On linear Volterra integral equations of convolution type
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- by John S. Lew
- Proc. Amer. Math. Soc. 35 (1972), 450-456
- DOI: https://doi.org/10.1090/S0002-9939-1972-0308699-8
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Erratum: Proc. Amer. Math. Soc. 43 (1974), 490.
Abstract:
Let A be the set of all complex-valued locally integrable functions defined on $[0, + \infty )$, and let T be the topology for A determined by the seminorms ${t_r}(f) = \smallint _0^r {|f(x)|dx}$ for $r = 1,2, \cdots$ , so that A is a topological algebra under pointwise addition, complex scalar multiplication, and Laplace convolution. Then the map $f \to f’$ from each element to its quasi-inverse is a homeomorphism of (A, T) onto itself. For each f, g in A the equation $v = f + g ^\ast v$ has a unique solution in A which depends T-continuously on f, g, and is the T-limit of Picard approximations. The set of all f in A with $f’$ in ${L^1}[0, + \infty )$ is a set of first category in (A, T) but an open subset of A with the metric ${\left \| {f - g} \right \|_1}$. For each series $\sum \nolimits _{n = 1}^\infty {{p_n}{z^n}}$ converging in some neighborhood of $z = 0$, and each element f in A, the series $\sum \nolimits _{n = 1}^\infty {{p_n}{f^{ \ast n}}}$ converges in T to some element ${p^ \ast }(f)$ in A.References
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
- Sandy Grabiner, The use of formal power series to solve finite convolution integral equations, J. Math. Anal. Appl. 30 (1970), 415–419. MR 259515, DOI 10.1016/0022-247X(70)90172-1
- Gregers Krabbe, Ratios of Laplace transforms, Mikusiński operational calculus, Math. Ann. 162 (1965/66), 237–245. MR 188729, DOI 10.1007/BF01360913 V. Lovitt, Linear integral equations, McGraw-Hill, New York, 1924.
- Jan Mikusiński, Operational calculus, International Series of Monographs on Pure and Applied Mathematics, Vol. 8, Pergamon Press, New York-London-Paris-Los Angeles; Państwowe Wydawnictwo Naukowe, Warsaw, 1959. MR 0105594
- R. K. Miller, On Volterra integral equations with nonnegative integrable resolvents, J. Math. Anal. Appl. 22 (1968), 319–340. MR 227707, DOI 10.1016/0022-247X(68)90176-5
- Richard K. Miller and Alan Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math. Anal. 2 (1971), 242–258. MR 287258, DOI 10.1137/0502022 A. Naĭmark, Normed rings, GITTL, Moscow, 1956; English transl., Noordhoff, Groningen, 1959. MR 19, 870; MR 22 #1824.
- Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
- F. G. Tricomi, Integral equations, Pure and Applied Mathematics, Vol. V, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957. MR 0094665
- Lucien Waelbroeck, Le calcul symbolique dans les algèbres commutatives, C. R. Acad. Sci. Paris 238 (1954), 556–558 (French). MR 73950
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 450-456
- MSC: Primary 45D05
- DOI: https://doi.org/10.1090/S0002-9939-1972-0308699-8
- MathSciNet review: 0308699