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MORE ON THE INVERSE OF THE SUM OF SET-VALUED
MAPPINGS WITH APPLICATIONS TO VARIATIONAL
INCLUSIONS
BOUALEM ALLECHE AND VICENŢIU D. RĂDULESCU
Abstract. In this paper, we deal with the inverse of the sum of setvalued non necessarily Lipschitzian mappings. First, we develop results
on the behavior of fixed points sets of set-valued pseudo-contraction
mappings. Then, we investigate the notions related to the Aubin property and make use of connections between the involved set-valued mappings to establish results on the inverse of the sum of two set-valued
non necessarily Lipschitzian mappings similar to those in the literature
generalizing Lyusternik and Graves theorems. By means of proximal
convergence, we apply our results to the sensitivity analysis of variational inclusions.
1. Introduction
Studies about the inverse of the sum of set-valued mappings have drawn in
last years the attention of many authors and constitute today an important
and active research field. One of the principal motivations of such studies is
related to the existence of solutions of variational inclusions. Recall that a
variational inclusion (or a generalized equation) is a problem of the form
find x ∈ X such that y ∈ A (x) ,
(VI)
where A is a set-valued mapping acting between two Banach spaces X and
Y , and y ∈ Y is a given point. In many cases, the point y could be of
the form f (x) where f is a single-valued function from X to Y or of the
form f (p, x) with p a parameter leading to an important class of variational
inclusions called parameterized generalized equations.
It is well known that this problem serves as a general framework for describing in a unified manner various problems arising in nonlinear analysis
and in other areas in mathematics including optimization problems and variational inequality problems, for more information on the subject with survey
of old and recent developments, we refer to [17] and the references therein.
In the simple case of a single-valued mapping A, problem (VI) reduces
to a simply functional equation, and it is then related to the surjectivity of
the involved single-valued mapping. From the same point of view, in the
2010 Mathematics Subject Classification. 49J53, 47J22, 49J40, 49K40.
Key words and phrases. Set-valued mapping, Pseudo-Lipschitzian, Aubin property,
Fixed point, Proximal convergence.
1
2
BOUALEM ALLECHE AND VICENŢIU D. RĂDULESCU
case of set-valued mappings, the problem is also related to the surjectivity
of the involved set-valued mapping in the analogue sense. The pioneering
work in this direction is the well-known Banach open mapping theorem which
guarantees that a continuous mapping acting between Banach spaces is open
if and only if it is surjective.
Among various generalizations of the Banach open mapping theorem,
there are the famous works by Lyusternik [25] and Graves [21] for nonlinear
Fréchet differentiable functions. Also, many investigations about the solution mappings by means of classical differentiability or by the concepts of
generalized differentiation have been carried out and several results for variational inclusions have been obtained. This direction has given rise to the rich
theory of what is known by the theory of implicit functions for parameterized
generalized equations, see [17, 32, 23]) and the references therein.
Another point of view in the generalization of the Banach open mapping
theorem, which has roots in the Milyutin’s covering mapping theorem, is
what is known in the literature under the name of the openness with linear
rate or the covering property, see [12]. This approach makes use of a constant
like that appearing in the Banach open mapping theorem and has led to
studying the regularity properties of the inverse of set-valued mappings which
has produced many results with applications to different kinds of variational
inclusions in the infinite dimensional settings. Recently, this direction has
attracted a special attention of several authors, see for instance [4, 10, 11,
13, 14, 18, 19] and the references therein.
In this paper, we investigate the regularity properties and deal mainly with
the Lipschitzian property of the inverse of the sum of two set-valued mappings. As the inverse of the inverse of a set-valued mapping is the set-valued
mapping itself, and since the inverse of a Lipschitzian set-valued mapping
need not be Lipschitzian, we wonder why always consider set-valued Lipschitzian mappings if we want to obtain that the inverse of their sum is
Lipschitzian. This leads to say that nothing can prevent a set-valued nonLipschitzian mapping to have an inverse set-valued mapping which is Lipschitzian. However, going back to the Banach open mapping theorem, we
understand that this question has roots in the fact that the inverse of a
surjective linear and continuous mapping acting between Banach spaces is
Lipschitzian, and by linearity, the mapping itself is Lipschitzian. Of course,
this property is not conserved by set-valued mapping. Motivated by this
question, we follow the approach of [10] to study the Lipschitzian property
of the sum of two set-valued mappings. Rather than set-valued Lipschitzian
mappings, we investigate here weaker conditions on the involved set-valued
mappings such as the property of being set-valued pseudo-Lipschitzian mapping and related properties. Note that we are aware that, as mentioned in
[10, 13], the global properties are quite different from their local counterparts. In this paper, some additional conditions connecting between them
the two involved set-valued mappings are considered.
INVERSE OF THE SUM OF SET-VALUED MAPPINGS
3
The paper is structured as follows. In the next section, we give the necessary background to deal with set-valued mappings in the settings of metric
spaces. By new arguments, we first obtain some results on the behaviors of
fixed points sets of set-valued pseudo-contraction mappings. This investigation is based on other conditions for which our results are similar to those
in this direction obtained in the literature, and more recently in [2, 6, 7, 27].
In the third section, we deal with the inverse of the sum of set-valued nonnecessarily Lipschitzian mappings. We introduce some notions defined from
the properties of set-valued pseudo-Lipschitzian mappings and by techniques
inspired from [10, 11], we obtain some properties by means of our results on
the behavior of fixed point sets of set-valued pseudo-contraction mappings.
Under weakened conditions of being Lipschitzian but with additional conditions on the existence of fixed points, we obtain that the inverse of the sum
of two set-valued mappings is Lipschitzian. In the last section, we make use
of the proximal convergence to develop techniques that allow us to obtain
results on the sensitivity analysis of variational inclusions.
2. Notations and preliminary results
Throughout this paper, (X, d) stands for a metric space. Given x ∈ X
and r > 0, we denote by B (x, r) (resp. B (x, r)) the open (resp. closed) ball
around x with radius r.
The distance from a point x ∈ X to a nonempty subset A of X is given
by
d (x, A) = inf{d (x, y) | y ∈ A}.
As usual, d(x, ∅) = +∞. For A 6= ∅, the open ball around A with radius r is
denoted by
[
B (A, r) = {x ∈ X | d (x, A) < r} =
B (u, r) .
u∈A
For two subsets A and B of X, the excess of A over B with respect to d is
denoted by ed (A, B) (or by e (A, B) if no confusion may occur) and defined
by
ed (A, B) = sup d (x, B) .
x∈A
In particular, for two subsets A and B of X, we adopt the conventions
ed (∅, B) = 0 and ed (A, ∅) = +∞ if A 6= ∅.
