Some related classical results
An open mapping f : X → Y between topological spaces is defined at a point x if, for every open set V containing x, there is an open set W in Y containing f(x) such that W is a subset of f(V) The Banach's open mapping principle provides a sufficient condition for a continuous linear operator to be classified as an open mapping.
Theorem 1.1.1 (Banach–Schauder’s open mapping theorem [13,77]) Let X, Y beBanach spaces and A : X → Y be a continuous linear operator onto Y, that is
A(X) = Y Then A is an open mapping.
Remark 1.1.1 A is an open mapping if and only if there exists r > 0 such that rBY ⊂A(BX).
Definition 1.1.1 (Banach constant [48]) Let X, Y be Banach spaces and
A: X → Y be a continuous linear operator The quantity
C(A) = sup{r ≥0 : rB Y ⊂ A(B X )} is called the Banach constant of A.
From the Banach’s open mapping theorem, we obtain the metric regularity of the linear single-valued mapping F.
Proposition 1.1.1 Let X, Y be Banach spaces and F : X → Y be a continuous linear operator Then the following statements are equivalent:
(ii) F is an open mapping.
(iii) There exists k >0 such that d(x, F −1 (y))≤ k.d(y, F(x)), ∀x ∈X, y ∈Y (1.1)
Proof The implication (i)⇒(ii) is obvious by the Banach’s open mapping theorem.
To demonstrate the implication (ii) ⇒ (i), consider an arbitrary element y in Y If y equals zero, then it follows that 0 = F(0) is contained in F(X) For the case where y is non-zero, select a positive r such that the norm of y is bounded by r, ensuring that y belongs to the image of F applied to the ball around X Consequently, there exists a z in the ball around X such that the norm of y is equal to F(z), leading to the conclusion that y can be expressed as a transformation of z, thereby proving Y is a subset of F(X) and confirming that F is surjective For the transition from (ii) to (iii), given that F is an open mapping, there exists a radius r such that the ball around Y is included in the image of the ball around X under F By selecting arbitrary elements x in X and y in Y, if y equals F(x), the condition (iii) is satisfied for all k In cases where y does not equal F(x), the difference can be expressed in terms of a point in the image of F, establishing that y can be represented as a transformation involving F and confirming the relationship outlined in (iii).
It follows that y = F(u) +F(x) =F(u+x), so u+x∈F −1 (y) One has d(x, F ư1 (y))≤ kxưu+xk ≤ kuk
To show (iii)⇒ (ii), we assume that there existsk >0 such that (iii) occurs Then, from (1.1), by taking x= 0 and y ∈ k 1 BY, we have d(0, F −1 (y))≤kd(y, F(0)), so kF −1 (y)k ≤ kkyk ≤ k 1 k = 1 In consequence, F −1 (y) ∈ B X implies
1 kB Y ⊂F(B X ) Thus, F is an open mapping.
The Lyusternik–Graves theorem demonstrates the metric regularity of a differentiable nonlinear single-valued mapping F A single-valued mapping F is considered continuously Fréchet differentiable when its Fréchet derivative ∇F(x) exists and is a continuous function.
Theorem 1.1.2 (Lyusternik–Graves theorem [25]) Let X, Y be Banach spaces and F : X → Y be a continuously Fr´echet differentiable mapping The following assertions are equivalent:
(i) There exist τ, ε >0 such that d(x, F −1 (y))≤ τ d(y, F(x)), ∀(x, y) ∈B(¯x, ε)×B(¯y, ε) with y¯=F(¯x).(ii) F 0 (¯x) is surjective.
Basic tools from variational analysis and nonsmooth analysis
Ekeland’s variational principles
Ekeland's variational principle, introduced in 1974, is a fundamental technique in variational analysis with significant applications across various fields such as optimization, nonsmooth analysis, economics, and control theory This principle serves as a crucial tool in our research.
Let \( f \) be an extended real-valued function that can take on both real values and ±∞ The domain of \( f \), denoted as \( \text{dom} f \), consists of all \( x \in X \) for which \( |f(x)| < ∞ \) The epigraph of \( f \), represented as \( \text{epif} \), includes pairs \( (x, r) \in X \times \mathbb{R} \) where \( f(x) \leq r \) A function \( f \) is considered proper if its domain is non-empty, and it is lower semicontinuous at a point \( x_0 \in \text{dom} f \) if \( f(x_0) \leq \liminf_{x \to x_0} f(x) \) Furthermore, \( f \) is classified as lower semicontinuous if it meets this criterion at every point in its domain Conversely, \( f \) is defined as upper semicontinuous if the function \( -f \) is lower semicontinuous.
Let us recall that a lower semicontinuous function on a compact metric space
The compactness of the set X is essential for the infimum of a lower semicontinuous function f to be achieved at a specific point ¯x; without it, a minimizer generally does not exist However, by the definition of the infimum, there is always a sequence {x n } n∈ N where f(x n ) approaches the infimum When the infimum is finite, it can be concluded that for any ε > 0, there exists a point ¯x such that f(¯x) is less than the infimum of f plus ε.
The Ekeland variational principle demonstrates that in a complete space, even without compactness, a slight perturbation of a function can yield a minimizer that is near a given point.
Theorem 1.2.1 ([16, 36]) Let (X, d) be a complete metric space and let f : X → R ∪ {+∞} be a proper lower semicontinuous function bounded from below Suppose that ε > 0 and z ∈X satisfy f(z) 0 and z ∈X satisfy f(z) 0 there exists y ∈X such that
Subdifferentials and some calculus rules
The notion of gradientof a function f : X → Ron a Banach spaceX, denoted
The gradient of a function \( f \), denoted as ∇f, is essential across various scientific disciplines Geometrically, it represents the slope of the tangent line to the function at a specific point \( \bar{x} \) In the case of a convex function, this tangent line remains entirely below the epigraph of the function A function \( f: X \to \mathbb{R} \cup \{+\infty\} \ defined on a vector space is classified as convex if it satisfies the inequality \( f(\lambda x + (1 - \lambda) y) \leq \lambda f(x) + (1 - \lambda) f(y) \) for all \( x, y \) in the domain of \( f \) and for \( \lambda \) in the interval \( (0, 1) \).
The concept of a subgradient extends the idea of a gradient for convex functions Specifically, a subgradient at a point ¯x represents the slope of a continuous affine function that lies beneath the epigraph of the convex function f and intersects it at ¯x The collection of all subgradients at this point forms the subdifferential of the function f.
In convex functions, the subdifferential serves as an effective alternative to the gradient, existing even at points of non-differentiability It shares key characteristics with the differential, including adherence to the Fermat Principle.
The condition "0 ∈ ∂f(x) iff x is a minimizer of f" highlights the significance of exact calculus rules, such as the sum rule, which states that "∂(f + g)(x) = ∂f(x) + ∂g(x)." While many functions in optimization are convex, there are instances where generalized subdifferentials are necessary for nonconvex functions Various types of generalized subdifferentials are tailored to specific contexts, and in this discussion, we will focus on the Fréchet subdifferential, which serves as a natural extension of the classical Fréchet derivative and subdifferential in convex analysis, particularly for nonsmooth functions.
The Fréchet subdifferentials, introduced over forty years ago, gained significant attention with the advent of “fuzzy rules” in the 1980s These subdifferentials have proven to be valuable tools for analyzing nonsmooth functions within Asplund spaces, a crucial subset of general Banach spaces This framework is broad enough to encompass various variational and optimization problems encountered in practical applications For further insights, readers can refer to the works of Mordukhovich and Kruger.
X into an extend real line R∪ {+∞}, finite at ¯x.
∂fb (¯x) x ∗ ∈X ∗ : lim inf x→¯ x f(x)−f(¯x)− hx ∗ , x−xi¯ kx−xk¯ ≥0
(1.2) is called the Fr´echet subdifferential of f at ¯x Its elements are sometimes referred to as Fr´echet subgradients.
In some cases, it is convenient to use the following modification of subdifferentials depending on a parameter ε≥0
∂b ε f(¯x) x ∗ ∈X ∗ : lim inf x→¯ x f(x)−f(¯x)− hx ∗ , x−xi¯ kx−xk¯ ≥ −ε
∂b ε f(¯x) is called the Fr´echet ε-subdifferential of f at ¯x If ε = 0, the set above coincides with the subdifferential defined by (1.2).
Thelimiting subdifferential(or,Mordukhovich subdifferential)at ¯xis constructed as the limit of Fr´echet ε-subdifferential, i.e.,
∂b ε f(x), where for a mapping F : X ⇒X ∗ , the notation
The notation \( x^* \) with \( x^* k \in F(x_k) \) for all \( k \in \mathbb{N} \) represents the sequential Painlevé–Kuratowski upper/outer limit concerning the norm topology of \( X \) and the weak* topology of \( X^* \) In this context, \( x \to f(\bar{x}) \) indicates that for a function \( f: X \to \mathbb{R} \), \( f(x) \) converges to \( f(\bar{x}) \) as \( x \) approaches \( \bar{x} \) Similarly, \( x \to \Omega(\bar{x}) \) signifies that \( x \) converges to \( \bar{x} \) with \( x \) belonging to the subset \( \Omega \subset X \) Additionally, the symbol \( F: X \rightharpoonup Y \) denotes that \( F \) is a set-valued mapping between metric spaces.
X, Y”, that is a correspondence associates every x a set F(x), possibly empty.
A Banach space is classified as an Asplund space if every convex continuous function is generically Fréchet differentiable Key characterizations of Asplund spaces include the fuzzy sum rule for Fréchet subdifferentials and the exact characterization for limiting subdifferentials, as established by researchers Fabian and Mordukhovich.
Lemma 1.2.1([56]) LetX be an Asplund space, and letf 1 , , f n : X →R∪ {+∞}, and any x 0 ∈ domf 1 ∩ ã ã ã ∩ domf n such that f 1 is lower semicontinuous and f 2 ,ã ã ã , f n are Lipschitzian in a neighbourhood of x 0 If f 1 + f 2 + ã ã ã + f n attains a local minimum at x 0 , then for any ε > 0, there exist x i ∈ x 0 + εB X , with |f i (x i )−f i (x 0 )|< ε, i ∈ {1,ã ã ã , n}, such that
Lemma 1.2.2 ([56]) Let X be a Banach space, and f 1 , f 2 : X → R∪ {+∞} be proper lower semicontinuous functions Let x 0 ∈ domf 1 be such that f 2 is locallyLipschitz at x 0 Then the following statements hold
(i) ∂(fˆ 1 +f 2 )(x 0 ) = ˆ∂f 1 (x 0 )+f 2 0 (x 0 ) whenf 2 is continuously Fr´echet differentiable at x 0 , where f 2 0 (x 0 ) denotes the derivative of f 2 at x 0
(ii) If X is an Asplund space then ∂(f 1 +f 2 )(x 0 ) ⊂∂f 1 (x 0 ) +∂f 2 (x 0 ).
The following lemma gives a chain rule for the Fr´echet subdifferential of the composition of a differentiable and a lower semicontinuous function.
Lemma 1.2.3 ([62]) Let X be an Asplund space If f : X → R ∪ {+∞} is a lower semicontinuous function and γ is differentiable on (0,+∞) then for every x∈domf such that f(x)> 0, one has
Coderivatives of set-valued mappings
Introduced by Mordukhovich in 1980, the coderivative concept serves as a general duality framework for the classic derivative in mappings between Banach spaces It is a valuable tool for advancing the dual approach in optimization and equilibrium problems For an in-depth exploration of these concepts and their applications, readers are encouraged to consult Mordukhovich's standard book.
Let X, Y be metric spaces For every set-valued mapping F : X ⇒ Y, we associate two sets, the graph and the domain:
TheinverseofF is the mappingF −1 : Y ⇒ X defined byF −1 (y) ={x∈ X : y ∈ F(x)}. Then,
A set-valued mapping F is closed valued if for any x the set F(x) is closed;
A mapping F is considered closed if its graph forms a closed set At a specific point on the graph, F is termed locally closed if there exists a closed neighborhood around that point, where the intersection with the graph remains closed Furthermore, F is classified as locally closed if this condition holds true for every point on the graph.
