Iterative Optimization in Inverse Problems

232 Iterative Optimization in Inverse Problems

16.9 The Split Common Null-Point Problem

The split common null-point problem (SCNPP) [69] is related to the

SMVIP. Let H

1

and H

2

be real Hilbert spaces. Let B

i

: H

1

→ 2

H

1

,for

i =1, ..., p,andF

j

: H

2

→ 2

H

2

,forj =1, ..., r, be set-valued mappings, and

A

j

: H

1

→H

2

be bounded linear operators. The SCNPP is the following:

find a point x

∗

in H

1

such that

0 ∈∩

p

i=1

B

i

(x

∗

),

and such that, for y

∗

j

= A

j

(x

∗

), we have

0 ∈∩

r

j=1

F

j

(y

∗

j

).

16.10 Exercises

Ex. 16.1 In R

2

,letA and B be the closed circles with radius one cen-

tered at (−1, 0) and (1, 0), respectively. Show that N

A∩B

((0, 0)) = R

2

,while

N

A

((0, 0)) + N

B

((0, 0)) is the x-axis.