Existence of integrals and the solution of integral equations
Authors: Helton Jon C. 
Published: July 2010

Functions are from R to N or

R
×
R

R \times R

to N, where R denotes the real numbers and N denotes a normed complete ring. If S, T and G are functions from

R
×
R

R \times R

to N, each of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S left-parenthesis p Superscript minus Baseline comma p right-parenthesis comma upper S left-parenthesis p Superscript minus Baseline comma p Superscript minus Baseline right-parenthesis comma upper T left-parenthesis p Superscript minus Baseline comma p right-parenthesis">

S
(

p
−

,
p
)
,
S
(

p
−

,

p
−

)
,
T
(

p
−

,
p
)

S({p^ - },p),S({p^ - },{p^ - }),T({p^ - },p)

and

T
(

p
−

,

p
−

)

T({p^ - },{p^ - })

exists for

a
>
p
⩽
b

a > p \leqslant b

, each of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S left-parenthesis p comma p Superscript plus Baseline right-parenthesis comma upper S left-parenthesis p Superscript plus Baseline comma p Superscript plus Baseline right-parenthesis comma upper T left-parenthesis p comma p Superscript plus Baseline right-parenthesis">

S
(
p
,

p
+

)
,
S
(

p
+

,

p
+

)
,
T
(
p
,

p
+

)

S(p,{p^ + }),S({p^ + },{p^ + }),T(p,{p^ + })

and

T
(

p
+

,

p
+

)

T({p^ + },{p^ + })

exists for

a
⩽
p
>
b

a \leqslant p > b

, G has bounded variation on [a, b] and

∫
a
b

G

\smallint _a^bG

exists, then each of
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript a Superscript b Baseline upper S left-bracket upper G minus integral upper G right-bracket upper T and integral Subscript a Superscript b Baseline upper S left-bracket 1 plus upper G minus product left-parenthesis 1 plus upper G right-parenthesis right-bracket upper T">

∫
a
b

S

[

G
−
∫
G

]

T

and

∫
a
b

S

[

1
+
G
−
∏

(
1
+
G
)

]

T

\int _a^b S \left [ {G - \int G } \right ]T\quad {\text {and}}\quad \int _a^b {S\left [ {1 + G - \prod {(1 + G)} } \right ]} \;T

\]
exists and is zero. These results can be used to solve integral equations without the existence of integrals of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript a Superscript b Baseline StartAbsoluteValue upper G minus integral upper G EndAbsoluteValue equals 0 and integral Subscript a Superscript b Baseline StartAbsoluteValue 1 plus upper G minus product left-parenthesis 1 plus upper G right-parenthesis EndAbsoluteValue equals 0 period">

∫
a
b

|

G
−
∫
G

|

=
0

and

∫
a
b

|

1
+
G
−
∏

(
1
+
G
)

|

=
0.

\int _a^b {\left | {G - \int G } \right | = 0} \quad {\text {and}}\quad \int _a^b {\left | {1 + G - \prod {(1 + G)} } \right |} = 0.

\]
This is demonstrated by solving the linear integral equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals h left-parenthesis x right-parenthesis plus left-parenthesis upper L upper R right-parenthesis integral Subscript a Superscript x Baseline left-parenthesis f upper G plus f upper H right-parenthesis">

f
(
x
)
=
h
(
x
)
+
(
L
R
)

∫
a
x

(
f
G
+
f
H
)

f(x) = h(x) + (LR)\int _a^x {(fG + fH)}

\]
and the Riccati integral equations
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals w left-parenthesis x right-parenthesis plus left-parenthesis upper L upper R upper L upper R right-parenthesis integral Subscript a Superscript x Baseline left-parenthesis f upper H plus upper G f plus f upper K f right-parenthesis">

f
(
x
)
=
w
(
x
)
+
(
L
R
L
R
)

∫
a
x

(
f
H
+
G
f
+
f
K
f
)

f(x) = w(x) + (LRLR)\int _a^x {(fH + Gf + fKf)}

\]
without the existence of the previously mentioned integrals.