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**A generalized retarded Gronwall-like inequality in two variables and applications to BVP.**
*(English)*
Zbl 1193.26014

Summary: We establish a generalized retarded integral inequality of Gronwall-like type in two variables, which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals without assumption of monotonicity. Using our result we can solve both the integral inequality in [W.S. Cheung, Nonlinear Anal., Theory Methods Appl. 64, No. 9 (A), 2112–2128 (2006; Zbl 1094.26011)] and the one in [R.P. Agarwal, S. Deng, W. Zhang, Appl. Math. Comput. 165, No. 3, 599–612 (2005; Zbl 1078.26010)]. We apply our result to a boundary value problem of a partial differential equation for boundedness, uniqueness and continuous dependence.

### MSC:

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

35G30 | Boundary value problems for nonlinear higher-order PDEs |

### Keywords:

integral inequality; monotonization; stronger nondecreasing; boundary value problem; boundedness; uniqueness
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\textit{W.-S. Wang}, Appl. Math. Comput. 191, No. 1, 144--154 (2007; Zbl 1193.26014)

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### References:

[1] | Agarwal, R.P.; Deng, S.; Zhang, W., Generalization of a retarded Gronwall-like inequality and its applications, Appl. math. comput., 165, 599-612, (2005) · Zbl 1078.26010 |

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

[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., A generalization of a lemma of Bellman and its application to uniqueness problem of differential equation, Acta math. acad. sci. hung., 7, 81-94, (1956) · Zbl 0070.08201 |

[5] | Cheung, W.S., Some new nonlinear inequalities and applications to boundary value problems, Nonlinear anal., 64, 2112-2128, (2006) · Zbl 1094.26011 |

[6] | Dafermos, C.M., The second law of thermodynamics and stability, Arch. rat. mech. anal., 70, 167-179, (1979) · Zbl 0448.73004 |

[7] | Dannan, F., Integral inequalities of gronwall – bellman – bihari type and asymptotic behavior of certain second order nonlinear differential equations, J. math. anal. appl., 108, 151-164, (1985) · Zbl 0586.26008 |

[8] | Dragomir, S.S.; Kim, Y.H., Some integral inequalities for functions of two variables, Electr. J. diff. eqns., 2003, 10, 1C13, (2003) |

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

[10] | Kim, Y.H., On some new integral inequalities for functions in one and two variables, Acta math. sinica, 21, 423-434, (2005) · Zbl 1084.26013 |

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

[12] | Massalitina, E.V., On the perow integro-summable inequality for functions of two variables, Ukrainian math. J., 56, 1864-1872, (2004) |

[13] | Medina, R.; Pinto, M., On the asymptotic behavior of solutions of a class of second order nonlinear differential equations, J. math. anal. appl., 135, 399-405, (1988) · Zbl 0668.34056 |

[14] | Mitrinović, D.S.; Pečarić, J.E.; Fink, A.M., Inequalities involving functions and their integrals and derivatives, (1991), Kluwer Academic Dordrecht · Zbl 0744.26011 |

[15] | Ou-Yang, L., The boundedness of solutions of linear differential equations \(y'' + A(t) y = 0\), Shuxue jinzhan, 3, 409-415, (1957) |

[16] | Pachpatte, B.G., On some new inequalities related to certain inequalities in the theory of differential equations, J. math. anal. appl., 189, 128-144, (1995) · Zbl 0824.26010 |

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

[18] | Pinto, M., Integral inequalities of bihari-type and applications, Funkcial. ekvac., 33, 387-430, (1990) · Zbl 0717.45004 |

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