Let Ω be a nonempty subset of a real Banach space X Given x ∈ Ω and ε≥ 0 The set of ε-normals to Ω at x is given by
Nb ε (x; Ω) :( x ∗ ∈ X ∗ : lim sup u →x Ω hx ∗ , u−xi ku−xk ≤ε
When ε equals zero, the elements of the Fréchet normal cone, denoted as Nb(x; Ω), represent the Fréchet normals to the set Ω at the point x If the point x is not part of the set Ω, then Nb(x; Ω) is defined as an empty set Notably, Nb(x; Ω) corresponds precisely to the Fréchet subdifferential of the indicator function i Ω associated with the set Ω.
Let ¯x∈Ω One calls x ∗ ∈ X ∗ a Mordukhovich (or, limiting/ basic) normal to Ω at ¯ x if there are sequences ε k ↓0, x k → Ω x¯ and x ∗ k ω
→ x ∗ such that x ∗ k ∈Nb εk (x k ; Ω) for all k ∈ N The collection of such normals
Nb ε (x; Ω) is said to be the Mordukhovich (or limiting /basic) normal cone to Ω at ¯x Put
For given (¯x,y)¯ ∈ GraphF, we define the Fr´echet coderivative of F at (¯x,y)¯ as a multifunction Db ∗ F(¯x,y) :¯ Y ∗ ⇒ X ∗ with the values
It follows from the definition thatDb ∗ F(¯x,y)(y¯ ∗ ) =∅for ally ∗ ∈Y ∗ if (¯x,y)¯ ∈/ GraphF. Similarly, theMordukhovich coderivative (limiting coderivative)ofF at (¯x,y)¯ ∈ GraphF is a set-valued mapping D ∗ F(¯x,y) :¯ Y ∗ ⇒ X ∗ defined by
We put D ∗ F(¯x,y)(y¯ ∗ ) = ∅ for all y ∗ ∈ Y ∗ if (¯x,y)¯ ∈/ GraphF The relationships between coderivatives and derivatives of single-valued differentiable mappings are established by Mordukhovich ([56], Theorem 1.38) If f : X → Y is Fr´echet differentiable at ¯x, then
If, moreover, f is strictly Fr´echet differentiable at ¯x, then
A function f is considered strictly Fréchet differentiable at a point ¯x if it is Fréchet differentiable at that point and there exists a neighborhood around ¯x Within this neighborhood, for every point x, the equation f(x+h)−f(x)−f 0 (¯x)h equals r(x, h)khk holds true, where the norm kr(x, h)k approaches zero as both x approaches ¯x and h approaches zero.
Duality mappings
The concept of duality mapping, initially introduced by Beurling and Livingston in a geometric context, was later slightly generalized by Asplund, who characterized these mappings through the subdifferential of convex functions For further reading, standard texts by Cioranescu and Chidume provide comprehensive insights into this topic.
Definition 1.2.1 ([21]) Let φ : R+ → R+ be a given continuous and strictly increasing function such that φ(0) = 0 and lim t→+∞φ(t) = +∞ The mapping
J φ (x) :={u ∗ ∈ X ∗ : hx, u ∗ i=kxkku ∗ k;ku ∗ k=φ(kxk)} (1.5) is called the duality map with function φ, where X is any normed space In the particular case φ(t) =t, the duality map J φ is called the normalized duality map.
As a consequence of the following lemma, we will work on normed linear spaces to ensure that for each x ∈X, J φ (x) is not empty.
Lemma 1.2.4 In a normed linear space X, for every function φ, J φ (x) is not empty for any x in X.
In this proof, we establish that for x = 0, there exists u* = 0 in J φ(x) For non-zero x in X, the expression xφ(kxk) is also non-zero According to the Hahn–Banach theorem, we can find an x* in X* such that kx* k = 1 and hx*, xi = kxk By defining u as xφ(kxk), we derive that ku* k = φ(kxk)kx* k = φ(kxk) Consequently, we find that hx, u* i = hx, φ(kxk)kx* ki = hxφ(kxk), kx* ki = φ(kxk)hx, x* i.
= φ(kxk)kxk= kxkφ(kxk)kx ∗ k. Therefore, u ∗ ∈ J φ (x).
Lemma 1.2.5 ([21]) Let φ be a function as in Definition 1.2.1 and à(t) Z t 0 φ(s)ds.
Then à is a convex function on R+.
Theorem 1.2.3 ([6]) If J φ is a duality mapping with function φ, then
Li and Mordukhovich introduced extended versions of the duality mapping concept, termed q-duality mapping and its normalized enlargements, to establish adequate coderivative conditions for H¨older metric subregularity.
In this subsection, we present these concepts for the more general cases in which the function à is differentiable in R + Then from (1.5), the à-duality mapping
J à (x) ={à 0 (kxk)z ∗ : z ∗ ∈ J(x)} for all x∈ X\{0}, where the duality mapping J : X ⇒ X ∗ for a Banach space X is defined by J(0) =S X ∗ , and
J(x) ={x ∗ ∈X ∗ : kx ∗ k= 1 and hx ∗ , xi=kxk} whenever x∈ X\{0}.
In particular, in the case of X being a Hilbert space, we have J(x) x kxk if x6= 0.
Remark 1.2.1 Note that the set J(x), for every x ∈ X, defined as above agrees with to the convex subdifferential of the norm function k ã k at x, i.e.,
J(x) =∂k ã k(x) ={x ∗ ∈X ∗ : kuk ≥ kxk+hx ∗ , u−xi, for all u∈X}.
Given ε ≥0, the normalized ε-enlargement of the à-duality mapping is
Strong slope and some error bound results
The concept of infinitesimal in metric spaces is represented by a quantity known as (strong) slope, which was introduced by De Giorgi, Marino, and Tosques in 1980 This notion has since become a valuable tool in various areas of analysis within metric spaces, particularly in characterizing error bounds—an essential regularity property that estimates the distance of a point from the solution set Building on the theory of error bounds, researchers Ioffe and Azé et al have demonstrated that the existence of an error bound can be leveraged to derive metric regularity results for multifunctions.
Definition 1.2.2 ([22,65]) Let X be a metric space, and f : X → R∪ {+∞}. The symbol [f(x)]+ stands for max(f(x),0).
(i) The quantity defined by |∇f|(x) = 0 if x is a local minimum of f; otherwise
|∇f|(x) = lim sup u→x,u6=x f(x)−f(u) d(x, u) is called the local slope of the function f at x∈ domf.
[f(x)−f(u)] + d(x, u) is called the nonlocal slope of the function f at x∈ domf.
For x /∈ domf, we set |∇f|(x) =|Γf|(x) = +∞ Obviously, |∇f|(x) ≤ |Γf|(x) for all x∈ X.
In geometric terms, the slope of a function at a specific point represents the steepest descent of that function from that point Specifically, when X is a normed space and the function f is Fréchet differentiable at x, the magnitude of the gradient |∇f|(x) is equal to the norm of the derivative kf'(x)k for all x in X.
Example 1.2.1 Let f : R →R be given by f(x) :
Since f attains the (global) minimum atx = 0, |∇f|(0) =|Γf|(0) = 0 For x 6= 0, f is differentiable, so we have |∇f|(x) = 2x if x > 0 and |∇f|(x) = 1 if x < 0. Furthermore, if x >0, one has
|xưu|, if u < 0; thus, in the case of u≥ 0,
|xưu| ≤ sup u6=x, u 0 \) such that the inequality \( \tau d(x, [f \leq \alpha]) \leq (f(x) - \alpha)^{+} \) holds for all \( x \in X \).
In [10], Az´e and Corvellec provide a characterization of global error bounds for lower semicontinuous functions defined on a complete metric space in terms of slope.
Theorem 1.2.4 ([10]) Let X be a complete metric space, f : X → R∪ {+∞} be a lower semicontinuous function, and α ∈R, β ∈R∪ {+∞} with α < β. a) Assume that [f < β]6= ∅ and τ := inf
Then τ d(x,[f ≤γ])≤ (f(x)−γ) + , ∀α ≤ γ < β,∀x∈ [f < β] (1.7) b) Conversely, assume that (1.7) holds for some τ >0, then inf
In the context of applying error bounds to the metric regularity of multifunctions, a condition for establishing a linear error bound in parametric scenarios is presented in [12, Theorem 3.1] Prior to outlining this theorem, we revisit the definition of epi-upper semicontinuity for a family of functions.
Definition 1.2.3 ([12]) Let X be a metric space and P be a topological space.
We say that a function f : X ×P → R ∪ {+∞} is epi-upper semicontinuity at (¯x,p)¯ ∈X ×P if f˜(¯x,p)¯ ≤f(¯x,p),¯ where ˜f(¯x,p) is defined by¯ f˜(¯x,p) := sup¯ ε>0 inf δ>0 sup p∈B(¯ p,δ) inf x∈B(¯ x,ε) f(x, p).
In fact, f˜(¯x,p) = sup¯ ε>0 lim sup p→ p ¯ inf x∈B(¯ x,ε)f(x, p)
Theorem 1.2.5 ([12,64]) Let X be a complete metric space, P be a metric space, f :X×P →R∪ {+∞}, (x 0 , p 0 ) ∈X×P with f(x 0 , p 0 ) = 0, b∈(0,+∞], V 0 be a neighborhood of x 0 , N 0 be a neighborhood of p 0 , and σ >0 Assume that
(1) f is epi-upper semicontinuous at (x 0 , p 0 ),
(2) for every p∈N 0 , f p := f(ã, p) is lower semicontinuous on V 0 ,
Then, for every ε >0 small enough, there exists a neighborhood N of p 0 such that
[f ≤ 0]∩B(x 0 , ε) 6=∅ for every p∈ N (1.8) and f(x, p)−c ≥σd(x,[f p ≤ c]∩B(x 0 ,2ε)) (1.9) for every 0≤ c < b, every p∈N, and every x∈ [c < f p < b]∩B(x 0 , ε).
Remark 1.2.2 In the similar argument, we can see that the conclusion of Theorem 1.2.5 still holds if the local slope of f p at x, |∇f p |(x), in the assumption
(3) is replaced by the global slope of f p at x, |Γf p |(x).
To apply the aforementioned result, it is essential to estimate the slope The following proposition provides a lower estimation of the local slope through the use of a subdifferential operator, which is defined in the context of this discussion (see, e.g., [10,48]).
In this article, we explore a Banach space \(X\) and the associated metric \(d^*\) of its topological dual space \(X^*\) We introduce an abstract subdifferential operator \(\partial^*\) that relates to any lower semicontinuous function \(f: X \to \mathbb{R} \cup \{+\infty\}\) and any point \(x \in X\) The subset \(\partial^* f(x)\) of the dual space \(X^*\) is defined as empty (\(\partial^* f(x) = \emptyset\)) when \(x\) is not in the domain of \(f\), and it adheres to specific properties that govern its behavior.
(P2) if g : X → R is convex and Lipschitz continuous, and if ¯x ∈ domf is a local minimum point of f + g then, for every ε > 0 there exist x, y ∈ X, x ∗ ∈ ∂ ∗ f(x), and y ∗ ∈ ∂ ∗ g(y) such that kx−xk ≤¯ ε, ky−xk ≤¯ ε, f(x) ≤f(¯x) +ε, and kx ∗ +y ∗ k ≤ε.
In the case of X being an Asplund space, the Fr´echet and Mordukhovich subdifferentials satisfy (P1) and (P2) (see, e.g., [59]).
Proposition 1.2.1 ([9]) Let X be a Banach space and ∂ ∗ be a subdifferential operator such that properties (P1) and (P2) are satisfied Then, for every lower semicontinuous function f : X → R∪ {+∞} and every x ∈X, we have
As an immediate consequence of Theorem1.2.4and Proposition1.2.1, we thus have the following corollary:
Corollary 1.2.1 ([9]) LetX be a Banach space and ∂ ∗ be a subdifferential operator such that (P1) and (P2) hold, f : X → R ∪ {+∞} be a lower semicontiuous function, and α ∈ R, β ∈ R∪ {+∞} with α < β and [f < β] 6= ∅ Assume that, for some τ >0, one has inf x∈[α0 such that
The upper bound surF(¯x|y) of such¯ r is the modulus of the surjection of F near (¯x,y) If no such¯ r, ε exist, we set surF(¯x|y) = 0;¯
(ii) metrically regular near (¯x,y) if there are¯ K >0, ε >0 such that d(x, F −1 (y)) ≤Kd(y, F(x)) if d(x,x)¯ < ε, d(y,y)¯ < ε.
