# Stability Criteria and Calculus Rules via Conic Contingent Coderivatives in
Banach Spaces

Boris S. Mordukhovich Department of Mathematics, Wayne State University
Detroit MI 48202 (e-mail: aa1086@wayne.edu). Research of this author was
partly supported by the U.S. National Science Foundation under grants
DMS-1808978 and DMS-2204519, by the Australian Research Council under grant
DP-190100555, and by Project 111 of China under grant D21024. Pengcheng Wu
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong
Kong (e-mail: pcwu0725@163.com). Xiaoqi Yang Department of Applied
Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
(mayangxq@polyu.edu.hk). Research of this author was partly supported by the
Research Grants Council of Hong Kong (PolyU 15205223).

Abstract. This paper addresses the study of novel constructions of variational
analysis and generalized differentiation that are appropriate for
characterizing robust stability properties of constrained set-valued
mappings/multifunctions between Banach spaces important in optimization theory
and its applications. Our tools of generalized differentiation revolves around
the newly introduced concept of $\varepsilon$-regular normal cone to sets and
associated coderivative notions for set-valued mappings. Based on these
constructions, we establish several characterizations of the central stability
notion known as the relative Lipschitz-like property of set-valued mappings in
infinite dimensions. Applying a new version of the constrained extremal
principle of variational analysis, we develop comprehensive sum and chain
rules for our major constructions of conic contingent coderivatives for
multifunctions between appropriate classes of Banach spaces.
Keywords: variational analysis, generalized differentiation, well-posedness,
relative Lipschitzian stability, constrained systems, normal cones and
coderivatives, calculus rules
Mathematics Subject Classifications (2010): 49J53, 49J52, 49K40

##  1 Introduction

This paper is devoted to developing adequate machinery of variational analysis
and generalized differentiation to study robust well-posedness properties of
set-valued mappings subject to constraints. The importance of such properties
and their applications to optimization, control, and related areas has been
highly recognized over the years, while the main attention was paid to
unconstrained mappings in finite and infinite dimensions; see, e.g., the books
[1, 2, 5, 7, 8, 11, 12] with the extensive bibliographies therein. The study
and applications of such properties for set-valued mappings relative to given
sets has begun quite recently in [6, 10, 13], where the challenges in the
study of the constrained properties in comparison with their unconstrained
counterparts have been well realized.

In this paper, we focus on investigating the relative Lipschitz-like property
of set-valued mappings between Banach spaces, which in fact is equivalent to
the relative metric regularity and relative linear openness of the inverses;
see [10]. We address here infinite-dimensional settings of Banach spaces,
although the major results obtained below are new even in finite dimensions.
Besides mathematical reasons, the main motivation for our research comes from
the fact that infinite-dimensional problems naturally arise in many areas of
applied mathematics and practical modeling such as dynamic optimization and
equilibria, stochastic programming, finance, systems control, economics etc.

To accomplish our goals, we introduce a novel type of normals to sets labeled
as the regular $\varepsilon$-normal cone. It is essentially different from the
“collection of $\varepsilon$-normals” (which is not a cone) overwhelmingly
used in infinite-dimensional variational analysis; see [7]. Based on this new
normal cone and invoking also the contingent cone to constraint sets, we
define various coderivative constructions for set-valued mappings and show
that these newly introduced coderivatives lead us to complete
characterizations (of the neighborhood and pointbased types) of the relative
Lipschitz-like property of constrained multifunctions with the precise
calculations of the exact Lipschitzian bound.

The efficient usage of the established stability criteria in the case of
structured multifunctions, which appear in constrained problems and
applications, requires well-developed calculus rules. We derive here
comprehensive pointbased chain and sum rules for conic contingent
coderivatives, the limiting constructions from the obtained characterizations
of the relative Lipschitz-like property for general multifunctions. Our proofs
are mainly based on the fundamental extremal principle of variational analysis
and its new relative version needed for the study of constrained set systems.

The rest of the paper is structured as follows. In Section 2, we introduce the
$\varepsilon$-regular normal cone and recall the needed preliminaries from
variational analysis and generalized differentiation. Sections 3 and 4 are
devoted, respectively, to establishing neighborhood and pointbased necessary
conditions, sufficient conditions, and characterizations of the relative
Lipschitz-like property. In Sections 5 and 6, we derive pointbased chain and
sum rules, respectively, for the newly introduced conic contingent
coderivatives with establishing the novel relative extremal principle and
relative fuzzy intersection rule for set systems in Banach spaces. The
concluding Section 7 summarizes the main achievements of the paper and
discusses some directions of the future research.

##  2  $\varepsilon$-Regular Normal Cone and Preliminaries

First we recall some standard notations used in this paper. Given a Banach
space $X$ (which is assumed throughout below unless otherwise stated) and its
topological dual $X^{*}$, the norms of $X$ and $X^{*}$ are denoted by
$\|\cdot\|_{X}$ and $\|\cdot\|_{X^{*}}$, respectively. Denote the closed unit
balls of $X$ and $X^{*}$ by $B_{X}$ and $B_{X^{*}}$, respectively, and use
$\mathbb{S}:=\\{x\in X~{}|~{}\|x\|_{X}=1\\}$ for the unit sphere. For $x\in X$
and $x^{*}\in X^{*}$, the symbol $\langle x^{*},x\rangle$ indicates the
canonical pairing between $X$ and $X^{*}$. We use the notation $``x_{k}\to x"$
for the strong convergence of sequences, while $``x_{k}\rightharpoonup x"$
denotes the weak convergence, i.e., $\langle x^{*},x_{k}\rangle\to\langle
x^{*},x\rangle$ for all $x^{*}\in X^{*}$. The notation
$``x_{k}^{*}\rightharpoonup^{*}x^{*}"$ stands for the weak∗ convergence, i.e.,
$\langle x_{k}^{*},x\rangle\to\langle x^{*},x\rangle$ for any $x\in X$. The
symbol $``x_{k}\stackrel{{\scriptstyle\Omega}}{{\longrightarrow}}x"$ tells us
that $x_{k}\to x$ with $x_{k}\in\Omega$ as $k\to\infty$. The collection of
natural numbers is denoted by ${\rm I\\!N}:=\\{1,2,\ldots\\}$.

Recall that a multifunction $F\colon X\rightrightarrows Y$ between Banach
spaces is positively homogeneous if $0\in F(0)$ and $F(\lambda x)=\lambda
F(x)$ for all $x\in X$ and $\lambda>0$. The norm of a positively homogeneous
multifunction is defined by

$\|F\|:=\sup\big{\\{}\|y\|\;\big{|}\;y\in F(x),\;\|x\|\leq 1\big{\\}}.$ (2.1)

Given a set $\Omega\subset X$ with $\bar{x}\in\Omega$, the symbol
$T(\bar{x};\Omega)$ stands for the tangent/contingent cone to $\Omega$ at
$\bar{x}$. That is, $w\in T(\bar{x};\Omega)$ if there exist sequences
$t_{k}\downarrow 0$ and $\\{w_{k}\\}\subset X$ such that $w_{k}\to w$ as
$k\to\infty$ and $\bar{x}+t_{k}w_{k}\in\Omega$ whenever $k\in{\rm I\\!N}$.
Fixing any $\varepsilon\geq 0$, we now introduce our major new notion of the
$\varepsilon$-regular normal cone to $\Omega$ at $\bar{x}\in\Omega$ by

$\widehat{N}_{\varepsilon}^{c}(\bar{x};\Omega):=\Big{\\{}x^{*}\in
X^{*}~{}\Big{|}~{}\limsup\limits_{x\stackrel{{\scriptstyle\Omega}}{{\longrightarrow}}\bar{x}}\frac{\langle
x^{*},x-\bar{x}\rangle}{\|x-\bar{x}\|_{X}}\leq\varepsilon\|x^{*}\|_{X^{*}}\Big{\\}},$
(2.2)

where “$c$” stands for “cone”. Observe that
$\widehat{N}_{\varepsilon}^{c}(\bar{x};\Omega)$ is indeed a cone for any
$\varepsilon\geq 0$, in contrast to the $\varepsilon$-normal set to $\Omega$
at $\bar{x}$ defined by

$\widehat{N}_{\varepsilon}(\bar{x};\Omega):=\Big{\\{}x^{*}\in
X^{*}~{}\Big{|}~{}\limsup\limits_{x\stackrel{{\scriptstyle\Omega}}{{\longrightarrow}}\bar{x}}\frac{\langle
x^{*},x-\bar{x}\rangle}{\|x-\bar{x}\|_{X}}\leq\varepsilon\Big{\\}},$ (2.3)

which is broadly used in variational analysis; see [7]. When $\varepsilon=0$,
both sets in (2.2) and (2.3) reduce to the regular normal cone to $\Omega$ at
$\bar{x}$ denoted by $\widehat{N}(\bar{x};\Omega)$. It is well known that the
latter notion is associated with the regular subdifferential of an extended-
real-valued function $\varphi\colon X\to\overline{{\rm
I\\!R}}:=(-\infty,\infty]$ at $\bar{x}$ with $\varphi(\bar{x})<\infty$ given
by

$\widehat{\partial}\varphi(\bar{x}):=\Big{\\{}x^{*}\in
X^{*}~{}\Big{|}~{}\limsup\limits_{x\to\bar{x}}\frac{\varphi(x)-\varphi(\bar{x})-\langle
x^{*},x-\bar{x}\rangle}{\|x-\bar{x}\|_{X}}\geq 0\Big{\\}}.$ (2.4)

Consider further a multifunction $S:X\rightrightarrows Y$ between Banach
spaces with the graph

$\mbox{\rm gph}\,S:=\big{\\{}(x,y)\in X\times Y\;\big{|}\;y\in S(x)\big{\\}}$

and denote by ${\cal N}(z)$ the collection of (open) neighborhoods around a
point $z$.

######  Definition 2.1.

A mapping $S:X\rightrightarrows Y$ has the Lipschitz-like property relative to
$\Omega$ around $(\bar{x},\bar{y})\in\mbox{\rm gph}\,S$ with
$\bar{x}\in\Omega$ if there are neighborhoods $V\in\mathcal{N}(\bar{x})$ and
$W\in\mathcal{N}(\bar{y})$ together with a constant $\kappa\geq 0$ such that

$S(x^{\prime})\cap W\subset S(x)+\kappa\|x^{\prime}-x\|_{X}B_{X}\;\mbox{ for
all }\;x,x^{\prime}\in\Omega\cap V.$

The exact Lipschitzian bound of $S$ relative to $\Omega$ at
$(\bar{x},\bar{y})$ is defined by

$\displaystyle\operatorname{lip}_{\Omega}S(\bar{x},\bar{y}):=\inf\big{\\{}\kappa\geq
0$ $\displaystyle~{}\big{|}~{}\exists
V\in\mathcal{N}(\bar{x}),W\in\mathcal{N}(\bar{y})~{}\operatorname{such~{}that}$
$\displaystyle S(x^{\prime})\cap W\subset
S(x)+\kappa\|x^{\prime}-x\|_{X}B_{X}\;\mbox{ for all
}\;x,x^{\prime}\in\Omega\cap V\big{\\}}.$

The duality mapping $J\colon X\rightrightarrows X^{*}$ between a Banach space
$X$ and its dual is given by

$J(x):=\big{\\{}x^{*}\in X^{*}~{}\big{|}~{}\langle
x^{*},x\rangle=\|x\|_{X}^{2},~{}\|x^{*}\|_{X^{*}}=\|x\|_{X}\big{\\}}\;\mbox{
for all }\;x\in X.$ (2.5)

We clearly have $J(\lambda x)=\lambda J(x)$ whenever $\lambda\in\mathbb{R}$
and get by [1, Proposition 2.12] that $J(x)\not=\emptyset$ for all $x\in X$.
It is well known from geometric theory of Banach spaces (see, e.g., [4]) that
any reflexive Banach space $X$ admits a renorming $\|\cdot\|$ such that both
$X$ and $X^{*}$ are locally uniformly convex and $\|\cdot\|$ is Fréchet
differentiable at each nonzero point of $X$. Unless otherwise stated, in what
follows we always consider reflexive Banach spaces with such a renorming.
Recall from [3, Proposition 8] that the duality mapping (2.5) in reflexive
spaces is single-valued and bicontinuous.

######  Proposition 2.2.

Let $X$ be a reflexive Banach space, let $\Omega\subset X$ be a closed and
convex set, and let $x^{*}\in J(T(\bar{x};\Omega))$. Then for any
$\varepsilon>0$, there exists $\delta>0$ such that we have

$x^{*}\in J\big{(}T(x;\Omega)\big{)}+\varepsilon B_{X^{*}}\;\mbox{ whenever
}\;x\in\Omega\cap(\bar{x}+\delta B_{X}).$

###### Proof.

Suppose on the contrary that the claimed inclusion fails and then find a
number $\varepsilon_{0}>0$ and a sequence
$x_{k}\stackrel{{\scriptstyle\Omega}}{{\longrightarrow}}\bar{x}$ as
$k\to\infty$ such that

$x^{*}\notin J\big{(}T(x_{k};\Omega)\big{)}+\varepsilon_{0}B_{X^{*}}\;\mbox{
for all }\;k\in\mathbb{N}.$ (2.6)

By the aforementioned properties of the duality mapping, we deduce from
$J^{-1}(x^{*})\in T(\bar{x};\Omega)$ and the well-known relationship

$T(\bar{x};\Omega)\subset
s\mbox{-}\liminf\limits_{x\stackrel{{\scriptstyle\Omega}}{{\longrightarrow}}\bar{x}}T(x;\Omega),$

valid for convex sets $\Omega$, that there exists a sequence of $z_{k}\in
T(x_{k};\Omega)$ such that $z_{k}\to J^{-1}(x^{*})$ for all $k\in\mathbb{N}$.
Letting $x_{k}^{*}:=J(z_{k})$, we get that

$x_{k}^{*}\in
J\big{(}T(x_{k};\Omega)\big{)}~{}~{}\operatorname{and}~{}~{}x_{k}^{*}\to
x^{*}\;\mbox{ as }\;k\to\infty,$

which contradicts (2.6) and completes the proof of the proposition. ∎

##  3 Neighborhood Conditions for Relative Lipschitz-like Multifunctions

The goal of this section is to derive necessary conditions, sufficient
conditions, and complete characterizations of the Lipschitz-like property for
set-valued mappings $S\colon X\to Y$ relative to sets $\Omega$ in terms of the
$\varepsilon$-regular normal cone (2.2) to the graph of the constrained
mapping $S|_{\Omega}$ with the usage of the contingent cone $T_{\Omega}(x)$ to
the constraint set $\Omega$. According to our general scheme (cf. [7, 10]),
such a construction can be viewed as the $\varepsilon$-coderivative of
constrained mapping in question; see Definition 4.2(i). Note the conditions
derived in this section in the case of set-valued between reflexive Banach
spaces are neighborhood, i.e., they involve points nearby the reference one.

First we present a neighborhood necessary condition for the Lipschitz-like
property of multifunctions relative to arbitrary constraint sets.

######  Lemma 3.1.

Let $S:X\rightrightarrows Y$ be a set-valued mapping between reflexive Banach
spaces $X$ and $Y$, let $\Omega\subset X$ be a nonempty set, and let
$(\bar{x},\bar{y})\in\operatorname{gph}S|_{\Omega}$. If $S$ has the Lipschitz-
like property relative to $\Omega$ around $(\bar{x},\bar{y})$ with constant
$\kappa\geq 0$, then for any $\varepsilon\in(0,\frac{1}{\sqrt{\kappa^{2}+1}})$
there exists $\delta>0$ such that we have the condition

$\widehat{N}_{\varepsilon}^{c}\big{(}(x,y);\mbox{\rm
gph}\,S|_{\Omega}\big{)}\cap[J\big{(}T(x;\Omega)\big{)}\times\\{0\\}]=\\{(0,0)\\}$
(3.1)

whenever $(x,y)\in\mbox{\rm gph}\,S|_{\Omega}\cap[(\bar{x}+\delta
B_{X})\times(\bar{y}+\delta B_{Y})]$.

