ABSTRACT
Conditions for the operator 1 +K +N to be invertible or Fredholm in the space C,(L,) (1 ~ p ~ 00) may be found in KALITVIN [1994, 1997, 1997a, 1997b]. Let
(20.32) Nz(t) =fot n(t, T)Z(T) dT.
(20.33)
Theorem 20.2. Suppose that the integral operator (20.26) is compact in L,([-I, 1]) (1 ~ p < 00) or in C([-1,1J), and the operators (20.24) and (20.32) are compact in C([O, aJ). Then the operator K + N acts in Ct(L,) or C(D), respectivelJl, and the equalities
= O'I(K +N) =O'tl(K +N) =O'(M) are true. D The fact that the operator K + N acts in C,(D) or C(D) is obvious. The compactness of N is a consequence of the compactness of the operators (20.24) and (20.32). The equalities
follow from Theorem 20.1 and the stability of Fredholmness and the index with respect to compact perturbations. It remains to prove that O'(K +N) = O'(M). The inclusion O'(M) ~ O'(K +N) is obvious. Let A ¢ O'(M). Then the equation AZ - (K + N)z = f has the form
(I - i,i)~(~1 - M)z = <l.,i - N)~Mx +f. Since the operator (1 - i,i)~(~1 - M) is invertible, this equation is equivalent to the equation x = Az +9, where
and 9 =(I - i1)-I~(~I - M)-1f. Moreover, since (I - i1)-I(i1N) is a compact integral operator of Volterra type in C«(O, aD, its spectral radius is zero. Consequently, A has also spectral radius zero, and thus the equation x = Ax+9 admits a unique solution. But then the equation ~x - (K +N)x = f has a unique solution as well which means that ~ ¢ u(K +N).•
Theorem 20.2 implies, in particular, that the operator ~I+K +N is Fredholm of index zero if and only if -~ ¢ O'(M). Since -1 ¢ O'(M), the operator I +K+N in equation (20.30) is certainly Fredholm of index zero. Consequently, the Fredholm alternative applies to equation (20.30). We remark that equation (20.30) occurs also in the mathema.tical modelling of some axially symmetric contact problems for non-uniformly aging visco-elastic foundations (MANZHIROV [1983, 1985]). The corresponding Volterra-Fredholm equations have the same properties as the operator K +N in Theorem 20.2. However, there are other axially symmetric contact problems which lead to completely different partial integral operators of Volterra-Fredholm type. We will consider such problems in Subsections 20.4 and 20.6. Again, under the hypotheses of Theorem 20.2 the invertibility, the Fredholmness, and the Fredholmness with index zero of I +K +N in C,(L2) are all equivalent to the condition -1 ¢ O'(M). In the important particular case net, T) = 'et, T), this condition also implies the integral representation
for the solution x. The condition -1 ~ O'(M) is again satisfied automatically, since the kernel m = m(s, 0') in (20.30) defines a selfadjoint positive definite operator in L 2 ([-1,1]). Approximate solutions x" of equation (20.30) may be constructed in ra.ther the same way as we have done in Subsection 20.1 for the equation (20.12). Let X(ti,S) denote the value of the solution x of the
system (20.30) for t = ti (i = 0,1, ... , n - 1), and let
Equation (20.30) may then be approximated by the system of integral equations
Since -1 rt er(M}, for each n we get a unique solution 2:o{s)", ., 2:n{s) of this system. Putting this solution into (20.34) we get the approximate solution (i = 0,1, ... , n - 1)
for equation (20.30). If the kernels I and n in the equations of continuous media and mixed problems ofevolutionary type are continuous, the sequences (2:n}n and (rn}n obtained in this way converge in the space with mixed norm [Loo +- L2] (see Subsection 12.2) to the exact solution. We point out that our method for the approximate solution of the equations (20.28) and (20.21) is different from that proposed in ALEKSANDROV-KoVALENKO [1980] or KOVALENKO [1981]. For numerical purposes the following approach is suitable. Let {to, ... , In} be an equidistant partition of [0, a], and {so, ••. ,1m } an equidistant
partition of [-1, 1]. Replace the original equation by the approximating system
This system may be solved successively for i = 0, 1, ... ,n, and then the solution is extended by linear interpolation. In this way, one obtains continuous approximate solutions to the original equation which are arbitrarily close to the exact solution.
