{'fraction_non_alphanumeric': 0.08067494659086795, 'fraction_numerical': 0.014882137813665496, 'mean_word_length': 3.7089514066496165, 'pattern_counts': {'":': 0, '<': 16, '<?xml version=': 0, '>': 34, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Quest for Simple and Unified Proofs in Regularity Theory: Perturbation Stability

Ioffe's criterion and various reformulations of it have become a~standard tool in proving theorems guaranteeing metric regularity of a (set-valued) mapping. First, we demonstrate that one should always use directly the so-called general criterion which follows, for example, from Ekeland's variational principle, and that there is no need to make a detour through the slope-based consequences of this general statement. Second, we argue that when proving perturbation stability results, in the spirit of Lyusternik-Graves theorem, there is no need to employ the concept of a lower semi-continuous envelope even in the case of an incomplete target space. The gist is to use the"correct"function to which Ekeland's variational principle is applied; namely, the distance function to the graph of the set-valued mapping under consideration. This approach originates in the notion of graphical regularity introduced by L. Thibault, which is equivalent to the property of metric regularity. Our criteria cover also both metric subregularity and metric semiregularity, which are weaker properties obtained by fixing one of the points in the definition of metric regularity.


Introduction
Metric regularity, linear openness, and Aubin property of the inverse of a given set-valued mapping are three equivalent properties playing a fundamental role in modern variational analysis and have been broadly covered in several monographs [6,8,25,38,39,41]. We formulate all the results in terms of the first mentioned property and call it simply regularity. Although our notation is fairly standard, in case of any difficulty, the reader can consult the end of this introduction. We distinguish two main settings in which regularity statements for a set-valued mapping F : X ⇒ Y from a metric space (X, d) into another metric space (Y, ̺) are proved: (S 1 ) the graph of F is (locally) complete; (S 2 ) the domain space X is complete and the graph of F is (locally) closed.
For example, (S 1 ) occurs when both the domain space and the target space are complete and the graph of F is closed in X × Y , which often can be assumed, without any loss of generality, by using [38,Proposition 2.40] provided that all the assumptions translate to the completions of the spaces and the graph. There are several ways of proving sufficient conditions for regularity in the literature. Let us mention the most popular ones (from our point of view at least): (A) Iterative processes. This (constructive) method finds a solution of the inclusion [equation] as the limit of a sequence of iterates generated by a process resembling Newton's method or the contraction mapping iteration, see, for example, [30], [22], [25, Proof I of Theorem 1A.1, p. 12]. One can say that these proofs are the longest ones but work without any difficulty in the setting (S 2 ), that is, when Y is not necessarily complete [25, Proof I of Theorem 5E.1, p. 309].
A full version of this principle ensures that the inequality in (1) is strict provided thatû = u. However, we show that this is not needed for applications in regularity theory. Theorem 1.1 is usually proved via an iterative process generating a (countable) sequence converging to a desired point.
(C) Application of slope-based conditions. These conditions are usually derived either from the EVP directly or via a detour through error bounds [33]. The concept of the (local strong) slope of a function ϕ : X → R ∪{∞} at a point x ∈ X, with ϕ(x) < ∞, was introduced in De Giorgi, Marino, and Tosques [21]; and this quantity is defined by where [a] + := max{a, 0}, a ∈ R; when ϕ(x) = ∞ then |∇ϕ|(x) := ∞. This approach, going back to works by D. Azé and J.N. Corvellec [7], can be found, for example, in [1], [31], [32], [33]; see [38] for more details concerning the historical background. Especially in the setting (S 2 ), additional concepts such as lower semi-continuous envelopes are employed in the literature, which is tangling the proofs.
