Page 670 - Handbook Of Integral Equations

◦
2 . Right Regularization. Consider Eq. (24) and the corresponding regularized equation
˜
KK[ω(t)] = f(t), (29)
which is obtained by substitution
˜
K[ω(t)] = ϕ(t). (30)
If ω j (t) is a solution of Eq. (29), then formula (30) gives the corresponding solution of the original
equation
˜
ϕ j (t)= K[ω j (t)].
Hence, the right regularization cannot lead to irrelevant solutions.
Conversely, assume that ϕ k (t) is a solution of the original equation. In this case a solution of the
regularized equation (29) can be obtained as a solution of the nonhomogeneous singular equation
˜
K[ω(t)] = ϕ k (t);

however, this solution may be unsolvable. Thus, the right regularization can lead to loss of solutions.
˜
We have no loss of solutions if Eq. (30) is solvable for each right-hand side. In this case the operator K
will be an equivalent right regularizer.
˜
3 . The Equivalent Regularization. The operator K = K ∗◦ is an equivalent regularizer for any index;
◦
for ν ≥ 0, we must apply left regularization, while for ν ≤ 0 we must use right regularization.
In the latter case we obtain an equation for a new function ω(t), and if it is determined, then
we can construct all solutions to the original equation in antiderivatives, and it follows from the
properties of the right regularization that no irrelevant solutions can occur.
For the other methods of equivalent regularization, see the references at the end of this section.

13.4-5. Fredholm Theorems
Let a complete singular integral equation be given:

K[ϕ(t)] = f(t). (31)

THEOREM 1. The number of solutions of the singular integral equation (31) is finite.
THEOREM 2. A necessary and sufficient solvability condition for the singular equation (31) is

f(t)ψ j (t) dt =0, j =1, ... , m, (32)
L
where ψ 1 (t), ... , ψ m (t) is a maximal finite set of linearly independent solutions of the transposed
homogeneous equation K [ψ(t)]=0. (Since the functions under consideration are complex, it
∗
follows that condition (32) is not the orthogonality condition for the functions f(t) and ψ j (t).)
THEOREM 3. The difference between the number n of linearly independent solutions of the
singular equation K[ϕ(t)]=0 and the number m of linearly independent solutions of the transposed
equation K [ψ(t)] = 0 depends on the characteristic part of the operator K only and is equal to its
∗
index, i.e.,
n – m = ν. (33)

Corollary. The number of linearly independent solutions of characteristic equations is minimal
among all singular equations with given index ν.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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