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Strongly regular points of mappings
Fixed Point Theory and Algorithms for Sciences and Engineering volume 2021, Article number: 14 (2021)
Abstract
In this paper, we use a robust lower directional derivative and provide some sufficient conditions to ensure the strong regularity of a given mapping at a certain point. Then, we discuss the Hoffman estimation and achieve some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the coefficient of the error bound.
1 Introduction
Let f be a mapping acting between the normed spaces \(\mathbb{X}\) and \(\mathbb{Y}\), whose norms are denoted by the same symbol \(\Vert \cdot \Vert \). To estimate the approximate solutions of the equation \(y=f(x)\), we seek an error bound
locally, for all \((x,y)\) near \((\bar{x}, \bar{y} = f(\bar{x}))\), or globally, for all x and y, where κ is some positive constant. The infimum of such κ is called the modulus of regularity of f. For instance, when \(f:\mathbb{R}\to \mathbb{R}\) is smooth and verifies \(f^{\prime }(\bar{x})\neq 0\), it is easily observed that the modulus of regularity of f at x̄ is exactly \(\vert f^{\prime }(\bar{x})\vert ^{1}\). A first approach to the concept of regularity goes back to a celebrated fundamental result proved in 1934 by Lyusternik [1]:
Theorem 1.1
(Lyusternik, [1])
Let f be a mapping from a Banach space \(\mathbb{X}\) to a Banach space \(\mathbb{Y}\). Suppose that f is Fréchet differentiable in a neighborhood of x̄ and that its derivative \(f^{\prime }(x)\) is continuous at x̄ and \(f^{\prime }(\bar{x})\) is surjective. Then, for every \(\varepsilon >0\), there exists \(r>0\) such that
whenever
In other words, the tangent manifold to \(f^{1}(0)\) is equal to \(\bar{x} +\mathop{\operatorname{Ker}}f^{\prime }(\bar{x})\), where \(\mathop{\operatorname{Ker}}f^{\prime }(\bar{x})\) is the set of those x such that \(f^{\prime }(\bar{x})(x)=0\).
We refer to Dontchev [2] for a nice overview on the Lyusternik theorem and to the fact that the Lyusternik theorem can be easily obtained from the Graves theorem. We also refer to the forthcoming book by Thibault [3].
Theorem 1.2
(Graves, [4])
Let \(\mathbb{X}\) and \(\mathbb{Y}\) be Banach spaces, \(\bar{x}\in \mathbb{X,}\) and \(f:\mathbb{X}\to \mathbb{Y}\) be a \(C^{1}\)mapping whose derivative \(f^{\prime }(\bar{x})\) is onto. Then, there exist a neighborhood \(\mathbb{U}\) of x̄ and a constant \(c > 0\) such that for every \(x\in \mathbb{U}\) and \(\tau > 0 \) with \(\mathbb{ B}(\bar{x}, \tau ) \subset \mathbb{U}\),
Ioffe and Tihomirov showed in [5] that the original Lyusternik proof may lead to a stronger result and proved that if \(f^{\prime }(\bar{x})\) is surjective, then there are \(\kappa > 0\) and \(\delta > 0\) such that
Ioffe’s remark leads to a standard definition:
Definition 1.1
Point \(\bar{x}\in \mathbb{X} \) is said to be a regular point of a mapping \(f:\mathbb{X}\to \mathbb{Y}\) if the relation (1.1) is satisfied.
In this note, we will call x̄ a strongly regular point of f if the inequality
holds locally, for all x belonging to a neighborhood of x̄, where \(\kappa >0\) is a positive constant. Next, we will provide sufficient conditions for x̄ to be a strongly regular point. Our results allow us to estimate the constant κ in (1.2). Then, we apply our results to the Hoffman estimate and obtain some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the upper limit of the error. In particular, for a finitedimensional space \(\mathbb{X}\) and a linear (continuous) mapping \(A:\mathbb{X}\rightarrow \mathbb{X}\), we prove that the estimate
holds for all \(x\in \mathbb{X}\) (Corollary 3.2 below). We can easily see that this estimate is sharp for injective linear mappings, in the sense that, if A is an injective linear mapping and
then there exists some \(x\in \mathbb{X}\) such that \(\x\=\mathop{\operatorname{dist}}(x,\mathop{\operatorname{Ker}}A)>\mu ^{1} \Vert A(x)\Vert \).
