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Regularized solutions for abstract Volterra equations. (English) Zbl 0952.45005
The author shows the existence, the uniqueness, and some qualitative properties of solutions for the abstract Volterra equation $$u(t) = f(t) + \int_{0}^{t}a(t-s)Au(s) ds,\quad t\in [0,T]$$ on a complex Banach space $X$ by means of an extended notion of resolvent, where $A$ is a closed linear unbounded operator with domain $D(A)$, $a \not=0$ is a scalar kernel, and $f \in C([0,T],X)$.

45N05Abstract integral equations, integral equations in abstract spaces
45D05Volterra integral equations
47D03(Semi)groups of linear operators
Full Text: DOI
[1] Arendt, W.: Vector valued Laplace transforms and Cauchy problems. Israel J. Math. 59, 327-352 (1987) · Zbl 0637.44001
[2] Arendt, W.; Kellermann, H.: Integrated solutions of Volterra integrodifferential equations and applications. (1987) · Zbl 0675.45017
[3] B. Bäumer, and, F. Neubrander, Existence and uniqueness of solutions of ordinary linear differential equations in Banach spaces, preprint. [See, http://math.lsu.edu/Tiger, Notes: Preprints in Evolution Equations and Related Fields of Analysis.]
[4] Chang, J. C.; Shaw, S. Y.: Rates of approximation and ergodic limits of resolvent families. Arch. math. 66, 320-330 (1996) · Zbl 0859.47027
[5] Cioranescu, I.; Lumer, G.: On $K(t)$-convoluted semigroups. (1995)
[6] El-Mennaoui, O.; Keyantuo, V.: Trace theorems for holomorphic semigroups and the second order Cauchy problem. Proc. amer. Math. soc. 124, 1445-1458 (1996) · Zbl 0852.47017
[7] Hieber, M.: Laplace transforms and ${\alpha}$-times integrated semigroups. Forum math. 3, 595-612 (1991) · Zbl 0766.47013
[8] Hieber, M.: Integrated semigroups and differential operators on lp spaces. Math. ann., 1-16 (1991) · Zbl 0724.34067
[9] Hieber, M.: Lp spectra of pseudodifferential operators generating integrated semigroups. Trans. amer. Math. soc. 347, 4023-4035 (1995) · Zbl 0847.47027
[10] Kellermann, H.; Hieber, M.: Integrated semigroups. J. funct. Anal. 84, 160-180 (1989) · Zbl 0689.47014
[11] M. Kim, Abstract Volterra Equations, Ph.D. thesis, Louisiana State University, Baton Rouge, 1995.
[12] Lizama, C.: On an extension of the Trotter--Kato theorem for resolvent families of operators. J. integral equations appl. 2, 269-280 (1990) · Zbl 0739.47016
[13] Lizama, C.: On Volterra equations associated with a linear operator. Proc. amer. Math. soc. 118, 1159-1166 (1993) · Zbl 0781.45013
[14] Lizama, C.: A mean ergodic theorem for resolvent operators. Semigroup forum 47, 227-230 (1993) · Zbl 0799.47024
[15] Oka, H.: Integrated resolvent operators. J. integral equations appl. 7, 193-232 (1995) · Zbl 0846.45005
[16] Oka, H.: Linear Volterra equations and integrated solution families. Semigroup forum 53, 278-297 (1996) · Zbl 0862.45017
[17] Prüss, J.: Positivity and regularity of hyperbolic Volterra equations in Banach spaces. Math. ann. 279, 317-344 (1987) · Zbl 0608.45007
[18] Prüss, J.: Evolutionary integral equations and applications. Monographs in mathematics 87 (1993) · Zbl 0784.45006
[19] Rhandi, A.: Multiplicative perturbations of linear Volterra equations. Proc. amer. Math. soc. 119, 493-501 (1993) · Zbl 0791.45005
[20] Srivastava, H. M.; Buschman, R. G.: Theory and applications of convolution integral equations. (1992) · Zbl 0755.45002