## Differential and Integral Equations

- Differential Integral Equations
- Volume 14, Number 7 (2001), 883-896.

### On solvability of degenerate nonstationary differential-difference equations in Banach spaces

Angelo Favini and Larisa Vlasenko

#### Abstract

The differential problem $$ A(t)u'(t)+B(t)u(t)+M(t)u'(t-\omega)+ N(t)u(t-\omega)=f(t) , \ \ \ t \geq 0, $$ $u(t)=g(t)$, $-\omega \leq t \leq 0,$ where $g$ is a given strongly continuous, $X$-valued function on $[-\omega,0]$, $f$ is strongly continuous from $[0,\infty)$ into $Y$, and $A(t)$, $B(t)$, $M(t)$, $N(t)$ are closed linear operators from the complex Banach space $X$ into the complex Banach space $Y$, is studied. Solvability on $[0,T]$ with $T < \infty$ is considered, too. Moreover, the case where $A(t)$, $B(t)$ are independent of $t$ and $\lambda = 0$ is a multiple pole for $(\lambda B+A)^{-1}$ is investigated.

#### Article information

**Source**

Differential Integral Equations, Volume 14, Number 7 (2001), 883-896.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356123196

**Mathematical Reviews number (MathSciNet)**

MR1828329

**Zentralblatt MATH identifier**

1017.35112

**Subjects**

Primary: 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx]

Secondary: 35G10: Initial value problems for linear higher-order equations 35G15: Boundary value problems for linear higher-order equations 35K65: Degenerate parabolic equations 35R10: Partial functional-differential equations

#### Citation

Favini, Angelo; Vlasenko, Larisa. On solvability of degenerate nonstationary differential-difference equations in Banach spaces. Differential Integral Equations 14 (2001), no. 7, 883--896. https://projecteuclid.org/euclid.die/1356123196