The distance between A and B with respect to d is denoted by hd (A, B)
(or by h (A, B) if no confusion may occur) and is defined by
hd (A, B) = max {e (A, B) , e (B, A)} .
Restricted to the closed subsets, h is an (extended real-valued) metric called
the Pompeiu-Hausdorff metric. This notion was introduced in the Ph.D.
thesis of the Romanian mathematician D. Pompeiu [31]. In 1914, Hausdorff
[22] studies the notion of set distance in the natural setting of metric spaces
and quotes Pompeiu as the author of this notion. Moreover, Hausdorff also
4
BOUALEM ALLECHE AND VICENŢIU D. RĂDULESCU
P
establishes that the distances defined with “ " (Pompeiu) or with “max"
(Hausdorff) are equivalent.
Let (X, dX ) and (Y, dY ) be two metric spaces. In the sequel, a set-valued
mapping T from X to Y will be denoted by T : X ⇒ Y . The domain of T
denoted by dom (T ) is the set
dom (T ) = {x ∈ X | T (x) 6= ∅}
and the graph of T denoted by graph (T ) is the set
graph (T ) = {(x, y) ∈ X × Y | y ∈ T (x)} .
If the graph of a set-valued mapping T : X ⇒ Y is closed then T has closed
values. The converse holds under additional conditions, in particular if T is
upper semicontinuous, see for example [1, 8].
For a subset A of X, we denote T (A) = ∪x∈A T (x). For a subset B of Y ,
we write
T −1 (B) = {x ∈ X | B ∩ T (x) 6= ∅}
while T −1 (y) stands for T −1 ({y}), if y ∈ Y .
The Lipschitz continuity (with respect to the Pompeiu-Hausdorff metric)
is one of the most popular properties of set-valued mappings. This property will be studied for the inverse mappings of set-valued mapping in next
section.
Let M be a subset of X and T : X ⇒ Y be a set-valued mapping. Recall
that T is said to be L-Lipschitzian on M if
hdY (T (x1 ) , T (x2 )) ≤ λdX (x1 , x2 )
∀x1 , x2 ∈ M.
If L ∈ [0, 1), then T is called L-contraction on M . If T is L-Lipschitzian
or L-contraction on the whole space X, we say that T is L-Lipschitzian or
L-contraction, respectively.
Following [6], a set-valued mapping T : X ⇒ X is said to be L-pseudoLipschitzian with respect to a subset M of X if
e (T (x1 ) ∩ M, T (x2 )) ≤ Ld (x1 , x2 )
∀x1 , x2 ∈ M.
If L ∈ [0, 1), then T is called L-pseudo-contraction with respect to M .
An important property of set-valued Lipschitzian mappings has been proved
in [10, 11] to deal with the properties of the inverse of the sum of two setvalued mappings. That is, if T : X ⇒ Y is L-Lipschitzian and A, B ⊂
dom (T ), then
edY (T (A) , T (B)) ≤ LedX (A, B) .
This property being not adapted to pseudo-contraction set-valued mappings,
a similar definition will be presented later to deal with the properties of the
inverse of the sum of two non necessarily set-valued Lipschitzian mappings.
In the sequel, the fixed points set of a set-valued mapping T : X ⇒ X
will be denoted by Fix (T ), that is,
Fix (T ) = {x ∈ X | x ∈ T (x)} .
INVERSE OF THE SUM OF SET-VALUED MAPPINGS
5
Results on the existence of fixed points for contraction mappings and
set-valued contraction mappings as well as for set-valued pseudo-contraction mappings (with additional conditions) abound in the literature. Also,
existence of fixed points is a subject which is not limited to contraction
set-valued mappings and in this spirit, we will not make use of classical
conditions assuring existence of fixed points for set-valued mappings. More
precisely, we will always consider two set-valued mappings and assume that
the fixed points sets of the involved set-valued mappings are nonempty and
linked in such a way that a result on the behaviors of their fixed points sets is
derived. And instead of conditions on the distance between the images of the
set-valued mappings as considered in some recent papers (see for instance
[6, 27]), we impose here conditions only on those for the fixed points.
However, in certain situations as in the following result, our assumptions
will also involve indirectly known conditions for existence of fixed points for
set-valued pseudo-contraction mappings. In the proof of this result, we will
avoid classical procedures usually employed when dealing with the behaviors
of fixed points sets of set-valued mappings. We use here the following more
precise version of a well-known result on existence of fixed points of set-valued
pseudo-contraction mappings, see [9, 15, 24, 29]. This version is enhanced
in the sense that not only the completeness is assumed only on the closed
ball, but more particulary, only the values of the restriction of the set-valued
mapping on the closed ball are assumed to be nonempty and closed. Of
course the proof follows, step by step, the arguments used in [15] which are
based on techniques having roots in the Banach contraction principle. A
proof using arguments based on a weak variant of the Ekeland variational
principle has also been carried out recently in [9, 30].
Lemma 2.1. Let (X, d) be a metric space. Let x
¯ ∈ X and α > 0 be such
x, α) is a complete metric subspace. Let λ ∈ [0, 1) and T : X ⇒ X
that B (¯
be a set-valued mapping with nonempty closed values on B (¯
x, α) such that
(1) d (¯
x, T (¯
x)) < (1 − λ) α and
x, α) , T (x0 ) ≤ λd (x, x0 )
(2) e T (x) ∩ B (¯
∀x, x0 ∈ B (¯
x, α).
Then, T has a fixed point in B (¯
x, α).
Now, we derive the following result on the behavior of fixed points sets of
set-valued mappings which should be compared to [6, Proposition 2.4], [7,
Proposition 2.4], [20, Proposition 4.5] and more recently to [2, Theorem 3.1].
This result is important for dealing with the properties of the inverse of the
sum of two set-valued mappings.
Theorem 2.2. Let (X, d) be a metric space. Let x0 ∈ X and r > 0 be such
that B (x0 , r) is a complete metric subspace. Let λ ∈ (0, 1) and 0 < β <
(1 − λ) r and let T, S : X ⇒ X be two set-valued mappings such that
(1) T is λ-pseudo-contraction with respect to B (x0 , r) and has nonempty
closed values on B (x0 , r);
6
BOUALEM ALLECHE AND VICENŢIU D. RĂDULESCU
(2) S has nonempty fixed points set and for every x ∈ Fix (S),
d (x, x0 ) < β
and
d (x, T (x)) < λβ.
