Applied Mathematical Sciences, Vol. 4, 2010, no. 26, 1289 - 1298
A Generalization of the Implicit
Function Theorem
Elvio Accinelli
Facultad de Economia de la UASLP
Av. Pintores S/N, Fraccionamiento Burocratas del Estado
CP 78263 San Luis Potosi, SLP Mexico
[email protected]
Abstract
In this work, we generalize the classical theorem of the implicit function, for functions whose domains are open subset of Banach spaces in
to a Banach space, to the case of functions whose domains are convex
subset, not necessarily open, of Banach spaces in to a Banach space.
We apply this theorem to show that the excess utility function of an
economy with infinitely many commodities, is a differentiable mapping.
Keywords: Implicit Function Theorem, convex subsets, Banach spaces
1
Introduction
The purpose of this work is to show that the implicit function theorem can be
generalized to the case of functions whose domains are defined as a cartesian
product of two convex subsets S, and W not necessarily open, of a cartesian
product X × Y of Banach spaces, in to a Banach space Z.
To prove our main theorem we introduce the concept of Gateaux derivative.
However, it is enough the existence of the Gateaux derivatives of a given
function, only in admissible directions. Let S ⊂ X be a subset of a Banach
space X, and x ∈ S. So, we will say that the vector h ∈ X is admissible for
x ∈ S if and only if x + h ∈ S. In order to define the Gateaux derivative of a
function in admissible directions in a point x, the only necessary condition is
the convexity of the domain of the function. Note that if S is a convex subset
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E. Accinelli
of a Banach space and if h is admissible for x, then, all vector αh, ∀α ∈ [0, 1]
is admissible too. Then, if f : S → Y is a function, it is possible to define,
for each h admissible at x ∈ S, a new function φh : [0, 1] → Y given by
φh (α) = f (x + αh) ∀α ∈ [0, 1].
The mathematical definitions and the main steps to prove our generalized
theorem, are introduced following [Zeidler, E.] chapter 4, vol. 1, where the
implicit function theorem is proved, but for open subsets, of Banach spaces.
The generalization of the implicit function theorem, has many important
applications in economics, in particular in the case of economies with infinitely
many goods, including the cases where the consumption sets have empty interior. This is for instance, the case where the bundle set of an economy, is given
[X], 1 ≤ p < ∞ (i.e, the set of the real functions
by the positive cone of L+
pp
f : X → R for which |f | < ∞ for any representative f ). The positive cone
of these spaces, are convex subset, but unfortunately, with empty interior. In
the last section 5 we use this generalization to prove that de excess utility function is a differentiable map, even in the case where the consumption subset is
the positive cone with empty interior, of a Banach space.
We first of all need is a generalization of the classical derivative concept
for maps between Banach spaces (B-spaces), to the case where the domain of
the function is a convex, no necessarily, open subset, of a Banach space. This
will be closely related with the classical, concept of the Gateaux derivative (Gderivative) given for instance in [Zeidler, E.]. The purpose of the next section
is to introduce such generalization.
In section 4 we give the fundamental theorem of this work, finally we introduce some economical applications, in particular our generalization include
the case where the economy is defined in the positive cone of a Banach space.
Recall that in all case a consumer with utility u need to compare points in his
consumption set, and generally this is a convex subset (with empty interior)
S of a Banach space so, if x and y are in this set then, for all 0 ≤ α ≤ 1, the
vector z = x + αh ∈ S where h = (x − y) i.e, αh are admissible vectors at x..
Then, we can define the Gateaux derivative in each admissible direction given
at x.
A generalization of the implicit function theorem
2
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Formal Definitions
We begin with some notation. Let X and Y be Banach spaces. For each x ∈ X
the symbol x represents the norm of the vector x. The norm in the B-spaces
X and Y can be different but, without loss of generality we consider the same
norms in both spaces. Let f : S → Y be a mapping between the convex subset
S ⊂ X and the B-space Y. For a point x ∈ S we consider the −neighborhood
U(x) ⊂ X, i.e, as U(x) = {y ∈ X : x − y ≤ } and the relative neighborhood
V (x) = U(x) ∩ S. We will use o(h) to describe those expressions which are
higher than the first order in h as h → 0. We write L(X, Y ) for the class of all
maps T : X → Y, T linear and continuous, where X and Y are B-spaces.
Definition 1. (Admissible vector) Let S ⊂ X be a convex subset of a Banch
space X. We will say that a vector h ∈ X is admissible for x ∈ S if and only
if x + h ∈ S.
We introduce the notation:
Sx = {h ∈ X : x + h ∈ S} ,
to denote the set of all admissible vectors for each x ∈ S.
