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By Theorem 2.7 and the last two relations in (4.1), we have that has exactly generalized zeros in and (422).

where and are real and continuous functions in over over are arranged as and an eigenfunction corresponding to has exactly generalized zeros in the open interval.

(ii) If, then it is noted that is eigenvalue of (1.1) with (2.14) if and only if Hence, is an eigenfunction with respect to By Theorem 2.7, has exactly generalized zeros in and (429)  .

If, then it is noted that is eigenvalue of (1.1) with (2.14) if and only if Hence, is an eigenfunction with respect to By Theorem 2.7, has exactly generalized zeros in and (429).

As it was already the case before the projection, this expression does not exactly generalizes that of the pairing between like-particles case.

The oscillation theorem for (E0) then says that the j-th eigenfunction has exactly j generalized zeros in the interval (0, N + 1]. The generalized zeros are defined as follows, see [7, 8]. A sequence x = { x k } k = 0 N + 1 has a generalized zero in (k, k + 1], if x k ≠ 0 and r k x k x k + 1 ≤ 0. (1.3).

It remains to connect the above global oscillation theorem with the traditional statement saying that the j-th eigenfunction has exactly j generalized zeros in the interval (0, N + 1]. We will see that under some additional assumption the statement of this result remains exactly the same when we replace the eigenfunctions of (E0) by its finite eigenfunctions.

Then n2(λ j ) = j and from (2.22) we get n1(λ j ) = j, i.e., x j)has exactly j generalized zeros in (0, N + 1]. The proof is complete. In the last part of this section we present certain results on the existence of finite eigenvalues of (E0).

As an application of our new oscillation theorem we prove that the j-th finite eigenfunction has exactly j generalized zeros in the interval (0, N + 1], which is a discrete analogue of a traditional statement in the continuous time theory.

Again, we give an exactly distribution free solution for generalized Behrens Fisher problem.

Finally, note also that many known fixed-point theorems could be extended exactly the same way we generalized Schauder's to Theorem 3.1.