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Extra info for Advanced Complex Calculus

Example text

The proof of (i) is immediate. 1) are given by t fx(s)ds u(t, t0, u0) = u0e° ,t> tQ. OO If (a) holds, then letting N(t0) = JX(s)ds, we get u(t,t0,uQ) < *o uQe ° . As a result, for any e > 0, we have, choosing S(tQ) t0. 1) is (h0, /i)-equistable. 1) follows oo immediately. Since / \(s)ds = — oo, we have JX(s)ds lim u(t, t0, u0) = uQlim e ° =0. 1) is Chapter 1 33 (h0, /i)-equiasymptotically stable. 1) is uniformly asymptotically stable. 1) is uniformly stable. Let e > 0 be given.

1) is uniformly stable. 1). 1). h(t0,x(t0)) t0 such that V i . s ( ' i ) ) = *» M*2i»('a)) = (t,x(t)) 6 S(h,e)r\Sc(h0,6) e and for i € [tut2]. 1, we obtain V^(*,a:(*))<7(*»*i,«o)»*€[*i»*2]. *i,«o) < 6 (0, which is a contradiction. 1-1) is (h0,h)- uniformly stable and the proof is complete. 5. As we have seen, the use of comparison principle provides a unified approach and generalizes several stability results into one framework. However, a direct analysis of the right-hand side of the comparison equation can sometimes yield sharper results.

Thus the theorem is proved. A stability property can be considered as a family of properties depending on some parameters. Consequently, when we employ a single Lyapunov function to prove a given stability property, the Lyapunov function used is assumed to play the role for every choice of these parameters. As a result, if we utilize a family of Lyapunov functions instead of one, it is natural to expect that each member of the family has to satisfy weaker requirements. 2. 5: Assume that (i) h0) h G T and h0 is uniformly finer than h; (ii) for every r\ > 0, there exists a function V^ G C[S(h, p) D Sc(h0, TJ), R + ] such that V^t, x) is locally Lipschitzian x and satisfies b(h(t,x)) < v(t,x) < «(V*.

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