The infimum of such K denoted by regF(¯x|y) is the¯ modulus of metric regularity of F near (¯x,y) If no such¯ K, ε exist, we set regF(¯x|¯y) =∞;
(iii) pseudo-Lipschitz or has the Aubin property near (¯x,y) if there are¯ K, ε > 0 such that d(y, F(x))≤ Kd(x, u) if d(x,x)¯ < ε, d(y,y)¯ < ε, y ∈F(u), d(u,x)¯ < ε.
The infimum of such K denoted by lipF(¯x|¯y) is the Lipschitz modulus of F near (¯x,y) If no such¯ K, ε exist, we set lipF(¯x|y) =¯ ∞.
All three properties of the definition refer to the same phenomenon More specifically, one has the next proposition.
Proposition 1.3.1([51]) LetF be a set-valued mapping between any pair of metric spaces X, Y and (¯x,y)¯ ∈ GraphF Then the following properties are equivalent:
(i) F is open at a linear rate near (¯x,y).¯
(ii) F is metrically regular near (¯x,y).¯
(iii) F −1 has the Aubin property near (¯y,x).¯
Moreover, under the convention that 0ã ∞ = 1, surF(¯x|y)¯ ãregF(¯x|y) = 1;¯ regF(¯x|y) = lip¯ F −1 (¯y|¯x).
Nonlocal metric regularity
Local regularity refers to the fulfillment of regular inequalities within a neighborhood of a point ¯x, whereas in the nonlocal scenario, the regularity domain is fixed and cannot be altered The nonlocal approach serves as a robust tool for addressing various practical issues, including the establishment of existence theorems within a specified set.
In the context of metric spaces X and Y, with open subsets U of X and V of Y, we define a function F mapping from X to Y Additionally, we consider extended real-valued functions γ and δ defined on X and Y, which take non-negative values, potentially including infinity, within the respective open sets U and V.
Definition 1.3.2 ([51], nonlocal regularity properties) We say that a set-valued mapping F is
(i) γ-open (or γ-covering) at a linear rate on U ×V if there is a r >0 such that
B(F(x), rt)∩V ⊂F(B(x, t)) if x ∈ U and t < γ(x) Denote by sur γ F(U|V) the supremum bound of such r If no such r exists, set sur γ F(U|V) = 0 We shall call sur γ F(U|V) the modulus of γ-openness of F on U ×V.
A function F is considered γ-metrically regular on the set U × V if there exists a constant K > 0 such that the distance between a point x in U and the preimage of a point y in V, denoted as F^(-1)(y), is less than or equal to K times the distance between y and F(x), given that Kd(y, F(x)) is less than γ(x) The infimum of such constants K is referred to as reg γ F(U|V), and if no such constant exists, it is assigned a value of infinity This concept is termed the modulus of γ-metric regularity of F on U × V.
The δ-pseudo-Lipschitz property on U × V is defined by the existence of a constant K > 0, such that for any x in U and y in V, the inequality d(y, F(x)) ≤ Kd(x, u) holds when Kd(x, u) < δ(y) and y belongs to F(u) The notation lip δ F(U|V) represents the infimum of such constants K; if no such K can be found, then lip δ F is set to infinity This concept is referred to as the δ-Lipschitz modulus of F on U × V.
The functions γ and δ are not necessary for local regularity since the neighborhood around (¯x, y) can be adjusted freely However, for the fixed sets ¯U and V, these functions are crucial as they define the extent of our reach from any point to the solution set F −1 (y) Consequently, they are referred to as regularity horizon functions Utilizing these functions allows us to manage the distance between the exact solution and the approximate solution, ensuring that this distance can be made arbitrarily small.
The equivalence theorem is a key outcome of regularity theory, highlighting the metric equivalence among the three defined regularity phenomena.
Theorem 1.3.1 ([51]) The following three properties are equivalent for any pair of metric spaces X, Y, any F : X ⇒ Y, any U ⊂ X and V ⊂ Y and any extended real-valued function γ : X → R+ which is positive on U:
(i) F is γ-open at a linear rate on U ×V.
(ii) F is γ-metrically regular on U ×V.
(iii) F −1 has γ-pseudo-Lipschitz property on V ×U.
Moreover, under the convention that 0ã ∞ = 1, sur γ F(U|V)ãreg γ F(U|V) = 1, reg γ F(U|V) = lip γ F −1 (V|U).
Nonlinear metric regularity
In Chapter 2 of Ioffe's work, regularity models are discussed where the radius of balls centered at x in X does not correspond to the radius of the neighborhood of F(x) covered by the image of these balls under the function F Nevertheless, applying techniques analogous to those used in the linear case yields consistent results for these models This involves substituting rt in Definition 1.3.2 (i) with a specific nonlinear function, which is a nonnegative modulus function that is strictly increasing on the interval [0, +∞), satisfies à(0) = 0, and approaches +∞ as t approaches +∞.
Definition 1.3.3 ([51]) Given an F : X ⇒ Y, where as usual X andY are metric spaces Let as before, U ⊂ X and V ⊂ Y be open sets, γ(ã) be a function on X which is positive onU, andδ(ã) be a function onY which is positive onV Assume finally that we are given three gauge functions à(ã), ν(ã) and η(ã).
(i) F is γ- open on U ìV with functional modulus not smaller than à(ã) if the inclusion
(ii) F is γ-metrically regular on U ×V with functional modulus not greater than ν(ã) if the inequality d(x, F −1 (y))≤ ν(d(y, F(x))) holds whenever x∈ U, y∈ V, ν(d(y, F(x))) < γ(x).
(iii) F is δ- Hăolder on U ìV with functional modulus not greater than η(ã) if the inequality d(y, F(x))≤η(d(x, u)) holds provided x∈U, y ∈V ∩F(u) and η(d(x, u))< δ(y).
The equivalence theorem is also extended to this model.
Theorem 1.3.2 ([51]) Let à be given a gauge function The following properties are equivalent:
(i) F is γ-open on U ìV with functional modulus not smaller than à.
(ii) F is γ-metrically regular on U ×V with functional modulus not greater than à −1
(iii) F −1 is γ-Hăolder on V ìU with functional modulus not greater than à −1
In this article, we focus on a specific category of gauge functions, particularly the significant class of nonlinear functions represented as à(t) = rt^k This exploration enables us to advance the understanding of nonlinear regularity of order k.
Fix as usual an F : X ⇒ Y with complete graph, open sets U ⊂ X, V ⊂ Y and a function γ on X, and let k ≥ 1.
Definition 1.3.4 ([51], regularity of order k) We say that
(i) F is γ-open of order k on U ×V if there is an r > 0 such that
Denote by sur (k) γ F(U|V) the upper bound of such r If no such r exists, set sur (k) γ F(U|V) = 0 We shall call sur (k) γ F(U|V) the γ-rate of surjection of order k of F on U ×V.
(ii) F is γ-metrically regular of order k on U ×V if there is a κ >0 such that d(x, F −1 (y))≤κ(d(y, F(x))) 1 k if x∈ U, y∈ V, κd(y, F(x))< γ(x).
Denote by reg (k) γ F(U|V) the infimum of such κ If no such κ exists, set reg (k) γ F(U|V) = ∞ We shall call reg (k) γ F(U|V) the modulus (or rate) of γ-metric regularity of order k of F on U ×V.
(iii) F −1 is γ-H¨older of order k on V ×U if there is a κ > 0 such that d(x, F −1 (y))≤κ(d(y, v)) 1 k if (x, y) ∈U ×V, v ∈F(x), κd(y, v) < γ(x).
Denote by hol (k) γ F −1 (V|U) the infimum of such κ If no such κ exists, set hol (k) γ F −1 (V|U) =∞ We shall call hol (k) γ F −1 (V|U) the γ-H¨older modulus of order k of F −1 on V ×U.
Theorem 1.3.2 also guarantees an equivalence of the phenomena: γ-open of order k of F, γ-metric regularity of order k of F and γ-H¨older of order k of F −1 as well as a connection of the rates sur (k) γ F(U|V), reg (k) γ F(U|V) and hol (k) γ F −1 (V|U).
We define the γ-rate of surjection of order k at (¯x,y), denoted as ¯ sur (k) γ F(¯x|y), as the upper bound of ¯ r > 0 Additionally, the γ-rate of metric regularity of order k at (¯x,y), referred to as reg¯ (k) γ F(¯x|y), is the infimum of ¯ κ ≥ 0 Furthermore, the γ-Hölder modulus of order κ at (¯x,y), labeled as hol¯ (k) γ F −1 (¯y|x), is defined as the infimum of ¯ κ ≥ 0, ensuring that certain conditions hold for some ε > 0 within neighborhoods U = B(¯x, ε) and V = B(¯y, ε), with γ(x) ≡ ∞ The relationship between these local regularity rates is established through a specific proposition.
Proposition 1.3.2 ([51]) For any set-valued mapping F : X ⇒ Y and any (¯x,y)¯ ∈ GraphF, one has reg (k) F(¯x|y)¯ ã(sur (k) F(¯x|y))¯ k 1 = 1; reg (k) F(¯x|y) = hol¯ (k) F −1 (¯y|x).¯
Metric regularity criteria in metric spaces
Ioffe outlines the necessary and sufficient conditions for regularity, specifically through Theorem 1.4.2 and Theorem 1.4.3 Notably, Theorem 1.4.2 will be fundamental in deriving various qualitative and quantitative characterizations of regularity.
Throughout this section and subsequent chapters we work with the following two functions associated with F : X ⇒ Y and y ∈ Y as follows: ψ(x, y)
Let ξ > 0 be given, we define ξ-metric on X×Y by d ξ ((x, y),(x 0 , y 0 )) = max{d(x, x 0 ), ξd(y, y 0 )}.
Assume that U ⊂X is an open set and γ is a positive function on U, we set
Theorem 1.4.1 establishes a general regularity criterion involving open sets U in X and V in Y, along with a function F that has a complete graph in the product metric It specifies conditions under which, for any y in V and for any pair (x, v) in U and F(x) respectively, if the distance d(y, v) is less than rγ(x), there exists a different pair (u, w) in the graph of F This pair satisfies the inequality ψy(u, w) ≤ ψy(x, v) - rdξ((x, v), (u, w)), indicating a structured relationship between these points.
Then for any (x, v) ∈ GraphF with x ∈ U with d(y, v) ≤ rt < rγ(x), there is an u ∈ B(x, t) such that y ∈ F(u) In particular, F is γ-regular on U × V with sur γ F(U|V) ≥ r, provided the assumption of the theorem is satisfied for any y ∈ V.
Conversely, if F is γ-regular on U × V, then for any positive r < sur γ F(U|V), any ξ > 0 such that rξ < 1, any x ∈ U, v ∈ F(x) and any y ∈ V satisfying
0< d(y, v)< rγ(x), there is a pair (u, w) ∈GraphF different from(x, v) such that (1.10) holds.
According to our established criteria, the sufficient condition for γ-regularity necessitates that x is an element of U γ, while the necessary condition only requires x to belong to U This distinction between necessary and sufficient conditions presents challenges in verifying the regularity of the set-valued mapping F To eliminate this discrepancy, we introduce the regularity horizon function γ(x) := m(x), where m(x) is defined as d(x, X\U).
Milyutin regularity is a key concept related to the horizon function, indicating that m(x) is greater than zero for all x in an open subset U of X, which implies that U m equals U Consequently, employing the horizon function m(x) in a nonlocal context allows us to focus on specific points in the analysis.
U We only mention in the following Milyutin regularity when considering issues related to nonlocal metric regularity.
Theorem 1.4.2 establishes the first criterion for Milyutin regularity, stating that if the graph of F is complete, a necessary and sufficient condition for F to be Milyutin regular on U × V with sur m F ≥ r is the existence of a ξ > 0 Specifically, for any x ∈ U, v ∈ F(x), and y ∈ V where 0 < d(y, v) < rm(x), there must exist a point (u, w) in Graph F, distinct from (x, v), such that the condition in (1.10) is satisfied.
