Uniqueness criteria for solutions of singular boundary value problems
Authors: Dunninger D. R. , Levine Howard A. 
Published: June 2010

In this paper we consider the equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel u left-parenthesis t right-parenthesis plus left-parenthesis k slash t right-parenthesis u prime left-parenthesis t right-parenthesis plus upper A u left-parenthesis t right-parenthesis equals 0 comma 0 greater-than t greater-than upper T comma u left-parenthesis upper T right-parenthesis equals 0 comma EndLayout">

(

1

)

u
(
t
)
+
(
k

/

t
)

u
′

(
t
)
+
A
u
(
t
)
=
0
,

0
>
t
>
T
,

u
(
T
)
=
0
,

\begin{equation}\tag {$1$} u(t) + (k/t)u’(t) + Au(t) = 0,\quad 0 > t > T,\quad u(T) = 0,\end{equation}

where

u
:
(
0
,
T
)
→
D
(
A
)
⊂
B

u:(0,T) \to D(A) \subset B

is a Banach space valued function taking values in a dense subdomain

D
(
A
)

D(A)

of the Banach space B. Here A is a closed (possibly unbounded) linear operator on

D
(
A
)

D(A)

while k is a real constant. The differential equation is an abstract Euler-Poisson-Darboux equation. We give necessary and sufficient conditions on the point spectrum of A to insure uniqueness of the strong solution

u
≡
0

u \equiv 0

as well as sufficient conditions on the point spectrum to insure uniqueness of weak solutions. u is only required to satisfy (a)

t
k

‖

u
′

(
t
)

‖

→

0
+

{t^k}\left \| {u’(t)} \right \| \to {0^ + }

as

t
→

0
+

t \to {0^ + }

if

k
>
1

k > 1

, (b)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t Superscript k Baseline double-vertical-bar u prime left-parenthesis t right-parenthesis double-vertical-bar plus t Superscript k plus 1 Baseline double-vertical-bar u left-parenthesis t right-parenthesis double-vertical-bar right-arrow 0">

t
k

‖

u
′

(
t
)

‖

+

t

k
+
1

‖

u
(
t
)

‖

→
0

{t^k}\left \| {u’(t)} \right \| + {t^{k + 1}}\left \| {u(t)} \right \| \to 0

as

t
→

0
+

,
0
>
k
⩽
1

t \to {0^ + },0 > k \leqslant 1

, (c)

t

‖

u
′

(
t
)

‖

+

‖

u
(
t
)

‖

→
0

t\left \| {u’(t)} \right \| + \left \| {u(t)} \right \| \to 0

as

t
→

0
+

,
k
>
0

t \to {0^ + },k > 0

. The operator A need not possess a complete set of eigenvectors nor need one have a backward uniqueness theorem available for (1) for the Cauchy final value problem. Our techniques extend to the n-axially symmetric abstract equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel sigma-summation Underscript i equals 1 Overscript n Endscripts left-bracket partial-differential squared u slash partial-differential t Subscript i Superscript 2 Baseline plus left-parenthesis k Subscript i Baseline slash t Subscript i Baseline right-parenthesis partial-differential u slash partial-differential t Subscript i Baseline right-bracket plus upper A u equals 0 period EndLayout">

(

2

)

∑

i
=
1

n

[

∂
2

u

/

∂

t
i
2

+
(

k
i

/

t
i

)
∂
u

/

∂

t
i

]
+
A
u
=
0.

\begin{equation}\tag {$2$} \sum \limits _{i = 1}^n {[{\partial ^2}u/\partial t_i^2 + ({k_i}/{t_i})\partial u/\partial {t_i}] + Au = 0.} \end{equation}

The proofs rest upon an application of the Hahn-Banach Theorem and the consequent separation properties of

B
∗

{B^\ast }

, the dual of B, as well as the completeness properties of the eigenfunctions of certain Bessel equations associated with (1).