###### Proof.

Having the imposed relative Lipschitz-like property gives us $\delta>0$ with

$S(x)\cap(\bar{y}+2\delta B_{Y})\subset
S(x^{\prime})+\kappa\|x-x^{\prime}\|_{X}B_{X}\;\mbox{ for all
}\;x,x^{\prime}\in\Omega\cap(\bar{x}+2\delta B_{X}).$ (3.2)

Picking $\varepsilon\in(0,\frac{1}{\sqrt{\kappa^{2}+1}})$, fix
$(x,y)\in\text{gph}~{}S|_{\Omega}\cap[(\bar{x}+\delta
B_{X})\times(\bar{y}+\delta B_{Y})]$ and $x^{*}\in
T(x;\Omega)\setminus\\{0\\}$. By definition of the contingent cone, there
exist $t_{k}\downarrow 0$ and $x_{k}^{*}\to x^{*}$ as $k\to\infty$ such that

$x_{k}:=x+t_{k}x_{k}^{*}\in\Omega\;\mbox{ for all }\;k\in{\rm I\\!N},$

which implies the existence of $k_{0}\in{\rm I\\!N}$ ensuring the estimates

$t_{k}<\frac{\delta}{2\|x^{*}\|_{X^{*}}}~{}~{}\text{and}~{}~{}\|x_{k}^{*}-x^{*}\|_{X^{*}}<\frac{\|x^{*}\|_{X^{*}}}{4}\;\mbox{for
all }\;k>k_{0}.$

This clearly leads us to the inequalities

$\displaystyle\|x_{k}-\bar{x}\|_{X}$
$\displaystyle\leq\|x_{k}-x\|_{X}+\|x-\bar{x}\|_{X}<\frac{\delta}{2\|x^{*}\|_{X^{*}}}\|x_{k}^{*}\|_{X^{*}}+\delta$
$\displaystyle\leq\frac{\delta}{2\|x^{*}\|_{X^{*}}}(\|x^{*}\|_{X^{*}}+\|x^{*}-x_{k}^{*}\|_{X^{*}})+\delta<2\delta.$

It follows from (3.2) that for all natural numbers $k>k_{0}$ we get

$y\in S(x)\cap(\bar{y}+2\delta B_{Y})\subset S(x_{k})+\kappa\|x-x_{k}\|_{X}.$

Therefore, there exist sequences $y_{k}\in S(x_{k})$ and
$\kappa_{k}\downarrow\kappa$ such that

$\|y_{k}-y\|_{Y}\leq\kappa_{k}\|x_{k}-x\|_{X},$

which implies in turn that

$\|(x_{k},y_{k})-(x,y)\|_{X\times
Y}=\sqrt{\|x_{k}-x\|_{X}^{2}+\|y_{k}-y\|_{Y}^{2}}\leq\sqrt{\kappa_{k}^{2}+1}\|x_{k}-x\|_{X}.$

Furthermore, the convergence $x_{k}^{*}\to x^{*}$ as $k\to\infty$ allows us to
find $k_{1}>k_{0}$ such that

$\langle J(x^{*}),x_{k}^{*}\rangle\geq\frac{1}{2}\|x^{*}\|^{2}>0\;\mbox{ for
all }\;k>k_{1}.$

Thus for all $k>k_{1}$ we have

$\displaystyle\frac{\langle(J(x^{*}),0),(x_{k},y_{k})-(x,y)\rangle}{\|x^{*}\|_{X^{*}}\|(x_{k},y_{k})-(x,y)\|_{X\times
Y}}\geq\frac{t_{k}\langle
J(x^{*}),x_{k}^{*}\rangle}{\sqrt{\kappa_{k}^{2}+1}\|x^{*}\|_{X^{*}}\|x_{k}-x\|_{X}}$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{\kappa_{k}^{2}+1}}\frac{\langle
J(x^{*}),x_{k}^{*}\rangle}{\|x^{*}\|_{X^{*}}\|x_{k}^{*}\|_{X}}\to\frac{1}{\sqrt{\kappa^{2}+1}}>\varepsilon~{}~{}\text{as}~{}k\to\infty,$

which yields
$(J(x^{*}),0)\notin\widehat{N}_{\varepsilon}^{c}\big{(}(x,u);\operatorname{gph}S|_{\Omega}\big{)}$.
This verifies (3.1) due to the arbitrary choice of
$(x,y)\in\operatorname{gph}~{}S|_{\Omega}\cap[(\bar{x}+\delta
B_{X})\times(\bar{y}+\delta B_{Y})]$ and $x^{*}\in
T(x;\Omega)\setminus\\{0\\}$. ∎

The next lemma provides a refinement of (3.1), which is valid in the same
setting for $S$ and $\Omega$ as in Lemma 3.1, while being useful to establish
the neighborhood sufficient condition for the relative Lipschitz-like property
in Lemma 3.3 below.

######  Lemma 3.2.

Let $S$, $\Omega$, and $(\bar{x},\bar{y})$ be the same as in Lemma 3.1. If
there exist numbers $\varepsilon\in(0,1)$ and $\delta>0$ such that (3.1)
holds, then whenever $\kappa>\sqrt{\frac{1}{\varepsilon^{2}}-1}$ we find
$\gamma\in(0,\varepsilon)$ ensuring the fulfillment of the estimate

$\|x^{*}\|\leq\kappa\|y^{*}\|\;\mbox{ when
}\;(x^{*},y^{*})\in\widehat{N}_{\gamma}^{c}\big{(}(x,y);\mbox{\rm
gph}\,S|_{\Omega}\big{)}\cap[J\big{(}T(x;\Omega)\big{)}\times Y^{*}]$ (3.3)

for all $(x,y)\in\operatorname{gph}S|_{\Omega}\cap[(\bar{x}+\delta
B_{X})\times(\bar{y}+\delta B_{Y})]$.

###### Proof.

Set $\gamma:=\varepsilon-\frac{1}{\sqrt{\kappa^{2}+1}}$ and verify that (3.3)
holds. Suppose on the contrary that there exist
$(x,y)\in\operatorname{gph}S|_{\Omega}\cap[(\bar{x}+\delta
B_{X})\times(\bar{y}+\delta B_{Y})]$ and
$(x^{*},y^{*})\in\widehat{N}_{\gamma}^{c}\big{(}(x,y);\operatorname{gph}S|_{\Omega}\big{)}\cap[J\big{(}T(x;\Omega)\big{)}\times
Y^{*}]$ such that

$\|x^{*}\|_{X^{*}}>\kappa\|y^{*}\|_{Y^{*}}.$

Considering further the pairs

$(x_{N}^{*},y_{N}^{*}):=\frac{\kappa^{2}}{\kappa^{2}+1}(x^{*},y^{*})~{}~{}~{}\text{and}~{}~{}~{}(x_{T}^{*},y_{T}^{*}):=(x^{*},0)-(x_{N}^{*},y_{N}^{*}),$

we easily check the relationships

$\displaystyle\|(x_{T}^{*},y_{T}^{*})\|_{X^{*}\times Y^{*}}=$
$\displaystyle\left\|(x^{*},0)-\frac{\kappa^{2}}{\kappa^{2}+1}(x^{*},y^{*})\right\|_{X^{*}\times
Y^{*}}$ (3.4) $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{(\kappa^{2}+1)^{2}}\|x^{*}\|_{X^{*}}^{2}+\frac{\kappa^{4}}{(\kappa^{2}+1)^{2}}\|y^{*}\|_{Y^{*}}^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\sqrt{\kappa^{2}+1}}\|x^{*}\|_{X^{*}},$
$\|(x_{N}^{*},y_{N}^{*})\|_{X^{*}\times Y^{*}}\leq\|x^{*}\|_{X^{*}}.$ (3.5)

Then we deduce from (3.4) the inequalities

$\displaystyle\langle(x_{T}^{*},y_{T}^{*}),(x^{\prime},y^{\prime})-(x,y)\rangle\leq$
$\displaystyle\|(x_{T}^{*},y_{T}^{*})\|_{X^{*}\times
Y^{*}}\|(x^{\prime},y^{\prime})-(x,y)\|_{X\times Y}$ $\displaystyle\leq$
$\displaystyle\frac{1}{\sqrt{\kappa^{2}+1}}\|x^{*}\|_{X^{*}}\|(x^{\prime},y^{\prime})-(x,y)\|_{X\times
Y},$

which, being combined with (3.5) and
$(x_{N}^{*},y_{N}^{*})\in\widehat{N}_{\gamma}^{c}\big{(}(x,u);\operatorname{gph}S|_{\Omega}\big{)}$,
lead us to

$\displaystyle\limsup\limits_{(x^{\prime},y^{\prime})\stackrel{{\scriptstyle\text{gph}~{}S|_{\Omega}}}{{\longrightarrow}}(x,y)}\frac{\langle(x^{*},0),(x^{\prime},y^{\prime})-(x,y)\rangle}{\|(x^{\prime},y^{\prime})-(x,y)\|_{X\times
Y}}$ $\displaystyle=$
$\displaystyle\limsup\limits_{(x^{\prime},y^{\prime})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(x,y)}\frac{\langle(x_{N}^{*},y_{N}^{*})+(x_{T}^{*},y_{T}^{*}),(x^{\prime},y^{\prime})-(x,y)\rangle}{\|(x^{\prime},y^{\prime})-(x,y)\|_{X\times
Y}}$ $\displaystyle\leq$
$\displaystyle\limsup\limits_{(x^{\prime},y^{\prime})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(x,y)}\frac{\langle(x_{N}^{*},y_{N}^{*}),(x^{\prime},y^{\prime})-(x,y)\rangle}{\|(x^{\prime},y^{\prime})-(x,y)\|_{X\times
Y}}+\frac{1}{\sqrt{\kappa^{2}+1}}\|x^{*}\|_{X^{*}}$ $\displaystyle\leq$
$\displaystyle\Big{(}\gamma+\frac{1}{\sqrt{\kappa^{2}+1}}\Big{)}\|x^{*}\|_{X^{*}}=\varepsilon\|x^{*}\|_{X^{*}},$

where the last inequality follows from the definition of $\varepsilon$-normal
cone. The latter yields
$(x^{*},0)\in\widehat{N}_{\varepsilon}^{c}\big{(}(x,y);\operatorname{gph}S|_{\Omega}\big{)}$,
which clearly contradicts (3.1) and hence verifies the claimed condition
(3.3). ∎

Having Lemma 3.2 in hand and employing fundamental tools of variational
analysis and generalized differentiation, we show next that the neighborhood
condition (3.3) is actually sufficient for the relative Lipschitz-like
property. Observe that, in contrast to the necessary conditions in Lemmas 3.1
and 3.2, we now require that the graph of $S$ is locally closed, and the set
$\Omega$ is closed and convex.

######  Lemma 3.3.

Let $S\colon X\rightrightarrows Y$ be a set-valued mapping between reflexive
Banach spaces, and let $\Omega\subset X$ be a closed and convex set. Fix
$(\bar{x},\bar{y})\in\operatorname{gph}S|_{\Omega}$, and assume that
$\operatorname{gph}S$ is locally closed around $(\bar{x},\bar{y})$. If there
exist $\delta>0$, $\kappa\geq 0$, and $\gamma\in(0,1)$ such that condition
(3.3) holds for all
$(x,y)\in\operatorname{gph}~{}S|_{\Omega}\cap[(\bar{x}+\delta
B_{X})\times(\bar{y}+\delta B_{Y})]$, then $S$ has the Lipschitz-like property
relative to $\Omega$ around $(\bar{x},\bar{y})$ with constant $\kappa$.

###### Proof.

Assume without loss of generality that $\operatorname{gph}S|_{\Omega}$ is
closed in $X\times Y$ and define the positive number
$\sigma:=\delta/4(\kappa+1)$. Arguing by contradiction, suppose that $S$ does
not have the Lipschitz-like property relative to $\Omega$ around
$(\bar{x},\bar{y})$ with constant $\kappa$. Then there are
$x^{\prime},x^{\prime\prime}\in\Omega\cap(\bar{x}+\sigma B_{X})$ with
$x^{\prime}\not=x^{\prime\prime}$ and $y^{\prime\prime}\in
S(x^{\prime\prime})\cap(\bar{y}+\sigma B_{Y})$ such that

$d(y^{\prime\prime};S(x^{\prime}))>\kappa\|x^{\prime}-x^{\prime\prime}\|_{X}:=\beta$
(3.6)

via the distance function in $Y$. Clearly, we have $0<\beta\leq
2\kappa\sigma$. Let us now show that condition (3.3) from Lemma 3.2 fails in
this case whenever $\gamma\in(0,1)$. Define the extended-real-valued function
$\varphi:X\times Y\to\overline{{\rm I\\!R}}$ by

$\varphi(x,y)=\|x-x^{\prime}\|_{X}+\delta_{\operatorname{gph}S|_{\Omega}}(x,y),\quad(x,y)\in
X\times Y,$

via the indicator function of the set $\mbox{\rm gph}\,S_{\Omega}$ and observe
that $\varphi$ is lower semicontinuous (l.s.c.) by the closedness of gph
$S|_{\Omega}$), that $\inf\varphi$ is finite, and that

$\varphi(x^{\prime\prime},y^{\prime\prime})\leq\inf\varphi+\frac{\beta}{\kappa}.$

Equip the product space $X\times Y$ with the norm

$p(x,y):=\kappa\varrho\|x\|_{X}+\|y\|_{Y},\;\mbox{ where }\;\varrho:={\rm
min}\Big{\\{}\frac{\gamma}{4},\frac{1}{2k}\Big{\\}},$ (3.7)

and apply the fundamental Ekeland variational principle (see, e.g., [7,
Theorem 2.26]) to $\varphi$ on $X\times Y$ with the norm (3.7). This gives us
a pair $(\widetilde{x},\widetilde{y})\in X\times Y$ such that

$\displaystyle
p(\widetilde{x}-x^{\prime\prime},\widetilde{y}-y^{\prime\prime})\leq\beta,$
(3.8)
$\displaystyle\varphi(\widetilde{x},\widetilde{y})\leq\varphi(x^{\prime\prime},u^{\prime\prime}),\;\mbox{
and}$ (3.9)
$\displaystyle\varphi(\widetilde{x},\widetilde{y})\leq\varphi(x,y)+\frac{1}{\kappa}p(x-\widetilde{x},y-\widetilde{y})\;\mbox{
for all }\;(x,y)\in X\times Y.$ (3.10)

It immediately follows from (3.8) that

$\|\widetilde{y}-y^{\prime\prime}\|_{Y}\leq\beta.$ (3.11)

So we have $\widetilde{x}\not=x^{\prime}$, since otherwise

$d(y^{\prime\prime};S(x^{\prime}))=d(y^{\prime\prime};S(\widetilde{x}))\leq\|\widetilde{y}-y^{\prime\prime}\|_{Y}\leq\beta,$

which contradicts (3.6). Furthermore, using the continuity of the duality
mapping (2.5) due to the reflexivity of $X$ yields the existence of $\nu>0$
such that

$\|J(x^{\prime}-x)-J(x^{\prime}-\widetilde{x})\|_{X^{*}}\leq\frac{\varrho}{4}\|x^{\prime}-\widetilde{x}\|_{X}\;\mbox{
for all }\;x\in\widetilde{x}+\nu B_{X}.$ (3.12)

Put $\eta:={\rm
min}\\{\frac{\varrho}{4}\|x^{\prime}-\widetilde{x}\|_{X},\frac{\delta}{4},\varrho,\gamma\\}$
and deduce from (3.10) and [10, Proposition 3.3(ii)] that there exist
$(x_{i},y_{i})\in\operatorname{gph}S|_{\Omega}\cap[(\widetilde{x}+\eta
B_{X})\times(\widetilde{y}+\eta B_{Y})]$, $i=1,2,3$, such that

$(0,0)\in\widehat{\partial}\psi_{1}(x_{1},y_{1})+\widehat{\partial}\psi_{2}(x_{2},y_{2})+\widehat{N}\big{(}(x_{3},y_{3});\operatorname{gph}S|_{\Omega}\big{)}+\eta[B_{X^{*}}\times
B_{Y^{*}}],$

where both $\psi_{1}$ and $\psi_{2}$ are the Lipschitz continuous convex
functions defined by

$\psi_{1}(x,y):=\|x-x^{\prime}\|_{X}~{}~{}\mbox{and}~{}~{}\psi_{2}(x,y):=\frac{1}{\kappa}p(x-\widetilde{x},y-\widetilde{y}).$

Since we clearly have $x_{1}\not=x^{\prime}$ due to
$\widetilde{x}\not=x^{\prime}$ and
$\|x_{1}-\widetilde{x}\|_{X}<\|x^{\prime}-\widetilde{x}\|_{X}$, it follows
from [9, Example 3.36]) that

$\widehat{\partial}\psi_{1}(x_{1},y_{1})=\bigg{(}\frac{J(x_{1}-x^{\prime})}{\|x_{1}-x^{\prime}\|_{X}},0\bigg{)}~{}~{}\operatorname{and}~{}~{}\widehat{\partial}\psi_{2}(x_{2},y_{2})\subset\varrho
B_{X^{*}}\times\frac{1}{\kappa}B_{Y^{*}}.$

Thus there exists $(u_{1},v_{1})\in B_{X^{*}}\times B_{Y^{*}}$ such that

$(x^{*},y^{*})\in\widehat{N}\big{(}(x_{3},y_{3});\operatorname{gph}S|_{\Omega}\big{)}\;\mbox{
with }$ (3.13)
$x^{*}:=\frac{J(x_{1}-x^{\prime})}{\|x^{\prime}-x_{1}\|_{X}}-(\varrho+\eta)u_{1}~{}~{}\mbox{and}~{}~{}y^{*}:=\left(\frac{1}{\kappa}+\eta\right)v_{1}.$

This gives us by taking (3.12) into account that

$\displaystyle\left\|\frac{J(x^{\prime}-x_{i})}{\|x^{\prime}-x_{i}\|_{X}}-\frac{J(x^{\prime}-\widetilde{x})}{\|x^{\prime}-\widetilde{x}\|_{X}}\right\|_{X^{*}}$
$\displaystyle\leq$
$\displaystyle\left\|\frac{J(x^{\prime}-x_{i})}{\|x^{\prime}-x_{i}\|_{X}}-\frac{J(x^{\prime}-x_{i})}{\|x^{\prime}-\widetilde{x}\|_{X}}\right\|_{X^{*}}+\left\|\frac{J(x^{\prime}-x_{i})}{\|x^{\prime}-\widetilde{x}\|_{X}}-\frac{J(x^{\prime}-\widetilde{x})}{\|x^{\prime}-\widetilde{x}\|_{X}}\right\|_{X^{*}}$
$\displaystyle\leq$
$\displaystyle\frac{\|x_{i}-\widetilde{x}\|_{X}}{\|x^{\prime}-\widetilde{x}\|_{X}}+\frac{\varrho}{4}\leq\frac{\varrho}{2}~{}~{}\mbox{for}~{}~{}i=1,3,$

which shows therefore that

$\left\|\frac{J(x^{\prime}-x_{1})}{\|x^{\prime}-x_{1}\|_{X}}-\frac{J(x^{\prime}-x_{3})}{\|x^{\prime}-x_{3}\|_{X}}\right\|_{X^{*}}\leq\varrho.$

