A NOTE ON GLOBAL IMPLICIT FUNCTION
THEOREM
Global Implicit Function Theorem
MIHAI CRISTEA
University of Bucharest
Faculty of Mathematics
Str. Academiei 14, R-010014,
Bucharest, Romania
EMail: mcristea@fmi.unibuc.ro
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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01 May, 2006
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Communicated by:
G. Kohr
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2000 AMS Sub. Class.:
26B10, 46C05.
Key words:
Global implicit function, Boundary behaviour of a maximal implicit function.
Abstract:
We study the boundary behaviour of some certain maximal implicit function. We
give estimates of the maximal balls on which some implicit functions are defined
and we consider some cases when the implicit function is globally defined. We
extend in this way an earlier result from [3] concerning an inequality satisfied by
the partial derivatives ∂h
and ∂h
of the map h which verifies the global implicit
∂x
∂y
function problem
h (t, x) = h (a, b) , x (a) = b.
Received:
21 April, 2006
Accepted:
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The implicit function theorem is a classical result in mathematical analysis. Local
versions can be found in [1],[8], [10], [13], [15], [17], [18] and some papers deal with
some global versions (see [2], [3], [9], [16]).
We first give some local versions of the implicit function theorem, using our local
homeomorphism theorem from [4].
Theorem 1. Let E, F be Banach spaces, dim F < ∞, U ⊂ E open, V ⊂ F
open, h : U × V → F continuous such that there exists K ⊂ U × V countable
such that h is differentiable on (U × V ) \ K and ∂h
(x, y) ∈ Isom (F, F ) for every
∂y
(x, y) ∈ (U × V ) \ K and let A = Pr1 K ⊂ U . Then, for every (a, b) ∈ U × V
there exists r, δ > 0 and a unique continuous map ϕ : B (a, r) → B (b, δ) such that
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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ϕ (a) = b and h (x, ϕ (x)) = h (a, b) for every x ∈ B (a, r)
and ϕ is differentiable on B (a, r) \A.
Proof. Let (a, b) ∈ U × V be fixed and f : U × V → E × F be defined by
f (x, y) = (x, h (x, y) + b − h (a, b)) for (x, y) ∈ U × V.
Also, let T : U × V → E × F ,
T (x, y) = (0, y − h (x, y) + h (a, b) − b) for (x, y) ∈ U × V.
Then ImT ⊂ F , hence T is compact and we see that f = I − T, f is differentiable on (U × V ) \ K and f 0 (x, y) ∈ Isom (E × F, E × F ) for every (x, y) ∈
(U × V ) \ K. Using the local inversion theorem from [4], we see that f is a local
homeomorphism on U × V. Let W ∈ V ((a, b)) and δ > 0 be such that
f |B (a, δ) × B (b, δ) : B (a, δ) × B (b, δ) → W
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is a homeomorphism and let
g = (g1 , g2 ) : W → B (a, δ) × B (b, δ)
be its inverse. We take ` > 0 such that Q = B (a, b) × (b, `) ⊂ W and let r =
min {`, δ}. We have
(x, z) = f (g (x, z))
= f (g1 (x, z) , g2 (x, z))
= (g1 (x, z) , h (g1 (x, z) , g2 (x, z)) + b − h (a, b))
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
for every (x, z) ∈ Q, hence
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x = g1 (x, z) ,
h (x, g2 (x, z)) = z + h (a, b) − b
for x ∈ B(a, r), z ∈ B(b, `).
We define now ϕ : B (a, r) → B (b, δ) by ϕ (x) = g2 (x, b) for every x ∈
B (a, r) and we see that h (x, ϕ (x)) = h (a, b) for every x ∈ B (a, r). We have
f (a, b) = (a, b) = f (a, ϕ (a)) and using the injectivity of f on B (a, δ) × B (b, δ) ,
we see that ϕ (a) = b. Also, if ψ : B (a, r) → B (b, δ) is continuous and ψ (a) =
b, h (x, ψ (x)) = h (a, b) for every x ∈ B (a, r) , then f (x, ϕ (x)) = (x, b) =
f (x, Ψ (x)) for every x ∈ B (a, r) and using again the injectivity of the map f
on B (a, δ) × B (b, δ), we find that ϕ = ψ on B (a, r).
Let now x0 ∈ B (a, r) \A. Then (x0 , b) = f (x0 , β), with (x0 , β) ∈ (B(a, r) ×
B(b, δ)\K), hence f is differentiable in (x0 , β) , f 0 (x0 , β) ∈ Isom (E × F, E × F )
and since f is a homeomorphism on B(a, r) × B(b, δ), it results that g is also differentiable in (x0 , b) = f (x0 , β) and g 0 (x0 , b) = [f 0 (x0 , β)−1 ], and we see that ϕ is
differentiable in x0 .
