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A Generalization of the Implicit Function Theorem

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A Generalization of the Implicit Function Theorem 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 func- tion, 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 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 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 inte- rior. This is for instance, the case where the bundle set of an economy, is given + ≤ ∞ by the positive cone of Lp [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 func- tion 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 (G- derivative) 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 in- troduce 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 1291 2 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 ≤ ≤ k = βv, where β = α h and v = h is admissible for all 0 β 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 − differentiable at x in the admissible direction h ∈Sx, if there exist a map T ∈ L(X, Y ) such that: f(x + βv) − f(x)=βTv + 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 dGf(x, v)=f (x)v. Where f (x)v = Tv. 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(x + βv) − f(x) f (x)v = lim (2) β→0 β h 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 lin- earization. Assume that f : W (x) ⊂ V (x) → L(X, Y ), f (x + βw)=f (x)+f (x)w + o(β},β→ 0. (3) k ∈W ∈ Where w = k , and k 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)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: (n) f (x)h1...hn to symbolize the nth-derivative in x for h1, ..., hn ∈Sx, and we set (n) n (n) f (x)h = f (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: n−1 1 k k f(x + h)=f(x)+ f (x)h + Rn. (4) i=1 k We have the following generalized Taylor’s Theorem: 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- (n) spaces and h ∈Sx. If f ,f , ..., f exist as G-derivatives in the admissible direction h then 1 n Rn≤ sup {f (x + βw,w) n 0≤β≤1 h where w = h . Proof: We set φ(t)=f(x + th), for h ∈Ax and 0 ≤ t ≤ 1 and we obtain: φk(0) = f (k)(x)h...h.
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