Define $\hat{x}^{*}:=\frac{J(x^{\prime}-x_{3})}{\|x^{\prime}-x_{3}\|_{X}}$ and
$\hat{y}^{*}:=\left(\frac{1}{\kappa}-\eta\right)v_{1}$. Then we get

$\|\hat{x}^{*}\|_{X^{*}}>\kappa\|\hat{y}^{*}\|_{Y^{*}}$ (3.14)

and find $u_{2}\in B_{X^{*}}$ satisfying

$\hat{x}^{*}=x^{*}+(\varrho+\eta)u_{1}+\varrho
u_{2}~{}~{}\operatorname{and}~{}~{}\hat{y}^{*}=y^{*}-2\eta v_{1}.$ (3.15)

Observe that $\hat{x}^{*}\in J(T(x_{3};\Omega))$ by the convexity of $\Omega$
and the homogeneity of $J$. It follows from (3.9) that
$(\widetilde{x},\widetilde{u})\in\text{gph}~{}S|_{X}$, and so
$\|\widetilde{x}-x^{\prime}\|\leq\|x^{\prime\prime}-x^{\prime}\|$. Employing
further the triangle inequality yields

$\displaystyle\|x_{3}-\bar{x}\|_{X}\leq$
$\displaystyle\|x_{3}-\widetilde{x}\|_{X}+\|\widetilde{x}-x^{\prime}\|_{X}+\|x^{\prime}-\bar{x}\|_{X}$
$\displaystyle\leq$
$\displaystyle\eta+\|x^{\prime\prime}-\bar{x}\|_{X}+2\|x^{\prime}-\bar{x}\|_{X}\leq\frac{\delta}{4}+3\sigma<\delta,$

which being combined with (3.11) tells us that

$\|y_{3}-\bar{y}\|_{Y}\leq\|y_{3}-\widetilde{y}\|_{Y}+\|\widetilde{y}-y^{\prime\prime}\|_{Y}+\|y^{\prime\prime}-\bar{y}\|_{Y}\leq\eta+\beta+\sigma<\delta.$

This shows that
$(x_{3},y_{3})\in\operatorname{gph}~{}S|_{\Omega}\cap[(\bar{x}+\delta
B_{X})\times(\bar{y}+\delta B_{Y})]$ and thus verifies (3.3) for
$(x,y):=(x_{3},y_{3})$. Furthermore, it follows from (3.13) and (3.15) that

$\displaystyle\limsup\limits_{(x,y)\stackrel{{\scriptstyle\text{gph}~{}S|_{\Omega}}}{{\longrightarrow}}(x_{3},y_{3})}\frac{\langle(\hat{x}^{*},\hat{y}^{*}),(x,y)-(x_{3},y_{3})\rangle}{\|(x,y)-(x_{3},y_{3})\|_{X\times
Y}}$ $\displaystyle=$
$\displaystyle\limsup\limits_{(x,y)\stackrel{{\scriptstyle\text{gph}~{}S|_{\Omega}}}{{\longrightarrow}}(x_{3},y_{3})}\frac{\langle(x^{*}+(\varrho+\eta)u_{1}+\varrho
u_{2},y^{*}-2\eta
v_{1}),(x,y)-(x_{3},y_{3})\rangle}{\|(x,y)-(x_{3},y_{3})\|_{X\times Y}}$
$\displaystyle\leq$
$\displaystyle\limsup\limits_{(x,y)\stackrel{{\scriptstyle\text{gph}~{}S|_{\Omega}}}{{\longrightarrow}}(x_{3},y_{3})}\frac{\langle(x^{*},y^{*}),(x,y)-(x_{3},y_{3})\rangle}{\|(x,y)-(x_{3},y_{3})\|_{X\times
Y}}+4\varrho$ $\displaystyle\leq$
$\displaystyle\gamma\leq\gamma\|(\hat{x}^{*},\hat{y}^{*})\|_{X^{*}\times
Y^{*}},$

which justifies
$(\hat{x}^{*},\hat{y}^{*})\in\widehat{N}_{\gamma}^{c}\big{(}(x_{3},y_{3});\operatorname{gph}S|_{\Omega}\big{)}$.
Combining the latter with the estimate in (3.14) contradicts (3.3) and hence
completes the proof of the claimed sufficiency. ∎

Unifying now the above lemmas brings us to the complete neighborhood
characterizations of the relative Lipschitz-like property of set-valued
mappings with the precise calculation of the exact Lipschitzian bound.

######  Theorem 3.4.

Let $S:X\rightrightarrows Y$ be a set-valued mapping between reflexive Banach
spaces, and let $\Omega\subset X$ be a closed and convex set. Assume that the
set $\operatorname{gph}S|_{\Omega}$ is closed around some point
$(\bar{x},\bar{y})\in\operatorname{gph}S|_{\Omega}$. Then given
$\varepsilon_{0}\in(0,1)$ and $\kappa_{0}\geq 0$ with
$\varepsilon_{0}=\frac{1}{\sqrt{\kappa_{0}^{2}+1}}$, the following statements
are equivalent:

(ii) For any $\kappa>\kappa_{0}$, $S$ has the Lipschitz-like property relative
to $\Omega$ with constant $\kappa$.

(ii) For any $\varepsilon\in(0,\varepsilon_{0})$, there exists $\delta>0$ such
that (3.1) holds.

(iii) For any $\kappa>\kappa_{0}$, there exist $\nu\in(0,\nu_{0})$ and
$\delta>0$ such that (3.3) holds.
Moreover, the exact Lipschitz bound of $S$ relative to $\Omega$ around
$(\bar{x},\bar{y})$ is calculated by

$\displaystyle\operatorname{lip}_{\Omega}S(\bar{x}|\bar{y})\hskip-5.69046pt$
$\displaystyle=$ $\displaystyle\hskip-5.69046pt\lim\limits_{\delta\to
0}\inf\bigg{\\{}\sqrt{\frac{1}{\varepsilon^{2}}-1}~{}\bigg{|}~{}\widehat{N}_{\varepsilon}^{c}\big{(}(x,y);\operatorname{gph}S|_{\Omega}\big{)}\times[J(T(x;\Omega))\times\\{0\\}]=\\{(0,0)\\},$
$\displaystyle\hskip 28.45274pt\mbox{ for all
}\;(x,y)\in\operatorname{gph}S|_{\Omega}\times[(\bar{x}+\delta
B_{X})\times(\bar{y}+\delta B_{Y})]\\}\bigg{\\}}.$ (3.16)

As we see, the exact bound formula (3.16) makes use of the parameter
$\varepsilon$ in the $\varepsilon$-regular normal cone. An explanation of
(3.16) can be as follows. Let
$\kappa_{0}:=\operatorname{lip}_{\Omega}S(\bar{x}|\bar{y})\geq 0$. We know
that $\varepsilon$ in the definition of
$\widehat{N}_{\varepsilon}^{c}\big{(}(x,y);\operatorname{gph}S|_{\Omega}\big{)}$
cannot be greater than 1, since otherwise the normal cone is the entire space.
When $\delta$ in (3.16) is getting smaller, so is the neighborhood
$(\bar{x}+\delta B_{X})\times(\bar{y}+\delta B_{Y})$. Thus the range of the
$\varepsilon$’s that satisfies the condition
$\widehat{N}_{\varepsilon}^{c}\big{(}(x,y);\operatorname{gph}S|_{\Omega}\big{)}\times[J(T(x;\Omega))\times\\{0\\}]=\\{(0,0)\\}$
is getting larger. Therefore, the “inf” is a decreasing function of $\delta$
with a reachable lower bound $\kappa_{0}$.

Theorem 3.4 tells us that condition (3.1) provides a complete neighborhood
characteristics of the relative Lipschitz-like property of set-valued mappings
when the constraint set $\Omega$ is closed and convex. However, such s
characterization cannot be achieved if the $\varepsilon$-regular normal cone
(2.2) is replaced by the set of $\varepsilon$-normals (2.3). The following
example confirms this statement even in the case of the simplest single-valued
mapping $S\colon{\rm I\\!R}\to{\rm I\\!R}$ and the closed interval $\Omega$ in
${\rm I\\!R}$.

######  Example 3.5.

Let $S:\mathbb{R}\to\mathbb{R}$ be defined by $S(x):=x$ for all
$x\in\mathbb{R}$, and let $\Omega:=[0,1]$. Clearly, $S$ has the Lipschitz-like
property relative to $\Omega$ around $(0,0)$. In this case, $J$ is the
identity mapping, $T(0;\Omega)=[0,\infty)$, and

$\widehat{N}_{\varepsilon}((0,0);\operatorname{gph}S|_{\Omega})=\big{\\{}(x^{*},y^{*})\in{\rm
I\\!R}^{2}~{}\big{|}~{}x^{*}+y^{*}\leq\sqrt{2}\varepsilon\big{\\}}$

for any $\varepsilon>0$. Therefore, we have the expression

$\widehat{N}_{\varepsilon}((0,0);\operatorname{gph}S|_{\Omega})\cap[T(0;\Omega)\times\\{0\\}]=[0,\sqrt{2}\varepsilon]\times\\{0\\},$

which tells us that the counterpart of (3.1) fails for the set of
$\varepsilon$-normals (2.3).

##  4 Pointbased Coderivative Characterizations of the Relative Lipschitz-like
Property

The usage of the neighborhood characterizations of the relative Lipschitz-like
property obtained in Section 3 may be challenging to apply, since they involve
the consideration of neighborhood points. Much more preferable are pointbased
criteria, which are expressed precisely at the reference point. To derive such
pointbased results, in this section we introduce new coderivative
constructions, which enjoy the major pointbased calculus rules developed in
the subsequent Sections 5 and 6.

Recall first the notions of (sequential) outer and inner limits for
multifunctions acting between Banach spaces that are used in what follows.

######  Definition 4.1.

Let $S:X\rightrightarrows Y$ be a set-valued mapping between Banach spaces,
and let $\bar{x}\in{\rm dom}\,S:=\\{x\in X\;|\;S(x)\neq\emptyset\\}$. Then we
have:

(i) The strong and weak∗ $($if $Y=Z^{*}$ for some Banach space Z$)$ outer
limits are defined, respectively, by

$\displaystyle s\mbox{-}\mathop{{\rm Lim}\,{\rm
sup}}_{x\to\bar{x}}S(x):=\big{\\{}y\in Y~{}\big{|}~{}\exists
x_{k}\to\bar{x}~{}\operatorname{and}~{}y_{k}\in
S(x_{k})~{}\operatorname{such~{}that}~{}y_{k}\to y\big{\\}},$ $\displaystyle
w^{*}\mbox{-}\mathop{{\rm Lim}\,{\rm sup}}_{x\to\bar{x}}S(x):=\big{\\{}y\in
Y~{}\big{|}~{}\exists x_{k}\to\bar{x}~{}\operatorname{and}~{}y_{k}\in
S(x_{k})~{}\operatorname{such~{}that}~{}y_{k}\rightharpoonup^{*}y\big{\\}}.$

(ii) The strong inner limit is described by

$s\mbox{-}\mathop{{\rm Lim}\,{\rm inf}}_{x\to\bar{x}}S(x):=\big{\\{}y\in
Y~{}\big{|}~{}\forall x_{k}\to\bar{x},~{}\exists y_{k}\in
S(x_{k})~{}\operatorname{such~{}that}~{}y_{k}\to y\big{\\}}.$

Next we define the coderivative notions needed for our characterizations. Note
that the $\varepsilon$-coderivative construction from Definition 4.2(i) was
actually used in Section 3, where we preferred to deal directly with the
$\varepsilon$-regular normal cone. Similarly to [10], observe that, in
comparing with the recent projectional coderivative studied in [6] via
projections of normals to the tangent cone of the constraint set, here we are
motivated by the fact that the intersection of the $\varepsilon$-regular
normal cone with the contingent cone would be much easier to deal with than to
find projections.

######  Definition 4.2.

Let $S:X\rightrightarrows Y$ be a set-valued mapping between Banach spaces,
let $\Omega$ be a nonempty subset of $X$, and let
$(\bar{x},\bar{y})\in\operatorname{gph}S|_{\Omega}$. We have the following
constructions:

(i) Given $\varepsilon\geq 0$, the $\varepsilon$-conic regular contingent
coderivative of $S$ relative to $\Omega$ at $(\bar{x},\bar{y})$ is defined as
a multifunction
$\widehat{D}_{\Omega,\varepsilon}^{c*}S(\bar{x}|\bar{y}):Y^{*}\rightrightarrows
X^{*}$ with the values

$\begin{array}[]{ll}\widehat{D}_{\Omega,\varepsilon}^{c*}S(\bar{x}|\bar{y})(y^{*}):=\big{\\{}x^{*}\in
X^{*}~{}\big{|}&(x^{*},-y^{*})\in\widehat{N}_{\varepsilon}^{c}\big{(}(\bar{x},\bar{y});\operatorname{gph}S|_{\Omega}\big{)}\\\
&\cap\big{[}J\big{(}T(\bar{x};\Omega)\big{)}\times
Y^{*}\big{]}\big{\\}}.\end{array}$ (4.1)

(ii) The conic normal contingent coderivative of $S$ relative to $\Omega$ at
$(\bar{x},\bar{y})$ is a multifunction
$\widetilde{D}_{\Omega}^{c*}S(\bar{x}|\bar{y}):Y^{*}\rightrightarrows X^{*}$
defined by

$\widetilde{D}_{\Omega}^{c*}S(\bar{x}|\bar{y})(\bar{y}^{*}):=w^{*}\mbox{-}\mathop{{\rm
Lim}\,{\rm
sup}}_{\stackrel{{\scriptstyle\stackrel{{\scriptstyle(x,y)\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y})}}{{y^{*}\rightharpoonup^{*}\bar{y}^{*}}}}}{{\varepsilon\downarrow
0}}}\widehat{D}_{\Omega,\varepsilon}^{c*}S(x|y)(y^{*}).$ (4.2)

That is, (4.2) is the collection of such $\bar{x}^{*}\in X^{*}$ for which
there exist sequences $\varepsilon_{k}\downarrow 0$,
$(x_{k},y_{k})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y})$,
and $(x_{k}^{*},y_{k}^{*})\rightharpoonup^{*}(\bar{x}^{*},\bar{y}^{*})$ with

$(x_{k}^{*},-y_{k}^{*})\in\widehat{N}^{c}_{\varepsilon_{k}}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{\Omega}\big{)}\cap\big{[}J\big{(}T(x_{k};\Omega)\big{)}\times
Y^{*}\big{]},\quad k\in\mathbb{N}.$ (4.3)

(iii) The conic mixed contingent coderivative of $S$ relative to $\Omega$ at
$(\bar{x},\bar{y})$ is a multifunction
$D_{\Omega}^{c*}S(\bar{x}|\bar{y}):Y^{*}\rightrightarrows X^{*}$ defined by

$D_{\Omega}^{c*}S(\bar{x}|\bar{y})(\bar{y}^{*}):=w^{*}\mbox{-}\mathop{{\rm
Lim}\,{\rm
sup}}_{\stackrel{{\scriptstyle\stackrel{{\scriptstyle(x,y)\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y})}}{{y^{*}\longrightarrow\bar{y}^{*}}}}}{{\varepsilon\downarrow
0}}}\widehat{D}_{\Omega,\varepsilon}^{c*}S(x|y)(y^{*}).$ (4.4)

That is, (4.4) is the collection of such $\bar{x}^{*}\in X^{*}$ for which
there exist sequences $\varepsilon_{k}\downarrow 0$,
$(x_{k},y_{k})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y})$,
$y_{k}^{*}\to\bar{y}^{*}$ and $x_{k}^{*}\rightharpoonup^{*}\bar{x}^{*}$ with
$(x^{*}_{k},y^{*}_{k})$ satisfying (4.3).

(iv) We say that $S$ is coderivatively normal relative to $\Omega$ at
$(\bar{x},\bar{y})$ if

$\|D^{c*}_{\Omega}S(\bar{x},\bar{y})\|=\|\widetilde{D}^{c*}_{\Omega}S(\bar{x},\bar{y})\|$

via the coderivative norms as positively homogeneous mappings (2.1).

Note that the $\varepsilon$-conic regular contingent coderivative from
Definition 4.2(i) differs from the $\varepsilon$-contingent relative
coderivative from [10, Definition 2.4(i)] by using the $\varepsilon$-regular
normal cone (2.2) instead of the $\varepsilon$-normal set (2.3). As shown in
the previous Section 3, the usage of the new $\varepsilon$-coderivative allows
us to establish neighborhood characterizations of the relative Lipschitz-like
property of set-valued mappings, which is not the case for the
$\varepsilon$-construction from [10]; see Example 3.5.

On the hand, the next lemma tells us that the limiting constructions from
Definition 4.2(ii,iii) agrees with the corresponding limiting coderivatives
from [10, Definition 2.4] in the general Banach space setting. Note that this
does not diminish the value of our new coderivative constructions. First of
all, they are simpler to deal with and allow us to essentially simplify the
proofs of the stability results from [10]; see, e.g., Theorem 4.5 below.
Furthermore, it is most important for variational theory and applications that
the newly introduced limiting coderivatives enjoy the fundamental calculus
rules derived in Sections 5 and 6. These results are also new for the
coderivative constructions from [10] for which calculus rules have not been
developed earlier.