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Theorem 2. Let E be an infinite dimensional Banach space, dim F < ∞, U ⊂ E,
V ⊂ F be open sets, h : U ×V → F be continuous such that there exists K ⊂ U ×V ,
∞
[
K=
Kp
p=1
with Kp compact sets for p ∈ N such that h is differentiable on (U × V ) \K, there
exists ∂h
on U × V and ∂h
(x, y) ∈ Isom (F, F ) for every (x, y) ∈ U × V and let
∂y
∂y
A = Pr1 K. Then, for every (a, b) ∈ U × V there exists r, δ > 0 and a unique
continuous implicit function ϕ : B (a, r) → B (b, δ) differentiable on B (a, r) \A
such that ϕ (a) = b and h (x, ϕ (x)) = h (a, b) for every x ∈ B (a, r) .
Proof. We apply Theorem 11 of [8]. We see that in an infinite dimensional Banach
space E, a set K which is a countable union of compact sets is a "thin" set, i.e.,
int K = φ and B\K is connected and simply connected for every ball B from E.
Also, since A is a countable union of compact sets, we see that int A = φ.
If E, F are Banach spaces and A ∈ L (E, F ), we let
kAk = sup kA (x)k
kxk=1
and
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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` (A) = inf kA (x)k
kxk=1
and if D ⊂ E, λ > 0, we let
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λD = {x ∈ E| there exists y ∈ D such that x = λy}.
If X, Y are Banach spaces, D ⊂ X is open, x ∈ D and f : D → Y is a map, we let
D+ f (x) = lim sup
y→x
kf (y) − f (x)k
ky − xk
and we say that f is a light map if for every x ∈ D and every U ∈ V (x), there exists
Q ∈ V (x) such that Q ⊂ U and f (x) ∈
/ f (∂Q) .
Remark 1. We can replace in Theorem 1 and Theorem 2 the condition "dim F <
on U × V and it is continuous on U × V and ∂h
(x, y) ∈
∞ " by "There exists ∂h
∂y
∂y
Isom (F, F ) for every (x, y) ∈ U × V " to obtain the same conclusion, and this is the
classical implicit function theorem. Also, keeping the notations from Theorem 1 and
Theorem 2, we see that if (α, β) ∈ B (a, r)×B (b, δ) is such that h (α, β) = h (a, b),
then β = ϕ (α) .
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
We shall use the following lemma from [7].
Lemma 3. Let a > 0, f : [0, a] → [0, ∞) be continuous and let ω : [0, ∞) → [0, ∞)
be continuous such that ω > 0 on (0, ∞) and
Z c
|f (b) − f (c)| ≤
ω (f (t)) dt for every 0 < b < c ≤ a.
b
Then, if
m = inf f (t) ,
M = sup f (t) ,
t∈[0,a]
t∈[0,a]
it results that
Global Implicit Function Theorem
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Z
M
m
ds
≤ a.
ω (s)
We obtain now the following characterization of the boundary behaviour of the
solutions of some differential inequalities.
Theorem 4. Let E, F be Banach spaces, U ⊂ E a domain, K ⊂ U at most countable, ϕ : U → F continuous on U and differentiable on U \K such that there exists
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ω : [0, ∞) → [0, ∞) continuous with kϕ0 (x)k ≤ ω (kϕ (x)k) for every x ∈ U \K.
Then, if α ∈ ∂U and C ⊂ U is convex such that α ∈ C, either there exists
lim ϕ (x) = ` ∈ F or x→α
lim kϕ (x)k = ∞
x→α
x∈C
x∈C
or, if ω > 0 on (1, ∞) and
∞
Z
1
ds
= ∞,
ω (s)
there exists
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
lim ϕ (x) = ` ∈ F.
x→α
x∈C
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If ω > 0 on (0, 1) and
Z
0
1
ds
=∞
ω (s)
and there exists α ∈ U such that ϕ (α) = 0, it results that ϕ (x) = 0, for every
x ∈ U.
Proof. Replacing, if necessary, ω by ω + λ for some λ > 0, we can suppose that
ω > 0 on [0, ∞). Let α ∈ ∂U, C ⊂ U convex such that α ∈ C and let q : [0, 1) → C
be a path such that
lim q (t) = α
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t→1
and there exists L > 0 such that Dq+ (t) ≤ L for every t ∈ [0, 1). Then
kq (s) − q (t)k ≤ L · (s − t)
for every 0 ≤ t < s < 1 and let 0 ≤ c < d < 1 be fixed,
A = co (q ([c, d]))
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and ε > 0. Let g : A → R be defined by
g (z) = ω (kϕ (z)k) for every z ∈ A.
