A special integral and a Gronwall inequality

Helton, Burrell W.

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

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

A special integral and a Gronwall inequality

Author: Burrell W. Helton
Journal: Trans. Amer. Math. Soc. 217 (1976), 163-181
MSC: Primary 26A42; Secondary 26A86
MathSciNet review: 0407215
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper considers a special integral $(I)\smallint _a^b(fdg + H)$ which is a subdivision-refinement-type limit of the approximating sum

$\displaystyle \sum\limits_1^n {\{ f({t_i})[g({x_i}) - g({x_{i - 1}})] + H({x_{i - 1}},{x_i})\} ,}$

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

$\displaystyle (R)\smallint _x^y(fdg + H - {A^ - }) = (L)\smallint _x^y(fdg + H + {A^ + })$

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.

[1] S. C. Chu and F. T. Metcalf, On Gronwall’s inequality, Proc. Amer. Math. Soc. 18 (1967), 439–440. MR 0212529 (35 #3400), http://dx.doi.org/10.1090/S0002-9939-1967-0212529-0
[2] Ben Dushnik, On the Stieltjes integral, Dissertation, University of Michigan, 1931.
[3] Burrell W. Helton, Integral equations and product integrals, Pacific J. Math. 16 (1966), 297–322. MR 0188731 (32 #6167)
[4] Burrell W. Helton, A product integral representation for a Gronwall inequality, Proc. Amer. Math. Soc. 23 (1969), 493–500. MR 0248310 (40 #1562), http://dx.doi.org/10.1090/S0002-9939-1969-0248310-8
[5] Burrell W. Helton, The solution of a nonlinear Gronwall inequality, Proc. Amer. Math. Soc. 38 (1973), 337–342. MR 0310185 (46 #9287), http://dx.doi.org/10.1090/S0002-9939-1973-0310185-7
[6] J. V. Herod, A Gronwall inequality for linear Stieltjes integrals, Proc. Amer. Math. Soc. 23 (1969), 34–36. MR 0249557 (40 #2802), http://dx.doi.org/10.1090/S0002-9939-1969-0249557-7
[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 0291742 (45 #833), http://dx.doi.org/10.1090/S0002-9939-1972-0291742-2
[8] Wayne W. Schmaedeke and George R. Sell, The Gronwall inequality for modified Stieltjes integrals, Proc. Amer. Math. Soc. 19 (1968), 1217–1222. MR 0230864 (37 #6422), http://dx.doi.org/10.1090/S0002-9939-1968-0230864-8
[9] H. L. Smith, On the existence of the Stieltjes integral, Trans. Amer. Math. Soc. 27 (1925), no. 4, 491–515. MR 1501324, http://dx.doi.org/10.1090/S0002-9947-1925-1501324-5
[10] 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 0298188 (45 #7240), http://dx.doi.org/10.1090/S0002-9939-1972-0298188-1
[11] D. Willett, A linear generalization of Gronwall’s inequality, Proc. Amer. Math. Soc. 16 (1965), 774–778. MR 0181726 (31 #5953), http://dx.doi.org/10.1090/S0002-9939-1965-0181726-3
[12] 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 0185175 (32 #2644)
[13] Fred M. Wright and James D. Baker, On integration-by-parts for weighted integrals, Proc. Amer. Math. Soc. 22 (1969), 42–52. MR 0245750 (39 #7056), http://dx.doi.org/10.1090/S0002-9939-1969-0245750-8
[14] 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 0283147 (44 #380), http://dx.doi.org/10.1090/S0002-9939-1971-0283147-4
[15] Yue-sheng Li, The bound, stability and error estimates for the solution of nonlinear differential equations, Chinese Math. 3 (1962), 34–41. MR 0150406 (27 #405)

Bibliography