ABSTRACT
To illustrate these theorems, let us consider the linear integral equation
+ 10' m(s, O')Z(t, 0') dO' = I(t,s) which occurs in the mechanics of continuous media (see e.g. ALEKSANDROV-KoVALENKO [1980, 1986] and KOVALENKO [1981]). Here we have T = [-1,1],5 = [O,a], and the kernell has the form
where ~ and 8 are a parameters (which have a mechanical and geometric meaning), and i is a positive continuous function satisfying
i(z) =A +O(z:l) (z - 0), Izli(z) =B +O(z-I) (z - 00). Moreover, the kernel m is either continuous or weakly singular, and the function I has the form
I(t,s) = ft(t) +J:r(s) +th(s) (ft E Lp(T), J:r,h E Lp(5». Equation (15.36) may be studied in the space Lp(T X S). Under the hypotheses given above, the operators i and fA are compact in Lp(T) and Lp(S), respectively, and thus the results of § 14 apply. The spectrum of the compact operator L consists of 0 and either a finite number, or a sequence converging to 0, of eigenvalues ~. On the other hand, the spectrum of fA contains only O. The operator L is selfadjoint and positive definite in L:I(T). Therefore, its spectrum is contained in the nonnegative real axis. Since 0 e O'(L) in any of the spaces Lp(T), and the other points of spectrum form a discrete set, by a well-known results of HALBERG-TAYLOR [1956], the spectrum of the operator i in Lp(T) is the same for any p, i.e. consists of 0 and a finite number or a sequence (converging to 0) of positive eigenva.- lues. Thus, the spectrum of the corresponding operator K coincides in this case with that of the operator ij in particular, a.ll elements of
