A note on existence and stability of periodic and almost periodic solutions of quasilinear equations with ”maxima”

Nataliya Bantsur Institute of Mathematics, NAS of Ukraine

Let us consider differential equation with ”maxima”

y′(t) = −δy(t) + p max_{s∈[t−h,t]} y(x) + f(t)  (1)

with initial conditions

y(t) = φ(t), t ∈ [−h, 0],

where f(t) ∈ C(R^{n}), f(t + T) = f(t), y(t) and φ(t) are continuous functions with values in R^{n}, p and δ are some positive constants.

We investigate problems relation to existence and stability of periodic and almost periodic solutions of Eq.(1). Note, that analogocal problems for some other types of differential equations with ”maxima” were studied in [1-3].

The sufficient conditions for existence, uniqueness and stability of periodic (almost periodic) solutions of Eq.(1) were obtained.

References:

1. Bantsur N.R., Trofimchuk E.P. Existence and stability of the periodic and almost periodic solutions of quasilinear system with maxima // Ukr. Math. J. - 1998. - 50, N 6. - P. 747-754.

2. Liz E., Trofimchuk S. Existence ans stability of almost periodic solutions for quasilinear delay systems and the Halanay inequality // J. Math. Anal. and Appl. 248, 625-644 (2000).

3. Pinto M., Trofimchuk S. Stability and existence of multiple periodic solutions for a quasilinear differential equation with maxima // Preceedings of the Royal Society of Edinburgh, 130A, 1103-1118 (2000).