A fixed point theorem, a perturbed differential equation, and a multivariable Volterra integral equation
Authors: Lovelady David Lowell 
Published: July 2010

A fixed point theorem is obtained for an equation of the form

u
=
T
[
p
,
f
+
G
[
u
]
]

u = T[p,f + G[u]]

. This theorem is then applied to a functionally perturbed ordinary differential equation of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u prime left-parenthesis t right-parenthesis equals f left-parenthesis t right-parenthesis plus upper A left-parenthesis t comma u left-parenthesis t right-parenthesis right-parenthesis plus upper G left-bracket u right-bracket left-parenthesis t right-parenthesis semicolon u left-parenthesis 0 right-parenthesis equals p">

u
′

(
t
)
=
f
(
t
)
+
A
(
t
,
u
(
t
)
)
+
G
[
u
]
(
t
)
;
u
(
0
)
=
p

u’(t) = f(t) + A(t,u(t)) + G[u](t);u(0) = p

, and, as a consequence of this, Fredholm integrodifferential equations of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z prime left-parenthesis t right-parenthesis equals f left-parenthesis t right-parenthesis plus phi left-parenthesis t comma z left-parenthesis t right-parenthesis right-parenthesis plus integral Subscript 0 Superscript normal infinity Baseline alpha left-parenthesis t comma s right-parenthesis omega left-parenthesis z left-parenthesis s right-parenthesis right-parenthesis d s comma z left-parenthesis 0 right-parenthesis equals p">

z
′

(
t
)
=
f
(
t
)
+
ϕ
(
t
,
z
(
t
)
)
+

∫
0
∞

α
(
t
,
s
)
ω
(
z
(
s
)
)
d
s
,
z
(
0
)
=
p

z’(t) = f(t) + \phi (t,z(t)) + \int _0^\infty {\alpha (t,s)\omega (z(s))ds,z(0) = p}

. Applications are also made to a multivariable Volterra integral system
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row u 1 left-parenthesis s comma t right-parenthesis equals g 1 left-parenthesis s comma t right-parenthesis plus integral Subscript 0 Superscript s Baseline integral Subscript 0 Superscript t Baseline upper F left-bracket u 1 left-parenthesis x comma y right-parenthesis comma u 2 left-parenthesis x comma y right-parenthesis comma u 3 left-parenthesis x comma y right-parenthesis right-bracket d y d x 2nd Row u 2 left-parenthesis s comma t right-parenthesis equals g 2 left-parenthesis s comma t right-parenthesis plus integral Subscript 0 Superscript t Baseline upper F left-bracket u 1 left-parenthesis s comma y right-parenthesis comma u 2 left-parenthesis s comma y right-parenthesis comma u 3 left-parenthesis s comma y right-parenthesis right-bracket d y 3rd Row u 3 left-parenthesis s comma t right-parenthesis equals g 3 left-parenthesis s comma t right-parenthesis plus integral Subscript 0 Superscript s Baseline upper F left-bracket u 1 left-parenthesis x comma t right-parenthesis comma u 2 left-parenthesis x comma t right-parenthesis comma u 3 left-parenthesis x comma t right-parenthesis right-bracket d x comma EndLayout">

u
1

(
s
,
t
)
=

g
1

(
s
,
t
)
+

∫
0
s

∫
0
t

F

[

u
1

(
x
,
y
)
,

u
2

(
x
,
y
)
,

u
3

(
x
,
y
)

]

d
y

d
x

u
2

(
s
,
t
)
=

g
2

(
s
,
t
)
+

∫
0
t

F

[

u
1

(
s
,
y
)
,

u
2

(
s
,
y
)
,

u
3

(
s
,
y
)

]

d
y

u
3

(
s
,
t
)
=

g
3

(
s
,
t
)
+

∫
0
s

F

[

u
1

(
x
,
t
)
,

u
2

(
x
,
t
)
,

u
3

(
x
,
t
)

]

d
x
,

\begin {array}{*{20}{c}} {{u_1}(s,t) = {g_1}(s,t) + \int _0^s {\int _0^t {F\left [ {{u_1}(x,y),{u_2}(x,y),{u_3}(x,y)} \right ]dy\;dx} } } \hfill \\ {{u_2}(s,t) = {g_2}(s,t) + \int _0^t {F\left [ {{u_1}(s,y),{u_2}(s,y),{u_3}(s,y)} \right ]} \;dy} \hfill \\ {{u_3}(s,t) = {g_3}(s,t) + \int _0^s {F\left [ {{u_1}(x,t),{u_2}(x,t),{u_3}(x,t)} \right ]} \;dx,} \hfill \\ \end {array}

