# On Nonsmooth Global Implicit Function Theorems for Locally Lipschitz Functions from Banach Spaces to Euclidean Spaces

Authors: Guy Degla; Cyrille Dansou; Fortuné Dohemeto
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

In this paper, we establish a generalization of the Galewski-Rădulescu nonsmooth global implicit function theorem to locally Lipschitz functions defined from infinite dimensional Banach spaces into Euclidean spaces. Moreover, we derive, under suitable conditions, a series of results on the existence, uniqueness, and possible continuity of global implicit functions that parametrize the set of zeros of locally Lipschitz functions. Our methods rely on a nonsmooth critical point theory based on a generalization of the Ekeland variational principle.


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## Body

## 1. Introduction

Many mathematical models involving real or vector-valued functions stand as equations of the form(1)fx=0.For complex phenomena, the unknownx is often a vector-variable x=x1,x2,⋯,xn belonging to ℝn or to an abstract Banach space having a direct sum V1⊕V2⊕⋯⊕Vn. It may even happen that equation (1) is just a state equation depending in fact on a parameter (or a control) h. In this case, it takes the form
(2)Fx,h=0,and the most aspiring aim of mathematical analysis is to know the local or global structure of the solution setF−10 by finding out whether it is nonempty, discrete, a graph or a manifold, etc.The essence of the implicit function theorem in mathematical analysis is to ascertain if the solutions to an equation involving parameters exist and may be viewed locally as a function of those parameters and to know a priori which properties this function might inherit from those of the data. Geometrically, implicit function theorems provide sufficient conditions under which the solution set in some neighborhood of a given solution is the graph of some function. The well-known implicit function theorems deal with a continuous differentiability hypothesis and in such cases are equivalent to inverse function theorems (see [1]). It was originally conceived (in the complex variable form in a pioneering work by Lagrange) over two centuries ago to tackle celestial mechanics problems. Subsequently, it attracted Cauchy who managed to provide its rigorous version and became its discoverer. Later, the generalization of this implicit function theorem to the case of finitely many real variables was proved for the first time by Dini. In this way, the classical theory of implicit functions started with single variables and have progressed through multiple real variables to equations in infinite dimensional spaces, e.g., functional equations involving integral or differential operators. Nowadays, most categories of smooth functions have virtually their own version of the implicit function theorem, and there are special versions adapted to Banach spaces and algebraic geometry and to various types of geometrically degenerate situations. Some of these (such as Nash-Moser implicit function theorem) are quite sophisticated and have been used in amazing ways to solve important open problems (in Riemannian manifolds, partial differential equations, functional analysis, …) [1]. There are also in the literature [2, 3] some implicit numerical schemes used to approximate the solutions of certain differential equations and that could be regarded as implicit functions in sequence spaces.Nevertheless, there are interesting phenomena governed by parametric equations with nonsmooth data which need to be stressed and are more and more attracting researchers. Indeed, the implicit function theorems for nondifferentiable functions are less known but are regaining interest in the literature due to their importance in applied sciences that deal with functions having less regularity than smoothness. Few versions have been stated in Euclidean spaces for functions that are continuous with respect to all their variables and (partially) monotone with respect to some of their variables [4, 5].Recently, Galewski and Rădulescu [6] proved a generalized global implicit function theorem for locally Lipschitz function F:ℝn×ℝp⟶ℝn, by using a nonsmooth Palais-Smale condition and a coercivity condition. Their proof is essentially based on the fact that a locally Lipschitz function in a finite dimension is almost everywhere differentiable with respect to the Lebesgue measure according to Rademacher’s theorem [7]. It is known that Rademacher’s theorem for locally Lipschitz functions has no direct infinite dimensional extension. This justifies all difficulties to have conditions of existence of local or global implicit function in the case of locally Lipschitz function defined on infinite dimensional space (see [8]). Several works have been done to overcome these difficulties. For example, the papers [9, 10] provided conditions for surjectivity and inversion of locally Lipschitz functions between Banach spaces under assumptions formulated in terms of pseudo-Jacobian.In this work, our aim is to establish under suitable conditions a global implicit function theorem for locally Lipschitz mapF:X×Y⟶H, where X,Y are real Banach spaces and H is a real Euclidean space, and to provide conditions under which this implicit function is continuous. This extends Theorem 30 of Galewski and Rădulescu to the locally Lipschitz functions in infinite dimension with a very relatively simple method compared to those used for this purpose. Knowing that there exist noncoercive functions satisfying the h-condition (see Definition 18 and Remark 19), we work in this paper under the h-condition using a variational approach and applying a recent nonsmooth version of Mountain Pass Theorem, namely, Theorem 27.The contribution of this work is quadruple:(i)
An improvement of the classical Clarke’s implicit function Theorem24 for function F:ℝn×ℝp⟶ℝn by replacing ℝp by any Banach space Y (Remark 26). Consequently, by considering the approach used in [6] (Theorem 4) and Remark 26, we prove our first main result (Theorem 31) on the existence and uniqueness of global implicit function theorem for equation Fx,y=0, where F:ℝn×Y⟶ℝn with Y a Banach space(ii)
The proof of the continuity of the implicit function based on a simple additional hypothesis, Theorem35(iii)
The weakening of the coercivity assumption used in [6] by considering a compactness type condition called h-condition in [11](iv)
By our Lemmas42 and 43, we obtain Theorem 38 on the existence and uniqueness of global implicit functions under the h-condition on the function x↦Fx,yα with 0<α<2. This is a generalization of the result (49) in the nonsmooth case. It also generalizes the result [12] (Theorem 3.6) in the C1 caseThis article is organized as follows. In Section2, we recall some preliminary and auxilliary results on Clarke’s generalized gradient, Clarke’s generalized Jacobian, and the h-condition for locally Lipschitz functions. Section 3 is devoted to our main results established under the h-condition, on the existence and uniqueness of global implicit function for equation Fx,y=0, where F is defined from ℝn×Y to ℝn and Y is a Banach space, namely, Theorems 31, 35, 38, 39, and 40. In Section 4, we give an example of a function satisfying our conditions of existence of implicit function but not the conditions of Theorem 1 of 6 which we have extended. This is the energy functional defined in (139), of a certain differential inclusion problem involving the p-Laplacian [13].


