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METRIC REGULARITY OF THE SUM OF MULTIFUNCTIONS AND APPLICATIONS ´ HUYNH VAN NGAI, NGUYEN HUU TRON, AND MICHEL THERA Abstract. In this work, we use the theory of error bounds to study of metric regularity of the sum of two multifunctions, as well as some important properties of variational systems. We use an approach based on the metric regularity of epigraphical multifunctions. Our results subsume some recent results by Durea and Strugariu in [12].

1. Introduction The study of a wide range of problems from variational analysis, optimization, variational inequalities, and many other areas in mathematics leads to generalized equations of the form (1)

0 ∈ F (x, p),

where F : X × P ⇒ Y is a multifunction, X, Y are metric spaces, and P is a metric space considered as the space of parameters. A typical example of equation (1) is given by a parametrized system of inequalities/equalities. Indeed, let us consider a mapping f : Rm × Rn → Rk+d with f (x, p) = (f1 (x, p), · · · , fk (x, p), fk+1 (x, p), · · · , fk+d (x, p)), and if we set F (x, p) = f (x, p) − Rk− × {0}d ), then the system (S) consisting of those points x for which fi (x, p) ≤ 0, i ∈ {1, · · · , k}, fi (x, p) = 0, i ∈ {k + 1, · · · , d}, can be rewritten in the form (1). Let us notice also that equation (1) subsumes the important subcase of parametrized inclusions of the type: (2)

0 ∈ H(x) + f (x, p),

where H : X ⇒ Y is a set-valued mapping and f : X × P → Y is a mapping. For a given parameter p ∈ P , we denote by S(p) the set of solutions of (2) , i.e., (3)

S(p) := {x ∈ X : 0 ∈ F (x) + f (x, p)}.

Let us consider the perturbed optimization problem (P) min g(x) − hp, xi x∈C

Date: Version of december 24, 2011. 2000 Mathematics Subject Classification. 47H04 and 49J53 and 90C31. Key words and phrases. Error bound and Perturbation stability and Metric regularity and Implicit multifunction and Generalized equations. Research of the first author has been supported by NAFOSTED annd LIA FORMATH VIETNAM (Laboratoire international associ´e. Research of the second author has been supported by R´egion Limousin. Research of the third author has been partially supported by the Australian Research Council, Project DP110102011 and by the ECOS-SUD Project C10E08. 1

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´ HUYNH VAN NGAI, NGUYEN HUU TRON, AND MICHEL THERA

where g : Rn → R is a convex C 1 −function, and p ∈ Rn is a given parameter.The first order optimality condition of problem (P) is given by ∇g(x) − p ∈ NC (x), where NC (x) = {p ∈ Rn : hp, y − xi ≤ 0 ∀y ∈ C} stands for the normal cone to C at x. Setting f (x, p) = p − ∇g(x), the first order optimality condition is given by 0 ∈ f (x, p) + NC (x) and appears as a special case of equation (2), as well as are parametrized variational inequalities, i.e., the problem of finding x ∈ C such that hf (x, p), u − xi ≥ 0 for all u ∈ C. The study of variational properties and the stability of the solutions of equation (1) has attracted a lot of interest from an important number of authors and we refer the reader to the monographs by Mordukhovich [23], Rockafellar & Wets [33], Dontchev & Rockafellar [11] and the references therein. Let us first provide definitions and properties of some essential notions from set-valued analysis that will be used throughout this paper. In what follows X, Y etc., unless specified otherwise, are metric spaces and we use the same symbol d(·, ·) to denote the distance in all of them or ¯ ρ) between a point x to a subset S of one of them : d(x, S) := inf u∈S d(x, u). By B(x, ρ) and B(x, we denote the open and closed balls of radius ρ around x, while if X is a Banach space, we use ¯X for the open and the closed unit balls, respectively. By a multifunction the notations BX , B (set-valued mapping) S : X ⇒ Y , we mean a mapping from X into the subsets (possibly empty) of Y . We denote by gph S the graph of S, that is the set {(x, y) ∈ X × Y : y ∈ S(x)} and by D(S) := {x ∈ X : S(x) 6= ∅} the domain of S. When S has a closed graph, we say that S is a closed multifunction. Since various types of multifunctions arise in a considerable number of models ranging from mathematical programs, through game theory and to control and design problems, they represent probably the most developed class of objects. A number of useful regularity properties have been introduced and investigated (see [11], [33] and the references therein). Among them, the most popular is that of metric regularity ([6], [9], [11], [18], [19], [20], [21], [23], [28], [29], [33]), the root of which can be traced back to the classical Banach open mapping theorem and the subsequent fundamental results of Lyusternik and Graves. S is said to be metrically regular around (¯ x, y¯) ∈ gph S with modulus τ > 0 if there exist neighborhoods U, V of x ¯, y¯, respectively such that, for every (x, y) ∈ U × V, d(x, S −1 (y)) ≤ τ d(y, S(x)). A classical illustration of this concept concerns the case when S is a bounded linear continuous operator. Then, metric regularity of S amounts to saying that S is surjective. Another well known concept is the Pseudo-Lipschitz property, also called Aubin property (see [2]): let S : X ⇒ Y be a set-valued mapping and fix (¯ x, y¯) ∈ gph S. S is said to have the Aubin property around (¯ x, y¯) if there exists a constant κ ≥ 0 and neighborhoods U ∈ N (¯ x) ⊂ X of x ¯ and V ∈ N (¯ y ) ⊂ Y of y¯ such that (4)

¯Y S(x0 ) ∩ V ⊂ S(x) + κd(x, x0 )B

for all x, x0 ∈ U.

The Lipschitz modulus of S for y¯ at x ¯, denoted by lip(S; (¯ x, y¯)), is defined by (5) lip(S; (¯ x, y¯)) = inf {κ ∈ R+ : ∃ U ∈ N (¯ x), V ∈ N (¯ y ) such that condition (4) is satisfied}.

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The concept of openness or covering (at a linear rate) is also widely used: one says that S : X ⇒ Y is open at linear rate τ > 0 around (¯ x, y¯) ∈ gph S if there exist neighborhoods U, V of x ¯, y¯, respectively and a positive number ε > 0 such that, for every (x, y) ∈ gph S ∩ (U × V) and every ρ ∈ (0, ε), B(y, ρτ ) ⊂ S(B(x, ρ)). We refer to Aubin [2], Dmitruk, Milyutin & Osmolovsky [10], Ioffe [18], Kruger [22], Mordukhovich [23], Penot [29], Rockafellar & Wets [33] and the references therein for different developments of these notions. The following relation is well established: (6)

Metric regularity ⇐⇒ Covering ⇐⇒ Aubin property of the inverse.

In this paper, we are especially interested in metric regularity of the sum of two multifunctions. The starting point of the study is the paper by J. Arag´on Artacho, A. L. Dontchev, M. Gaydu, M. H. Geoffroy, and V. M. Veliov [1], in which it is proved that if F is metrically regular and if we perturb F by a mapping g(·, ·), Lipschitz with respect to x, uniformly in p, with a suitable Lipschitz constant, then the perturbed mapping F (·) + g(·, p) is metrically regular for every p near p¯. When we perturb a metrically regular multifunction F by a set-valued mapping G which is Lipschitz-like, the perturbed mapping F +G fails in general to be metrically regular. However, if for example the so-called “ sum-stable” property (introduced below) holds, then metric regularity as well as the Aubin property of the variational system remains. Recently Durea & Strugariu [12] considered the sum of two set-valued mappings and obtained a result very similar to openness of the sum of two set-valued mappings. They also gave some applications to the generalized variational system. Motivated by the ideas and results from [12], we attack these problems by using a different approach and with rather different assumptions. Indeed, using an approach based on the theory of error bounds, we study metric regularity of a special multifunction called the epigraphical multifunction associated to F and G. This intermediate result allows us to study metric regularity/ linear openness of the sum of two set-valued mappings, as well as metric regularity of the general variational system avoiding the strong assumption of the closedness of the multifunction F + G. From the point of view of applications to optimization (sensitivity analysis, convergence analysis of algorithms, and penalty functions methods), one of the most important regularity properties seems to be that of error bounds providing an estimate for the distance of a point from the solution set. This theory was initiated by the pioneering work by Hoffman [16]. However, it has been pointed out to the authors by J.-B. Hiriart-Urruty recently that traces of the error bounds property can be found in a work by P.C. Rosenbloom [32], published in 1951. Applications of the theory of error bounds to the investigation of metric regularity of multifunctions have been recently studied and developed by many authors, including for instance [24], [14], [4], [5], [28], [26]. The paper is structured as follows. In section 2, we recall some recent results on error bounds of parametrized systems and give, sometimes with some modifications, characterizations of metric regularity of multifunctions given in Huynh & Th´era [28], and Huynh, Nguyen & Th´era [24]. In the next section, in the context of Asplund spaces, we estimate the strong slope of the function ϕE ((x, k), y), (see page 5 below for the definition) and give sufficient conditions as well as a point-based condition for metric regularity of the epigraphical multifunction under a coderivative condition. In the last section, we study Robinson metric regularity and Aubin property of a generalized variational system.

´ HUYNH VAN NGAI, NGUYEN HUU TRON, AND MICHEL THERA

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2. Metric regularity of epigraphical multifunctions via error bound Let us remind some basic notions used in the paper. Let f : X → R ∪ {+∞} be a given extended-real-valued function. As usual, Dom f := {x ∈ X : f (x) < +∞} denotes the domain of f . We recall the concept of error bounds that is one of the most important regularity properties. We set (7)

S := {x ∈ X :

f (x) ≤ 0}.

We use the symbol [f (x)]+ to denote max{f (x), 0}. We shall say that the system (7) admits an error bound if there exists a real c > 0 such that  (8) d(x, S) ≤ c f (x)]+ for all x ∈ X. For x0 ∈ S, we shall say that system (7) has an error bound at x0 , when there exist reals c > 0 and ε > 0 such that relation (8) is satisfied for all x around x0 , i.e., in an open ball B(x0 , ε). Here and in what follows the convention 0 · (+∞) = 0 is used. We now consider a parametrized inequality system, that is, the problem of finding x ∈ X such that (9)

f (x, p) ≤ 0,

where f : X × P → R ∪ {+∞} is an extended-real-valued function, X is a complete metric space and P is a metric space. We denote by S(p) the set of solutions of system (9): S(p) := {x ∈ X :

f (x, p) ≤ 0}.

Recall from De Giorgi, Marino & Tosques [8], that the strong slope |∇f |(x) of a lower semicontinuous function f at x ∈ Dom f is the quantity defined by |∇f |(x) = 0 if x is a local minimum of f, and f (x) − f (y) |∇f |(x) = lim sup , d(x, y) y→x,y6=x otherwise. For x ∈ / Dom f, we set |∇f |(x) = +∞. In what follows, given a multifunction F : X ⇒ Y , we make use of the lower semicontinuous envelope (x, y) 7→ ϕF (x, y) of the function (x, y) 7→ d(y, F (x)), i.e., for (x, y) ∈ X × Y, ϕF (x, y) := lim inf d(v, F (u)) = lim inf d(y, F (u)). (u,v)→(x,y)

u→x

In the sequel, we use the notation Fp for F (·, p) and ϕp for ϕFp and the metric defined on the cartesian product X × Y by d((x, y), (u, v)) = max{d(x, u), d(y, v)},

(x, y), (u, v) ∈ X × Y.