The following second criterion gives us a characterization of Milyutin regularity via the lower semicontinuous envelope function ϕ y (x).
Theorem 1.4.3 establishes a second criterion for Milyutin regularity in the context of a complete metric space X, where U is a subset of X and V is an open set in Y It states that a function F: X ⇒ Y, possessing a closed graph, is Milyutin regular on U × V with sur m F(U|V) greater than r if, for every x in U and y in V satisfying 0 < ϕ y (x) < rm(x), there exists a point u in X such that ϕ y (u) is less than or equal to ϕ y (x) minus rd(x, u).
The Milyutin regularity is beneficial not only in nonlocal contexts but also for determining the necessary and sufficient conditions for local regularity Specifically, when considering sets U = B(¯x, ε) and V = B(¯y, ε), the Milyutin regularity of the function F on the product space U × V is equivalent to the local regularity of F at the point (¯x, y)¯ within the graph of F This leads to the criteria established in Theorem 1.4.2.
Theorem 1.4.4 ([51], criteria for local regularity) Let F : X ⇒ Y and (¯x,y)¯ ∈ GraphF Then F is regular near (¯x,y)¯ with surF(¯x|y)¯ ≥ r > 0 if and only if one of the following two properties holds for U =B(¯x, ε), V =B(¯y, ε) with some ε > 0.
(i) GraphF is locally complete and there is a ξ > 0 such that for any x ∈ U, v ∈ F(x) and any y ∈ V, y 6= v there is a pair (u, w) ∈ GraphF such that (1.10) holds;
(ii) X is complete and for any x ∈domF ∩U and any y ∈ V, y /∈F(x) there is an u ∈domF such that (1.11) holds.
The general regularity criterion is extended to the nonlinear case, considering open sets U in X and V in Y In this context, γ(x) is defined as a Lipschitz function on X with a Lipschitz constant of 1, accompanied by a specified gauge function à(t).
Theorem 1.4.5 ([51]) Let F : X ⇒ Y with complete graph in the product metric
X × Y If there is a ξ > 0 such that for any x ∈ U γ , y ∈ V, v ∈ F(x) with
0< d(y, v)< à(γ(x)) there is a pair (u, w) ∈ GraphF, (u, w) 6= (x, v) such that à −1 (d(y, w))≤ à −1 (d(y, v))−d ξ ((x, v),(u, w)), (1.12) then F is γ-open on U ìV with functional modulus not smaller than à.
Conversely, F is γ-open on U ×V with functional modulus not smaller than à, then there is a ξ > 0 satisfyingξτ ≤ à −1 (τ) for allτ ∈[0, à(γ(x))]and all x ∈U such that (1.12) holds.
1.5 The infinitesimal criteria for metric regularity in metric spaces
The characterizations outlined in Theorems 1.4.1, 1.4.2, and 1.4.3 present a general framework that makes verifying the metric regularity of set-valued mappings F challenging To address this complexity, Ioffe, along with Ngai, Tron, and Théra, introduced infinitesimal characterizations that facilitate the calculation and estimation of regularity rates more effectively.
Given ρ ∈ R, α > 0, set [ρ] α = |ρ| α signρ and define the function [f] α by [f] α (x) = [f(x)] α We further define the slope of order k of f as the slope of [f] 1 k :
Using |∇ (k) ξ ψ y | the k-slope of ψ y with respect to the ξ-metric in X × Y, here ψ y (x, v) = d(y, v) + i Graph F (x, v), Ioffe [51] established the slope criterion for regularity of order k Recall that the indicator function of a set Ω⊂ X is i Ω (x)
Theorem 1.5.1 (Slope regularity criterion of order k [51]) Let F : X ⇒ Y with complete graph in the product metric X×Y, let U and V be open subsets ofX and
Y respectively, and let γ(ã) be a function on X with Lipschitz constant one which is positive on U Let finally k ≥1 Suppose there is an r > 0 such that
In particular, if for some (¯x,y)¯ ∈GraphF there is an ε > 0 such that (1.13) holds whenever d(x,x)¯ < ε, d(y,y)¯ < ε, v ∈F(x), d(y, v) < ε, then sur (k) F(¯x|y)¯ ≥ r.
Metric regularity on a fixed set: definitions and characterizations
This chapter introduces innovative nonlinear regularity models for set-valued mappings, expanding upon the foundational concepts established by Ioffe It explores characterizations of these regular models within complete metric spaces, utilizing variational analysis tools such as local slope, nonlocal slope, and coderivative The discussion is grounded in the research presented in previous studies.
In this chapter, we focus on metric spaces, often emphasizing their completeness, as Ekeland’s principle necessitates it We denote the distance using the symbol d(ã,ã), which should not lead to any confusion regarding the space in consideration Additionally, we follow the standard convention of assigning a distance of +∞ to the empty set.
Definitions and equivalence of the nonlinear metric regularity concepts 39
This section explores the concept of metric regularity in relation to a modulus function \( à \), which is defined as a non-negative, strictly increasing function on the interval \([0, +\infty)\) Key properties of this function include that as \( t \) approaches 0 from the right, \( à(t) \) converges to \( à(0) = 0 \), and as \( t \) approaches infinity, \( à(t) \) diverges to infinity Additionally, we introduce a gauge function \( γ : X \rightarrow \mathbb{R}^+ \cup \{+\infty\} \) We will also revisit Definition 1.3.3 of nonlinear metric regularity as proposed by Ioffe.
In the context of metric spaces X and Y, let U and V be open subsets of these spaces, and consider a function F mapping from X to Y A gauge function γ, which is positive on U, is used to define the concept of (à, γ)-metric regularity Specifically, F is considered (à, γ)-metrically regular on U ì V if there exists a positive real number κ such that the distance between a point x and the preimage of a point y under F is bounded by κ times the function à evaluated at the distance between y and F(x) This condition holds for all pairs (x, y) within U ì V, provided that the distance is less than γ(x).
In contrast to local regularity, the domains U and V play a crucial role in the definition provided An example illustrates that, in nonlocal scenarios, the regular domain cannot be altered arbitrarily.
Example 2.1.1 Let X =Y = R, F(x) = {x,3}, U = (0,1), V = (0,2) Then, F is 1-metrically regular on U ×V with modulus 1 but F is not γ-regular on U ×V 0 with V 0 = (0,3) for any γ.
Indeed, take x ∈ U, y ∈ V such that 0 < d(y, F(x)) < γ(x) Then, one has d(y, F(x)) < 1, which follows that d(y, F(x)) = |x− y| So, d(x, F −1 (y)) |x−y| = d(y, F(x)) It means that F is 1-metrically regular on U ×V However,
F is not γ-metrically regular on U × V 0 for any γ because for x ∈ U, there exists t > 0 such that 0 < x + t < 1, then (3 − t,3) ⊂ B(F(x), t) ∩ V 0 but
We can extend this definition to a general setW ⊂ X×Y Given now a subset
W ofX×Y For everyy ∈ Y, we associate it to the setW y ={x∈X : (x, y) ∈W}, and for every x ∈ X, we associate it to the set W x = {y ∈ Y : (x, y) ∈ W} Then, we denote P X W := ∪ y∈Y W y , and P Y W := ∪ x∈X W x It is easy to see that when
The relationship W = U × V indicates that the sets W y (where y ∈ V) and P X W correspond to U, while the sets W x (where x ∈ U) and P Y W correspond to V By appropriately modifying this definition, we can generalize Definition 2.1.1 of regularity to apply to any set W within the product space X×Y.
Definition 2.1.2 ([79]) Let X, Y be metric spaces, W be a subset of X×Y, and
F : X ⇒ Y Let γ be a gauge function on X which is positive on P X W F is said to be (à, γ)-metrically regular on W if there is a real number κ > 0 such that d(x, F −1 (y))≤ κà(d(y, F(x))), (2.1) for all (x, y) ∈W with
In fact, in the definition above, parameters ”κ” in (2.1) and ”κ” in (2.2) can be taken differently as shown in the following proposition.
Proposition 2.1.1 states that for metric spaces X and Y, and a subset W of the Cartesian product X × Y, a set-valued mapping F: X ⇒ Y is considered (à, γ)-metrically regular on W if there exist positive constants κ and r Specifically, for every pair (x, y) in W where the distance from y to F(x) is greater than 0 but less than γ(x), the distance from x to the inverse image of y under F is bounded by κ times the distance from y to F(x).
To establish the necessary and sufficient conditions, we observe that if there exist constants κ and r greater than zero, such that for all pairs (x, y) in the set W, the inequality 0 < rà(d(y, F(x))) < γ(x) holds, then it follows that d(x, F −1(y)) ≤ κà(d(y, F(x))) By defining ˜κ as the maximum of r and κ, we can derive Definition 2.1.2 with modulus ˜κ Specifically, for (x, y) in W satisfying 0 < κà(d(y, F˜(x))) < γ(x), it also leads to 0 < rà(d(y, F(x))) < γ(x) Consequently, we can conclude that d(x, F −1(y)) is bounded by κà(d(y, F(x))) and further by κà(d(y, F˜(x))).
This chapter presents an alternative version of the (à, γ)-metric regularity, which can be more convenient than Definition 2.1.2 This approach allows for flexibility in choosing the constant r in the gauge condition, differing from the modulus constant κ.
Definition 2.1.3 Let X, Y be metric spaces, W be a subset of X × Y, and
F : X ⇒Y Let γ be a gauge function on X which is positive on P X W F is said to be (à, γ)-metrically regular on W with constant κ > 0 if there is a real number r > 0 such that d(x, F −1 (y))≤ κà(d(y, F(x))) for all (x, y) ∈W with 0< rà(d(y, F(x))) < γ(x).
Note that this concept covers many well-known regularity models in the literature For instance, if in Definition 2.1.3,
(i) we take W = U × {y}¯ with ¯y ∈ Y and U ⊂ Y be an open of X, one has the following property: d(x, F −1 (¯y)) ≤κà(d(¯y, F(x))) for all x ∈ U satisfying 0 < rà(d(¯y, F(x))) < γ(x) which is called to be (à, γ)-subregularity;
In this article, we define the function \( à(t) = t^\alpha \) where \( \alpha \) is in the range \( (0,1] \) We consider the set \( W = (U \cap [\bar{x} + B(u, \delta)]) \setminus \{\bar{y}\} \), where \( (\bar{x}, \bar{y}) \) is a point in the graph of the function \( F \), \( \delta > 0 \), and \( u \) is an element of \( X \) Given a neighborhood \( U \) of \( \bar{x} \), we aim to establish the concept of directional Hölder subregularity of \( F \) in the direction of \( u \), as recently introduced by Ngai, Tron, and Tinh.
When r = κ, the version of (à, γ)-metric regularity on W in the sense of Ioffe is stated as follows.
Definition 2.1.4 Let X, Y be metric spaces, W be a subset of X × Y, and
F : X ⇒Y Let γ be a gauge function on X which is positive on P X W F is said to be (à, γ, κ)-metrically regular on W with constant κ > 0 if d(x, F −1 (y))≤ κà(d(y, F(x))) provided that (x, y) ∈W with 0< κà(d(y, F(x))) < γ(x).
Similar to the local case, we next introduce equivalent versions of the regularity such as (à, γ)-Hăolder property and (à, γ)-openness of set-valued mappings.
Definition 2.1.5 Let X, Y be metric spaces, W be a subset of X × Y, and
F : X ⇒Y Let γ be a gauge function on X which is positive on P X W. Denote W −1 := {(y, x) : (x, y) ∈ W} F −1 : Y ⇒ X is (à, γ)-Hăolder on W −1 with constant κ >0 if there is a real number r > 0 such that d(x, F −1 (y))≤ κà(d(y, v)) holds for all (x, y) ∈W, v ∈ F(x), and 0< rà(d(y, v)) < γ(x).
Definition 2.1.6 Let X, Y be metric spaces, W be a subset of X × Y, and
F : X ⇒Y Let γ be a gauge function on X which is positive on P X W F is (à, γ)-open on W with constant κ >0 if there is a real numberr > 0 such that the conclusion
B(F(x), à(rt))∩W x ⊂ F(B(x, κrt)) fullfils for all x∈P X W and 0 < t < γ(x).