## 2. Preliminaries and Auxilliary Results

LetU be a nonempty open subset of a Banach space X and let f:U⟶ℝ be a function. We recall that f is Lipschitz if there exists some constant K>0 such that for all y and z in U, we have
(3)fy−fz≤Ky−z.Forx∈U, f is said to be locally Lipschitz at x if there exists an open neighborhood V⊂U of x on which the restriction of f is Lipschitz. We will say that f is locally Lipschitz on U if f is locally Lipschitz at every point x∈U. We recall that any convex function has this property in Euclidean spaces.Definition 1.
Letf:U⊂X⟶ℝ be a locally Lipschitz function. Let x∈U and v∈X\0. The generalized directional derivative of f at x in the direction v, denoted by f0x;v, is defined by
(4)f0x;v≔limsupw⟶xt⟶0+fw+tv−fwt.Observe at once thatf0x;v is a (finite) number for all v∈X\0.Indeed, letx∈V⊂U and let K>0 be such that (3) holds for all y,z∈V, with V bounded (without loss of generality). Let wmm>0⊂X be a sequence such that wm⟶x and tm a sequence of 0;+∞ such that tm⟶0. For v∈X\0, as m⟶+∞, the vectors wm+tmv will belong to V. Indeed, by boundedness of V, there exists ρ>0 such that x−y<ρ⇒y∈V. Then, for m large enough, we have
(5)wm+tmv−x≤wm−x+tmv<ρ2+ρ2=ρ.Thus, there existsm0>0 such that for all m>m0, we have
(6)fwm+tmv−fwmtm≤Kv.It follows from (3) and (6) that for all v∈X,
(7)f0x,v≤Kv.Remark 2.
Iff is locally Lipschitz and Gâteaux differentiable at x, then its Gâteaux differential fG′x at x coincides with its generalized gradient. That is,
(8)f0x;v=fG′x⋅vforallv∈X.Proposition 3.
The functionv↦f0x;v is positively homogeneous and subadditive.Proof.
The homogeneity is an immediate consequence of Definition1. We prove the subadditivity. Let v and z be in X. Then,
(9)f0x;v+z=limsupw⟶xt⟶0+fw+tv+tz−fwt≤limsupw⟶xt⟶0+fw+tz+tv−fw+tzt+limsupw⟶xt⟶0+fw+tz−fwt≤limsupr⟶xt⟶0+fr+tv−frt+limsupw⟶xt⟶0+fw+tz−fwt,r≔w+tz=f0x;v+f0x;z.From the previous Proposition3 and the Hahn-Banach theorem [14] (p. 62), it follows that there exists at least one linear function ξ∗:X⟶ℝ satisfying
(10)f0x;v≥ξ∗,vfor allv∈X. From (10) and (7) also rewritten with −v, we obtain
(11)ξ∗,v≤Kvfor allv∈X. Thus, ξ∗∈X∗ (as usual, X∗ denotes the (continuous) dual of X and <.,.> is the duality pairing between X and X∗). Thus, we can give the following definition.Definition 4.
Letf:U⊂X⟶ℝ be locally Lipschitz at a point x∈U. Clarke’s generalized gradient of f at x, denoted ∂fx, is the (nonempty) set of all ξ∗∈X∗ satisfying (10), i.e.,
(12)∂fxs≔ξ∗∈X∗:∀v∈X,f0x;v≥ξ∗,v.We refer to [15–17] for some of the fundamental results in the calculus of generalized gradients. In particular, we shall need the following.Proposition 5 (see [18], Chang).
Iff:U⟶ℝ is a convex function, then Clarke’s generalized gradient of f at x, defined in (12), coincides with the subdifferential of f in the sense of convex analysis.Proposition 6 (see [11], Chen).
LetX be a real Banach space and f:X⟶ℝ be a locally Lipschitz function. Then, the function γ:X⟶ℝ defined by
(13)γu≔minx∗∈∂fux∗,forallu∈X,
is well defined and lower semicontinuous.Proposition 7 (see [15], Proposition 6).
Ifx0 is a minimizer of f, then 0∈∂fx0.Remark 8.
LetX be an infinite dimensional Banach space and f:X⟶ℝp be a locally Lipschitz mapping. For any finite dimensional subspace of X, it makes sense to talk about Clarke’s generalized Jacobian of the function fL:L∍x↦fx∈ℝp at every point x∈L.Notation 9.
. For a locally Lipschitz functionf:ℝn⟶ℝp and x∈ℝn, we consider the set Ωfx defined by Ωfx≔xmmsequenceinℝn such that xm⟶x and f is differentiable at xm.LetX,Z be two Banach spaces such that dim Z=n<∞. Let F:X⟶Z be a locally Lipschitz mapping and L a finite dimensional subspace of X. For x∈L, we denote by ∂FLx Clarke’s generalized Jacobian at a point x, of the restriction of F to L, namely, the function
(14)FL:L⟶Z;x↦Fx.LetY be a Banach space and consider a function F:ℝn×Y⟶ℝp which is locally Lipschitz. For any x¯,y¯∈ℝn×Y, ∂xFx¯,y¯ denotes Clarke’s generalized Jacobian at a point x¯ of the function
(15)F⋅,y¯:ℝn⟶ℝp,x↦Fx,y¯.LetX,Y,Z be three Banach spaces with dim Z<∞ and F:X×Y⟶Z a locally Lipschitz function. For any finite dimensional subspace L of X and for every x¯,y¯∈L×Y, ∂xFLx¯,y¯ will denote Clarke’s generalized Jacobian of the function F~:L∍x↦F~x≔Fx,y¯∈Z at a point x¯.Theorem 10 (Rademacher).
Letf:ℝn⟶ℝ be a locally Lipschitz function. Then, f is almost everywhere differentiable with respect to Lebesgue measure.According to Rademacher’s Theorem10, we have the following.Proposition 11 (see [19], Clarke).
Letf:ℝn⟶ℝ be a locally Lipschitz function and x∈ℝn. If ∂fx denotes the set defined by (12), then
(16)∂fx=colimm⟶+∞f′xm:xmm∈ℕ∈Ωfx.Note that, sincef is almost everywhere differentiable with respect to Lebesgue measure, there exists a sequence xmm∈ℕ⊂ℝn such that xm⟶x, and for any m∈ℕ, f is differentiable at xm. So, Ωfx≠∅. In addition for any xmm∈ℕ∈Ωfx and for any v∈ℝn, we have
(17)f′xm⋅v≤Kv,where K is the Lipschitz constant of f. This means that f′xmm is bounded in Lℝn,ℝ which has a finite dimension. Then, there exists a subsequence f′xσmm of f′xmm that converges to some x∗∈Lℝn,ℝ. That is,
(18)limm⟶+∞f′xσm=x∗.Thus, the convex hull of such limits in (18) is ∂fx.Even if the functionf is defined from ℝn to ℝp, regarding (17) and (18) component by component, we notice that the set defined by (16) is nonempty, compact, and convex in Lℝn,ℝp (see [20] (Definition 1)). Thus, this characterization of ∂fx stated in Proposition 11 is extended to locally Lipschitz functions defined from ℝn to ℝp. In this case, ∂fx is called Clarke’s generalized Jacobian of the function f at a point x.Definition 12.
Letf:ℝn⟶ℝp be a locally Lipschitz mapping and x∈ℝn. Clarke’s generalized Jacobian of f at x also denoted by ∂fx is defined as follows:
(19)∂fx=colimm⟶+∞f′xm:xmm∈ℕ∈Ωfx.The following notions will also be useful in the sequel.Definition 13.
Letf:ℝn⟶ℝp be a locally Lipschitz mapping and x∈ℝn with n≥p. We say that ∂fx is of maximal rank if for all x∗∈∂fx, x∗ is surjective.Definition 14.
LetX be a metric space. A function f:X⟶ℝ is said to be (sequentially) lower semicontinuous at a point x∈X, if for all sequence xmm∈ℕ⊂X such that xm⟶x, we have the inequality
(20)fx≤liminfm⟶+∞fxm.If for all sequencexmm∈ℕ⊂X such that xm⇀x, (20) holds; we say that f is weakly sequentially lower semicontinuous at x.Remark 15.
LetX be a normed vector space and xmm a sequence of X. If x∈X, then
(21)xm⟶x⇒xm⇀x.It follows that the weakly sequentially lower semicontinuity implies the sequentially lower semicontinuity. But the converse is not generally true. However, in the convex case, these two notions are equivalents.The following theorem is a generalization of Ekeland’s variational principle [21].Theorem 16 (see [21], J. Chen).
Leth:0,+∞⟶0,+∞ be a continuous nondecreasing function such that