\]
and, as a corollary to this, a differential equation of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row StartFraction partial-differential squared Over partial-differential s partial-differential t EndFraction u left-parenthesis s comma t right-parenthesis equals f left-parenthesis s comma t right-parenthesis plus upper F left-bracket u left-parenthesis s comma t right-parenthesis comma StartFraction partial-differential Over partial-differential s EndFraction u left-parenthesis s comma t right-parenthesis comma StartFraction partial-differential Over partial-differential t EndFraction u left-parenthesis s comma t right-parenthesis right-bracket comma 2nd Row u left-parenthesis s comma 0 right-parenthesis equals phi left-parenthesis s right-parenthesis comma u left-parenthesis 0 comma t right-parenthesis equals psi left-parenthesis t right-parenthesis period EndLayout">

∂
2

∂
s
∂
t

u
(
s
,
t
)
=
f
(
s
,
t
)
+
F

[

u
(
s
,
t
)
,

∂

∂
s

u
(
s
,
t
)
,

∂

∂
t

u
(
s
,
t
)

]

,

u
(
s
,
0
)
=
ϕ
(
s
)
,
u
(
0
,
t
)
=
ψ
(
t
)
.

\begin {array}{*{20}{c}} \hfill {\frac {{{\partial ^2}}}{{\partial s\partial t}}u(s,t) = f(s,t) + F\left [ {u(s,t),\frac {\partial }{{\partial s}}u(s,t),\frac {\partial }{{\partial t}}u(s,t)} \right ],} \\ \hfill {u(s,0) = \phi (s),u(0,t) = \psi (t).} \\ \end {array}

\]
These last two equations are set in a Banach space so as to allow applications to integrodifferential equations such as
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row StartFraction partial-differential squared Over partial-differential s partial-differential t EndFraction u left-parenthesis s comma t comma z right-parenthesis equals f left-parenthesis s comma t comma z right-parenthesis plus upper H left-parenthesis z comma u left-parenthesis s comma t comma z right-parenthesis comma StartFraction partial-differential Over partial-differential s EndFraction u left-parenthesis s comma t comma z right-parenthesis comma StartFraction partial-differential Over partial-differential t EndFraction u left-parenthesis s comma t comma z right-parenthesis right-parenthesis 2nd Row plus integral Subscript 0 Superscript 1 Baseline upper K left-parenthesis z comma r comma u left-parenthesis s comma t comma r right-parenthesis comma StartFraction partial-differential Over partial-differential s EndFraction u left-parenthesis s comma t comma r right-parenthesis comma StartFraction partial-differential Over partial-differential t EndFraction u left-parenthesis s comma t comma r right-parenthesis right-parenthesis d r comma 3rd Row u left-parenthesis s comma 0 comma z right-parenthesis equals sigma left-parenthesis s comma z right-parenthesis comma u left-parenthesis 0 comma t comma z right-parenthesis equals tau left-parenthesis t comma z right-parenthesis period EndLayout">

∂
2

∂
s
∂
t

u
(
s
,
t
,
z
)
=
f
(
s
,
t
,
z
)
+
H

(

z
,
u
(
s
,
t
,
z
)
,

∂

∂
s

u
(
s
,
t
,
z
)
,

∂

∂
t

u
(
s
,
t
,
z
)

)

+

∫
0
1

K

(

z
,
r
,
u
(
s
,
t
,
r
)
,

∂

∂
s

u
(
s
,
t
,
r
)
,

∂

∂
t

u
(
s
,
t
,
r
)

)

d
r
,

u
(
s
,
0
,
z
)
=
σ
(
s
,
z
)
,
u
(
0
,
t
,
z
)
=
τ
(
t
,
z
)
.

\begin {array}{*{20}{c}} {\frac {{{\partial ^2}}}{{\partial s\partial t}}u(s,t,z) = f(s,t,z) + H\left ( {z,u(s,t,z),\frac {\partial }{{\partial s}}u(s,t,z),\frac {\partial }{{\partial t}}u(s,t,z)} \right )} \\ { + \int _0^1 {K\left ( {z,r,u(s,t,r),\frac {\partial }{{\partial s}}u(s,t,r),\frac {\partial }{{\partial t}}u(s,t,r)} \right )} \;dr,} \\ {u(s,0,z) = \sigma (s,z),u(0,t,z) = \tau (t,z).} \\ \end {array}

\]