######  Lemma 4.3.

Let $S:X\rightrightarrows Y$ be a set-valued mapping between Banach spaces,
let $\Omega\subset X$, and let
$(\bar{x},\bar{y})\in\operatorname{gph}S|_{\Omega}$. The following hold:

(i) We have $x^{*}\in D_{\Omega}^{c*}S(\bar{x}|\bar{y})(y^{*})$ if and only if
there exist sequences $\varepsilon_{k}\downarrow 0$,
$(x_{k},y_{k})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y})$,
and
$(x_{k}^{*},-y^{*})\in\widehat{N}_{\varepsilon_{k}}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{\Omega}\big{)}\cap\big{[}J\big{(}T(x_{k};\Omega)\big{)}\times
Y^{*}\big{]}$ such that $y_{k}^{*}\to y^{*}$ and
$x_{k}^{*}\rightharpoonup^{*}x^{*}$ as $k\to\infty$.

(ii) We have $x^{*}\in\widetilde{D}_{\Omega}^{c*}S(\bar{x}|\bar{y})(y^{*})$ if
and only if there exist sequences $\varepsilon_{k}\downarrow 0$,
$(x_{k},y_{k})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y})$,
and
$(x_{k}^{*},-y_{k}^{*})\in\widehat{N}_{\varepsilon_{k}}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{\Omega}\big{)}\cap\big{[}J\big{(}T(x_{k};\Omega)\big{)}\times
Y^{*}\big{]}$ such that $y_{k}^{*}\rightharpoonup^{*}y^{*}$ and
$x_{k}^{*}\rightharpoonup^{*}x^{*}$ as $k\to\infty$.

###### Proof.

To verify (i), consider the set

$\begin{array}[]{ll}K:=\big{\\{}&x^{*}\in
X^{*}~{}\big{|}~{}\operatorname{there~{}exist}~{}\varepsilon_{k}\downarrow
0,(x_{k},y_{k})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y}),\\\
&\operatorname{and}~{}(x_{k}^{*},-y_{k}^{*})\in\widehat{N}_{\varepsilon_{k}}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{M}\big{)}\cap\big{[}J\big{(}T(x_{k};\Omega)\big{)}\times
Y^{*}\big{]}\\\
&\operatorname{such~{}that}y_{k}^{*}\rightharpoonup^{*}y^{*}\;\mbox{ and
}\;x_{k}^{*}\rightharpoonup^{*}x^{*}\big{\\}}.\end{array}$ (4.5)

Pick any $x^{*}\in D_{\Omega}^{c*}S(\bar{x}|\bar{y})(y^{*})$ and find by
definition (4.4) sequences

$\varepsilon_{k}\downarrow
0,~{}~{}(x_{k},y_{k})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y}),\;\mbox{
and }$ (4.6)
$(x_{k}^{*},-y_{k}^{*})\in\widehat{N}_{\varepsilon_{k}}^{c}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{\Omega}\big{)}\cap\big{[}J\big{(}T(x_{k};\Omega)\big{)}\times
Y^{*}\big{]}$ (4.7)

for which we have the convergence

$y_{k}^{*}\to
y^{*}~{}~{}\operatorname{and}~{}~{}x_{k}^{*}\rightharpoonup^{*}x^{*}\;\mbox{
as }\;k\to\infty.$ (4.8)

It follows from (4.8) by the uniform boundedness principle that there is $L>0$
with

$\|(x_{k}^{*},-y_{k}^{*})\|_{X^{*}\times Y^{*}}\leq L\;\mbox{ for all
}\;k\in{\rm I\\!N}.$

This implies therefore by definition (2.2) that

$\limsup\limits_{(x,y)\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(x_{k},y_{k})}\frac{\langle(x_{k}^{*},-y_{k}^{*}),(x,y)-(x_{k},y_{k})\rangle}{\|(x,y)-(x_{k},y_{k})\|_{X\times
Y}}\leq\varepsilon_{k}\|(x_{k}^{*},-y_{k}^{*})\|_{X^{*}\times Y^{*}}\leq
L\varepsilon_{k}.$

Using the latter together with (4.7) ensures that

$(x_{k}^{*},-y_{k}^{*})\in\widehat{N}_{L\varepsilon_{k}}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{\Omega}\big{)}\cap\big{[}J\big{(}T(x_{k};\Omega)\big{)}\times
Y^{*}\big{]},\quad k\in{\rm I\\!N}.$ (4.9)

Combining now (4.5), (4.6), (4.8), and (4.9) brings us to $x^{*}\in K$ and
thus justifies the inclusion $D_{\Omega}^{c*}S(\bar{x}|\bar{y})(y^{*})\subset
K$.

To verify the opposite inclusion, fix $x^{*}\in K$ and find by definition
(4.5) sequences

$\varepsilon_{k}\downarrow
0,~{}~{}(x_{k},y_{k})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y}),\;\mbox{
and }$ (4.10)
$(x_{k}^{*},-y^{*})\in\widehat{N}_{\varepsilon_{k}}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{\Omega}\big{)}\cap\big{[}J\big{(}T(x_{k};\Omega)\big{)}\times
Y^{*}\big{]}$ (4.11)

satisfying the convergence properties in (4.8). In the case where
$(x_{k}^{*},-y_{k}^{*})\to(0,0)$, we get $(x^{*},y^{*})=(0,0)$ and obviously
have $x^{*}\in D_{\Omega}^{c*}S(\bar{x}|\bar{y})(y^{*})$. Thus assume without
loss of generality that there exists $M>0$ such that

$\|(x_{k}^{*},-y_{k}^{*})\|_{X^{*}\times Y^{*}}\geq M\;\mbox{ for all
}\;k\in{\rm I\\!N},$

which tells us by (4.11) and definition (2.3) that

$\limsup\limits_{(x,y)\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y})}\frac{\langle(x_{k}^{*},-y_{k}^{*}),(x,y)-(x_{k},y_{k})\rangle}{\|(x,y)-(x_{k},y_{k})\|_{X\times
Y}}\leq\varepsilon_{k}\leq\frac{\varepsilon_{k}}{M}\|(x_{k}^{*},-y_{k}^{*})\|_{X^{*}\times
Y^{*}}.$

By (2.2), the latter implies that

$(x_{k}^{*},-y_{k}^{*})\in\widehat{N}^{c}_{\varepsilon_{k}/M}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{\Omega}\big{)}.$

Employing (4.11) again brings us to the inclusion

$(x_{k}^{*},-y_{k}^{*})\in\widehat{N}^{c}_{\varepsilon_{k}/M}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{\Omega}\big{)}\cap\big{[}J\big{(}T(x_{k};\Omega)\big{)}\times
Y^{*}\big{]},$ (4.12)

which shows that $x^{*}\in D_{\Omega}^{c*}S(\bar{x}|\bar{y})(y^{*})$ by
combining (4.8), (4.10), and (4.12).

(ii) The proof of assertion (ii) is similar to (i). ∎

To proceed further with deriving pointbased characterizations of the relative
Lipschitz-like property, we formulate the following relative partial
sequentially normal compactness (relative PSNC) condition in the vein of [7]
for unconstrained multifunction and of [10] for constrained ones. Note that
the latter condition differs from the one given below by the usage of the
$\varepsilon$-normal set (2.3) instead of the $\varepsilon$-regular normal
cone (2.2). It follows from Lemma 4.3 that these two PSNC properties are in
fact equivalent. Note that they are obviously fulfilled if $X$ is finite-
dimensional.

######  Definition 4.4.

Given a set-valued mapping $S\colon X\rightrightarrows Y$ between Banach space
and an nonempty set $\Omega\subset X$, it is said that $S$ has the relative
PSNC property with respect to $\Omega$ at
$(\bar{x},\bar{y})\in\operatorname{gph}S|_{\Omega}$ if for any sequence
$(\varepsilon_{k},x_{k},y_{k},x_{k}^{*},y_{k}^{*})\in[0,\infty)\times\operatorname{gph}S|_{\Omega}\times
X^{*}\times Y^{*}$ satisfying the relationships

$\displaystyle\varepsilon_{k}\downarrow
0,\;(x_{k},y_{k})\to(\bar{x},\bar{y}),~{}(x_{k}^{*},-y_{k}^{*})\in\widehat{N}_{\varepsilon_{k}}^{c}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{\Omega}\big{)}\cap[J\big{(}T(x_{k};\Omega)\big{)}\times
Y^{*}],$ (4.13) $\displaystyle
x_{k}^{*}\rightharpoonup^{*}0,\;\operatorname{and}~{}y_{k}^{*}\to 0\;\mbox{ as
}\;k\to\infty,$

we have the strong $X^{*}$-convergence $\|x_{k}^{*}\|_{X^{*}}\to 0$ as
$k\to\infty$.

The next theorem presents pointbased necessary conditions, sufficient
conditions, and characterizations of the relative Lipschitz-like property of
multifunctions with evaluations of the exact relative Lipschitzian bound at
the point in question via the coderivative norms as positive homogeneous
mappings (2.1). By taking into account Lemma 4.3 and the discussions above,
the characterization part can be derived from [10, Theorem 3.6].

######  Theorem 4.5.

Let $\Omega\subset X$ and $(\bar{x},\bar{y})\in\operatorname{gph}S|_{\Omega}$,
where $S:X\rightrightarrows Y$ be a set-valued mapping between reflexive
Banach spaces. Consider the following statements:

(i) $S$ has the Lipschitz-like property relative to $\Omega$ around
$(\bar{x},\bar{y})$.

(ii) $S$ is PSNC at $(\bar{x},\bar{y})$ relative to $\Omega$ and
$\|D_{\Omega}^{c*}S(\bar{x}|\bar{y})\|<\infty$.

(iii) $S$ is PSNC at $(\bar{x},\bar{y})$ relative to $\Omega$ and
$D_{\Omega}^{c*}S(\bar{x}|\bar{y})(0)=\\{0\\}$.
Then we have implications (i)$\Longrightarrow$(ii)$\Longrightarrow$(iii). If
in addition the set $\Omega$ is closed and convex and the set
$\operatorname{gph}S$ is locally closed around $(\bar{x},\bar{y})$, then we
also have implication (iii)$\Longrightarrow$(i). Moreover, if in the latter
setting $\mbox{\rm dim}\,X<\infty$, that the exact Lipschitzian bound of $S$
relative to $\Omega$ around $(\bar{x},\bar{y})$ satisfies the estimate

$\operatorname{lip}_{\Omega}S(\bar{x},\bar{y})\leq\|\widetilde{D}_{\Omega}^{c*}S(\bar{x}|\bar{y})\|,$
(4.14)

while the lower bound estimate

$\|D_{\Omega}^{c*}S(\bar{x}|\bar{y})\|\leq\operatorname{lip}_{\Omega}S(\bar{x},\bar{y})$
(4.15)

holds for $S\colon X\rightrightarrows Y$ between any reflexive Banach spaces
$X$ and $Y$ relative to an arbitrary set $\Omega$. Finally, if we add to the
above assumptions on the fulfillment of (4.14) that $S$ is coderivative normal
relative to $\Omega$ at $(\bar{x},\bar{y})$, then

$\operatorname{lip}_{\Omega}S(\bar{x},\bar{y})=\|D_{\Omega}^{c*}S(\bar{x}|\bar{y})\|=\|\widetilde{D}_{\Omega}^{c*}S(\bar{x}|\bar{y})\|.$
(4.16)

######  Proof.

First we simultaneously verify implication (i)$\Longrightarrow$(ii) and the
lower estimate (4.15) of the exact Lipschitzian bound in the general setting
of the theorem. Assume that $S$ enjoys the Lipschitz-like property relative to
$\Omega$ around $(\bar{x},\bar{y})$ with constant $\kappa$ and take a sequence
$(\varepsilon_{k},x_{k},y_{k},x_{k}^{*},y_{k}^{*})\in(0,1)\times\operatorname{gph}S|_{\Omega}\times
X^{*}\times X^{*}\times Y^{*}$ satisfying (4.13). It follows from Lemma 3.2
that $\|(x_{k}^{*})\|_{X^{*}}\leq(\kappa+1)\|y_{k}^{*}\|_{Y^{*}}$ for large
$k\in{\rm I\\!N}$, and hence $x_{k}^{*}\to 0$ as $k\to\infty$, which yields
the PSNC property of $S$ at $(\bar{x},\bar{y})$ relative to $\Omega$.

Further, pick any $(x^{*},y^{*})\in X^{*}\times Y^{*}$ with $x^{*}\in
D_{\Omega}^{*}S(\bar{x}|\bar{y})(y_{*})$ and find by the coderivative
definition such sequences $\varepsilon_{k}\downarrow 0$,
$(x_{k},y_{k})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y})$,
and
$(x_{k}^{*},-y_{k}^{*})\in\widehat{N}^{c}_{\varepsilon_{k}}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{\Omega}\big{)}$
that $y_{k}^{*}\to y^{*}$ and $x_{k}^{*}\rightharpoonup^{*}x^{*}$ as
$k\to\infty$. Denoting $\kappa_{k}:=\kappa+k^{-1}$, it follows from Lemma 3.2
that $\|x_{k}^{*}\|_{X^{*}}\leq\kappa_{k}\|y_{k}^{*}\|_{Y^{*}}$ when $k$ is
sufficiently large. Therefore,

$\|x^{*}\|_{X^{*}}\leq\liminf\limits_{k\to\infty}\|x_{k}^{*}\|_{X_{k}^{*}}\leq\liminf\limits_{k\to\infty}\kappa_{k}\|y_{k}^{*}\|_{Y^{*}}=\kappa\|y^{*}\|_{Y^{*}},$

which tells us that $\|D_{\Omega}^{*}S(\bar{x}|\bar{y})\|\leq\kappa$ and thus
verifies (ii) together with the lower estimate of the exact Lipschitzian bound
in (4.15). Implication (ii)$\Longrightarrow$(iii) is obvious.

Next we proceed with the justification of (iii)$\Longrightarrow$(i) under the
additional assumptions made. Arguing by contradiction, suppose that $S$ does
not have the Lipschitz-like property relative to $\Omega$ around
$(\bar{x},\bar{y})$. It follows from Lemma 3.3 that there exist
$\varepsilon_{k}\downarrow 0$,
$(x_{k},y_{k})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y})$,
and
$(x_{k}^{*},-y_{k}^{*})\in\widehat{N}^{c}_{\varepsilon_{k}}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{\Omega}\big{)}$
for which the estimate

$\|x_{k}^{*}\|_{X^{*}}>k\|y_{k}^{*}\|_{Y^{*}}\;\mbox{ for all }\;k\in{\rm
I\\!N}$

is satisfied. We may assume without generality that $\|x_{k}^{*}\|_{X^{*}}=1$
and hence get that $y_{k}^{*}\to 0$ as $k\to\infty$. It follows from the
reflexivity of $X$ that the dual ball $B_{X^{*}}$ is weak∗ compact in $X^{*}$,
and hence there exists $x^{*}\in X^{*}$ such that
$x_{k}^{*}\rightharpoonup^{*}x^{*}$ along a subsequence. Applying the
condition $D_{\Omega}^{*}(\bar{x}|\bar{y})(0)=\\{0\\}$ gives us $x^{*}=0$,
i.e., $x_{k}^{*}\rightharpoonup^{*}0$. By the imposed PSNC property of $S$ at
$(\bar{x},\bar{x})$ relative to $\Omega$, this implies that $x_{k}^{*}\to 0$
as $k\to\infty$ along the selected subsequence, which contradicts
$\|x_{k}^{*}\|_{X^{*}}=1$ and thus gives us (i).

It remains to verify the upper bound estimate assuming in addition that
$\mbox{\rm dim}\,X<\infty$. Fix any
$\kappa>\|\widetilde{D}_{\Omega}^{*}S(\bar{x}|\bar{y})\|$ and show that we can
find $\delta>0$ so that (3.3) holds for any small $\varepsilon>0$. Supposing
the contrary yields the existence of sequences $\varepsilon_{k}\downarrow 0$,
$(x_{k},y_{k})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y})$
as $k\to\infty$, and

$(x_{k}^{*},y_{k}^{*})\in\widehat{N}^{c}_{\varepsilon_{k}}\big{(}(x_{k},y_{k});\operatorname{gph}S|_{\Omega}\big{)}\cap\big{[}J\big{(}T(x_{k};\Omega)\big{)}\times
Y^{*}\big{]}$

along which $\|x_{k}^{*}\|_{X^{*}}>\kappa\|y_{k}^{*}\|_{Y_{*}}$ whenever
$k\in{\rm I\\!N}$. We may assume that $\\{x_{k}^{*}\\}$ is bounded in $X$.
Since $\mbox{\rm dim}\,X<\infty$ and $Y$ is reflexive, there is a pair
$(x^{*},y^{*})\in X^{*}\times Y^{*}$ such that $x_{k}^{*}\to x^{*}$ and
$y_{k}^{*}\rightharpoonup^{*}y^{*}$ along some subsequences. Remembering that
the norm function is continuous in finite dimensions and lower semicontinuous
in the weak∗ topology of duals to Banach spaces, we get
$x^{*}\in\widetilde{D}_{\Omega}^{*}S(\bar{x}|\bar{y})(y^{*})$ and

$\|y^{*}\|_{Y^{*}}\leq\liminf\limits_{k\to\infty}\|y_{k}^{*}\|_{Y^{*}}\leq\kappa^{-1}=\kappa^{-1}\|x^{*}\|_{X^{*}},$

which contradicts the condition
$\kappa>\|\widetilde{D}_{\Omega}^{*}S(\bar{x}|\bar{y})\|$ and thus verifies
(4.14). Equality (4.16) is a consequence of (4.14), (4.15), and the assumed
coderivative normality. ∎

##  5 Chain Rules for Conic Contingent Coderivatives

In this section, we show that both conic mixed and normal contingent
coderivatives from Definition 4.2(ii,iii) for constrained multifunctions
between reflexive Banach spaces satisfy pointbased chain rules under a mild
qualification condition. This qualification condition is expressed in terms of
the relative mixed contingent coderivative (4.4) and holds, in particular,
under the relative well-posedness properties of the multifunctions in question
due to the coderivative characterizations of Section 4. Since chain rules have
never been developed for coderivative constructions from [10], the obtained
results are also new for those constructions. The proofs below mostly revolve
around the (approximate) extremal principle of variational analysis; see [7].

