Asymptotic convergence criteria of solutions of delayed functional differential equations. (English) Zbl 1025.34062

Diblík, Josef

J. Math. Anal. Appl. 274, No. 1, 349-373 (2002).

The author studies the scalar equation \(\dot x=f(t,x_t)\) under the assumption that every constant is its solution. Criteria and sufficient conditions for the convergence of solutions are found. It is proved that all solutions are convergent if and only if there exists at least one increasing and convergent solution. As application, criteria for some classes of linear and nonlinear equations are obtained.

Reviewer: Bolis Basit (Clayton)

MSC:

34K07	Theoretical approximation of solutions to functional-differential equations
34K25	Asymptotic theory of functional-differential equations
34K05	General theory of functional-differential equations

Keywords:

asymptotic convergence criteria; solutions; delayed functional-differential equations

References:

[1]	Arino, O.; Györi, I.; Pituk, M., Asymptotically diagonal delay differential systems, J. math. anal. appl., 204, 701-728, (1996) · Zbl 0876.34078
[2]	Arino, O.; Pituk, M., Convergence in asymptotically autonomous functional differential equations, J. math. anal. appl., 237, 376-392, (1999) · Zbl 0936.34064
[3]	Arino, O.; Pituk, M., More on linear differential systems with small delays, J. differential equations, 170, 381-407, (2001) · Zbl 0989.34053
[4]	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
[5]	Bellman, R.; Cooke, K.L., Differential – difference equations, Mathematics in science and engineering, (1963), Academic Press New York · Zbl 0118.08201
[6]	Čermák, J., The asymptotic bounds of solutions of linear delay systems, J. math. anal. appl., 225, 373-388, (1998) · Zbl 0913.34063
[7]	Diblı́k, J., A criterion for convergence of solutions of homogeneous delay linear differential equations, Ann. polon. math., 72, 2, 115-130, (1999) · Zbl 0953.34065
[8]	Diblı́k, J., Asymptotic equilibrium for homogeneous delay linear differential equations with L-perturbation term, Nonlinear anal., 30, 3927-3933, (1997) · Zbl 0896.34065
[9]	Diblı́k, J., Asymptotic representation of solutions of equation \(ẏ(t)=β(t)[y(t)−y(t−τ(t))]\), J. math. anal. appl., 217, 200-215, (1998) · Zbl 0892.34067
[10]	Györi, I.; Pituk, M., L2-perturbation of a linear delay differential equation, J. math. anal. appl., 195, 415-427, (1995) · Zbl 0853.34070
[11]	Györi, I.; Pituk, M., Comparison theorems and asymptotic equilibrium for delay differential and difference equations, Dynamic systems appl., 5, 277-302, (1996) · Zbl 0859.34053
[12]	Hale, J.; Lunel, S.V., Introduction to functional differential equations, (1993), Springer-Verlag Berlin
[13]	Hartman, Ph., Ordinary differential equations, (1982), Birkhäuser Basel
[14]	Krisztin, T., On the convergence of solutions of functional differential equations with infinite delay, J. math. anal. appl., 109, 509-521, (1985) · Zbl 0586.34061
[15]	Lakshmikantham, V.; Leela, S., Differential and integral inequalities, vol. I, (1969), Academic Press New York · Zbl 0177.12403
[16]	Murakami, K., Asymptotic constancy for systems of delay differential equations, Nonlinear anal., 30, 4595-4606, (1997) · Zbl 0959.34058
[17]	Razumikhin, B.S., Stability of hereditary systems, (1988), Nauka Moscow, (in Russian) · Zbl 0145.32203
[18]	Rybakowski, K.P., Ważewski’s principle for retarded functional differential equations, J. differential equations, 36, 117-138, (1980) · Zbl 0407.34056
[19]	Ważewski, T., Sur un principe topologique de l’examen de l’allure asymptotique des intégrales des équations différentielles ordinaires, Ann. soc. polon. math., 20, 279-313, (1947) · Zbl 0032.35001
[20]	Slater, G.L., The differential-difference equation w′(s)=g(s)[w(s−1)−w(s)], Proc. roy. soc. Edinburgh sect. A, 78, 51-55, (1977) · Zbl 0368.34022
[21]	Zhang, S.N., Asymptotic behaviour and structure of solutions for equation \(ẋ(t)=p(t)[x(t)−x(t−1)]\), J. anhui university (natural science edition), 2, 11-21, (1981), (in Chinese)