A relative neighborhood of x ∈ S will be symbolized by Vx (). Recall that
Vx () is defined as Bx ()∩S where Bx () = {z ∈ X : z − x < } . To simplify
we use sometimes, V (x) = VX ().
Note that if x and y are points in the convex subset S and if h = (y − x)
then, for all 0 ≤ α ≤ 1 the vector k = αh is admissible for x. So, the vector
h
is admissible for all 0 ≤ β ≤ h.
k = βv, where β = αh and v = h
Definition 2. Let f : V (x) → Y be a map. Consider the subset S, a convex
subset (not necessarily open) of a B-space X, let Y be a B-space and V (x) a
relative neighborhood of x ∈ S. We say that the map is G − dif f erentiable at
x in the admissible direction h ∈ Sx , if there exist a map T ∈ L(X, Y ) such
that:
f (x + βv) − f (x) = βT v + o(β), β → 0
(1)
h
and v = h
, and β in some neighborhood oh zero. The map T is called the
G-derivative of f at x in the (admissible) direction h, if T does not depend on
the admissible direction we define the G-differential dG f (x, v) = f (x)v. Where
f (x)v = T v.
E. Accinelli
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Since f (x) = T is a linear operator it follows that f (x)h = βf (x)v.
Equation (1) show that the derivative is defined through a linearization, and
f (x) is uniquely defined by the equivalent expression:
f (x + βv) − f (x)
β→0
β
f (x)v = lim
(2)
h
. If we set φ(β) = f (x + βv) then for the G-derivative 2 implies
where v = h
that: φ (0) = f (x)v.
The derivative of f at x i.e, f (x) is obtained by definition from the linearization. Assume that f : W (x) ⊂ V (x) → L(X, Y ),
f (x + βw) = f (x) + f (x)w + o(β}, β → 0.
(3)
k
, and k ∈ Wx for x ∈ W (x). To simplify assume that S = W (x).
Where w = k
So, f (x) is a continuous linear operator from X into L(X, Y ), i.e, f (x) ∈
L(X, L(X, Y )). We will abbreviate this writing f (x)kh. From the fact that:
f (x)kh ≤ f (x)hk
it follows that f (x) is a bounded bilinear map, such that for each h ∈ X
f (x)h : X → L(X, Y ).
Higher derivatives can be defined in the same way, the image sets of
the derivatives acquire a more and more complicated structure. We set:
f (n) (x)h1 ...hn to symbolize the nth-derivative in x for h1 , ..., hn ∈ Sx , and
we set
f (n) (x)hn = f (n) (x)h1 ...hn
Definition 3. Let X and Y be B-spaces. Let F : D(F ) ⊂ X × Y → Z be a
mapping, where D(F ) = S × Y and S ⊂ X is a convex subset of the B-space
X. Let x be fixed and set g(y) = F (x, y), if g has G-derivative at y we say
that g (y) = Fy (x, y) is the partial G-derivative of F with respect to the second
variable at (x, y).
3
Generalized Taylor’s Theorem
In generalizing the classical Taylor series formula we will consider:
f (x + h) = f (x) +
n−1
1 k
f (x)hk + Rn .
k
i=1
We have the following generalized Taylor’s Theorem:
(4)
A generalization of the implicit function theorem
1293
Theorem 1. Let S be a convex subset of X. Consider the mapping f : V (x) ⊂
X → Y where V (x) is a relative neighborhood of x ∈ S, let X and Y be Bspaces and h ∈ Sx . If f , f , ..., f (n) exist as G-derivatives in the admissible
direction h then
1
sup {f n (x + βw, w)
Rn ≤
n 0≤β≤1
where w =
h
.
h
Proof: We set φ(t) = f (x + th), for h ∈ Ax and 0 ≤ t ≤ 1 and we obtain:
φ (0) = f (k) (x)h...h. k
4
The Implicit Function Theorem
The classical implicit function theorem, shows that under well behaved conditions for a given function F, if there exists a solution (x0 , y0 ), for the equation
F (x, y) = 0 then there exists a mapping x → y(x) such that y(x0 ) = y0 and
F (x, y(x)) = 0 in a neighborhood of (x0 , y0 ). The determinative condition is
that the inverse operator [Fy (x0 , y0)]−1 : Z → Y exists as a continuous linear
operator. In our case F : S × W → Z ⊆ X × Y, where S and W are respectively convex subsets, no necessarily open,X and Y, are Banach spaces. By
the notation Fy (x, y) we symbolize the G-derivative of F with respect to the
second variable at (x, y) ∈ S × W in all admissible direction.