Then A is compact and convex and g is uniformly continuous on A, hence we can
find δε0 > 0 such that
|g (z1 ) − g (z2 )| ≤ ε for z1 , z2 ∈ A
Global Implicit Function Theorem
with kz1 − z2 k ≤ δε0 . Since q : [c, d] → C is uniformly continuous, we can find
δε > 0 such that kq (t) − q (s)k ≤ δε0 if s, t ∈ [c, d] are such that |s − t| ≤ δε . Let
now
∆ = (c = t0 < t1 < · · · < tm = d) ∈ D ([c, d])
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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be such that k∆k ≤ δε . Using Denjoi-Bourbaki’s theorem we have
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|kϕ (q (d))k − kϕ (q (c))k| ≤ kϕ (q (d)) − ϕ (q (c))k
m−1
X
≤
ϕ (q (tk+1 )) − ϕ (q (tk ))
≤
k=0
m−1
X
k(q (tk+1 ) − q (tk ))k ·
k=0
m−1
X
≤L·
≤L·
k=0
m−1
X
k=0
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sup
kϕ0 (z)k
z∈[q(tk ),q(tk+1 )] \ K
(tl+1 − tk ) ·
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sup
ω (kϕ (z)k)
z∈[q(tk ),q(tk+1 )]
(tl+1 − tk ) · (ω (kϕ(q (tk )k + ε) .
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Letting k∆k → 0 and then ε → 0, we obtain
| kϕ (q (d))k − kϕ (q (c))k| ≤ kϕ (q (d)) − ϕ (q (c))k
Z d
(1)
≤L·
ω (kϕ (q (t))k) dt for 0 ≤ c < d < 1.
c
If
m = inf kϕ (q (t))k ,
t∈[0,1)
Global Implicit Function Theorem
M = sup kϕ (q (t))k ,
Mihai Cristea
t∈[0,1)
vol. 8, iss. 4, art. 100, 2007
we obtain from Lemma 3 and (1)
Z
M
(2)
m
ds
≤ L.
ω (s)
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Let now zp → α be such that
1
kzp − αk ≤ P ,
2
zp ∈ C for p ∈ N
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and suppose that there exists ρ > 0 such that kϕ (zp )k ≤ ρ for every p ∈ N. We take
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0 = t0 < t1 < · · · < tk < tk+1 < · · · < 1
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such that tk % 1 and we define q : [ 0, 1) → C by
zk (tk+1 − t) + zk+1 (t − tk )
for t ∈ [tk , tk+1 ] ,
q (t) =
tk+1 − tk
Then
D+ q (t) = ck =
kzk+1 − zk k
for t ∈ [tk , tk+1 ]
tk+1 − tk
Close
k ∈ N.
and taking tk =
k
k+1
for k ∈ N, we see that ck → 0. Then
αp = sup D+ q (t) = sup ck → 0
k≥p
t∈[tp ,1)
and let
ap = inf kϕ (q (t))k ,
bp = sup kϕ (q (t))k for p ∈ N.
t∈[tp ,1)
t∈[tp ,1)
Global Implicit Function Theorem
Mihai Cristea
Using (2) we obtain that
vol. 8, iss. 4, art. 100, 2007
Z
bp
ap
ds
≤ αp for p ∈ N
ω (s)
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and let p0 ∈ N be such that
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∞
Z
αp <
ρ
ds
for p ≥ p0 .
ω (s)
Suppose that there exists p ≥ p0 such that bp = ∞. Then, since q (tk ) = zk , we see
that
ak ≤ kϕ (q (tk ))k = kϕ (zk )k ≤ ρ for k ∈ N,
hence
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Z
0<
ρ
∞
ds
≤
ω (s)
Z
bp
ap
ds
≤ αp <
ω (s)
and we have reached a contradiction.
It results that bp < ∞ for p ≥ p0 and let
Kp = sup ω (t)
t∈[0,bp ]
Z
ρ
∞
ds
ω (s)
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for p ≥ p0 . Then Kp < ∞ and we see from (1) that
kϕ (q (d)) − ϕ (q (c))k ≤ αp · Kp · |d − c|
for tp ≤ c < d < 1 and p ≥ p0 , and this implies that
lim ϕ (q (t)) = ` ∈ F.
t→1
Global Implicit Function Theorem
It results that
Mihai Cristea
lim ϕ (zp ) = lim ϕ (q (tp )) = ` ∈ F.
vol. 8, iss. 4, art. 100, 2007
lim kϕ (x)k = ∞
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p→∞
p→∞
Now, if the case
x→α
x∈C
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does not hold, there exists ρ > 0 and xp → α, xp ∈ C with
||ϕ(xp )|| ≤ ρ and ||xp − α|| ≤
1
2p
for every p ∈ N, and from what we have proved before, it results that
lim ϕ (xp ) = ` ∈ F.