(D) Application of a fixed point theorem. A particular choice of the principle depends on the regularity statement we want to prove. In case of Graves' theorem [30], the usual Banach contraction mapping principle [25,Theorem 1A.3] does the job, cf. [25, Proof II of Theorem 1A. 1, p. 19]; when considering the mappings F + g, with a smooth single-valued part g and a set-valued part F , we can choose the contraction mapping principle for set-valued mappings [25,Theorem 5E.2] proved by A.L. Dontchev and W.W. Hager in 1994. As the proofs of both the preceding fixed point theorems rely on the iterative procedure one can expect to obtain shorter proofs of the regularity statement in question. Similarly to (A), this approach works without any difficulty in the setting (S 2 ), see [25,Proof of Theorem 5E.5]. Under certain compactness assumptions, Brouwer's or Kakutani's fixed point theorem can be used instead of contraction mapping principles. In finite-dimensions, one can obtain results that do not seem to follow from the infinite-dimensional ones, see [18] for more details. However, when considering the mappings F + g, with a non-smooth single-valued part g in infinite dimensions, it seems to be necessary to combine Brouwer's fixed point theorem (or Glicksberg's extension of Kakutani's fixed point theorem) and the iterative procedure (A) (or Ioffe's regularity criterion (E) described below), see [11,12,14].
(E) Application of general Ioffe-type regularity criteria. This approach, extensively used in [38], is presented in this note. It encapsulates both the iterative procedure as in (A) and Ekeland's variational principle from (B). In fact, one can use any of these two methods to get these criteria, see [14, Proposition 2.1] going back to [29] for the first and [38, Theorem 2.45] going back to [34] for the latter. Similarly to (B) and (C), the key staring point is to choose an appropriate underlying lower semi-continuous function, see [38, pp. 60-65] for several criteria formulated in terms of various functions described below. Given a single-valued continuous mapping f from a complete metric space (X, d) into a metric space (Y, ̺), one applies Theorem 1.1 to the function ϕ := ̺(y, f (·)), where y is the right-hand side of the equation f (x) = y the solution of which we are searching for; hence no completeness assumption is imposed on the target space, cf. the setting (S 2 ). There are several ways of handling set-valued mappings. Assume that a right-hand side y ∈ Y is given and we want to find an x ∈ X solving the inclusion F (x) ∋ y. First, one can apply the criterion for single-valued mappings to the restriction of the canonical projection from X × Y onto Y to the graph of F , that is, the assignment gph F ∋ (u, v) −→ v ∈ Y , e.g., see [34,Proposition 3]; arriving at the statement in the setting (S 1 ). Equivalently, one can apply Theorem 1.1 to the function gph F ∋ (u, v) −→ ϕ(u, v) := ̺(y, v). In [14], it was demonstrated that, in this way, one gets short and transparent proofs of well-known regularity statements by using three ingredients only: the definition of the approximating object (e.g., the one-sided directional derivative, the graphical derivative, the contingent variation, etc.), the triangle inequality, and the criterion.
Second, in the setting (S 2 ), one could consider the function X ∋ u −→ ϕ(u) := dist(y, F (u)) which, unfortunately, is not lower semi-continuous in general. To cure this drawback, the authors work with a lower semi-continuous envelope of ϕ which brings an extra intricacy to the proof.
Third, a much less explored choice is the distance function generated by the graph of F , that is, the function X ∋ u −→ ϕ(u) := dist (u, y), gph F , which is Lipschitz continuous with the constant 1. This originates in the notion of graphical regularity introduced by L. Thibault [42], which is equivalent to the property of metric regularity. The strength of this approach can be seen from [28] by M. Fabian, A.D. Ioffe, and J. Revalski, where a separable reduction of local metric regularity is established. We demonstrate that, in this way, one can avoid conditions based on local slopes as well as lower semi-continuous envelopes when proving stability of metric regularity under additive perturbations in the setting (S 2 ).
To conclude general comments, we express a personal authors' impression. Whenever one reads a proof relying on slope-based conditions, there is a place in the proof, where one can directly apply a suitable regularity criterion to get a shorter transparent proof, that is, the following detour does not seem to be necessary: Ekeland's variational principle =⇒ the error bound property =⇒ the slope-based condition for regularity =⇒ a particular regularity statement.
