Differential Equations in Abstract Spaces by V. Lakshmikantham

By V. Lakshmikantham

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Since A does not depend on time, the operator T(r)does not depend on s. The stability requirement implies that T(r) is a continuous operator on X . The solution u ( t r ) at time t r is given by T ( t r ) uo. At time r the solution is T ( r )uo. Therefore taking this as initial data t units of time later, the solution becomes u ( t + r ) = T ( r ) [ T ( ru) O ] . From the uniqueness requirement and assuming that D ( A ) is dense in A', we obtain the semigroup property + + T(t+s) = T ( t )T(s), t,s + > 0.

1) is nonincreasing in t for t 2 0. yo, xo E D ( A ) . 3. Let A on D ( A ) be the infinitesimal generator of a strongly continuous semigroup { T ( t ) } ,t 2 0. L e t 8 [0, co)+ X be a (strongly) continuously differentiable function. 2) has the unique solution s(t) = T(t)s,. + T ( r - s ) j ( s ) ds, t 2 0. 2. 2). Obviously x(0) = xo. 2. The Infinitesimal Generator Define the function = p ( s ) f ( t-s) ds. 1 that the Riemann integral Ji T ( s ) f ( t-s)ds exists. We shall first prove that g(r) is (strongly) differentiable.

Since A,, I - B is one to one, we obtain xB= x, E D ( A ) and the claim is proved. 1 is therefore complete. Next we prove there exist real numbers M and o such that IIT(t)II < Mexp(ot), t 2 0. As proved in Claim 6, every 1. > o is in p (A) and for I > o r m From the resolvent formula (see Appendix VIII) one obtains I,p > R ( i . ; A )- R ( p ; A ) = ( p - I ) R ( L ; A ) R ( p , A ) , 0. Consequently the analyticity of R ( 2 ;A ) for 1. E p ( A ) yields ( d / d . ) R ( i - ; A )= lim[R(I;A) - R ( p ; A ) ] / ( I - p ) P-1 = -R(A;A)’.

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