Let us now recall a result given in Huynh, Nguyen and Th´era [24]. This result which is valid for all y in a neighborhood of y¯, instead of y¯ only, gives a necessary and sufficient condition for metric regularity of implicit multifunctions. For given y¯ ∈ Y , we denote S(¯ y , p) := {x ∈ X : y¯ ∈ F (x, p)}. Theorem 1. Let X be a complete metric space and Y be a metric space. Let P be a topological space and suppose that the set-valued mapping F : X × P ⇒ Y satisfies the following conditions for some (¯ x, y¯, p¯) ∈ X × Y × P : (a) x ¯ ∈ S(¯ y , p¯); (b) the multifunction p ⇒ F (¯ x, p) is lower semicontinuous at p¯;

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(c) for any p near p¯, the set-valued mapping x ⇒ F (x, p) is a closed multifunction. Let τ ∈ (0, +∞), be fixed. Then, the following three statements (i), (ii), (iii) are equivalent. Moreover, (iv) ⇒ (iii), and (iv) ⇔ (iii) provided Y is a normed space. (i) There exists a neighborhhood U × V × W ⊆ X × P × Y of (¯ x, p¯, y¯) such that U ∩ S(y, p) 6= ∅ for any (p, y) ∈ V × W and d(x, S(y, p)) ≤ τ d(y, F (x, p))

for all

(x, p, y) ∈ U × V × W;

(ii) There exists a neighborhhood U × V × W ⊆ X × P × Y of (¯ x, p¯, y¯) such that U ∩ S(y, p) 6= ∅ for any (p, y) ∈ V × W and d(x, S(y, p)) ≤ τ ϕp (x, y)

for all

(x, p, y) ∈ U × V × W;

(iii) There exist a neighborhood U × V × W ⊆ X × P × Y of (¯ x, p¯, y¯) and a real γ ∈ (0, +∞) such that for any (x, p, y) ∈ U × V × W with y ∈ / F (x, p) and ϕp (x, y) < γ, any ε > 0, and any sequence {xn }n∈N ⊆ X converging to x with lim d(y, F (xn , p)) = lim inf d(y, F (u, p)),

n→∞

u→x

there exists a sequence {un }n∈N ⊆ X with lim inf n→∞ d(un , x) > 0 such that (10)

lim sup n→∞

d(y, F (xn , p)) − d(y, F (un , p)) 1 > ; d(xn , un ) τ +ε

(iv) There exist a neighborhood U × V × W ⊆ X × P × Y of (¯ x, p¯, y¯) and a real γ > 0 such that 1 (11) |∇ϕp (·, y)|(x) ≥ for all (x, p, y) ∈ U × W × V with ϕp (x, y) ∈ (0, γ). τ Given two multifunctions F, G : X ⇒ Y , (Y is a normed linear space) we define the epigraphical multifunction associated with F and G as the multifunction E(F,G) : X × Y ⇒ Y defined by  F (x) + k, if k ∈ G(x), E(F,G) (x, k) = ∅, otherwise. For given y ∈ Y, we set (12)

SE(F,G) (y) := {(x, k) ∈ X × Y : y ∈ E(F,G) (x, k)}.

The lower semicontinuous envelope ((x, k), y) 7→ ϕE ((x, k), y) of the distance function d(y, E(F,G) (x, k)) is defined for (x, k, y) ∈ X × Y × Y by ϕE ((x, k), y) := lim inf d(w, E(F,G) (u, v)) (u,v,w)→(x,k,y) ( lim inf d(y, F (u) + k) if k ∈ G(x) u→x = +∞ otherwise. The next lemma is useful. Lemma 2. Assume that F : X ⇒ Y and G : X ⇒ Y are closed multifunctions. Then, the epigraphical multifunction E(F,G) has a closed graph and for each y ∈ Y, (13) SE(F,G) (y) = {(x, k) ∈ X×Y :

ϕE ((x, k), y) = 0} = {(x, k) ∈ X×Y :

k ∈ G(x), y ∈ F (x)+k}.

´ HUYNH VAN NGAI, NGUYEN HUU TRON, AND MICHEL THERA

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Proof. Observe that if F : X ⇒ Y and G : X ⇒ Y are closed multifunctions, then so is the epigraphical multifunction E(F,G) . Let us prove (13). Obviously, for each y ∈ Y if (x, k) ∈ SE(F,G) (y) then ϕE ((x, k), y) = 0. Conversely, suppose that ϕE ((x, k), y) = 0. Then, k ∈ G(x) and there exists a sequence {xn } → x such that d(y − k, F (xn )) → 0. By the closedness of the graph of F , one has that y − k ∈ F (x), i.e., y ∈ F (x) + k. Hence, (x, k) ∈ SE(F,G) (y) establishing the proof.  By virtue of to Lemma 2, we adapt Theorem 1 to the multifunction E(F,G) . Lemma 3. Let X be a complete metric space, let Y be a Banach space and let F, G : X ⇒ Y ¯ y¯) ∈ X × Y × Y such that y¯ ∈ F (¯ ¯ k¯ ∈ G(¯ be closed multifunctions. Suppose that (¯ x, k, x) + k, x). Let τ ∈ (0, +∞), be fixed. Then, the following statements are equivalent: ¯ y¯) such that (U ×V)∩SE (i) There exists a neighborhood U ×V ×W ⊆ X ×Y ×Y of (¯ x, k, (y) 6= (F,G) ∅ for any y ∈ W and d((x, k), SE(F,G) (y)) ≤ τ ϕE ((x, k), y)

for all

(x, k, y) ∈ U × V × W;

¯ y¯) and a real γ ∈ (0, +∞) such (ii) There exist a neighborhood U × V × W ⊆ X × Y × Y of (¯ x, k, that for any (x, k, y) ∈ U × V × W with y ∈ / F (x) + k, k ∈ G(x) and ϕE ((x, k), y) < γ, any ε > 0, and any sequences {xn }n∈N ⊆ X converging to x, {kn }n∈N ⊆ Y converging to k with lim d(y − kn , F (xn )) = lim inf d(y − k, F (u)),

n→∞

u→x

there exist sequences {un }n∈N ⊆ X, {zn }n∈N ⊆ Y with lim inf n→∞ d((un , zn ), (x, k)) > 0 such that 1 d(y − kn , F (xn )) − d(y − zn , F (un )) > ; (14) lim sup d((x , u ), (k , z )) τ + ε n→∞ n n n n ¯ y¯) and a real γ > 0 such that (iii) there exist a neighborhood U × V × W of (¯ x, k, |∇ϕE ((·, ·), y)|(x, k) ≥

1 for all (x, k, y) ∈ U × V × W with ϕE ((x, k), y) ∈ (0, γ). τ

Proposition 4. Let X be a complete metric space, Y be a Banach space and let F, G : X ⇒ Y ¯ y¯) ∈ X × Y × Y is such that y¯ ∈ F (¯ ¯ k¯ ∈ G(¯ be closed multifunctions. Suppose that (¯ x, k, x) + k, x). Consider the following statements: ¯ y¯) and τ > 0 such that (i) there exist a neighborhood U × V × W of (¯ x, k, d((x, k), SE(F,G) (y)) ≤ τ ϕE ((x, k), y)

for all

(x, k, y) ∈ U × V × W;

¯ y¯) and τ > 0 such that (ii) there exist a neighborhood U × V × W of (¯ x, k, d(x, (F + G)−1 (y)) ≤ τ d(y, F (x) + G(x) ∩ V) for all (x, y) ∈ U × W; ¯ y¯ − k) ¯ and ε, τ > 0 such that for every (iii) there exist a neighborhood U × V × W of (¯ x, k, (x, k, z) ∈ U × V × W, k ∈ G(x), z ∈ F (x), and ρ ∈ (0, ε), (15)

B(k + z, ρτ −1 ) ⊂ (F + G)(B(x, ρ)). Then one has the following implications: (i) ⇒ (ii) ⇔ (iii). Proof. For (i) ⇒ (ii). By (i), there exist δ1 , δ2 , δ3 > 0 such that for every ε > 0 and for every ¯ δ2 )∩G(x)]×B(¯ (x, k, y) ∈ B(¯ x, δ1 )×[B(k, y , δ3 ), there is (u, z) ∈ X×Y with y ∈ F (u)+z, z ∈ G(u) such that d((x, k), (u, z)) < (1 + ε)τ ϕE ((x, k), y).

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Consequently, d(x, u) ≤ max{d(x, u), kk − zk} < (1 + ε)τ d(y, F (x) + k). Noticing that y ∈ F (u) + G(u), i.e., u ∈ (F + G)−1 (y); it follows that d(x, (F + G)−1 (y)) < (1 + ε)τ d(y, F (x) + k). In conclusion, we have that ¯ δ2 )) d(x, (F + G)−1 (y)) < (1 + ε)τ d(y, F (x) + G(x) ∩ B(k,

for all

(x, y) ∈ B(¯ x, δ1 ) × B(¯ y , δ3 ).

Hence, taking the limit as ε > 0 goes to 0 yields the desired conclusion. ¯ δ2 ) × B(¯ For (ii) ⇒ (iii). Suppose that (ii) holds for the neighborhood B(¯ x, δ1 ) × B(k, y , δ3 ) with δ1 , δ2 , δ3 > 0 and τ > 0. Choose ρ1 = δ1 , ρ2 = 1/4 min{δ2 , δ3 }, ρ3 = 1/4δ3 , ε < τ δ3 /2. ¯ ρ2 ) × B(¯ ¯ ρ3 ), k ∈ G(x), z ∈ F (x), we take y ∈ Then, for (x, k, z) ∈ B(¯ x, ρ1 ) × B(k, y − k, −1 B(k + z, ρτ ). Consequently, ky − k − zk < ρτ −1 , and ¯ + kk¯ − y¯ + zk, ky − y¯k ≤ ky − k − zk + kk − kk −1 < ρτ + ρ2 + ρ3 , < ετ −1 + δ3 /4 + δ3 /4, < δ3 /2 + δ3 /2 = δ3 . Therefore, we have that ¯ δ2 )) ≤ ky − k − zk < ρτ −1 . d(y, F (x) + G(x) ∩ B(k, Hence, d(x, (F + G)−1 (y)) < τ ρτ −1 = ρ. Let γ > 0 with d(x, (F + G)−1 (y)) + γ < ρ. Find u ∈ (F + G)−1 (y), i.e., y ∈ (F + G)(u) such that d(x, u) < d(x, (F + G)−1 (y)) + γ. Thus, d(x, u) < ρ. It follows that y ∈ (F + G)(B(x, ρ)). ¯ ρ2 ) × B(¯ For (iii) ⇒ (ii). Suppose that (iii) holds for the neighborhood B(¯ x, ρ1 ) × B(k, y , ρ3 ) with ρ1 , ρ2 , ρ3 > 0 and τ > 0, ε > 0. Take ρ1 , ρ3 smaller if neccesary and consider a positive real η sufficiently small so that the ¯ ρ2 )) + η satisfies the conclusion of (iii) together with y ∈ quantity ρ := τ d(y, F (x) + G(x) ∩ B(k, −1 B(k + z, ρτ ). Then, there is a u ∈ B(x, ρ) such that y ∈ (F + G)(u), that is, u ∈ (F + G)−1 (y). Thus, ¯ ρ2 )) + η. d(x, (F + G)−1 (y)) ≤ d(x, u) < ρ = τ d(y, F (x) + G(x) ∩ B(k, Since η > 0 is arbitrary, the proof is complete. 

