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5 years ago

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

124 Theorem 2.4 Let X be

124 Theorem 2.4 Let X be (D-reflexive with respect to Ao, and let B be a bounded perturbation of Ao. Then the operator Az = Ao°*z + Bx with 7)(A) = {z E 7)(A~*)IAx E X} is the generator of a strongly continuous semigroup {T(t)} and the variation-e(-constants.formula holds. 3 The shift semigroup T(t)x = To(t)z + To°*(~ - T)BT(r)x dr (2.1) We repeat some of the material presented in [Die87]. Let ( be a given n x n real-matrix valued function of bounded variation such that ~(0) = 0 for 0 < 0 and ((0) = ((h) for # _> h > 0. Here and in the following we assume that all bounded variation functions are normalized such that they are right continuous on (0, h), zero on (-oo, 0] and constant on [h, oo). Let g be a C k mapping, k > 1, of X = C([-h, 0]; R ") into R '~ such that g(0) = 0 and Dg(O) = 0. We consider the nonlinear RFDE with initial condition &(t) = ~0 h d((v)x(t-v)+g(zt), (3.1) x(0)=~(0) -h

is generated by 125 V(A~) = {f : f(t)= fCOT)T f~gCv)drfor t>O, where gENBV and g( h)=O} a~f = g. From the general theory we know that X ® = ;D(A~). In the case at hand this results in Lemma 3.3 X ® = {f : f(t)= f(0+)+f~g(r)dr for t> 0, where g E LI(R+) and g(o) = 0 for a > h} D(A°o) = {I : f(t) = f(O+) + f~ g(r) dr for t > O, where g e Ac(rt+) and g(a) = 0 for ¢r >_ h} Elements of X ® axe completely described by f(0+) E R '= and g E LI([0, h]; R"). In other words, the space X ® is isometrically isomorphic to R n x LI([0, h]; R n) equiped with the norm In these coordinates we have IICc, g)ll = Icl=. + IlgllL,. Lemma 3.4 The semigroup To°(t)(e,g)= (e+ f~ g(r)dr, g(t + .)) is generated by D(AOo) = {(e,g) : g E AC(R+)} and A°o(e,g)= (g(0),~)). We represent X o* by R" × Leo(j0, h]; R n) equiped with the norm and the pairing I1( a, +)ll = sup{I'll,,", 114'IILo.} o h ((c,g), (.,~)) = e. + g(~)~(-r) d~. Lemma 3.5 The semigroup To@*(t)(a, ~b) = (a, q~'), where by definition is generated by ~(r)= f d~(t+r) if t+rO, VCAo °') = {Ca, ~) : ~ ~ Zip(a) }, A0o.(., ~) = (0, ~). Here Lip(or) denotes the subset of Le°(R+; R n) whose elements contain a Lipschitz continuous function which assumes the value a at r = 0. Taking the closure of D(A°o *) we lose the Lipschitz condition but the continuity remains. Lemma 3.6 X ®® = {(~, ~b) I ~b is continuous and d~(O) = ot } ~ X.

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