A Generalization of the Implicit. Function Theorem

Transcription

1 Applied Mathematical Sciences, Vol. 4, 2010, no. 26, A Generalization of the Implicit Function Theorem Elvio Accinelli Facultad de Economia de la UASLP Av. Pintores S/N, Fraccionamiento Burocratas del Estado CP San Luis Potosi, SLP Mexico elvio.accinelli@eco.uaslp.mx 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

2 1290 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 φ h : [0, 1] Y given by φ h (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 by the positive cone of L + p [X], 1 p< (i.e, the set of the real functions f : X R for which f p < 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.

3 A generalization of the implicit function theorem 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 ɛ neighborhood U(x) X, i.e, as U(x) ={y X : x y ɛ} and the relative neighborhood V (x) =U(x) S. We will use o() 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: S x = {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 V x (ɛ). Recall that V x (ɛ) is defined as B x (ɛ) S where B x (ɛ) ={z X : z x <ɛ}. To simplify we use sometimes, V (x) =V X (ɛ). 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 k = βv, where β = α and v = h is admissible for all 0 β. 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 differentiable at x in the admissible direction h S x, if there exist a map T L(X, Y ) such that: f(x + βv) f(x) =βtv + o(β), β 0 (1) 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 d G f(x, v) =f (x)v. Where f (x)v = Tv.

4 1292 E. Accinelli 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 f(x + βv) f(x) (x)v = lim (2) β 0 β where v = h. If we set φ(β) =f(x + βv) then for the G-derivative 2 implies 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) Where w = k k, and k W x for x W (x). To simplify assume that S = W (x). 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) k 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)h 1...h n to symbolize the nth-derivative in x for h 1,..., h n S x, and we set f (n) (x)h n = f (n) (x)h 1...h n 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) =F y (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 We have the following generalized Taylor s Theorem: i=1 1 k f k (x)h k + R n. (4)

5 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 B- spaces and h S x. If f,f,..., f (n) exist as G-derivatives in the admissible direction h then R n 1 n sup {f n (x + βw,w) 0 β 1 where w = h. Proof: We set φ(t) =f(x + th), for h A x and 0 t 1 and we obtain: φ k (0) = f (k) (x)h...h. 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 (x 0,y 0 ), for the equation F (x, y) = 0 then there exists a mapping x y(x) such that y(x 0 )=y 0 and F (x, y(x)) = 0 in a neighborhood of (x 0,y 0 ). The determinative condition is that the inverse operator [F y (x 0,y 0 )] 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 F y (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(x 0,y 0 )=U X (x 0 ) U Y (y 0 ) X Y is an open neighborhood of (x 0,y 0 ) and V X Y (x 0,y 0 )=V X (x 0 ) V Y (y 0 ) represents the relative neighborhood, defined by V X (x 0 )=S U X (x 0 ) a relative neighborhood of x 0 and V Y (y 0 )=W U Y (y 0 ) a relative neighborhood of y 0. The subsets U X (x 0 ) X and U Y (y 0 ) Y are open neighborhoods of x 0 and y 0 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 : V X Y (x 0,y 0 ) X Y Z is defined in the relative neighborhood V X Y (x 0,y 0 ) of (x 0,y 0 ) S W and F (x 0,y 0 )=0.

6 1294 E. Accinelli (ii) F y (x, y) exists as a partial G-derivative on V X Y (x 0,y 0 ) in all admissible directions and F y (x 0,y 0 ):Y Z is bijective. (iii) F and F y are continuous at (x 0,y 0 ). Then the following statement are true: 1. Existence and uniqueness. For all h S x0 there exist positive numbers r 0 and r such that for every α r 0 and x = x 0 + αh there is exactly one y(x) W for which y(x) y 0 r and F (x, y(x)) = Continuity. If F is continuous in a relative neighborhood V X Y (x 0,y 0 ) of (x 0,y 0 ) then y(x) is continuous in a relative neighborhood V X (x 0 ) of x Continuous differentiability If F is a C m map 1 m on a a relative neighborhood V X Y (x 0,y 0 ) of (x 0,y 0 ) then y(x) is a C m map on a relative neighborhood V X (x 0 ) of x 0. Proof: Without loss of generality let x 0 = 0 and y 0 =0. Set g(x, y) F y (0, 0)y F (x, y). Where F y (0, 0) represents the partial G-derivative of F with respect to the second variable, in all admissible direction h W at zero, F (0,tw) F (0,0) i,e: F y (0, 0)w = lim t 0 where w = h. t The equation F (x, y) = 0 is equivalent to the equation: y = F y (0, 0) 1 y = T x y. Note that T x 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 = α and v = h then i,t follows that x = βv. Let r be a positive number, consider y and z vectors in W, such that y, z r, and β<r. By the continuity at (0, 0) of F and F y, Taylor s theorem implies that: g(x, y) g(x, z) sup g y (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 r 0 and r we obtain the bounds T x y F y (0, 0) 1 g(x, y)