Our work is outlined as follows. In Sect. 1, we recall the famous Lyusternik theorem and survey briefly its relationship with the concept of metric regularity. In Sect. 2, we first introduce the notion of homogeneous continuity of mappings. Then, using an appropriate notion of lower directional derivative, we achieve some results ensuring in finite dimension that for a given mapping a point is strongly regular. Finally, in Sect. 3, we focus our attention on Hoffman’s estimate of approximate solutions of finite systems of linear inequalities and prove some similar estimates.
2 Sufficient conditions of regularity via generalized derivative
Throughout the paper, we use standard notations. For a normed space \(\mathbb{X}\), we denote its norm by \(\Vert \cdot \Vert \) and by \(\mathbb{X}^{*}\) its (continuous) dual. The symbol \(\mathbb{S}\) stands for the unit sphere, that is, the set of all points of \(\mathbb{X}\) of norm one, while \(\mathbb{B}(x,r)\) and \(\overline{\mathbb{B}}(x,r)\) denote, respectively, the open and closed balls centered at x with radius r. Some other notations are introduced as and when needed.
2.1 Homogeneous continuity
We begin with the following definition.
Definition 2.1
Let \(\mathbb{X}\) and \(\mathbb{Y}\) be normed spaces and \(\mathbb{E}\subset \mathbb{X}\). The mapping \(f:\mathbb{X}\rightarrow \mathbb{Y}\) is said to be homogeneously continuous at \(\bar{x}\in \mathbb{X}\) on \(\mathbb{E}\) if for every \(\epsilon >0\) there exist \(\delta >0\) and \(0<\beta \leq 1\) such that
for all \(0< t\leq \beta \) and all \(x,y\in \mathbb{E}\).
We are going to provide some sufficient conditions under which a mapping f is homogeneously continuous. Let us recall that a mapping \(f:\mathbb{X}\rightarrow \mathbb{Y}\) is said to be locally Lipschitz around \(\bar{x}\in \mathbb{X}\) if there exist a neighborhood \(\mathbb{O}\) of x̄ and a real number \(\lambda >0\) such that
for all \(x,y\in \mathbb{O}\).
Lemma 2.1
Suppose that \(\mathbb{X}\) and \(\mathbb{Y}\) are normed spaces. If \(f:\mathbb{X}\rightarrow \mathbb{Y}\) is locally Lipschitz around \(\bar{x}\in \mathbb{X}\), then f is homogeneously continuous at x̄ on some closed ball \(\overline{\mathbb{B}}(0,r)\).
Proof
By hypothesis, there exist a constant \(\lambda >0\) and a neighborhood \(\mathbb{O}\) of x̄ in \(\mathbb{X}\) such that (2.1) holds for all \(x,y\in \mathbb{O}\). Choose \(r>0\) such that \(\overline{\mathbb{B}}(\bar{x},r)\subset \mathbb{O}\). It follows that
for all \(x,y\in \overline{\mathbb{B}}(0,r)\) and all \(0\leq t\leq 1\). Now for each \(\epsilon >0\) take \(0<\delta <\epsilon \lambda ^{1}\). It follows that
for all \(x,y\in \overline{\mathbb{B}}(0,r)\) and all \(0< t\leq 1\). This completes the proof. □
Proposition 2.1
Let \(\mathbb{X}\) and \(\mathbb{Y}\) be normed spaces, \(f:\mathbb{X}\rightarrow \mathbb{Y}\) be a mapping, \(\mathbb{E}\) be a subset of \(\mathbb{X}\) equipped with the topology induced by the norm and \(\bar{x}\in \mathbb{X}\). If the bifunction \(f_{\mathbb{E}}:\mathbb{E} \times (0,1]\rightarrow \mathbb{Y}\), defined by
is uniformly continuous (\(\mathbb{E} \times (0,1]\) equipped with the product topology with the usual linear operations of vector addition and scalar multiplication), then f is homogeneously continuous at x̄ on \(\mathbb{E}\).
