ABSTRACT
Observe that, if the kernels 12, m2 and n are continuous, then the operators K2 and K2 + N2 certainly act in C(D) and the equalities (20.47) and (20.48) hold true.
20.6. A unified approach to some equations of mechanics The operators considered in Subsections 20.2 • 20.5 may be written in the general form
+d(t,s) loti' n(t,r)m(I,O')z(r,O')dO'dr «t,s) E D = [O,T) X [O,b]) with continuous functions a = a(s), e = e(t) and d = de', s), and measurable kernels 1= 'et, r), m =m(I, 0') and net, r). The spectral properties of the operator (20.50) may be studied by the methods discussed in the preceding two subsections. Consider the Volterra integral operators
(20.51)
Put
and E(c) = {c(t) : 0 ~ t ~ T} as before. In the following theorem we use the terminology of Sections 4 and 12.