(22)∫0∞ds1+hs=+∞.LetM be a complete metric space, x0∈M fixed, f:M⟶ℝ∪∞ a lower semicontinuous function, not identically +∞, and bounded from below. Then, for every ε>0, and y∈M such that
(23)fy<infMf+ε,and every λ>0, there exists some point z∈M such that
(24)fz<fy,dz,x0≤r0+r¯,fx≥fz−ελ1+hdx0,zdx,z,∀x∈M,where r0=dx0,y and r¯ is such that
(25)∫r0r0+r¯ds1+hs≥λ.By Theorem16, one has the following.Theorem 17 (see [21], J. Chen).
LetX be a Banach space, h:0,+∞⟶0,+∞ be a continuous nondecreasing function such that
(26)∫0∞ds1+hs=+∞
andf:X⟶ℝ a locally Lipschitz function, bounded from below. Then, there exists a minimizing sequence zmm of f such that
(27)f0zm;v−zm1+hzm≥−εmv−zm,∀v∈X,where εm⟶0+ as m⟶+∞.Proof.
For each positive integerm, choose ym∈Y be such that
(28)fym≤infMf+εm.Takex0=0,X=M, and λ=1 in Theorem 16. Then, there exists zm∈X such that
(29)fzm≤fym,zm≤ym+r¯,fx≥fzm−εm1+hzmx−zm,∀x∈X,where r¯ is such that
(30)∫ymym+r¯ds1+hs≥1.Consequently, for eachx∈X, one has
(31)infε>0δ>0supw<ε0<t<δfzm+w−tx−zm−fzm+wt=infδ>0sup0<t<δfzm+tx−zm−fzmt≥−εmx−zm1+hzm.Hence,f0zm;v−zm1+hzm≥−εmv−zm, for all v∈X.Moreover, obviously,zmm is a minimizing sequence of f.Definition 18.
LetX be a Banach space, f:X⟶ℝ be bounded from below, locally Lipschitz function, and h:0,+∞⟶0,+∞ be continuous nondecreasing function such that
(32)∫0∞ds1+hs=+∞.We say thatumn≥0⊂X is a h-sequence of f if fumm is bounded and f0um;v−um1+hum≥−εmv−um, for all v∈X, where εm⟶0+. We say that f satisfies the h-condition if any h-sequence of f possesses a convergent subsequence.Remark 19.
Sometimes, the following version ofh-condition is also used: Any sequence umm⊂X such that fumm is bounded and
(33)limm⟶∞γum1+hum=0
possesses a convergent subsequence, whereγ is defined in Proposition 6. This condition is equivalent to that of Definition 18.Remark 20.
A coercive function defined onℝn satisfies the h-condition regardless of h. But a function satisfying the h-condition is not necessary coercive. Indeed, Section 4 is devoted to the exposition of an example of a noncoercive function satisfying the h-condition. It is the function defined in (139).The following is the Weierstrass theorem.Lemma 21 (see [13], Lemma 2.1).
Assume thatf:X⟶ℝ is functional on a reflexive Banach space X which is weakly lower semicontinuous and coercive. Then, there exists x∗∈X such that fx∗=minx∈Xfx.Better, by virtue of Theorem17, we can prove the following result.Theorem 22.
LetX be a Banach space, h:0,+∞⟶0,+∞ a continuous nondecreasing function such that
(34)∫0∞ds1+hs=+∞,
andf:X⟶ℝ a locally Lipschitz function and bounded from below. If f satisfies the h-condition, then f achieves its minimum at some critical point z∈X of f.Proof.
By virtue of Theorem17, there exists a minimizing sequence zmm of f and
(35)f0zm;v−zm1+hzm≥−εmv−zmforallv∈X
whereεm⟶0+. Since f satisfies the h-condition, zmm has a convergent subsequence in X. We can assume that zm⟶z in X. Consequently, by the continuity of f,
(36)fz=limm⟶+∞fzm=infx∈Xfx.By Remark19 and the lower continuity of γ, we know γz=0.Theorem 23 (see [22], Clarke).
Letf:ℝn⟶ℝn be a locally Lipschitz mapping such that the Clarke generalized Jacobian ∂fx0 of f at a point x0∈ℝn is of maximal rank. Then, there exist neighborhoods U and V of x0 and fx0, respectively, and a Lipschitz function g:V⟶U such that fgu=u for all u∈U and gfu=v for all v∈V.The following result is Clarke’s implicit function theorem which will be very useful.Theorem 24 (see [6], Clarke].
Assume thatF:ℝn×ℝp⟶ℝn is a locally Lipschitz mapping on a neighborhood of a point x0,y0 such that Fx0,y0=0. Assume further that ∂xFx0,y0 is of maximal rank. Then there exists a neighborhood V⊂ℝp of y0 and a Lipschitz function G:V⟶ℝn such that for every y in V, it holds
(37)FGy,y=0,Gy0=x0.Remark 25.
It would be important to point out that the Clarke implicit function Theorem24 is a corollary of the Clarke inverse function Theorem 23 that can be found in the book [23]. Indeed, as it is done for example in [24] on page 256, when we put
(38)F~:ℝn×ℝp⟶ℝn×ℝp,x,y↦Fx,y,y.F~ is locally Lipschitz in a neighborhood of x0,y0. Moreover, when the Jacobian matrix DF~ exists, it is of the form
(39)DxFDyF0nIp,and it follows that the Clarke generalized Jacobian∂F~x0,y0 of F~ at the point x0,y0 is of maximal rank. Then, by Theorem 4 D.3 of [23], there exist U⊂ℝn×ℝp, V≔FU⊂ℝn×ℝp, and f:V⟶U which is inverse of F~ on U. Obviously, f has the form fx,y=ϕx,y,y, where ϕ:ℝn×ℝp⟶ℝn. Therefore,
(40)x,y∈U,Fx,y=0⇔f0,y=ϕ0,y,y=x,y⇔x=ϕ0,y.Thus, we can writeGy=ϕ0,y.Ifℝp is replaced by any infinite dimensional Banach space Y in Theorem 24, Clarke’s generalized Jacobian of the function F~ above cannot be defined. In other words, we will no longer be in finite dimension to be able to apply Theorem 1 in Clarke’s work [22].This remark is very important in the rest of the work.Remark 26.
LetY be an infinite dimensional Banach space and F:ℝn×Y⟶ℝn be a locally Lipschitz mapping on a neighborhood of a point x0,y0 such that Fx0,y0=0. Assume that ∂xFx0,y0, Clarke’s generalized Jacobian is of maximal rank. Then, there exists V⊂Y, subset containing y0, and a Lipschitz mapping φ:V⟶U⊂ℝn such that for every y∈V, we have
(41)Fφy,y=0,φy0=x0.Moreover, we have the following equivalence:(42)x,y∈U×V,Fx,y=0⇔x=φy.Indeed, letM be a finite dimensional subspace of Y with y0∈M and dimM=mm<∞. We consider the map
(43)F~:ℝn×M⟶ℝn,x,y↦Fx,y.Obviously,F~ is locally Lipschitz mapping, and ∂xF~x0,y0=∂xFx0,y0 is of maximal rank. Then, by Theorem 24, there exist V⊂M, open in M and containing y0, U⊂ℝn, open containing x0 and a locally Lipschitz mapping φ:V⟶U such that conditions (41) and (42) hold.Here is another result that will serve us in this work.Theorem 27 (see [11], J. Chen).
Leth:0,+∞⟶0,+∞ be a continuous nondecreasing function such that
(44)∫0∞ds1+hs=+∞.X is a reflexive Banach space and J:X⟶ℝ is a locally Lipschitz function. Assume that there exists u0∈X,u1∈X and a bounded open neighborhood Ω of u0 such that u1∉Ω and
(45)infx∈∂ΩJx>maxJu0,Ju1.LetM≔g∈C0,1,X:g0=u0,g1=u1 and c≔infg∈Mmaxs∈0,1Jgs. If J satisfies the h-condition, then c is a critical value of J and c>maxJu0,Ju1.Lemma 28.
LetX be a normed vector space and H be a Hilbert space equipped with the inner product ,. Let f:X⟶H be a locally Lipschitz mapping. Then, the function φ:X⟶ℝ defined by
(46)φx=fx,fx=fxH2
is locally Lipschitz.Theorem 29 (see [16], Clarke).
LetX be a normed vector space, f:X⟶ℝn be locally Lipschitz function near x∈X, and h:ℝn⟶ℝ be a given C1 function. Then,
(47)∂h∘fx⊂∇hfx∂fx.Theorem 30 (see [6], Theorem 1).
Assume thatF:ℝn×ℝp⟶ℝn is a locally Lipschitz mapping such that
a1 for any y∈ℝp the functional φy:ℝn⟶ℝ given by
(48)φyx=12Fx,y2
is coercive, i.e.,limx⟶∞φyx=+∞
a2 for any x,y∈ℝn×ℝp, the set ∂xFx,y is of maximal rank
Then, there exists a unique locally Lipschitz functionf:ℝp⟶ℝn such that equation Fx,y=0 and x=fy are equivalent in the set ℝn×ℝp.