We will use the following notation: U(x0 , y0) = UX (x0 ) × UY (y0 ) ⊂ X ×
Y is an open neighborhood of (x0 , y0 ) and VX×Y (x0 , y0) = VX (x0 ) × VY (y0 )
represents the relative neighborhood, defined by VX (x0 ) = S∩UX (x0 ) a relative
neighborhood of x0 and VY (y0 ) = W ∩ UY (y0 ) a relative neighborhood of y0 .
The subsets UX (x0 ) ⊂ X and UY (y0 ) ⊂ Y are open neighborhoods of x0 and
y0 respectively.
Theorem 2. (Generalized Implicit Function Theorem) Let S ⊂ X be
a convex subset of the B-space X, and let W ⊂ Y be a convex subset of the
B-space Y, and consider a function F : S × W → Z, where Z is a B-space. All
the B-spaces are defined over R.
Suppose that
(i) the mapping F : VX×Y (x0 , y0 ) ⊂ X × Y → Z is defined in the relative
neighborhood VX×Y (x0 , y0) of (x0 , y0) ∈ S × W and F (x0 , y0 ) = 0.
E. Accinelli
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(ii) Fy (x, y) exists as a partial G-derivative on VX×Y (x0 , y0 ) in all admissible
directions and Fy (x0 , y0 ) : Y → Z is bijective.
(iii) F and Fy are continuous at (x0 , y0).
Then the following statement are true:
1. Existence and uniqueness. For all h ∈ Sx0 there exist positive numbers
r0 and r such that for every α ≤ r0 and x = x0 + αh there is exactly one
y(x) ∈ W for which y(x) − y0 ≤ r and F (x, y(x)) = 0.
2. Continuity. If F is continuous in a relative neighborhood VX×Y (x0 , y0) of
(x0 , y0 ) then y(x) is continuous in a relative neighborhood VX (x0 ) of x0 .
3. Continuous differentiability If F is a C m −map 1 ≤ m ≤ ∞ on a a
relative neighborhood VX×Y (x0 , y0 ) of (x0 , y0 ) then y(x) is a C m −map on
a relative neighborhood VX (x0 ) of x0 .
Proof: Without loss of generality let x0 = 0 and y0 = 0. Set g(x, y) ≡
Fy (0, 0)y − F (x, y). Where Fy (0, 0) represents the partial G-derivative of F
with respect to the second variable, in all admissible direction h ∈ W at zero,
(0,0)
h
where w = h
.
i,e: Fy (0, 0)w = limt→0 F (0,tw)−F
t
The equation F (x, y) = 0 is equivalent to the equation:
y = Fy (0, 0)−1y = Tx y.
Note that Tx does not denote a partial derivative.
Let h ∈ X be an admissible vector at x ∈ S then, for all 0 ≤ α ≤ 1, the
vector αh is admissible, i.e, x + αh ∈ S is admissible. Consider β = αh and
h
then i,t follows that x = βv. Let r be a positive number, consider y
v = h
and z vectors in W, such that y, z ≤ r, and β < r.
By the continuity at (0, 0) of F and Fy , Taylor’s theorem implies that:
g(x, y) − g(x, z) ≤ sup gy (x, z + τ (y − z)y − z = o(τ )y − z, r → 0.
0<τ <1
Since g(0, 0) = 0 and g is continuous at (0, 0) it follows that:
g(x, y) ≤ g(x, y) − g(x, 0) + g(x, 0) = 0(r)y + o(r), r → 0.
For sufficiently small positive r0 and r we obtain the bounds
Tx y ≤ Fy (0, 0)−1g(x, y)
A generalization of the implicit function theorem
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and
Tx y − Tx z ≤ o(r)Fy (0, 0)−| y − z
for all y, z ∈ W such that y ≤ r, z ≤ r, and for arbitrary x ∈ S : x ≤ r0 .
The Banach fixed point theorem applied to the map Tx : M → M where
M = {y ∈ Y : y ≤ r} with x fixed, immediately yield conclusions (1) and
(2).
To see (3), let h be admissible for x ∈ S. So, x + αh ∈ S ∀α ∈ [0, 1]. Let us
h
. The conclusion (3) follows from the equalities:
define v = h
0 = F (x + βv, y(x + βv)) − F (x, y(x)) =
= Fx (x, y(x))βv + Fy (x, y(x))[y(x + βv)) − y(x)] + o(β), β → 0,
so,
y(x + βv)) − y(x) = −βFy (x, y(x))−1 Fx (x, y(x))v + o(β).
Therefore the G-derivative of y(x) exists, and
y (x)v = −Fy (x, y(x))−1Fx (x, y(x))v.
So we obtain the conclusion for the case m = 1.
If m = 2 consider G(x) = Fy (x, y(x))−1 y (x)v + Fx (x, y(x))v = 0 for x =
x0 + βv being h admissible for x0 and 0 ≤ αh ≤ r0 . Let k be admissible for
x some the expression for y (x) comes from taking derivative in G(x + tk) = 0,
with respect to t at t = 0.