Then, T has a nonempty fixed points set and
1
e(Fix (S) , Fix (T )) ≤
sup e (S (x) , T (x)) .
1 − λ x∈B(x0 ,r)
Proof. Fix ε > 0 and put
(
)
1
λβ
α = min
sup e (S (x) , T (x)) + ε,
.
1 − λ x∈B(x0 ,r)
1−λ
Let x
¯ ∈ Fix (S) be an arbitrary element.
Claim 1. We prove that B (¯
x, α) ⊂ B (x0 , r). To do this, let x ∈ B (¯
x, α).
Then, from assumption (2), we have
d (x, x0 ) ≤ d (x, x
¯) + d (¯
x, x0 )
1
<α+β ≤
λβ + β < λr + (1 − λ) r = r.
1−λ
Claim 2. We have d (¯
x, T (¯
x)) < (1 − λ) α. Indeed, since x
¯ ∈ Fix (S), then
by assumption (2), d (¯
x, T (¯
x)) < λβ. Also,
d (¯
x, T (¯
x)) ≤ d (S (¯
x) , T (¯
x)) ≤
sup
e (S (x) , T (x)) ,
x∈B(x0 ,r)
and since d (¯
x, T (¯
x)) is finished, then
d (¯
x, T (¯
x)) <
sup
e (S (x) , T (x)) + (1 − λ) ε.
x∈B(x0 ,r)
Thus d (¯
x, T (¯
x)) < (1 − λ) α.
It results by Claim 1 and assumption (1) that T has nonempty closed
values on B (¯
x, α) and for every x, x0 ∈ B (¯
x, α),
e T (x) ∩ B (¯
x, α) , T x0 ≤ e T (x) ∩ B (x0 , r) , T x0 ≤ λd x, x0 .
x, α) and then,
Now all the conditions of Lemma 2.1 are satisfied for T on B (¯
∗
T has a fixed point x ∈ B (¯
x, α).
It results that
1
d (x, Fix (T )) ≤ d (x, x∗ ) ≤ α ≤
sup e (S (x) , T (x)) + ε
1 − λ x∈B(x0 ,r)
This inequality being valid for any x ∈ Fix (S), we obtain
e (Fix (S) , Fix (T )) ≤
1
sup e (S (x) , T (x)) + ε.
1 − λ x∈B(x0 ,r)
Letting ε go to zero, we complete the proof.
INVERSE OF THE SUM OF SET-VALUED MAPPINGS
7
Corollary 2.3. Suppose that the conditions of Theorem 2.2 are verified,
where T and S can be inverted. Then
1
h(Fix (S) , Fix (T )) ≤
sup h (S (x) , T (x)) .
1 − λ x∈B(x0 ,r)
Remark 1. It is worthwhile emphasizing the importance of the above result
which allows to replace the excess by the Pompeiu-Hausdorff metric in the
conclusion of Theorem 2.2. To our knowledge, even if all the fixed points
sets of the involved set-valued mappings are in B (x0 , r), there does not seem
to be any result in the literature dealing with set-valued pseudo-contraction
mappings which provides such a conclusion, see for comparison [6, Proposition 2.4].
In the following example we give two set-valued mappings satisfying the
conditions of Theorem 2.2 with respect to each other. Though some conditions are relaxed, this example provides us a situation where the PompeiuHausdorff metric can be used in the conclusion of Theorem 2.2.
Example 1. According to Theorem 2.2, let X = R2 , x0 = (0, 0), r = 1 and
λ = √12 .
Let T : R2 ⇒ R2 be the set-valued mapping defined by
i
( h
x
0, |x|
if k (x, y) k < 1,
2 ×
2 ∪ [3, +∞[
T ((x, y)) =
2
{2x} × 0, x
if k (x, y) k ≥ 1.
Clearly, T has nonempty closed values on B (x0 , r) and the images of points
of B (x0 , r) are not necessarily included in B (x0 , r). And since, for every
(x1 , y1 ) and (x2 , y2 ) in B (x0 , r), we have
1
e (T ((x1 , y1 )) ∩ B (x0 , r) , T ((x2 , y2 ))) ≤ √ |x1 − x2 | ,
2
then, T is λ-pseudo-contraction with respect to B (x0 , r). We note that T is
not Lipschitzian on R2 and Fix (T ) = {(0,
0)}.
q √
Now, take any α ∈ 0, 2 λ5 1 − λ and define S : R2 ⇒ R2 by
h
i
|x|
×
0,
∪
[3,
+∞[
if k (x, y) k < 1,
2
2
if k (x, y) k ≥ 1.
{2x} × 0, x3
(
x+α
S ((x, y)) =
The set-valued mapping S has nonempty closed values on B (x0 , r) and the
images of points of B (x0 , r) are not necessarily included in B (x0 , r). Also, it
is λ-pseudo-contraction with respect to B (x0 , r) and Fix (S) = {α}× 0, α22 .
Finally, S is not Lipschitzian
on R2 .
√ We put β = 1 − λ < (1 − λ) r and we will verify the other conditions
of Theorem 2.2.
8
BOUALEM ALLECHE AND VICENŢIU D. RĂDULESCU
(1) For the unique fixed point (0, 0) of T , we have
α
d ((0, 0) , S ((0, 0))) =
2r
√ λ
≤2
1 − λ < λ (1 − λ) = λβ.
5
(2) For any (α, γ) ∈ Fix (S), we have
r
√
√ √ α2
5
2
d ((α, γ) , (0, 0)) ≤ α +
=
α= λ 1− λ <β
4
2
and
r
α α α2 α 2
d ((α, γ) , T ((α, γ))) ≤ d (α, γ) ,
,
=
+ γ−
2 2
4
2
r √
λ
α
1 − λ < λ (1 − λ) = λβ.
≤ ≤
2
5
We conclude the section by the following easily verified result. This corollary will be useful in the sequel.
Proposition 2.4. Under assumptions of Theorem 2.2, we have
1
e(Fix (S) ∩ B, Fix (T )) ≤
sup e (S (x) ∩ B, T (x)) ,
1 − λ x∈B(x0 ,r)
for every subset B of X such that B ∩ Fix (S) 6= ∅.
3. On the inverse of the sum of two set-valued mappings
In this section we will be concerned with the properties of the inverse of
the sum of two set-valued mappings.