This note is organized as follows. In Section 2, we recall various versions of regularity on a fixed set and provide sufficient conditions for these properties in settings (S 1 ) and (S 2 ), respectively. In Section 3, we apply general criteria to show the stability of regularity under additive perturbations by particular single-valued as well as set-valued mappings. In the last section, we briefly discuss several possible generalizations and provide an outlook for future work.
Notation and terminology. We use the convention that inf ∅ := ∞ and sup ∅ := 0, because we work with non-negative quantities; and that 0 · ∞ = 1. If a set is a singleton we identify it with its only element, that is, we write a instead of {a}. By f : X → Y we denote a single-valued mapping f acting from a set X into another set Y ; while F : X ⇒ Y denotes a mapping from X into Y which may be set-valued, i.e., the correspondence determined by a subset of X × Y called the graph of F and denoted by gph F ; the value of F at an x ∈ X is the set F (x) := {y ∈ Y : (x, y) ∈ gph F }; the domain of F is the set dom F := {x ∈ X : F (x) = ∅}; and the inverse of F is the mapping Y ∋ y −→ {x ∈ X : y ∈ F (x)} =: In a metric space (X, d), by B X [x, r] and B X (x, r) we denote the closed ball and the open ball centered at an x ∈ X with a radius r ∈ (0, ∞]; in particular, B X [x, ∞] = B X (x, ∞) = X; the distance from a point x ∈ X to a set C ⊂ X is dist (x, C) := inf{d(x, u) : u ∈ C}; and the diameter of C is defined as diam C = sup{d(u, x) : u, x ∈ C}. The set C is said to be closed [complete] around an x ∈ C provided that there is an r > 0 such that the set We consider only the vector (linear) spaces X the scalar field of which is R.
for each x, u ∈ U; to have Aubin property around a point (x,ȳ) ∈ gph F with the constant ℓ provided that there are α > 0 and β > 0 such that F has Aubin property on B X (x, α) × B Y (ȳ, β) with the constant ℓ; and to be Hausdorff-Lipschitz on a set C ⊂ X with the constant ℓ provided that F has Aubin property on C × Y with the constant ℓ.

General Regularity Criteria
2.1. Regularity on a fixed set. There are several distinct concepts of regularity on a prescribed set. Given a set W ⊂ X × Y , we denote by W X and W Y the projections of W onto X and Y , respectively, defined by W X := {x ∈ X : (x, y) ∈ W for some y ∈ Y } and W Y := {y ∈ Y : (x, y) ∈ W for some x ∈ X}; and for a fixed y ∈ Y , we set W ·,y := {x ∈ X : (x, y) ∈ W }. Let us start with two basic concepts in the spirit of [25, p. 328].
Definition 2.1. Let (X, d) and (Y, ̺) be metric spaces. Consider a κ > 0, nonempty sets V ⊂ Y and W ⊂ X × Y , and a set-valued mapping F : X ⇒ Y . The mapping F is said to be (I) V −restrictedly regular on W with the constant κ provided that Given a point (x,z) ∈ gph F , the mapping F is said to be regular around (x,z) with the constant κ provided that there is a neighborhood W of (x,z) in X × Y such that F is regular on W with the constant κ.
Remark 2.2. Let us emphasize that one has to impose additional assumptions on V , W , and F in order to obtain non-trivial properties. For example, (I) holds for each F with F (x) ∩ V = ∅ for each x ∈ W X ; and the same is true in (II) when dom F ∩ W X = ∅. Obviously, if F is regular on W with the constant κ, then F is V −restrictedly regular on W with the constant κ for each non-empty V ⊂ Y . On the other hand, F is V −restrictedly regular on W with the constant κ provided that the mapping F (x) : There is a substantial difference between properties (I) and (II) as the following easy but leading example shows.