Proof
Let \(\epsilon >0\). By hypothesis, there exist \(\delta, \beta >0\) such that for all \(x,y\in \mathbb{E}\) with \(\Vert xy\Vert <\delta \) and all \(s,h\in (0,1]\) with \(\vert sh\vert <\beta \) we have \(\Vert f_{\mathbb{E}}(x,s) f_{\mathbb{E}}(y,h)\Vert < \epsilon \). It follows that
for all \(x,y\in \mathbb{E}\) with \(\Vert xy\Vert <\delta \) and all \(0< t\leq 1\). Thus
for all \(x,y\in \mathbb{E}\) with \(\Vert xy\Vert <\delta \) and all \(0< t\leq 1\). This completes the proof. □
2.2 Generalized derivatives
We recall the definitions of the Hadamard and Gateaux derivatives: The Hadamard directional derivative \(f_{H}^{\prime }(\bar{x})(\nu )\) of f at x̄ in direction ν is defined as
where \((\nu _{n}) \) and \((t_{n})\) are any sequences such that \(\nu _{n}\to \nu \) and \(t_{n}\to 0^{+}\).
The Gateaux directional derivative \(f_{G}^{\prime }(\bar{x})(\nu )\) of f at x̄ in direction ν is defined by
The following facts are well known:

Hadamard differentiability is a stronger notion than Gateaux differentiability, see, e.g., [6]; when f is Hadamard differentiable at x̄, it is Gateaux (directional) differentiable at x̄ and, moreover, \(f_{G}^{\prime }(\bar{x})\) is continuous;

For locally Lipschitz mappings in normed spaces, Hadamard and Gateaux directional derivatives coincide.
The following corollary uses Hadamard differentiability and provides another sufficient condition for a mapping f to be homogeneously continuous.
Corollary 2.1
Let \(\mathbb{X}\) and \(\mathbb{Y}\) be normed spaces, \(f:\mathbb{X}\rightarrow \mathbb{Y}\) be a continuous mapping, \(\mathbb{E}\) be a compact subset of \(\mathbb{X}\) (equipped with the topology induced by the norm) and \(\bar{x}\in \mathbb{X}\). If the Hadamard directional derivative of f at x̄ in every direction \(\nu \in \mathbb{E}\) exists, then f is homogeneously continuous at x̄ on \(\mathbb{E}\).
Proof
Define the bifunction \(\bar{f}_{\mathbb{E}}:\mathbb{E} \times [0,1]\rightarrow \mathbb{Y}\) as
Since f is continuous and the Hadamard directional derivative of f at x̄ in every direction \(\nu \in \mathbb{E}\) exists, the bifunction \(\bar{f}_{\mathbb{E}}\) is continuous. Since \(\mathbb{E} \times [0,1]\) is compact, \(\bar{f}_{\mathbb{E}}\) is uniformly continuous. It follows that the bifunction \(f_{\mathbb{E}}:\mathbb{E} \times (0,1]\rightarrow \mathbb{Y}\), defined by
is uniformly continuous. Now apply Proposition 2.1. □
The following proposition illustrates our main motivation for introducing the homogeneously continuous mappings.
