ABSTRACT
In Banach spaces X and Y we deal with a closed linear operator A with a dense domain, the elements Uo, Ul E X and the mappings
F: [0, T] x X x X x Y 1-+ X
and
B: X 1-+ Y,
It is required to recover a pair of the functions
524 8. Inverse Problems for Equations of Second Order
which comply with the following relations:
(8.1.1) u"(t) =A u(t) + F( t, u(t), u'(t), p(t») ,
(8.1.2)
(8.1.3)
B u(t) =1/J(t) ,
The basic restriction imposed on the operator A is connected with the requirement for the linear Cauchy (direct) problem
(8.1.4)
(8.1.5)
u"(t) = A u(t) + F(t) , 0 ~ t ~ T,
u(O) = Uo, u'(O) = UI •
to be well-posed. Therefore, the operator A is supposed to generate a strongly continuous cosine function C(t), that is, an operator function which is defined for all t E R with values in the space £(X) is continuous on the real line R in the strong topology of the space £(X) and is subject to the following two conditions:
(1) C(O) =I; (2) C(t + s) + C(t - s) = 2C(t) C(s) for all t, s E R. It should be noted that the operator A can be recovered from its
cosine function as a strong second derivative at zero
Ax = C"(O) x
with the domain V(A) = {x: C(t) x E C2(R)}. Recall that the Cauchy problem (8.1.4)-(8.1.5) is uniformly well-posed if and only if the operator A generates a strongly continuous cosine function. In the sequel we will exploit some facts concerning the solvability of the Cauchy problem and relevant elements of the theory of cosine functions. For more detail we recommend to see Fattorini (1969a,b), Ivanov et al. (1995), Kisynski (1972), Kurepa (1982), Lutz (1982), Travis and Webb (1978), Vasiliev (1990).
