Embedding Stieltjes-Volterra integral equations in Stieltjes integral equations

Gibson, William L.

doi:10.1090/S0002-9947-1977-0430695-X

Embedding Stieltjes-Volterra integral equations in Stieltjes integral equations
HTML articles powered by AMS MathViewer

by William L. Gibson PDF

Trans. Amer. Math. Soc. 227 (1977), 263-277 Request permission

Abstract:

J. A. Reneke has shown that the linear Stieltjes-Volterra integral equations studied by D. B. Hinton can be transformed into Stieltjes integral equations of the type studied by J. S. Mac Nerney. By taking advantage of the nonlinear nature of Mac Nerney’s results, Reneke was able to extend Hinton’s existence theorem to a nonlinear setting. In this paper, we use Reneke’s embedding technique to generalize several other of Hinton’s results, and we characterize completely, in the linear case, the range of Reneke’s embedding transformation.

References

D. B. Hinton, A Stieltjes-Volterra integral equation theory, Canadian J. Math. 18 (1966), 314–331. MR 188733, DOI 10.4153/CJM-1966-035-3
Alvin J. Kay, Nonlinear integral equations and product integrals, Pacific J. Math. 60 (1975), no. 1, 203–222. MR 399781, DOI 10.2140/pjm.1975.60.203
J. S. MacNerney, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148–173. MR 144179
J. S. MacNerney, A linear initial-value problem, Bull. Amer. Math. Soc. 69 (1963), 314–329. MR 146613, DOI 10.1090/S0002-9904-1963-10905-2
J. S. MacNerney, Note on successive approximations, Rend. Circ. Mat. Palermo (2) 12 (1963), 87–90. MR 166594, DOI 10.1007/BF02855910
J. S. MacNerney, A nonlinear integral operation, Illinois J. Math. 8 (1964), 621–638. MR 167815
James A. Reneke, A product integral solution of a Stieltjes-Volterra integral equation, Proc. Amer. Math. Soc. 24 (1970), 621–626. MR 252999, DOI 10.1090/S0002-9939-1970-0252999-5
James A. Reneke, Product integral solutions for hereditary systems, Trans. Amer. Math. Soc. 181 (1973), 483–493. MR 324340, DOI 10.1090/S0002-9947-1973-0324340-8