On integral inequalities of Gronwall-Bellman type

Young, Eutiquio C.

doi:10.1090/S0002-9939-1985-0792275-6

On integral inequalities of Gronwall-Bellman type
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by Eutiquio C. Young

Proc. Amer. Math. Soc. 94 (1985), 636-640

DOI: https://doi.org/10.1090/S0002-9939-1985-0792275-6

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Abstract:

A sharp bound is given for solutions of an integral inequality of the Gronwall-Bellman type. The bound which is the exact solution of the corresponding integral equation is obtained by reducing the equation to a system of differential equations.

References

J. Abramowich, On Gronwall and Wendroff type inequalities, Proc. Amer. Math. Soc. 87 (1983), no. 3, 481–486. MR 684643, DOI 10.1090/S0002-9939-1983-0684643-6
Paul R. Beesack, Gronwall inequalities, Carleton Mathematical Lecture Notes, No. 11, Carleton University, Ottawa, Ont., 1975. MR 0486735
Jagdish Chandra and B. A. Fleishman, On a generalization of the Gronwall-Bellman lemma in partially ordered Banach spaces, J. Math. Anal. Appl. 31 (1970), 668–681. MR 420363, DOI 10.1016/0022-247X(70)90016-8
Jagdish Chandra and Paul W. Davis, Linear generalizations of Gronwall’s inequality, Proc. Amer. Math. Soc. 60 (1976), 157–160 (1977). MR 422735, DOI 10.1090/S0002-9939-1976-0422735-3
S. C. Chu and F. T. Metcalf, On Gronwall’s inequality, Proc. Amer. Math. Soc. 18 (1967), 439–440. MR 212529, DOI 10.1090/S0002-9939-1967-0212529-0
V. B. Headley, A multidimensional nonlinear Gronwall inequality, J. Math. Anal. Appl. 47 (1974), 250–255. MR 346111, DOI 10.1016/0022-247X(74)90020-1
B. G. Pachpatte, A note on Gronwall-Bellman inequality, J. Math. Anal. Appl. 44 (1973), 758–762. MR 335721, DOI 10.1016/0022-247X(73)90014-0
Donald R. Snow, Gronwall’s inequality for systems of partial differential equations in two independent variables, Proc. Amer. Math. Soc. 33 (1972), 46–54. MR 298188, DOI 10.1090/S0002-9939-1972-0298188-1
Wolfgang Walter, Differential and integral inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 55, Springer-Verlag, New York-Berlin, 1970. Translated from the German by Lisa Rosenblatt and Lawrence Shampine. MR 0271508
D. Willett, A linear generalization of Gronwall’s inequality, Proc. Amer. Math. Soc. 16 (1965), 774–778. MR 181726, DOI 10.1090/S0002-9939-1965-0181726-3
Eutiquio C. Young, Gronwall’s inequality in $n$ independent variables, Proc. Amer. Math. Soc. 41 (1973), 241–244. MR 320493, DOI 10.1090/S0002-9939-1973-0320493-1
A. D. Ziebur, On the Gronwall-Bellman lemma, J. Math. Anal. Appl. 22 (1968), 92–95. MR 223629, DOI 10.1016/0022-247X(68)90162-5