##
**Solvability of nonlinear integral equations of Volterra type.**
*(English)*
Zbl 1259.45006

Summary: This paper deals with the existence of continuous bounded solutions for a rather general nonlinear integral equation of Volterra type and discusses also the existence and asymptotic stability of continuous bounded solutions for another nonlinear integral equation of Volterra type. The main tools used in the proofs are some techniques in analysis and the Darbo fixed point theorem via measures of noncompactness. The results obtained in this paper extend and improve essentially some known results in the recent literature. Two nontrivial examples that explain the generalizations and applications of our results are also included.

### MSC:

45G10 | Other nonlinear integral equations |

45D05 | Volterra integral equations |

45M05 | Asymptotics of solutions to integral equations |

45M10 | Stability theory for integral equations |

47H10 | Fixed-point theorems |

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

### Keywords:

continuous bounded solutions; nonlinear integral equation of Volterra type; asymptotic stability; Darbo fixed point theorem; measures of noncompactness
PDF
BibTeX
XML
Cite

\textit{Z. Liu} et al., Abstr. Appl. Anal. 2012, Article ID 932019, 17 p. (2012; Zbl 1259.45006)

Full Text:
DOI

### References:

[1] | R. P. Agarwal, D. O’Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. · Zbl 1194.01052 |

[2] | T. A. Burton, Volterra Integral and Differential Equations, vol. 167, Academic Press, Orlando, Fla, USA, 1983. · Zbl 0565.70028 |

[3] | D. O’Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, vol. 445, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998. · Zbl 1185.34026 |

[4] | M. R. Arias, R. Benítez, and V. J. Bolós, “Nonconvolution nonlinear integral Volterra equations with monotone operators,” Computers and Mathematics with Applications, vol. 50, no. 8-9, pp. 1405-1414, 2005. · Zbl 1088.45004 |

[5] | J. Banaś and A. Chlebowicz, “On existence of integrable solutions of a functional integral equation under Carathéodory conditions,” Nonlinear Analysis, vol. 70, no. 9, pp. 3172-3179, 2009. · Zbl 1168.45005 |

[6] | J. Banaś and B. C. Dhage, “Global asymptotic stability of solutions of a functional integral equation,” Nonlinear Analysis, vol. 69, no. 7, pp. 1945-1952, 2008. · Zbl 1154.45005 |

[7] | J. Banaś and K. Goebel, “Measures of noncompactness in banach spaces,” in Lecture Notes in Pure and Applied Mathematics, vol. 60, Marcel Dekker, New York, NY, USA, 1980. · Zbl 0441.47056 |

[8] | J. Banaś, J. Rocha, and K. B. Sadarangani, “Solvability of a nonlinear integral equation of Volterra type,” Journal of Computational and Applied Mathematics, vol. 157, no. 1, pp. 31-48, 2003. · Zbl 1026.45006 |

[9] | J. Banaś and B. Rzepka, “On existence and asymptotic stability of solutions of a nonlinear integral equation,” Journal of Mathematical Analysis and Applications, vol. 284, no. 1, pp. 165-173, 2003. · Zbl 1029.45003 |

[10] | J. Banaś and B. Rzepka, “An application of a measure of noncompactness in the study of asymptotic stability,” Applied Mathematics Letters, vol. 16, no. 1, pp. 1-6, 2003. · Zbl 1015.47034 |

[11] | J. Banaś and B. Rzepka, “On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation,” Applied Mathematics and Computation, vol. 213, no. 1, pp. 102-111, 2009. · Zbl 1175.45002 |

[12] | T. A. Burton and B. Zhang, “Fixed points and stability of an integral equation: nonuniqueness,” Applied Mathematics Letters, vol. 17, no. 7, pp. 839-846, 2004. · Zbl 1066.45002 |

[13] | A. Constantin, “Monotone iterative technique for a nonlinear integral equation,” Journal of Mathematical Analysis and Applications, vol. 205, no. 1, pp. 280-283, 1997. · Zbl 0878.45006 |

[14] | B. C. Dhage, “Local asymptotic attractivity for nonlinear quadratic functional integral equations,” Nonlinear Analysis, vol. 70, no. 5, pp. 1912-1922, 2009. · Zbl 1173.47056 |

[15] | W. G. El-Sayed, “Solvability of a neutral differential equation with deviated argument,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 342-350, 2007. · Zbl 1115.34076 |

[16] | X. L. Hu and J. R. Yan, “The global attractivity and asymptotic stability of solution of a nonlinear integral equation,” Journal of Mathematical Analysis and Applications, vol. 321, no. 1, pp. 147-156, 2006. · Zbl 1108.45006 |

[17] | J. S. Jung, “Asymptotic behavior of solutions of nonlinear Volterra equations and mean points,” Journal of Mathematical Analysis and Applications, vol. 260, no. 1, pp. 147-158, 2001. · Zbl 0985.45007 |

[18] | Z. Liu and S. M. Kang, “Existence of monotone solutions for a nonlinear quadratic integral equation of Volterra type,” The Rocky Mountain Journal of Mathematics, vol. 37, no. 6, pp. 1971-1980, 2007. · Zbl 1147.45005 |

[19] | Z. Liu and S. M. Kang, “Existence and asymptotic stability of solutions to a functional-integral equation,” Taiwanese Journal of Mathematics, vol. 11, no. 1, pp. 187-196, 2007. · Zbl 1145.45003 |

[20] | D. O’Regan, “Existence results for nonlinear integral equations,” Journal of Mathematical Analysis and Applications, vol. 192, no. 3, pp. 705-726, 1995. · Zbl 0862.45006 |

[21] | C. A. Roberts, “Analysis of explosion for nonlinear Volterra equations,” Journal of Computational and Applied Mathematics, vol. 97, no. 1-2, pp. 153-166, 1998. · Zbl 0932.45007 |

[22] | M. A. Taoudi, “Integrable solutions of a nonlinear functional integral equation on an unbounded interval,” Nonlinear Analysis, vol. 71, no. 9, pp. 4131-4136, 2009. · Zbl 1203.45004 |

[23] | G. Darbo, “Punti uniti in trasformazioni a codominio non compatto,” Rendiconti del Seminario Matematico della Università di Padova, vol. 24, pp. 84-92, 1955. · Zbl 0064.35704 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.