p→∞
If ap ∈ C, kap − αk ≤
1
2p
for every p ∈ N, then
lim ϕ (ap ) = `1 ∈ F.
p→∞
Let z2p = xp , z2p+1 = ap for p ∈ N. We see that
lim ϕ (zp ) = `2 ∈ F,
p→∞
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hence
` = lim ϕ (xp ) = lim ϕ (z2p ) = `2
p→∞
p→∞
and
`1 = lim ϕ (ap ) = lim ϕ (z2p+1 ) = `2 ,
p→∞
p→∞
hence ` = `1 = `2 . We have proved that if ap ∈ C, kap − αk ≤
then
lim ϕ (ap ) = `.
1
2p
p→∞
for every p ∈ N,
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
We show now that if ap ∈ C, ap → α, then
lim ϕ (ap ) = `.
p→∞
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Indeed, if this is false, there exists ε > 0 and (apk )k∈N such that
for every k ∈ N. Let apkq
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kϕ (apk ) − `k > ε
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be a subsequence such that
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q∈N
1
apkq − α < q
2
for every q ∈ N. From what we have proved before it results that
lim ϕ apkq = `
q→∞
and we have reached a contradiction.
We have therefore proved that either
lim ϕ (x) = ` ∈ F,
x→α
x∈C
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or
lim kϕ(x)k = ∞.
x→α
x∈C
Suppose now
Z
∞
1
ds
=∞
ω (s)
and let α ∈ ∂U . We take x ∈ C and let q : [0, 1) → C be defined by
q (t) = (1 − t) x + tα
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
for t ∈ [0, 1) and
m = inf kϕ (q (t))k ,
M = sup kϕ (q (t))k .
t∈[0,1)
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t∈[0,1)
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Since D+ q (t) = kx − αk for every t ∈ (0, 1), we see from (2)
Z M
Z M
ds
ds
≤
≤ kx − αk
kxk ω (s)
m ω (s)
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and this implies that M < ∞. Let
b = sup ω (t) .
t∈[0,M ]
Then b < ∞ and using (1), we see that
kϕ (q (d)) − ϕ (q (c))k ≤ b · kx − αk · |d − c|
for 0 ≤ c < d < 1 and this implies that
lim ϕ (x) = ` ∈ F.
x→α
x∈C
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It results that the case
lim kϕ (x)k = ∞
x→α
x∈C
cannot hold, hence
lim ϕ (x) = ` ∈ F.
x→α
Suppose now that
Z
1
ds
=∞
0 ω (s)
and that there exists α ∈ U such that ϕ (α) = 0. Let r = d (α, ∂U ) , y ∈ B (α, r)
and q : [0, 1] → B (α, r) , q (t) = (1 − t) α + ty for t ∈ [0, 1]. Then D+ q (t) =
ky − αk for t ∈ [0, 1] and let
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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m = inf kϕ (q (t))k
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t∈[0,1]
and
M = sup kϕ (q (t))k .
t∈[0,1]
Then m = 0 and we see from (2) that
Z M
ds
≤ ky − αk .
ω (s)
0
This implies that M = 0 and hence ϕ (y) = 0. We proved that ϕ ≡ 0 on B (α, r)
and since U is a domain, we see that ϕ ≡ 0 on U.
Remark 2. We proved that if ϕ is as in Theorem 2 and
Z ∞
ds
= ∞,
ω (s)
1
then ϕ has angular limits in every point α ∈ ∂U.
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We now obtain the following characterization of the boundary behaviour of some
implicit function.
Theorem 5. Let E, F be Banach spaces, U ⊂ E a domain, K ⊂ U × F such that
A = Pr1 K is at most countable and let h : U × F → F be continuous on U × F ,
differentiable on (U × F ) \K such that
∂h
`
(x, y) > 0 on (U × F ) \K
∂y
and there exists ω : [0, ∞) → [0, ∞) continuous such that
∂h
∂h
∂x (x, y) ` ∂y (x, y) ≤ ω (kyk)
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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for every (x, y) ∈ (U × F ) \K. Suppose that ϕ : U → F is continuous on U ,
differentiable on U \A, ϕ (a) = b and
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h (x, ϕ (x)) = h (a, b)
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for every x ∈ U.
Then, if α ∈ ∂U and C ⊂ U is convex such that α ∈ C, either
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lim kϕ (x)k = ` ∈ F,
x→α
x∈C
or
lim kϕ (x)k = ∞.
x→α
x∈C
Also, if ω > 0 on (1, ∞) and
Z
1
∞
ds
= ∞,
ω (s)
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then
lim ϕ (x) = ` ∈ F.
x→α
x∈C
Proof. We see that if x ∈ U \ A, then (x, ϕ (x)) ∈ (U × F ) \ K, hence h is
differentiable in (x, ϕ (x)) and we have
∂h
∂h
(x, ϕ (x)) +
(x, ϕ (x)) · ϕ0 (x) = 0
∂x
∂y
and we see that
∂h
∂h
∂h
0
0
(x, ϕ (x)) ≤ (x, ϕ (x)) (ϕ (x)) = (x, ϕ (x))
kϕ (x)k · `
.