Proposition 2.2
Let \(\mathbb{X}\) and \(\mathbb{Y}\) be normed vector spaces, \(f:\mathbb{X}\rightarrow \mathbb{Y}\) be a mapping, \(\mathbb{E}\) be a subset of \(\mathbb{X}\) and \(\bar{x}\in \mathbb{X}\). If f is homogeneously continuous at x̄ on \(\mathbb{E}\), then there exist \(\delta >0\) and \(\beta >0\) such that
for all \(x,y\in \mathbb{E}\) with \(\Vert xy\Vert <\delta \) and all \(0< t\leq \beta \).
Proof
The proof is obvious; we therefore omit it. □
For a mapping \(f:\mathbb{X}\rightarrow \mathbb{Y}\), we consider the following notions of lower directional derivatives which are crucial to our approach:
Note that we have
for every \(\nu \in \mathbb{X}\). We shall observe that if \(\inf_{\nu \in \mathbb{S}}f_{l}^{\prime }(\bar{x})(\nu )>0\) and f is homogeneously continuous at x̄ on \(\mathbb{S}\), then f satisfies the property (1.2) above.
2.3 Main results
Throughout the remaining part of the discussion, unless specified otherwise, we assume that \(\mathbb{X}\) is a finitedimensional space and \(\mathbb{Y}\) is an arbitrary normed space. We now are completely ready to state the main theorem of the paper. For a positive scalar \(\alpha \in \mathbb{R}\), let
Theorem 2.1
Let \(f:\mathbb{X}\rightarrow \mathbb{Y}\) be homogeneously continuous at \(\bar{x}\in \mathbb{X}\) on \(\mathbb{S}_{\alpha }\) for some positive scalar α. If there exists some \(\kappa >0\) such that \(\inf_{\nu \in \mathbb{S}_{\alpha }}f_{l}^{\prime }(\bar{x})(\nu )>\kappa \), then there exists \(\delta >0\) such that
for all \(x\in \mathbb{B}(\bar{x},\delta )\). In other words, x̄ is a strongly regular point of f.
Proof
Let \(\kappa <\gamma <\inf_{\nu \in \mathbb{S}_{\alpha }}f_{l}^{\prime }( \bar{x})(\nu )\) and \(\epsilon:= \gamma \kappa \). Hence, for all \(\nu \in \mathbb{S}_{\alpha }\) there exists \(0< r_{\nu }\leq 1\) such that
Since f is homogeneously continuous at x̄ on \(\mathbb{S}_{\alpha }\), there exist \(\theta >0\) and \(\beta >0\) such that
for all \(\nu, \mu \in \mathbb{S}_{\alpha }\) and all \(0< t\leq \beta \), by Proposition 2.2. Let \(\hat{r}_{\nu }:=\min \{\theta,\beta, r_{\nu }\}\) for all \(\nu \in \mathbb{S}_{\alpha }\). Clearly, \(\mathbb{S}_{\alpha }\subset \bigcup_{\nu \in \mathbb{S}_{\alpha }} \mathbb{B}(\nu, \hat{r}_{\nu }) \). The compactness of \(\mathbb{S}_{\alpha }\) implies that there exist \(\nu _{1}, \nu _{2}, \ldots, \nu _{m}\in \mathbb{S}_{\alpha }\) such that \(\mathbb{S}_{\alpha }\subset \bigcup_{k=1}^{m} \mathbb{B}(\nu _{k}, \hat{r}_{\nu _{k}}) \). Now let \(x\in \mathbb{B}(\bar{x},\alpha \hat{\delta })\setminus \lbrace \bar{x} \rbrace \) and \(\nu:=\frac{\alpha }{\x\bar{x}\}(x\bar{x})\), where \(\hat{\delta }:=\min \{\hat{r}_{\nu _{k}}: 1\leq k\leq m\}\). Then, \(\nu \in \mathbb{S}_{\alpha }\) and therefore \(\nu \in \mathbb{B}(\nu _{s}, \hat{r}_{\nu _{s}})\) for some \(1\leq s\leq m\). It follows that \(\\nu \nu _{s}\<\theta \) and \(\alpha ^{1}\x\bar{x}\<\beta \). By (2.4), we deduce that
Hence
since \(\alpha ^{1}\x\bar{x}\< r_{\nu _{s}}\). It follows that
for all \(x\in \mathbb{B}(\bar{x},\alpha \hat{\delta })\). Letting \(\delta:=\alpha \hat{\delta }\) completes the proof. □
Corollary 2.2
Let \(f:\mathbb{X}\rightarrow \mathbb{Y}\) be homogeneously continuous at \(\bar{x}\in \mathbb{X}\) on \(\mathbb{S}\). If there exists some \(\kappa >0\) such that \(\inf_{\nu \in \mathbb{S}}f_{0}^{\prime }(\bar{x})(\nu )>\kappa \), then there exists \(\delta >0\) such that
for all \(x\in \mathbb{B}(\bar{x},\delta )\).