## 3. Main Results

The following is a generalization of the global implicit function theorem of [6] to the case of locally Lipschitz functions from Banach spaces to Euclidean spaces.Theorem 31.
LetY be a real Banach space and F:ℝn×Y⟶ℝn be a locally Lipschitz function. Suppose that
(1)
for everyy∈Y, the function φy defined by(49)φyx=12Fx,y2
satisfies theh-condition, where h:0,+∞⟶0,+∞ is a continuous nondecreasing function such that
(50)∫0∞ds1+hs=+∞(2)
for allx,y∈ℝn×Y, ∂xFx,y is of maximal rank
Then, there exists a unique functionf:Y⟶ℝn such that the equation “x,y∈ℝn×YandFx,y=0” are equivalent to x=fy. Moreover, for any finite dimensional subspace L of Y, f is locally Lipschitz on L.Proof.
Lety∈Y. We prove that there exists a unique element xy∈ℝn such that Fxy,y=0. Indeed, φy is locally Lipschitz and satisfies the h-condition. Then, by Theorem 22, there is xy∈ℝn such that minℝnφy=φyxy. Since φy=g∘F⋅,y:ℝn⟶ℝ and by assumption (1), the function F⋅,y:ℝn∍x↦Fx,y∈ℝn is a locally Lipschitz mapping and g:ℝn∍x↦1/2x2=1/2x,xℝn∈ℝ, it follows from Lemma 28 that φy is locally Lipschitz. Then, by Proposition 7, we have 0∈∂ϕyxy. Moreover, according to Theorem 29, we have
∂φyxy⊂∇gF⋅,yxy∘∂F⋅,yxy=∇gFxy,y∘∂xFxy,y=∇gFxy,y∘x∗:x∗∈∂xFxy,y.Thus, there existsx∗∈∂xFxy,y such that ∇gFxy,y∘x∗=0, i.e.,
(51)∀v∈ℝn,∇gFxy,yx∗v=Fxy,y,x∗v=0.By assumption (2)x∗ℝn=ℝn. It follows that Fxy,y=0.About the uniqueness ofxy∈ℝn such that Fxy,y=0, we argue by contradiction supposing that there exists x1≠xy in ℝn with Fx1,y=Fxy,y=0. We use Remark 26. Thus, we set e=x1−xy, and we define the mapping ψy:ℝn⟶ℝ by
(52)ψyx≔φyx+xy=12Fx+xy,y2.We haveψy0=ψye=0. Consider ψy on the boundary ∂B0,ρ of the ball B0,ρ⊂ℝn with some 0<ρ<e. By assumption (2) and Remark 26, we conclude that there exist V⊂Y containing y (not necessary open in Y, but open in some finite dimensional subspace L⊂Y), an open subset U⊂ℝn containing xy, and a function ξ:V⟶U such that the following equivalence holds:
(53)x,y∈U×V,Fx,y=0⇔x=ξy.ψy is also a locally Lipschitz function (so continuous), and ∂B0,ρ is compact (by the fact that it is closed and bounded). Then, ∃x¯∈∂B0,ρ such that
(54)ψyx¯=min∂B0,ρψy.We claim that there exists at least oneρ>0,ρ<e such that minx=ρψy>0. Otherwise, we would have
(55)∀0<ρ<e,minx=ρψy=0;this means that for all nonnegativeρ<e, there exists x¯∈ℝn,x¯=ρ such that ψyx¯=0. Since U is open around xy, there exists 0<ε<e such that
(56)x−xy≤ε⇒x∈U.Letx¯∈ℝn,x¯=ε:ψyx¯=0. Then,
(57)x¯+xy−xy=x¯=ε,ψyx¯=12Fx¯+xy,y2=0⇔Fx¯+xy,y=0.By (56) and (57), we have x¯+xy∈U,x¯+xy≠xy and Fx¯+xy,y=0. It follows from (53) that x¯+xy=ξy. Thus, x¯+xy and xy are two different elements of U with x¯+xy=ξy=xy, what is impossible. As conclusion,
(58)∃ρ<e,infx=ρψy>0=maxψy0,ψye.The functionψy is locally Lipschitz and satisfies the h-condition (because φy satisfies this condition). Then, by (58) and Theorem 24 applied to J=ψy, we note that ψy has a generalized critical point v which is different from 0 and e since the corresponding critical value ψyv holds
(59)ψyv>maxψy0,ψye=0.We have also(60)0∈∂φyxy+v⊂∇gF⋅,yxy+v∘∂F⋅,yxy+v=∇gFxy+v,y∘∂xFxy+v,y.This implies thatFxy+v,y=0⇔ψyv=0. This contradiction with (59) confirms that for every y∈Y, there exists a unique xy∈ℝn such that Fxy,y=0, and we can set fy=xy. Of course, according to Remark 26, we can say that for any finite dimensional subspace L of Y, f is locally Lipschitz on L.An example of function satisfying the assumptions of Theorem31 for which Y is a Banach space is F:ℝ×Y⟶ℝ defined by
(61)Fx,y=2x+x+y.Indeed,F defined in (61) is locally Lipschitz function which is not differentiable and for any y∈Y if we consider the function φy:ℝ⟶ℝ defined by
(62)φyx=12Fx,y2=12x+2x+y2,thenφy is coercive and consequently satisfies the h-condition. Moreover, for any x,y∈ℝ×Y, the partial generalized gradient ∂xFx,y defined as follows
(63)∂xFx,y=3,ifx>0,1,ifx<0,1,3,ifx=0,is of maximal rank. Namely, for anyx,y∈ℝ×Y,0∉∂xFx,y⊂ℝ. Indeed, a straightforward argument shows that
(64)Fx,y=0⇔x=−y.With the conclusion about the regularity off in Theorem 31, we cannot expect in general the continuity of f on the whole Y. Here is a counterexample.Remark 32.
Let us setY=ℓ1, where ℓ1 stands for the space of real sequences umm∈ℕ such that ∑m=0∞um<∞, endowed with the nonequivalent norms:
(65)um1=∑m=0∞um,um2=∑m=0∞um21/2.Indeed, form∈ℕ∗, we define Xm≔1,1/2,1/3,⋯,1/m,0,⋯∈ℓ1. We claim that Xmm is a bounded sequence with respect to the norm ⋅2 which is unbounded with respect to ⋅1. For m∈ℕ∗, we have
(66)Xm2=∑k=1m1k21/2,Xm1=∑k=1m1k.Then,(67)limm⟶+∞Xm2=∑k=1+∞1k21/2<+∞,(68)limm⟶+∞Xm1=∑k=1+∞1k=+∞.Now, let us consider the canonical injectionI:ℓ1,⋅2⟶ℓ1,⋅1. It is obvious by (67) that I is not continuous on ℓ1. However, for any finite dimensional subspace L of ℓ1, since the restriction of these norms on L are equivalent on L, it follows that IL:L1,⋅2⟶L1,⋅1 is Lipschitz.We add some technical hypothesis to those of Theorem31 in order to obtain the continuity of the implicit function f.Definition 33.
LetX,Y be two normed vector spaces. We say that a function F:X×Y⟶ℝ is coercive with respect to x (the first variable), locally uniformly with respect to y (the second variable), if for any y¯∈Y, there exists an open neighborhood V of y¯ in Y such that
(69)limx⟶∞infy∈VFx,y=+∞.Lemma 34.
LetE be a Euclidean space. Then, every bounded sequence with a unique limit point is convergent.Proof.