∂y
∂y
∂x
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Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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It results that
∂h
∂h
kϕ (x)k ≤ (x, ϕ (x)) `
(x, ϕ (x)) ≤ ω (kϕ (x)k)
∂x
∂y
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0
for every x ∈ U \ A and we now apply Theorem 4.
Remark 3. If E is an infinite dimensional Banach space and
K=
∞
[
Kp with Kp ⊂ E
M (K, y) = {w ∈ E| there exists t > 0 and x ∈ K such that w = tx}
is also a countable union of compact sets and hence a "thin" set. Keeping the notations from Theorem 4, we see that the basic inequality
z∈[z1 ,z2 ]
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are compact sets for every p ∈ N and y ∈ E, then the set
kϕ (z1 ) − ϕ (z2 )k ≤ sup ω (kϕ (z)k)
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p=1
(3)
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if [z1 , z2 ] ⊂ U
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is also valid for K a countable union of compact sets and ϕ as in Theorem 4.
If S
dim E = n and K ⊂ E has a σ-finite (n − 1)-dimensional measure (i.e.
K= ∞
p=1 Kp , with mn−1 (Kp ) < ∞ for every p ∈ N, where mq is the q-Hausdorff
measure from Rn ), a theorem of Gross shows that if H ⊂ E is a hyperplane and
P : E → H is the projection on H, then P −1 (z) ∩ K is at most countable with the
possible exception of a set B ⊂ H, with mn−1 (B) = 0. Applying as in Theorem
4 the theorem of Denjoi and Bourbaki on each interval from P −1 (z) ∩ K for every
z ∈ H \ B and using a natural limiting process, we see that if dim E = n and K ⊂
E has a σ-finite (n − 1)-dimensional measure, then the inequality (3) also holds. It
is easy see now that Theorem 4 and Theorem 5 hold if the set K, respectively the set
A = Pr1 K are chosen to be a countable union of compact sets if dim E = ∞ and
having σ-finite (n − 1)-dimensional measure if dim E = n.
The following theorem is the main theorem of the paper and it gives some cases
when the implicit function is globally defined or some estimates of the maximal balls
on which some implicit function is defined.
We say that a domain D from a Banach space is starlike with respect to the point
a ∈ D if [a, x] ⊂ D for every x ∈ D, and if D is a domain in the Banach space E
and a ∈ D. We set
Da = {x ∈ D| [a, x] ⊂ D} .
Theorem 6. Let E, F be Banach spaces, dim F < ∞, D ⊂ E a domain, K ⊂
D × F at most countable, and A = Pr1 K.Also, let h : D × F → F be continuous
on D × F and differentiable on (D × F ) \K such that
∂h
`
(x, y) > 0 on (D × F ) \K.
∂y
In addition, there exists ω : [0, ∞) → [0, ∞) continuous such that ω > 0 on (0, ∞)
Global Implicit Function Theorem
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vol. 8, iss. 4, art. 100, 2007
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and
∂h
∂h
∂x (x, y) ` ∂y (x, y) ≤ ω (kyk)
for every (x, y) ∈ (D × F ) \K. Then, if (a, b) ∈ D × F and
Z ∞
ds
Qa,b = Da ∩ B a,
,
kbk ω (s)
there exists a unique continuous map ϕ : Qa,b → F , differentiable on Qa,b \A such
that h (x, ϕ (x)) = h (a, b) for every x ∈ Qa,b . If D is starlike with respect to a and
Z ∞
ds
= ∞,
ω (s)
1
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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Contents
then Qa,b = D and ϕ : D → F is globally defined on D.
Proof. Let z ∈ Qa,b and let B = {x ∈ [a, z]| there exists an open, convex domain
Dx ⊂ Qa,b such that [a, x] ⊂ Dx } and a continuous implicit function ϕx : Dx → F,
differentiable on Dx \A such that ϕx (a) = b and h (u, ϕx (u)) = h (a, b) for every
u ∈ Dx . We see that B is open, and from Theorem 1, B 6= ∅. We show that B is
a closed set. Let xp ∈ B, xp → x, and we can suppose that xp ∈ [a, x) for every
p ∈ N, and let
∞
[
Q=
Dxp .