If m ≥ 3 the argument is similar. 5
An applications of the Implicit Function Theorem
Let E = {X+ , ui, wi , I} be a pure exchange economy. Suppose that the consumption set X+ is the positive cone of a Banach space X. Let ui be the
utility function of the i − th consumer. Assume that these are real valued and
ui (x) ∈ C 2 (X+ ), ∀i ∈ I, in the Gateaux sense, and that u (x) is a bilinear
form positive definite for all x ∈ X+ . The initial endowments are denoted by
wi ∈ X++ , where by X++ we represent the strictly positive cone of X. Let I
be a finite set of index, one for each consumer.
E. Accinelli
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Definition 4. (Feasible allocation) We say that an allocation x = (x1 , ..., xn ) ∈
X+n is feasible if and only if ni=1 xi = ni=1 xi .
We consider the social utility function U : Sn × X+n → R defined by
U(λ, x) =
n
λi ui (xi ),
(5)
i=1
where by Sn we symbolize the simplex n−dimensional:
n
λi = 1, λ1 > 0, ∀i ∈ I .
Sn = λ ∈ R n :
i=1
Each λ ∈ Sn represents a distribution of relative social weights on the
agents of the economy. For each λ ∈ Sn we consider the social utility function:
Uλ : F → R defined by
Uλ (x) =
n
λi ui (xi ).
(6)
i=1
where x = (x1 , ..., xn ) is a feasible allocation..
As it is well known (see [Accinelli, E. Plata, E. )]) a feasible allocation x∗
is a Pareto optimal allocation if and only if there exists λ ∈ Sn such that x∗
solves
maxx ni=1 λi ui (xi )
(7)
n
n
i=1 xi =
i=1 wi .
So, for a fixed λ the solution x∗ = x(λ) of this problem is a Pareto optimal
allocation.
Assume conditions on the utilities of the consumers such that individually
rational Pareto optimal allocations are in X++ so, each individually rational
Pareto optimal allocation verify the equation system:
λi ui(xi ) − γ = 0 i = 1, ..., n;
n
i=1
(8)
xi − W = 0.
Where W = ni=1 wi , and γ ∈ X ∗ (where by X ∗ is symbolized the continuous
dual of X) is the Lagrange multiplier.
A generalization of the implicit function theorem
1297
Consider the functions Fi : Sn × X+ × X ∗ → X ∗ defined by
Fi (λi , xi , γ) = λi ui (xi ) − γ, i = 1, ..., n
and Fn+1 : X+n × X++ → X defined by
Fn+1 (x, w) =
n
(xi − wi ).
i=1
n
Assume that (λ0 , x0 , γ 0 , w 0 ) ∈ Sn × X+n × X ∗ × X++
verify:
Fi (λ0 , x0 , γ 0 ) = 0
Fn+1 (x0 , w 0 ) = 0.
n
, and a =
We introduce the notation A = X+n × X ∗ and B = Sn × X++
∗ n
n
(x, γ), b = (λ, w). Let F : A × B → (X ) × X be the vectorial function
defined by
F (x, γ, λ, w) = (F1 (λ1 , x1 , γ), ..., Fn (λn , xn , γ), Fn+1 (x, w)) .
Note that
F (x0 , γ 0 , λ0 , w 0) = 0.
Note that Fa (x0 , γ 0 , λ0 , w 0 ) : X n × X ∗ → X n × X ∗ is a bijection then,
from the implicit function theorem, it follows that there exist a function
f : Sn × X n → X n × X ∗ such that f (λ, w) = (x(λ, w), γ(λ, w)) verifying
F (x(λ, w), γ(λ, w), λ, w) = 0 and x0 = x(λ0 , w 0 ), γ 0 = γ(λ0 , w 0 ) for all λ ∈ Sn
and w ∈ X++ such that, |λ − λ0 | ≤ r and w − w 0 ≤ r.
So for each w ∈ X++ we can define the Negishi map (λ, x(λ)) as a differentiable manifold, that resume the relation between efficiency and social
welfare. Our main result is that the excess utility function e : Sn × Rn defined
by e = (e1 , ..., en ) where ei (λ) = λi u (xi (λ))(xi (λ) − wi ) is a G-differentiable
function.
References
[Accinelli, E. Plata, E. )] “Las crisis sociales y las singularidades: Los fundamentos microeconómicos de las crisis sociales” . Ensayos Revista de
Economı́a Vol.27,/2 pp. 49-84, (noviembre 2008).
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E. Accinelli
[Zeidler, E.] Nonlinear Functional Analysis and its Applications. Springer Verlag 1986.
Received: January, 2010