7 A generalization of the implicit function theorem 1295 and T x y T x z o(r) F y (0, 0) y z for all y, z W such that y r, z r, and for arbitrary x S : x r 0. The Banach fixed point theorem applied to the map T x : 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 define v = h. The conclusion (3) follows from the equalities: 0=F (x + βv,y(x + βv)) F (x, y(x)) = = F x (x, y(x))βv + F y (x, y(x))[y(x + βv)) y(x)] + o(β), β 0, so, y(x + βv)) y(x) = βf y (x, y(x)) 1 F x (x, y(x))v + o(β). Therefore the G-derivative of y(x) exists, and y (x)v = F y (x, y(x)) 1 F x (x, y(x))v. So we obtain the conclusion for the case m =1. If m = 2 consider G(x) =F y (x, y(x)) 1 y (x)v + F x (x, y(x))v = 0 for x = x 0 + βv being h admissible for x 0 and 0 α r 0. 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 +,u i,w i,i} be a pure exchange economy. Suppose that the consumption set X + is the positive cone of a Banach space X. Let u i be the utility function of the i th consumer. Assume that these are real valued and u i (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 w i X ++, where by X ++ we represent the strictly positive cone of X. Let I be a finite set of index, one for each consumer.

8 1296 E. Accinelli Definition 4. (Feasible allocation) We say that an allocation x =(x 1,..., x n ) X+ n is feasible if and only if n i=1 x i = n i=1 x i. We consider the social utility function U : S n X n + R defined by U(λ, x) = n λ i u i (x i ), (5) i=1 where by S n we symbolize the simplex n dimensional: { } n S n = λ R n : λ i =1,λ 1 > 0, i I. i=1 Each λ S n represents a distribution of relative social weights on the agents of the economy. For each λ S n we consider the social utility function: U λ : F Rdefined by U λ (x) = n λ i u i (x i ). (6) i=1 where x =(x 1,..., x n ) 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 S n such that x solves max x n i=1 λ iu i (x i ) n i=1 x i = n i=1 w i. (7) 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: λ i u i (x i) γ =0 i =1,..., n; n i=1 x i W =0. (8) Where W = n i=1 w i, and γ X (where by X is symbolized the continuous dual of X) is the Lagrange multiplier.

9 A generalization of the implicit function theorem 1297 Consider the functions F i : S n X + X X defined by and F n+1 : X n + X ++ X defined by F i (λ i,x i,γ)=λ i u i (x i) γ, i =1,..., n F n+1 (x, w) = n (x i w i ). i=1 Assume that (λ 0,x 0,γ 0,w 0 ) S n X n + X X n ++ verify: F i (λ 0,x 0,γ 0 )=0 F n+1 (x 0,w 0 )=0. We introduce the notation A = X n + X and B = S n X n ++, and a = (x, γ), b =(λ, w). Let F : A B (X ) n X n be the vectorial function defined by F (x, γ, λ, w) =(F 1 (λ 1,x 1,γ),..., F n (λ n,x n,γ),f n+1 (x, w)). Note that F (x 0,γ 0,λ 0,w 0 )=0. Note that F a (x0,γ 0,λ 0,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 : S n 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 x 0 = x(λ 0,w 0 ),γ 0 = γ(λ 0,w 0 ) for all λ S n and w X ++ such that, λ λ 0 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 : S n R n defined by e =(e 1,..., e n ) where e i (λ) =λ i u (x i (λ))(x i (λ) w i ) 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 Economía Vol.27,/2 pp , (noviembre 2008).

10 1298 E. Accinelli [Zeidler, E.] Nonlinear Functional Analysis and its Applications. Springer Verlag Received: January, 2010