Letxmm≥0 be a sequence of E which has a unique limit point x¯∈E. This implies that any subsequence of xmm≥0 has a subsequence converging to x¯ by the Bolzano Weierstrass theorem. We argue by contradiction assuming that xmm≥0 is not convergent. Then, there exists ε>0 such that
(70)foranyk>0,thereexistsmk≥k,withxmk−x¯>ε.
xmkk must have a subsequence xmkii such that
(71)xmki⟶x¯,
which contradicts (70).Theorem 35.
LetY be a real Banach space and F:ℝn×Y⟶ℝn be locally Lipschitz. Suppose that
(1)
the functionχ:ℝn×Y⟶ℝ defined by(72)χx,y=12Fx,y2is coercive with respect to x, locally uniformly with respect to y(2)
for allx,y∈ℝn×Y, ∂xFx,y is of maximal rank
Then, there exists a unique functionf:Y⟶ℝn such that
(73)x,y∈ℝn×Y,Fx,y=0⇔x=fy.
Moreover,f is continuous on the whole Y.Proof.
Lety∈Y. We consider the function φy:ℝn⟶ℝ defined by
(74)φyx≔12Fx,y2.Sinceφy is coercive (because χ is coercive with respect to x, locally uniformly with respect to y), it follows that for any continuous nondecreasing function h:0,+∞⟶0,+∞ such that
(75)∫0∞ds1+hs=+∞,φy satisfies the h-condition. Moreover, F is locally Lipschitz. So, by Theorem 31, we conclude that there exists a unique global implicit function f:Y⟶ℝn such that
(76)x,y∈ℝn×Y,Fx,y=0⇔x=fy.It remains to show thatf is continuous on the whole Y. For this, let ymm∈ℕ⊂Y be sequence such that
(77)ym⟶y¯∈Y.For allm∈ℕ, Ffym,ym=0. This implies that the sequence χxm,ymm=Ffym,ymm is bounded. Since χ is coercive with respect to x, locally uniformly with respect to y, there exists an open subset Q⊂Y containing y¯ such that
(78)limx⟶∞infy∈Qχx,y=limx⟶∞infy∈Q12Fx,y2=+∞.In addition, by the convergence ofym to y¯, there exists m0∈ℕ such that
(79)m>m0⇒ym∈Q.So,(80)form>m0,12Fxm,ym2≥infy∈Q12Fxm,y2.According to (78) and (80), we conclude that the sequence xmm∈ℕ≔fym is bounded in ℝn. Let x¯ be a limit point of xmm. Thus, there exists a convergent subsequence xmkk of xm such that
(81)xmk⟶x¯∈ℝn.On the other hand, for allk∈ℕ, Fxmk,ymk=0. Then, it follows from (77), (81), and the continuity of the function F that
(82)0=limk⟶+∞Fxmk,ymk=Fx¯,y¯.Thus, we have(83)Fx¯,y¯=0⇔x¯=fy¯.So,xmm has a unique limit point x¯=fy¯. So, by Lemma 34,
(84)xm⟶x¯,that is,(85)fym⟶fy¯.From (77) and (85), f is continuous on Y.As a consequence of our Theorem31, we have the following nonsmooth global inverse function theorem.Theorem 36.
Assumef:ℝn⟶ℝn is a locally Lipschitz mapping such that
(1)
for anyy∈ℝn, there exists a continuous nondecreasing function h:ℝ+⟶ℝ+ such that(86)∫0∞ds1+hs=+∞,
and the functionalφy:ℝn⟶ℝ defined by
(87)φyx=12fx−y2
satisfies theh-condition
(2)
for anyx∈ℝn, we have that ∂fx is of maximal rank
Then,f is a global homeomorphism on ℝn and f−1 is locally Lipschitz.Corollary 37 (see [25], Hadamard-Palais].
LetX,Y be finite dimensional Banach spaces. Assume that f:X⟶Y is a C1-mapping such that
(1)
limx⟶∞fx=∞(2)
for anyx∈X, f′x is invertible
thenf is a diffeomorphism.Question. Is it still possible the conclusion of Theorem31 under the assumption of the h-condition on the function τy:x↦Fx,yα, where α is a positive constant different from 2?In fact, according to our two Lemmas42 and 43 and Corollary 44, it is enough to assume that τy is locally Lipschitz in the case 0<α<2. Else, this additional hypothesis is not need in the case α>2.Therefore, we have the following result from Theorem31.Theorem 38.
LetY be a real Banach space and F:ℝn×Y⟶ℝn be a locally Lipschitz mapping. Suppose that
(1)
for anyy∈Y, there exists 0<α<2, so that the function τy:ℝn⟶ℝ(88)τyx=Fx,yα
is locally Lipschitz and satisfies theh-condition, where h:0,+∞⟶0,+∞ is a continuous nondecreasing function such that
(89)∫0∞ds1+hs=+∞(2)
for allx,y∈ℝn×Y, ∂xFx,y is of maximal rank
Then, there exists a unique functionf:Y⟶ℝn such that the equation "x,y∈ℝn×Y and Fx,y=0" are equivalent to x=fy. Moreover, for any finite dimensional subspace L of Y, f is locally Lipschitz on L.Proof.
Lety∈Y. We notice that τy=2φyα/2, where φy is defined by (49). Since τy satisfies the h-condition, it follows from Lemma 43 and Corollary 44 that φy satisfies the h-condition. Thus, we achieve the proof by using Theorem 31.But, what happens if we replaceℝn by any Banach space X in the domain of the function F?Theorem 39.
LetX,Y be Banach spaces and Z be Euclidean space such that dim Z=n<∞. Let F:X×Y⟶Z be locally Lipschitz function. Assume that
(1)
for ally∈Y, the function φy:X⟶ℝ defined by(90)ϕyx=12Fx,y2
satisfies theh-condition, where h:0,+∞⟶0,+∞ is a continuous nondecreasing function such that
(91)∫0∞ds1+hs=+∞(2)
for any finite dimensional subspaceL of X with dim L=n and for all x,y∈L×Y, ∂xFLx,y is of maximal rank
Then,(92)x,y∈X×Y,Fx,y=0⇔x=0.Proof.
We use Theorem31 in order to prove this result. Firstly, we prove that there exists a unique global implicit function f:Y⟶X such that x,y∈X×Y and Fx,y=0 are equivalent from x=fy. After that, we will claim that f≡0 on Y.
Lety∈Y. Since φy is bounded form below, locally Lipschitz, and satisfies the h-condition, we see by Theorem 22 that φy has a minimum which is achieved at a critical point xy∈X. Let L be a finite dimensional subspace of X such that xy∈L and dimL=n.Consider functionsF~:L×Y⟶Z and φ~y:L⟶ℝ defined, respectively, by
(93)F~x,y=Fx,y,φ~yx=φyx=12Fx,y2=12F~x,y2.By assumption (1), the functionF~ is locally Lipschitz and φ~y is then locally Lipschitz as composition of F~ by C1 function g where
(94)g:Z⟶ℝ;x↦12x2=12x,xZ.Likewise,φ~y satisfies the h-condition. It follows that φ~y has a minimum on L. Since xy∈L, it is obvious that minx∈Lφ~yx=φxy. Thus, by Theorem 29, we obtain