Recall that if (Y, d) is a linear metric space, the distance d is said to be
shift invariant metric if
for all y, y 0 , z ∈ X.
d y + z, y 0 + z = d y, y 0
Let A and B be two subsets of a linear metric space (Y, d) with shift
invariant metric d, and a, b, b0 ∈ Y . It is shown in [10, 11] that
ed (A + a, B + a) ≤ h (A, B) and ed A + b, A + b0 ≤ d b, b0 .
Following [5], recall that a set valued mapping T : X ⇒ Y , where (X, dX )
and (Y, dY ) are two metric spaces, is said to be pseudo-Lipschitzian around
(x, y) ∈ graph (T ) if there exist a constant L and neighborhoods Mx of x
and My of y such that
edY (T (x1 ) ∩ My , T (x2 )) ≤ LdX (x1 , x2 )
∀x1 , x2 ∈ Mx .
As a definition, we say in this case that T is L-pseudo-Lipschitzian around
(x, y) on Mx with respect to My . The notion of being pseudo-Lipschitzian
around (x, y) is called the Aubin property when Mx and My are closed balls
around x and y respectively. It is well-known that the Aubin property of the
set-valued mapping T turns out to be equivalent to the metric regularity of
INVERSE OF THE SUM OF SET-VALUED MAPPINGS
9
the set-valued mapping T −1 , see [19, 33, 23, 26, 16] for more details on the
notion of metric regularity and its applications to variational problems.
As mentioned before, if T : X ⇒ Y is L-Lipschitzian, then by [10,
Lemma 2], we have
edY (T (A) , T (B)) ≤ LedX (A, B)
for all subsets A and B of dom (T ). This property, which seems to be not
conserved for set-valued pseudo-Lipschitzian mappings, plays an essential
role for the results obtained in [10]. However, it will be replaced in the sequel
by an adapted version for set-valued pseudo-Lipschitzian mappings and will
play a key role in the results concerning the properties of the inverse of the
sum of two non necessarily Lipschitzian set-valued mappings.
Let T : X ⇒ Y be a set-valued mapping, M and M 0 two nonempty subsets
of dom (T ), y0 ∈ Y and r > 0. We say that T is fully L-pseudo-Lipschitzian
on M for M 0 with respect to B (y0 , r) if for any two nonempty subsets A and
B of M , we have
edY (T (A) ∩ B (y0 , r) , T (B)) ≤ LedX A ∩ M 0 , B .
It results immediately from the definition that any set-valued fully L-pseudoLipschitzian on M for M 0 with respect to B (y0 , r) such that M is a neighborhood of x and (x, y0 ) ∈ graph (T ) is L-pseudo-Lipschitzian around (x, y0 )
on Mx with respect to B (y0 , r).
Conversely, any set-valued L-Lipschitzian mapping T : X ⇒ Y is fully
L-pseudo-Lipschitzian on M for M 0 with respect to any subset of Y , for any
two subsets M and M 0 of dom (T ) such that M ⊂ M 0 .
Here, we show that the example considered above provides us with a setvalued non-Lipschitzian mapping which verifies our notion of set-valued fully
pseudo-Lipschitzian mapping.
Example 2. Let T : R2 ⇒ R2 be the set-valued mapping defined by
i
( h
|x|
x
×
0,
∪
[3,
+∞[
if k (x, y) k < 1,
2
2
T ((x, y)) =
2
{2x} × 0, x
if k (x, y) k ≥ 1.
The set-valued mapping T is not Lipschitzian on R2 . However, T is fully √12 pseudo-Lipschitzian on B ((0, 0) , 1) for B ((0, 0) , 1) with respect to B ((0, 0) , 1).
More generally, we have the following result for set-valued pseudo-Lipschitzian
mappings which can be compared to [10, Lemma 2] where the proof is similar. Also, we conclude that this notion for set-valued mapping of being
fully pseudo-Lipschitzian fits very well with the existing notions of being
Lipschitzian, pseudo-Lipschitzian and so on.
Proposition 3.1. Let T : X ⇒ Y be a set-valued L-pseudo-Lipschitzian
around (x, y) on Mx with respect to My , and suppose that Mx 6= ∅ is contained in dom (T ). Then, for any nonempty subsets A and B contained in
Mx , we have
edY (T (A) ∩ My , T (B)) ≤ LedX (A, B) .
10
BOUALEM ALLECHE AND VICENŢIU D. RĂDULESCU
In particular, T is fully L-pseudo-Lipschitzian on Mx for M 0 with respect to
My , for any subset M 0 of dom (T ) containing Mx .
Proof. Let A and B be nonempty and contained in Mx . Then,
edY (T (A) ∩ My , T (B)) =
sup
dy (u, T (B))
u∈T (A)∩My
= sup
sup
inf dy (u, T (y))
x∈A u∈T (x)∩My y∈B
≤ sup inf
sup
x∈A y∈B u∈T (x)∩My
dy (u, T (y))
= sup inf edy (T (x) , T (y))
x∈A y∈B
≤ L sup inf dX (x, y) = LedX (A, B) .
x∈A y∈B
Since obviously LedX (A, B) = LedX (A ∩ M 0 , B), then T is fully L-pseudoLipschitzian on Mx for M 0 with respect to My .
As in Theorem 2.2 of the last section, the two set-valued mappings involved in all the following results on the inverse of the sum of two set-valued
mappings, will be also connected between them by some additional conditions related to the existence of fixed points. We formulate this connection
in the following definition.
Let F, G : X ⇒ Y be two set-valued mapping, x0 ∈ X, y0 ∈ Y , B ⊂ Y ,
α > 0 and β > 0. In the sequel, we say that F is (α, β)-compatible with
respect to G on B for x0 and y0 if the following conditions hold:
(FP1) for every y ∈ B, there exists xy ∈ X such that (y − G (xy )) ∩
(F (xy ) − y0 ) 6= ∅;
(FP1) whenever x is such that (y − G (x)) ∩ (F (x) − y0 ) 6= ∅ for
some
y ∈ B, then dX (x, x0 ) < β and dX x, F −1 (y 0 + y0 − G (x)) < αβ,
for every y 0 ∈ B with y 0 6= y.
In comparison with [10, Theorem 3], we formulate now the following inverse set-valued mapping result for the sum of two set-valued mappings where
the conditions of being L-Lipschtzian are replaced by some local conditions
such as the condition of being pseudo-Lipschitzian.