p=1
We define Ψ : Q → F by Ψ (u) = ϕxp (u) for u ∈ Dxp and p ∈ N and the definition
is correct. Indeed, if p, q ∈ N, p 6= q let Upq = Dxp ∩ Dxq . Then Upq is a nonempty,
open and convex set, hence it is a nonempty domain. If
Vpq = u ∈ Upq |ϕxq (u) = ϕxp (u) ,
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we see that a ∈ Vpq , hence Vpq 6= ∅ and we see that Vpq is a closed set in Upq . Using
the property of the local unicity of the implicit function from Theorem 1, we obtain
that Vpq is also an open set. Since Upq is a domain, it results that Upq = Vpq and
hence that Ψ is correctly defined. We also see immediately that Ψ (a) = b, and Ψ is
continuous on Q and differentiable on Q \ A.
Let q : [0, 1] → Q be defined by
q (t) = (1 − t) a + tx for t ∈ [0, 1] .
Global Implicit Function Theorem
Mihai Cristea
Then
D+ q (t) = kx − ak <
Z
∞
||b|
ds
.
ω (s)
vol. 8, iss. 4, art. 100, 2007
Title Page
Let
m = inf kΨ (q (t))k ,
t∈[0,1)
M = sup kΨ (q (t))k .
Contents
t∈[0,1)
As in Theorem 5, we see that
kΨ0 (u)k ≤ ω (kΨ (u)k) for every u ∈ Q \A,
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hence
kΨ (z1 ) − Ψ (z2 )k ≤ sup ω (kΨ (z)k)
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z∈[z1 ,z2 ]
if [z1 , z2 ] ⊂ Q. This implies that relations (1) and (2) from Theorem 4 also hold and
we see that
Z M
ds
≤ kx − ak .
m ω (s)
Then
Z M
Z M
ds
ds
≤
≤ kx − ak
kbk ω (s)
m ω (s)
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and this implies that M < ∞, hence
` = sup ω (t) < ∞.
t∈[0,M ]
Using (1) and Theorem 4, we see that
kΨ (q (d)) − Ψ (q (c))k ≤ ` · kx − ak · |d − c| for every 0 ≤ c < d < 1
Global Implicit Function Theorem
and this implies that
lim
Ψ (u) = w ∈ F.
u→x
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
u∈Im q
Using Theorem 1, we can find r, δ > 0 and a unique continuous implicit function
Ψx : B (x, r) → B (w, δ), differentiable on B (x, r) \ A such that
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Ψx (x) = w and h (u, Ψx (u)) = h (a, b)
Contents
for every u ∈ B (x, r). Let 0 < ε < r and pε ∈ N be such that
kxp − xk < ε and kΨ (xp ) − wk < δ
for p ≥ pε and let p ≥ pε be fixed. Since
ϕxp (xp ) = Ψ (xp ) ∈ B (w, δ) ,
we see from Remark 1 that ϕxp (xp ) = Ψx (xp ) and hence the set
U = u ∈ Dxp ∩ B (x, ε) |ϕxp (u) = Ψx (u)
is nonempty. We also see that Dxp ∩B (x, ε) is an open, nonempty, convex set, hence
it is a domain and U is an open, closed and nonempty subset of Dxp ∩ B (x, ε), and
this implies that
U = Dxp ∩ B (x, ε) .
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Let U0 = Dxp ∪ B (x, ε). We can now correctly define Φ : U0 → F by
Φ (u) = ϕxp (u) if u ∈ Dxp
and
Φ (u) = Ψx (u) if u ∈ B (x, ε)
and we see that Φ is continuous on U0 , differentiable on U0 \ A, Φ (a) = b and
h (u, Φ (u)) = h (a, b) for every u ∈ U0 . It results that x ∈ B, hence B is also a
closed set and since [a, z] is a connected set, we see that B = [a, z] .
We have therefore proved that for every z ∈ Qa,b there exists a convex domain Dz
such that [a, z] ⊂ Dz and a unique continuous implicit function ϕz : Dz → F ,
differentiable on Dz \ A such that
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
Title Page
ϕz (a) = b, h (u, ϕz (u)) = h (a, b) for every u ∈ Dz .
We now define ϕ : Qa,b → F by ϕ (x) = ϕz (x) for x ∈ Dz and we see, as before,
that the definition is correct, that ϕ (a) = b, ϕ is continuous on Qa,b , differentiable
on Qa,b \ A and h (x, ϕ (x)) = h (a, b) for every x ∈ Qa,b .
