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.
1. Introduction
Many mathematical models involving real or vector-valued functions stand as equations of the form
For complex phenomena, the unknown is often a vector-variable belonging to or to an abstract Banach space having a direct sum . It may even happen that equation (1) is just a state equation depending in fact on a parameter (or a control) . In this case, it takes the form
and the most aspiring aim of mathematical analysis is to know the local or global structure of the solution set 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 , 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 map , where are real Banach spaces and 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 -condition (see Definition 18 and Remark 19), we work in this paper under the -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 Theorem 24 for function by replacing by any Banach space (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 , where with a Banach space(ii)The proof of the continuity of the implicit function based on a simple additional hypothesis, Theorem 35(iii)The weakening of the coercivity assumption used in [6] by considering a compactness type condition called -condition in [11](iv)By our Lemmas 42 and 43, we obtain Theorem 38 on the existence and uniqueness of global implicit functions under the -condition on the function with . This is a generalization of the result (49) in the nonsmooth case. It also generalizes the result [12] (Theorem 3.6) in the case
This article is organized as follows. In Section 2, we recall some preliminary and auxilliary results on Clarke’s generalized gradient, Clarke’s generalized Jacobian, and the -condition for locally Lipschitz functions. Section 3 is devoted to our main results established under the -condition, on the existence and uniqueness of global implicit function for equation , where is defined from to and 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 -Laplacian [13].
2. Preliminaries and Auxilliary Results
Let be a nonempty open subset of a Banach space and let be a function. We recall that is Lipschitz if there exists some constant such that for all and in , we have
For , is said to be locally Lipschitz at if there exists an open neighborhood of on which the restriction of is Lipschitz. We will say that is locally Lipschitz on if is locally Lipschitz at every point . We recall that any convex function has this property in Euclidean spaces.
Definition 1. Let be a locally Lipschitz function. Let and . The generalized directional derivative of at in the direction , denoted by , is defined by
Observe at once that is a (finite) number for all .
Indeed, let and let be such that (3) holds for all , with bounded (without loss of generality). Let be a sequence such that and a sequence of such that . For , as , the vectors will belong to . Indeed, by boundedness of , there exists such that . Then, for large enough, we have
Thus, there exists such that for all , we have
It follows from (3) and (6) that for all ,
Remark 2. If is locally Lipschitz and Gâteaux differentiable at , then its Gâteaux differential at coincides with its generalized gradient. That is,
Proposition 3. The function is positively homogeneous and subadditive.
Proof. The homogeneity is an immediate consequence of Definition 1. We prove the subadditivity. Let and be in . Then,
From the previous Proposition 3 and the Hahn-Banach theorem [14] (p. 62), it follows that there exists at least one linear function satisfying
for all . From (10) and (7) also rewritten with , we obtain
for all . Thus, (as usual, denotes the (continuous) dual of and <.,.> is the duality pairing between and ). Thus, we can give the following definition.
Definition 4. Let be locally Lipschitz at a point . Clarke’s generalized gradient of at , denoted , is the (nonempty) set of all satisfying (10), i.e.,
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). If is a convex function, then Clarke’s generalized gradient of at , defined in (12), coincides with the subdifferential of in the sense of convex analysis.
Proposition 6 (see [11], Chen). Let be a real Banach space and be a locally Lipschitz function. Then, the function defined by is well defined and lower semicontinuous.
Proposition 7 (see [15], Proposition 6). If is a minimizer of , then .
Remark 8. Let be an infinite dimensional Banach space and be a locally Lipschitz mapping. For any finite dimensional subspace of , it makes sense to talk about Clarke’s generalized Jacobian of the function at every point .
Notation 9. . For a locally Lipschitz function and , we consider the set defined by such that and is differentiable at .
Let be two Banach spaces such that dim . Let be a locally Lipschitz mapping and a finite dimensional subspace of . For , we denote by Clarke’s generalized Jacobian at a point , of the restriction of to , namely, the function
Let be a Banach space and consider a function which is locally Lipschitz. For any , denotes Clarke’s generalized Jacobian at a point of the function
Let be three Banach spaces with dim and a locally Lipschitz function. For any finite dimensional subspace of and for every , will denote Clarke’s generalized Jacobian of the function at a point .