(95)0∈∂φ~yxy=∂Lφyxy⊂∇gF~xy,y∘∂xF~xy,y=∇gFxy,y∘∂xFLxy,y.This means that there existsx∗∈∂xFLxy,y such that
(96)∀h∈L,Fxy,y,x∗h=0.Thus, we conclude by assumption (2) thatF~xy,y=Fxy,y=0. It is clear by Theorem 31 that xy is the only solution of the equation Fxy,y=0 in L. But the question is the uniqueness of this solution in all X. Even though it is unpredictable, we will answer yes to this question in the following. About uniqueness of xy∈X such that Fxy,y=0, we argue by contradiction.Suppose that there existsx1≠xy such that Fx1,y=Fxy,y=0. We choose then another finite dimensional subspace of X (that we note again L here to keep the same notation) which contains both x1 and xy, such that dim L=n. We consider the same functions defined in (93), but with the subspace L that we choose here. We find that minx∈Lφ~yx=φ~yxy and by [15] (Proposition 6), Theorem 29, and assumption (2), we conclude that F~xy,y=0. Considering the function ψ~y defined as in (52) by
(97)ψ~yx≔φ~yx+xy=12Fx+xy,y2and following the same approach in the proof of Theorem31, we come to the contradiction. Thus, there exists a unique global implicit function f:Y⟶X such that Ffy,y=0 for all y∈Y.It remains to be shown thatf≡0 on Y. Indeed, since X is infinite dimensional Banach space, it is possible to find two n-dimensional subspaces L1 and L2 of X such that L1∩L2=0X. Let F1:L1×Y⟶Z and F2:L2×Y⟶Z be the function defined by
(98)F1x,y=Fx,y,F2x,y=Fx,y.By assumptions (1) and (2), both functionsF1 and F2 verify the assumptions of Theorem 31. Consequently, there exist two functions φ1:Y⟶L1 and ϕ2:Y⟶L2 such that ∀y∈Y,Fφ1y,y=Fφ2y,y=0. Then, according to the uniqueness of xy∈X such that Fxy,y=0, we have φ1y=fy=φ2y∈L1∩L1. Thus, fy=0,∀y∈Y.In virtue of Lemma43, the following theorem is a consequence of Theorem 39.Theorem 40.
LetX,Y be Banach spaces and Z be Euclidean space such that dimZ=n<∞. Let F:X×Y⟶Z be locally Lipschitz function. Assume that
(1)
for everyy∈Y, there exists 0<α<2 such that the function τy:X⟶ℝ defined by(99)τyx=Fx,yα
is locally Lipschitz and satisfies theh-condition, where h:0,+∞⟶0,+∞ is a continuous nondecreasing function such that
(100)∫0∞ds1+hs=+∞(2)
for any finite dimensional subspaceL of X and for any x,y∈L×Y, ∂xFLx,y is of maximal rankThen.(101)x,y∈X×Y,Fx,y=0⇔x=0.Remark 41.
Ifα>2, it is useless to add the locally Lipschitz condition to the h-condition for the first assumption of Theorems 38 and 40. Indeed, in Theorem 38, for example, since φy is locally Lipschitz and α/2>1, it follows from Lemma 43 that τy=2φyα/2 is also locally Lipschitz. Moreover, φy satisfies the h-condition.Lemma 42.
Letg:ℝn⟶ℝ+ be a locally Lipschitz function. Let α>1. If gα is locally Lipschitz, then for any x,v∈ℝn with v≠0, we have
(102)gα0x;v=α⋅gxα−1g0x;v.Proof.
Letwmm be a sequence in ℝn and tmm⊂0;+∞ another sequence such that
(103)wm⟶x,tm⟶0+.For fixedm, the function μ:Im⟶ℝ;t↦tα is differentiable, where
(104)Im≔θgam+1−θgbm;θ∈0,1,witham=wm+tmv,bm=wm.Now, it is known that there existscm∈Im such that
(105)gamα−gbmα=μgam−μgbm=μ′cmgam−gbm=μ′θmgam+1−θmgbm⋅gam−gbm=α⋅θmgam+1−θmgbmα−1⋅gam−gbm=Km⋅gwm+tmv−gwm,where
(106)Km=α⋅θmgwm+tmv+1−θmgwmα−1,withθm∈0,1. Then, we have
(107)gαwm+tmv−gαwmtm=Km⋅gwm+tmv−gwmtm.Sinceg is continuous, there exists a neighborhood V of x and K>0 such that
(108)gz≤K,∀z∈V.It follows from the convergence ofwm,tm to x,0 and the continuity of g and (108) that
(109)limm⟶+∞Km=α⋅gxα−1.By (107), (109), and the fact that
(110)limsupw⟶xt⟶0+gw+tv−gwt=g0x;v,we conclude that(111)gα0x;v=α⋅gxα−1g0x;v.Lemma 43.
Letg:ℝn⟶ℝ+ be a locally Lipschitz function. Let α>1 and h:0,+∞⟶0,+∞ a continuous nondecreasing function such that
(112)∫0∞ds1+hs=+∞.Then,gαx≔gxα is locally Lipschitz function. Moreover, g satisfies the h-condition if and only if gα satisfies the h-condition.Proof.
Letx¯∈X. There exist V∍x¯ open subset of ℝn and k>0 such that
(113)gx−gy≤kx−y,∀x,y∈V.Letρ>0 such that B¯ρx¯≔x∈ℝn:x¯−x≤ρ⊂V. For x,y∈B¯ρ, as in the previous Lemma 42, there exists θ∈0,1 such that we have
(114)gαx−gαy=α⋅gx−gy⋅θgx+1−θgyα−1.g is continuous, and B¯ρ is compact. Let
(115)M=maxz∈B¯ρx¯gz.Then, we have(116)θgx+1−θgyα−1≤Mα−1.It follows from (113), (114), and (116) that we have
(117)gαx−gαy≤kMα−1x−y,∀x,y∈Bρx¯,where Bρx¯≔x∈ℝn:x¯−x<ρ⊂V is open. Then, gα is locally Lipschitz.For the second part of Lemma43, just specify that umm≥0 is a h-sequence of g if and only if umm≥0 is a h-sequence of gα.Letvmm≥0⊂ℝn be a h-sequence of g. Then, there exist q>0, τmm⊂0,+∞ with τm⟶0+, such that
(118)gvm≤q,∀m≥0,(119)g0vm;v−vm1+hvm≥−τmv−vm,∀v∈ℝn.It follows from (118) that gvmα is bounded. From Lemma 42 and inequality (119), we deduce that
(120)gα0vm;v−vm1+hvm≥−τ¯mv−vm,∀v∈ℝn,τ¯m≔τmαqα−1⟶0+.Thus,vmm≥0 is a h-sequence of gα.Conversely, letumm≥0⊂ℝn be a h-sequence of gα. Then, there exists p>0 such that
(121)gumα≤p,∀m≥0,(122)gα0um;v−um1+hum≥−εmv−um,∀v∈ℝn,εm⟶0+.It follows from (121) that there exists p¯>0 such that
(123)gum≤p¯,∀m≥0.By Lemma42, (122), and (123), we have
(124)g0um;v−um1+hum≥−δmv−um,∀v∈ℝn,δm≔εmαp¯α−1⟶0+.Therefore,umm≥0 is also a h-sequence of g. Then, for any α>1, g satisfies the h-condition if and only if gα satisfies the h-condition.But what about0<α<1?Corollary 44.
Letg:ℝn⟶ℝ+ be a function and 0<α<1 such that gα is locally Lipschitz. Then, g is locally Lipschitz and for any continuous nondecreasing function h:ℝ+⟶ℝ+ such that
(125)∫0∞ds1+hs=+∞.g satisfies the h-condition if and only if gα satisfies the h-condition.Proof.
We notice thatg=gα1/α and 1/α>1. Then, we apply Lemma 43.