Proof
Apply Theorem 2.1 and (2.2). □
Corollary 2.3
Suppose that \(f:\mathbb{X}\rightarrow \mathbb{Y}\) is locally Lipschitz around \(\bar{x}\in \mathbb{X}\). If there exists some \(\kappa >0\) such that \(\inf_{\nu \in \mathbb{S}}f_{l}^{\prime }(\bar{x})(\nu )>\kappa \), then there exists \(\delta >0\) such that (2.5) holds for all \(x\in \mathbb{B}(\bar{x},\delta )\).
Proof
By Lemma 2.1, f is homogeneously continuous at x̄ on some closed ball \(\overline{\mathbb{B}}(0,r)\). It follows that f is homogeneously continuous at x̄ on \(\mathbb{S}_{r}\) (since \(\mathbb{S}_{r} \subset \overline{\mathbb{B}}(0,r)\)). The condition \(\inf_{\nu \in \mathbb{S}}f_{l}^{\prime }(\bar{x})(\nu )>\kappa \) implies that \(\inf_{\nu \in \mathbb{S}_{r}}f_{l}^{\prime }(\bar{x})(\nu )>r\kappa \). Now apply Theorem 2.1. □
Corollary 2.4
Let \(f:\mathbb{X}\rightarrow \mathbb{Y}\) be a continuous mapping and \(\bar{x}\in \mathbb{X}\). Assume that the Hadamard directional derivative of f at x̄ in every direction \(\nu \in \mathbb{S}\) exists. If there exists some \(\kappa >0\) such that \(\inf_{\nu \in \mathbb{S}}\Vert f_{H}^{\prime }(\bar{x})(\nu )\Vert > \kappa \), then there exists \(\delta >0\) such that (2.5) holds for all \(x\in \mathbb{B}(\bar{x},\delta )\).
Proof
By hypothesis, \(f_{H}^{\prime }(\bar{x})(\nu )\) exists for every \(\nu \in \mathbb{S}\). By continuity of f and \(\Vert \cdot \Vert \), it follows that
for every \(\nu \in \mathbb{S}\). It follows that \(\inf_{\nu \in \mathbb{S}}f_{l}^{\prime }(\bar{x})(\nu )>\kappa \). Since \(\mathbb{X}\) is finite dimensional, \(\mathbb{S}\) is compact. Hence, f is homogeneously continuous at \(\bar{x}\in \mathbb{X}\) on \(\mathbb{S}\), by Corollary 2.1. Now apply Theorem 2.1. □
The following example has been considered in [7] (Example 2.1). We shall prove that the origin is a regular point of the involved mapping f once again by Theorem 2.1.