Theorem 10 (Rademacher). Let be a locally Lipschitz function. Then, is almost everywhere differentiable with respect to Lebesgue measure.
According to Rademacher’s Theorem 10, we have the following.
Proposition 11 (see [19], Clarke). Let be a locally Lipschitz function and . If denotes the set defined by (12), then
Note that, since is almost everywhere differentiable with respect to Lebesgue measure, there exists a sequence such that , and for any , is differentiable at . So, . In addition for any and for any , we have where is the Lipschitz constant of . This means that is bounded in which has a finite dimension. Then, there exists a subsequence of that converges to some . That is,
Thus, the convex hull of such limits in (18) is .
Even if the function is defined from to , regarding (17) and (18) component by component, we notice that the set defined by (16) is nonempty, compact, and convex in (see [20] (Definition 1)). Thus, this characterization of stated in Proposition 11 is extended to locally Lipschitz functions defined from to . In this case, is called Clarke’s generalized Jacobian of the function at a point .
Definition 12. Let be a locally Lipschitz mapping and . Clarke’s generalized Jacobian of at also denoted by is defined as follows:
The following notions will also be useful in the sequel.
Definition 13. Let be a locally Lipschitz mapping and with . We say that is of maximal rank if for all , is surjective.
Definition 14. Let be a metric space. A function is said to be (sequentially) lower semicontinuous at a point , if for all sequence such that , we have the inequality
If for all sequence such that , (20) holds; we say that is weakly sequentially lower semicontinuous at .
Remark 15. Let be a normed vector space and a sequence of . If , then
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). Let be a continuous nondecreasing function such that
Let be a complete metric space, fixed, a lower semicontinuous function, not identically , and bounded from below. Then, for every , and such that and every , there exists some point such that where and is such that
By Theorem 16, one has the following.
Theorem 17 (see [21], J. Chen). Let be a Banach space, be a continuous nondecreasing function such that and a locally Lipschitz function, bounded from below. Then, there exists a minimizing sequence of such that where as .
Proof. For each positive integer , choose be such that
Take , and in Theorem 16. Then, there exists such that where is such that
Consequently, for each , one has
Hence, , for all .
Moreover, obviously, is a minimizing sequence of .
Definition 18. Let be a Banach space, be bounded from below, locally Lipschitz function, and be continuous nondecreasing function such that
We say that is a -sequence of if is bounded and , for all , where . We say that satisfies the -condition if any -sequence of possesses a convergent subsequence.
Remark 19. Sometimes, the following version of -condition is also used: Any sequence such that is bounded and possesses a convergent subsequence, where is defined in Proposition 6. This condition is equivalent to that of Definition 18.
Remark 20. A coercive function defined on satisfies the -condition regardless of . But a function satisfying the -condition is not necessary coercive. Indeed, Section is devoted to the exposition of an example of a noncoercive function satisfying the -condition. It is the function defined in (139).
The following is the Weierstrass theorem.
Lemma 21 (see [13], Lemma 2.1). Assume that is functional on a reflexive Banach space which is weakly lower semicontinuous and coercive. Then, there exists such that .
Better, by virtue of Theorem 17, we can prove the following result.
Theorem 22. Let be a Banach space, a continuous nondecreasing function such that and a locally Lipschitz function and bounded from below. If satisfies the -condition, then achieves its minimum at some critical point of .
Proof. By virtue of Theorem 17, there exists a minimizing sequence of and where . Since satisfies the -condition, has a convergent subsequence in . We can assume that in . Consequently, by the continuity of ,
By Remark 19 and the lower continuity of , we know .
Theorem 23 (see [22], Clarke). Let be a locally Lipschitz mapping such that the Clarke generalized Jacobian of at a point is of maximal rank. Then, there exist neighborhoods and of and , respectively, and a Lipschitz function such that for all and for all .
The following result is Clarke’s implicit function theorem which will be very useful.
Theorem 24 (see [6], Clarke]. Assume that is a locally Lipschitz mapping on a neighborhood of a point such that . Assume further that is of maximal rank. Then there exists a neighborhood of and a Lipschitz function such that for every in , it holds
Remark 25. It would be important to point out that the Clarke implicit function Theorem 24 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
is locally Lipschitz in a neighborhood of . Moreover, when the Jacobian matrix exists, it is of the form
and it follows that the Clarke generalized Jacobian of at the point is of maximal rank. Then, by Theorem 4 D.3 of [23], there exist , , and which is inverse of on . Obviously, has the form , where . Therefore,
Thus, we can write .