The three properties mentioned are equivalent, as demonstrated in the subsequent proposition A comparable outcome regarding the (à, γ)-metric regularity on a box can be found in Ioffe's works, specifically in [16, Theorem 5.5.9] and [46, Theorem 1].
Proposition 2.1.2 Let F : X ⇒ Y and W be an open subset of X×Y Then the following statements are equivalent:
(i) F is (à −1 , γ)-open on W with constant κ.
(ii) F is (à, γ)-regular on W with constant κ −1
(iii) F −1 is (à, γ)-Hăolder on W −1 with constant κ −1
Proof Firstly, we shall prove (i) ⇒ (ii) Since F is (à −1 , γ)-open on W with constant κ then there exists r > 0 such that
Let (x, y) be an element of W such that the distance from y to F(x) is bounded by a function related to γ(x) This implies that the distance d(y, F(x)) is less than a certain threshold By introducing a small positive value ε and defining t accordingly, we ensure that t remains less than γ(x) Consequently, y is located within a specific neighborhood of F(x) and belongs to W This leads to the conclusion that there exists a point u in the vicinity of x such that y is the image of u under the function F As a result, the distance between x and the preimage of y under F can be controlled, ultimately yielding a bound that relates this distance to the distance from y to F(x) when ε approaches zero.
The implication (ii) ⇒(iii) is trivial To show (iii) ⇒ (i), by the hypothesis, there exists r > 0 such that d(y, F(x))≤κàd(x, u) holds for all (x, y) ∈W, y ∈ F(u) and 0 < ràd(x, u) < γ(x).
(2.4) Let now x ∈ P X W, 0 < t < γ(x), and let y ∈ W x and y ∈ B(F(x), à −1 (r −1 t)). Then (x, y) ∈ W and d(y, v) ≤ à −1 (r −1 t) for some v ∈ F(x) Since à is increasing, à(d(y, v))≤ à(à −1 (r −1 t)) =r −1 t It follows that rà(d(y, v))≤ t < γ(x) and (x, y) ∈ W, x ∈ F −1 (v) So, by (2.4), d(x, F −1 (y)) ≤κ −1 à(d(y, v))< κ −1 r −1 t. Hence, there exists u ∈ F −1 (y) such that d(x, u) < κ −1 r −1 t, that is y ∈ F(B(x, κr −1 t)) Consequently,
Remark 2.1.2 (i) In the case of γ ≡ ∞, we remove the gauge condition in the above definitions.
By substituting the modulus function with a linear function in the form of à(t) = t, we can redefine the concepts of γ-metrically regular and (γ, κ)-metrically regular, which are also referred to as Lipschitz-type metrically regular on a general set W ⊂ X × Y.
Characterizations of nonlinear metric
The following propositions give the necessary/sufficent conditions of γ-metric regularity through the nonlocal and local slope of the lower envelope function associated to the set-valued mapping F.
Let X, Y be metric spaces and the metric on X×Y be defined by d((x 1 , y 1 ),(x 2 , y 2 )) =d(x 1 , x 2 ) +d(y 1 , y 2 ), ∀(x 1 , y 1 ),(x 2 , y 2 )∈ X×Y.
Let F : X ⇒ Y be given Note that the distance function associated to
The function F, denoted as d(ã, F(ã)), is typically not lower semicontinuous, even when its graph is closed However, variational analysis often necessitates that the function in question be lower semicontinuous As a result, rather than utilizing the distance function directly, we frequently adopt the lower semicontinuous envelope, represented as ϕ(x, y), of the function defined by d(y, F(x)) This envelope is expressed mathematically for (x, y) in X × Y as ϕ F (x, y) := lim inf.
We use in what follows the function ϕ F y which is always lower semicontinous on X. Namely, the property of this function will be given in the next lemma.
Lemma 2.2.1 For each y ∈ Y, the lower semicontinuous envelop function ϕ F y (ã) associated to a set-valued mapping F is the largest lower semicontinuous function not greater than d(y, F(ã)).
The relationship between the closed set epiϕ F y (ã) and the lower semicontinuity of the function ϕ F y is established through the equation epiϕ F y (ã) = epid(y, F(ã)) This implies that ϕ F y (ã) is less than or equal to d(y, F(ã)) Assuming g is a lower semicontinuous function with g(ã) ≤ d(y, F(ã)), we find that the closed set epig(ã) contains epiϕ F y (ã), leading to the conclusion that g(ã) is also less than or equal to ϕ F y (ã).
As the approach to the local regularity in the papers [63,64,66,68], we make use of the following fact:
Fact 1 Let F : X ⇒ Y be a closed multifunction, i.e., its graph is a closed set in
If \( x \in F^{-1}(y) \), then \( 0 \leq \varphi_F^y(x) \leq d(y, F(x)) = 0 \), leading to \( \varphi_F^y(x) = 0 \) Conversely, if \( \varphi_F^y(x) = 0 \), there exists a sequence \( \{x_n\}_{n \in \mathbb{N}} \subset X \) converging to \( x \) such that \( d(y, F(x_n)) \) approaches 0 Consequently, we can find a sequence \( \{v_n\}_{n \in \mathbb{N}} \) where \( v_n \in F(x_n) \) and \( d(y, v_n) \to 0 \) Since the graph of \( F \) is closed, it follows that \( (x, y) \in \text{Graph} F \), implying \( x \in F^{-1}(y) \).
The following lemma permits us to examine the (à, γ)-metric regularity of a multifunction by using the lower semicontinuous envelope of the distance function to its images.
In the context of metric spaces X and Y, and a subset W of X×Y, a function F: X ⇒ Y is considered (à, γ)-metrically regular on W with a constant κ if there exist positive real numbers r and κ such that the inequality d(x, F⁻¹(y)) ≤ κ(à◦ϕ F y)(x) holds for all (x, y) in W, provided that 0 < r(à◦ϕ F y)(x) < γ(x) Conversely, if F is (à, γ)-metrically regular on W with the same constant κ, and if the set Wy is open for every y in Y while γ is lower semicontinuous, then there exists a positive number r such that the aforementioned inequality is satisfied.
Proof It is easy to see that if (2.6) fullfils, then F is (à, γ)-metrically regular on
W with constant κ Conversely, suppose that F is (à, γ)-metrically regular on W with constant κ, i.e., there exists a real number r >0 such that d(x, F −1 (y))≤ κà(d(y, F(x))), (2.7) for all (x, y) ∈W with 0< rà(d(y, F(x))) < γ(x).
Let (x, y) be an element of W where 0 < r(à ◦ ϕ F y )(x) < γ(x) Consider a sequence {x n } in X that converges to x, with the distance d(y, F(x n )) approaching ϕ F y (x) as n approaches infinity This implies that à(d(y, F(x n ))) converges to (à ◦ ϕ F y )(x) as n increases Due to the openness of W, it follows that (x n, y) is contained in W for sufficiently large n Therefore, for large n, we can conclude that the properties of the sequence are maintained within the set W.
0< rà(d(y, F(x n ))) < γ(x n ) It follows from (2.7) that d(x n , F −1 (y))≤ κà(d(y, F(x n ))).
Let n tends to ∞ in this inequality, one gets d(x, F −1 (y))≤ κ(à◦ϕ F y )(x).
By using the global slope, we establish the necessary/ sufficient conditions for the nonlinear regularity in the theorem below.
Theorem 2.2.1 Let X be a complete metric space, Y be a metric space, and
W ⊂X×Y be a nonempty subset Let F : X ⇒Y be a closed set-valued mapping. Let à: R+ → R+ be a continuous modulus function Then,
(i) Assume that γ is lower semicontinuous If W is open and F is (à, γ)- metrically regular on W with constant κ, then for each (x, y) ∈ W such that
(ii) Conversely, assume further that γ : X → R+ is a Lipschitz continuous function with constant 1 If there is a real number κ > 0 such that limδ↓0 inf
(2.8) then F is (à, γ)-metrically regular on W with constant κ.
If F is (à, γ)-metrically regular on W with a constant κ, then according to Lemma 2.2.2, there exists a positive number δ such that for any (x, y) in W, if 0 < δ(à◦ϕ F y )(x) < γ(x), it follows that d(x, F −1 (y)) ≤ κ(à◦ϕ F y )(x) For a fixed (x, y) and any arbitrary ε > 0, one can find a point u in F −1 (y) such that d(x, u) ≤ (κ + ε)(à◦ϕ F y )(x) Consequently, this leads to the relationship d(x, u) ≤ (κ + ε)(à◦ϕ F y )(x) = (κ + ε)[(à◦ϕ F y )(x) − (à◦ϕ F y )(u)].
|Γ(à◦ϕ F y )|(x) ≥ 1 κ+ε. Since ε > 0 is arbitrary, one gets
(ii) If (2.8) happens, then there is δ > 0 such that for every (x, y) ∈ X × Y satisfying 0 κ λ. Let now (x, y) ∈W be such that 0< τ à(d(y, F(x))) < γ(x) Then,
00 Then, from (ii) and the Lipschitz continuity of γ, it follows that
Thus, by the definition of λ, d(x, z) ≤ λ
Since (τ λ) −1 < κ −1 , one obtains from (iii) that Γf(z) = Γ(à◦ϕ F y )(z) ≤(τ λ) −1 < κ −1 (2.13)
One sees that (2.11),(2.12), and (2.13) contradict (2.10).
So, f(z) = 0 which is equivalent to ϕ F y (z) = 0 It implies that z ∈ F −1 (y). Then, by (ii), we have 0 ≤f(x)−(τ λ) −1 d(x, z) It follows that
Consequently, for all (x, y) ∈W with 0 < τ à(d(y, F(x))) < γ(x), one has d(x, F −1 (y))≤ d(x, z) ≤λτ à(d(y, F(x))).
Since τ λ is arbitrarily closed to κ, one obtains that d(x, F −1 (y))≤ κà(d(y, F(x))).
So, F is (à, γ)-metrically regular on W with constant κ The proof is completed.
From the estimation |Γ(à◦ϕy)|(x) ≥ |∇(à◦ϕy)|(x), the following immediate corollary gives us a sufficient condition for the (à, γ)-metric regularity by using the local slope.
Corollary 2.2.1 Let X be a complete metric space, Y be a metric space, and
W ⊂X×Y be a nonempty subset Let F : X ⇒Y be a closed set-valued mapping. Suppose that γ : X → R + is a Lipschitz function with constant 1 and à :R + → R + is a continuous modulus function If there exists κ > 0 such that limδ↓0 inf
> κ −1 then F is (à, γ)-metrically regular on W with constant κ.
Regarding Definition 2.1.4 of the (à, γ, κ)-metric regularity, we obtain the following theorem.
Theorem 2.2.2 With the assumptions as in Corollary 2.2.1, if there is a real number κ >0 such that
|Γ(à◦ϕ F y )|(x)> κ −1 ,∀x ∈(W y ) γ , y∈ P Y W,0< κ(à◦ϕ F y )(x) < γ(x), where (W y ) γ := ∪ x∈W y B(x, γ(x)), then F is (à, γ, κ)-metrically regular on W with constant κ.
Proof Let (x, y) ∈W, 0 < κà(d(y, F(x)))< γ(x) Then, 0 0 sufficiently small such that 0 < τ +ε < κ −1 Applying Ekeland’s variational principle to the lower semicontinuous function f(u) := (à◦ϕ F y )(u) on the complete metric space
X, we can find a point z ∈ X such that
We shall prove that f(z) = 0 Indeed, suppose that f(z) >0 Then, from (ii) and the Lipschitzness of γ, it follows that
By (i), one hasd(x, z)≤ τ τ +εγ(x) < γ(x) which impliesz ∈ (W y ) γ Moreover, one obtains from (iii) that|Γ(à◦ϕ F y )|(z) ≤τ+ε < κ −1 This contradicts the assumption.
So, f(z) = 0 which is equivalent to ϕ F y (z) = 0 It means that z ∈ F −1 (y) Then, by (ii), we have
0 =f(z) ≤f(x)−τ d(x, z), which yields τ d(x, z) ≤ f(x) It follows that d(x, F −1 (y))≤ τ −1 à(d(y, F(x))), and as τ is close arbitrarily to κ −1 , the theorem is proved.