In the next result, we give conditions for the sum of two metrically regular mappings F, G to remain metrically regular. Before stating this result, we need to recall the so-called “locally sum-stable” property introduced by Durea & Strugariu [12].

´ HUYNH VAN NGAI, NGUYEN HUU TRON, AND MICHEL THERA

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Definition 5. Let F, G be two multifunctions and (¯ x, y¯, z¯) ∈ X × Y × Y such that y¯ ∈ F (¯ x), z¯ ∈ G(¯ x). We say that the pair (F, G) is locally sum-stable around (¯ x, y¯, z¯) if for every ε > 0, there exists δ > 0 such that for every x ∈ B(¯ x, δ) and every w ∈ (F + G)(x) ∩ B(¯ y + z¯, δ), there are y ∈ F (x) ∩ B(¯ y , ε) and z ∈ G(x) ∩ B(¯ z , ε) such that w = y + z. A simple case which ensures the local sum-stability of (F, G) is as follows. Proposition 6. Let F : X ⇒ Y, G : X ⇒ Y be two multifunctions and (¯ x, y¯, z¯) ∈ X × Y × Y such that y¯ ∈ F (¯ x), z¯ ∈ G(¯ x). If G(¯ x) = {¯ z } and G is upper semicontinuous at x ¯, then the pair (F, G) is locally sum-stable around (¯ x, y¯, z¯). Proof. Since G is upper semicontinuous at x ¯, for every ε > 0 there exists δ > 0 such that G(x) ⊂ G(¯ x) + B(0, ε/2) = z¯ + B(0, ε/2) = B(¯ z , ε/2),

for all x ∈ B(¯ x, δ).

Set η := min{δ, ε/2} and take x ∈ B(¯ x, η) and w ∈ (F + G)(x) ∩ B(¯ y + z¯, η). Then, there are y ∈ F (x), z ∈ G(x) such that w = y + z and w ∈ B(¯ y + z¯, η). Clearly, z ∈ B(¯ z , ε/2) ⊂ B(¯ z , ε). Moreover, ky − y¯k = kw − z − y¯k ≤ kw − y¯ − z¯k + kz − z¯k < η + ε/2 ≤ ε/2 + ε/2 = ε. Consequently, w = y + z, y ∈ F (x) ∩ B(¯ y , ε), z ∈ G(x) ∩ B(¯ z , ε). Hence we have established that (F, G) is locally sum-stable around (¯ x, y¯, z¯). Proposition 7. Let X be a complete metric space, Y be a Banach space and let F, G : X ⇒ Y ¯ y¯) ∈ X × Y × Y is such that y¯ ∈ F (¯ ¯ k¯ ∈ G(¯ be closed multifunctions. Suppose that (¯ x, k, x) + k, x). ¯ k) ¯ and there exist a neighborhood U × V If the pair (F, G) is locally sum-stable around (¯ x, y¯ − k, of (¯ x, y¯) and τ, θ > 0 such that (16)

¯ θ)) d(x, (F + G)−1 (y)) ≤ τ d(y, F (x) + G(x) ∩ B(k,

for all

(x, y) ∈ U × V,

then F + G is metrically regular around (¯ x, y¯) with modulus τ. ¯ then F + G is metrically As a result, if G is upper semicontinuous at x ¯ and G(¯ x) = {k}, regular around (¯ x, y¯) with modulus τ. Proof. Suppose that (16) holds for every (x, y) ∈ B(¯ x, δ1 ) × B(¯ y , δ2 ) for some δ1 , δ2 > 0. Since ¯ ¯ (F, G) is locally sum-stable around (¯ x, y¯ − k, k), there exists δ > 0 such that for every x ∈ B(¯ x, δ) ¯ θ) and z ∈ G(x) ∩ B(k, ¯ θ) and every w ∈ (F + G)(x) ∩ B(¯ y , δ), there are y ∈ F (x) ∩ B(¯ y − k, such that w = y + z. Taking δ smaller if necessary, we can assume that δ < δ1 . Fix (x, y) ∈ B(¯ x, δ/2) × B(¯ y , δ/2). We consider two following cases: Case 1. d(y, F (x) + G(x)) < δ/2. Fix γ > 0 , small enough in order to have d(y, F (x) + G(x)) + γ < δ/2,

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and take t ∈ F (x)+G(x) such that ky −tk < d(y, F (x)+G(x))+γ. Hence we have ky −tk < δ/2, and since we also have ky − y¯k < δ/2, this yields kt − y¯k ≤ ky − tk + ky − y¯k < δ/2 + δ/2 = δ. It follows that t ∈ [F (x) + G(x)] ∩ B(¯ y , δ). ¯ ¯ ¯ θ) and Since (F, G) is locally sum-stable around (¯ x, y¯ − k, k), there are y ∈ F (x) ∩ B(¯ y − k, ¯ z ∈ G(x) ∩ B(k, θ) such that t = y + z. Consequently, ¯ θ) + G(x) ∩ B(k, ¯ θ) ⊂ F (x) + G(x) ∩ B(k, ¯ θ). t ∈ F (x) ∩ B(¯ y − k, Therefore, ¯ θ)) ≤ ky − tk, d(y, F (x) + G(x) ∩ B(k, from which we derive ¯ θ)) ≤ d(y, F (x) + G(x)) + γ, d(y, F (x) + G(x) ∩ B(k, and therefore, as γ is arbitrarily small, we obtain that ¯ θ)) ≤ d(y, F (x) + G(x)). d(y, F (x) + G(x) ∩ B(k, By (16), one gets that d(x, (F + G)−1 (y)) ≤ τ d(y, F (x) + G(x)). Since (x, y) is arbitrary in B(¯ x, δ/2) × B(¯ y , δ/2), this yields d(x, (F + G)−1 (y)) ≤ τ d(y, F (x) + G(x)), for all (x, y) ∈ B(¯ x, δ/2) × B(¯ y , δ/2). Case 2. If d(y, F (x) + G(x)) ≥ δ/2. Choose δ sufficiently small so that τ δ/4 < δ1 . For every (x, y) ∈ B(¯ x, τ δ/4) × B(¯ y , δ/4) and −1 any ε > 0, by (16), there exists u ∈ (F + G) (y) such that d(¯ x, u) < (1+ε)τ d(y, F (¯ x)+G(¯ x)) ≤ (1+ε)τ ky− y¯k < (1+ε)τ δ/2 ≤ (1+ε)τ /2d(y, F (x)+G(x)). So, d(x, u) ≤ d(x, x ¯) + d(¯ x, u) < τ δ/4 + (1 + ε)τ /2d(y, F (x) + G(x)) < τ /2d(y, F (x) + G(x)) + (1 + ε)τ /2d(y, F (x) + G(x)). Taking the limit as ε > 0 goes to 0, it follows that d(x, (F + G)−1 (y)) ≤ τ d(y, F (x) + G(x)). So, d(x, (F + G)−1 (y)) ≤ τ d(y, F (x) + G(x)), for all (x, y) ∈ B(¯ x, τ δ/4) × B(¯ y , δ/4). The proof is complete.



The following theorem establishes metric regularity of the multifunction E(F,G) as well as metric regularity of a metrically regular set-valued mapping perturbed by a Lipschitz-like one.

´ HUYNH VAN NGAI, NGUYEN HUU TRON, AND MICHEL THERA

10

Theorem 8. Let X be a complete metric space, let Y be a Banach space and let F, G : X ⇒ Y ¯ y¯) ∈ X × Y × Y is such that y¯ ∈ F (¯ ¯ k¯ ∈ G(¯ be closed multifunctions. Suppose that (¯ x, k, x) + k, x), ¯ F is metrically regular around (¯ x, y¯ − k) with modulus τ > 0 and G is Lipschitz-like around ¯ with modulus λ > 0 with τ λ < 1. Suppose that the product space X × Y is endowed with (¯ x, k) the metric defined by d((x, k), (u, z)) = max{d(x, u), kz − kk/λ}. ¯ y¯) with modulus (τ −1 − λ)−1 . Then E(F,G) is metrically regular around (¯ x, k, ¯ k), ¯ then If in addition we suppose that the pair (F, G) is locally sum-stable around (¯ x, y¯ − k, −1 −1 F + G is metrically regular around (¯ x, y¯) with modulus (τ − λ) . ¯ with modulus λ > 0, there exist Proof. Since by assumption G is Lipschitz-like around (¯ x, k) δ1 , δ2 > 0 such that ¯ δ1 ) ⊂ G(x2 ) + λkx1 − x2 kB ¯Y , for all x1 , x2 ∈ B(¯ (17) G(x1 ) ∩ B(k, x, δ2 ). ¯ with modulus τ > 0, there exist Furthermore, since F is metrically regular around (¯ x, y¯ − k) δ3 , δ4 > 0 and a real γ > 0 such that 1 ¯ δ4 ) with ϕF (x, y) ∈ (0, γ). for all (x, y) ∈ B(¯ x, δ3 ) × B(¯ y − k, (18) |∇ϕF (·, y)|(x) ≥ τ So, for any ε > 0, there exists u ∈ B(x, δ3 ), u 6= x such that ϕF (x, y) − ϕF (u, y) 1 > . d(x, u) τ + ε/2 Taking δ1 , δ3 smaller if neccesary, we can assume that δ1 < δ4 , and δ3 < δ2 . Then, for every ¯ δ1 ) × B(¯ (x, k, y) ∈ B(¯ x, min{δ2 , δ3 }/2) × B(k, y , δ4 − δ1 ) with y − k ∈ / F (x), k ∈ G(x), any ε > 0 and any sequence {xn }n∈N ⊆ X converging to x, {kn }n∈N ⊆ X converging to k with kn ∈ G(xn ), and lim d(y − kn , F (xn )) = lim inf d(y − k, F (u)), n→∞

u→x

we deduce that 1 ϕF (x, y − k) − ϕF (u, y − k) ¯ δ4 )), > , (since y − k ∈ B(¯ y − k, (19) d(x, u) τ + ε/2 and lim d(y − k, F (xn )) = lim d(y − kn , F (xn )) = lim inf d(y − k, F (u)) = ϕF (x, y − k).

n→∞

n→∞

u→x

On the other hand, by definition of the function ϕE , there is a sequence {un }n∈N ⊆ X converging to u such that lim d(y − k, F (un )) = ϕF (u, y − k). n→∞

Because u ∈ B(x, δ3 ), x ∈ B(¯ x, min{δ2 , δ3 }/2), {un }n∈N → u, for n large enough, one has that ¯ δ1 ) and {kn }n∈N ⊆ X converges to k, for n large enough, un ∈ B(¯ x, δ2 ). Similarly, since k ∈ B(k, ¯ one has that kn ∈ B(k, δ1 ). Therefore, by (17), and (19), there exists zn ∈ G(un ) such that kzn − kn k ≤ λd(xn , un ).