We begin with evaluating the $\varepsilon$-regular normal cone to inverse
images of sets under single-valued continuous mappings that holds in any
Banach spaces.

######  Lemma 5.1.

Let $f:X\to Y$ be a single-valued continuous mapping between Banach spaces,
and let $\Theta$ be a nonempty subset of $Y$. Take some $\bar{x}\in X$ and
denote $\bar{y}:=f(\bar{x})\in\Theta$. Then for any $\varepsilon>0$, we have
the inclusion

$\widehat{N}_{\varepsilon}^{c}\big{(}\bar{x};f^{-1}(\Theta)\big{)}\subset\big{\\{}x^{*}\in
X^{*}~{}\big{|}~{}(x^{*},0)\in\widehat{N}_{\varepsilon}^{c}\big{(}(\bar{x},\bar{y});(X\times\Theta)\cap\operatorname{gph}f\big{)}\big{\\}}.$

###### Proof.

Fix $x^{*}\in\widehat{N}_{\varepsilon}^{c}(\bar{x};f^{-1}(\Theta))$ and get by
Definition (2.2) of the $\varepsilon$-regular normals that

$\limsup\limits_{x\stackrel{{\scriptstyle
f^{-1}(\Theta)}}{{\longrightarrow}}\bar{x}}\frac{\langle
x^{*},x-\bar{x}\rangle}{\|x-\bar{x}\|_{X}}\leq\varepsilon\|x^{*}\|_{X^{*}}.$

The continuity of $f$ and the convergence $x\stackrel{{\scriptstyle
f^{-1}(\Theta)}}{{\longrightarrow}}\bar{x}$ yields
$f(x)\stackrel{{\scriptstyle\Theta}}{{\longrightarrow}}f(\bar{x})=\bar{y}$.
Thus

$\displaystyle\limsup\limits_{(x,y)\stackrel{{\scriptstyle[X\times\Theta]\cap\operatorname{gph}f}}{{\longrightarrow}}(\bar{x},\bar{y})}\frac{\langle(x^{*},0),(x,y)-(\bar{x},\bar{y})\rangle}{\|(x,y)-(\bar{x},\bar{y})\|_{X\times
Y}}\leq$ $\displaystyle\limsup\limits_{x\stackrel{{\scriptstyle
f^{-1}(\Theta)}}{{\longrightarrow}}\bar{x}}\frac{\langle(x^{*},0),(x,f(x))-(\bar{x},f(\bar{x}))\rangle}{\sqrt{\|x-\bar{x}\|_{X}^{2}+\|f(x)-f(\bar{x})\|_{Y}^{2}}}$
$\displaystyle\leq$ $\displaystyle\limsup\limits_{x\stackrel{{\scriptstyle
f^{-1}(\Theta)}}{{\longrightarrow}}\bar{x}}\frac{\langle
x^{*},x-\bar{x}\rangle}{\|x-\bar{x}\|_{X}}\leq\varepsilon\|x^{*}\|_{X^{*}},$

which yields
$(x^{*},0)\in\widehat{N}_{\varepsilon}^{c}\big{(}(\bar{x},\bar{y});(X\times\Theta)\cap\operatorname{gph}f\big{)}$
and verifies the claimed inclusion. ∎

Next we recall the notions of inner semicontinuity and inner semicompactness
for set-valued mappings relative to sets. These notions are extensions of
those from [7] to the case of constrained multifunctions.

######  Definition 5.2.

Let $S:X\rightrightarrows Y$ be a set-valued mapping between Banach spaces,
$\Omega\subset X$, $\bar{x}\in\operatorname{dom}S\cap\Omega$, and
$(\bar{x},\bar{y})\in\operatorname{gph}S|_{\Omega}$. Then we say that:

(i)

$S$ is inner semicontinuous relative to $\Omega$ at $(\bar{x},\bar{y})$ if for
every sequence
$x_{k}\stackrel{{\scriptstyle\operatorname{dom}S\cap\Omega}}{{\longrightarrow}}\bar{x}$
there exists $y_{k}\in S(x_{k})$ converging to $\bar{y}$ as $k\to\infty$.

(ii)

We say that $S$ is inner semicompact relative to $\Omega$ at
$(\bar{x},\bar{y})$ if for every sequence
$x_{k}\stackrel{{\scriptstyle\operatorname{dom}S\cap\Omega}}{{\longrightarrow}}\bar{x}$
there exists $y_{k}\in S(x_{k})$ that contains a convergent subsequence.

To establish general chain rules for the relative contingent coderivatives
from Definition 4.2(ii,iii), we now introduce an auxiliary coderivative
construction and a related PSNC property used in deriving the desired results.

######  Definition 5.3.

Let $S:X\rightrightarrows Y$ be a set-valued mapping between Banach spaces,
and let $\Theta$ be a nonempty subset of $Y$. Denote
${\mathfrak{F}}(x):=S(x)\cap\Theta$ and fix
$(\bar{x},\bar{y})\in\operatorname{gph}\mathfrak{F}$. Then:

(i) The complete mirror contingent coderivative of $S$ relative to $\Theta$ at
$(\bar{x},\bar{y})$ is the multifunction
$D_{\Theta}^{*-1}S(\bar{x}|\bar{y}):Y^{*}\rightrightarrows X^{*}$ meaning that
$x^{*}\in D_{\Theta}^{*-1}S(\bar{x}|\bar{y})(y^{*})$ if and only if there
exist sequences $\varepsilon_{k}\downarrow 0$,
$(x_{k},y_{k})\stackrel{{\scriptstyle\operatorname{gph}\mathfrak{F}}}{{\longrightarrow}}(\bar{y},\bar{x})$
as $k\to\infty$, and

$(x_{k}^{*},-y_{k}^{*})\in\widehat{N}_{\varepsilon_{k}}^{c}\big{(}(x_{k},y_{k});\operatorname{gph}\mathfrak{F}\big{)}\cap\big{[}X^{*}\times
J\big{(}T(y_{k};\Theta)\big{)}\big{]},\quad k\in\mathbb{N},$

such that $x_{k}^{*}\rightharpoonup^{*}x^{*}$ and $y_{k}^{*}\to y^{*}$ as
$k\to\infty$.

(ii) We say that $S$ is completely mirror PSNC relative to $\Omega$ at
$(\bar{x},\bar{y})\in\operatorname{gph}S|_{\Omega}$ if for any sequence
$(\varepsilon_{k},x_{k},y_{k},x_{k}^{*},y_{k}^{*})\in(0,1)\times\operatorname{gph}\mathfrak{F}\times
X^{*}\times Y^{*}$ satisfying

$\displaystyle\varepsilon_{k}\downarrow
0,(x_{k},y_{k})\to(\bar{x},\bar{y}),~{}(x_{k}^{*},-y_{k}^{*})\in\widehat{N}_{\varepsilon_{k}}^{c}\big{(}(x_{k},y_{k});\operatorname{gph}\mathfrak{F}\big{)}\cap\big{[}X^{*}\times
J\big{(}T(y_{k};\Theta)\big{)}\big{]},$ $\displaystyle
x_{k}^{*}\rightharpoonup^{*}0,\;\operatorname{and}~{}y_{k}^{*}\to 0\;\mbox{ as
}\;k\to\infty,$

the strong $X^{*}$-convergence $\|x_{k}^{*}\|_{X^{*}}\to 0$ as $k\to\infty$
holds.

Here are our main chain rules for both relative contingent coderivatives from
Definition 4.2(ii,iii) that are formulated via the refined constructions from
Definition 5.3. For brevity, we confine ourselves to the case of the inner
semicontinuity of the auxiliary mapping in Theorem 5.4(i), while both
properties from Definition 5.2 are used in the coderivative sum rules derived
in the next section.

######  Theorem 5.4.

Let $S_{1}:X\rightrightarrows Y$ and $S_{2}:Y\rightrightarrows Z$ be set-
valued mappings between reflexive Banach spaces, and let $\Omega$ be a closed
and convex subset of $X$. Consider the composition $S:=S_{2}\circ S_{1}\colon
X\rightrightarrows Z$ and define the auxiliary multifunction $G\colon X\times
Z\rightrightarrows Y$ by

$G(x,z):=S_{1}(x)\cap S_{2}^{-1}(z)=\big{\\{}y\in S_{1}(x)~{}\big{|}~{}z\in
S_{2}(y)\big{\\}}\;\mbox{ for }\;x\in X,\;z\in Z.$ (5.1)

Fixing $(\bar{x},\bar{z})\in\operatorname{gph}S|_{\Omega}$ and $\bar{y}\in
G(\bar{x},\bar{z})$, suppose that $\Omega$ is convex and locally closed around
$\bar{x}$, that the graphs of $S_{1}$ and $S_{2}$ are locally closed around
$(\bar{x},\bar{y})$ and $(\bar{y},\bar{z})$, respectively, and that $G$ is
inner semicontinuous relative to $\Omega\times Z$ at
$(\bar{x},\bar{z},\bar{y})$. Then the following assertions are satisfied:

(i) Assume that either $S_{2}$ is PSNC at $(\bar{y},\bar{z})$ or $S_{1}^{-1}$
is completely mirror PSNC relative to $\Omega$ at $(\bar{y},\bar{x})$, and
that the relative mixed coderivative qualification condition

$D^{*}S_{2}(\bar{y}|\bar{z})(0)\cap\big{(}-D_{\Omega}^{*-1}S_{1}^{-1}(\bar{y}|\bar{x})(0)\big{)}=\\{0\\}$
(5.2)

holds. Then for all $z^{*}\in Z^{*}$ we have the inclusions

$D_{\Omega}^{c*}S(\bar{x}|\bar{z})(z^{*})\subset\widetilde{D}_{\Omega}^{c*}S_{1}(\bar{x}|\bar{y})\circ
D^{*}S_{2}(\bar{y}|\bar{z})(z^{*}),$ (5.3)
$\widetilde{D}_{\Omega}^{c*}S(\bar{x}|\bar{z})(z^{*})\subset\widetilde{D}_{\Omega}^{c*}S_{1}(\bar{x}|\bar{y})\circ\widetilde{D}^{*}S_{2}(\bar{y}|\bar{z})(z^{*}).$
(5.4)

(ii) If $S_{2}:=f\colon Y\to Z$ is single-valued and strictly differentiable
at $\bar{y}$, then we have the enhanced inclusion for the relative mixed
contingent coderivative:

$D_{\Omega}^{c*}S(\bar{x}|\bar{z})(z^{*})\subset
D_{\Omega}^{c*}S_{1}(\bar{x}|\bar{y})\big{(}\nabla
f(\bar{y})^{*}z^{*})\big{)}\;\mbox{ whenever }\;z^{*}\in Z^{*}.$ (5.5)

###### Proof.

To justify assertion (i), we just verify (5.3) while observing that the proof
of (5.4) is similar and thus can be omitted. To proceed with (5.3), pick any
$x^{*}\in D_{\Omega}^{c*}S(\bar{x}|\bar{z})(z^{*})$ and find sequences
$\varepsilon_{k}\downarrow 0$,
$(x_{k},z_{k})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{z})$,
and

$(x_{k}^{*},-z_{k}^{*})\in\widehat{N}_{\varepsilon_{k}}^{c}\big{(}(x_{k},z_{k});\operatorname{gph}S|_{\Omega}\big{)}\cap[J(T(x_{k};\Omega)\big{)}\times
Z^{*}],\quad k\in\mathbb{N},$ (5.6)

along which we have the convergence $z_{k}^{*}\to
z^{*}~{}~{}\operatorname{and}~{}~{}x_{k}^{*}\rightharpoonup^{*}x^{*}$ as
$k\to\infty$. Since $G$ is inner semicontinuous relative to $\Omega\times Z$
at $(\bar{x},\bar{y},\bar{z})$, there are $y_{k}\in G(x_{k},z_{k})$ with
$y_{k}\to\bar{y}$.

Let $C:=\\{(x,y,z)\in X\times Y\times Z~{}|~{}y\in S_{1}|_{\Omega}(x),~{}z\in
S_{2}(y)\\}$. It follows from (5.6) that

$\displaystyle\limsup\limits_{(x,y,z)\stackrel{{\scriptstyle
C}}{{\longrightarrow}}(x_{k},y_{k},z_{k})}\frac{\langle(x_{k}^{*},0,-z_{k}^{*}),(x,y,z)-(x_{k},y_{k},z_{k})\rangle}{\|(x,y,z)-(x_{k},y_{k},z_{k})\|_{X\times
Y\times Z}}$ $\displaystyle=$
$\displaystyle\limsup\limits_{(x,y,z)\stackrel{{\scriptstyle
C}}{{\longrightarrow}}(x_{k},y_{k},z_{k})}\frac{\langle(x_{k}^{*},-z_{k}^{*}),(x,z)-(x_{k},z_{k})\rangle}{\sqrt{\|x-x_{k}\|_{X}^{2}+\|y-y_{k}\|_{Y}^{2}+\|z-z_{k}\|_{Z}^{2}}}$
$\displaystyle\leq$
$\displaystyle\limsup\limits_{(x,z)\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(x_{k},z_{k})}\frac{\langle(x_{k}^{*},-z_{k}^{*}),(x,z)-(x_{k},z_{k})\rangle}{\sqrt{\|x-x_{k}\|_{X}^{2}+\|z-z_{k}\|_{Z}^{2}}}$
$\displaystyle\leq\varepsilon_{k}\|(x_{k}^{*},0,-z_{k}^{*})\|_{X^{*}\times
Y^{*}\times Z^{*}},$

which readily verifies the inclusion

$(x_{k}^{*},0,-z_{k}^{*})\in\widehat{N}^{c}_{\varepsilon_{k}}\big{(}(x_{k},y_{k},z_{k});C\big{)},\quad\mbox{for~{}all}~{}k\in\mathbb{N}.$

Let $Q:=\operatorname{gph}S_{1}|_{\Omega}\times\operatorname{gph}S_{2}$.
Defining the single-valued continuous mapping $h\colon X\times Y\times Z\to
X\times\times Y\times Y\times Z$ by $h(x,y,z):=(x,y,y,z)$, we have
$C=h^{-1}(Q)$. Denote further

$\displaystyle\Omega_{1}:=X\times Y\times Z\times
Q,~{}~{}\Omega_{2}:=\operatorname{gph}h,\;\mbox{ and}$ (5.7) $\displaystyle
w_{k}:=(x_{k},y_{k},z_{k},x_{k},y_{k},y_{k},z_{k}),\;\;w_{k}^{*}:=(x_{k}^{*},0,-z_{k}^{*},0,0,0,0).$

Applying Lemma 5.1 to $h$ and $Q$ tells us that

$w_{k}^{*}\in\widehat{N}_{\varepsilon_{k}}^{c}(w_{k};\Omega_{1}\cap\Omega_{2}),\quad
k\in\mathbb{N}.$ (5.8)

It follows from (5.6) and Proposition 2.2 that there exists $\delta_{k}>0$
such that

$x_{k}^{*}\in J\big{(}T(x;\Omega)\big{)}+\varepsilon_{k}B_{X^{*}}\;\mbox{ for
all }\;x\in\Omega\cap(x_{k}+\delta_{k}B_{X}).$ (5.9)

With the notation $\xi_{k}:={\rm min}\\{\delta_{k},\varepsilon_{k}\\}$,
$W:=X\times Y\times Z\times X\times Y\times Y\times Z$, and
$W^{*}:=X^{*}\times Y^{*}\times Z^{*}\times X^{*}\times Y^{*}\times
Y^{*}\times Z^{*}$, it is easy to see that there exists $\ell>0$ such that

$\|w_{k}^{*}\|_{W^{*}}\leq\ell\;\mbox{ for all }\;k\in\mathbb{N}.$ (5.10)

By (5.8), it follows from [10, Proposition 3.3(i)] whose proof is based on the
approximate extremal principle (see [7, Lemma 3.1]), that there exist
sequences

$\hat{w}_{k}\in\Omega_{1}\cap(w_{k}+\xi_{k}B_{W}),~{}~{}\widetilde{w}_{k}\in\Omega_{2}\cap(w_{k}+\xi_{k}B_{W}),$
(5.11) $\hat{w}_{k}^{*}\in\widehat{N}(\hat{w}_{k};\Omega_{1}),\;\mbox{ and
}\;\widetilde{w}_{k}^{*}\in\widehat{N}(\hat{w}_{k};\Omega_{2})$ (5.12)

such that for each $k\in\mathbb{N}$ we have the estimates

$\displaystyle\|\lambda_{k}w_{k}^{*}-\hat{w}_{k}^{*}-\widetilde{w}_{k}^{*}\|_{W^{*}}\leq
2(1+\ell)\varepsilon_{k}~{}~{}\operatorname{and}$ (5.13) $\displaystyle
1-(1+\ell)\varepsilon_{k}\leq\max\big{\\{}\lambda_{k},\|\hat{w}_{k}^{*}\|_{W^{*}}\big{\\}}\leq
1+(1+\ell)\varepsilon_{k}.$