The preceding theorem remains valid when substituting the global slope with the local strong slope, as noted in Corollary 2.2.1 Consequently, a direct corollary of Theorem 2.2.2 emerges by replacing the global slope with the local slope for the regular domain defined by the box-type form W = U × V ⊂ X × Y.
Corollary 2.2.2 With the same assumptions as in the preceding theorem, for a nonempty subset W := U ×V ⊂ X ×Y, setting U γ := ∪ x∈U B(x, γ(x)), if there exists κ >0 such that
|∇(à◦ϕ F y )|(x)≥ κ −1 , ∀x∈ U γ , y ∈ V,0< κ(à◦ϕ F y )(x) < γ(x), then F is (à, γ, κ)-metrically regular on U ìV with constant κ.
Remark 2.2.1 Note that by applying the similar arguments as in the above characterizations, with the gauge functionàdefined byà(t) =t, the characterizations for the Lipschitz-type metric regularity are also indicated.
The usefulness of the characterizations is demonstrated in the following example.
Example 2.2.1 Let X = Y = R, F(x) = x 3 , U = V = (−1,1), à(t) = t 1 3 , and γ(x) =|x| Then, F is (à, γ)-metrically regular on U ìV with modulus 1.
Indeed, due to F is a closed mapping and defined in finite space R, so the distance function (x, y) → d(y, F(x)) is lower semicontinuous and consequently, ϕ y (x) =d(y, F(x)) So, for everyy ∈(−1,1),x ∈U γ =∪ x∈(−1,1) B(x,|x|) = (−2,2) such that 0 < (à ◦ ϕ y )(x) < |x|, i.e., 0 < |y − x 3 | 1 3 < |x|, which follows that
Thus, F is (à, γ)-metrically regular on (−1,1)ì(−1,1) with modulus 1.
The next simple example shows that the inverse of Corollary 2.2.2 does not hold in general.
Obviously,F has closed graph Then, F is 1-metrically regular with modulusτ = 1, however, the condition (5.26) does not hold.
Indeed, one has that for x ∈ U, y ∈ V such that d(y, F(x)) < γ(x), i.e., d(y, F(x)) < 1 and thus, d(y, F(x)) = |x −y|, and so d(x, F −1 (y)) = |x− y| d(y, F(x)) It means that F is 1-regular with modulus τ = 1 However, taking y ∈ V, x ∈U 1 =∪ u 0 ∈U B(u 0 ,1) = ∪ u 0 ∈(0,1) (u 0 −1, u 0 + 1) = (−1,2), x < 0 such that
In Theorem 1.5.1, as referenced in [46, Theorem 3.1] by Ioffe, a similar sufficient condition for metric regularity on the box U × V is established This is achieved through the use of the function ψ y: X × Y → R ∪ {+∞}, which is defined as ψ y(x, v) = d(y, v) if v ∈ F(x), and +∞ otherwise.
To apply Ekeland’s variational principle to the function ψ y defined on the product space X × Y, it is essential that either the image space Y or the graph of the multifunction F is complete In contrast, the slope sufficient conditions discussed in previous theorems utilize the function ϕ F y, which is defined in a single variable x within space X, thereby eliminating the need for the completeness assumption on the image space Additionally, in certain cases, analyzing the slopes of the function ϕ F y may prove to be more advantageous than examining those of ψ y.
When GraphF is complete, a lower estimate for |∇(à◦ϕ F y )| via |∇(à◦ψ y )| is given in the following lemma.
Lemma 2.2.3 Let X, Y be metric spaces Let the product space X ×Y endowed with a metric denoted by d(ã,ã) such that d((x, y),(u, v))≥ d(x, u) for all (x, y),(u, v) ∈X ×Y.
Suppose that F : X ⇒ Y is a multifunction between metric spaces X, Y with complete graph Then for (¯x,y)¯ ∈ X×Y, with x¯∈domϕ F y ¯ , and for any continuous modulus function à : R + → R + , one has
Proof Pick any m > |∇(à◦ϕ F y ¯ )|(¯x), then there is ε ∈]0,1[ such that
Picking a sequence of positive reals (ε n ) → 0 + with ε n ∈]0, ε/4[, then there is a sequence ((x n , y n )) with (x n , y n ) ∈GraphF, such that x n ∈ B(¯x, ε 2 n ); d(¯y, y n ) → ϕ F y ¯ (¯x);
As ϕ F y ¯ (x) ≤ ψ y ¯ (x, v), for all (x, v) ∈GraphF, from (2.14), one has
In the context of the complete GraphF, we apply Ekeland’s variational principle to the function \( g_n(x, v) = (à◦ψ y ¯)(x, v) + md((x, v), (x_n, y_n)) \) This leads us to identify a pair \( (u_n, v_n) \) within GraphF, satisfying the condition \( d((u_n, v_n), (x_n, y_n)) \leq ε_n \) Consequently, it follows that \( g_n(u_n, v_n) \leq g_n(x, v) + (m + 1)ε_n d((x, v), (u_n, v_n)) \) for all \( x \in B(¯¯ x, ε) \) and \( (x, v) \in GraphF \).
(à◦ψ y ¯ )(u n , v n )−(à◦ψ y ¯ )(x, v)≤ (m+ (m+ 1)ε n )d((x, v),(u n , v n )), for all x ∈ B(¯¯ x, ε), (x, v) ∈ GraphF Hence, |∇(à◦ψ y ¯ )|(u n , v n ) ≤ m+ (m+ 1)ε n Since (x n ) →x, d(¯¯ y, y n ) → ϕ F y ¯ (¯x), and d((u n , v n ),(x n , y n ))≤ ε n , one has (u n )→ x¯ and d(¯y, v n ) → ϕ F y ¯ (¯x), therefore the lemma is proved 2
Characterizations of nonlinear metric
In the context of Fréchet smooth target spaces, we derive coderivative characterizations by providing a lower estimate for the strong slope of the composite function à◦ϕ F y at point x, utilizing the defined quantity τ(x, y) It's important to note that a Banach space X qualifies as Fréchet smooth if its norm exhibits Fréchet differentiability at every point in X excluding zero.
In an Asplund space \(X\) and a Fréchet smooth Banach space \(Y\), if there exists a differentiable modulus function on \((0,+\infty)\), then for any closed set-valued mapping \(F: X \Rightarrow Y\), it holds that for every pair \((x, y) \in X \times Y\) where \(y\) is not an element of \(F(x)\), certain conditions are met.
|∇(à◦ϕ F y )|(x) ≥ τ(x, y), (2.15) where τ(x, y) is defined by τ(x, y) := lim η↓0
à 0 (ky−vk)kx ∗ k: x ∗ ∈Db ∗ F(u, v) J à η (v−y) (u, v) ∈ GraphF, v 6=y u∈B X (x, η) ky −vk ≤ d(y, F(u)) +η d(y, F(u))≤ϕ F y (x) +η
In this proof, we consider a point (x, y) in the Cartesian product X × Y, where the image set F(x) is non-empty and y is not an element of F(x) We define m as the magnitude of the gradient of the function à◦ϕ F y at point x Due to the lower semicontinuity of ϕ F y and the concept of local slope, for any small ε within the interval (0, ϕ F y (x)), there exists a corresponding η in (0, ε) such that the inequality 2η + ε < ϕ F y (x) holds This ensures that the distance from y to F(u) is greater than zero for all u within a ball of radius 5η centered at x, confirming that y is not in F(u) Additionally, it follows that m + ε is greater than or equal to the difference between the values of (à◦ϕ F y) at x and any z in the neighborhood B(x, η), scaled by the distance kx - zk.
Take into account of the continuity of à, one can take 0< δ < η 2 such that à ϕ F y (x) +δ
Take u ∈ B(x, η 4 2 ), v ∈ F(u) such that ky −vk ≤ ϕ F y (x) +δ Since the function à is strictly increasing on R + one gets à(ky −vk)≤ à ϕ F y (x) +δ
. Then, à(ky−vk) ≤(à◦ϕ F y )(z) + (m+ε)kz−xk+η 2 ∀z ∈B(x, η).¯
Therefore, for every (z, w) ∈B¯(x, η)×Y one has à(kyưvk) ≤à(kyưwk) +δ Graph F (z, w) + (m+ε)kzưuk+ (m+ε)η 2
By applying Ekeland’s variational principle to the lower semicontinuous function h: (z, w) 7→ à(ky ưwk) +δ Graph F (z, w) + (m+ε)kzưuk on ¯B(x, η)×Y, we can select (¯u,v)¯ ∈ (u, v) + η 4 B X×Y with (¯u,v)¯ ∈ GraphF such that h(¯u,¯v)≤ h(u, v), i.e., à(ky ưvk) + (m¯ +ε)kuưuk ≤¯ à(kyưvk) ≤ (à◦ϕ F y )(x) +η 2
+(m+ε+ 4)ηk(z, w)−(¯u,v)k¯ attains a minimum on ¯B(x, η)×Y at (¯u,¯v) It follows that
Hence, by Lemma 1.2.1, we can find (u 1 , v 1 ) ∈B((¯u,¯v), η)∩GraphF;
(0,0) ∈(u ? 1 , v ? 1 ) + (u ? 2 , v ? 2 ) + (m+ε+ 5)ηB X ? ×B Y ? Note that since ky−v 1 k ≥ ky−vk − kv 1 −vk ≥ ϕ F y (x)−ε−2η > 0 and Y is a Fr´echet smooth Banach space then
∂b(à(ky− ãk+δ Graph F (ã,ã)) (u 1 , v 1 ) = [{0}ìà 0 (ky−v 1 k)J(v 1 −y)]+Nb(GraphF,(u 1 , v 1 )).
It follows that there exist z ? ∈J(v 1 −y),(u ? 3 , v 3 ? ) ∈B X ? ×B Y ? such that
4 ) ⊂B(x,5η), one has the following estimation kà 0 (ky −v 1 k)z ? + (m+ε+ 5)ηv 3 ? k ≥ kà 0 (ky−v 1 k)z ? k −(m+ε+ 5)η
(this holds because à 0 (ky − v 1 k) > 0 and η > 0 arbitrarily small), which makes sense to define x ? := −u ? 2 −(m+ε+ 5)ηu ? 3 kà 0 (ky−v 1 k)z ? + (m+ε+ 5)ηv 3 ? k and y ? := à 0 (ky−v 1 k)z ? + (m+ε+ 5)ηv 3 ? kà 0 (ky−v 1 k)z ? + (m+ε+ 5)ηv 3 ? k.
Then, one obtains x ∗ ∈Db ∗ F(u 1 , v 1 )(y ∗ ), (2.16) kx ∗ k ≤ m+ε+ (m+ε+ 5)η à 0 (ky−v 1 k)−(m+ε+ 5)η, (2.17) ky ? k= 1, and y ? := à 0 (ky−v 1 k)z ? + (m+ε+ 5)ηv ? 3 kà 0 (ky−v 1 k)z ? + (m+ε+ 5)ηv ? 3 k (2.18)
It derives that y ? ∈J à ξ (v 1 −y) (2.19) with ξ := (m+ε+ 5)η → 0 as η → 0.
On the other hand, d(y, F(u 1 ))≤ ky−v 1 k
Asε, η >0 are arbitrarily small, by combining relations (2.16)–(2.20), we complete the proof.
In case Y is an Asplund space but not necessary a Fr´echet smooth Banach space, one obtains a weaker estimation as follows.
In the context of Asplund spaces X and Y, if we consider a modulus function that is Fréchet differentiable and locally Lipschitz continuous on the interval (0, +∞), then for any closed set-valued mapping F: X ⇒ Y, it holds that for each pair (x, y) in the product space X×Y, where y is not an element of F(x), certain properties can be established.
à 0 (ky −wk)kx ∗ k: x ∗ ∈ Dˆ ∗ F(u, v)(J à η (w−y)) (u, v) ∈GraphF, u ∈BX(x, η) w ∈B Y (v, η) ky−vk ≤d(y, F(u)) +η, d(y, F(u))≤ ϕ F y (x) +η
Proof The proof of this Lemma is completely similar to Lemma 2.3.1 by applying Lemma 1.2.2 to the function
(z, w)7→ à(kyưwk)+δ Graph F (z, w)+(m+ε)kzưuk+(m+ε+1)ηk(z, w)ư(u 1 , v 1 )k with noting that the function k(w) := à(ky −wk) is locally Lipschitz continuous on Y.