(20) and

d(y − k, F (xn )) − d(y − k, F (un )) 1 > . n→∞ d(x, u) τ +ε Thus, noticing that u 6= x, one has that lim

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d(y − k, F (xn )) − d(y − k, F (un )) d(xn , un ) n→∞ d(y − k, F (xn )) − d(y − k, F (un )) = lim n→∞ d(xn , un ) d(y − k, F (xn )) − d(y − k, F (un )) d(x, u) = lim n→∞ d(x, u) d(xn , un ) d(y − k, F (xn )) − d(y − k, F (un )) d(x, u) = lim lim n→∞ n→∞ d(xn , un ) d(x, u) 1 ≥ . τ +ε

lim sup

On the other hand, d(y − zn , F (un )) ≤ d(y − kn F (un )) + kkn − zn k.

(21)

¯ δ1 ) × From relations (20), (21), we deduce that for any (x, k, y) ∈ B(¯ x, min{δ2 , δ3 }/2) × B(k, B(¯ y , δ4 − δ1 ) with y − k ∈ / F (x), k ∈ G(x), and any ε > 0, any sequence {xn }n∈N ⊆ X converging to x, {kn }n∈N ⊆ X converging to k, there exists {(un , zn )}n∈N with lim inf d((un , zn ), (x, k)) = lim inf max{d(un , x), kzn − kk/λ} ≥ lim inf d(un , x) > 0, n→∞

n→∞

n→∞

(since 0 < d(x, u) ≤ d(un , x) + d(un , u) and un → u) such that d(y − kn , F (xn )) − d(y − zn , F (un )) lim sup d((xn , kn ), (un , zn )) n→∞ d(y − kn , F (xn )) − d(y − kn , F (un )) − kkn − zn k ≥ lim sup d((xn , kn ), (un , zn )) n→∞ d(y − kn , F (xn )) − d(y − kn , F (un )) − kkn − zn k = lim sup max{d(xn , un ), kkn − zn k/λ} n→∞ d(y − kn , F (xn )) − d(y − kn , F (un )) ≥ lim sup −λ max{d(xn , un ), kkn − zn k/λ} n→∞ d(y − kn , F (xn )) − d(y − kn , F (un )) 1 = lim sup −λ> − λ, d(x , u ) τ + ε n→∞ n n (since kzn − kn k/λ ≤ d(xn , un )). ¯ y¯) with By Lemma 3 ((i) ⇔ (ii)), one concludes that E(F,G) is metrically regular around (¯ x, k, modulus (τ −1 − λ)−1 . ¯ k), ¯ then by combining the hypothesis If the pair (F, G) is locally sum-stable around (¯ x, y¯ − k, with Proposition 7 and Proposition 4, we complete the proof.  Combining Proposition 4 and Theorem 8, we obtain the following corollary which is equivalent to the main result (Theorem 3.3) in [12], which is stated for the difference of an open mapping and a Lipschitz-like one. Corollary 9. Let X be a complete metric space, let Y be a Banach space and let F, G : X ⇒ Y ¯ y¯) ∈ X × Y × Y is such that y¯ ∈ F (¯ ¯ k¯ ∈ G(¯ be closed multifunctions. Suppose that (¯ x, k, x) + k, x)

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¯ with modulus τ > 0 and G is Lipschitz-like around and F is metrically regular around (¯ x, y¯ − k) ¯ ¯ y¯− k) ¯ (¯ x, k) with modulus λ > 0 with τ λ < 1. Then, there exist a neighborhood U ×V ×W of (¯ x, k, and ε, τ > 0 such that for every (x, k, z) ∈ U × V × W, k ∈ G(x), z ∈ F (x), and ρ ∈ (0, ε), B(k + z, ρτ −1 ) ⊂ (F + G)(B(x, ρ)). 3. Metric regularity of the epigraphical multifunction under coderivative conditions In this section, X, Y are assumed to be Asplund spaces, i.e., Banach spaces for which each separable subspace has a separable dual (in particular, any reflexive space is Asplund; see, e.g., [6] for more details). We recall some notations, terminology and definitions basically standard and conventional in the area of variational analysis and generalized differentials. As usual, k · k stands for the norm on X or Y indifferently and h·, ·i signifies for the canonical pairing between w?

X and its topological dual X ? with the symbol → indicating the convergence in the weak? topology of X ? and the symbol cl∗ standing for the weak? topological closure of a set. Given a set-valued mapping F : X ⇒ X ? between X and X ? , recall that the symbol n o w? (22) Lim sup F (x) := x? ∈ X ? ∃ xn → x ¯, ∃ x?n → x? with x?n ∈ F (xn ), n ∈ N x→¯ x

stands for the sequential Painlev´e-Kuratowski outer/upper limit of F as x → x ¯ with respect to the norm topology of X and the weak? topology of X ? . Let us give an extended-real-valued f

lower semicontinuous function f : X → R ∪ {+∞} and x ¯ ∈ X. The notation x → x ¯ means that with x → x ¯ with f (x) → f (¯ x). For ε ≥ 0, the ε-Fr´echet subdifferential of f at x ¯ ∈ Dom f is the set f (x) − f (¯ x) − hx? , x − x ¯i ≥ −ε}, (23) ∂ˆε f (¯ x) := {x? ∈ X ? : lim inf x→¯ x, x6=x ¯ kx − x ¯k and if x ¯∈ / Dom f , we set ∂ˆε f (¯ x) = ∅. ˆ (¯ When ε = 0, formula (23) defines the Fr´echet subdifferential ∂f x) of f at x ¯: f (x) − f (¯ x) − hx? , x − x ¯i ˆ (¯ ∂f x) = {x? ∈ X ? : lim inf ≥ 0}, x→¯ x, x6=x ¯ kx − x ¯k ˆ (¯ and ∂f x) = ∅ if x ¯∈ / Dom f. The notation ∂f (¯ x) is used to denote the limiting subdifferential of f at x ¯ ∈ Dom f . It is defined by ∂f (¯ x) := Lim sup ∂ˆε f (x), f

x→¯ x, ε↓0

i.e., ?

f w ∂f (¯ x) = {x? ∈ X ? : ∃εn ↓ 0, xn → x ¯, x?n → x? with x?n ∈ ∂ˆεn f (xn ) for all n ∈ N}.

The following formula holds: ˆ (x). ∂f (¯ x) := Lim sup ∂f f

x→¯ x

ˆ (¯ For a closed set C ⊂ X and x ¯ ∈ C, the Fr´echet normal cone to C at x ¯ is denoted N x; C) and is defined as the Fr´echet subdifferential of indicator function δC of C at x ¯, i.e., ˆ C (¯ ˆ (¯ N x; C) := ∂δ x),

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where δC (x) = 0 if x ∈ C, and δC (x) = +∞ if x ∈ / C. The limiting normal cone of C at x ¯ is defined and denoted by N (¯ x; C) = ∂δC (¯ x). Let us consider a closed multifunction F : X ⇒ Y and y¯ ∈ F (¯ x). The Fr´echet coderivative of ˆ ? F (¯ F at (¯ x, y¯) is the mapping D x, y¯) : Y ? ⇒ X ? defined by ˆ ? F (¯ ˆ ((¯ x? ∈ D x, y¯)(y ? ) ⇔ (x? , −y ? ) ∈ N x, y¯); gph F ), while the Mordukhovich (limiting) coderivative of F at (¯ x, y¯) is the mapping D? F (¯ x, y¯) : Y ? ⇒ X ? defined by x? ∈ D? F (¯ x, y¯)(y ? ) ⇔ (x? , −y ? ) ∈ N ((¯ x, y¯); gph F ). ˆ ((¯ Here, N x, y¯); gph F ) and N ((¯ x, y¯); gph F ) are the Fr´echet and the limiting normal cone to gph F at (¯ x, y¯), respectively. To obtain a point-based condition for metric regularity of multifunctions in infinite dimensional spaces, one often uses the so-called partial sequential normal compactness (PSNC) property. A multifunction F : X ⇒ Y is partially sequentially normally compact at (¯ x, y¯) ∈ gph F , if for any sequences {(xk , yk , x?k , yk? )} ∈ gph F × X ? × Y ? satisfying w?

ˆ ? (xk , yk )(y ? ), x? → 0, ky ? k → 0 (xk , yk ) → (¯ x, y¯), x?k ∈ D k k k one has kx?k k → 0 as k → ∞. Remark 10. Condition (PSNC) at (¯ x, y¯) ∈ gph F is satisfied if X, or Y is finite dimensional space, or F is Lipschitz-like around that point. In the following, we need a result on the metric inequality (see, e.g., Ioffe [18], Huynh &Th´era [25]). Let us recall that the sets {Ω1 , Ω2 } satisfy the metric inequality at x ¯ if there are τ > 0 and r > 0 such that d(x, Ω1 ∩ Ω2 ) ≤ τ [d(x, Ω1 ) + d(x, Ω2 )] for all x ∈ B(¯ x, r). Proposition 11. Let {Ω1 , Ω2 } be two closed subsets of X and fix x ¯ ∈ Ω1 ∩ Ω2 . Suppose that the following hypothesis (H) is satisfied at x ¯: ˆ (xik ; Ωi ) such that {xik }k∈N → x ¯, i=1, (H) for any sequences {xik }k∈N ⊂ Ωi , {x?ik }k∈N ⊂ N 2 and kx?1k + x?2k kk∈N → 0 implies x?1k → 0, x?2k → 0, then, the sets {Ω1 , Ω2 } satisfy the metric inequality at x ¯. Under this assumption, there is r > 0 such that for every ε > 0, and x ∈ B(¯ x, r), there exist x1 , x2 ∈ B(x, ε) such that (24)

ˆ (x; Ω1 ∩ Ω2 ) ⊂ N ˆ (x1 ; Ω1 ) + N ˆ (x2 ; Ω2 ) + εBX ? . N

Let us consider two multifunctions F, G : X ⇒ Y . To these multifunctions, we associate the two sets C1 := {(x, y, z) ∈ X × Y × Y : y ∈ G(x)} and C2 := {(x, y, z) ∈ X × Y × Y : z ∈ F (x)}. Remark 12. Hypothesis (H) can be restated for the sets {C1 , C2 } at (¯ x, y¯, z¯) ∈ C1 ∩ C2 as follows: (i) (H): for any sequences {(xk , yk )}k∈N ⊂ gph G, {(vk , zk )}k∈N ⊂ gph F, ˆ ? G(xk , yk )(y ? ), u? ∈ D ˆ ? F (vk , zk )(z ? ), x?k ∈ D k k k

