This paper presents several Lipschitz implicit function theorems for maps between Banach spaces. The results use semi-inner products as a means for understanding the geometry of normed spaces. The arguments rely on corresponding geometric properties of operators on normed spaces, and on Ekeland's variational principle, in contrast with classic results such as the inverse function theorems of Nash and Moser and their use of Newton iteration procedures (e.g., see Hamilton).
The literature on implicit function theorems is significant, with many important results using variational analysis to extend standard theorems, including the classic theorem of Clarke. Dontchev and Rockafellar provide a comprehensive development and extension of much of this work. Most of these results focus on extending strong differentiability notions from functions to set-valued maps to establish implicit mapping theorems for inclusions, or on using generalized derivatives to establish implicit function theorems for nonsmooth functions; for example, see [13], [3], [10], [11], [12], [15], [16], [23], among others. In contrast, the results here focus on the standard implicit function problem, and weaken differentiability properties of these functions by using geometric properties of operators based on semi-inner products.
Semi-inner products have been important in many problems regarding operators and geometry in Banach spaces, including characterizations of metric projections and best approximations, orthogonality, properties of linear operators, differential equations, and machine learning and pattern analysis, as illustrated in [22], [24], [25], [9], [26], [29], [30], among others. Here the use of semi-inner products allows the formulation of geometric properties of derivatives that can be used to ensure Lipschitz continuity in place of the strong differentiability conditions required in standard results. The results are illustrated with an application to a central problem in mathematical economics, characterizing Lipschitz behavior of Pareto optimal allocations in a model of trade over an infinite horizon, which in turn is the key property ensuring generic determinacy of equilibrium in economies with infinitely many commodities.
2. Preliminaries
Let X be a real Banach space and X* be its norm dual.
The main tools in the paper are generalizations of inner product for Banach spaces. For the real Banach space X, consider the semi-inner products between two elements x and y defined by equation and equation.
For Rn or for any Hilbert space, these semi-inner products coincide with the standard inner product (x,y). In a general Banach space, these share many of the important properties of an inner product; the main difference with an inner product is that these semi-inner products are not necessarily bilinear or symmetric. See Deimling, for example, for discussion. Some of the main properties of (.,.)+- are given in the lemma below.
Lemma 1
Deimling [8, Proposition 13.1]
Let X be a real Banach space. For all x, y, z ∈ X, (i) equation and equation; (ii) equation; (iii) If equation and equation, then equation; (iv) (.,.)+ is upper semicontinuous, (.,.)- is lower semicontinuous, and (.,y)+-  is continuous for each y ∈ X; (v) IfX∗ is strictly convex, then (·,·)+ = (·,·)−.
In particular, using (v) and a basic result of Lindenstrauss, whenever X is reflexive then it has an equivalent norm with respect to which (·, ·)+ = (·, ·)−. More generally, for any x, y ∈ X, (x, y)+ = −(−x, y)−.
Remark. Alternatively, following e.g., Lumer and Giles, a semi-inner product can be defined ab- stractly to be a function on X × X satisfying key properties of an inner product, including linearity in one component and a version of the Cauchy-Schwarz inequality. All of the definitions below could be formulated analogously for such an abstract semi-inner product. It is straightforward to see that all of the results of the paper would hold under these more general conditions, with the modification that the norm on X is given by the norm induced by the semi-inner product ∥x∥(·,·) = (x,x)12 , which need not coincide with the original norm on X. We return to discuss this point in more detail at the end of this section.
Throughout let T : X → X be a linear operator on X, and given an index set A,let T : = {T(a) : X → X : a ∈ A} be a family of linear operators on X. We start with some basic terminology. The operator T is bounded below if there exists b > 0 such that ∥Tx∥ ≥ b∥x∥ for each x ∈ X. The family T is uniformly boundedbelowifthereexistsb>0suchthat∥T(a)x∥≥b∥x∥foreachx∈Xandforeacha∈A,andT is uniformly bounded if there exists M > 0 such that ∥T(a)x∥ ≤ M∥x∥ for each x ∈ X and for each a ∈ A. For a subset B ⊆ X, co B denotes the convex hull of B, and coB denotes the closed convex hull of B. For r > 0 and x ∈ X, Br(x) := {y ∈ X : ∥y − x∥ < r}.
Of particular interest in this paper are analogues of negative definite linear operators defined by use of the semi-inner products (·, ·)±. Say the operator T : X → X is weakly + negative definite if equation and + negative definite if there exists c > 0 such that equation. Similarly, T is weakly − negative definite if equation and − negative definite if there exists c > 0 such that equation.
The family of linear operators T = {T(a) : X → X : a ∈ A} is weakly + negative definite if T(a) is weakly + negative definite for each a ∈ A, and uniformly + negative definite if there exists c > 0 such that equation. Similarly, T is weakly − negative definite if T (a) is weakly − negative definite for each a ∈ A, and uniformly − negative definite if there exists c > 0 such that equation.
The lack of symmetry of the semi-inner products (·,·)± leads to alternative definitions of "negative definite" to the ones given above. For a linear operator T : X → X, say T is weakly + dissipative if equation and + dissipative if there exists c > 0 such that equation. Similarly, T is weakly − dissipative if equation and − dissipative if there exists c > 0 such that equation. The family of linear operators T = {T(a) : X → X : a ∈ A} is weakly + dissipative if T(a) is weakly + dissipative for each a ∈ A, and uniformly + dissipative if there exists c > 0 such that equation. Similarly, T is weakly − dissipative if T(a) is weakly − dissipative for each a ∈ A, and uniformly − dissipative if there exists c > 0 such that equation. Throughout the notation ± indicates that the associated condition holds with respect to either (·,·)+ or (·, ·)−.
