The null space and the range of a convolution operator in a fading memory space

Author:
Olof J. Staffans

Journal:
Trans. Amer. Math. Soc. **281** (1984), 361-388

MSC:
Primary 34K25; Secondary 45F05

DOI:
https://doi.org/10.1090/S0002-9947-1984-0719676-X

MathSciNet review:
719676

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Abstract: We study the convolution equation

, as well as a perturbed version of

, namely

Here

is a

-valued function on

, and

and

are matrix-valued measures. If

and

are supported on

, with

atomic at zero, then

can be regarded as a linear, autonomous, neutral functional differential equation with infinite delay. However, most of the time we do not consider the ordinary Cauchy problem for the neutral equation, i.e. we do not suppose that

and

are supported on

, prescribe an initial condition of the type

, and require

and

to hold only for

. Instead we permit

and

to be of "Fredholm" type, i.e.

and

need not vanish on

, we restrict the growth rate of

and

at plus and minus infinity, and we look at the problem of the existence and uniqueness of solutions of

and

on the whole real line, satisfying conditions like

, where

is a constant, depending on

, and

is a predefined function. Some authors use the word "admissible" when discussing problems of this type. In the case when the homogeneous version of

has nonzero solutions, we decompose the solutions into components with different exponential growth rates, and give a priori bounds on the growth rates of the solutions. As an application of the basic theory, we look at the Cauchy problem for a neutral functional differential equation, and prove the existence of stable and unstable manifolds.

**[1]**
T. A. Burton, *Uniform stabilities of Volterra equations*, J. Differential Equations **36** (1980), 40-53. MR **571126 (81f:45015)**
**[2]**
J. Dieudonné, *Foundations of modern analysis*, Academic Press, New York, 1969. MR **0349288 (50:1782)**
**[3]**
J. K. Hale, *Theory of functional differential equations*, Springer-Verlag, New York, 1977. MR **0508721 (58:22904)**
**[4]**
G. S. Jordan, O. J. Staffans and R. L. Wheeler, *Local analyticity in weighted* -*spaces, and applications to stability problems for Volterra equations*, Trans. Amer. Math. Soc. **274** (1982), 749-782. MR **675078 (83k:45025)**
**[5]**
R. C. MacCamy and J. S. W. Wong, *Exponential stability for a nonlinear functional differential equation*, J. Math. Anal. Appl. **39** (1972), 699-705. MR **0313614 (47:2168)**
**[6]**
G. Seifert, *Almost periodic solutions for delay-differential equations with infinite delay*, J. Differential Equations **41** (1981), 416-425. MR **633826 (83a:34109)**
**[7]**
O. J. Staffans, *A nonlinear Volterra equation with rapidly decaying solutions*, Trans. Amer. Math. Soc. **258** (1980), 523-530. MR **558188 (81b:45023)**
**[8]**
-, *A neutral functional differential equation in a fading memory space*, J. Differential Equations **50** (1983). MR **719446 (85b:34085)**

**[1]**- T. A. Burton,
*Uniform stabilities of Volterra equations*, J. Differential Equations **36** (1980), 40-53. MR **571126 (81f:45015)**
**[2]**- J. Dieudonné,
*Foundations of modern analysis*, Academic Press, New York, 1969. MR **0349288 (50:1782)**
**[3]**- J. K. Hale,
*Theory of functional differential equations*, Springer-Verlag, New York, 1977. MR **0508721 (58:22904)**
**[4]**- G. S. Jordan, O. J. Staffans and R. L. Wheeler,
*Local analyticity in weighted* -*spaces, and applications to stability problems for Volterra equations*, Trans. Amer. Math. Soc. **274** (1982), 749-782. MR **675078 (83k:45025)**
**[5]**- R. C. MacCamy and J. S. W. Wong,
*Exponential stability for a nonlinear functional differential equation*, J. Math. Anal. Appl. **39** (1972), 699-705. MR **0313614 (47:2168)**
**[6]**- G. Seifert,
*Almost periodic solutions for delay-differential equations with infinite delay*, J. Differential Equations **41** (1981), 416-425. MR **633826 (83a:34109)**
**[7]**- O. J. Staffans,
*A nonlinear Volterra equation with rapidly decaying solutions*, Trans. Amer. Math. Soc. **258** (1980), 523-530. MR **558188 (81b:45023)**
**[8]**- -,
*A neutral functional differential equation in a fading memory space*, J. Differential Equations **50** (1983). MR **719446 (85b:34085)**

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DOI:
https://doi.org/10.1090/S0002-9947-1984-0719676-X

Article copyright:
© Copyright 1984
American Mathematical Society