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Remark 4. The result from Theorem 6 extends a global implicit function theorem
from [3]. The result from [3] also involves an inequality containing
∂h
∂h
∂x (x, y) and ` ∂y (x, y)
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and it says that if E, F are Banach spaces, h : E × F → F is a C 1 map such that
∂h
(x, y) ∈ Isom (F, F ) for every (x, y) ∈ E × F and there exists ω : [0, ∞) →
∂y
(0, ∞) continuous such that
∂h
∂h
1+
(x, y)
`
(x, y) ≤ ω (max (kxk , kyk))
∂x
∂y
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for every (x, y) ∈ E × F , then, for (x0, y0 ) ∈ E × F, z0 = h (x0 , y0 ) and
Z ∞
ds
,
r=
max(kx0 k,ky0 k) 1 + ω (s)
there exists a C 1 map ϕ : B (x0 , r) × B (z0 , r) → F such that h (x, ϕ (x, z)) = z
for every (x, z) ∈ B (x0 , r) × B (z0 , r). The main advantage of our new global
implicit function theorems is that these theorems hold even if the map h is defined
on a proper subset of E × F, namely, on a set D × F ⊂ E × F , where D ⊂ E is an
open starlike domain.
Example 1. A known global inversion theorem of Hadamard, Lévy and John says
that if E, F are Banach spaces, f : E → F is a C 1 map such that f 0 (x) ∈
Isom (E, F ) for every x ∈ E and there exists ω : [0, ∞) → (0, ∞) continuous
such that
Z ∞
ds
= ∞ and f 0 (x)−1 ≤ ω (kxk)
ω (s)
1
for every x ∈ E, then it results that f : E → F is a C 1 diffeomorphism (see [11],
[14], [12], [3], [7]). If E = F = Rn or if dim E = ∞ and f = I − T with
T compact, we can drop the continuity of the derivative on E and we can impose the
essential condition "f 0 (x) ∈ Isom (F, F )" with the possible exception of a "thin" set
(see [4], [5],[6]) and we will still obtain that f : E → F is a homeomorphism.
Now, let E, F be Banach spaces, D ⊂ E a domain, a ∈ D, b ∈ F, g : D → F
be differentiable on D, f : F → F be differentiable on F such that f 0 (y) ∈
Isom (F, F ) for every y ∈ F and there exists ω : [0, ∞) → (0, ∞) continuous
such that
0 −1 f (y) ≤ ω (kyk) for every y ∈ F.
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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Let h : D × F → F be defined by h (x, y) = f (y) − g (x) for x ∈ D, y ∈ F ,
r0 = d (a, ∂D) and suppose that
Mr = sup kg 0 (x)k < ∞ for every 0 < r ≤ r0 .
x∈B(a,r)
Then
∂h
0 −1 ∂h
0
`
f (y) ≤ Mr · ω (kyk)
(x,
y)
(x,
y)
≤
kg
(x)k
·
∂x
∂y
if (x, y) ∈ B (a, r) × F and 0 < r ≤ r0 and let
Z ∞
ds
1
·
for 0 < r ≤ r0 .
δr = min r,
Mr kbk ω (s)
Using Theorem 6, we see that there exists a unique differentiable map ϕ : B (a, δr ) →
F such that ϕ (a) = b and h (x, ϕ (x)) = h (a, b) for every x ∈ B (a, δr ), i.e.
f (ϕ (x)) = g (x) + h (a, b) for every x ∈ B (a, r) and every 0 < r ≤ r0 .
If
Z ∞
ds
r0 · Mr0 <
,
kbk ω (s)
then ϕ is defined on B (a, r0 ). Additionally, if D = B (a, r0 ), then ϕ is globally
defined on D and f ◦ ϕ = g + h (a, b) on D. In the special case D = E, g (x) = x
for every x ∈ E,
Z ∞
ds
= ∞ and b = f (a) ,
ω (s)
1
then f (ϕ (x)) = x for every x ∈ E, and ϕ is defined on E and is the inverse of f and
it results that f : F → F is a homeomorphism. In this way we obtain an alternative
proof of the Hadamard-Lévy-John theorem.
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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Remark 5. The global implicit function problem
h (t, x) = h (a, b) ,
x (a) = b
considered before has two basic properties:
1. It satisfies the differential inequality kϕ0 (x)k ≤ ω (kϕ (x)k) .
2. It has the property of the local existence and local unicity of the solutions
around each point (t0 , x0 ) .
This shows that by considering some other conditions of local existence and local unicity of the implicit function instead of the conditions from Theorem 1, we
can produce corresponding global implicit function results. Using the conditions of
local existence and local unicity from Theorem 11 of [8], we obtain the following
corresponding version of Theorem 6.