In [36], A.D. Ioffe defined an abstract version of regularity involving not only the set W as above but also containing a reference point and a radius of a ball around this point as extra parameters; unifying in this way almost all regularity concepts. However, we prefer a top-to-bottom approach, that is, to introduce these extra parameters at the time when (semi-)local statements are considered, in order to keep the presentation as transparent as possible. The property (II) covers, in particular, the so-called γ-regularity and Milyutin regularity. Definition 2.4. Let (X, d) and (Y, ̺) be metric spaces. Consider constants δ > 0 and κ > 0, a non-empty set W ⊂ X × Y , a set-valued mapping F : X ⇒ Y , and a function γ : ; then we speak about Milyutin regularity. (2) Any F having property in (a) has property in (b) for any δ ∈ (0, 1/κ]; and the same is true for Milyutin regularities. Note that, setting γ(x) := ∞, x ∈ W X , property in (a) is nothing else but regularity of F on W with the constant κ from (II). However, we prefer to consider (a) as a particular case of (II) because the reversed process would give weaker sufficient conditions in the following subsections.

2.2.
Mappings with a complete graph. A stronger concept of regularity than (II) in the setting (S 1 ) was studied in [36], see also Section 4. Therefore we focus on the restricted regularity and start with single-valued mappings.
Proposition 2.6. Let (X, d) be a complete metric space and (Y, ̺) be a metric space. Consider a κ > 0, non-empty sets V ⊂ Y and W ⊂ X × Y , and a continuous single-valued mapping g : X → Y , defined on whole X and such that g(W X ) ∩ V = ∅. Then g is V -restrictedly regular on W with the constant κ provided that for each u ∈ X and each y ∈ W Y such that 0 < κ̺(y, g(u)) ≤ κ̺(y, g(x)) − d(u, x) for some x ∈ W ·,y with g(x) ∈ V , there is aû ∈ X, better than u, satisfying κ̺(y, g(û)) < κ̺(y, g(u)) − d(û, u).
Remembering that x ∈ W ·,y is such that g(x) ∈ V and that y ∈ W Y , using (6), we infer that 0 < κ̺(y, g(u)) ≤ κ̺(y, g(x)) − d(u, x). Hence the assumption containing (5), yields aû ∈ X such that d(û, u) < κ̺(y, g(u)) − κ̺(y, g(û)). This in combination with (7) implies that d(û, u) < κ̺(y, g(u))−κ̺(y, g(û)) ≤ κ̺(y, g(û))+d(û, u)−κ̺(y, g(û)) = d(u,û), a contradiction. Hence y = g(u). Now, we use the "projection trick" described in the introduction. Let (X, d) and (Y, ̺) be metric spaces and let λ > 0. Denote by ξ a (compatible) metric on X × Y defined by Proposition 2.7. Let (X, d) and (Y, ̺) be a metric spaces. Consider constants λ > 0 and κ > λ, non-empty sets V ⊂ Y and W ⊂ X × Y , and a set-valued mapping F : X ⇒ Y such that gph F is complete and F (W X ) ∩ V = ∅. Let us equip X ×Y by the metric ξ in (8). Then F is V −restrictedly regular on W with the constant κ provided that for each (u, v) ∈ gph F and each y ∈ W Y such that Proof. We follow [36,Proposition 7] and the proof of it. Let X := gph F and We apply Proposition 2.6 with (X, d) := ( X, ξ) and a continuous mapping (9), which is (5) with d := ξ for our g. Applying Proposition 2.6, we get that In the literature, the second inequality in the constraint on (u, v) and y is commonly omitted, that is, only v = y is used. However, this immediately excludes the possibility to get effective statements for W X = {x}, that is, for the property of semiregularity, see [15] for a detailed treatment and a list of relevant references.
The set Γ represents the set of the admissible right-hand sides y for which we want to solve the inclusion F (x) ∋ y; while Λ, ranging between {ȳ} and F (x), determines the strength of the estimates on the distance fromx to the solution set F −1 (y). Taking Γ := B Y (ȳ, r/κ) and Λ := {ȳ}, with fixed r > 0 and y ∈ F (x), we arrive at [15, Proposition 4.2(i)].

2.3.