If is replaced by any infinite dimensional Banach space in Theorem 24, Clarke’s generalized Jacobian of the function 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. Let be an infinite dimensional Banach space and be a locally Lipschitz mapping on a neighborhood of a point such that . Assume that , Clarke’s generalized Jacobian is of maximal rank. Then, there exists , subset containing , and a Lipschitz mapping such that for every , we have
Moreover, we have the following equivalence:
Indeed, let be a finite dimensional subspace of with and dim. We consider the map
Obviously, is locally Lipschitz mapping, and is of maximal rank. Then, by Theorem 24, there exist , open in and containing , , open containing and a locally Lipschitz mapping such that conditions (41) and (42) hold.
Here is another result that will serve us in this work.
Theorem 27 (see [11], J. Chen). Let be a continuous nondecreasing function such that
is a reflexive Banach space and is a locally Lipschitz function. Assume that there exists and a bounded open neighborhood of such that and
Let and . If satisfies the -condition, then is a critical value of and .
Lemma 28. Let be a normed vector space and be a Hilbert space equipped with the inner product . Let be a locally Lipschitz mapping. Then, the function defined by is locally Lipschitz.
Theorem 29 (see [16], Clarke). Let be a normed vector space, be locally Lipschitz function near , and be a given function. Then,
Theorem 30 (see [6], Theorem 1). Assume that is a locally Lipschitz mapping such that
for any the functional given by
is coercive, i.e.,
for any , the set is of maximal rank
Then, there exists a unique locally Lipschitz function such that equation and are equivalent in the set .
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. Let be a real Banach space and be a locally Lipschitz function. Suppose that (1)for every , the function defined bysatisfies the -condition, where is a continuous nondecreasing function such that (2)for all , is of maximal rankThen, there exists a unique function such that the equation “” are equivalent to . Moreover, for any finite dimensional subspace of , is locally Lipschitz on .
Proof. Let . We prove that there exists a unique element such that . Indeed, is locally Lipschitz and satisfies the -condition. Then, by Theorem 22, there is such that . Since and by assumption (1), the function is a locally Lipschitz mapping and , it follows from Lemma 28 that is locally Lipschitz. Then, by Proposition 7, we have . Moreover, according to Theorem 29, we have
Thus, there exists such that , i.e.,
By assumption (2) . It follows that .
About the uniqueness of such that , we argue by contradiction supposing that there exists in with . We use Remark 26. Thus, we set , and we define the mapping by
We have . Consider on the boundary of the ball with some . By assumption (2) and Remark 26, we conclude that there exist containing (not necessary open in , but open in some finite dimensional subspace ), an open subset containing , and a function such that the following equivalence holds:
is also a locally Lipschitz function (so continuous), and is compact (by the fact that it is closed and bounded). Then, such that
We claim that there exists at least one such that . Otherwise, we would have
this means that for all nonnegative , there exists such that . Since is open around , there exists such that
Let . Then,
By (56) and (57), we have and . It follows from (53) that . Thus, and are two different elements of with , what is impossible. As conclusion,
The function is locally Lipschitz and satisfies the -condition (because satisfies this condition). Then, by (58) and Theorem 24 applied to , we note that has a generalized critical point which is different from and since the corresponding critical value holds
We have also
This implies that . This contradiction with (59) confirms that for every , there exists a unique such that , and we can set . Of course, according to Remark 26, we can say that for any finite dimensional subspace of , is locally Lipschitz on .
An example of function satisfying the assumptions of Theorem 31 for which is a Banach space is defined by
Indeed, defined in (61) is locally Lipschitz function which is not differentiable and for any if we consider the function defined by
then is coercive and consequently satisfies the -condition. Moreover, for any , the partial generalized gradient defined as follows
is of maximal rank. Namely, for any . Indeed, a straightforward argument shows that
With the conclusion about the regularity of in Theorem 31, we cannot expect in general the continuity of on the whole . Here is a counterexample.