Example 2.1
Consider the mapping \(f:\mathbb{R}\rightarrow \mathbb{R}\) defined as
We have \(\mathbb{S}=\lbrace \pm 1\rbrace \) and therefore f is homogeneously continuous at 0 on \(\mathbb{S}\). One may easily verify that
It follows that \(\inf_{s\in \mathbb{S}}f_{l}^{\prime }(0)(s)=\frac{2}{\pi }>0\). Hence, if \(0<\kappa <\frac{2}{\pi }\), then there exists \(\delta >0\) such that
for all \(\vert x\vert <\delta \), by Theorem 2.1. Hence, 0 is a strongly regular point of f. Since f is continuous, thus the subset \(f^{1}(0)\) is closed and therefore the distance function \(\mathop{\operatorname{dist}}(\cdot,f^{1}(0))\) is Lipschitz around 0 (see [8, p. 11]). Hence, 0 is a regular point of f.
3 Hoffman’s estimate for the distance to the set of solutions to a system of linear inequalities
Theorem 3.1
Let \(x_{i}^{*}, i= 1, 2, \dots, k\) be a finite family of linear forms on \(\mathbb{X}^{*}\). Set
Then, there exists \(\kappa >0\) such that
where \(\Phi (x):=\max \{\langle x_{i}^{*}, x\rangle, i= 1, 2,\dots, k\} \) and \([\Phi (x)]_{+}:= \max (\Phi (x),0)\).
Hoffman’s result is considered as the starting point of the theory of error bounds, theory that has been extended over the years to different contexts. We refer to [3, 11–13] and the references therein for the discussion of the fundamental role played by Hoffman bounds and more generally by error bounds in mathematical programming. As described, for example, in [14], they are used, for instance, in convergence properties of algorithms, in sensitivity analysis, in designing solution methods for nonconvex quadratic problems. When \(\mathbb{C}:=\lbrace x\in \mathbb{X}: A(x)=0, \langle x_{i}^{*},x \rangle \leq 0, i=1,2,\dots, k\rbrace \) where \(x_{i}^{*}\in X^{*}\), \(i=1,2,\dots, k\), are some given functionals and \(A:\mathbb{X}\rightarrow \mathbb{Y}\) is a linear (continuous) mapping, we have the following result.
Theorem 3.2
(Ioffe, 1979, [15])
There exists some \(\kappa >0\) such that
for all \(x\in \mathbb{X}\).
Now let \(\mathbb{G}:=\mathop{\operatorname{Ker}}A\cap ( \bigcap_{i=1}^{k} \mathop{\operatorname{Ker}}x_{i}^{*} )\). Then, Theorem 3.2 yields the following result.
Corollary 3.1
There exists some \(\kappa ^{\prime }>0\) such that
for all \(x\in \mathbb{X}\).
In this section, we apply Theorem 2.1 and establish similar estimates. We prove that there exists \(\bar{\kappa }>0\) such that
for all \(x\in \mathbb{X}\), where \(L:\mathbb{X}\rightarrow \mathbb{X}\) is a linear mapping with \(\mathop{\operatorname{Ker}}L=\mathbb{G}\). Our results also allow us to evaluate the constant κ̄. The details are as follows.