By combining Theorem 2.2.1 and Lemma 2.3.1, 2.3.2, one obtains a characterization for (à, γ)-metric regularity on fixed sets via the coderivative.
Theorem 2.3.1 establishes that for Asplund spaces X and Y, with a differentiable and locally Lipschitz continuous modulus function à defined on (0, +∞), and a closed set-valued mapping F: X ⇒ Y, the presence of a Lipschitz function γ with a constant of 1 is essential Furthermore, it requires the existence of a positive real number κ, ensuring that the limit as δ approaches 0 of the infimum is well-defined.
If ∆(x, y) is defined as in Lemma 2.3.2, then F is (à, γ)-metrically regular on W with constant κ Moreover, if Y is a Fréchet smooth Banach space, the conclusion remains valid if ∆(x, y) is substituted with τ(x, y) as defined in Lemma 2.3.1.
Remark 2.3.1 (i) The sufficient conditions for the local metric regularity involving some coderivatives were well established in the literature (see, e.g.,
[11,39,48,56,63,66]) The preceding theorem gives a non-local version of sufficient conditions based on coderivatives.
(ii) In the case of X, Y being finite dimensional spaces, obviously the quantity τ(x, y) is determined more simply as follows. τ(x, y) = inf à 0 (ky −vk)kx ∗ k :x ∗ ∈D ∗ F(x, v) v−y kv−yk
, where P roj F (x) y is the projection of y on F(x) defined by
{z ∈ F(x) :ky−zk =d(y, F(x))}, andD ∗ F(x, v)(y ∗ ) denotes the limiting coderivative of F at (x, v) ∈ GraphF.
Below we give some illustrated examples.
Example 2.3.1 Let us consider F : R n ⇒ R defined by F(x) = f(x) + R+, where f : R n → R given by f(x) = kxk 2 , x ∈ R n Using Theorem 2.3.1, we shall verify that F is (à(t) = t 1/2 ,+∞)-regular on W = R n ì[0,+∞) with modulus τ = 1 Firstly, for the gauge γ(x) := kxk, for any (x, y) ∈ W with y /∈ F(x), i.e.,
0≤y < kxk 2 , ϕ y (x) 1/2 = (kxk 2 −y) 1/2 ≤ kxk, as P roj F (x) y =f(x), and
2(kxk 2 −y) 1/2 2kxk ≥ 1, for all (x, y) ∈W,with 0< ϕ y (x) 1/2 ≤ γ(x) =kxk.Thus according to Theorem 2.3.1,
The function F is (t 1/2, k ã k)-metrically regular on the set W with a modulus of 1, satisfying the inequality d(x, F −1 (y)) ≤ d(y, F(x)) for all pairs (x, y) in W where 0 < d(y, F(x)) < 1/2 < kxk Additionally, for pairs (x, y) in W where d(y, F(x)) ≥ 1/2 ≥ kxk, the condition holds as 0 is included in F −1 (y), leading to the conclusion that d(x, F −1 (y)) ≤ kx − 0k ≤ d(y, F(x)) 1/2 Therefore, it can be concluded that F is (t 1/2, +∞)-metrically regular on W.
Example 2.3.2 Consider the function f : R n → R defined by f(x)
Clearly this function is locally surjective around (0,0) ∈ R n × R We shall verify the following two statements.
(i) The function f is not (à, γ)-metrically regular around (0,0) for any modulus function à : R+ → R+, and any gauge function γ : R n → R+ being positive on a neighborhood of 0∈ R n
Indeed, for anyε >0,pick an integerksufficiently large such that 1/(2kπ) < ε. Consider the function α :R → R, defined by α(t)
Then α 0 (0) = 0, and for t >0, α 0 (t) = 2tsin(1/t)−cos(1/t); α 00 (t) = (2−1/t 2 ) sin(1/t)−(2/t) cos(1/t).
Since α 0 (1/(2kπ)) = −1 0, there is some ¯t ∈ (1/(2kπ + π/2),1/2kπ) such that α 0 (¯t) = 0 Moreover, since α 00 (t) f(¯x) =α(¯t), one has d(¯x, f −1 (y)) ≥min{(2kπ) −1 −¯t,¯t−(2kπ+π/2) −1 }.
For any sequence \( (y_j) \) converging to \( f(\bar{x}) \) with \( y_j > f(\bar{x}) \) for all \( j \), and for any modulus function \( \alpha: \mathbb{R}^+ \to \mathbb{R}^+ \), it follows that \( \lim_{j \to \infty} \alpha(|y_j - f(\bar{x})|) d(\bar{x}, f^{-1}(y_j)) = 0 \) This demonstrates that \( f \) is not \( (\alpha, \gamma) \)-metrically regular on \( \epsilon B_R^n \cap [-\epsilon, \epsilon] \) for any \( \epsilon > 0 \), any modulus function \( \alpha \), and any gauge function \( \gamma \) that is positive on \( \epsilon B_R^n \).
(ii) f is (t 1/2 ,+∞)-metrically regular on
Firstly, invoking Theorem 2.3.1 and Remark 2.3.1, we show that f is (t 1/2 ,k ã k)-metrically regular on W Indeed, for x 6= 0, y ∗ ∈R,
Thus for y ∗ either 1 or −1, kD ∗ f(x)(y ∗ )k=kxk
Observe that for any δ > 0 sufficiently small, for any x ∈ R n \ {0} with kxk ≤ 1 +δ, and |sin(1/kxk)| < √
2/2 + 2δ, there is γ = π/4 + O(δ), such that 1/kxk ≥ π−γ Therefore,
≥ 3π/4−11/5−O(δ), (2.25) for all x ∈ R n with kxk ≤ 1 + δ, |sin(1/kxk)| < √
2/2 + 2δ For given δ ∈(0,1/8) for y ∈ P R W; d(x, W y ) < δ/2ã kxk with 0 < |y − f(x)| 1/2 < δ/2ã kxk, then obviously 0 0 such that d(x, f −1 (y))≤κ|y−f(x)| 1/2 , (2.26) for all (x, y) ∈ W, with |y −f(x)| 1/2 < r −1 kxk Let now (x, y) ∈ W with
|y−f(x)| 1/2 ≥ r −1 kxk Set y 1 = max{f(x) : kxk ≤ 1} We prove the following claim.
Claim There is τ >0 such that for all y ∈[y 0 , y 1 ], one has d(0, f −1 (y))≤τ|y| 1/2
For y ∈ [y 0 , y 1 ], then f −1 (y)∩BR n 6= ∅, due to the continuity of the function f So for |y| ≥ 1/(2π +π/2) 2 , one has d(0, f −1 (y)) ≤ 1 ≤ 2π+π/2 1 |y| 1/2 For
0 \frac{1}{\|xy\|^2}(2k\pi + \frac{5\pi}{2})^2 > (2k)^2\).
= − (3π) 1 2 < y < f(0), kx y k< 3π 2 , and sin kx 1 y k < 0, therefore there is k ∈ N ∗ such that kx y k ∈ 2kπ 1 , 2kπ−π 1
49, and one obtains d(0, f −1 (y))≤ 7 2 |y| 1/2 The claim is proved.
For (x, y) ∈ W with |y − f(x)| 1/2 ≥ r −1 kxk, since P R W ⊆ [y 0 , y 1 ], and
≤ r(1 + 2 −1/4 τ)|y −f(x)| 1/2 This together with relation (2.26) show thatf is (t 1/2 ,+∞)-metrically regular on W.
Milyutin-type regularity and applications
This chapter focuses on two key results: first, it presents theorems that estimate the metric regularity modulus of perturbed mappings under composite perturbations in metric spaces, which includes a specific case of additive perturbations of set-valued mappings between Banach spaces by Lipschitz mappings Secondly, it introduces a fixed point theorem applicable to the composition of two mappings from space X.
Y and the other from Y to X is obtained by the use of the above perturbation theorems The results mentioned here were appeared in the papers [69,79,81].
In this chapter, we explore the concepts of γ-metrically regular and (γ, κ)-metrically regular within the context of metric spaces X and Y, where W is a nonempty subset of their Cartesian product X × Y The regularity horizon function γ: X → R+ is defined by two essential properties that are crucial for our analysis.
It is easy to see that the regularity horizon function associated to the Milyutin regularity m(x) := d(x, X\P X W) satisfies (3.1) and (3.2).
Perturbation stability of Milyutin-type
Firstly, we recall the definitions of γ-metrically regular and (γ, κ)-metrically regular as in Definitions 2.1.3, 2.1.4 for the function γ mentioned above.
A set-valued mapping F : X ⇒ Y is defined as γ-Milyutin regular on a set W with a constant κ if there exists a positive real number r such that for all pairs (x, y) in W, the inequality d(x, F −1 (y)) ≤ κd(y, F(x)) holds, provided that 0 < rd(y, F(x)) < γ(x) The lower bound reg γ F(W) of κ represents the modulus of γ-Milyutin regularity of F on W If such a constant κ cannot be found, then reg γ F(W) is defined to be infinite.
A set-valued mapping \( F: X \Rightarrow Y \) is considered (γ, κ)-Milyutin regular on a set \( W \) with a constant \( κ > 0 \) if the condition \( d(x, F^{-1}(y)) \leq κ d(y, F(x)) \) holds for all pairs \( (x, y) \) in \( W \), provided that \( 0 < κ d(y, F(x)) < γ(x) \) The infimum of all such constants \( κ \) is referred to as reg(γ, κ) F(W), representing the modulus of (γ, κ)-Milyutin regularity of \( F \) on \( W \).
If no such κ exists, we set reg (γ,κ) F(W) =∞.
The difference between these two concepts is that in Definition 3.1.1, the constantκin (3.3) and the constantrin the gauge condition 0< rd(y, F(x))< γ(x) could be different; meanwhile, in Definition 3.1.2, they are equal.
Before stating main results, it is essential to suggest the definition of closure of a mapping as follows.
Definition 3.1.3 Let X, Y be metric spaces and F : X ⇒ Y Denote by F the closure of the mapping F which has GraphF = GraphF, i.e., F : X ⇒ Y defined by
Then, the lower semicontinuous envelop function of the distance function associated to F defined by ϕ F y (x) := lim inf u→x d(y, F(u)), is agreed with the one associated to F as in the following lemma.
Lemma 3.1.1 Let X, Y be metric spaces and F :X ⇒ Y Then, for each y ∈Y, ϕ F y (x) =ϕ F y (x), for all x ∈X.
Proof Take (x, y) ∈ X × Y Then, ϕ F y (x) ≤ ϕ F y (x) due to F(x) ⊂ F(x) If ϕ F y (x) = +∞ then ϕ F y (x) ≤ ϕ F y (x) is obvious In the case of ϕ F y (x) < +∞, there exists {u n } → x, v n ∈ F(u n ) such that d(y, v n ) → ϕ F y (x) From the definition of
F, for each n ∈ N ∗ , there exist k(n) ≥ n, z k(n) ∈X and w k(n) ∈F(z k(n) ) such that d(z k(n) , u n ) < n 1 and d(w k(n) , v n ) < n 1 It follows that d(z k(n) , x) ≤ d(z k(n) , u n ) +d(u n , x)≤ 1 n +d(u n , x).
Letting n→ ∞, one obtains z k(n) → x Furthermore, one has d(y, F(z k(n) ))≤ d(y, w k(n) ) ≤d(y, v n ) +d(v n , w k(n) ) ≤d(y, v n ) + 1 n.
By taking lim inf on both sides of the inequality in above when n tends to infinity, we deduce that ϕ F y (x) ≤ϕ F y (x).
Based on the assumptions of the function γ, the findings from the previous section are applicable to both γ-Milyutin regularity and (γ, κ)-Milyutin regularity, as outlined in the subsequent theorems.