Mappings with a complete domain and a closed graph. Let (X, d) and (Y, ̺) be metric spaces and let κ > 0. Denote by ω a (compatible) metric on X × Y defined by We start with Ioffe-type conditions for regularity in (II) in the setting (S 2 ). Proposition 2.9. Let (X, d) be a complete metric space and (Y, ̺) be a metric space. Consider constants κ > 0 andκ ≥ κ, a set W ⊂ X × Y , and a set-valued mapping F : X ⇒ Y , with a closed graph, such that dom F ∩ W X = ∅. Let us equip X × Y by the metric ω in (11). Then F is regular on W with the constant κ provided that for each u ∈ X and each y ∈ W Y such that for some x ∈ W ·,y , there is aû ∈ X, better than u, satisfying Proof. Put α := κ/κ ∈ (0, 1]. Fix any (x, y) ∈ W . If y ∈ F (x), then we are done. Assume that y / ∈ F (x). The function (X, d) ∋ u −→ ϕ(u) := dist (u, y), gph F is Lipschitz continuous with the constant 1 and ϕ(u) ≤ κ dist y, F (u) for each u ∈ X. Also ϕ(x) > 0 since the graph of F is closed. Theorem 1.1, with d := α d, yields a u ∈ X such that (14) ϕ By (14), we have In particular, . To get the inequality in (3) with κ :=κ, it suffices to show that u ∈ F −1 (y). Assume, on the contrary, that y / ∈ F (u). Since (x, y) ∈ W , we have y ∈ W Y and x ∈ W ·,y . As the graph of F is closed, (16) yields that which is (12). Thus there is aû ∈ X such that α d(û, u) < ϕ(u) − ϕ(û). This in combination with (15) implies that a contradiction. Hence y ∈ F (u).
The above statement is a quantitative non-local version of the sufficiency part of [28,Theorem 2], where the second inequality in (12) is omitted, and covers γ-regularity and, in particular, Milyutin regularity; while the corresponding statement in the setting (S 1 ), with W := U × V , is [37, Theorem 3.1].
Corollary 2.10. Let X, Y , κ,κ, W , F , and ω be as in Proposition 2.9. Consider a function γ : X → [0, ∞] which is positive on dom F ∩ W X . Then F is γregular on W with the constantκ provided that for each u ∈ X and each y ∈ W Y such that 0 <κ dist (u, y), gph F < κ γ(x) − d(u, x) for some x ∈ W ·,y , there is aû ∈ X, better than u, satisfying (13).
Pick any u ∈ X and any y ∈ Ω Y ⊂ W Y such that (12) holds true for some x ∈ Ω ·,y ⊂ W ·,y . Then Thus there is aû ∈ X satisfying (13). Proposition 2.9 says that F is regular on Ω with the constantκ, that is, F is γ-regular on W with the constantκ.
In case of Milyutin regularity, the above statement provides a conclusion for the interior points of W X only. Consider a set U ⊂ X with a non-empty interior, a non-empty set V ⊂ Y , and a point (x,ȳ) ∈ X × Y . The choice W := U × {ȳ} is allowed and provides conditions for the property called subregularity while the choice W := {x} × V yields a void statement, that is, the property of semiregularity is not covered. On the other hand, Proposition 2.9 gives an analogue of Corollary 2.8.
Corollary 2.11. Let (X, d) be a complete metric space and (Y, ̺) be a metric space. Consider constants κ > 0 andκ ≥ κ, a non-empty set Γ ⊂ Y , a point x ∈ X, and a set-valued mapping F : X ⇒ Y , with a closed graph, such that dom F ∋x. Let us equip X×Y by the metric ω in (11). Assume that for each u ∈ X and each y ∈ Γ such that 0 < dist (u, y), gph F ≤ κ dist y, F (x) − κ κ d(u,x), there is aû ∈ X, better than u, satisfying (13). Then (10), with κ :=κ and Λ := F (x), holds true.

Perturbation Stability
3.1. Mappings with a complete graph. In [36], there is studied the stability of a stronger version of regularity than (II), cf. Section 4, under single-valued additive perturbations in the setting (S 1 ). Therefore we focus on the stability of the regularity in (I) under particular set-valued additive perturbations. We start with a semi-local result having several seemingly different reincarnations. In the proof, we shall employ a truncated version of the so-called epigraphical multifunction from [31] which allows to avoid the assumption that the sum under consideration has a closed graph.
