More generalised discrete Gronwall inequalities. (English) Zbl 0578.26004
Summary: The most general linear discrete Gronwall inequality is first considered, and best possible solutions of the inequality are obtained by elementary methods which are discrete analogues of those used for Volterra integral equations. Upper bounds are also obtained for solutions in case the kernel is of a generalised Abel type, improving and extending recent work of S. McKee [Z. Angew. Math. Mech. 62, 429-434 (1982; Zbl 0524.26013)].
MSC:
| 26D10 | Inequalities involving derivatives and differential and integral operators |
| 39A12 | Discrete version of topics in analysis |
| 45D05 | Volterra integral equations |
Citations:
Zbl 0524.26013References:
| [1] | Gronwall Inequalities, Carleton Mathematical Lecture Notes No. 11, May 1975. |
| [2] | Chandra, J. Math. Anal. Appl. 31 pp 668– (1970) |
| [3] | Hull, Numer. Math. 2 pp 30– (1960) |
| [4] | Jones, SIAM Journ. 12 pp 43– (1964) |
| [5] | Some results for Abel-Volterra integral equations of the second kind, in: ; (eds.), Treatment of Integral Equations by Numerical Methods, Academic Press, London, New York 1982, pp. 273–282. |
| [6] | Some results for Abel-Volterra integral equations of the second kind, in: ; (eds.), Treatment of Integral Equations by Numerical Methods, Academic Press, London, New York 1982, pp. 459–461. |
| [7] | Lees, SIAM Journ. 7 pp 167– (1959) |
| [8] | Lees, Trans. Amer. Math. Soc. 94 pp 58– (1960) |
| [9] | On the stability of linear multistep methods for Volterra integral equations of the second kind, in: ; (eds.), Treatment of Integral Equations by Numerical Methods, Academic Press, London, New York 1982, pp. 233–238. |
| [10] | McKee, ZAMM 62 pp 429– (1982) |
| [11] | A review of linear multistep methods and product integration methods and their convergence analysis for first kind Voltera integral equations, in: ; (eds.), Treatment of Equations by Numerical Methods, Academic Press, London, New York 1982, pp. 153–161. |
| [12] | Analytic Inequalities, Springer, Berlin-Heidelberg-New York 1970. · Zbl 0199.38101 · doi:10.1007/978-3-642-99970-3 |
| [13] | The numerical solution of nonlinear integral equàtions and relateds topics, in: (ed.), Nonlinear Integral Equaltions, University of Wisconsion Press, Madiscon 1964, pp. 215–318. |
| [14] | Pachpatte, J. Indian Math. Soc. 37 pp 147– (1973) |
| [15] | Popenda, Fasc. Math. 11 pp 143– (1979) |
| [16] | Ordinary Differential Equations, Wiley, New York-London 1971. |
| [17] | Sugiyama, Bull. Sci. Engr. Research Lab. Waseda Univ. 45 pp 140– (1969) |
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