Theorem 3.2. Let (X, dX ) be a metric space, (Y, dY ) be a linear metric
space with shift-invariant metric, r > 0, x0 ∈ X and y0 ∈ Y be such that
B (x0 , r) is a complete metric subspace. Let F, G : X ⇒ Y be two set-valued
mappings satisfying the following assumptions
(1) B (x0 , r) ⊂ dom (G), G (x0 ) is a bounded set with diameter d0 , and
there exist α > 0, z0 ∈ G (x0 ) and a neighborhood Mz0 of z0 such that
G is α-pseudo-Lipschitzian
around (x0 , z0 ) on B (x0 , r) with respect
to Mz0 , and G B (x0 , r) ⊂ B (G (x0 ) , αr + ε), for every ε > 0;
(2) B (x0 , r) ⊂ dom (F ), F is upper semicontinuous and has the following
properties
(a) there exists δ > 0 such that B (y0 , δ + αr + d0 ) ⊂ F (B (x0 , r));
INVERSE OF THE SUM OF SET-VALUED MAPPINGS
11
(b) there exists K > 0 such that F −1 is fully K-pseudo-Lipschitzian
on B (y0 , δ + αr + d0 ) for Mz0 with respect to B (x0 , r);
(c) αK < 1.
(3) there exists β > 0 such that β < (1 − αK) r, and F is (α, β)compatible with respect to G on B (G (x0 ) , δ) for x0 and y0 .
K
Then, B (G (x0 ) + y0 , δ) ⊂ dom (F + G)−1 and (F + G)−1 is 1−αK
-Lipschitzian
on B (G (x0 ) + y0 , δ).
Proof. Let y ∈ B (G (x0 ) , δ) be fixed and define the set-valued mapping
Ty : X ⇒ X by
Ty (x) = F −1 (y + y0 − G (x))
= {t ∈ X | ∃z ∈ G (x) , y + y0 − z ∈ F (t)} .
Remark that we always have
Fix (Ty ) = (F + G)−1 (y + y0 ) .
Indeed, we have
x ∈ Ty (x) ⇐⇒ ∃z ∈ G (x) , y + y0 − z ∈ F (x)
⇐⇒ ∃z ∈ G (x) , y ∈ F (x) − y0 + z
⇐⇒ y ∈ (F + G) (x) − y0
⇐⇒ x ∈ (F + G)−1 (y + y0 ) .
Now, it follows by condition (3) that there exist xy ∈ X, yG ∈ G (xy ) and
yF ∈ F (xy ) such that y − yG = yF − y0 . Therefore,
y = yF + yG − y0 ∈ (F + G) (xy ) − y0
and then xy ∈ (F + G)−1 (y + y0 ). This proves in particular that
B (G (x0 ) + y0 , δ) ⊂ dom (F + G)−1 ,
and since xy ∈ Ty (xy ), then Fix (Ty ) 6= ∅.
Before going further in the proof and state the different next claims, we
need first to prove the following fact
y + y0 − G (x) ⊂ B (y0 , δ + αr + d0 )
∀x ∈ B (x0 , r) .
Let x ∈ B (x0 , r) and z ∈ G (x). Since dY is a shift-invariant metric, it
suffices to verify that dY (y, z) < δ + d0 + αr. Let yx0 ∈ G (x0 ) be such that
dY (y, yx0 ) < δ and put ε = δ − dY (y, yx0 ) > 0. Let uε,z ∈ G (x0 ) be such
that
ε
dY (uε,z , z) < αr + .
2
12
BOUALEM ALLECHE AND VICENŢIU D. RĂDULESCU
Then, we obtain
dY (y, z) ≤ dY (y, yx0 ) + dY (yx0 , uε,z ) + dY (uε,z , z)
ε
< dY (y, yx0 ) + d0 + αr +
2
ε
= δ − ε + d0 + αr + < δ + d0 + αr.
2
Now, we turn to verify the conditions of Theorem 2.2 to any couple of
set-valued mappings Ty with y ∈ BY (G (x0 ) , δ).
Claim 1. The set-valued mapping Ty has nonempty closed values on B (x0 , r).
Indeed, let x ∈ B (x0 , r), then y + y0 − z ∈ B (y0 , δ + αr + d0 ), for every
z ∈ G (x). By Assumption (2a), F −1 (y + y0 − z) 6= ∅, for every z ∈ G (x).
This yields in particular that Ty (x) 6= ∅, for every x ∈ B (x0 , r). Moreover,
by the upper semicontinuity of F and since y + y0 − G (x) is closed, then
Ty (x) = F −1 (y + y0 − G (x)) is closed, for every x ∈ B (x0 , r).
Claim 2. The set-valued mapping Ty is αK-pseudo-contraction with respect
to B (x0 , r). Indeed; for x1 , x2 ∈ B (x0 , r), we know from above that y + y0 −
G (x1 ) and y + y0 − G (x2 ) are contained in B (y0 , δ + αr + d0 ), and then,
dX (Ty (x1 ) ∩ B (x0 , r) , Ty (x2 )) =
edX F −1 (y + y0 − G (x1 )) ∩ B (x0 , r) , F −1 (y + y0 − G (x2 )) ≤
KedY ((y + y0 − G (x1 )) ∩ Mz0 , (y + y0 − G (x2 ))) =
KedY (G (x1 ) ∩ Mz0 , G (x2 )) ≤ αKdX (x1 , x2 ) .
Claim 3. To verify the condition (2) of Theorem 2.2, take y, y 0 ∈ BY (G (x0 ) , δ),
y 6= y 0 and suppose x ∈ Fix (Ty ). Then, by condition (3), dX (x, x0 ) <
β and dX x, F −1 (y 0 + y0 − G (x)) < αβ. That is, dX (x, x0 ) < β and
dX (x, Ty (x)) < αβ which are required.
K
It remains now to verify that the set-valued mapping (F + G)−1 is 1−αK
Lipschitzian on BY (G (x0 ) + y0 , δ). To do this, we will apply Theorem 2.2
(and more precisely, Proposition 2.4) to the set-valued mappings Ty with
y ∈ BY (G (x0 ) , δ). Note first that we have Fix (Ty ) ⊂ B (x0 , r), for every
y ∈ BY (G (x0 ) , δ).
For z, z 0 ∈ BY (G (x0 ) + y0 , δ), let z = y + y0 and z 0 = y 0 + y0 with
y, y 0 ∈ (G (x0 ) , δ). We have
edX (F + G)−1 (z) , (F + G)−1 z 0 = edX Fix (Ty ) , Fix Ty0
and
edX Fix (Ty ) , Fix Ty0
= edX Fix (Ty ) ∩ B (x0 , r) , Fix Ty0
1
≤
sup edX Ty (x) ∩ B (x0 , r) , Ty0 (x) .