Thereforev ∈ E(û,ŵ). It remains to prove (19). By (22), we get that Multiplying this by λ and taking into account also (22), we infer thatκ̺(y, v) > max{d(û, u), ̺(ŵ, w)/ℓ, λ̺(v, v)} and that which both together reveal that (19) holds true. By Proposition 2.7, for each (x, w, y) Corollary 3.2. Let (X, d) be a complete metric space and (Y, ̺) be a linear complete metric space with a translation-invariant metric. Consider constants κ > 0 and ℓ > 0 such that κℓ < 1 and set-valued mappings F , G : X ⇒ Y having closed graphs. If F is regular on X × Y with the constant κ and G is Hausdorff-Lipschitz on X with the constant ℓ, then F + G is regular on X × Y with any constantκ > κ/(1 − κℓ).
Second, we derive an improved version of [1,Theorem 3.2]. Note that the original slope-based proof is rather complicated, e.g., the EVP is used twice.
Instead of the condition on the diameter one can assume the so-called local sum-stability introduced by M. Durea and R. Strugariu [27].
Using the above notion, we obtain [31, Theorem 8]: Corollary 3.7. Let (X, d) be a metric space and (Y, ̺) be a linear metric space with a translation-invariant metric. Consider constants κ > 0 and ℓ > 0 such that κℓ < 1, a point (x,z,w) ∈ X × Y × Y , and set-valued mappings F , G : X ⇒ Y , withz ∈ F (x) andw ∈ G(x), such that the sets gph F and gph G are complete around (x,z) and (x,w), respectively. If F is regular around (x,z) with the constant κ, G has Aubin property around (x,w) with the constant ℓ, and the pair (F, G) is sum-stable around (x,z,w), then F +G is regular around (x,z +w) with any constantκ > κ/(1 − κℓ).

3.2.
Mappings with a complete domain and a closed graph. In this subsection, we investigate the stability of the second type of regularity under additive perturbations. We start with single-valued perturbations.
Theorem 3.8. Let (X, d) be a complete metric space and (Y, ̺) be a linear metric space with a translation-invariant metric. Consider constants κ > 0 and ℓ > 0 such that κℓ < 1, a non-empty set Ω ⊂ X × Y , a function γ : X → [0, ∞] which is both positive and Lipschitz continuous on Ω X with the constant 1, a setvalued mapping F : X ⇒ Y with a closed graph, and a single-valued mapping g : X → Y which is continuous on X. Assume that the set is such that dom F ∩ W X is non-empty. If F is γ-regular on Ω with the constant κ and g is Lipschitz continuous on Ω X with the constant ℓ, then F +g is γ-regular on W with any constantκ > κ/(1 − κℓ).
We immediately obtain the stability of Milyutin regularity.
Theorem 3.9. Let (X, d) be a complete metric space and (Y, ̺) be a linear metric space with a translation-invariant metric. Consider positive constants r, ε, κ, and ℓ such that κℓ < 1, non-empty open sets U ⊂ X and V ⊂ Y , a setvalued mapping F : X ⇒ Y with a closed graph, and a continuous single-valued mapping g : X → Y such that the set is non-empty. If F is Milyutin regular on U ×V with the constants r and κ and g is Lipschitz continuous on U with the constant ℓ, then for eachκ > κ/(1−κℓ) the mapping F + g is Milyutin regular on W ε with the constants δ := min{κr, ε, 1}/κ andκ.
Then Ω X = U. Let W be defined by (26). Asκδ ≤ 1, we have B X (x, γ(x)) ⊂ B X (x, dist(x, X \ U)) ⊂ U for each x ∈ U. Hence Theorem 3.8 implies that F + g is γ-regular on W with the constantκ. Note that W ε ⊂ W becauseκδ ≤ ε and that dist(x, X \ (W ε ) X ) ≤ dist(x, X \ U) for each x ∈ X because (W ε ) X ⊂ U. Therefore F + g is Milyutin regular on W ε with the constants δ andκ.
