Exponential dichotomies for linear differential equations with delays: Periodic and autonomous equations. (English) Zbl 0559.34071
The author considers a homogeneous linear differential equation with delays in a Banach space with a so-called ”memory with bounded recall” under natural conditions. The presence of an exponential dichotomy is examined when the equation is inhomogeneous and periodic. The autonomous case is also detailed.
Reviewer: W.Muniz Oliva
MSC:
| 34K20 | Stability theory of functional-differential equations |
| 34A30 | Linear ordinary differential equations and systems |
Keywords:
homogeneous linear differential equation with delays; Banach space; memory with bounded recall; exponential dichotomy; autonomous caseReferences:
| [1] | Coffman, C. V.; Schäffer, J. J., Dichotomies for linear difference equations, Math. Ann., 172, 139-166 (1967) · Zbl 0189.40303 |
| [2] | Coffman, C. V.; Schäffer, J. J., Linear differential equations with delays. Existence, uniqueness, growth, and compactness under natural Carathéodory conditions, J. Differential Equations, 16, 26-44 (1974) · Zbl 0258.34059 |
| [3] | Coffman, C. V.; Schäffer, J. J., The structure of translation-invariant memories, J. Differential Equations, 16, 428-459 (1974) · Zbl 0266.34077 |
| [4] | Coffman, C. V.; Johnson, B. E.; Schäffer, J. J., The structure of translation-invariant memories: two addenda, J. Differential Equations, 20, 270-281 (1976) · Zbl 0336.34061 |
| [5] | Edwards, R. E., Fourier Series (1967), New York: Holt Rinehart and Winston, New York · Zbl 0189.06602 |
| [6] | Hille, E.; Phillips, R. S., Functional Analysis and Semi-groups, American Mathematical Society Colloquium Publications, Vol. 31 (1957), Providence: American Mathematical Society, Providence · Zbl 0078.10004 |
| [7] | Massera, J. L.; Schäffer, J. J., Linear differential equations and function spaces (1966), New York: Academic Press, New York · Zbl 0202.14701 |
| [8] | Riesz, F.; Sz.-Nagy, B., Functional Analysis (1955), New York: F. Ungar, New York |
| [9] | Schäffer, J. J., Ecuaciones diferenciales lineales con coeficientes constantes en espacios de Banach, Univ. Repúbl. Fac. Ingen. Agrimens. Montevideo Publ. Inst. Mat. Estadíst., 3, 105-110 (1958) · Zbl 0084.34202 |
| [10] | Schäffer, J. J., Dichotomies for linear differential equations with delays: the Carathéodory case, Ann. Mat. Pura Appl., (4), 104, 297-311 (1975) · Zbl 0315.34084 |
| [11] | Schäffer, J. J., On the structure of translation-invariant core-memories, J. Differential Equations, 37, 225-237 (1980) · Zbl 0479.34025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