The structures in (5.7) and [10, Lemma 2.6] lead us to the representations

$\begin{array}[]{ll}\hat{w}_{k}=(\hat{x}_{1k},\hat{y}_{1k},\hat{z}_{1k},\hat{x}_{2k},\hat{y}_{2k},\hat{y}_{3k},\hat{z}_{2k}),\;\widetilde{w}_{k}=(\widetilde{x}_{k},\widetilde{y}_{k},\widetilde{z}_{k},\widetilde{x}_{k},\widetilde{y}_{k},\widetilde{y}_{k},\widetilde{z}_{k}),\\\
\hat{w}_{k}^{*}=(0,0,0,\hat{x}_{k}^{*},\hat{y}_{1k}^{*},\hat{y}_{2k}^{*},\hat{z}_{k}^{*}),\;\widetilde{w}_{k}^{*}=(-\widetilde{x}_{k}^{*},-\widetilde{y}_{1k}^{*}-\widetilde{y}_{2k}^{*},-\widetilde{z}_{k}^{*},\widetilde{x}_{k}^{*},\widetilde{y}_{1k}^{*},\widetilde{y}_{2k}^{*},\widetilde{z}_{k}^{*}).\end{array}$
(5.14)

Since $w_{k}\to(\bar{x},\bar{y},\bar{z},\bar{x},\bar{y},\bar{y},\bar{z})$ and
$\xi_{k}\to 0$, we get

$(\hat{x}_{2k},\hat{y}_{2k},\hat{y}_{3k},\hat{z}_{2k})\to(\bar{x},\bar{y},\bar{y},\bar{z})\;\mbox{
as }\;k\to\infty.$ (5.15)

It follows now from (5.7), (5.12), (5.14), and [7, Proposition 1.2] that

$(\hat{x}_{k}^{*},\hat{y}_{1k}^{*},\hat{y}_{2k}^{*},\hat{z}_{k}^{*})\in\widehat{N}\big{(}(\hat{x}_{2k},\hat{y}_{2k},\hat{y}_{3k},\hat{z}_{2k});Q\big{)},$

which, being combined with the definition of $Q$, yields

$\begin{array}[]{ll}(\hat{x}_{k}^{*},\hat{y}_{1k}^{*})\in\widehat{N}\big{(}(\hat{x}_{2k},\hat{y}_{2k});\operatorname{gph}S_{1}|_{\Omega}\big{)},\\\
(\hat{y}_{2k}^{*},\hat{z}_{k}^{*})\in\widehat{N}\big{(}(\hat{y}_{3k},\hat{z}_{2k});\operatorname{gph}S_{2}\big{)},\quad
k\in\mathbb{N}.\end{array}$ (5.16)

In particular, we arrive at the inclusion

$(\hat{y}_{1k}^{*},0)\in[\widehat{N}\big{(}(\hat{y}_{2k},\hat{x}_{2k});\operatorname{gph}S_{1}|_{\Omega}^{-1}\big{)}+{\|\hat{x}_{k}^{*}\|_{X^{*}}}B_{X^{*}\times
Y^{*}}]\cap\big{[}Y^{*}\times J\big{(}T(\hat{x}_{2k};\Omega)\big{)}\big{]}.$
(5.17)

Next we claim that there exists a positive number $\lambda_{0}$ such that
$\lambda_{k}\geq\lambda_{0}$ for all $k$. Indeed, supposing the contrary gives
us without loss of generality that $\lambda_{k}\downarrow 0$ as $k\to\infty$.
Then it follows from (5.10) and (5.13) that

$\hat{w}_{k}^{*}+\widetilde{w}_{k}^{*}\to
0,~{}~{}\|\hat{w}_{k}^{*}\|_{W^{*}}\to
1,~{}~{}\operatorname{and}~{}~{}\|\widetilde{w}_{k}^{*}\|_{W^{*}}\to 1.$
(5.18)

Employing further (5.14) and (5.18) shows that

$\hat{x}_{k}^{*}\to 0,~{}\hat{y}_{1k}^{*}+\hat{y}_{2k}^{*}\to
0,~{}\hat{z}_{k}^{*}\to
0,~{}~{}\operatorname{and}~{}~{}(\|\hat{y}_{1k}^{*}\|_{Y^{*}},\|\hat{y}_{2k}^{*}\|_{Y^{*}})\to(1/2,1/2).$
(5.19)

This tells us therefore that

$\|\hat{y}_{1k}^{*}\|_{Y^{*}},\|\hat{y}_{2k}^{*}\|_{Y^{*}}\geq\frac{1}{4}~{}~{}\mbox{for
sufficiently large }\;k,$ (5.20)

and that there exists $y^{*}\in Y^{*}$ for which

$\hat{y}_{1k}^{*}\rightharpoonup^{*}y^{*},~{}~{}\hat{y}_{2k}^{*}\rightharpoonup^{*}-y^{*}\;\mbox{
as }\;k\to\infty.$ (5.21)

It follows from (5.17) and (5.20) that

$(\hat{y}_{1k}^{*},0)\in\widehat{N}_{4\|\hat{x}_{k}^{*}\|_{X^{*}}}^{c}\big{(}(\hat{y}_{2k},\hat{x}_{2k});\operatorname{gph}S_{1}|_{\Omega}^{-1}\big{)}\cap\big{[}Y^{*}\times
J\big{(}T(\hat{x}_{2k};\Omega)\big{)}\big{]}.$

Combining the latter with (5.15), (5.16), (5.19), and (5.21) brings us to

$y^{*}\in
D^{*}S_{2}(\bar{y}|\bar{z})(0)\cap(-D_{\Omega}^{*-1}S_{1}^{-1}(\bar{y}|\bar{x})(0)),$

which ensures by the qualification condition (5.2) that $y^{*}=0$. Moreover,
the imposed requirements on either $S_{2}$ is PSNC at $(\bar{y},\bar{z})$, or
$S_{1}^{-1}$ is completely mirror PSNC relative to $\Omega$ at
$(\bar{y},\bar{x})$ tell us that

$\mbox{either }\;\hat{y}_{1k}^{*}\to 0,\;\mbox{ or }\;\hat{y}_{2k}^{*}\to
0\;\mbox{ as }\;k\to\infty,$

respectively. This contradicts (5.20) and hence verifies the claimed existence
of $\lambda_{0}>0$.

Thus we can assume without loss of generality that $\lambda_{k}=1$ for all
$k\in\mathbb{N}$ and then deduce from (5.7), (5.13), and (5.14) that

$\displaystyle\|(\hat{x}_{k}^{*},\hat{y}_{1k}^{*},\hat{y}_{2k}^{*},\hat{z}_{k}^{*})+(\widetilde{x}_{k}^{*},\widetilde{y}_{1k}^{*},\widetilde{y}_{2k}^{*},\widetilde{z}_{k}^{*})\|_{X^{*}\times
Y^{*}\times Y^{*}\times Z^{*}}$
$\displaystyle+\|(x_{k}^{*},0,-z_{k}^{*})-(-\widetilde{x}_{k}^{*},-\widetilde{y}_{1k}^{*}-\widetilde{y}_{2k}^{*},-\widetilde{z}_{k}^{*})\|_{X^{*}\times
Y^{*}\times Z^{*}}\leq 2(1+\ell)\varepsilon_{k},$

which brings us to the estimate

$\|x_{k}^{*}-\hat{x}_{k}^{*}\|_{X^{*}}+\|z_{k}^{*}+\hat{z}_{k}^{*}\|_{Z^{*}}+\|\hat{y}_{1k}^{*}+\hat{y}_{2k}^{*}\|_{Y^{*}}\leq
2(1+\ell)\varepsilon_{k}$ (5.22)

and ensures together with (5.16) and (5.22) that

$\displaystyle(x_{k}^{*},-\hat{y}_{2k}^{*})\in\widehat{N}\big{(}(\hat{x}_{2k},\hat{y}_{2k});{\rm
gph}\,S_{1}|_{\Omega}\big{)}+2(1+\ell)\varepsilon_{k}B_{X^{*}\times Y^{*}},$
$\displaystyle(\hat{y}_{2k}^{*},-z_{k}^{*})\in\widehat{N}\big{(}(\hat{y}_{3k},\hat{z}_{2k});{\rm
gph}\,S_{2}\big{)}+2(1+\ell)\varepsilon_{k}B_{Y^{*}\times Z^{*}}.$

Furthermore, we deduce from (5.6), (5.9), and (5.11) that there exists
$x_{2k}^{*}\in J\big{(}T(\hat{x}_{2k};\Omega)\big{)}$ satisfying
$\|x_{2k}^{*}-x_{k}^{*}\|_{X^{*}}\leq\varepsilon_{k}$, and hence

$\begin{array}[]{ll}(x_{2k}^{*},-\hat{y}_{2k}^{*})\in&[\widehat{N}\big{(}(\hat{x}_{2k},\hat{y}_{2k});\operatorname{gph}S_{1}|_{\Omega}\big{)}+{2(2+\ell)\varepsilon_{k}}B_{X^{*}\times
Y^{*}}]\\\ &\cap\big{[}J\big{(}T(\hat{x}_{2k};\Omega)\big{)}\times
Y^{*}\big{]}.\end{array}$ (5.23)

It follows from (5.13) and (5.14) that the sequence $\\{\hat{y}_{2k}^{*}\\}$
is bounded. Hence selecting a subsequence if necessary gives us $y^{*}\in
Y^{*}$ such that $\hat{y}_{2k}^{*}\rightharpoonup^{*}y^{*}$ as $k\to\infty$.
Thus we just need to show that
$x^{*}\in\widetilde{D}_{\Omega}^{c*}S_{1}(\bar{x}|\bar{y})(y^{*})$ and
$y^{*}\in D^{*}S_{2}(\bar{y}|\bar{z})(z^{*})$. If there exists $\alpha>0$ such
that $\|(x_{2k}^{*},-\hat{y}_{2k}^{*})\|_{X^{*}\times Y^{*}}>\alpha$ for all
$k$, then deduce from (5.23) that

$(x_{2k}^{*},-\hat{y}_{2k}^{*})\in\widehat{N}_{2(2+\ell)\varepsilon_{k}/\alpha}^{c}\big{(}(\hat{x}_{2k},\hat{y}_{2k});\operatorname{gph}S_{1}|_{\Omega}\big{)}\cap\big{[}J\big{(}T(\hat{x}_{2k};\Omega)\big{)}\times
Y^{*}\big{]},$

which implies that
$x^{*}\in\widetilde{D}_{\Omega}^{c*}S_{1}(\bar{x}|\bar{y})(y^{*})$. If
$(x_{2k}^{*},-\hat{y}_{2k}^{*})\to(0,0)$ as $k\to\infty$, then $x^{*}=0$,
$y^{*}=0$, and the claim is trivial. Similarly, we also get that $y^{*}\in
D^{*}S_{2}(\bar{y}|\bar{z})(z^{*})$ and thus complete the proof of assertion
(i).

Next we verify assertion (ii). Observe that the strict differentiability of
$S_{2}=f$ ensures that the qualification condition (5.2) and the PSNC property
of $S_{2}$ are satisfied automatically. To proceed with the proof of (5.5),
pick any $x^{*}\in D_{\Omega}^{c*}S(\bar{x}|\bar{z})(z^{*})$ and find

$\varepsilon_{k}\downarrow
0,~{}~{}(x_{k},z_{k})\stackrel{{\scriptstyle\operatorname{gph}S|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{z}),\quad\mbox{and}$
$(x_{k}^{*},-z_{k}^{*})\in\widehat{N}_{\varepsilon_{k}}^{c}\big{(}(x_{k},z_{k});\operatorname{gph}S|_{\Omega}\big{)}\cap[J(T(x_{k};\Omega)\big{)}\times
Z^{*}]$

with the convergence of the dual sequences

$z_{k}^{*}\to
z^{*}~{}~{}\operatorname{and}~{}~{}x_{k}^{*}\rightharpoonup^{*}x^{*}\;\mbox{
as }\;k\to\infty.$

Since $G$ from (5.1) is inner semicontinuous relative to $\Omega\times Z$ at
$(\bar{x},\bar{z},\bar{y})$ with $\bar{y}\in G(\bar{x},\bar{z})$, there exist
elements $y_{k}\in G(x_{k},z_{k})$ such that $y_{k}\to\bar{y}$ as
$k\to\infty$. Taking $S_{2}=f$ in the proof of assertion (i), we find

$(\hat{x}_{k},\hat{y}_{k})\in(x_{k},y_{k})+\xi_{k}B_{X\times
Y},~{}~{}(\widetilde{y}_{k},\widetilde{z}_{k})\in(y_{k},z_{k})+\xi_{k}B_{Y\times
Z},$ (5.24)

and
$(\hat{x}_{k}^{*},\hat{y}_{k}^{*})\in(x_{k}^{*}+\varepsilon_{k}B_{X^{*}})\times
Y^{*}$ such that

$\begin{array}[]{ll}(\hat{x}_{k}^{*},-\hat{y}_{k}^{*})\in[\widehat{N}\big{(}(\hat{x}_{k},\hat{y}_{k});\operatorname{gph}S_{1}|_{\Omega}\big{)}+{(2+\ell)\varepsilon_{k}}B_{X^{*}\times
Y^{*}}]\cap\big{[}J\big{(}T(\hat{x}_{k};\Omega)\big{)}\times Y^{*}\big{]},\\\
(\hat{y}_{k}^{*},-z_{k}^{*})\in\widehat{N}\big{(}(\widetilde{y}_{k},\widetilde{z}_{k});\operatorname{gph}f\big{)}+(1+\ell)\varepsilon_{k}(B_{Y^{*}}\times
B_{Z^{*}}).\end{array}$ (5.25)

We clearly have the convergence $\hat{x}_{k}^{*}\rightharpoonup^{*}x^{*}$.
Utilizing this together with (5.24) gives us

$(\hat{x}_{k},\hat{y}_{k})\to(\bar{x},\bar{y})~{}~{}\operatorname{and}~{}~{}(\widetilde{y}_{k},\widetilde{z}_{k})\to(\bar{y},\bar{z})\;\mbox{
as }\;k\to\infty.$

It follows from (5.25) that there exist pairs

$\begin{array}[]{ll}(\widetilde{y}_{k}^{*},-\widetilde{z}_{k}^{*})\in(\hat{y}_{k}^{*},-z_{k}^{*})+(1+\ell)\varepsilon_{k}B_{Y^{*}\times
Z^{*}}\;\mbox{ with}\\\
(\hat{y}_{k}^{*},-\hat{z}_{k}^{*})\in\widehat{N}\big{(}(\widetilde{y}_{k},\widetilde{z}_{k});\operatorname{gph}f\big{)},\quad
k\in\mathbb{N}.\end{array}$ (5.26)

Thus [10, Lemma 2.6] gives us a sequence $\rho_{k}\downarrow 0$ ensuring that
the estimate

$\|\widetilde{y}_{k}^{*}-\nabla
f(\bar{y})^{*}\widetilde{z}_{k}^{*}\|_{Y^{*}}\leq\rho_{k}$ (5.27)

holds for sufficiently large $k$. It follows from (5.26) and (5.27) that

$\widetilde{z}_{k}^{*}\to
z^{*}~{}~{}\operatorname{and}~{}\widetilde{y}_{k}^{*}\to\nabla
f(\bar{y})z^{*}\;\mbox{ as }\;k\to\infty.$

Passing to the limit in (5.25) with using the established convergence brings
us to

$x^{*}\in D_{\Omega}^{c*}S_{1}(\bar{x}|\bar{y})(\nabla f(\bar{y})^{*}z^{*}),$

which verifies (5.5) and thus completes the proof of the theorem. ∎

Observe that assertions (i) and (ii) of Theorem 5.4 are mutually independent.
Indeed, inclusion (5.5) for the mixed contingent coderivative of the
composition in (ii) is expressed via the mixed one for $S_{1}$ while not via
its normal counterpart as in (5.3).

To conclude this section, we recall from [10] the notions of relative metric
regularity and relative linear openness of multifunctions.

######  Definition 5.5.

Let $S\colon X\rightrightarrows Y$ be set-valued mapping between Banach
spaces, let $\Omega\subset X$, and let $(\bar{x},\bar{y})\in\mbox{\rm
gph}\,S|_{\Omega}$. Then we say that:

(i) $S$ is is metrically regular relative to $\Omega$ around
$(\bar{x},\bar{y})$ if there exist neighborhoods $U$ of $\bar{x}$ and $V$ of
$\bar{y}$ together with a number $\kappa\geq 0$ such that

$d(x^{\prime};S^{-1}(y))\leq\kappa d(y;S(x^{\prime}))\;\mbox{ for all
}\;x^{\prime}\in\Omega\cap U\;\mbox{ and }\;y\in S(\Omega)\cap V.$

(ii) $S$ is linearly open relative to $\Omega$ around $(\bar{x},\bar{y})$ if
there exist neighborhoods $U$ of $\bar{x}$ and $V$ of $\bar{y}$ together with
a number $\mu\geq 0$ such that

$\big{(}S(x^{\prime})+\varepsilon B_{Y}\big{)}\cap S(\Omega)\cap V\subset
S\big{(}x^{\prime}+\mu\varepsilon B_{X})\;\mbox{ for all
}\;x^{\prime}\in\Omega\cap U.$

It is known that the relative metric regularity and relative linear openness
of $S$ around $(\bar{x},\bar{y})$ are equivalent to each other (with different
moduli) being also equivalent to the relative Lipschitz-like property of the
inverse $S^{-1}$ around $(\bar{y},\bar{x})$. Thus the properties from
Definition 5.5 admit the corresponding coderivative characterizations; cf.
[10, Theorem 3.9]. This allows us to obtain the following consequence of
Theorems 5.4 and 4.5.

######  Corollary 5.6.

In the general setting of Theorem 5.4, suppose that either $S_{1}|_{\Omega}$
is metrically regular/linearly open around $(\bar{y},\bar{x})$, or $S_{2}$ is
Lipschitz-like around $(\bar{y},\bar{z})$. Then both conic contingent
coderivative chain rules in (5.3) and (5.4) hold.