Proposition 3.1
Let \(A:\mathbb{X}\rightarrow \mathbb{Y}\) be a linear mapping and \(x_{i}^{*}\in \mathbb{X}^{*}\), \(i=1,2,\dots, k\) be given. Suppose that \(L:\mathbb{X}\rightarrow \mathbb{X}\) is a linear mapping such that \(\mathop{\operatorname{Ker}}L=\mathbb{G}\). Then
where γ is a positive real number given by
Proof
Let us consider the finitedimensional quotient space \(\mathcal{M}:= \frac{X}{\mathbb{G}}\), and denote by \([x]\) the equivalence class containing x in \(\mathcal{M}\), that is, \([x]:= x+\mathbb{G}\). We note \(\Vert [x]\Vert:= \inf \{\x+y\: y\in \mathbb{G}\}\). Denote by \(\mathbb{S}_{\mathcal{M}}\) the unit sphere of \(\mathcal{M}\) (i.e., the elements of \(\mathcal{M}\) of norm one). Obviously,
Consider the continuous linear mapping \(\overline{L}:\mathcal{M}\rightarrow \mathbb{X}\) defined as \(\overline{L}([x]):=L(x)\) for all \([x]\in \mathcal{M}\). Also for each \(1\leq i\leq k\) define \(\langle [x_{i}]^{*},[ x]\rangle:=\langle x_{i}^{*},x\rangle \) for all \([x]\in \mathcal{M}\). Obviously, each \([x_{i}]^{*}\) belongs to \(\mathbb{G}^{\perp }\) (the orthogonal complement of \(\mathbb{G}\)), and hence belongs to the dual of \(\mathcal{M}\) (which is isometrically isomorphic to \(\mathbb{G}^{\perp }\) [16]). Set
We have \(\mathop{\operatorname{Ker}}\overline{L}=\mathop{\operatorname{Ker}}L=\mathbb{G}\) and therefore \(\overline{\mathbb{C}}=\lbrace [0]\rbrace \). Now define the mapping \(f:\mathcal{M}\rightarrow \mathbb{R}\) as
We show that the conditions of Theorem 2.1 for f at \([\bar{x}]=[0]\) are all satisfied. For all \([\nu ]\in \mathbb{S}_{\mathcal{M}}\), one has
Hence, f is homogeneously continuous at \([0]\) on \(\mathbb{S}_{\mathcal{M}}\), by Corollary 2.1. We also have
The continuity of f implies that the mapping \(f\vert _{\mathbb{S}_{\mathcal{M}}}\) (the restriction of f to \(\mathbb{S}_{\mathcal{M}}\)) attains its minimum at some \([\nu _{0}]\in \mathbb{S}_{\mathcal{M}}\). Then, \([\nu _{0}]\notin \overline{\mathbb{C}}\) (note that \(\overline{\mathbb{C}}=\lbrace [0]\rbrace \)) and therefore
It follows that \(\inf_{[\nu ]\in \mathbb{S}_{\mathcal{M}}}f_{l}^{\prime }([0])([\nu ])>0\). Using (3.7), we obtain
Thus \(\gamma >0\). Now let \(0<\kappa <\gamma \). Theorem 2.1 implies that there exists some \(\delta >0\) such that
for all \([x]\in \mathbb{B}_{\mathcal{M}}([0],\delta )\). Since f is sublinear,
for all \([x]\in \mathcal{M}\). It follows that
For all \(x\in \mathbb{X}\). Letting \(\kappa \rightarrow \gamma \) in (3.8), we obtain the desired result. □
Remark 3.1
The existence of the linear mapping \(L:\mathbb{X}\rightarrow \mathbb{X}\) discussed in Proposition 3.1 is straightforward. Indeed, \(\mathbb{G}\) is a closed subspace of \(\mathbb{X}\) and \(\mathbb{X}\) is separable, thus there exists a (continuous) linear mapping \(L:\mathbb{X}\rightarrow \mathbb{X}\) with \(\mathop{\operatorname{Ker}}L=\mathbb{G}\) (see [17]). Of course, one can easily define L directly (without using [17]). To see this, suppose that \(\mathop{\operatorname{dim}}\mathbb{X}=n\) and let \(\lbrace e_{1}, \dots, e_{j}\rbrace \) be a linearly independent basis for the vector space \(\mathbb{G}\). By linear algebra, we can extend \(\lbrace e_{1}, \dots, e_{j}\rbrace \) to get a linearly independent basis for \(\mathbb{X}\) (since \(\mathbb{G}\) is a subspace of \(\mathbb{X}\)). Let us denote this basis by \(\lbrace e_{1}, \dots,e_{j},e_{j+1},\dots, e_{n}\rbrace \). Now for every \(x:=x_{1}e_{1}+\cdots +x_{n}e_{n}\in \mathbb{X}\), define the mapping \(L:\mathbb{X}\rightarrow \mathbb{X}\) as
One can easily check that L is welldefined, linear, and \(\mathop{\operatorname{Ker}}L=\mathbb{G}\).