## 4. Example of Noncoercive Function Satisfying the (h)-Condition

To illustrate that the compactness condition allowing to obtain the existence of a global implicit function in our main results is weaker than that used in Theorem30, we provide in this section an example of a noncoercive and locally Lipschitz function satisfying the h-condition. We follow the idea used by Chen and Tang in [13] (Theorem 3.3).Let1<p<∞. Define
(126)Lp0,T;ℝN=u∈L10,T;ℝN:∫0Tutpdt<∞,with the norm(127)up=∫0Tupdt1/p.Foru∈Lloc10,T;ℝN, u′ is said to be the weak derivative of u, if u′∈Lloc10,T;ℝN and
(128)∫0Tu′ϕdt=−∫0Tuϕ′dt,∀ϕ∈C0∞0,T;ℝN.Let(129)W01,p0,T;ℝN=u∈Lp0,T;ℝN:u0=uT,u′∈Lp0,T;ℝN.W01,p0,T;ℝN is a reflexive Banach space (see [13]) with the norm
(130)uW01,p0,T;ℝN=∫0Tup+u′pdt1/p.Remark 45 (see [13]).
We have the following direct decomposition ofW01,p0,T;ℝN(131)W01,p0,T;ℝN=ℝN⊕V,whereV=v∈W01,p0,T;ℝN:∫0Tvtdt=0.Consider now the following functional:(132)Ju=∫0T1pu′pdt,u∈W01,p0,T;ℝN.We know that (see [26]) J∈C1W01,p0,T;ℝN,ℝ and p-Laplacian operator u↦u′p−2u′′ is the derivative operator of J in the weak sense. That is,
(133)A=J′:W01,p0,T;ℝN⟶W01,p0,T;ℝN∗,(134)Au,v=∫0Tu′tp−2u′t,v′tℝNdt,u,v∈W01,p0,T;ℝN.Proposition 46 (see [27], Fan and Zhao).
J′ is a mapping of S+, i.e., if
(135)um⇀u,limsupm⟶∞J′um−J′u,um−u≤0,then umm has a convergent subsequence in W01,p0,T;ℝN.For everyu∈W01,p0,T;ℝN, set
(136)u¯=1T∫0Tutdt,(137)u~t=ut−u¯.We have the following Poincare-Wirtinger inequality (see [28]):
(138)∃a>0suchthatu~∞≤au′p,∀u∈W01,p0,T;ℝN.We consider the functionalϕ:W01,p0,T;ℝN⟶ℝ defined by
(139)ϕu=∫0T1pu′pdt−∫0Tjt,udt,u∈W01,p0,T;ℝN,where jt,u:0,T×ℝN⟶ℝ is the norm ⋅ on ℝN defined by
(140)jt,u=u=∑i=1Nui21/2,foru≔u1,u2,⋯,uN∈ℝN.Leths=s. Following the same approach as done by Chen and Tang in [13], we will show that the function φ satisfies the h-condition and is noncoercive on all of W01,p0,T;ℝN.We show that the functionj satisfies the following assumptions:
(1)
For allu∈ℝN,t↦jt,u is measurable(2)
For almost allt∈0,T,u↦jt,u is locally Lipschitz(3)
For everyr>0, there exists αr∈L10,T such that for almost all t∈0,T,u≤r and all w∈∂jt,u, we have w≤αrt, where ∂jt,s is Clarke’s generalized gradient of j with respect to the variable s(4)
There exist0<μ<p and M>0 such that for almost all t∈0,T and all u≥M, we have(141)j0t,u;u<μjt,u(5)
jt,u⟶+∞ uniformly for almost all t∈0,T as u⟶∞Obviously, the functionj defined in (140) satisfies conditions (1), (2), and (5). In addition, for t,u∈0,T×ℝN, we have the following:
(1)
Ifu≠0,∂jt,u=u/u(2)
Ifu=0,∂jt,0=w∈ℝN:y≥w,y,foranyy∈ℝN=B¯0,1That is,(142)∂jt,0=B¯0,1≔y∈ℝN:y≤1.Indeed, fort∈0,T, the function jt,⋅ is convex. Then, Clarke’s generalized gradient ∂jt,u of jt,⋅ at a point u coincides with the subdifferential of jt,⋅ in convex analysis sense (see Proposition 5). We recall also that the norm in Hibert space is Fréchet differentiable at any point u≠0.Thus,(143)w≤1,forallw∈∂jt,u.Consequently, according to (143), for every r>0, and αrt=1, then αr∈L10,T and
(144)u≤r⇒w≤αr,∀w∈∂jt,u.Thus, the functionj satisfies the assumption 3.On other hand, since1<p, taking μ=1+p/2, we have
(145)0<1<μ<p.Moreover, forM>0, we have
(146)u≥M⇒u≠0,(147)μjt,u=μu.Then,(148)j0t,u;u=uu,u=u.It follows from (145), (146), and (148) that
(149)u≥M⇒j0t,u;u<μjt,u.Thus, the functionj satisfies the assumption (4).Under the previous assumptions,φ is locally Lipschitz (see [13] (Theorem 3.3)).
(1)
By (130), for any u∈ℝN, we have(150)u=T1/pu,∫0T1pu′pdt=0,ϕu=−∫0Tudt=−Tu.Then,(151)limu⟶+∞u∈ℝNϕu=limu⟶+∞−Tu=−∞.Thus,φ is not coercive.
(2)
Letumm≥1 be a h-sequence of φ, i.e., there exists M1>0 such that(152)φum≤M1,(153)limm⟶+∞1+umγum=0,where
(154)γum=minw∗∈∂ϕumw∗.Without loss of generality, we suppose thatum≠0,∀m≥1.According to Proposition6, let um∗∈∂ϕum such that um∗=γum.By definition (134) of operator A, we have
(155)um∗=Aum−wm,withwm∈∂jt,um.From the second assertion of (152), we have
(156)um∗,um=∫0Tum′tpdt−∫0Twmt,umtdt≤εm,εm↓0.Thus, it follows from Definition4 and inequality (156) that
(157)∫0Tum′tpdt−∫0Tj0t,umt;umtdt≤εm.Sinceum≠0, according to (148), the inequality (157) implies
(158)∫0Tum′tpdt−∫0Tumtdt≤εm.From the first assertion of (152), we have
(159)−μp∫0Tum′tpdt+∫0Tμumtdt≤μM1.It follows from (158) and (159) that
(160)1−μp∫0Tum′tpdt+∫0Tμ−1umtdt≤Mm,withMm=εm+μM1. By (160), we have
(161)1−μp∫0Tum′tp≤Mm,m≥1,Mm⟶μM1.By (161), there exists M0>0 such that for t∈0,T and m≥1,
(162)um′t≤M0.From (161) and the Poincare-Wirtinger inequality (138), u~m is bounded in W01,p0,T;ℝN. By exploiting (152) once again, we use (136) to have
(163)1p∫0Tu~m′pdt−∫0Tumtdt≤M1,m≥1.Sinceu~m is bounded, it follows from (163) that there exists M2>0 such that
(164)∫0Tumtdt≤M2,∀m≥1.Thus, there existsM3>0 such that for t∈0,T and m≥1,
(165)umt≤M3.By (162) and (165), we infer that umm≥1⊂W01,p0,T;ℝN is bounded, and so by passing to a subsequence if necessary, we may assume that
(166)um⇀uinW01,p0,T;ℝN,um⟶uinC00,T;ℝN.Next, we will prove thatum⟶uinW01,p0,T;ℝN. By Proposition 46, it suffices to prove that the following inequality holds:
(167)limm⟶∞Aum−Au,um−u≤0.In fact, from the choice of the sequenceumm≥1, we have
(168)um∗,um≤εm↓0.Then, by (155), we have
(169)Aum,um−u−∫0Twmt,umt−utℝNdt≤εm,∀m≥1.By (3),wm⊂L10,T is bounded and
(170)limm⟶∞∫0Twmt,umt−utℝNdt=0.Then,(171)limsupm⟶∞Aum,um−u≤0.So, we have(172)limsupm⟶∞Aum−Au,um−u≤0,εm↓0.


