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1 results

  • ON MAXIMAL REGULARITY AND SEMIVARIATION OF COSINE OPERATOR FUNCTIONS

  • D.-K. CHYAN, S.-Y. SHAW, S. PISKAREV
  • Journal:
    Journal of the London Mathematical Society / Volume 59 / Issue 3 / June 1999
    Published online by Cambridge University Press:
    01 June 1999, pp. 1023-1032
    Print publication:
    June 1999

  • It is proved that a cosine operator function C(·), with generator A, is locally of bounded semivariation if and only if u″(t) = Au(t)+f(t), t>0, u(0), u′(0)∈D(A), has a strong solution for every continuous function f, if and only if the function ∫t0∫t−s0C(τ)f(s)dτds, t>0, is twice continuously differentiable for every continuous function f, that is, C(·) has the maximal regularity property if and only if A is a bounded operator. Some other characterisations of bounded generators of cosine operator functions are also established in terms of their local semivariations.