The above statement is a quantitative version of [40,Theorem 4.2]; and if r := 1/κ and ε = 1, it reduces to [40,Theorem 4.3]. Two separate proofs of these statements in [40] are based on two different slope sufficient conditions (with two independent proofs again). Despite this, both the proofs are slightly longer, request the use of sequences, and calculating several limits, etc. On the contrary, we use very definitions of the properties of F , g, and ̺; the triangle inequality; and the criterion. Now, we prove an analogue of Theorem 3.1 having several seemingly different reincarnations again.
It is well-known that the second parameter r can be neglected in the conclusion of Theorem 3.10 in exchange for a possibly smaller neighborhood of the reference point, cf. [38, Exercise 2.12].
If ̺(y, v) < r, then (x, y) ∈ Ω; and we get the same inequality immediately.
Second, we derive an analogue of Corollary 3.4 in the setting (S 2 ).
If G(x) = {w} we cover [33,Corollary 4.5]. Note that Corollary 3.4 and Corollary 4.5 in [33] are consequences of two general approximation results with two separate proofs based on two different slope sufficient conditions (with two independent proofs again); and request also F to have a closed graph and Y to be a normed space. The following elementary example, see also [25,Example 5I.1] for an excessively elaborated one, shows a well-known fact that the sum of two set-valued mappings, one of which is locally regular while the latter is locally Hausdorff-Lipschitz, does not need to be regular around the reference point; in particular, [33,Corollary 4.8] is wrong.
Remember that given a β > 1 = diam G(x), there are no α > 0 and κ > 0 such that F is regular on (−α, α) × (−β, β) with the constant κ. Hence the condition on the diameter in Corollary 3.4 and Corollary 3.13 cannot be improved.
Finally, we obtain an analogue of Corollary 3.7, that is, of [31,Theorem 8] in the setting (S 2 ). Corollary 3.15. Let (X, d) be a complete metric space and (Y, ̺) be a linear metric space with a translation-invariant metric. Consider constants κ > 0 and ℓ > 0 such that κℓ < 1, a point (x,z,w) ∈ X × Y × Y , and set-valued mappings F , G : X ⇒ Y , withz ∈ F (x) andw ∈ G(x), such that the sum H := F + G has a closed graph. If F is regular around (x,z) with the constant κ, G has Aubin property around (x,w) with the constant ℓ, and the pair (F, G) is sumstable around (x,z,w), then H is regular around (x,z +w) with any constant κ > κ/(1 − κℓ).
Proof. Letκ > κ/(1 − κℓ) be arbitrary. Find a β > 0 such that F is regular on B X (x, β) × B Y (z, 2β) with the constant κ and that G has Aubin property 5. One can consider the set W in the last displayed inequality as the graph of another mapping G : X ⇒ Y . We arrive at the (equivalent) property: for each x ∈ dom G we have that (37) dist x, F −1 (y) ∩ G −1 (y) ≤ κ inf{̺(y, v) : (y, v) ∈ G(x) × F (x)}.
In particular, if this holds for an x ∈ dom G ∩ dom F , then the set of the coincidence points Ξ := {ξ ∈ X : F (ξ) ∩ G(ξ) = ∅} of F and G is nonempty and we have an estimate to the distance from x to Ξ. In [3], results on coincidence points of two (set-valued) mappings can be found, which can be proved either by the iterative procedure or by using the EVP. Let us emphasize that this approach, in contrast to Theorem 3.1, requests that one of the mappings F and G has a complete graph while the other has a closed graph only, see also [24,Theorem 3] and compare it with Corollary 3.2. Roughly speaking, the criterion for the property in (36) is obtained by applying the EVP to the function gph F ∋ (u, v) −→ ϕ(u, v) := ̺(y, v) + dist((u, y), W ), with a given y ∈ W Y , provided that F has a complete graph [36]. If W := gph G, then the roles of F and G can be interchanged which does not effect (37). However, any serious discussion on the relation between these two concepts takes several pages, see also the umbrella theorem [37, Theorem 6.1] corresponding to the setting (S 1 ).