Remark 32. Let us set , where stands for the space of real sequences such that , endowed with the nonequivalent norms:
Indeed, for , we define . We claim that is a bounded sequence with respect to the norm which is unbounded with respect to . For we have
Then,
Now, let us consider the canonical injection . It is obvious by (67) that is not continuous on . However, for any finite dimensional subspace of , since the restriction of these norms on are equivalent on , it follows that is Lipschitz.
We add some technical hypothesis to those of Theorem 31 in order to obtain the continuity of the implicit function .
Definition 33. Let be two normed vector spaces. We say that a function is coercive with respect to (the first variable), locally uniformly with respect to (the second variable), if for any , there exists an open neighborhood of in such that
Lemma 34. Let be a Euclidean space. Then, every bounded sequence with a unique limit point is convergent.
Proof. Let be a sequence of which has a unique limit point . This implies that any subsequence of has a subsequence converging to by the Bolzano Weierstrass theorem. We argue by contradiction assuming that is not convergent. Then, there exists such that must have a subsequence such that which contradicts (70).
Theorem 35. Let be a real Banach space and be locally Lipschitz. Suppose that (1)the function defined byis coercive with respect to , locally uniformly with respect to (2)for all , is of maximal rankThen, there exists a unique function such that Moreover, is continuous on the whole .
Proof. Let . We consider the function defined by
Since is coercive (because is coercive with respect to , locally uniformly with respect to ), it follows that for any continuous nondecreasing function such that
satisfies the -condition. Moreover, is locally Lipschitz. So, by Theorem 31, we conclude that there exists a unique global implicit function such that
It remains to show that is continuous on the whole . For this, let be sequence such that
For all , . This implies that the sequence is bounded. Since is coercive with respect to , locally uniformly with respect to , there exists an open subset containing such that
In addition, by the convergence of to , there exists such that
So,
According to (78) and (80), we conclude that the sequence is bounded in . Let be a limit point of . Thus, there exists a convergent subsequence of such that
On the other hand, for all , . Then, it follows from (77), (81), and the continuity of the function that
Thus, we have
So, has a unique limit point . So, by Lemma 34,
that is,
From (77) and (85), is continuous on .
As a consequence of our Theorem 31, we have the following nonsmooth global inverse function theorem.
Theorem 36. Assume is a locally Lipschitz mapping such that (1)for any , there exists a continuous nondecreasing function such thatand the functional defined by satisfies the -condition (2)for any , we have that is of maximal rankThen, is a global homeomorphism on and is locally Lipschitz.
Corollary 37 (see [25], Hadamard-Palais]. Let be finite dimensional Banach spaces. Assume that is a -mapping such that (1)(2)for any , is invertiblethen is a diffeomorphism.
Question. Is it still possible the conclusion of Theorem 31 under the assumption of the -condition on the function , where is a positive constant different from 2?
In fact, according to our two Lemmas 42 and 43 and Corollary 44, it is enough to assume that is locally Lipschitz in the case . Else, this additional hypothesis is not need in the case .
Therefore, we have the following result from Theorem 31.
Theorem 38. Let be a real Banach space and be a locally Lipschitz mapping. Suppose that (1)for any , there exists , so that the function is locally Lipschitz and satisfies the -condition, where is a continuous nondecreasing function such that (2)for all , is of maximal rankThen, there exists a unique function such that the equation and are equivalent to . Moreover, for any finite dimensional subspace of , is locally Lipschitz on .
Proof. Let . We notice that , where is defined by (49). Since satisfies the -condition, it follows from Lemma 43 and Corollary 44 that satisfies the -condition. Thus, we achieve the proof by using Theorem 31.
But, what happens if we replace by any Banach space in the domain of the function ?
Theorem 39. Let be Banach spaces and be Euclidean space such that dim . Let be locally Lipschitz function. Assume that (1)for all , the function defined bysatisfies the -condition, where is a continuous nondecreasing function such that (2)for any finite dimensional subspace of with dim and for all , is of maximal rankThen,
Proof. We use Theorem 31 in order to prove this result. Firstly, we prove that there exists a unique global implicit function such that and are equivalent from . After that, we will claim that on .
Let . Since is bounded form below, locally Lipschitz, and satisfies the -condition, we see by Theorem 22 that has a minimum which is achieved at a critical point . Let be a finite dimensional subspace of such that and dim.