##  6 Relative Extremal Principle and Sum Rules for Conic Contingent
Coderivatives

The primary goal of this section is to establish, for the first time in the
literature, pointbased sum rules for both normal and mixed conic contingent
coderivatives from Definition 4.2(ii,iii), and hence for their equivalent
versions from [10]. To derive such rules, we develop two major results of the
undoubted own interest for variational analysis and applications, which go far
beyond coderivative calculus. They are the relative extremal principle for
systems of closed sets and the relative fuzzy intersection rule for
$\varepsilon$-regular normals whose proof is based on the relative extremal
principle.

We start with the definition of relative extremal points of set systems, where
$\Pi_{X}(\Lambda)$ signifies the projection of a set $\Lambda\subset X\times
Y$ onto the space $X$.

######  Definition 6.1.

Let $\Lambda_{1},\Lambda_{2}\subset X\times Y$ and $\Omega\subset X$ be
nonempty subsets of Banach spaces. We say that
$(\bar{x},\bar{y})\in\Lambda_{1}\times\Lambda_{2}$ is a local extremal point
of the set system $\\{\Lambda_{1},\Lambda_{2}\\}$ relative to $\Omega$ if
there exist sequence $\\{b_{k}\\}\subset X$ and a neighborhood $U$ of
$(\bar{x},\bar{y})$ such that $b_{k}\to 0$ as $k\to\infty$,
$\Pi_{X}(\Lambda_{1}\cup\Lambda_{2})\subset\Omega$, and

$\big{(}\Lambda_{1}+(0,b_{k})\big{)}\cap\Lambda_{2}\cap U=\emptyset\;\mbox{
for all~{}large }\;k\in\mathbb{N}.$

Here is the relative extremal principle for closed sets in reflexive Banach
spaces.

######  Theorem 6.2.

Let $X$ and $Y$ be two reflexive Banach spaces, let $\Omega$ be a closed and
convex subset of $X$, and let $(\bar{x},\bar{y})$ be a local extremal point of
the system of closed sets $\\{\Lambda_{1},\Lambda_{2}\\}$ relative to
$\Omega$. Then for any $\varepsilon>0$, there exist pairs
$(x_{i},y_{i})\in\Lambda_{i}\cap[(\bar{x}+\varepsilon
B_{X})\times(\bar{y}+\varepsilon B_{Y})]$ and $(x_{i}^{*},y_{i}^{*})\in
X^{*}\times Y^{*}$, $i=1,2$, such that we have the relationships

$\displaystyle(x_{i}^{*},y_{i}^{*})\in\big{[}\widehat{N}\big{(}(x_{i},y_{i});\Lambda_{i}\big{)}+\varepsilon(B_{X^{*}}\times
B_{Y^{*}})\big{]}\cap\big{[}J\big{(}T(x_{i};\Omega)\big{)}\times
Y^{*}\big{]},\quad i=1,2,$ (6.1)
$\displaystyle(x_{1}^{*},y_{1}^{*})+(x_{2}^{*},y_{2}^{*})=0,\;\mbox{ and
}\;\|(x_{1}^{*},y_{1}^{*})\|_{X^{*}\times
Y^{*}}+\|(x_{2}^{*},y_{2}^{*})\|_{X^{*}\times Y^{*}}=1.$

###### Proof.

Take a neighborhood $U$ of $(\bar{x},\bar{y})$ from Definition 6.1 such that
for any $\varepsilon>0$ there exists $b\in Y$ with
$\|b\|_{Y}\leq\varepsilon^{3}/2$ and $(\Lambda_{1}+(0,b))\cap\Lambda_{2}\cap
U=\emptyset$. Assume for simplicity that $U=X\times Y$ and that
$\varepsilon<1/2$. Then considering the function

$\varphi(z):=\|(x_{1},y_{1})-(x_{2},y_{2})+(0,b)\|_{X\times
Y}~{}~{}\operatorname{for}~{}z=(x_{1},y_{1},x_{2},y_{2})\in X\times Y\times
X\times Y,$

we conclude that $\varphi(z)>0$ on $\Lambda_{1}\times\Lambda_{2}$, and hence
$\varphi$ is Fr$\acute{\text{e}}$chet differentiable at any point
$z\in\Lambda_{1}\times\Lambda_{2}$ due to our choice of the equivalent norm on
the reflexive space $X\times Y$; see Section 2. In what follows, we always use
the product norm

$\|z\|:=(\|x_{1}\|_{X}^{2}+\|y_{1}\|_{Y}^{2}+\|x_{2}\|_{X}^{2}+\|y_{2}\|_{Y}^{2})^{1/2}\;\mbox{
on }\;X\times Y\times X\times Y,$

which is obviously Fr$\acute{\text{e}}$chet differentiable at nonzero points
of $X\times Y\times X\times Y$.

Let $z_{0}:=(\bar{x},\bar{y},\bar{x},\bar{y})$ and observe that the set

$W(z_{0}):=\\{z\in\Lambda_{1}\times\Lambda_{2}~{}|~{}\varphi(z)+\varepsilon\|z-z_{0}\|^{2}/2\leq\varphi(z_{0})\\}$

is nonempty and closed. Furthermore, for each $z:=(x_{1},y_{1},x_{2},y_{2})\in
W(z_{0})$ we get

$\|x_{1}-\bar{x}\|_{X}^{2}+\|y_{1}-\bar{y}\|_{Y}^{2}+\|x_{2}-\bar{x}\|_{X}^{2}+\|y_{2}-\bar{y}\|_{Y}^{2}\leq
2\varphi(z_{0})/\varepsilon=2\|b\|_{Y}/\varepsilon\leq\varepsilon^{2},$ (6.2)

which implies that $W(z_{0})\subset(\bar{x}+\varepsilon
B_{X})\times(\bar{y}+\varepsilon B_{Y})\times(\bar{x}+\varepsilon
B_{X})\times(\bar{y}+\varepsilon B_{Y})$. Next we inductively construct
sequences of vectors $z_{k}\in\Lambda_{1}\times\Lambda_{2}$ and nonempty
closed sets $W(z_{k})$, $k\in\mathbb{N}$, as follows. Given $z_{k}$ and
$W(z_{k})$, $k=0,1,\ldots$, select $z_{k+1}\in W(z_{k})$ satisfying

$\varphi(z_{k+1})+\varepsilon\sum\limits_{j=0}^{k}\frac{\|z_{k+1}-z_{j}\|^{2}}{2^{j+1}}<\inf\limits_{z\in
W(z_{k})}\left\\{\varphi(z)+\varepsilon\sum\limits_{j=0}^{k}\frac{\|z-z_{j}\|^{2}}{2^{j+1}}\right\\}+\frac{\varepsilon^{3}}{2^{3k+2}}.$

Construct further the set $W(z_{k+1})$ by

$W(z_{k+1}):=\left\\{z\in\Lambda_{1}\times\Lambda_{2}~{}|~{}\varphi(z)+\varepsilon\sum\limits_{j=0}^{k+1}\frac{\|z-z_{j}\|^{2}}{2^{j+1}}\leq\varphi(z_{k+1})+\varepsilon\sum\limits_{j=0}^{k}\frac{\|z_{k+1}-z_{j}\|^{2}}{2^{j+1}}\right\\}$

and then define the function

$\phi(z):=\varphi(z)+\varepsilon\sum\limits_{j=0}^{\infty}\frac{\|z-z_{j}\|^{2}}{2^{j+1}}.$
(6.3)

Following the proof of [7, Theorem 2.10], we know that there exists
$\bar{z}:=(\bar{x}_{1},\bar{y}_{1},\bar{x}_{2},\bar{y}_{2})\in X\times Y\times
X\times Y$ such that $z_{k}\to\bar{z}$ as $k\to\infty$ and that $\bar{z}$ is a
minimum point of $\phi$ over the set $\Lambda_{1}\times\Lambda_{2}$. Hence the
function $\psi(z):=\phi(z)+\delta_{\Lambda_{1}\times\Lambda_{2}}(z)$ achieves
at $\bar{z}$ its minimum over $X\times Y\times X\times Y$, which immediately
yields $0\in\hat{\partial}\psi(\bar{z})$. Note that $\phi$ is
Fr$\acute{\text{e}}$chet differentiable at $\bar{z}$ due to
$\varphi(\bar{z})\not=0$ and the smoothness of $\|\cdot\|^{2}$. Now applying
the sun rule of [7, Proposition 1.107(i)], we get

$-\nabla\phi(\bar{z})\in\widehat{N}(\bar{z};\Lambda_{1}\times\Lambda_{2})=\widehat{N}\big{(}(\bar{x}_{1},\bar{y}_{1});\Lambda_{1}\big{)}\times\widehat{N}\big{(}(\bar{x}_{2},\bar{y}_{2});\Lambda_{2}\big{)}.$

It follows from (6.3) that
$\nabla\phi(\bar{z})=(u_{1}^{*},v_{1}^{*},u_{2}^{*},v_{2}^{*})\in X^{*}\times
Y^{*}\times X^{*}\times Y^{*}$, where

$\displaystyle(u_{1}^{*},v_{1}^{*}):=(x^{*},y^{*})+\varepsilon\sum\limits_{j=0}^{\infty}\frac{\big{(}J(\bar{x}_{1}-x_{1j}),J(\bar{y}_{1}-y_{1j})\big{)}}{2^{j}},$
$\displaystyle(u_{2}^{*},v_{2}^{*}):=-(x^{*},y^{*})+\varepsilon\sum\limits_{j=0}^{\infty}\frac{\big{(}J(\bar{x}_{2}-x_{2j}),J(\bar{y}_{2}-y_{2j})\big{)}}{2^{j}}$

with $(x_{1j},y_{1j},x_{2j},y_{2j})=z_{j}$, $j=0,1,\ldots$, and where

$(x^{*},y^{*}):=\left(\frac{J(\bar{x}_{1}-\bar{x}_{2})}{\sqrt{\|\bar{x}_{1}-\bar{x}_{2}\|_{X}^{2}+\|\bar{y}_{1}-\bar{y}_{2}+b\|_{Y}^{2}}},\frac{J(\bar{y}_{1}-\bar{y}_{2}+b)}{\sqrt{\|\bar{x}_{1}-\bar{x}_{2}\|_{X}^{2}+\|\bar{y}_{1}-\bar{y}_{2}+b\|_{Y}^{2}}}\right).$

The latter tells us that $\|(x^{*},y^{*})\|_{X^{*}\times Y^{*}}=1$. It follows
from (6.2) that

$\displaystyle\big{\|}\big{(}J(\bar{x}_{i}-x_{ij}),J(\bar{y}_{i}-y_{ij})\big{)}\big{\|}_{X^{*}\times
Y^{*}}=\sqrt{\|\bar{x}_{i}-x_{ij}\|_{X}^{2}+\|\bar{y}_{i}-y_{ij}\|_{Y}^{2}}$
$\displaystyle\leq$
$\displaystyle\sqrt{\|\bar{x}_{i}-\bar{x}\|_{X}^{2}+\|\bar{y}_{i}-\bar{y}\|_{Y}^{2}}+\sqrt{\|\bar{x}-x_{ij}\|_{X}^{2}+\|\bar{y}-y_{ij}\|_{Y}^{2}}\leq
2\varepsilon<1$

for $j=0,1,\ldots$ and $i=1,2$. Therefore,

$\sum\limits_{j=0}^{\infty}\frac{\big{\|}\big{(}J(\bar{x}_{1}-x_{1j}),J(\bar{y}_{1}-y_{1,j})\big{)}\big{\|}_{X^{*}\times
Y^{*}}}{2^{j}}\leq 2,\quad i=1,2.$

Putting $(x_{i},y_{i}):=(\bar{x}_{i},\bar{y}_{i})$ and
$(x_{i}^{*},y_{i}^{*}):=(-1)^{i}(x^{*},y^{*})/2$ for $i=1,2$, we arrive at the
relationships in (6.1) by $J(\bar{x}_{2}-\bar{x}_{1})\in
J(T(\bar{x}_{1};\Omega))$ and $J(\bar{x}_{1}-\bar{x}_{2})\in
J(T(\bar{x}_{2};\Omega))$ due to the convexity of $\Omega$ and the
construction of $(x^{*},y^{*})$, which completes the proof. ∎

The next theorem establishes a relative fuzzy intersection rule for closed
sets in Hilbert spaces. The proof is based on the relative extremal principle
from Theorem 6.2 while using the convexity-preserving property of the duality
mapping, which holds in Hilbert spaces; see Remark 6.4 for more discussions.
Note that in Hilbert spaces, the duality mapping reduce to the identity
operator.

######  Theorem 6.3.

Let $X$ and $Y$ be Hilbert spaces, let $\Theta_{1},\Theta_{2}\subset X\times
Y$ be locally closed around $(\bar{x},\bar{y})\in\Theta_{1}\cap\Theta_{2}$,
and let $\Pi_{X}(\Theta_{1}\cup\Theta_{2})\subset\Omega$ for some closed and
convex set $\Omega\subset X$. Fix $\varepsilon\in(0,1)$ and pick
$(x^{*},y^{*})\in\widehat{N}_{\varepsilon}^{c}\big{(}(\bar{x},\bar{y});\Theta_{1}\cap\Theta_{2}\big{)}\cap[T(\bar{x};\Omega)\times
Y]$. Then for any $\nu>0$, there exist $\lambda\geq 0$,
$(x_{i},y_{i})\in\Theta_{i}\cap[(\bar{x}+\nu B_{X})\times(\bar{y}+\nu
B_{Y})]$, and
$(x_{i}^{*},y_{i}^{*})\in[\widehat{N}\big{(}(x_{i},y_{i});\Theta\big{)}+(\|(x^{*},y^{*})\|_{X\times
Y}+\nu)\varepsilon(B_{X}\times B_{Y})]\cap[T(x_{i};\Omega)\times Y]$, $i=1,2$,
such that

$\lambda(x^{*},y^{*})-(x_{1}^{*},y_{1}^{*})-(x_{2}^{*},y_{2}^{*})\in\nu(B_{X}\times
B_{Y})\;\mbox{ and }\;\max\\{\lambda,\|(x_{1}^{*},y_{1}^{*})\|_{X\times
Y}\\}=1.$ (6.4)

###### Proof.

Take any $\nu>0$ and
$(x^{*},y^{*})\in\widehat{N}_{\varepsilon}^{c}\big{(}(\bar{x},\bar{y});\Theta_{1}\cap\Theta_{2}\big{)}\cap[T(\bar{x};\Omega)\times
Y]$, and denote $L:=\|(x^{*},y^{*})\|_{X\times Y}$. Then there exists a
neighborhood $U$ of $(\bar{x},\bar{y})$ such that

$\langle x^{*},x-\bar{x}\rangle+\langle
y^{*},y-\bar{y}\rangle-(L+\frac{1}{2}\nu)\varepsilon\|(x,y)-(\bar{x},\bar{y})\|_{X\times
Y}\leq 0$ (6.5)

for all $(x,y)\in\Theta_{1}\cap\Theta_{2}\cap U$. Define subsets of $X\times
Y\times\mathbb{R}$ by

$\displaystyle\Lambda_{1}:=\big{\\{}(x,y,\alpha)\in X\times
Y\times\mathbb{R}~{}\big{|}$ $\displaystyle~{}(x,y)\in\Theta_{1},~{}\alpha\geq
0\big{\\}};$ $\displaystyle\Lambda_{2}:=\big{\\{}(x,y,\alpha)\in X\times
Y\times\mathbb{R}~{}\big{|}$
$\displaystyle~{}(x,y)\in\Theta_{2},~{}\alpha\leq\langle
x^{*},x-\bar{x}\rangle+\langle y^{*},y-\bar{y}\rangle$
$\displaystyle-(L+\frac{1}{2}\nu)\varepsilon\|(x,y)-(\bar{x},\bar{y})\|_{X\times
Y}\big{\\}}.$

Observe that $(\bar{x},\bar{y},0)\in\Lambda_{1}\cap\Lambda_{2}$ and that the
sets $\Lambda_{i}$ are locally closed around $(\bar{x},\bar{y},0)$. Moreover,
it is easy to check that

$\Lambda_{1}\cap\big{(}\Lambda_{2}-(0,0,\nu)\big{)}\cap(U\times\mathbb{R})=\emptyset~{}~{}\operatorname{for~{}all}~{}\nu>0$

due to (6.5) and the structure of $\Lambda_{i}$, $i=1,2$. Thus
$(\bar{x},\bar{y},0)$ is a local extremal point relative to $\Omega$ of the
set system $\\{\Lambda_{1},\Lambda_{2}\\}$. For any $\tau\in(0,\frac{1}{2})$,
it follows from Lemma 2.2 that there exists a positive number $\rho$ such that

$x^{*}\in T(x;\Omega)+\tau\varepsilon B_{X}\;\mbox{ for all
}\;x\in\Omega\cap(\bar{x}+2\rho B_{X}).$ (6.6)

Applying Theorem 6.2 to the set system $\\{\Lambda_{1},\Lambda_{2}\\}$, we
find $(x_{i},y_{i},\alpha_{i})\in\Lambda_{i}$ and
$(x_{i}^{*},y_{i}^{*},\lambda_{i})\in\widehat{N}\big{(}(x_{i},y_{i},\alpha_{i});\Lambda_{i}\big{)}\cap[T(x_{i};\Omega)\times
Y\times\mathbb{R}+\tau\varepsilon(B_{X}\times B_{Y}\times\mathbb{R})]$,
$i=1,2$, such that

$\displaystyle\|x_{1}^{*}+x_{2}^{*}\|_{X}^{2}+\|y_{1}^{*}+y_{2}^{*}\|_{Y}^{2}+|\lambda_{1}+\lambda_{2}|^{2}<\tau^{2}\varepsilon^{2},$
(6.7)
$\displaystyle\sqrt{\|x_{i}^{*}\|_{X}^{2}+\|y_{i}^{*}\|_{Y}^{2}+|\lambda_{i}|^{2}}\in\left(\frac{1}{2}-\tau\varepsilon,\frac{1}{2}+\tau\varepsilon\right),$
$\displaystyle\|x_{i}-\bar{x}\|_{X}^{2}+\|y_{i}-\bar{y}\|_{Y}^{2}+|\lambda_{i}|^{2}<{\rm
min}\\{\tau^{2}\varepsilon^{2},\rho^{2}\\},\quad i=1,2.$

It is easy to deduce from the above that $\lambda_{1}\leq 0$,
$(x_{1}^{*},y_{1}^{*})\in\widehat{N}\big{(}(x_{1},y_{1});\Theta_{1}\big{)}\cap[T(x_{1};\Omega)\times
Y+\tau\varepsilon(B_{X}\times B_{Y})]$, and

$\limsup\limits_{(x,y,\alpha)\stackrel{{\scriptstyle\Lambda_{2}}}{{\longrightarrow}}(x_{2},y_{2},\alpha_{2})}\frac{\langle
x_{2}^{*},x-x_{2}\rangle+\langle
y_{2}^{*},y-y_{2}\rangle+\lambda_{2}(\alpha-\alpha_{2})}{\sqrt{\|x-x_{2}\|_{X}^{2}+\|y-y_{2}\|_{Y}^{2}+|\alpha-\alpha_{2}|^{2}}}\leq
0.$ (6.8)

We get from the structure of $\Lambda_{2}$ that $\lambda_{2}\geq 0$ and

$\alpha_{2}\leq\langle x_{2}^{*},x_{2}-\bar{x}\rangle+\langle
y_{2}^{*},y_{2}-\bar{y}\rangle-(L+\frac{1}{2}\nu)\varepsilon\|(x_{2},y_{2})-(\bar{x},\bar{y})\|_{X\times
Y}.$ (6.9)

If inequality (6.9) is strict, then (6.8) yields $\lambda_{2}=0$ and
$(x_{2}^{*},y_{2}^{*})\in\widehat{N}\big{(}(x_{2},y_{2});\Theta_{2}\big{)}\cap[T(x_{2};\Omega)\times
Y+\tau\varepsilon(B_{X}\times B_{Y})]$. In this case, we arrive by using (6.7)
at (6.4) with $\lambda=0$.

