Stability in part of the variables of “partial” equilibria of systems with aftereffect

On partial stability of retarded functional differential systems

“…Following [6], we let x(t) = x(t, t0, φ) denote the value of x(t0, φ) at time t. Just as in [6], we can prove the following assertion [5]: if for each bounded closed subset S in R+×C y1 h, the operator Y(t, φy1, φy2, φz) maps the set S×C y2 ×C z in to a bounded set (in R n ), then the inequality |y1(t, t0, φ)| ≤ h1 < h means that the y1-components of the corresponding solutions of system (1) are determined for all t ≥ t0. In this case, the (y2,z)-components of the solutions can be determined only on a finite time interval t  [t0-τ, β), β < +∞, and |y2(t, t0, φ)| + |z(t, t0, φ)| → ∞ as t → β.…”

“…Definition [5]. A "partial" equilibrium position y = 0 of system (1) is y1-stable for a large values of φz1 and on the whole with respect to φz2, if for any ε > 0, t0 ≥ 0 and for any given number L > 0 there is δ(ε, t0, L) > 0 such that from φ  D it follows that |y1(t; t0, φ)| < ε for all t ≥ t0.…”

“…The Razumikhin approach is used. The method of Lyapunov-Krasovskii functionals for solving of this problem was used in [5].…”

On uniform partial asymptotic stability of retarded functional differential systems

“…In this article the problem of asymptotic stability with respect to a part of the variables of the "partial" equilibrium position is considered for nonlinear retarded systems of functional differential equations. Condition of uniform asymptotic stability of this type is obtained within the method of Lyapunov-Krasovskii functionals; this condition supplemented a number of existing results [4,5].…”

“…Following [6], we let x(t) = x(t, t0, φ) denote the value of x(t0, φ) at time t. We additionally assume [4,5] that the solutions are (y2, z)-continuable, namely, the solutions of the system are determined for t  t0 such that |y1(t, t0, φ)| < h. In this case, if |y1(t, t0, φ)| ≤ h1 < h for all t ≥ t0, then the corresponding functions x(t, t0, φ) are determined for all t ≥ t0.…”