This work is concerned with differential equations with delays, on [0, ∞[, of the form\(\dot u\)+Mu=r, and the corresponding homogeneous equation\(\dot u\)+Mu=0, where u and r take values in a Banach space E, and M is a « memory ». The main results relate the existence of dichotomies or exponential dichotomies of the solutions of the homogeneous equation (a kind of conditional stability) to the admissibility of certain pairs of function spaces for the inhomogeneous equation. The « memory » M is quite general: it is a linear mapping that takes continuous E-valued functions on [−1, ∞[ into locally integrable functions on [0, ∞[ in such a way that Mu on [a, b] ⊂ [0, ∞[ depends only on u on [a −1, b], and satisfies a reasonable boundedness condition.
Differential Equation Banach Space Linear Mapping Function Space Integrable Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
C. V. Coffman - J. J. Schäffer,Linear differential equations with delays: admissibility and conditional stability, Department of Mathematics, Carnegie-Mellon University, Report 70-2, Pittsburgh, Pennsylvania, 1970.Google Scholar
C. V. Coffman -J. J. Schäffer,Linear differential equations with delays: admissibility and conditional exponential stability, J. Differential Equations,9 (1971), pp. 521–535.CrossRefMathSciNetGoogle Scholar
C. V. Coffman -J. J. Schäffer,Linear differential equations with delays: existence, uniqueness, growth, and compactness under natural Carathéodory conditions, J. Differential Equations,16 (1974), pp. 26–44.CrossRefMathSciNetGoogle Scholar
J. L. Massera -J. J. Schäffer,Linear differential equations and function spaces, Academic Press, New York, 1966.Google Scholar
J. J. Schäffer,Linear differential equations with delays: admissibility and conditional exponential stability, II, J. Differential Equations,10 (1971), pp. 471–484.CrossRefzbMATHMathSciNetGoogle Scholar