A Nonlinear Inequality and Evolution Problems

Assume that g(t) ≥ 0, and ġ(t) ≤ −γ(t)g(t) + α(t, g(t)) + β(t), t ≥ 0; g(0) = g0; ġ := dg dt , on any interval [0, T ) on which g exists and has bounded derivative from the right, ġ(t) := lims→+0 g(t+s)−g(t) s . It is assumed that γ(t), and β(t) are nonnegative continuous functions of t defined on R+ := [0,∞), the function α(t, g) is defined for all t ∈ R+, locally Lipschitz with respect to g uniformly with respect to t on any compact subsets [0, T ], T < ∞, and non-decreasing with respect to g, α(t, g1) ≥ α(t, g2) if g1 ≥ g2. If there exists a function μ(t) > 0, μ(t) ∈ C(R+), such that α t, 1 μ(t) + β(t) ≤ 1 μ(t) γ(t) − μ̇(t) μ(t) , ∀t ≥ 0; μ(0)g(0) ≤ 1, then g(t) exists on all of R+, that is T = ∞, and the following estimate holds: 0 ≤ g(t) ≤ 1 μ(t) , ∀t ≥ 0. If μ(0)g(0) < 1, then 0 ≤ g(t) < 1 μ(t) , ∀t ≥ 0. A discrete version of this result is obtained. The nonlinear inequality, obtained in this paper, is used in a study of the Lyapunov stability and asymptotic stability of solutions to differential equations in finite and infinite-dimensional spaces.