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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The solution of a nonlinear Gronwall inequality


Author: Burrell W. Helton
Journal: Proc. Amer. Math. Soc. 38 (1973), 337-342
DOI: https://doi.org/10.1090/S0002-9939-1973-0310185-7
MathSciNet review: 0310185
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Abstract: This paper extends some of the earlier results of J. V. Herod, W. W. Schmaedeke and G. R. Sell, and B. W. Helton and shows that, under the given conditions,

(1) there is a function $ u$ satisfying the inequality

$\displaystyle f(x) \leqq h(x) + (RL)\int_a^x {(fG + fH)} $

such that, if $ f$ satisfies the given inequality, then $ f(x) \leqq u(x)$; and

(2) there is a function $ u$ satisfying the inequality

$\displaystyle 0 < f(x) \leqq k + (RL)\int_a^x {[(f{G_1} + {f^{pn}}{G_2}) + (f{H_1} + {f^{pn}}{H_2})],} $

where $ n$ is a positive integer and $ p = \pm 1$ and $ pn \ne 1$, such that, if $ f$ satisfies the given inequality, then $ f(x) \leqq u(x)$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0310185-7
Keywords: Integrals, product integrals, Gronwall inequality, integral equations
Article copyright: © Copyright 1973 American Mathematical Society