Consider functions and defined, respectively, by
By assumption (1), the function is locally Lipschitz and is then locally Lipschitz as composition of by function where
Likewise, satisfies the -condition. It follows that has a minimum on . Since , it is obvious that . Thus, by Theorem 29, we obtain
This means that there exists such that
Thus, we conclude by assumption (2) that . It is clear by Theorem 31 that is the only solution of the equation in . But the question is the uniqueness of this solution in all . Even though it is unpredictable, we will answer yes to this question in the following. About uniqueness of such that , we argue by contradiction.
Suppose that there exists such that . We choose then another finite dimensional subspace of (that we note again here to keep the same notation) which contains both and , such that dim . We consider the same functions defined in (93), but with the subspace that we choose here. We find that and by [15] (Proposition 6), Theorem 29, and assumption (2), we conclude that . Considering the function defined as in (52) by
and following the same approach in the proof of Theorem 31, we come to the contradiction. Thus, there exists a unique global implicit function such that for all .
It remains to be shown that on . Indeed, since is infinite dimensional Banach space, it is possible to find two -dimensional subspaces and of such that . Let and be the function defined by
By assumptions (1) and (2), both functions and verify the assumptions of Theorem 31. Consequently, there exist two functions and such that . Then, according to the uniqueness of such that , we have . Thus, .
In virtue of Lemma 43, the following theorem is a consequence of Theorem 39.
Theorem 40. Let be Banach spaces and be Euclidean space such that dim. Let be locally Lipschitz function. Assume that (1)for every , there exists such that the function defined byis locally Lipschitz and satisfies the -condition, where is a continuous nondecreasing function such that (2)for any finite dimensional subspace of and for any , is of maximal rank
Then.
Remark 41. If , it is useless to add the locally Lipschitz condition to the -condition for the first assumption of Theorems 38 and 40. Indeed, in Theorem 38, for example, since is locally Lipschitz and , it follows from Lemma 43 that is also locally Lipschitz. Moreover, satisfies the -condition.
Lemma 42. Let be a locally Lipschitz function. Let . If is locally Lipschitz, then for any with , we have
Proof. Let be a sequence in and another sequence such that
For fixed , the function is differentiable, where
Now, it is known that there exists such that where
with . Then, we have
Since is continuous, there exists a neighborhood of and such that
It follows from the convergence of to and the continuity of and (108) that
By (107), (109), and the fact that
we conclude that
Lemma 43. Let be a locally Lipschitz function. Let and a continuous nondecreasing function such that
Then, is locally Lipschitz function. Moreover, satisfies the -condition if and only if satisfies the -condition.
Proof. Let . There exist open subset of and such that
Let such that . For , as in the previous Lemma 42, there exists such that we have
is continuous, and is compact. Let
Then, we have
It follows from (113), (114), and (116) that we have where is open. Then, is locally Lipschitz.
For the second part of Lemma 43, just specify that is a -sequence of if and only if is a -sequence of .
Let be a -sequence of . Then, there exist , with , such that
It follows from (118) that is bounded. From Lemma 42 and inequality (119), we deduce that
Thus, is a -sequence of .
Conversely, let be a -sequence of . Then, there exists such that
It follows from (121) that there exists such that
By Lemma 42, (122), and (123), we have
Therefore, is also a -sequence of . Then, for any , satisfies the -condition if and only if satisfies the -condition.
But what about ?
Corollary 44. Let be a function and such that is locally Lipschitz. Then, is locally Lipschitz and for any continuous nondecreasing function such that
satisfies the -condition if and only if satisfies the -condition.
Proof. We notice that and . Then, we apply Lemma 43.
4. Example of Noncoercive Function Satisfying the ()-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 Theorem 30, we provide in this section an example of a noncoercive and locally Lipschitz function satisfying the -condition. We follow the idea used by Chen and Tang in [13] (Theorem 3.3).
Let . Define
with the norm
For , is said to be the weak derivative of , if and
Let
is a reflexive Banach space (see [13]) with the norm
Remark 45 (see [13]). We have the following direct decomposition of
Consider now the following functional:
We know that (see [26]) and -Laplacian operator is the derivative operator of in the weak sense. That is,
Proposition 46 (see [27], Fan and Zhao). is a mapping of , i.e., if then has a convergent subsequence in .