Theorem 7. Let E, F be Banach spaces, dim E = ∞, dim F < ∞, D ⊂ E a
domain, K ⊂ D × F ,
∞
[
K=
Kp
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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p=1
with Kp compact sets for every p ∈ N, A = Pr1 K, h : D × F → F continuous on
D × F and differentiable on (D × F ) \K such that
∂h
`
(x, y) > 0 on (D × F ) \K,
∂y
and there exists ω : [0, ∞) → [0, ∞) continuous such that ω > 0 on (0, ∞) and
∂h
∂h
∂x (x, y) ` ∂y (x, y) ≤ ω (kyk)
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for every (x, y) ∈ (D × F ) \K. Suppose that the map y → h (x, y) is a light map
on F for every x ∈ D. Then, if a, b ∈ D × F and
Z ∞
ds
Qa,b = Da ∩ B a,
,
kbk ω (s)
there exists a unique continuous implicit function ϕ : Qa,b → F , differentiable on
Qa,b \A such that ϕ (a) = b and h (x, ϕ (x)) = h (a, b) for every x ∈ Qa,b and if D
is starlike with respect to a and
Z ∞
ds
= ∞,
ω (s)
1
then Qa,b = D and ϕ : D → F is globally defined on D.
Remark 6. The condition "the map y → h (x, y) is a light map on F for every x ∈ D"
is satisfied if ∂h
exists on D × F and
∂y
∂h
`
(x, y) > 0 for every (x, y) ∈ D × F.
∂y
Using the conditions of local existence and local unicity from Theorem 7 of [8],
we obtain the following global implicit function theorem.
Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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Theorem 8. Let n ≥ 2, D ⊂ Rn be a domain, h : D × Rm → Rm be differentiable
and let K ⊂ D × Rm ,
∞
[
K=
Kp
p=1
with Kp closed sets such that mn−2 (Pr1 Kp ) = 0 for every p ∈ N, A = Pr1 K,
such that ∂h
(x, y) ∈ Isom (Rm , Rm ) for every (x, y) ∈ (D × Rm ) \K and the map
∂y
Close
y → h (x, y) is a light map on Rm for every x ∈ D. Suppose that there exists
ω : [0, ∞) → [0, ∞) continuous such that ω > 0 on (0, ∞) and
∂h
∂h
`
(x, y) ≤ ω (kyk) for every (x, y) ∈ (D × F ) \K.
∂x (x, y)
∂y
Then, if (a, b) ∈ D × F and
Qa,b
Z
= Da ∩ B a,
∞
kbk
ds
ω (s)
Global Implicit Function Theorem
,
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
there exists a unique continuous implicit function ϕ : Qa,b → F , differentiable on
Qa,b \A such that ϕ (a) = b and h (x, ϕ (x)) = h (a, b) for every x ∈ Qa,b , and if D
is starlike with respect to a and
Z ∞
ds
= ∞,
ω (s)
1
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then Qa,b = D and ϕ : D → F is globally defined on D.
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Proof. We see that mm+n−2 (Kp ) = 0 for every p ∈ N and A has σ-finite (n − 1)dimensional measure. We now apply the local implicit function theorem from Theorem 7 of [8], Remark 3 and the preceding arguments.
Page 25 of 28
Using the classical implicit function theorem, we obtain the following global
implicit function theorem
Theorem 9. Let E, F be Banach spaces, D ⊂ E a domain, h : D × F → F be
continuous such that ∂h
exists on D × F , it is continuous on D × F and
∂y
∂h
(x, y) > 0 for every (x, y) ∈ D × F.
`
∂y
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Also, let K ⊂ D × F be such that A = Pr1 K is a countable union of compact
sets if dim E = ∞ and has σ-finite (n − 1)-dimensional measure if dim E = n.
Additionally, suppose that h is differentiable on (D × F ) \K and there exists ω :
[0, ∞) → [0, ∞) continuous such that ω > 0 on (0, ∞) and
∂h
∂h
`
(x, y) ≤ ω (kyk)
∂x (x, y)
∂y
Global Implicit Function Theorem
for every (x, y) ∈ (D × F ) \K. Then, if (a, b) ∈ D × F and
Z ∞
ds
Qa,b = Da ∩ B a,
,
kbk ω (s)
there exists a unique continuous implicit function ϕ : Qa.b → F, differentiable on
Qa,b \A such that ϕ (a) = b and h (x, ϕ (x)) = h (a, b) for every x ∈ Qa,b . If D is
starlike with respect to a and
Z ∞
ds
= ∞,
ω (s)
1
then Qa,b = D and ϕ : D → F is globally defined.
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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References
[1] A.V. ARUTYNOV, The implicit function theorem and abnormal points, Doklady. Math., 60(2) (1999), 231–234.
[2] I. BLOT, On global implicit function, Nonlinear Analysis TMA, 17 (1992),
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Global Implicit Function Theorem
Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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[10] V.F. DEMYANOV, Theorem of an implicit function in quasidifferential calculus, J. Math. Sciences, 78(5) (1996), 556–562.
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Mihai Cristea
vol. 8, iss. 4, art. 100, 2007
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