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**Explicit bounds on certain integral inequalities.**
*(English)*
Zbl 0996.26008

In this paper, the author establish some explicit bounds on the retarded Gronwall-Bellman-Bihari type integral inequalities. In addition, two variable generalizations of these inequalities are obtained and applied to the study of boundedness, uniqueness and continuous dependence of the solutions of the initial boundary value problem for hyperbolic partial delay differential equations.

Reviewer: James Adedayo Oguntuase (Abeokuta)

### MSC:

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

26D15 | Inequalities for sums, series and integrals |

35R10 | Partial functional-differential equations |

### Keywords:

explicit bounds; retarded Gronwall-Bellman-Bihari type integral inequalities; two independent variable generalizations; differential and integral equations; partial derivatives; hyperbolic delay differential equations
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\textit{B. G. Pachpatte}, J. Math. Anal. Appl. 267, No. 1, 48--61 (2002; Zbl 0996.26008)

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### References:

[1] | Bainov, D.; Simeonov, P., Integral inequalities and applications, (1992), Kluwer Academic Dordrecht · Zbl 0759.26012 |

[2] | Beckenbach, E.F.; Bellman, R., Inequalities, (1961), Springer-Verlag New York · Zbl 0513.26003 |

[3] | Bellman, R., The stability of solutions of linear differential equations, Duke math. J., 10, 643-647, (1943) · Zbl 0061.18502 |

[4] | Bihari, I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta. math. acad. sci. hungar., 7, 81-94, (1965) · Zbl 0070.08201 |

[5] | Gronwall, T.H., Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. math., 20, 292-296, (1919) · JFM 47.0399.02 |

[6] | Lipovan, O., A retarded Gronwall-like inequality and its applications, J. math. anal. appl., 252, 389-401, (2000) · Zbl 0974.26007 |

[7] | Martyniuk, A.A.; Gutowski, R., Integral inequalities and stability of motion, (1979), Naukova Dumka Kiev |

[8] | Medved, M., Nonlinear singular integral inequalities for functions in two and n independent variables, J. inequalities and appl., 5, 287-308, (2000) · Zbl 0985.26010 |

[9] | Pachpatte, B.G., Inequalities for differential and integral equations, (1998), Academic Press New York · Zbl 1032.26008 |

[10] | Pachpatte, B.G., On some new inequalities related to a certain inequality arising in the theory of differential equations, J. math. anal. appl., 251, 736-751, (2000) · Zbl 0987.26010 |

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