Convergence in asymptotically autonomous functional differential equations. (English) Zbl 0936.34064

The authors consider linear and nonlinear perturbations of a linear autonomous functional-differential equation which has infinitely many equilibria. They give sufficient conditions under which the solutions to the perturbed equation tend to the equilibria to the unperturbed equation at infinity. As a consequence, they obtain sufficient conditions for systems of delay differential equations to have an asymptotic equilibrium.


34K25 Asymptotic theory of functional-differential equations
34K20 Stability theory of functional-differential equations
Full Text: DOI


[1] Arino, O.; Győri, I., Stability results based on Gronwall type inequalities for some functional differential systems, Differential equations, Colloq. math. soc. János bolyai, 47, (1987), p. 37-59 · Zbl 0624.34065
[2] Atkinson, F.V.; Haddock, J.R., Criteria for asymptotic constancy of solutions of functional differential equations, J. math. anal. appl., 91, 410-423, (1983) · Zbl 0529.34065
[3] Diblı́k, J., Asymptotic equilibrium for a class of delay differential equations, (), 137-143 · Zbl 0901.34072
[4] Győri, I.; Pituk, M., L2-perturbation of a linear delay differential equation, J. math. anal. appl., 195, 415-427, (1995) · Zbl 0853.34070
[5] Győri, I.; Pituk, M., Comparison theorems and asymptotic equilibrium for delay differential and difference equations, Dynamic syst. appl., 5, 277-303, (1996) · Zbl 0859.34053
[6] Hale, J.K., Theory of functional differential equations, (1977), Springer-Verlag New York · Zbl 0425.34048
[7] Hale, J.K.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer-Verlag New York · Zbl 0787.34002
[8] Philos, C.G.; Tsamatos, P.C., Asymptotic equilibrium of retarded differential equations, Funkcial. ekvac., 26, 281-293, (1983) · Zbl 0551.34040
[9] Pituk, M., On the limits solutions of functional differential equations, Math. bohemica, 118, 53-66, (1993) · Zbl 0778.34056
[10] Zaanen, A.C., Integration, (1967), North-Holland Amsterdam · Zbl 0175.05002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.