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Multiplicative perturbations of linear Volterra equations

Author: Abdelaziz Rhandi
Journal: Proc. Amer. Math. Soc. 119 (1993), 493-501
MSC: Primary 47N20; Secondary 45D05, 47D03
MathSciNet review: 1169047
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Abstract: We prove that the following problems are essentially equivalent:

\begin{displaymath}\begin{array}{*{20}{c}} {{{[\operatorname{VO} ]}_{CT}}} & {\q... ...uad v(t) = y + \int_0^t {a(t - s)TCv(s)\,ds,} } \\ \end{array} \end{displaymath}

where $ T$ is an unbounded closed linear operator in a Banach space $ X$ with dense domain $ D(T),\;C$ is a bounded linear operator on $ X$, and $ a \in L_{\operatorname{loc} }^1([0,\infty ),\mathbb{R})$, which is exponentially bounded. We give some applications of our abstract theorem to second-order differential operators on the line.

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  • [ArKe] W. Arendt and H. Kellermann, Integrated solutions of Volterra integro-differential equations and applications, Proceeding of the Conference on Volterra Integrodifferential Equations in Banach Spaces and Applications, Trento 1987, Pitman Research Notes Math. Ser., vol. 190, Longman Sci. Tech., Harlow, 1989. MR 1018871 (90h:47068)
  • [Br] H. Brezis, Analyse fonctionnelle, Masson, Paris, 1983. MR 697382 (85a:46001)
  • [DaIa] G. DaPrato and M. Iannelli, Linear integro-differential equations in Banach space, Rend. Sem. Math. Univ. Padova 62 (1980), 207-219. MR 582951 (82c:45021)
  • [DeScha1] W. Desch and W. Schappacher, On relatively bounded perturbations of linear $ {C_0}$-semigroups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), 327-341. MR 764949 (86b:47069)
  • [DeScha2] -, Some generation results for perturbed semigroups, Semigroup Theory and Applications (Trieste, 1987), Lecture Notes in Pure and Appl. Math., vol. 116, Dekker, New York, 1989, pp. 125-152. MR 1009392 (90k:47081)
  • [Fa] H. O. Fattorini, Second order linear differential equations in Banach spaces, Univ. of California at Los Angeles, 1985. MR 797071 (87b:34001)
  • [Ki] J. Kisynski, On cosine operator functions and one-parameter groups of operators, Studia Math. 44 (1972), 93-105. MR 0312328 (47:890)
  • [Na] R. Nagel, One-parameter semigroups of positive operators, Lecture Notes in Math., vol. 1184, Springer-Verlag, Berlin, Heidelberg, and New York, 1986. MR 839450 (88i:47022)
  • [Pr] J. Prüss, Linear Volterra equations in Banach space and applications (to appear).
  • [Rh1] A. Rhandi, Positive perturbations of linear Volterra equations and sine functions of operators, J. Integral Equations Appl. 4 (1992), 409-420. MR 1184715 (93k:47068)
  • [Rh2] -, Perturbations positives des equations d'evolution et applications, Thése de Doctorat de l'Université de Franche-Comté Besançon, France, 1990.
  • [Wa] M. Watanabe, A new proof of the generation theorem of cosine families in Banach spaces, Houston J. Math. 10 (1984), 285-290. MR 744914 (86f:47025)
  • [WaSe] M. Watanabe and H. Serizawa, Perturbation for cosine families in Banach spaces, Houston J. Math. 12 (1986), 117-124. MR 855797 (87k:34104)

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Article copyright: © Copyright 1993 American Mathematical Society

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