Convergence of the solutions of the equation \(\dot y(t) = \beta(t)[y(t-\delta)-y(t-\tau)]\) in the critical case. (English) Zbl 1125.34059

Diblík, Josef; Ružičková, Miroslava

J. Math. Anal. Appl. 331, No. 2, 1361-1370 (2007).

This paper deals with the asymptotic behavior of a first order linear homogeneous differential equation with double delay of the form
\[ y'(t)=\beta(t)[y(t-\delta)-y(t-\tau)], \]
where \(\delta\) and \(\tau\) are positive with \(\tau>\delta\); \(\beta\in C([t_0-\tau,\infty),\mathbb R^+)\). The authors especially deal with the so called critical case with respect to the function \(\beta\) which separates the case when all solutions are convergent and the case when there exist divergent solutions. For coefficients below the critical function, a strictly increasing and bounded solution is constructed, which characterizes the asymptotic convergence of all solutions.

Reviewer: Meng Fan (Changchun)

MSC:

34K25	Asymptotic theory of functional-differential equations
34K12	Growth, boundedness, comparison of solutions to functional-differential equations
34K06	Linear functional-differential equations

Keywords:

convergent solution; two delayed arguments

References:

[1]	Arino, O.; Pituk, M., Convergence in asymptotically autonomous functional differential equations, J. math. anal. appl., 237, 376-392, (1999) · Zbl 0936.34064
[2]	Arino, O.; Pituk, M., More on linear differential systems with small delays, J. differential equations, 170, 381-407, (2001) · Zbl 0989.34053
[3]	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
[4]	Bereketoglu, H.; Pituk, M., Asymptotic constancy or nonhomogeneous linear differential equations with unbounded delays, Discrete contin. dyn. syst., suppl. vol., 100-107, (2003) · Zbl 1071.34080
[5]	Čermák, J., The asymptotic bounds of solutions of linear delay systems, J. math. anal. appl., 225, 373-388, (1998) · Zbl 0913.34063
[6]	Diblík, J., Asymptotic convergence criteria of solutions of delayed functional differential equations, J. math. anal. appl., 274, 349-373, (2002) · Zbl 1025.34062
[7]	Diblík, J., Asymptotic representation of solutions of equation \(\dot{y}(t) = \beta(t) [y(t) - y(t - \tau(t))]\), J. math. anal. appl., 217, 200-215, (1998) · Zbl 0892.34067
[8]	Diblík, J.; Růžičková, M., Exponential solutions of equation \(\dot{y}(t) = \beta(t) [y(t - \delta) - y(t - \tau)]\), J. math. anal. appl., 294, 273-287, (2004) · Zbl 1058.34099
[9]	Györi, I.; Pituk, M., Comparison theorems and asymptotic equilibrium for delay differential and difference equations, Dynam. systems appl., 5, 277-302, (1996) · Zbl 0859.34053
[10]	Krisztin, T., A note on the convergence of the solutions of a linear functional differential equations, J. math. anal. appl., 145, 17-25, (1990) · Zbl 0693.45012
[11]	Murakami, K., Asymptotic constancy for systems of delay differential equations, Nonlinear anal., 30, 4595-4606, (1997) · Zbl 0959.34058
[12]	Vulich, B.Z., Short course of theory of functions of a real variable (an introduction to the integral theory), (1973), Nauka, (in Russian)