14

´ HUYNH VAN NGAI, NGUYEN HUU TRON, AND MICHEL THERA

such that if (xk , yk ) → (¯ x, y¯), (vk , zk ) → (¯ x, z¯), ? ? kxk + uk k → 0, yk? → 0, zk? → 0, then x?k → 0, u?k → 0, as k → 0. It holds whenever one of following conditions is fulfilled: (ii) F −1 or G−1 is Lipschitz-like around (¯ z, x ¯) and (¯ y, x ¯), respectively; (iii) either F is PSNC at (¯ x, z¯) or G is PSNC at (¯ x, y¯), and D? F (¯ x, z¯)(0) ∩ −D? G(¯ x, y¯)(0) = {0}. Proof. Observe that if F −1 or G−1 is Lipschitz-like around (¯ z, x ¯) and (¯ y, x ¯), respectively, then assumption (H) always holds (see for instance Mordukhovich [23]). We now assume that (iii) holds. Take {(xk , yk )}k∈N ⊂ gph G, {(vk , zk )}k∈N ⊂ gph F, ˆ ? F (vk , zk )(z ? ), ˆ ? G(xk , yk )(y ? ), u? ∈ D x?k ∈ D k k k such that (xk , yk ) → (¯ x, y¯), (vk , zk ) → (¯ x, z¯), ? ? kxk + uk k → 0, yk? → 0, zk? → 0. If the sequences {x?k }, {u?k } are unbounded, we can assume that kx?k k → ∞, ku?k k → ∞, and

x?k w? ? u?k w? ? →x , ? →u . kx?k k kuk k

Then, yk? /kx?k k → 0 and zk? /ku?k k → 0. Consequently, x? ∈ D? G(¯ x, y¯)(0), u? ∈ D? F (¯ x, z¯)(0). On the other hand, u? + x? = 0, (since kx?k + u?k k → 0). It follows that u? ∈ D? F (¯ x, z¯)(0) ∩ −D? G(¯ x, y¯)(0). ? ? Therefore, by assumption, this yields x = u = 0. Hence, x?k u?k → 0, or → 0, (by PSNC property of F or G). kx?k k ku?k k x?

u?

This contradicts the fact kxk? k , and ku?k k are in the unit sphere SY ? of Y ? . So, the sequences k k {x?k }, {u?k } are bounded. Without any loss of generality, we can assume that w?

w?

x?k → x? , u?k → u? .

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It follows that x? ∈ D? G(¯ x, y¯)(0), u? ∈ D? F (¯ x, z¯)(0). Moreover, x? + u? = 0. Hence, u? ∈ D? F (¯ x, z¯)(0) ∩ −D? G(¯ x, y¯)(0). Therefore, by assumption, we obtain x? = u? = 0, and x?k → 0, or, u?k → 0, (by PSNC property of F or G). The proof is complete.  The following lemma gives an estimation for the strong slope of the function ϕE ((x, k), y). ¯ k) ¯ ∈ X × F (¯ Lemma 13. Let (¯ x, y¯ − k, x) × G(¯ x) be given. Assume that the sets {C1 , C2 } ¯ y¯ − k). ¯ Then there exists ρ > 0 such that for all defined as above satisfy hypothesis (H) at (¯ x, k, ¯ (x, k, y) ∈ B((¯ x, k, y¯), ρ) with y ∈ / F (x) + k, k ∈ G(x) as well as d(y, F (x) + k) < ρ, one has    (u, w) ∈ gph F, (v, z) ∈ gph G, u, v ∈ B(x, δ)             ? ∈D     ˆ ? G(v, z)(y ? ), ky ? k = 1, z ∈ B(k, δ) u             ? ? ? ? ? ? ˆ ? x ∈ D F (u, w)(y + z ) + u , z ∈ δB ? Y |∇ϕE ((·, ·), y)|(x, k) ≥ lim inf kx k : .    δ↓0  d(y, F (u) + k) ≤ ϕE ((x, k), y) + δ                 kw + k − yk ≤ d(y, F (u) + k) + δ        ? ? |hy + z , w + k − yi − d(y, F (u) + k)| < δ ¯ y¯ − k) ¯ then it is‘also satisfied at all points Proof. Obviously, if (H) is satisfied at (¯ x, k, ¯ ¯ ¯ y¯− (u, v, w) ∈ X ×G(u)×F (u) near (¯ x, k, y¯− k), say (u, v, w) ∈ X ×G(u)×F (v)∩BX×Y ×Y ((¯ x, k, ¯ 3ρ). Let (x, k, y) ∈ BX×Y ×Y ((¯ ¯ y¯ − k), ¯ ρ) be such that y ∈ k), x, k, / F (x) + k, k ∈ G(x) and d(y, F (x) + k) < ρ. Set |∇ϕE ((·, ·), y)|(x, k) := m. By the lower semicontinuity of ϕE as well as the definition of the strong slope, for each ε ∈ (0, ϕE ((x, k), y)), there is η ∈ (0, ε) with 4η + ε < ϕE ((x, k), y) and 1 − (m + ε + 3)η > 0 such that d(y, F (u) + k) ≥ ϕE ((x, k), y) − ε, for all u ∈ B(x, 4η) and m+ε≥

ϕE ((x, k), y) − ϕE ((z, k 0 ), y) max{kx − zk, kk − k 0 k}

¯ η), k 0 ∈ B(k, ¯ η) ∩ G(x). for all z ∈ B(x,

Consequently, ϕE ((x, k), y) ≤ ϕE ((z, k 0 ), y)+(m+ε)kz−xk+(m+ε)kk−k 0 k

¯ η), k 0 ∈ B(k, ¯ η)∩G(x). for all z ∈ B(x,

Take u ∈ B(x, η 2 /4), v ∈ F (u) such that ky − k − vk ≤ ϕE ((x, k), y) + η 2 /4. Taking into account that ϕE ((z, k 0 ), y) ≤ d(y, F (z) + k 0 ) with k 0 ∈ G(z), then ϕE ((z, k 0 ), y) ≤ ky − k 0 − wk with w ∈ F (z) and k 0 ∈ G(z). It follows that ϕE ((z, k 0 ), y) ≤ ky − k 0 − wk + δC2 (z, k 0 , w) + δC1 (z, k 0 , w). From the inequality, ky − k − vk ≤ ϕE ((z, k 0 ), y) + (m + ε)kz − xk + η 2 /4, we obtain that ky − k − vk ≤ ky − k 0 − wk + δC2 (z, k 0 , w) + δC1 (z, k 0 , w) + (m + ε)kz − uk + (m + ε)η + η 2 /4, ¯ η) × Y, k 0 ∈ B(k, ¯ η). Ekeland variational principle to the function for all (z, w) ∈ B(x, (z, k 0 , w) 7→ ky − k 0 − wk + δC2 (z, k 0 , w) + δC1 (z, k 0 , w) + (m + ε)kz − uk

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¯ η) × B(k, ¯ η) × Y, we can select (u1 , k1 , w1 ) ∈ (u, k, v) + η BX×Y ×Y with (u1 , k1 , w1 ) ∈ on B(x, 4 C2 ∩ C1 such that ky − k1 − w1 k ≤ ky − k − vk(≤ ϕE ((x, k), y) + η 2 /4);

(25) and the function

(z, k 0 , w) 7→ ky−k 0 −wk+δC2 (z, k 0 , w)+δC1 (z, k 0 , w)+(m+ε)kz−uk+(m+ε+1)ηk(z, k 0 , w)−(u1 , v1 , w1 )k ¯ η)× B(k, ¯ η)×Y at (u1 , k1 , w1 ). Hence, using the sum rule for Fr´echet attains a minimum on B(x, subdifferentials, we can find (u2 , k2 , w2 ), (u4 , k4 , w4 ) ∈ BX×Y ×Y ((u1 , k1 , w1 ), η); (u3 , k3 , w3 ) ∈ BX×Y ×Y ((u1 , k1 , w1 ), η)∩C2 ∩C1 ; such that ˆ − · − ·k(u2 , k2 , w2 ), (0, k2? , w2? ) ∈ ∂ky ˆ C (·, ·, ·) + δC (·, ·, ·))(u3 , k3 , w3 ), (u? , k ? , w? ) ∈ ∂(δ 3

3 3 ? (u4 , 0, 0)

2

1

ˆ ∈ ∂((m + ε)k · −uk)(u4 , k4 , w4 )

and ¯X ? × B ¯Y ? × B ¯Y ? ]. (0, 0, 0) ∈ (0, k2? , w2? ) + (u?3 , k3? , w3? ) + (u?4 , 0, 0) + (m + ε + 2)η[B

(26)

Notice that ky − k2 − w2 k ≥ ky − v − kk − kw2 − vk − kk2 − kk

(27)

≥ ϕE ((x, k), y) − ε − (kw2 − w1 k + kw1 − vk) − (kk2 − k1 k + kk − k1 k) > ϕE ((x, k), y) − ε − 2η − 2η = ϕE ((x, k), y) − ε − 4η > 0. Then, by [[34]Theorem 2.8.3](see, also, [13][proof of Theorem 3.6]), we know that ˆ − · − ·k(u2 , k2 , w2 ) = {(0, y ? , y ? ) : y ? ∈ SY ? , hy ? , w2 + k2 − yi = ky − w2 − k2 k}. ∂ky Hence, w2? = k2? ∈ SY ? and hw2? , w2 + k2 − yi = ky − w2 − k2 k. ¯ y¯ − k), ¯ 3ρ), we take η smaller if necessary, Now, in order to have (u3 , k3 , w3 ) ∈ BX×Y ×Y ((¯ x, k, and by virtue of Proposition 11 one has ˆ ((u5 , k5 , w5 ); C2 ) + N ˆ ((u6 , k6 , w6 ); C1 ) + η[B ¯X ? × B ¯Y ? × B ¯Y ? ], (28) (u? , k ? , w? ) ∈ N 3

3

3

where (u5 , k5 , w5 ) ∈ C2 ∩ BX×Y ×Y ((u3 , k3 , w3 ), η), (u6 , k6 , w6 ) ∈ C1 ∩ BX×Y ×Y ((u3 , k3 , w3 ), η). From (26) and (28), one deduces that ˆ ((u5 , k5 , w5 ); C2 )+ (0, 0, 0) ∈ (0, k2? , w2? ) + N ¯Y ? ]. ˆ ((u6 , k6 , w6 ); C1 ) + (u? , 0, 0) + (m + ε + 3)η[B ¯X ? × B ¯Y ? × B N 4 ¯X ? × B ¯Y ? × B ¯Y ? ], (u? , k ? , 0) ∈ N ˆ ((u6 , k6 , w6 ); C1 ), i.e., Therefore, there exist (u?5 , k5? , w5? ) ∈ [B 6 6 ? ? ? ˆ u6 ∈ D G(u6 , k6 )(−k6 ) such that ˆ ((u5 , k5 , w5 ); C2 ). (−u?4 −(m+ε+3)ηu?5 −u?6 , −k2? −(m+ε+3)ηk5? −k6? , −w2? −(m+ε+3)ηw5? ) ∈ N It follows that −k2? − (m + ε + 3)ηk5? − k6? = 0, and ˆ ((u5 , w5 ); gph F ). (−u?4 − (m + ε + 3)ηu?5 − u?6 , −w2? − (m + ε + 3)ηw5? ) ∈ N

