2010/09/30 by A. G. Ramm, Ramm, A. G.
Mathematics · Physics and Astronomy · #26D10 #34G20 #37L05 #44J05 #47J35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CA #math.MP #msc:26D10 #msc:34G20 #msc:37L05 #msc:44J05 #msc:47J35
paper · pdf · doi:10.48550/arxiv.1009.6138
arxiv created 2010/09/30 · arxiv updated 2010/10/01
Assume that g(t)≥ 0, and g(t)≤ -γ(t)g(t)+α(t,g(t))+β(t), t≥ 0; g(0)=g0; g:=(dg)/(dt), on any interval [0,T) on which g exists and has bounded derivative from the right, g(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)∈ C1(\R+), such that α(t,(1)/(μ(t)))+β(t)≤ (1)/(μ(t))(γ(t)-(μ(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)≤ \frac 1μ(t), ∀ t≥ 0. If μ(0)g(0)< 1, then 0≤ g(t)< \frac 1μ(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.