1 − αK x∈B(x0 ,r)
INVERSE OF THE SUM OF SET-VALUED MAPPINGS
13
On the other hand, for every x ∈ B (x0 , r), we have
edX Ty (x) ∩ B (x0 , r) , Ty0 (x)
= edX F −1 (y + y0 − G (x)) ∩ B (x0 , r) , F −1 y 0 + y0 − G (x)
≤ KedY (y + y0 − G (x)) ∩ Mz0 , y 0 + y0 − G (x)
≤ KedY y − G (x) , y 0 − G (x) ≤ KdY y, y 0 .
Then,
K
dY y, y 0
1 − αK
K
=
dY z, z 0
1 − αK
which, by interchanging z and z 0 , completes the proof.
edX (F + G)−1 (z) , (F + G)−1 z 0 ≤
Remark 2. If G is a set-valued α-Lipschitzian
mapping on B (x0 , r) as in
[10, Theorem 3], then we have G B (x0 , r) ⊂ B (G (x0 ) , αr + ε), for every
ε > 0. Also, when F −1 is Lipschitzian, then we can apply Proposition 3.1
and choose Mz0 = Y in such a way that F −1 satisfies our condition of fully
pseudo-Lipschitzian. Condition 3 is of course related to the existence of fixed
points for the family of set-valued mappings Ty . This fact is guaranteed by
Nadler’s fixed point theorem in the case where the set-valued mappings Ty
are contraction as in [10, Theorem 3].
Now, we obtain the following result when f and g are single-valued mappings. As in Theorem 3.2, the mappings g and f −1 are supposed to be
connected by some properties and satisfying our notions in the same sense
as for set-valued mappings.
Proposition 3.3. Let (X, dX ) be a metric space, (Y, dY ) a linear metric
space with shift-invariant metric, r > 0, x0 ∈ X and y0 ∈ Y such that
B (x0 , r) is a complete metric subspace. Let f, g : B (x0 , r) ⇒ Y be two
single-valued mappings satisfying the following assumptions:
(1) there exist α > 0 and a neighborhood Mz0 of g (x0 ) such that g is
α-pseudo-Lipschitzian around (x0 , g (x0 )) on B (x0 , r) with respect to
Mz0 , and g B (x0 , r) ⊂ B (g (x0 ) , αr);
(2) f has the following properties
(a) there exists δ > 0 such that B (y0 , δ + αr) ⊂ f (B (x0 , r));
(b) there exists K > 0 such that f −1 is single-valued on the set
B (y0 , δ + αr) and fully K-pseudo-Lipschitzian on B (y0 , δ + αr)
for Mz0 with respect to B (x0 , r);
(c) αK < 1.
(3) there exists β > 0 such that β < (1 − αK) r, and f is (α, β)compatible with respect to g on B (g (x0 ) , δ) for x0 and y0 .
K
Then, (f + g)−1 is still a single-valued mapping and 1−αK
-Lipschitzian on
B (g (x0 ) + y0 , δ).
14
BOUALEM ALLECHE AND VICENŢIU D. RĂDULESCU
Proof. By applying Theorem 3.2, it remains only to prove that (f + g)−1 is
a single-valued mapping on B (g (x0 ) + y0 , δ).
Let y ∈ B (g (x0 ) , δ). We remark that under the notations of Theorem 3.2,
we have B (g (x0 ) + y0 , δ) ⊂ dom (f + g)−1 , and since f −1 is single-valued on
B (y0 , δ + αr), then Ty is single-valued on B (x0 , r). Also Ty is a contraction
on B (x0 , r), has at least a fixed point and Fix (Ty ) ⊂ B (x0 , r). Then, Ty
has a unique fixed point and therefore, (f + g)−1 is a single-valued mapping
on B (g (x0 ) + y0 , δ).
Now, we are going to obtain a result similar to the classical result due to
Graves on the inverse of continuous functions acting between Banach spaces.
The result of Graves generalizes that of Lyusternik (see [21, 25]) and both
have their roots in the Banach open mapping theorem, see also [17] for a
recent overview on the subject.
With some additional conditions, we first obtain the following result similar to Theorem 3.2.
Theorem 3.4. Suppose that the conditions of Theorem 3.2 are satisfied,
where instead of condition (2a) and condition (2b), we have
(a’) there exists δ > 0 such that B (y0 , δ + αr + d0 ) ⊂ F (B (x0 , r));
(b’) F −1 is fully K -pseudo-Lipschitzian on B (y0 , δ + αr + d0 ) for Mz0
with respect to B (x0 , r)
S
respectively, and B (G (x0 ) , δ) is replaced by u∈G(x0 ) B (u, δ) in condition (3).
S
K
Then, u∈G(x0 ) B (u + y0 , δ) ⊂ dom (F + G)−1 and (F + G)−1 is 1−αK
S
Lipschitzian on u∈G(x0 ) B (u + y0 , δ).
Proof. The proof follows step by step the proof
of Theorem 3.2 where instead
S
of taking y ∈ B (G (x0 ) , δ), we take y ∈ u∈G(x0 ) B (u, δ). The unique fact
S
which merits to be established is that for every y ∈ u∈G(x0 ) B (u, δ),
y + y0 − G (x) ⊂ B (y0 , δ + αr + d0 ) ∀x ∈ B (x0 , r) .
S
Let y ∈ u∈G(x0 ) B (u, δ) and take uy ∈ G (x0 ) such that y ∈ B (uy , δ).
Let x ∈ B (x0 , r) and z ∈ G (x). Since dY is a shift-invariant metric, it
suffices to verify that dY (y, z) ≤ δ + d0 + αr. Since dX (y, uy ) ≤ δ, let (%n )n
be an increasing sequence of positive numbers such that lim = 1 and
n→+∞
εn = δ − %n dX (y, uy ) > 0, for every n. Now, for every n, let yn,z ∈ G (x0 )
be such that
εn
dY (yn,z , z) < αr + .
2
Then, we obtain
dY (y, z) ≤ dY (y, uy ) + dY (uy , yn,z ) + dY (yn,z , z)
εn
< dY (y, uy ) + d0 + αr + ,
2
INVERSE OF THE SUM OF SET-VALUED MAPPINGS
15
and since lim εn = δ − dY (y, uy ), we have
n→+∞
dY (y, z) ≤ dY (y, uy ) + d0 + αr +
δ − dY (y, uy )
2
δ + dY (y, uy )
+ d0 + αr
2
≤ δ + d0 + αr
=
which completes the proof.