METRIC REGULARITY OF THE SUM OF MULTIFUNCTIONS AND APPLICATIONS

17

Consequently, ˆ ? F (u5 , w5 )(w? + (m + ε + 3)ηw? ). −k6? = k2? + (m + ε + 3)ηk5? and (−u?4 − (m + ε + 3)ηu?5 − u?6 ) ∈ D 2 5 Remark that kk6? k = k − k2? − (m + ε + 3)ηk5? k ≥ 1 − (m + ε + 3)η > 0. Hence, setting y ? := (k2? + (m + ε + 3)ηk5? )/kk2? + (m + ε + 3)ηk5? k, z ? := (w5? − k5? )(m + ε + 3)η/kk2? + (m + ε + 3)ηk5? k, x?1 := u?6 /kk2? + (m + ε + 3)ηk5? k, x?2 := (−u?4 − (m + ε + 3)ηu?5 )/kk2? + (m + ε + 3)ηk5? k, one obtains that ˆ ? G(u6 , k6 )(y ? ) and (x?2 − x?1 ) ∈ D ˆ ? F (u5 , w5 )(y ? + z ? ), x?1 ∈ D

(29) where,

ky ? k = 1, kz ? k ≤

(30)

2(m + ε + 3)η m + ε + (m + ε + 3)η := δ, kx?2 k ≤ . 1 − (m + ε + 3)η 1 − (m + ε + 3)η

On the other hand, according to relation (25) one has that (31) ϕE ((x, k), y) − ε ≤ d(y, F (u5 ) + k) ≤ ky − k − w5 k ≤ ky − k1 − w1 k + kw5 − w1 k + kk1 − kk ≤ ky − k − vk + η + 2η ≤ ϕE ((x, k), y) + η 2 /4 + 3η. Consequently, (32) hy ? + z ? , k + w5 − yi ≤ (1 + δ)ky − k − w5 k ≤ (1 + δ)d(y, F (u5 ) + k) + (1 + δ)(η 2 /4 + 3η + ε), whence (33)

hy ? + z ? , y − k − w5 i − d(y, F (u5 ) + k) ≤ δd(y, F (u5 ) + k) + (1 + δ)(η 2 /4 + 3η + ε).

Furthermore, hy ? + z ? , k + w5 − yi hw2? + (m + ε + 3)ηw5? , k + w5 − yi kk2? + (m + ε + 3)ηk5? k hw? , w2 + k2 − yi + hw2? , w5 − w2 i + hw2? , k − k2 i + (m + ε + 3)ηhw5? , w5 + k − yi = 2 kk2? + (m + ε + 3)ηk5? k kw2 + k2 − yk − 3η − 2η − (m + ε + 3)ηkw5 + k − yk ≥ , (1 + (m + ε + 3)η)

=

and by (27) and (31) we have that ky − k2 − w2 k ≥ ϕE ((x, k), y) − ε − 4η ≥ d(y, F (u5 ) + k) − η 2 /4 − 8η − ε. Along with (33), one

´ HUYNH VAN NGAI, NGUYEN HUU TRON, AND MICHEL THERA

18

deduces that hy ? , w5 + k − yi d(y, F (u5 ) + k) − η 2 /4 − 8η − ε − 4η − (m + ε + 3)η(d(y, F (u5 ) + k) + η 2 /4 + η + ε 1 + (m + ε + 3)η d(y, F (u5 ) + k)(1 − (m + ε + 3)η) − (η 2 /4 + η + ε)(1 + (m + ε + 3)η) − 11η = . 1 + (m + ε + 3)η

≥

So, (34)

hy ? , w5 + k − yi ≥

d(y, F (u5 ) + k)(1 − (m + ε + 3)η) − 11η − (η 2 /4 + η + ε). 1 + (m + ε + 3)η

As ε, η, δ > 0 are arbitrary small, by combining relations (29)-(34), we complete the proof.



Theorem 14. Let X, Y be Asplund spaces, and let F, G : X ⇒ Y be closed multifunctions. ¯ y¯) ∈ X × Y × Y is such that y¯ ∈ F (¯ ¯ k¯ ∈ G(¯ Suppose that (¯ x, k, x) + k, x) and the sets {C1 , C2 } ¯ ¯ satisfy the hypothesis (H) at (¯ x, k, y¯ − k). Let m > 0. If there exist a neighborhood U × V × W ¯ y¯) and γ > 0 such that for each (x, y, k) ∈ U × V × W with y ∈ of (¯ x, k, / F (x) + k, k ∈ G(x),    (u, w) ∈ gph F, (v, z) ∈ gph G, u, v ∈ B(x, δ)             ? ∈D     ˆ ? G(v, z)(y ? ), ky ? k = 1, z ∈ B(k, δ) u             ? ? ? ? ? ? ˆ ? x ∈ D F (u, w)(y + z ) + u , z ∈ δB ? Y m ≤ lim inf kx k :    δ↓0  d(y, F (u) + k) ≤ γ + δ                 kw + k − yk ≤ d(y, F (u) + k) + δ         ? ? |hy + z , w + k − yi − d(y, F (u) + k)| < δ ¯ y¯) such that then there exists a neighborhood U1 × V1 × W1 of (¯ x, k, md((x, k), SE(F,G) (y)) ≤ ϕE ((x, k), y)

for all

(x, k, y) ∈ U1 × V1 × W1 .

This theorem implies the following result: Theorem 15. Let X, Y be Asplund spaces, and let F, G : X ⇒ Y be closed multifunctions, and ¯ y¯) ∈ X × Y × Y be such that y¯ ∈ F (¯ ¯ k¯ ∈ G(¯ let (¯ x, k, x) + k, x). Let m > 0. If the sets {C1 , C2 } ¯ y¯ − k) ¯ and satisfy the hypothesis (H) at (¯ x, k,   ˆ ? F (x1 , w)(y ? + δBY ? ) + u? x? ∈ D ? (35) m< lim inf kx k : ˆ ? G(x2 , z)(y ? ), ky ? k = 1, F G u? ∈ D ¯ ¯ (x1 ,w)→(¯ x,¯ y −k),(x x,k),δ↓0 2 ,z)→(¯ F G ¯ (x2 , z) → ¯ mean that (¯ x, k) where the notations (x1 , w) → (¯ x, y¯ − k), ¯ (x2 , z) → (¯ ¯ and (x1 , w) ∈ gph F, (x2 , z) ∈ gph G, (x1 , w) → (¯ x, y¯ − k), x, k) ¯ y¯) such that then there exists a neighborhood U1 × V1 × W1 of (¯ x, k,

md((x, k), SE(F,G) (y)) ≤ ϕE ((x, k), y)

for all

(x, k, y) ∈ U1 × V1 × W1 .

The next result gives a point-based condition for metric regularity of the epigraphical multifunction. Theorem 16. Let X, Y be Asplund spaces, and let F, G : X ⇒ Y be closed multifunctions, and ¯ y¯) ∈ X × Y × Y be such that y¯ ∈ F (¯ ¯ k¯ ∈ G(¯ let (¯ x, k, x) + k, x). Suppose that ¯ ¯ (i) F or G is PSNC at (¯ x, y¯ − k) and (¯ x, k), respectively; ¯ ¯ (ii) D? F (¯ x, y¯ − k)(0) ∩ −D? G(¯ x, k)(0) = {0};

METRIC REGULARITY OF THE SUM OF MULTIFUNCTIONS AND APPLICATIONS

19

ˆ ? F (xn , yn − kn )(y ? + (1/n)BY ? ), v ? ∈ D ˆ ? G(xn , kn )(y ? ) such that (iii) for any u?n ∈ D n n n w?

ku?n + vn? k → 0, yn? → 0 it follows that yn? → 0; Under the condition that
¯ + D? G(¯ ¯ = {0}, Ker D? F (¯ x, y¯ − k) x, k) ¯ y¯). is metrically regular around (¯ x, k,

(?) the multifunction E(F,G)

Proof. We prove the result by contradiction. Suppose that E(F,G) fails to be metrically regular ¯ y¯). Then, by Theorem 15, there exist sequences around (¯ x, k, F G ¯ (xn , kn ) → ¯ (x? , u? , y ? , z ? ) ∈ X ? × X ? × Y ? × Y ? , (xn , yn − kn ) → (¯ x, y¯ − k), (¯ x, k), n n n n

with

ˆ ? F (xn , yn − kn )(yn? + zn? ) + u?n , x?n ∈ D ˆ ? G(xn , kn )(y ? ), u?n ∈ D n ? yn ∈ SY ? , zn? ∈ (1/n)BY ? ,

and x?n → 0. ˆ ? F (xn , yn − kn )(y ? + z ? ) such that x? = u? + v ? . Then there is vn? ∈ D n n n n n w?

Since Y is an Asplund space, we can assume that yn? → y ? ∈ Y ? . We consider the following cases: Case 1. The sequences {u?n }, {vn? } are unbounded. We can assume that ku?n k → ∞, kvn? k → ∞, and

u?n w? ? vn? w? ? →u , ? →v . ku?n k kvn k

Then, yn? /ku?n k → 0 and (yn? + zn? )/kvn? k → 0. Consequently, ¯ ¯ u? ∈ D? G(¯ x, k)(0), v ? ∈ D? F (¯ x, y¯ − k)(0). On the other hand, u? + v ? = 0, (sinceku?n + vn? k → 0). It follows that ¯ ¯ v ? ∈ D? F (¯ x, y¯ − k)(0) ∩ −D? G(¯ x, k)(0). Therefore, by (ii), we have that u? = v ? = 0. So, u?n vn? → 0, or → 0, (by PSNC property of F or G). ku?n k kvn? k This contradicts the fact

u?n ku?n k

and

? vn ?k kvn

belong to the unit sphere SY ? of Y ? . w?

w?

Case 2. The sequences {u?n }, {vn? } are bounded. Assume that u?n → u? , vn? → v ? . It follows that ¯ ? ), v ? ∈ D? F (¯ ¯ ? ). u? ∈ D? G(¯ x, k)(y x, y¯ − k)(y Moreover, u? + v ? = 0.

´ HUYNH VAN NGAI, NGUYEN HUU TRON, AND MICHEL THERA

20

So, ? ¯ ? ) + D? F (¯ ¯ ? )] = [D? G(¯ ¯ + D? F (¯ ¯ 0 ∈ [D? G(¯ x, k)(y x, y¯ − k)(y x, k) x, y¯ − k)](y ),

which means that ¯ + D? F (¯ ¯ y ? ∈ Ker[D? G(¯ x, k) x, y¯ − k)]. By (?), one has that y ? = 0. Now, by assumption, one gets yn? → 0 which contradicts to kyn? k = 1.



