Differential and Integral Equations

Nonautonomous integro-differential equations of hyperbolic type

Hirokazu Oka and Naoki Tanaka

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Abstract

This paper is devoted to two problems for the nonautonomous integrodifferential equation in a Banach space $X$ of hyperbolic type $$ \text{$u'(t) = A(t)u(t)+ \int_0^t B(t,s)u(s) ds + f(t)$ for $t\in[0,T]$, and $u(0) = u_0$.} $$ One is the problem of existence and uniqueness of classical solutions without assuming that the common domain of $A(t)$ is dense in $X$, and the other is the regularity problem in the case where the common domain of $A(t)$ is dense in $X$. The regularity result will play a role in developing an abstract theory which can be applied to the second-order integrodifferential equations with the third kind boundary conditions.

Article information

Source
Differential Integral Equations, Volume 8, Number 7 (1995), 1823-1831.

Dates
First available in Project Euclid: 12 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1368397760

Mathematical Reviews number (MathSciNet)
MR1347983

Zentralblatt MATH identifier
0826.45006

Subjects
Primary: 45N05: Abstract integral equations, integral equations in abstract spaces
Secondary: 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx]

Citation

Oka, Hirokazu; Tanaka, Naoki. Nonautonomous integro-differential equations of hyperbolic type. Differential Integral Equations 8 (1995), no. 7, 1823--1831. https://projecteuclid.org/euclid.die/1368397760


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