## Differential and Integral Equations

### A variation of constants formula for an abstract functional-differential equation of retarded type

#### Abstract

In this paper we obtain a variation of constants formula for the Cauchy problem associated to the nonhomogeneous retarded functional differential equation $$x'(t)=L(x_t)+f(t), \quad t\geq 0 , \tag{*}$$ where $L:C([-r,0];E)\longrightarrow E$ is a bounded linear operator and $E$ is a Banach space. Once the existence and uniqueness theorem is proved, we define a fundamental solution associated with this problem and we use it to derive a variation of constants formula. The known results on the decomposition of the space $C([-r,0];E)$ with respect to the homogeneous equation $x'(t)=L(x_t)$ allow us to obtain a similar decomposition for the solutions of $(*)$. Moreover, the projection of solutions on a certain invariant finite-dimensional subspace is characterized in terms of the semigroup defined by the homogeneous problem.

#### Article information

Source
Differential Integral Equations, Volume 9, Number 6 (1996), 1305-1320.

Dates
First available in Project Euclid: 6 May 2013

https://projecteuclid.org/euclid.die/1367846903

Mathematical Reviews number (MathSciNet)
MR1409930

Zentralblatt MATH identifier
0879.34072

Subjects
Primary: 34K15