Remark. When X∗ is strictly convex, (·,·)+ = (·,·)−, and the qualifier ± can be dropped everywhere above. Remark. When X is a Hilbert space, a linear operator T is (weakly) negative definite if and only if it is (weakly) dissipative.
The following two lemmas show that versions of negative definite and dissipative operators share many of the most important properties of standard negative definite operators in Hilbert spaces.
Lemma2.Let X be a real Banach space. Let T : X → X be a linear operator, and let T = {T(a) : X → X : a ∈ A} be a family of linear operators.
(i) If T is ± negative definite or ± dissipative, then T is bounded below.
(ii) If T is uniformly ± negative definite or uniformly ± dissipative, then T is uniformly bounded below.
Proof. The proofs will be for the negative definite case; the dissipative case is analogous. For part (i), there exists c > 0 such that (x,Tx)± ≤ −c∥x∥2 for all x ∈ X. So given x ∈ X, equation by Lemma 1(ii). Then ∥T x∥ ≥ c∥x∥. For part (ii), there exists c > 0 such that (x,Tx)± ≤ −c∥x∥2 for all T ∈ T and for all x ∈ X. So given T ∈ T and given x ∈ X, equation by Lemma 1(ii). Then ∥T x∥ ≥ c∥x∥.
Lemma 3. Let X be a real Banach space. Let T : X → X be a continuous, invertible linear operator. Let T = {T(a) : X → X : a ∈ A} be a family of continuous, invertible linear operators, and let T −1 = {T(a)−1 : X → X : a ∈ A}.
(i) If T is (weakly) ± negative definite, then T−1 is (weakly) ± dissipative.
(ii) If T is uniformly ± negative definite or uniformly ± dissipative, then T −1 is uniformly bounded.
(iii) If T is uniformly bounded and uniformly ± negative definite (uniformly ± dissipative), then T −1 is uniformly ± dissipative (uniformly ± negative definite).
Proof. For part (i), let y ∈ X and x = T−1y, so Tx=y. Then equation.
For part (ii), let T be uniformly ± negative definite; the proof for the dissipative case is analogous. Let c > 0 such that (x,Tx)± ≤ −c∥x∥2 for all T ∈ T . Let T−1 ∈ T −1 and y ∈ X. Set x = T−1y, so Tx = y. Then by Lemma 2(ii), ∥T x∥ ≥ c∥x∥. So equation which implies equation.
For part (iii), let T be uniformly ± negative definite; again the dissipative case is analogous. Let b > 0 such ∥Tx∥ ≤ b∥x∥ for all T ∈ T and all x ∈ X,and let c > 0 such that (x,Tx)± ≤ −c∥x∥2 for all T ∈ T and all x ∈ X. Then let T−1 ∈ T−1 and y ∈ X. Set x = T−1y,so Tx = y. Then equation. Thus T −1 is uniformly ± dissipative.
Say that T = {T(a) : X → X : a ∈ A} is convex ± negative definite if co T is ± negative definite, and uniformly convex ± negative definite if co T is uniformly ± negative definite. Similarly, say that T = {T(a) : X → X : a ∈ A} is convex ± dissipative if co T is ± dissipative, and uniformly convex ± dissipative if co T is uniformly ± dissipative. In the + dissipative case, the convex hull of T will inherit the properties of T , using Lemma 1, that is, T is + dissipative if and only if co T is + dissipative, and T is uniformly + dissipative if and only if co T is uniformly + dissipative. The lemma below records this and related results.
Lemma 4. Let X be a real Banach space. Let T1,T2 : X → X be linear operators, and let T1 = {T1(a) : X → X:a∈A} and T2 = {T2(a) : X → X : a ∈ A} be families of linear operators.
(i) If T1 and T2 are weakly + dissipative, then αT1 + βT2 is weakly + dissipative for any α, β ∈ R with α, β > 0.
(ii) If T1 is + dissipative and T2 is weakly + dissipative, then αT1 + βT2 is + dissipative for any α, β ∈ R with α,β > 0.
(iii) If T1 is uniformly + dissipative and T2 is weakly + dissipative, then T1 + T2 = {T1(a) + T2(a) : a ∈ A} is uniformly + dissipative.
(iv) If T is uniformly + dissipative, then co T is uniformly + dissipative. Similarly, if T is weakly + dissipative, then co T is weakly + dissipative.
Proof. For part (i), let α, β > 0. Then by Lemma 1(i), (iii), equation. For part (ii), let c1 > 0 such that (T1x,x)+ ≤ −c1∥x∥2 for all x ∈ X. Let α,β > 0. Then equation.
The argument for part (iii) is analogous to the proof of part (ii), noting that c1 > 0 can be chosen independent of T1 ∈ T1 above in this case.
For part (iv), suppose T is uniformly + dissipative, and let c > 0 such that (Tx,x)+ ≤ −c∥x∥2 for all T ∈ T . Then let T ∈ co T , so T = αiTi for some Ti ∈ T and some αi ≥ 0 with αi = 1. Then equation. The argument for the weakly + dissipative case is analogous. The final lemmas below record another useful property of + dissipative operators on product spaces.