Page 589 - Handbook Of Integral Equations

11.11-2. The Solution of the Wiener–Hopf Equations of the Second Kind

THEOREM 1. For Eq. (1) to have a unique solution of the class L + for an arbitrary f(x) ∈ L + ,it
is necessary and sufficient that the following conditions hold:
ˇ
1 – K(u) ≠ 0, –∞ < u < ∞, (15)
ˇ
ν = – Ind[1 – K(u)] = 0. (16)

THEOREM 2. If condition (15) holds, then the inequality ν >0 is necessary and sufficient for the
existence of nonzero solutions in the class L + of the homogeneous equation

∞
y(x) – K(x – t)y(t) dt = 0. (17)
0
The set of these solutions has a basis formed by ν functions ϕ k (x) (k =1, ... , ν) that tend to zero
as x →∞ and that are related as follows:
x x

ϕ k (x)= ϕ k+1 (t) dt, k =1, 2, ... , ν – 1, ϕ ν (x)= ψ(t) dt + C, (18)
0 0
where C is a nonzero constant and the functions ϕ k (t) and ψ(t) belong to L + .
THEOREM 3. If condition (15) holds and if ν >0, then for any f(x) ∈ L + Eq. (1) has infinitely
many solutions in L + .
However, if ν < 0, then, for a given f(x) ∈ L + , Eq. (1) has either no solutions from L + or a unique
solution. For the latter case to hold, it is necessary and sufficient that the following conditions be
satisfied:

∞
f(x)ψ k (x) dx =0, k =1, 2, ... , |ν|, (19)
0
where ψ k (x) is a basis of the linear space of all solutions of the transposed homogeneous equation

∞
ψ(x) – K(t – x)ψ(t) dt = 0. (20)
0
◦
1 . If conditions (15) and (16) hold, then there exists a unique factorization
ˇ
ˇ
ˇ
[1 – K(u)] –1 = M + (u)M – (u), (21)
and

∞ ∞
ˇ
ˇ
M + (u)=1 + R + (t)e iut dt, M – (u)=1 + R – (t)e –iut dt. (22)
0 0
The resolvent is defined by the formula
∞

R(x, t)= R + (x – t)+ R – (t – x)+ R + (x – s)R – (t – s) ds (23)
0
where 0 ≤ x < ∞,0 ≤ t < ∞, R + (x) = 0, and R – (x)=0 for x < 0, so that, for f(x) from L + , the
solution of the equation is determined by the expression
∞

y(x)= f(x)+ R(x, t)f(t) dt. (24)
0

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 572