ABSTRACT
Let A = L1®('\I - Mt )-lII with llz(s) = .9z(a). Since O'(L1) = {OJ, we get O'(A) =O'(Lt)O'([~I - Mt ]-I11) ={OJ as well (ICHINOSE [1978]). Consequently, equation (20.38) has a unique solution, and hence also the equation (~I - K l)Z = g. This shows that the operator
~I - K1 is invertible, hence ~ ¢ 0'(K1). In this way we have proved that O'(Kl) ~ O'(M1). We show now that no number ..\ < 0 may belong to the spectrum O'(Mt ). In fact, since the integral operator with kernel mis compact,
(20.39)
(20.40)
self-adjoint, and positive definite in L2([-1, I)), the same is true for the integral operator Ai with kernel m(8, (J) =same8, (J). Therefore Ml is a compact, self-adjoint, and positive definite operator in the space L 2([-1, 1]; w) with weight function w(s) = s, and thus its spectrum in this space contains only nonnegative real numbers. Since the spaces L2([-1, 1]; w) and L2([-1,1» are isomorphic, the spectrum of Ml in L2([-1, 1]) also contains only nonnegative real numbers. From this and (20.37) it follows that the operator >'1+Xl is invertible for positive A. Consequently, for A> 0 equation (20.35) has a unique solution in Ct(L2) or C(D) if 9 belongs to Ct(L2) or C(D), respectively. • Define operators Ll , Ml and Kl by
1 LIZ(t) = lot ll(t,r)z(r) dr, Mlz(s) = 11 ml(s,O')Z(O')do',
and - ItX I 2:(t,s) = a(s) 1
+L: ml(a, O')z(t, 0') dO', respectively. In the same way as we have proved (20.37) one may prove the following slightly more general result (KALITVIN [1997b)): Theorem 20.4. Suppose that the integral operator L1 in (20.39) acta in C([O, aJ) and is weakly compact, and the integral operator MI is compact in Lp([-l, 1] (1 S P < 00) or in C([-l, I)). Assume further that the function a = a(s) in the definition 0/ K1 is continuous on [-1,1]. Then the operator (20.40) acta in Ct(Lp ) and in C(D), and the equalities
Consider the integral equation (20.35) of the axially symmetric contact problem. In Theorem 20.4 we have proved the unique solvability of this equation for each ~ > O. For solving equation (20.35) explicitly it is useful to apply the Laplace transform with respect to the variable t. Thus, let X =X (P, s) and F =F(p, s) denote the Laplace transforms of x = x(t,s) and I = I(t,s), respectively. Equation (20.35) becomes then
(20.41) ~X(p,s)+ ....1?-jl m(s,u)X(p,u)clO' = -P-F(p,s), p+ s -1 p+ s
where m is defined through ml as above. In this way, we have reduced the problem to the solution of a Fredholm integral equation which depends on a parameter. For p ~ 0 it is useful to study this equation in the space L2([-1,1]jw) with weight w(p,s) = s(p+s). In fact, the integral operator with kernel m(s, u)p/(p+ s) is obviously compact, selfadjoint and positive definite in this weighted space. Denote by ~1 (p), ~2(P), ... its eigenvalues, and by el (p, s), e2(p, s), ... the corresponding normalized eigenfunctions. We have then
~X(p,s) = E x,l;(p)e,l;(p,s) k=1
and p 00
Putting this into equation (20.41) we get
(20.42) 00E ek(p,s)[X,l;(p)(~+ ~,I;(p» - 1,I;(p)] = O.
