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Lyapunov-type inequalities for certain nonlinear systems on time scales. (English) Zbl 1166.34005

The paper concerns Lyapunov inequalities for the nonlinear system

x Δ (t)=α 1 (t)x(σ(t))+β 1 (t)|u(t)| γ-2 u(t),
u Δ (t)=-β 2 (t)|x(σ(t))| α-2 x(σ(t))-α 1 (t)u(t)·

Here, 1/α+1/γ=1, σ is the backward jump on a scale time T, whereas x Δ and u Δ are the corresponding derivatives (see [M. Bohner and A. Peterson, Dynamic equations on time scales. An introduction with applications. Basel: Birkhäuser. (2001; Zbl 0978.39001)]). The results contain classical inequalities for differential (T=) and discrete (T=) systems.

34A99General theory of ODE
39A10Additive difference equations
34A34Nonlinear ODE and systems, general