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
DOI:
https://doi.org/10.1090/S0002-9947-1976-0407215-8
MathSciNet review:
0407215
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: This paper considers a special integral 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})\} ,} $](images/img2.gif)




![$ A(p,q) = [f(q) - f(p)][g(q) - g(p)],{A^ - }(p,q) = A({q^ - },q)$](images/img7.gif)

- [1] S. C. Chu and F. T. Metcalf, On Gronwall's inequality, Proc. Amer. Math. Soc. 18 (1967), 439-440. MR 35 #3400. MR 0212529 (35:3400)
- [2] Ben Dushnik, On the Stieltjes integral, Dissertation, University of Michigan, 1931.
- [3] B. W. Helton, Integral equations and product integrals, Pacific J. Math. 16 (1966), 297-322. MR 32 #6167. MR 0188731 (32:6167)
- [4] B. W. Helton, A product integral representation for a Gronwall inequality, Proc. Amer. Math. Soc. 23 (1969), 493-500. MR 40 #1562. MR 0248310 (40:1562)
- [5] -, The solution of a nonlinear Gronwall inequality, Proc. Amer. Math. Soc. 38 (1973), 337-342. MR 46 #9287. MR 0310185 (46:9287)
- [6] J. V. Herod, A Gronwall inequality for linear Stieltjes integrals, Proc. Amer. Math. Soc. 23 (1969), 34-36. MR 40 #2802. MR 0249557 (40:2802)
- [7] J. R. Kroll and K. 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 45 #833. MR 0291742 (45:833)
- [8] W. W. Schmaedeke and G. R. Sell, The Gronwall inequality for modified Stieltjes integrals, Proc. Amer. Math. Soc. 19 (1968), 1217-1222. MR 37 #6422. MR 0230864 (37:6422)
- [9] H. L. Smith, On the existence of the Stieltjes integral, Trans. Amer. Math. Soc. 27 (1925), 491-515. MR 1501324
- [10] D. R. Snow, Gronwall's inequality for systems of partial differential equations in two independent variables, Proc. Amer. Math. Soc. 33 (1972), 46-54. MR 45 #7240. MR 0298188 (45:7240)
- [11] D. W. Willett, A linear generalization of Gronwall's inequality, Proc. Amer. Math. Soc. 16 (1965), 774-778. MR 31 #5953. MR 0181726 (31:5953)
- [12] D. W. Willett and J. S. W. Wong, On the discrete analogues of some generalizations of Gronwall's inequality, Monatsh. Math. 69 (1965), 362-367. MR 32 #2644. MR 0185175 (32:2644)
- [13] F. M. Wright and J. D. Baker, On integration-by-parts for weighted integrals, Proc. Amer. Math. Soc. 22 (1969), 42-52. MR 39 #7056. MR 0245750 (39:7056)
- [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 44 #380. MR 0283147 (44:380)
- [15] Yue-sheng Li, The bound, stability and error estimates for the solution of nonlinear differential equations, Acta Math. Sinica 12 (1962), 32-39 = Chinese Math. 3 (1963), 34-41. MR 27 #405. MR 0150406 (27:405)
Retrieve articles in Transactions of the American Mathematical Society with MSC: 26A42, 26A86
Retrieve articles in all journals with MSC: 26A42, 26A86
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1976-0407215-8
Keywords:
Integrals,
Gronwall's inequality,
product integrals,
integration-by-parts,
bounded variation,
Smith mean integral,
Cauchy integrals,
Dushkin interior integral,
subdivision-refinement-limit
Article copyright:
© Copyright 1976
American Mathematical Society