The solution of a nonlinear Gronwall inequality

Helton, Burrell W.

doi:10.1090/S0002-9939-1973-0310185-7

The solution of a nonlinear Gronwall inequality
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Proc. Amer. Math. Soc. 38 (1973), 337-342 Request permission

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 \[ f(x) \leqq h(x) + (RL)\int _a^x {(fG + fH)} \] such that, if $f$ satisfies the given inequality, then $f(x) \leqq u(x)$; and (2) there is a function $u$ satisfying the inequality \[ 0 < f(x) \leqq k + (RL)\int _a^x {[(f{G_1} + {f^{pn}}{G_2}) + (f{H_1} + {f^{pn}}{H_2})],} \] where $n$ is a positive integer and $p = \pm 1$ and $pn \ne 1$, such that, if $f$ satisfies the given inequality, then $f(x) \leqq u(x)$.

References

Burrell W. Helton, Integral equations and product integrals, Pacific J. Math. 16 (1966), 297–322. MR 188731
Burrell W. Helton, A product integral representation for a Gronwall inequality, Proc. Amer. Math. Soc. 23 (1969), 493–500. MR 248310, DOI 10.1090/S0002-9939-1969-0248310-8
J. V. Herod, A Gronwall inequality for linear Stieltjes integrals, Proc. Amer. Math. Soc. 23 (1969), 34–36. MR 249557, DOI 10.1090/S0002-9939-1969-0249557-7
Wayne W. Schmaedeke and George R. Sell, The Gronwall inequality for modified Stieltjes integrals, Proc. Amer. Math. Soc. 19 (1968), 1217–1222. MR 230864, DOI 10.1090/S0002-9939-1968-0230864-8

Additional Information

Journal: Proc. Amer. Math. Soc. 38 (1973), 337-342
DOI: https://doi.org/10.1090/S0002-9939-1973-0310185-7
MathSciNet review: 0310185