Recall that the Banach open mapping theorem guarantees that a linear
continuous mapping A from a Banach space X to a Banach space Y is
surjective if and only it is an open mapping. In particular, if A is surjective
linear and continuous, then there exists K > 0 such that
BY (0, 1) ⊂ A (BX (0, K)) .
Corollary 3.5. Let (X, k.kX ) and (Y, k.kY ) be two Banach spaces. Denote
by A : X → Y a surjective linear and continuous mapping and let K be the
constant arising from the Banach open mapping theorem. Let r > 0 and
x0 ∈ X. Let g : X → Y be a single-valued mapping and suppose that the
following conditions are satisfied:
(1) there exist α > 0 and a neighborhood Mz0 of g (x0 ) such that g is
α-pseudo-Lipschitzian around (x0 , g (x0 )) on B (x0 , r) with respect to
Mz0 , and g B (x0 , r) ⊂ B (g (x0 ) , αr);
(2) f has the following properties
(a) F −1 is fully K-pseudo-Lipschitzian on B A (x0 ) , 1−αK
K r + αr
for Mz0 with respect to B (x0 , r);
(b) αK < 1.
(3) there exists β > 0 such that β < (1 − αK) r, and A is (α, β)compatible with respect to g on B g (x0 ) , 1−αK
K r for x0 and y0 .
1−αK
Then, B A (x0 ) + g (x0 ) , K r ⊂ (A + G) B (x0 , r) and (A + G)−1 is
K
1−αK
1−αK -Lipschitzian on B g (x0 ) + A (x0 ) , K r .
Proof. Let F = A|B(x0 ,r) and G = g|B(x0 ,r) . From the Banach open mapping
theorem,
r
B A (x0 ) ,
⊂ A B (x0 , r) = F B (x0 , r) .
K
The proof holds from Theorem 3.4 by taking δ =
d0 = 0.
1−αK
K r,
y0 = A (x0 ) and
We close this section by the following discussion about the conditions on
the mapping A which have been involved in the proof of Corollary 3.5. The
continuity of A is essential in the proof and it has been assumed in the
corollary. Another important property is the fact that A is open, which
16
BOUALEM ALLECHE AND VICENŢIU D. RĂDULESCU
is guaranteed by the Banach open mapping theorem. The openness of A
implies that there exists K > 0 such that
BY (0, 1) ⊂ A (BX (0, K))
and, thanks to the linearity of A, these two properties are equivalent and
both implies that A−1 is a set-valued Lipschitzian mapping. However, the
linearity of A and the Lipschitzian property of A−1 are not used in the proof
of Corollary 3.5. Instead of them, we need in fact that A−1 is fully pseudoLipschitzian in the sense defined in the paper.
On the other hand, the openness of A is not in reality subject which is
involved by the linearity of the mapping A acting between Banach spaces.
This idea has been already considered in the literature and especially, in
Convex Analysis without linearity where generalizations of some implicit
function theorems and other questions of Optimization have been obtained,
see [28]. See also [34] where a notion denoted by P L weaker than that of the
linearity has been recently defined and a generalization of the Banach open
mapping theorem has been derived. It is shown in particular, that every
surjective continuous mapping acting between Banach spaces and satisfying
the conditions of the notion P L is open.
In the same time, let us point out that even if the difference between
the local and the global property of being Lipschitzian have been brought
to light by several authors (see [10, 11, 13, 14]), in this paper, we make
use of some additional conditions on the involved set-valued mappings to
obtain that the inverse of their sum is Lipschitzian. It is also true that in
the case of the Banach open mapping theorem, the inverse of a surjective
linear and continuous mapping is Lipschitzian and the mapping itself is also
Lipschitzian by linearity. Otherwise, nothing can ensure that if the inverse
of a set-valued is Lipschitzian, then the set-valued itself is Lipschitzian. This
leads naturally to investigate further conditions on the involved set-valued
mapping and not necessarily the Lipschitzian property, in order to obtain
that the inverse of their sum is a set-valued Lipschitzian mapping.
4. Applications to variational inclusions
In this section we deal with the sensitivity analysis of variational inclusions. As in the approach considered in [10, 11], we will use our results
developed above and develop techniques related to the existence of solutions
of variational inclusions. These techniques are based here on the proximal
convergence.
Let (P, dP ) be a metric space (the set of parameters) and let A : P × X ⇒
Y be a set-valued mapping, where (X, dX ) is metric space and (Y, dY ) a linear
metric space with shift-invariant metric. For a fixed value of the parameter
p ∈ P , we consider the parameterized generalized equation:
find x ∈ dom (A (p, .)) such that 0 ∈ A (p, x) ,
(PGE)
INVERSE OF THE SUM OF SET-VALUED MAPPINGS
17
where its set of solutions is denoted by SA (p). For a set Z of P , we denote
SA (Z) = ∪p∈Z SA (p).
The regularity properties of the solution map p → SA (p) has been the
subject of study of many authors since it is related to the theory of implicit
functions and its applications for variational inclusions, see for instance,
[17, 18, 3] and the references therein.
In [10, 11] a measure of the sensitivity of the solutions with respect to small
changes in the problem’s data has been defined and applied to existence of
solutions of variational inclusions. In a similar way, for any p0 ∈ P , we define
the full condition number of A at p0 with respect to a subset W of X as the
extended real-valued number by
e (SA (Z) ∩ W, SA (Z 0 ))
e (Z, Z 0 )
Z,Z 0 →{p0 }
c∗f (A | p0 , W ) = lim sup
Z6=Z 0 ,Z6=∅
where the convergence is taken in the sense of the upper proximal convergence. That is, a net (Zγ )γ is upper proximal convergent to Z if lim e (Zγ , Z) =
γ
0, see [2, 8]. Then, we have
c∗f (A | p0 , W ) =
e (SA (Z) ∩ W, SA (Z 0 ))
0
0
inf sup
| Z, Z ⊂ B (p0 , ε) , Z 6= Z , Z 6= ∅ .