Remark 17. If X, Y are finite dimensional spaces, then conditions (i), (iii) hold true automat¯ or (¯ ¯ respectively. ically, while condition (ii) holds if F or G is Lipschitz-like at (¯ x, y¯ − k) x, k),

4. Applications to variational systems In this section, we use the above results to study some properties of variational systems of the form 0 ∈ F (x) + G(x, p),

(36)

where X is a complete metric space, Y is a Banach space, P is a topological space considered as a parameter space, F : X ⇒ Y, G : X × P ⇒ Y are given multifunctions. The solution set of (36) is defined by the notation below should also be defined S(F +G) (p) := {x ∈ X : 0 ∈ F (x) + G(x, p)},

(37) and we denote

S(F +G) (y, p) := {x ∈ X : y ∈ F (x) + G(x, p)}. For every (y, p) ∈ Y × P, SE(F,G) (y, p) = {(x, k) ∈ X × Y : y ∈ F (x) + k, k ∈ G(x, p)}, and, for every p ∈ P, SE(F,G) (p) = {(x, k) ∈ X × Y : 0 ∈ F (x) + k, k ∈ G(x, p)}. We say that the multifunction S(F +G) is Robinson metrically regular (see [30], [31]) around (¯ x, p¯) with modulus τ , if there exist neighborhoods U, V of x ¯, p¯, respectively, such that d(x, S(F +G) (p)) ≤ τ d(0, F (x) + G(x, p)), for all (x, p) ∈ U × V. We also recall that the multifunction G : X × P ⇒ Y is said to be Lipschitz-like around (¯ x, p¯, y¯) with y¯ ∈ G(¯ x, p¯) with respect to x, uniformly in p with constant κ > 0 if there is a neighborhood U × V × W of (¯ x, p¯, y¯) such that ¯Y for all x, u ∈ U, and for all p ∈ V. G(x, p) ∩ W ⊂ G(u, p) + κd(x, u)B The lower semicontinuous envelope (x, p, k, y) 7→ ϕp,E ((x, k), y) of the distance function d(y, E(F,G) ((x, p), k)) is defined by, for each (x, p, k, y) ∈ X × P × Y × Y ϕp,E ((x, k), y) := ( =

lim inf (u,v,w)→(x,k,y)

d(w, E(F,G) ((u, p), v))

lim inf d(y, F (u) + k) if k ∈ G(x, p) u→x

+∞

otherwise.

METRIC REGULARITY OF THE SUM OF MULTIFUNCTIONS AND APPLICATIONS

21

Lemma 18. Let X be a complete metric space and Y be a Banach space and let P be a topological space. Suppose that the set-valued mappings F : X ⇒ Y, G : X × P ⇒ Y satisfy the following ¯ p¯) ∈ X × Y × P : conditions for some (¯ x, k, ¯ (a) (¯ x, k) ∈ SE(F,G) (¯ p); (b) the set-valued mapping p ⇒ G(¯ x, p) is lower semicontinuous at p¯; (c) the set-valued mapping F is a closed multifunction, and for any p near p¯, the set-valued mapping x ⇒ G(x, p) is a closed multifunction. Then ¯ (i) for ever p near p¯, the epigraphical multifunction E(F,G) has closed graph, and, E(F,G) ((¯ x, ·), k) is lower semicontinuous at p¯; ¯ 0) is upper semicontinuous at p¯; (ii) the function p 7→ ϕp,E ((¯ x, k), (iii) for each (y, p) ∈ Y × P ; {(x, k) ∈ X × Y : ϕp,E ((x, k), y) = 0} = SE(F,G) (y, p). Proof. We only note that if the multifunction p ⇒ G(¯ x, p) is lower semicontinuous at p¯, then ¯ so is the mapping E(F,G) ((¯ x, ·), k). By using the strong slope of the lower semicontinuous envelope ϕp,E , one has the following result. Theorem 19. Let X be a complete metric space, Y be a Banach space and let P be a topological space. Suppose that the set-valued mappings F : X ⇒ Y, G : X × P ⇒ Y satisfy conditions ¯ p¯) ∈ X × Y × P . If there exist a neighborhood T1 × (a), (b), (c) from Lemma 18 around (¯ x, k, ¯ U1 × V1 × W1 of (¯ x, p¯, k, 0) and reals m, γ > 0 such that |∇ϕp,E ((·, ·), y)|(x, k) ≥ m for all (x, p, k, y) ∈ T1 × U1 × V1 × W1 with ϕp,E ((x, k), y) ∈ (0, γ), then there exists a neighborhood ¯ 0) such that T × U × V × W of (¯ x, p¯, k, md((x, k), SE(F,G) (y, p)) ≤ ϕp,E ((x, k), y), for all (x, p, k, y) ∈ T × U × V × W. Proof. Applying Theorem 1 and Lemma 18 for the mapping E(F,G) (·, ·), one obtains the proof.  Proposition 20. Let X be a complete metric space and Y be a Banach space and let P be a topological space. Suppose that the set-valued mappings F : X ⇒ Y, G : X × P ⇒ Y satisfy ¯ p¯) ∈ X ×Y ×P . If there exist a neighborhood conditions (a), (b), (c) from Lemma 18 around (¯ x, k, ¯ T × U × V × W ⊂ X × P × Y × Y of (¯ x, p¯, k, 0) and m > 0 such that md((x, k), SE(F,G) (y, p)) ≤ ϕp,E ((x, k), y)

for all

(x, p, k, y) ∈ T × U × V × W

then there exists θ > 0 such that ¯ θ)) md(x, S(F +G) (y, p)) ≤ d(y, F (x) + G(x, p) ∩ B(k,

for all

(x, p, y) ∈ T × U × W.

Therefore, ¯ θ)) md(x, S(F +G) (p)) ≤ d(0, F (x) + G(x, p) ∩ B(k,

for all

(x, p) ∈ T × U.

Proof. By the hypothesis, there exist a neighborhood T × U × V × W ⊂ X × P × Y × Y of ¯ 0) and m > 0 such that for every (x, p, k, y) ∈ T × U × V × W, it holds (¯ x, p¯, k, md((x, k), SE(F,G) (y, p)) ≤ ϕp,E ((x, k), y).

22

´ HUYNH VAN NGAI, NGUYEN HUU TRON, AND MICHEL THERA

¯ θ), with certain positive θ. Then, for every small ε > 0 and Here, we can assume V = B(k, ¯ θ) ∩ G(x, p)] × W, there is (u, z) ∈ SE for every (x, p, k, y) ∈ T × U × [B(k, (y, p), i.e., (F,G) y ∈ F (u) + z, z ∈ G(u, p) such that md(u, x) ≤ m max{d(u, x), kz − kk} < (1 + ε)d(y, F (x) + k). Noticing that u ∈ (F + G)−1 (y), we obtain that md(x, (F + G)−1 (y)) < (1 + ε)d(y, F (x) + k). Thus, ¯ θ)), md(x, (F + G)−1 (y)) ≤ (1 + ε)d(y, F (x) + G(x, p) ∩ B(k, or, ¯ θ)). md(x, S(F +G) (y, p)) ≤ (1 + ε)d(y, F (x) + G(x, p) ∩ B(k, Since this inequality does not depend on arbitrarily small ε > 0, we obtain that ¯ θ)) md(x, S(F +G) (y, p)) ≤ d(y, F (x) + G(x, p) ∩ B(k, for all (x, p, y) ∈ T × U × W. Taking y¯ = 0 and y = y¯, we obtain the second conclusion of the Theorem, and the proof is complete.  In the sequel we use for the parametrized case, the concept of locally sum-stability which was considered in the previous section. Definition 21. Let F : X ⇒ Y, G : X × P ⇒ Y be two multifunctions and (¯ x, p¯, y¯, z¯) ∈ X × P × Y × Y be such that y¯ ∈ F (¯ x), z¯ ∈ G(¯ x, p¯). We say that the pair (F, G) is locally sum-stable around (¯ x, p¯, y¯, z¯) if for every ε > 0 there exists δ > 0 and a neighborhood W of p¯ such that for every (x, p) ∈ B(¯ x, δ) × W and every w ∈ (F + G)(x) ∩ B(¯ y + z¯, δ), there are y ∈ F (x) ∩ B(¯ y , ε) and z ∈ G(x) ∩ B(¯ z , ε) such that w = y + z. A following simple case which ensures the locally sum-stability of the pair (F, G) is analogous to Proposition 6 . Proposition 22. Let F : X ⇒ Y, G : X × P ⇒ Y be two multifunctions and (¯ x, p¯, y¯, z¯) ∈ X × P × Y × Y such that y¯ ∈ F (¯ x), z¯ ∈ G(¯ x, p¯). If G(¯ x, p¯) = {¯ z } and G is upper semicontinuous at (¯ x, p¯), then the pair (F, G) is locally sum-stable around (¯ x, p¯, y¯, z¯). Proposition 23. Let X be a complete metric space, Y be a Banach space and let P be a topological pace. Suppose that the set-valued mappings F : X ⇒ Y, G : X × P ⇒ Y satisfy ¯ p¯) ∈ X ×Y ×P . If there exist a neighborhood conditions (a), (b), (c) from Lemma 18 around (¯ x, k, T × U of (¯ x, p¯) and θ, τ > 0 such that (38)

¯ θ)) d(x, S(F +G) (p)) ≤ τ d(0, F (x) + G(x, p) ∩ B(k,

for all

(x, p) ∈ T × U,

¯ k), ¯ then S(F+G) is Robinson metrically regular and (F, G) is locally sum-stable around (¯ x, p¯, −k, around (¯ x, p¯) with modulus τ . ¯ k) ¯ is The conclusion remains true if the assumption of local sum stability around (¯ x, p¯, −k, replaced by the following one: G(¯ x, p¯) = {¯ z } and G is upper semicontinuous at (¯ x, p¯).

