A Generalization of Volterras Derivative by Smith H. L.

By Smith H. L.

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Loc)={A: (5) i (to guarantee the stability (3) an additional condition on A is necessary). A number of resuits on the solvability of the Cauchy problem with operators A6Diss(E; ~) carry over to this case (Chambers and Oharu, Konishi, Oharu, Kutuzov, Khazan [71-73, 106, 151, 153, 265, 445], cf. 16). , to equations of the form du (t) / dt~ A (t) u (t) ( h e r e t h e semigroups a r e r e p l a c e d by e v o l u t i o n s y s t e m s ) , c f . p o i n t 2 . 1 7 . )-loc) [the corresponding function ~ can depend on p(z) and lllAzlll] and satisfy the conditions which guarantee the solvability of the Cauchy problem for the equation with "frozen coefficient" du(t)/dt~A(t, z)u(~) (cf.

Finally, in Smagin and Sobolevskii [127] the question of estimating ~(t) =sup]JV{OPx]J/ JJU{t)xHin terms of its known values for t = O and t = T is considered where P is a linear operator carrying D(A) into D(B); A and B act in different Hilbert spaces in general while iA and iB generate groups of class Co; the sup is taken over all x ~ 0. 2. 1. Introduction. We consider the Cauchy problem du(t)/dtoAu(t) (t~O), u(O)=x, (1) where A is a nonlinear and possibly multivalued operator in the Banach space* E, which assigns to each zED(A) a nonempty set A z c E .

Sauer [613] considered (38) by semigroup methods, giving up the requirement of closedhess or even closability of the operators B and C. Let F and E be two Banach spaces and B be a linear operator, acting from D(B)cF to E. A family S(t) (0 < t) of bounded linear operators from E to F is said to be a B-evolutlon, if S(O:[~cD(B) and S(t + s) = S(s)BS(t) for all s, t > 0. It follows from the definition that the operators U(t) = BS(t) form a semigroup in E. Let A be its complete infinitesimal operator.

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