Corollary 3.2
Let \(A:\mathbb{X}\rightarrow \mathbb{X}\) be a linear mapping. Then
for all \(x\in \mathbb{X}\).
Proof
Let \(\mathbb{X} =\mathbb{Y}\), and \(x_{i}^{*}\equiv 0\) for all \(1\leq i\leq k\). Then, \(\mathbb{G}=\mathop{\operatorname{Ker}}A\). Letting \(L:=A\) in Proposition 3.1 yields the result. □
Corollary 3.3
Let \(A:\mathbb{X}\rightarrow \mathbb{Y}\) and \(L:\mathbb{X}\rightarrow \mathbb{X}\) be linear mappings with \(\mathop{\operatorname{Ker}}L=\mathbb{G}\) and \(x_{i}^{*}\in \mathbb{X}^{*}\), \(i=1,2,\dots, k\), be some given functionals. Then
for every \(x\in \mathbb{C}_{\leq }\) (see (3.1) above).
Proof
Proposition 3.1 implies that
where
Now let \(x\in \mathbb{C}_{\leq }\). Thus \(\langle x_{i}^{*},x\rangle \leq 0\) for every \(1\leq i\leq k\). Hence \([\langle x_{i}^{*},x\rangle ]_{+} =0\) for every \(1\leq i\leq k\). Then, the above inequality yields
for every \(x\in \mathbb{C}_{\leq }\). This completes the proof. □
Finally, let us make a comparison between the two estimations (3.5) in Proposition 3.1 and (3.4) in Corollary 3.1 described above. First, note that an application of Corollary 3.1 (or a direct application of Theorem 3.2) with L (described in Proposition 3.1) in place of A and no inequalities immediately produces the following estimate:
for all \(x\in \mathbb{X}\), where \(\kappa _{0}\) is a constant. On the other hand, doing the same replacements in Proposition 3.1 (i.e., applying Proposition 3.1 with L in place of A and L in place of itself without the inequalities) yields
where
The question is: which of the above estimates (3.10) and (3.11) is better? To answer this question, we need to know the relationship between the coefficients \(\kappa _{0}\) and \(\gamma _{0}\) stated above. As long as the value of the constant \(\kappa _{0}\) in (3.10) is not known, we can’t say which of the estimates (3.10) and (3.11) produces a better result. We can just say that the estimate (3.11) technically is better, since it also allows us to estimate the unknown constant \(\kappa _{0}\) in (3.10). Indeed, \(\kappa _{0}\leq \frac{1}{\gamma _{0}}\).
Another question which may arise is: with the simple estimate (3.11) in hand, what is the necessity of using the estimate (3.5) in Proposition 3.1 (regarding the inequalities)? To answer this question, let’s take a closer look at the estimate (3.5). Indeed, by Proposition 3.1, we have
where
We observe that, on the one hand, \(\Vert L(x)\Vert \leq \Vert L(x)\Vert + \sum_{i=1}^{k}[\langle x_{i}^{*},x \rangle ]_{+}\) and, on the other hand, \(\frac{1}{\gamma }\leq \frac{1}{\gamma _{0}}\). As a result, we cannot generally compare the righthand sides of the estimates (3.11) and (3.12) to determine which is better. Corollary 3.3 says that when \(x\in \mathbb{C}_{\leq }\), it would be better to use (3.12).
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The authors would like to thank the anonymous referees whose meticulous reading helped us improve the presentation.
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This research benefited from the support of the FMJH Program PGMO and from the support of EDF.
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Dedicated to the memory of Jonathan Borwein, in recognition of his deep contributions to mathematics and of his lasting friendship. He was one of the pioneers of metric regularity.
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Abbasi, M., Théra, M. Strongly regular points of mappings. Fixed Point Theory Algorithms Sci Eng 2021, 14 (2021). https://doi.org/10.1186/s1366302100699z
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DOI: https://doi.org/10.1186/s1366302100699z