ε>0
e (Z, Z 0 )
Also, the extended real number K (A, δ|p0 , W ) is defined by
K (A, δ | p0 , W ) =
e (SA (Z) ∩ W, SA (Z 0 ))
0
0
| Z, Z ⊂ B (p0 , δ) , Z 6= Z , Z 6= ∅ .
sup
e (Z, Z 0 )
Clearly, the function δ 7→ K (A, δ | p0 , W ) is decreasing and we have, lim K (A, δ | p0 , W ) =
δ→0
c∗ (A | p0 , W ), for every p0 ∈ P .
By a similar proof as in [10, Proposition 3], we obtain the following result.
Proposition 4.1. If K (A, δ | p0 , W ) < +∞, then one of the following alternatives holds:
(1) there
every
(2) there
every
exists a neighborhood V (p0 ) of p0 such that SA (p) = ∅, for
p ∈ U (p0 );
exists a neighborhood V (p0 ) of p0 such that SA (p) 6= ∅, for
p ∈ U (p0 ).
In particular, if 0 < K (A, δ | p0 , W ) < +∞, then there exists a neighborhood V (p0 ) of p0 such that the solutions set of the parameterized generalized
equation (PGE) is nonempty, for every p ∈ V (p0 ).
In the sequel we focus on the special case where P = Y . We study the
parameterized generalized equation associated to A : Y × X ⇒ Y defined by
18
BOUALEM ALLECHE AND VICENŢIU D. RĂDULESCU
using a set-valued mapping F : X ⇒ Y as follows:
(
F (x) − p if x ∈ B (x0 , r) ,
A (p, x) =
∅
otherwise.
We remark that SA (Z) = F −1 (Z), for every subset Z of P and it results
that in this framework, the full condition number given above takes the more
explicit form
e F −1 (Z) ∩ W, F −1 (Z 0 )
∗
cf (A | p0 , W ) = lim sup
e (Z, Z 0 )
Z,Z 0 →{p0 }
Z6=Z 0 ,Z6=∅
In this settings when A is defined as above by a given set-valued mapping T :
X ⇒ Y , we will write in what follows c∗f (T | p0 , W ) and K (T, δ | p0 , B (x0 , r))
instead of c∗f (A | p0 , W ) and K (A, δ | p0 , B (x0 , r)) respectively.
Now, we obtain the following result on the existence of solutions of parameterized generalized equations.
Theorem 4.2. Let r > 0, x0 ∈ X and p0 ∈ Y be such that B (x0 , r)
is a complete metric subspace. Let G : X ⇒ Y be a set-valued mapping. Suppose that 0< c∗ (F | p0 , B (x0 , r)) < +∞ and choose δ such that
K F, δ | p0 , B (x0 , r) < +∞. Suppose further that the following conditions
are satisfied
d0 < δ,
(1) B (x0 , r) ⊂ dom (G), G (x0 ) isa bounded set with diameter
and there exist 0 < α < min
δ−d0
1
r , K (F,δ|p0 ,B(x0 ,r))
, z0 ∈ G (x0 )
and a neighborhood Mz0 containing B p0 , δ such that G is α-pseudoLipschitzian around (x0 , z0 ) on B (x0 , r) with respect to Mz0 , and
G B (x0 , r) ⊂ B (G (x0 ) , αr + ε), for every ε > 0;
(2) B (x0 , r) ⊂ dom (F ), F B (x0 , r) ∩ B p0 , δ − αr − d0 6= ∅ and F
is upper semicontinuous.
(3) there exists β > 0 such that β < 1 − αK F, δ | p0 , B (x0 , r) r, and
F is (α, β)-compatible with respect to G on the set B G (x0 ) , δ − αr − d0
for x0 and p0 .
K (F,δ|p0 ,B(x0 ,r))
Then, c∗ (F + G | p0 + y, B (x0 , r)) ≤ 1−αK F,δ|p ,B(x ,r) < +∞, for every
(
)
0
0
y ∈ G (x0 ).
Proof. Put δ = δ − αr − d0 > 0. Clearly αK F, δ | p0 , B (x0 , r) < 1. By
definition, for every subsets Z, Z 0 of B p0 , δ , we have
e F −1 (Z) ∩ B (x0 , r) , F −1 Z 0 ≤ K F, δ | p0 , B (x0 , r) e Z, Z 0
and then, F −1 is fully K F, δ | p0 , B (x0 , r) -pseudo-Lipschitzian on B p0 , δ
for Mz0 with respect to B (x0 , r). This also yields in particular that B p0 , δ − αr − d0 ⊂
F B (x0 , r) .
INVERSE OF THE SUM OF SET-VALUED MAPPINGS
19
It results by applying Theorem 3.2, that the set-valued mapping (F + G)−1
K (F,δ|p0 ,B(x0 ,r))
is 1−αK F,δ|p ,B(x ,r) -Lipschitzian on B G (x0 ) , δ . Now, by applying Propo(
)
0
0
sition 3.1 (see also [10, Lemma 2]), we obtain that for every y ∈ G (x0 ), we
have
c∗ (F + G | p0 + y, B (x0 , r))
e (F + G)−1 (Z) ∩ B (x0 , r) , (F + G)−1 (Z 0 )
= lim sup
e (Z, Z 0 )
Z,Z 0 →{p0 +y}
Z6=Z 0 ,Z6=∅
−1
e (F + G)
≤
sup
Z,Z 0 ⊂B (p0 +y,δ ),
(Z) , (F + G)
−1
(Z 0 )
e (Z, Z 0 )
Z6=Z 0 ,Z6=∅
≤
K F, δ | p0 , B (x0 , r)
1 − αK F, δ | p0 , B (x0 , r)
< +∞,
which completes the proof.
In conclusion, we have investigated in this paper the important properties
of the involved set-valued mappings in order to obtain that the inverse of
their sum is Lipschitzian. First, we have obtained some results on the behaviors of fixed points sets of set-valued mappings similar to the classical ones
but with new conditions and different proofs. Then, we have highlighted
the properties of set-valued pseudo-Lipschitzian mappings to deal with the
Lipschitzian property of the inverse of the sum of two set-valued mapping.
In our approach, we have also shown techniques based on handling subsets
rather than points which are usually employed in such studies. Finally, we
hope that the paper could bring to light other questions in the future and
stimulates for further investigations.
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Laboratoire de Mécanique, Physique et Modélisation Mathématique. Université de Médéa. Algeria
E-mail address: alleche.boualem@univ-medea.dz, alleche.boualem@gmail.com
Institute of Mathematics “Simion Stoilow" of the Romanian Academy, P.O.
Box 1-764, 014700 Bucharest, Romania
E-mail address: vicentiu.radulescu@math.cnrs.fr