For every , set
We have the following Poincare-Wirtinger inequality (see [28]):
We consider the functional defined by where is the norm on defined by
Let . Following the same approach as done by Chen and Tang in [13], we will show that the function satisfies the -condition and is noncoercive on all of .
We show that the function satisfies the following assumptions: (1)For all is measurable(2)For almost all is locally Lipschitz(3)For every , there exists such that for almost all and all , we have , where is Clarke’s generalized gradient of with respect to the variable (4)There exist and such that for almost all and all , we have(5) uniformly for almost all as
Obviously, the function defined in (140) satisfies conditions (1), (2), and (5). In addition, for , we have the following: (1)(2)
That is,
Indeed, for , the function is convex. Then, Clarke’s generalized gradient of at a point coincides with the subdifferential of in convex analysis sense (see Proposition 5). We recall also that the norm in Hibert space is Fréchet differentiable at any point .
Thus,
Consequently, according to (143), for every , and , then and
Thus, the function satisfies the assumption .
On other hand, since , taking , we have
Moreover, for , we have
Then,
It follows from (145), (146), and (148) that
Thus, the function satisfies the assumption (4).
Under the previous assumptions, is locally Lipschitz (see [13] (Theorem 3.3)). (1)By (130), for any , we have
Then,
Thus, is not coercive. (2)Let be a -sequence of , i.e., there exists such thatwhere
Without loss of generality, we suppose that .
According to Proposition 6, let such that .
By definition (134) of operator , we have
with .
From the second assertion of (152), we have
Thus, it follows from Definition 4 and inequality (156) that
Since , according to (148), the inequality (157) implies
From the first assertion of (152), we have
It follows from (158) and (159) that
with . By (160), we have
By (161), there exists such that for and ,
From (161) and the Poincare-Wirtinger inequality (138), is bounded in . By exploiting (152) once again, we use (136) to have
Since is bounded, it follows from (163) that there exists such that
Thus, there exists such that for and ,
By (162) and (165), we infer that is bounded, and so by passing to a subsequence if necessary, we may assume that
Next, we will prove that . By Proposition 46, it suffices to prove that the following inequality holds:
In fact, from the choice of the sequence , we have
Then, by (155), we have
By (3), is bounded and
Then,
So, we have
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 where is fixed; is an matrix which does not need to be positive definite, negative definite, or symmetric; and is a locally Lipschitz function.
Theorem 47. Let be an matrix. If is a locally Lipschitz mapping satisfying the following conditions: (1)For any , there exists a continuous nondecreasing function such thatand the functional defined by satisfies the -condition (2)For any and for every , is invertibleThen, problem (173) has a unique solution for fixed . Moreover, the map that assigns to each , the unique solution of problem (173) is locally Lipschitz.
Proof. Consider the function from to itself. By assumption (173) and Lemma 43, for any , the functional defined by satisfies the -condition. In addition, according to (173), is of maximal rank for any . Then, we achieve the proof applying Theorem 36.
Example of matrix and function satisfying conditions of Theorem 47.
Let us take a matrix
and the function defined from to by
Considering the Euclidean norm
we have
It follows that
On the other hand, for ,
Hence, for fixed , the function defined by
is coercive. Consequently, the function satisfies the -condition.
Let . (1)If and , then is differentiable at and is defined by
Thus, will be one of the following matrices:
In all these cases, we have . (2)If and, then is defined by
It follows that
Then, for , there exists such that (3)If and, then is the following:
It follows that
Then, for , there exists such that (4)If and , then is the following:
It follows that
Then, for , there exists such that (5)If and, then is the following:
It follows that
Then, for , there exists such that (6)If , then is the following:
It follows that
Then, for , there exists such that
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 functions by replacing by any Banach space . We have worked in this paper under the -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 Theorem 31 on the existence and uniqueness of the global implicit function theorem for equations , where is a locally Lipschitz function with 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 . 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 -condition on the function with .
Data Availability
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Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is funded by the “African Centers of Excellence in Mathematical and Computer Sciences and Applications” CEA-SMIA project grant. The authors C. Dansou and F. Dohemeto have been supported by the “African Center of Excellence in Mathematical and Computer Sciences, Informatics and Applications” CEA-SMIA project.