Differential and Integral Equations

Singular differential equations with delay

Angelo Favini, Luciano Pandolfi, and Hiroki Tanabe

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The differential equation $d(Mu(t))/dt=-Lu(t)+L_1u(t-1), $ $t\geq0, $ $u(t)=\varphi(t),$ $ -1\leq t\leq0$, for a given strongly continuous $X$-valued function $\varphi$ on $[-1,0]$ is studied, where $M, L, L_1$ are closed linear operators from the complex Banach space $X$ into itself, and $L$ is invertible. Though already in the finite dimensional case in general existence of continuous solutions on $[-1,\infty)$ may fail or it is possible to have continuous solutions only on a finite interval, we indicate classes of operators for which existence results analogous to the ones for regular equations $M=I$ hold. In particular, solutions are given explicitly by a recovery formula.

Article information

Differential Integral Equations, Volume 12, Number 3 (1999), 351-371.

First available in Project Euclid: 29 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx]
Secondary: 35R10: Partial functional-differential equations 47N20: Applications to differential and integral equations


Favini, Angelo; Pandolfi, Luciano; Tanabe, Hiroki. Singular differential equations with delay. Differential Integral Equations 12 (1999), no. 3, 351--371. https://projecteuclid.org/euclid.die/1367265216

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