On Volterra equations associated with a linear operator

On Volterra equations associated with a linear operator
HTML articles powered by AMS MathViewer

by Carlos Lizama PDF

Proc. Amer. Math. Soc. 118 (1993), 1159-1166 Request permission

In this work we define the Hille-Yosida space, in the sense of S. Kantorovitz, for a Volterra equation of convolution type.

References

P. L. Butzer, Die Anwendung des Operatorenkalküls von Jan Mikusiński auf lineare Integralgleichungen vom Faltungstypus, Arch. Rational Mech. Anal. 2 (1958), 114–128 (German). MR 104112, DOI 10.1007/BF00277923
Ioana Cioranescu, On the second order Cauchy problem associated with a linear operator, J. Math. Anal. Appl. 154 (1991), no. 1, 238–242. MR 1087971, DOI 10.1016/0022-247X(91)90083-C
G. Da Prato and M. Iannelli, Linear integro-differential equations in Banach spaces, Rend. Sem. Mat. Univ. Padova 62 (1980), 207–219. MR 582951

Laplace and Laplace-Stieltjes space

H. O. Fattorini, Second order linear differential equations in Banach spaces, North-Holland Mathematics Studies, vol. 108, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 99. MR 797071
R. Grimmer and J. Prüss, On linear Volterra equations in Banach spaces, Comput. Math. Appl. 11 (1985), no. 1-3, 189–205. Hyperbolic partial differential equations, II. MR 787436, DOI 10.1016/0898-1221(85)90146-4
Hernán R. Henríquez and Eduardo A. Hernández, On the abstract Cauchy problem in Fréchet spaces, Proc. Amer. Math. Soc. 115 (1992), no. 2, 353–360. MR 1081092, DOI 10.1090/S0002-9939-1992-1081092-1
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Shmuel Kantorovitz, The Hille-Yosida space of an arbitrary operator, J. Math. Anal. Appl. 136 (1988), no. 1, 107–111. MR 972586, DOI 10.1016/0022-247X(88)90118-7

On Hadamard correctness of the Cauchy problem for the equation of evolution

11

C. Lizama, On an extension of the Trotter-Kato theorem for resolvent families of operators, J. Integral Equations Appl. 2 (1990), no. 2, 269–280. MR 1045773, DOI 10.1216/JIE-1990-2-2-263
Carlos Lizama, A characterization of periodic resolvent operators, Results Math. 18 (1990), no. 1-2, 93–105. MR 1066645, DOI 10.1007/BF03323157

Uniform continuity and compactness of resolvent families of operators

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Jan Prüss, Positivity and regularity of hyperbolic Volterra equations in Banach spaces, Math. Ann. 279 (1987), no. 2, 317–344. MR 919509, DOI 10.1007/BF01461726