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A NONLINEAR INEQUALITY AND EVOLUTION PROBLEMS

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Author(s): A.G. RAMM

Journal: Journal of Inequalities and Special Functions
ISSN 2217-4303

Volume: 1;
Issue: 1;
Start page: 1;
Date: 2010;
Original page

Keywords: nonlinear inequality | Lyapunov stability | evolution problems | di®er- ential equations.

ABSTRACT
Assume that g(t) ¸ 0, andg_(t) · ¡°(t)g(t) + ®(t; g(t)) + ¯(t); t ¸ 0; g(0) = g0; g_ :=dgdt;on any interval [0; T) on which g exists and has bounded derivative from theright, g_(t) := lims!+0g(t+s)¡g(t)s . It is assumed that °(t), and ¯(t) arenonnegative continuous functions of t de¯ned on R+ := [0;1), the function®(t; g) is de¯ned for all t 2 R+, locally Lipschitz with respect to g uniformlywith respect to t on any compact subsets [0; T], T < 1, and non-decreasingwith respect to g, ®(t; g1) ¸ ®(t; g2) if g1 ¸ g2. If there exists a function¹(t) > 0, ¹(t) 2 C1(R+), such that®µt;1¹(t)¶+ ¯(t) ·1¹(t)µ°(t) ¡¹_ (t)¹(t)¶; 8t ¸ 0; ¹(0)g(0) · 1;then g(t) exists on all of R+, that is T = 1, and the following estimate holds:0 · g(t) ·1¹(t); 8t ¸ 0:If ¹(0)g(0) < 1, then 0 · g(t) < 1¹(t) ; 8t ¸ 0:A discrete version of this result is obtained.The nonlinear inequality, obtained in this paper, is used in a study ofthe Lyapunov stability and asymptotic stability of solutions to di®erentialequations in ¯nite and in¯nite-dimensional spaces.
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