theorem of Dhage is used in formulating the main results of this paper and the characterizations of solutions are obtained in the space of functions defined, continuous and bounded on unbounded intervals.

Reviewer: Kun Soo Chang (Seoul)

### MSC:

45M10 | Stability theory for integral equations |

45G10 | Other nonlinear integral equations |

47H08 | Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc. |