A variation-of-parameters inequality
Author:
David Lowell Lovelady
Journal:
Proc. Amer. Math. Soc. 26 (1970), 598-602
MSC:
Primary 34D05
DOI:
https://doi.org/10.1090/S0002-9939-1970-0435526-9
MathSciNet review:
0435526
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Abstract | References | Similar Articles | Additional Information
Abstract: Recent work in stability theory for linear ordinary differential equations has made much use of the inequalities which can be deduced from the variation-of-parameters formula. In this article it is shown that similar inequalities hold for nonlinear Stieltjes integral equations and hence, as a special case, for nonlinear differential equations in which the nonlinearity is of Lipschitz type.
- W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Co., Boston, Mass., 1965. MR 0190463
- Thomas G. Hallam, On the asymptotic growth of the solutions of a system of nonhomogeneous linear differential equations, J. Math. Anal. Appl. 25 (1969), 254–265. MR 234069, DOI https://doi.org/10.1016/0022-247X%2869%2990228-5
- Thomas G. Hallam, On asymptotic equivalence of the bounded solutions of two systems of differential equations, Michigan Math. J. 16 (1969), 353–363. MR 252766
- D. B. Hinton, A Stieltjes-Volterra integral equation theory, Canadian J. Math. 18 (1966), 314–331. MR 188733, DOI https://doi.org/10.4153/CJM-1966-035-3
- Ralph E. Lane, The integral of a function with respect to a function, Proc. Amer. Math. Soc. 5 (1954), 59–66. MR 59346, DOI https://doi.org/10.1090/S0002-9939-1954-0059346-3
- J. S. MacNerney, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148–173. MR 144179
- J. S. MacNerney, A nonlinear integral operation, Illinois J. Math. 8 (1964), 621–638. MR 167815
- R. H. Martin Jr., A bound for solutions of Volterra-Stieltjes integral equations, Proc. Amer. Math. Soc. 23 (1969), 506–512. MR 247394, DOI https://doi.org/10.1090/S0002-9939-1969-0247394-0 ---, Bounds for solutions to a class of nonlinear integral equations, (submitted).
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Additional Information
Keywords:
Bounds,
nonhomogeneous perturbations,
stability,
Stieltjes integral equations,
variation-of-parameters
Article copyright:
© Copyright 1970
American Mathematical Society