EQUATIONS OF ELASTIC HYPERSURFACES

4. BOUNDARY VALUE PROBLEMS: GENERAL RESULTS 125
Next we will prove the equivalence of conditions (i) and (iii).
By applying the decomposition (4.97) to the representation (4.104) we get
( ) ( d
B j u(t)=
dt
˜B d
j
dt
µ∑
=
since, due to Corollary 4.9,
j=0
A − ( d
dt
)
u(t) (4.105)
e λ− j t˜Bj ( d
dt + λ j
)
p j (t) , j = 0, . . . , µ − 1 ,
)
p j (t) ≡ 0 , j = 0, . . . , µ − 1 .
Thus, we can accept. without restricting generality, that ˜B j (λ) = B j (λ) and, therefore,
ord B j (λ) = m j ≤ µ − 1 , j = 0, . . . , µ − 1 . (4.106)
If we accept (iii) or (iv), the system { B j (λ) } µ−1
j=0
system { λ j} µ−1
, and there exists a unique solution to
j=0
is then linearly independent, as the
λ j =
m − −1
∑
k=0
namely the inverse
The representations
c jk B k (λ) =
m − −1
∑
k=0
m − −1
∑
m=0
c jk b km λ m , j = 0, . . . , m − − 1 , (4.107)
[
cjk
]m − ×m −
= [ b jk
] −1
m − ×m −
.
∣
d j := dj
m − −1 ∣∣∣t=0
dt u(t) ∑
( )
d = c j jk B k u(t)
dt ∣ =
t=0
k=0
m − −1
∑
k=0
j = 0, . . . , m − − 1 ,
c jk u k , (4.108)
which result from (4.107), make it possible to recalculate initial values in (4.92) and rewrite
IVP (4.92) in an equivalent form:
⎧ ( d
⎪⎨ A u(t) = f(t) , t ∈ R
dt)
+ ,
⎪⎩
d j
dt j u(t) ∣
∣∣∣t=0
= d j , j = 0, 1, . . . , m − − 1 .
(4.109)
Consider
u(t) := 1
m − −1
∑
∫
(−i) k d k
2πi
k=0
γ
A − m − −k−1 (τ)
e itτ dτ , (4.110)
A − (τ)