Now, given a function G: X ×Y → Z, U ⊂X, V ⊂ Y, and ε >0 We set
Theorem 3.1.1 Let X be a complete metric space and Y, Z be metric spaces Let
U and V be open subsets of X and Y, respectively Consider a set-valued mapping
F : X ⇒ Y with closed graph and a single-valued mapping G : X ×Y → Z Let γ : X →R+ satisfy assumptions (3.1) and (3.2) for W =U ×V We assume that the following statements hold true:
(a) F is γ-Milyutin regular on U×V with constant τ, i.e., there exists r >0 such that for all (x, y) ∈ U ×V with 0< rd(y, F(x))< γ(x), d(x, F −1 (y)) ≤τ d(y, F(x));
(b) G(x,ã) is metrically regular onY ìZ with modulus λ > 0uniformly in x∈U, i.e., d(y, G −1 x (z))≤λd(z, G(x, y)), ∀x∈ U,∀(y, z) ∈Y ×Z;
(c) G(ã, y) satisfies the Lipschitz condition with constant ` >0 such that `λτ 0, Φ is γ-Milyutin regular on W λ`ε with reg γ Φ(W λ`ε )≤ ((λτ) −1 −`) −1
Proof Firstly, we shall prove that limδ↓0 inf
1−δ 0, we establish the relevant conditions.
Theorem 3.1.2 Let X be a complete metric space and Y be a normed space Let
Let U and V be open subsets of spaces X and Y, respectively, and consider a closed set-valued mapping F: X ⇒ Y If a function γ: X → R+ meets specific assumptions for the product set W = U × V, and F exhibits γ-Milyutin regularity on U × V with a constant τ, then the relationship d(x, F^(-1)(y)) ≤ τ d(y, F(x)) holds for all (x, y) in U × V, provided 0 < rd(y, F(x)) < γ(x) Additionally, if g: X → Y is Lipschitz continuous on U with a constant λ > 0 such that τλ < 1, it follows that for any ε > 0, the regularity of the sum of F and g, denoted reg γ (F + g)(Ω λε), is bounded above by (τ - 1 - λ)^(-1) The proof relies on Theorem 2.2.1, necessitating the demonstration that lim δ↓0 inf is satisfied.
Indeed, choose 0 < δ < min{ 1+λ 1 , τ −1 }, δ(1+δ) 1−δ < ε, 1−δ δ < 1, and take (x, y) ∈X ×Y such that d(x,Ω λε y ) < δγ(x), y ∈ P Y Ω λε and 0 < ϕ F y +g (x) < δγ(x). Then there exists u∈Ω λε y such that d(x, u)< δγ(x) (3.8)
It follows that d(x, u)< δγ(u) +δd(x, u), which implies that d(x, u) < δ
Let now {u n } ⊂ X such that u n → x and d(y,(F +g)(u n ))→ ϕ F y +g (x) as n → ∞.
Thus, there exists n 0 ∈N such that for all n ≥n 0 ,
0< d(y−g(u n ), F(u n )) < δγ(u n ) and u n ∈ U( since u n → x∈ U - open). Using (3.8), the Lipschitz property of g and γ, we get kg(u n )−g(u)k ≤ λd(u n , u)
0< d(y−g(un), F(un))< δγ(un) < τ −1 γ(un) then from the Milyutin regularity of F on U ×V with modulus τ, one obtains d(u n , F −1 (y−g(u n )))≤ τ d(y−g(u n ), F(u n )), for all n ≥n 0
We now choose z n ∈F −1 (y −g(u n )), i.e., y−g(u n ) ∈ F(z n ) such that d(u n , z n )≤ τ + 1 n d(y−g(u n ), F(u n )) (3.9)
It follows that for all n≥ n 0 d(u n , z n ) ≤ τ + 1 n τ −1 γ(u n )
From here, one has z n ∈ U and thus, kg(z n )−g(u n )k ≤λd(u n , z n ) (3.10)
Since ϕ F+g y (x) > 0 and lim n→∞u n = x then lim inf n→∞ d(u n , z n ) > 0 Combining (3.9), and (3.10), one gets
Analysis similar to that in the proof of Theorem 3.1.1 actually shows that the version of the perturbation theorem for (γ, κ)-Milyutin regularity is also achieved.
Theorem 3.1.3 states that in a complete metric space X, with metric spaces Y and Z, and open subsets U and V, a closed set-valued mapping F from X to Y and a single-valued mapping G from the product space X × Y to Z are defined Additionally, a function γ from X to the positive reals is introduced, adhering to specific assumptions for the product space W = U × V The theorem is contingent upon certain conditions being met.
(a) F is (γ, τ)-Milyutin regular on U × V with constant τ, i.e., for all (x, y) ∈U ×V with 0< τ d(y, F(x))< γ(x), d(x, F −1 (y)) ≤τ d(y, F(x));
(b) G(x,ã) is metrically regular onY ìZ with modulus λ > 0uniformly in x∈U, i.e., d(y, G −1 x (z))≤λd(z, G(x, y)), ∀x∈ U,∀(y, z) ∈Y ×Z;
(c) G(ã, y) satisfies the Lipschitz condition with constant ` >0 such that `λτ 0 Note that d(z, G(u n , y n )) = 0 and by (3.12), (3.14), (3.15), similar arguments as in the proof of Theorem 3.1.1, one concludes that
1 λ(1 +ε n ) 2 (τ +ε n ) −`= (λτ) −1 −`, which follows from Lemma 3.1.1 that
According to Theorem 2.2.2, this completes the proof.
Corollary 3.1.1 Let X, Y be complete metric spaces, Z be a metric space, and
U, V be open subsets of X, Y, respectively Let a closed set-valued mapping
F : X ⇒Y and a single-valued mapping G: X×Y → Z be given Let γ : X → R+ satisfy assumptions (3.1) and (3.2) for W = U ×V Assume further that
(a) F is (γ, τ)-Milyutin regular on U × V with constant τ, i.e., for all (x, y) ∈U ×V with 0< τ d(y, F(x))< γ(x), d(x, F −1 (y)) ≤τ d(y, F(x));
(b) G(x,ã) is an isometry between Y and Z for all x ∈U;
(c) G(ã, y) satisfies the Lipschitz condition with constant ` > 0 such that `τ < 1 on U for all y ∈ Y, i.e., d(G(x, y), G(u, y)) ≤`d(x, u), ∀y ∈Y,∀x, u∈U.
Then, Φ is (γ, κ)-Milyutin regular on W λ` with reg (γ,κ) Φ(W λ` ) ≤ κ:= (τ −1 −`) −1 , where
Proof • Firstly, we see that the mapping Φ has closed graph Indeed, take a pair (x n , z n ) ∈ Graph Φ converging to (x, z) One has, z n ∈ Φ(x n ) = G(x n , F(x n )). Hence, we can pick a point y n ∈F(x n ) such as z n =G(x n , y n ) We have d(z n+m , z n ) =d(G(x n+m , y n+m ), G(x n , y n )
Since the sequences (x n) and (z n) converge, and (y n) is Cauchy in the complete metric space Y, there exists an element y ∈ Y such that y n converges to y Utilizing the closedness of the mappings F and G, we find that z = G(x, y), which implies that (x, z) belongs to the Graph(Φ) Consequently, this indicates that Graph(Φ) is closed in the product space X × Y, leading to the conclusion that Φ is equivalent to Φ.
• Secondly, G(x,ã) is metrically regular on Y ìZ with modulus 1 uniformly in x ∈ U Indeed, it follows from (b) that for x ∈ U and for all z ∈ Z, there is y 1 ∈ Y such that z = G(x, y 1 ) So, d(z, G(x, y)) =d(G(x, y 1 ), G(x, y)) =d(y, y 1 ) ≥d(y, G −1 x (z)).
That is the case which we desire.
• Finally, applying Theorem 3.1.3 for Φ ≡ Φ, one concludes that Φ is γ-Milyutin regular on W λ` with modulus (τ −1 −`) −1 , where
In the context of complete metric spaces X and Y, and a metric space Z, we consider a closed set-valued mapping F from X to Y and a single-valued mapping G from the product space X×Y to Z Given a point z¯ defined as G(¯x,y) where ¯y belongs to F(¯x), we define open subsets U and V in X and Y, respectively, as U = B(¯x, α) and V = B(¯y, β) Additionally, we introduce the function γ: X → R+ defined by γ(x) = d(x, X\U), which measures the distance from any point x in X to the complement of the open set U.
(a) F is (γ, τ)-Milyutin regular on U × V with constant τ, i.e., for all (x, y) ∈U ×V with 0< τ d(y, F(x))< γ(x), d(x, F −1 (y)) ≤τ d(y, F(x));
(b) G(x,ã) is an isometry between Y and Z for all x ∈U;
(c) G(ã, y) satisfies the Lipschitz condition with constant ` > 0 such that `τ < 1 on U for all y ∈ Y, i.e., d(G(x, y), G(u, y)) ≤`d(x, u), ∀y ∈Y,∀x, u∈U.
Then, Φ is γ 0 -Milyutin regular on U 0 ×V 0 with (τ −1 −`) −1 , where U 0 = B(¯x, α 0 ),
Proof According to Corollary 3.1.1, Φ is γ-Milyutin regular on W λ` with (τ −1 −`) −1 , hence it is also Milyutin regular on a set smaller than W 1 λ` of W λ` with a smaller gauge function γ 0 (x)≤ γ(x), here
To conclude the corollary, we must verify the relationship \( U_0 \times V_0 \subset W_1^{\lambda} \) Consider a pair \( (x, z) \in U_0 \times V_0 \) where \( d(x, x) < \alpha_0 \) and \( d(z, z) < \beta_0 \) Since \( x \in U_0 \), it suffices to demonstrate that \( B(G^{-1}(x)(z), \gamma_0(x)) \subset V \) Let \( y_0 \) be an element in \( B(G^{-1}(x)(z), \gamma_0(x)) \), which is equivalent to \( B(y, \gamma_0(x)) \) with \( y = G^{-1}(x)(z) \) It follows that \( d(y_0, y) \leq d(y_0, y) + d(y, y) < \gamma_0(x) + d(y, y) \).
≤ `γ 0 (x) +d(z,z) +¯ ld(x,x) =¯ lα 0 +β 0 =β, which turns out y 0 ∈V, that is B(G −1 x (z), `γ 0 (x)) ⊂V.
In Remark 3.1.1, it is noted that Ioffe's analogous result to Corollary 3.1.2 presents a slight variation in its assumptions; specifically, while both X and Y are required to be complete metric spaces in Corollary 3.1.2, Ioffe only requires the graph of the set-valued mapping F to be complete in the product space X × Y The proofs for these two results show minimal differences.
Application to fixed double-point problems
In this section, we use the result on the stability of Milyutin-type regularity under the composite perturbation in the previous section to study the problem:
In the context of metric spaces X and Y, we seek a pair (x, y) ∈ X × Y such that y belongs to F1(x) and x belongs to F2(y) This pair is referred to as the fixed double-point of the mappings F1 and F2, with the collection of all such pairs known as the fixed double-point set, denoted as DFix(F1, F2) Additionally, the fixed point set for a set-valued mapping from X to X is represented as Fix(F) A notable relationship exists between these sets: if (x, y) is an element of DFix(F1, F2), then x is in Fix(F2 ◦ F1) and y is in Fix(F1 ◦ F2) Furthermore, for r > 0 and s > 0, the metric d on the product space X × Y is defined by the equation d((x, v), (u, y)) = √r dX(x, u) + √s dY(v, y).
Before stating the main result, it is necessary to consider the following proposition.
Proposition 3.2.1 Let X, Y be metric spaces and F 1 : X ⇒ Y, F 2 :Y ⇒ X. Fix x¯ ∈ X, y¯ ∈ Y, v¯ ∈ Y, u¯ ∈ X together with some positive numbers α, β and for any x, y set U = B(¯x, α), V = B(¯y, β), γ(x) = [α−d(x,x)]¯ + , V 0 = B(¯v, β),
U 0 =B(¯u, α), δ(v) = [β − d(v,¯v)] + Consider the set-valued mapping
T : (x, v) ⇒ (F 1 (x), F 2 (v)) and the real numbers r > 0, s > 0 such that rs > 1.Assume that the following statements are satisfied:
(ii) F 2 is (δ, s −1 )-metrically regular on V 0 ×U 0
Then, T is (ρ,(√ rs) −1 )-metrically regular on (U ×V 0 )×(V ×U 0 ), where ρ(x, v) = min
Proof Take (x, v) ∈U ×V 0 , (y, u) ∈V ×U 0 such that d((y, u), T(x, v))