By Katz N.M.

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**Example text**

I R(u + iT)xll--+O uniformly relative to tT (x E ~(A». 14) tell us that IIR(A)xll--+O as IAI--+ co (xE~(A». 13) is valid for any x E E. 4. 15) I R(A) I ~ M l (1 holds for the resolvent. 17) + IAI> ~(A), I R (A) x I --+ 0 as (A = tT + iT. tT ~ W2) then IAI --+ co §1. THE CAUCHY PROBLEM 33 3. Construction of solutions of the Cauchy problem. 12). The difficulty with this procedure is that the integral in this formula is not as a rule absolutely convergent, since the resolvent cannot decrease more rapidly than 1/ IAI.

We require in addition that the resolvent be defined in the entire halfplane ReA ~ a, and moreover that the contour of integration may be contracted to the point Ao without changing its value. Then by the theorem of residues 1 x(O) = (l _ I)! 5. 22) ~ M(1 &S defined + I AI)k for some k ~ - 1. 1) The proof is obtained from the preceding considerations with l = [k] + 3. 2. 22) may be weakened somewhat and replaced by the following: I R(A) ~ ~ M(l + ITl)k (A = + iT, tT tT ~ a). 3. 18) the integral is taken on the line ReA = a.

We need only verify that the function x(t) satisfies the equation for t = O. It is right differentiable for t = O. Indeed, passing to the limit in the equation x(t) -X(E) = f'x/(t)dt as E- 0, we find that 28 I. FIRST-oRDER EQUATIONS WITH CONSTANT OPERATOR x(t) - X(O) = 1t X' (t) dt, from which it follows that x'(O) = lim x'(t). t_+o Using the fact that A is closed, we may now pass to the limit as t- + 0 in the equation x'(t) = Ax(t), and thus arrive at the equation x'(O) = Ax(O). The lemma is proved.