## 5. An Application

Inspired by the example result of Galewski-Rădulescu [6] (Theorem 7), we provide in this section an existence and uniqueness result for the problem
(173)Ax=Fx+ξ,where ξ∈ℝn is fixed; A is an n×n matrix which does not need to be positive definite, negative definite, or symmetric; and F:ℝn⟶ℝn is a locally Lipschitz function.Theorem 47.
LetA be an n×n matrix. If F:ℝn⟶ℝn is a locally Lipschitz mapping satisfying the following conditions:
(1)
For anyξ∈ℝn, there exists a continuous nondecreasing function h:ℝ+⟶ℝ+ such that(174)∫0∞ds1+hs=+∞,
and the functionalφξ:ℝn⟶ℝ defined by
(175)φξx=Ax−Fx−ξ
satisfies theh-condition
(2)
For anyx∈ℝn and for every T∈∂Fx, A−T is invertible
Then, problem (173) has a unique solution for fixed ξ∈ℝn. Moreover, the map that assigns to each ξ∈ℝn, the unique solution of problem (173) is locally Lipschitz.Proof.
Consider the functionfx=Ax−Fx from ℝn to itself. By assumption (173) and Lemma 43, for any ξ∈ℝn, the functional φξ:ℝn⟶ℝ defined by
(176)φξx=12fx−ξ2=12Ax−Fx−ξ2
satisfies theh-condition. In addition, according to (173), ∂fx=A−∂Fx is of maximal rank for any x∈ℝn. Then, we achieve the proof applying Theorem 36.Example of matrixA and function F satisfying conditions of Theorem 47.Let us take a matrix(177)A=−312−1and the functionF defined from ℝ2 to ℝ2 by
(178)Fu=x3+y,2x+x+y3,u=x,y.Considering the Euclidean norm(179)u=x2+y2forallu∈ℝ2,we have(180)x3,y32−12u32=x6+y6−14x2+y23=34x2+y2x2−y22≥0.It follows that(181)x3,y3≥12u3.On the other hand, foru∈ℝ2,
(182)Au≤A⋅u.From (181) and (182), we have
(183)Fu−Au−ξ=x3+y,2x+x+y3−Au−ξ≥x3,y3−0,2x−y,x−Au−ξ≥12u3−2u−u−A⋅u−ξ≥12u2−3−Au−ξ.Hence, for fixedξ∈ℝ2, the function φξ:ℝn⟶ℝ defined by
(184)φξu=Fu−Au−ξis coercive. Consequently, the functionφξ satisfies the h-condition.Letu=x,y∈ℝ2.
(1)
Ifx≠0 and y≠0, then F is differentiable at u and ∂Fu−A=JFu−A is defined by(185)∂Fu−A=3x2+3sgny−1sgnx3y2+1.Thus,∂Fu−A will be one of the following matrices:
(186)3x2+3013y2+1,3x2+3−213y2+1,3x2+3−2−13y2+1,3x2+30−13y2+1.In all these cases, we havedet∂Fu−A≠0.
(2)
Ifx<0andy=0, then ∂Fu is defined by(187)∂Fu=conv3x2−110,3x2110=3x2s10:−1≤s≤1.It follows that(188)∂Fu−A=3x2+3s−1−11:−1≤s≤1.Then, forT∈∂Fu, there exists s∈−1,1 such that
(189)detT−A=3x2+3+s−1=x2+s+2≥x2+1>0.(3)
Ifx>0andy=0, then ∂Fu is the following:(190)∂Fu=conv3x2130,3x2−130=3x2s30:−1≤s≤1.It follows that(191)∂Fu−A=3x2+3s−111:−1≤s≤1.Then, forT∈∂Fu, there exists s∈−1,1 such that
(192)detT−A=3x2+3−s−1=3x2+1+3−s≥3x2+1>0.(4)
Ifx=0 and y<0, then ∂Fu is the following:(193)∂Fu=conv0−133y2,0−113y2=0−1λ3y2:1≤λ≤3.It follows that(194)∂Fu−A=3−2λ−23y2+1:1≤λ≤3.Then, forT∈∂Fu, there exists λ∈1,3 such that
(195)detT−A=33y2+1+2λ−2=9x2+2λ−1>0.(5)
Ifx=0andy>0, then ∂Fu is the following:(196)∂Fu=conv0133y2,0113y2=01λ3y2:1≤λ≤3.It follows that(197)∂Fu−A=30λ−23y2+1:1≤λ≤3.Then, forT∈∂Fu, there exists λ∈1,3 such that
(198)detT−A=33y2+1>0.(6)
Ifu=0,0, then ∂Fu is the following:(199)∂Fu=conv0130,0−130,0−110,0110=0τs0:τ,s∈−1,1×1,3.It follows that(200)∂F0,0−A=3τ−1s−21:τ,s∈−1,1×1,3.Then, forT∈∂Fu, there exists τ,s∈−1,1×1,3 such that
(201)detT−A=3−s−2τ−1=s−11−τ+τ+2≥τ+2>0.


## 6. Conclusion

We have provided a general nonsmooth global implicit function theorem that yields Galewski-Rădulescu’s nonsmooth global implicit function theorem and a series of results on the existence, uniqueness, and possible continuity of global implicit functions for the zeros of locally Lipschitz functions. Our results deal with functions defined on infinite dimensional Banach spaces and thus generalize also classical Clarke’s implicit function theorem for functionsF:ℝn×ℝp⟶ℝn by replacing ℝp by any Banach space Y. We have worked in this paper under the h-condition which is weaker than the coercivity required in [6]. Our method is based on a variational approach and a recent nonsmooth version of Mountain Pass Theorem.More precisely, firstly, we have proved our Theorem31 on the existence and uniqueness of the global implicit function theorem for equations Fx,y=0, where F:ℝn×Y⟶ℝn is a locally Lipschitz function with Y a Banach space. Secondly, we observe that this extension to infinite dimension may not guarantee the continuity of the global implicit function. Thus, we provide an additional hypothesis on Theorem 31 in order to obtain the continuity of the implicit function f. Moreover, our Lemmas 42 and 43 allow us to prove other more general results on the existence and uniqueness of global implicit functions under the h-condition on the function x↦Fx,yα with 0<α<2.



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*Source: 1021461-2022-07-28.xml*