By W. Arendt (auth.), Franz Kappel, Wilhelm Schappacher (eds.)
Read or Download Infinite-Dimensional Systems: Proceedings of the Conference on Operator Semigroups and Applications held in Retzhof (Styria), Austria, June 5–11, 1983 PDF
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Extra info for Infinite-Dimensional Systems: Proceedings of the Conference on Operator Semigroups and Applications held in Retzhof (Styria), Austria, June 5–11, 1983
Integral equation, SIAM J. Math. Anal. : On Volterra equations of the first kind, J. of Integral Eq. , 3/4 (1980), 473-488. : On an integral equation Math. , 8 (1977), 950-970. : Nonlinear Volterra J. Diff. ,16 (1974), 373-393. L. Smith: Limits of Solutions Equations, Appl. Analysis, 7 (1977), 19-27. SIAM J. spa~e, of Nonlinear V o l t e r r a STABILITY OF N O N - A U T O N O M O U S EQUATIONS DELAY DIFFERENTIAL BY L I A P U N O V FUNCTIONALS K. L. Cooke Pomona Claremont, College CA 91711, USA 1.
1-al i=1 r: i < a -- V(t,x(-)) (1-r~)a~lb~x(hi)2 and h y p o t h e s i s ~ [-q - (1-~ -(l-a) = qx2 < - qx 2 (ii), )-1 we obtain [ ai]x 2 + 2 [ bixx(h i) [ a~lb~x(hi)2 I ai I1/2 x . )) 2 : i=i N : K. - (s)x(s)2ds i(t2 ) i hi(t1) [} [ K. 5) Example. for s < t 2 . -l(t2 ) _ = S:L_I ailbi(u)2h~(u)du. h i (t I ) follows from hypothesis (i) that ( • )) _< Ilxll 2 ~ a~ 1Y(t2-tl). the and the ~ = i/2. Theorem scalar : -c(t)x(t) N = i take it is s a t i s f i e d . Consider x'(t) may x(s) i Z(t2,x( Here of = )2 a~lbi(h~l(s))2ds tI I Thus, supremum i now The is from Burton's Theorem.
To p r o v e = ( f - A - l g , 0 ) ~ e K(E); from the o p e n m a p p i n g theorem. that thus Let n o w then K(x) and for all is t r i v i a l l y is onto the first : D~(~) # : (x,0) ~ : -(0,Ax) ~ : -J(Ax) let J(x) = (0,x) be an a r b i t r a r y = -(A-ix,0) is p r o v e d . 16) . of D~; we h a v e • K(D A) is a c o n s e q u e n c e of the i n t e r p o l a t i o n 57 3. S e v e r a l Consider hypotheses Operators now another of A We shall d e f i n e A-IB operator and such that an e x t e n s i o n and and d e n o t e w i t h the Same D o m a i n .
Infinite-Dimensional Systems: Proceedings of the Conference on Operator Semigroups and Applications held in Retzhof (Styria), Austria, June 5–11, 1983 by W. Arendt (auth.), Franz Kappel, Wilhelm Schappacher (eds.)