A special integral and a Gronwall inequality

Helton, Burrell W.

doi:10.1090/S0002-9947-1976-0407215-8

A special integral and a Gronwall inequality
HTML articles powered by AMS MathViewer

by Burrell W. Helton PDF

Trans. Amer. Math. Soc. 217 (1976), 163-181 Request permission

Abstract:

This paper considers a special integral $(I)\smallint _a^b(fdg + H)$ which is a subdivision-refinement-type limit of the approximating sum \[ \sum \limits _1^n {\{ f({t_i})[g({x_i}) - g({x_{i - 1}})] + H({x_{i - 1}},{x_i})\} ,} \] where ${x_{i - 1}} < {t_i} < {x_i}$. The author shows, with appropriate restrictions, that $(I)\smallint _a^b(fdg + H)$ exists if and only if \[ (R)\smallint _x^y(fdg + H - {A^ - }) = (L)\smallint _x^y(fdg + H + {A^ + })\] for $a \leqslant x < y \leqslant b$, where $A(p,q) = [f(q) - f(p)][g(q) - g(p)],{A^ - }(p,q) = A({q^ - },q)$ and ${A^ + }(p,q) = A(p,{p^ + })$. Furthermore, if either of the equivalent statements is true, then all the integrals are equal. These equivalent statements are used to prove an integration-by-parts theorem and to solve a Gronwall inequality involving this special integral. Product integrals are used in the solution of the Gronwall inequality.

References

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

On the Stieltjes integral

Burrell W. Helton, Integral equations and product integrals, Pacific J. Math. 16 (1966), 297–322. MR 188731, DOI 10.2140/pjm.1966.16.297
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
Burrell W. Helton, The solution of a nonlinear Gronwall inequality, Proc. Amer. Math. Soc. 38 (1973), 337–342. MR 310185, DOI 10.1090/S0002-9939-1973-0310185-7
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
Jeffrey R. Kroll and Keith P. Smith, An eigenvalue problem for the Stieltjes mean sigma-integral related to Gronwall’s inequality, Proc. Amer. Math. Soc. 33 (1972), 384–388. MR 291742, DOI 10.1090/S0002-9939-1972-0291742-2
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
H. L. Smith, On the existence of the Stieltjes integral, Trans. Amer. Math. Soc. 27 (1925), no. 4, 491–515. MR 1501324, DOI 10.1090/S0002-9947-1925-1501324-5
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
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
D. Willett and J. S. W. Wong, On the discrete analogues of some generalizations of Gronwall’s inequality, Monatsh. Math. 69 (1965), 362–367. MR 185175, DOI 10.1007/BF01297622
Fred M. Wright and James D. Baker, On integration-by-parts for weighted integrals, Proc. Amer. Math. Soc. 22 (1969), 42–52. MR 245750, DOI 10.1090/S0002-9939-1969-0245750-8
F. M. Wright, M. L. Klasi, and D. R. Kennebeck, The Gronwall inequality for weighted integrals, Proc. Amer. Math. Soc. 30 (1971), 504–510. MR 283147, DOI 10.1090/S0002-9939-1971-0283147-4
Yue-sheng Li, The bound, stability and error estimates for the solution of nonlinear differential equations, Chinese Math. 3 (1962), 34–41. MR 150406