Differential Equations in Abstract Spaces by V. Lakshmikantham

By V. Lakshmikantham

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By the uniform boundedness principle (see Appendix VI) we conclude that supn IIT(t,,)II co, which is a contradiction. 1, we have that w0 = (log I(T(t)ll)/texists and is a finite number or - co. For any 6 > coo there is a to such that ---f ---f -= (log II T(t>ll)lt < 6, t 2 to * < exp(60, t 2 to. Hence llT(t)ll In addition we know that IIT(t)ll is bounded on [ O , t o ] and the result follows. 2. The Infinitesimal Generator Let { T ( t ) } ,t 2 0, be a strongly continuous semigroup of operators in the Banach space X .

3 remains valid if, instead of assuming that f(f) is continuously differentiable, we assume that f ( r ) E D ( A ) for all r 2 0, andf'(r) and A f ( r ) are strongly continuous in t . 4. An operator A with domain D ( A ) dense in the Banach space X can be the infinitesimal generator of at most one strongly continuous semigroup { T ( t ) } ,t 2 0. Proof: Let { T ( t ) } ,f 2 0, and { S ( t ) } , t 2 0, be two strongly continuous semigroups of operators in X having A as infinitesimal generator. 1).

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. In fact + [g(t h) - g(r)]/h = h = f+ h I fis) + h-' T(s)J(t+ h - s) ds - h - 1 Lf(t sb T(s)f(t-s) ds +h - s) - f ( t -h)-J/hds L f i h T ( s ) f ( t + h - s )ds. Hence g'(t) exists and g'(t) = sO;(s)Y(r-s) ds + T(t)f(O). ) On the other hand, for h > 0 we have = [T(h)- Z]/h so' T(t- s)f(s) ds Since the limit on the left-hand side exists and also limh,o~:'h T(t+h-s) x f ( s ) d s = f ( r ) , it follows that limh+o+Ahyo T(t-s)f(s)ds exists and is equal to A Jb T ( t -s)f(s) ds.

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