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Proof. The proof of this proposition is very similar to that of Proposition 7. Here, we sketch the proof. Suppose that (38) holds for every (x, p) ∈ T × U. Here, we can assume that T = B(¯ x, δ), with some positive δ > 0. ¯ k), ¯ there exists δ > 0 such that for every Since (F, G) is locally sum-stable around (¯ x, p¯, −k, ¯ θ) and (x, p) ∈ B(¯ x, δ) × U and every w ∈ (F + G)(x) ∩ B(0, δ), there are y ∈ F (x) ∩ B(−k, ¯ z ∈ G(x) ∩ B(k, θ) such that w = y + z. Fix (x, p) ∈ B(¯ x, δ) × U. We consider two following cases: Case 1. d(0, F (x) + G(x, p)) < δ/2. Fix γ > 0, small enough so that d(0, F (x) + G(x, p)) + γ < δ/2, and take t ∈ F (x) + G(x, p) such that ktk < d(0, F (x) + G(x, p)) + γ. Hence we have ktk < δ/2, i.e., t ∈ B(0, δ/2) ⊂ B(0, δ). It follows that t ∈ [F (x) + G(x, p)] ∩ B(0, δ). ¯ θ) and z ∈ G(x, p) ∩ B(k, ¯ θ) such that t = y + z. Therefore, there are y ∈ F (x) ∩ B(−k, Consequently, ¯ θ) + G(x, p) ∩ B(k, ¯ θ) ⊂ F (x) + G(x, p) ∩ B(k, ¯ θ). t ∈ F (x) ∩ B(−k, It follows that ¯ θ)) ≤ ktk. d(0, F (x) + G(x, p) ∩ B(k, This yields ¯ θ)) < d(0, F (x) + G(x, p)) + γ, d(0, F (x) + G(x, p) ∩ B(k, and therefore, as γ > 0 is arbitrarily small, we derive that ¯ θ)) ≤ d(0, F (x) + G(x, p)). d(0, F (x) + G(x, p) ∩ B(k, By (38), one derives d(x, S(F +G) (p)) ≤ τ d(0, F (x) + G(x, p)),

for all

(x, p) ∈ B(¯ x, δ) × U.

Case 2. d(0, F (x) + G(x, p)) ≥ δ/2. According to condition (c), the multifunction p ⇒ G(¯ x, ·) is lower semicontinuous at p¯. It follows that the distance function d(0, F (¯ x) + G(¯ x, ·)) is upper semicontinuous at p¯, and thus, there exists a neighborhood W of p¯ such that d(0, F (¯ x) + G(¯ x, p) ≤ δ/4, for all p ∈ W. Shrinking W smaller if necessary, we can assume that W ⊂ U. Choosing 0 < δ1 < min{δ, τ δ/4}. For every (x, p) ∈ B(¯ x, δ1 ) × W, and for every small ε > 0, there exists u ∈ S(F +G) (p) such that d(¯ x, u) ≤ (1 + ε)τ d(0, F (¯ x) + G(¯ x, p)). So, d(x, u) ≤ d(x, x ¯) + d(¯ x, u) < δ1 + τ (1 + ε)d(0, F (¯ x) + G(¯ x, p)) < τ δ/4 + τ (1 + ε)δ/4 ≤ τ /2d(0, F (x) + G(x, p)) + τ /2(1 + ε)d(0, F (x) + G(x, p).

´ HUYNH VAN NGAI, NGUYEN HUU TRON, AND MICHEL THERA

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Taking the limit as ε > 0 goes to 0, it follows that d(x, S(F +G) (p)) ≤ τ d(0, F (x) + G(x, p)), establishing the proof. The following theorem establishes the Lipschitz property for the solution mapping SE(F,G) . Theorem 24. Let X be a complete metric space, Y be a Banach space, P be a topological space. Suppose that F : X ⇒ Y and G : X × P ⇒ Y are multifunctions satisfying conditions (a), (b), (c) in Lemma 18. ¯ with modulus τ > 0 and G is Lipschitz-like around If F is metrically regular around (¯ x, −k) ¯ with respect to x, uniformly in p with modulus λ > 0 such that τ λ < 1, then E(F,G) is met(¯ x, p¯, k) ¯ 0) with respect to (x, k), uniformly in p, with modulus (τ −1 − λ)−1 . rically regular around (¯ x, p¯, k, ¯ Moreover, assume in addition that P is a metric space, if G is Lipschitz-like around (¯ x, p¯, k) with respect to p, uniformly in x with modulus γ > 0 then SE(F,G) is Lipschitz-like around ¯ with modulus L = γ + (γ + 1)(τ −1 − λ)−1 ). Especially, SE ((0, p¯), (¯ x, k)) is Lipschitz-like (F,G) −1 −1 ¯ with modulus γ(1 + (τ − λ) ). around ((0, p¯), (¯ x, k)) Proof. The first part is the parametrized version of Theorem 8. Its proof is completely similar to the one of Theorem 8, and is omitted. For the second part, as E(F,G) is metrically regular ¯ 0) with respect to (x, k), uniformly in p, with modulus (τ −1 − λ)−1 , there exists around (¯ x, p¯, k, δ1 > 0 such that (39)

d((x, k), SE(F,G) (y, p)) ≤ (τ −1 − λ)−1 ϕp,E ((x, k), y),

¯ 0), δ1 ). for all (x, p, k, y) ∈ B((¯ x, p¯, k, ¯ with respect to p, uniformly in x with modulus γ > 0 Now, if G is Lipschitz-like around (¯ x, p¯, k) then there is δ2 > 0 such that ¯ δ2 ) ⊂ G(x, p0 ) + γd(p, p0 )B ¯Y , (40) G(x, p) ∩ B(k, for all p, p0 ∈ B(¯ p, δ2 ), for all x ∈ B(¯ x, δ2 ). Set α := min{δ1 /(γ + 1), δ2 }. ¯ α)]. Fix (y, p), (y 0 , p0 ) ∈ B(0, α) × B(¯ p, α). Take (x, k) ∈ SE(F,G) (y, p)) ∩ [B(¯ x, α) × B(k, ¯ α)], then Since (x, k) ∈ SE(F,G) (y, p)) ∩ [B(¯ x, α) × B(k, ¯ α). y ∈ F (x) + k, k ∈ G(x, p) and (x, k) ∈ B(¯ x, α) × B(k, Along with (40), we can find that k 0 ∈ G(x, p0 ) such that kk − k 0 k ≤ γd(p, p0 ) < γα, ¯ δ1 ). Therefore, by (39), one has which follows that k 0 ∈ B(k, d((x, k 0 ), SE(F,G) (y 0 , p0 )) ≤ (τ −1 − λ)−1 ϕp0 ,E ((x, k 0 ), y 0 ), ≤ (τ −1 − λ)−1 d(y 0 , F (x) + k 0 )), Hence, by noting that y ∈ F (x) + k, one deduces that (41)

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d((x, k), SE(F,G) (y 0 , p0 )) ≤ kk − k 0 k + d((x, k 0 ), SE(F,G) (y 0 , p0 )) ≤ γd(p, p0 ) + (τ −1 − λ)−1 d(y 0 , F (x) + k 0 )), ≤ γd(p, p0 ) + (τ −1 − λ)−1 (ky − y 0 k + kk − k 0 k) ≤ γ(1 + (τ −1 − λ)−1 )d(p, p0 ) + (τ −1 − λ)−1 ky − y 0 k and so ¯ α)] SE(F,G) (y, p)) ∩ [B(¯ x, α) × B(k, ¯X × B ¯Y , ⊆ SE(F,G) (y 0 , p0 ) + Ld((y 0 , p0 ), (y, p))B where, L = γ + (γ + 1)(τ −1 − λ)−1 , and by taking y = y 0 = 0 in relation (41), one also derives ¯ with modulus γ(1 + (τ −1 − λ)−1 ). The proof that SE(F,G) is Lipschitz-like around ((0, p¯), (¯ x, k)) is complete.  If we add the assumption that (F, G) is locally sum-stable, we obtain the Lipschitz property of S(F +G) . Theorem 25. Let X be a complete metric space and Y be a Banach space, P be a metric space. Suppose that F : X ⇒ Y and G : X × P ⇒ Y satisfy conditions (a), (b), (c) in Lemma 18. Moreover, assume that ¯ k); ¯ (i) (F, G) is locally sum-stable around (¯ x, p¯, −k, ¯ with modulus τ > 0; (ii) F is metrically regular around (¯ x, −k) ¯ with respect to x, uniformly in p with modulus λ > 0 (iii) G is Lipschitz-like around (¯ x, p¯, k) such that τ λ < 1; ¯ with respect to p, uniformly in x with modulus γ > 0. (iv) G is Lipschitz-like around (¯ x, p¯, k) Then S(F +G) is Robinson metrically regular around (¯ x, p¯) with modulus (τ −1 − λ)−1 . Moreover, S(F +G) is Lipschitz-like around (¯ x, p¯) with constant γ(τ −1 − λ)−1 . Proof. Applying Proposition 24, Proposition 23 and Proposition 20, respectively, we obtain that S(F +G) is Robinson metrically regular around (¯ x, p¯) with modulus (τ −1 − λ)−1 . Thus, there exists δ1 > 0 such that d(x, S(F +G) (p)) ≤ (τ −1 − λ)−1 d(0, F (x) + G(x, p)), for all (x, p) ∈ B((¯ x, p¯), δ1 ). ¯ with respect to p, uniformly in x On the other hand, since G is Lipschitz-like around (¯ x, p¯, k) with modulus γ > 0, we can find δ2 > 0 such that ¯ δ2 ) ⊂ G(x, p0 ) + γd(p, p0 )B ¯Y , G(x, p) ∩ B(k, for all p, p0 ∈ B(¯ p, δ2 ), for all x ∈ B(¯ x, δ2 ). ¯ k), ¯ there is δ3 > 0 such Moreover, since the pair (F, G) is locally sum-stable around (¯ x, p¯, −k, that for every (x, p) ∈ B(¯ x, δ3 ) × B(¯ p, δ3 ) and every w ∈ [F (x) + G(x, p)] ∩ B(0, δ3 ), there are ¯ δ2 ), z ∈ G(x, p) ∩ B(k, ¯ δ2 ) such that w = y + z. Set y ∈ F (x) ∩ B(−k, α := min{δ1 , δ2 , δ3 }. p, p0

Take ∈ B(¯ p, α), and x ∈ S(F +G) (p) ∩ B(¯ x, α), i.e., 0 ∈ F (x) + G(x, p) and x ∈ B(¯ x, α). Moreover, we observe that for every w ∈ [F (x) + G(x, p)] ∩ B(0, α), ¯ δ2 )) + G(x, p) ∩ B(k, ¯ δ2 ) ⊆ F (x) + G(x, p0 ) + γd(p, p0 )B ¯Y . w ∈ F (x) ∩ B(−k, Thus, ¯Y . [F (x) + G(x, p)] ∩ B(0, α) ⊆ F (x) + G(x, p0 ) + γd(p, p0 )B

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´ HUYNH VAN NGAI, NGUYEN HUU TRON, AND MICHEL THERA

Since 0 ∈ F (x) + G(x, p), and also 0 ∈ [F (x) + G(x, p)] ∩ B(0, α), thus ¯Y . 0 ∈ F (x) + G(x, p0 ) + γd(p, p0 )B It follows that there is w ∈ F (x) + G(x, p0 ) such that kwk ≤ γd(p, p0 ). Therefore, d(x, S(F +G) (p0 )) ≤ (τ −1 − λ)−1 d(0, F (x) + G(x, p0 )) ≤ (τ −1 − λ)−1 kwk ≤ γ(τ −1 − λ)−1 d(p, p0 ). So, ¯X , S(F +G) (p) ∩ B(¯ x, α) ⊆ S(F +G) (p0 ) + γ(τ −1 − λ)−1 d(p, p0 )B establishing the proof.



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