It remains to consider the case of equality in (6.9). This gives us
$(x,y,\alpha)\in\Lambda_{2}$ with

$\alpha=\langle x^{*},x-\bar{x}\rangle+\langle
y^{*},y-\bar{y}\rangle-(L+\frac{1}{2}\nu)\varepsilon\|(x,y)-(\bar{x},\bar{y})\|_{X\times
Y},~{}~{}(x,y)\in\Theta_{2}\setminus\\{(x_{2},y_{2})\\}.$

Substituting the latter into (6.8), we find a neighborhood $V$ of
$(x_{2},y_{2})$ such that

$\langle x_{2}^{*},x-x_{2}\rangle+\langle
y_{2}^{*},y-y_{2}\rangle+\lambda_{2}(\alpha-\alpha_{2})\leq\tau\varepsilon(\|(x,y)-(\bar{x},\bar{y})\|_{X\times
Y}+|\alpha-\alpha_{2}|)$ (6.10)

for all $(x,y)\in\Theta_{2}\cap V$ with the corresponding number $\alpha$
satisfying

$\alpha-\alpha_{2}=\langle x^{*},x-x_{2}\rangle+\langle
y^{*},y-y_{2}\rangle+(L+\frac{1}{2}\nu)\varepsilon(\|(x_{2},y_{2})-(\bar{x},\bar{y})\|_{X\times
Y}-\|(x,y)-(\bar{x},\bar{y})\|_{X\times Y}).$

The usage of the triangle and Cauchy–Schwarz inequalities ensures that

$|\alpha-\alpha_{2}|\leq(L+L\varepsilon+\frac{1}{2}\nu\varepsilon)\|(x,y)-(x_{2},y_{2})\|_{X\times
Y}.$

Observe that the left-hand side $\vartheta$ in (6.10) can be represented as

$\displaystyle\vartheta=$ $\displaystyle\langle
x_{2}^{*}+\lambda_{2}x^{*},x-x_{2}\rangle+\langle
y_{2}^{*}+\lambda_{2}y^{*},y-y_{2}\rangle$
$\displaystyle+\lambda_{2}(L+\frac{1}{2}\nu)\varepsilon(\|(x_{2},y_{2})-(\bar{x},\bar{y})\|_{X\times
Y}-\|(x,y)-(\bar{x},\bar{y})\|_{X\times Y}),$

and thus (6.10) implies the estimate

$\displaystyle\langle x_{2}^{*}+\lambda_{2}x^{*},x-x_{2}\rangle+\langle
y_{2}^{*}+\lambda_{2}y^{*},y-y_{2}\rangle$
$\displaystyle\leq[\lambda_{2}(L+\frac{1}{2}\nu)+\tau(1+L+L\varepsilon+\nu\varepsilon)]\varepsilon\|(x,y)-(x_{2},y_{2})\|_{X\times
Y}$

for all $(x,y)\in\Theta_{2}\cap V$. This gives us the inclusion

$\displaystyle(x_{2}^{*}+\lambda_{2}x^{*},y_{2}^{*}+\lambda_{2}^{*}y^{*})\in\widehat{N}_{\mu\varepsilon}\big{(}(x_{2},y_{2});\Theta_{2}\big{)}$
(6.11)
$\displaystyle\operatorname{with}~{}\mu:=\lambda_{2}(L+\frac{1}{2}\nu)+\tau(1+L+L\varepsilon+\nu\varepsilon).$

Employing Lemma 2.2 ensures the existence of $\rho^{\prime}\in(0,\rho)$ such
that

$x_{2}^{*}\in T(x;\Omega)+\tau\varepsilon B_{X}\;\mbox{ for all
}\;x\in\Omega\cap(x_{2}+\rho^{\prime}B_{X}).$ (6.12)

It follows from the representation of $\varepsilon$-normals in (6.11) taken
from [7, page 222] that there exists a pair
$(u,v)\in\Theta_{2}\cap[(x_{2},y_{2})+\rho^{\prime}(B_{X}\times B_{Y})]$ such
that

$(x_{2}^{*}+\lambda_{2}x^{*},y_{2}^{*}+\lambda_{2}y^{*})\in\widehat{N}\big{(}(u,v);\Theta_{2}\big{)}+(\mu+\tau)\varepsilon(B_{X}\times
B_{Y}).$

Then we get $x_{2}^{*}\in T(u;\Omega)+\tau\varepsilon$ from (6.12), $x^{*}\in
T(u;\Omega)+\tau\varepsilon$ from (6.6) and the third inequality in (6.7).
Denoting $\eta:=\max\\{\lambda_{2},\|(x_{2}^{*},y_{2}^{*})\|_{X\times Y}\\}$
gives us

$\frac{1}{4}-\frac{\tau\varepsilon}{2}<\eta<\frac{1}{2}+\nu\varepsilon$

by (6.7) with $\max\\{\lambda_{2},1/5\\}\leq\eta<3/4$ when $\tau$ is
sufficiently small. Setting

$\lambda:=\frac{\lambda_{2}}{\eta},~{}~{}(u_{1}^{*},v_{1}^{*}):=-\frac{1}{\eta}(x_{2}^{*},y_{2}^{*}),~{}~{}(u_{2}^{*},v_{2}^{*}):=\frac{1}{\eta}(x_{2}^{*}+\lambda_{2}x^{*},y_{2}^{*}+\lambda_{2}y^{*}),$

we have that $\lambda\geq 0$,
$\max\\{\lambda,\|u_{1}^{*}\|_{X}+\|v_{1}^{*}\|_{Y}\\}=1$, and
$\lambda(x^{*},y^{*})=(u_{1}^{*},v_{1}^{*})+(u_{2}^{*},v_{2}^{*})$. Moreover,
$(u_{2}^{*},v_{2}^{*})\in\widehat{N}\big{(}(u,v);\Theta_{2}\big{)}+(\mu+\tau)\varepsilon/\eta(B_{X}\times
B_{Y})$ and

$(u_{1}^{*},v_{1}^{*})=\frac{1}{\eta}(x_{1}^{*},y_{1}^{*})-\frac{1}{\eta}(x_{1}^{*}+x_{2}^{*},y_{1}^{*}+y_{2}^{*})\in\widehat{N}\big{(}(x_{1},x_{2});\Theta_{1}\big{)}+\tau\varepsilon/\eta(B_{X}\times
B_{Y})$

due to (6.7). This allows us to conclude that

$\displaystyle
u_{1}^{*}=\frac{x_{1}^{*}}{\eta}-\frac{x_{1}^{*}+x_{2}^{*}}{\eta}\in
T(x_{1};\Omega)+\frac{2\tau\varepsilon}{\eta}B_{X},$ (6.13) $\displaystyle
u_{2}^{*}=\frac{x_{2}^{*}}{\eta}+\frac{\lambda_{2}x^{*}}{\eta}\in
T(u;\Omega)+\frac{\lambda_{2}+1}{\eta}\tau\varepsilon B_{X},$ (6.14)

where (6.13) comes from the first inequality in (6.7) by $x_{1}^{*}\in
T(x_{1};\Omega)+\tau\varepsilon B_{X}$, while (6.14) follows from the fact
that $x^{*},x_{2}^{*}\in T(u;\Omega)+\tau\varepsilon B_{X}$ and the convexity
of $T(u;\Omega)$. This completes the proof of the theorem by taking $\tau$ to
be sufficiently small. ∎

######  Remark 6.4.

In the proof of Theorem 6.3, we deduce the inclusion in (6.14) from the
convexity of $J(T(u;\Omega))=T(u;\Omega)$ in the Hilbert space $X$. Note that
in reflexive spaces, the image set $J(T(u;\Omega))$ may be not convex when
$\Omega$ is convex.

Now we come to the last theorem of this paper providing the pointbased sum
rule for both conic contingent coderivatives from Definition 4.2(ii,iii).

######  Theorem 6.5.

Let $S_{i}:X\rightrightarrows Y$, $i=1,2$, be set-valued mappings between
Hilbert spaces, let $\Omega\subset X$ be a closed and convex set, and let
$(\bar{x},\bar{y})\in\operatorname{gph}(S_{1}+S_{2})|_{\Omega}$. Consider the
mapping $G\colon X\times Y\rightrightarrows Y^{2}$ defined by

$G(x,y):=\big{\\{}(y_{1},y_{2})\in Y\times Y~{}\big{|}~{}y_{1}\in
S_{1}(x),~{}y_{2}\in S_{2}(x),~{}y_{1}+y_{2}=y\big{\\}}.$

Fix $(\bar{y}_{1},\bar{y}_{2})\in G(\bar{x},\bar{y})$, and let $G$ be inner
semicontinuous relative to $\Omega\times Y$ at
$(\bar{x},\bar{y},\bar{y}_{1},\bar{y}_{2})$. Assume that the graphs of $S_{1}$
and $S_{2}$ are locally closed around $(\bar{x},\bar{y}_{1})$ and
$(\bar{x},\bar{y}_{2})$, respectively, that either $S_{1}$ is PSNC relative to
$\Omega$ at $(\bar{x},\bar{y}_{1})$ or $S_{2}$ is PSNC relative to $\Omega$ at
$(\bar{x},\bar{y}_{2})$, and that $\\{S_{1},S_{2}\\}$ satisfies the mixed
coderivative qualification condition

$D_{\Omega}^{c*}S_{1}(\bar{x}|\bar{y}_{1})(0)\cap(-D_{\Omega}^{c*}S_{2}(\bar{x}|\bar{y}_{2}))(0)=\\{0\\}.$
(6.15)

Then for all $y^{*}\in Y$, we have the inclusions

$D_{\Omega}^{c*}(S_{1}+S_{2})(\bar{x}|\bar{y})(y^{*})\subset
D_{\Omega}^{c*}S_{1}(\bar{x}|\bar{y}_{1})(y^{*})+D_{\Omega}^{c*}S_{2}(\bar{x}|\bar{y}_{2})(y^{*}),$
(6.16)
$\widetilde{D}_{\Omega}^{c*}(S_{1}+S_{2})(\bar{x}|\bar{y})(y^{*})\subset\widetilde{D}_{\Omega}^{c*}S_{1}(\bar{x}|\bar{y}_{1})(y^{*})+\widetilde{D}_{\Omega}^{c*}S_{2}(\bar{x}|\bar{y}_{2})(y^{*}).$
(6.17)

###### Proof.

We verify only (6.16), since the proof of (6.17) is similar. Take any
$(x^{*},y^{*})\in X\times Y$ with $x^{*}\in
D_{\Omega}^{c*}(S_{1}+S_{2})(\bar{x}|\bar{y})(y^{*})$ and find sequences

$\varepsilon_{k}\downarrow
0,\;(x_{k},y_{k})\stackrel{{\scriptstyle\operatorname{gph}(S_{1}+S_{2})|_{\Omega}}}{{\longrightarrow}}(\bar{x},\bar{y}),\;\mbox{
and}$
$(x_{k}^{*},-y_{k}^{*})\in\widehat{N}_{\varepsilon_{k}}^{c}\big{(}(x_{k},y_{k});\operatorname{gph}(S_{1}+S_{2})|_{\Omega}\big{)}\cap[T(x_{k};\Omega)\times
Y]$

satisfying the convergence conditions

$x_{k}^{*}\rightharpoonup^{*}x^{*}~{}~{}\operatorname{and}~{}~{}y_{k}^{*}\to
y^{*}\;\mbox{ as }\;k\to\infty.$

By the inner semicontinuity of $G$ relative to $\Omega\times Y$ at
$(\bar{x},\bar{y},\bar{y}_{1},\bar{y}_{2})$, there exists a sequence
$(y_{1k},y_{2k})\to(\bar{y}_{1},\bar{y}_{2})$ with $(y_{1k},y_{2k})\in
G(x_{k},y_{k})$ for all $k\in\mathbb{N}$. Define the sets

$\Theta_{i}:=\big{\\{}(x,y_{1},y_{2})\in X\times Y\times
Y~{}\big{|}~{}(x,y_{i})\in\operatorname{gph}S_{i}|_{\Omega}\big{\\}}~{}~{}\operatorname{for}~{}i=1,2,$

which are locally closed around $(\bar{x},\bar{y}_{1},\bar{y}_{2})$ since the
graphs of $S_{i}$ are assumed to be locally closed around
$(\bar{x},\bar{y}_{i})$, $i=1,2$. We have
$(x_{k},y_{1k},y_{2k})\in\Theta_{1}\cap\Theta_{2}$ and easily check that

$(x_{k}^{*},-y_{k}^{*},-y_{k}^{*})\in\widehat{N}_{\varepsilon_{k}}^{c}\big{(}(x_{k},y_{1k},y_{2k});\Theta_{1}\cap\Theta_{2}\big{)}\cap[T(x_{k};\Omega)\times
Y\times Y]~{}~{}\operatorname{for~{}all}~{}k\in\mathbb{N}.$ (6.18)

We know that the sequence $\\{(x_{k}^{*},-y_{k}^{*})\\}$ is bounded and hence
find $L>0$ such that $\|(x_{k}^{*},-y_{k}^{*})\|_{X\times Y}\leq L$ for all
$k\in\mathbb{N}$. Applying the fuzzy rule from Theorem 6.3 to the intersection
in (6.18) along some $\varepsilon_{k}\downarrow 0$ gives us $\lambda_{k}\geq
0$,
$(\widetilde{x}_{ik},\widetilde{y}_{ik})\in\operatorname{gph}S_{i}|_{\Omega}$,
and

$(x_{ik}^{*},y_{ik}^{*})\in\big{[}\widehat{N}\big{(}(x_{k},y_{ik});\operatorname{gph}S_{i}|_{\Omega}\big{)}+(L+1)\varepsilon_{k}(B_{X}\times
B_{Y})\big{]}\cap[T(x_{ik};\Omega)\times Y]$

such that $\|(\widetilde{x}_{ik},\widetilde{y}_{ik})-(x_{k},y_{ik})\|_{X\times
Y}\leq\varepsilon_{k}$, $i=1,2$, and

$\|\lambda_{k}(x_{k}^{*},-y_{k}^{*},-y_{k}^{*})-(x_{1k}^{*}+x_{2k}^{*},-y_{1k}^{*},-y_{2k}^{*})\|_{X\times
Y\times Y}\leq\varepsilon_{k}$ (6.19)

with $\max\\{\lambda_{k},\|(x_{1k}^{*},y_{1k}^{*})\|_{X\times Y}\\}=1$. We now
claim that there exists $\lambda_{0}>0$ for which $\lambda_{k}>\lambda_{0}$
whenever $k\in\mathbb{N}$. Indeed, assuming the contrary tells us that
$\lambda_{k}\downarrow 0$, and then (6.19) yields $y_{ik}^{*}\to 0$, $i=1,2$,
and $x_{1k}^{*}+x_{2k}^{*}\to 0$ as $k\to\infty$. Furthermore, we have
$\|(x_{ik}^{*},-y_{ik}^{*})\|_{X\times Y}>\frac{1}{2}$ for all large $k$ and
$i=1,2$, which ensures that

$(x_{ik}^{*},-y_{ik}^{*})\in\widehat{N}_{2(L+1)\varepsilon_{k}}^{c}\big{(}(x_{k},y_{ik});\operatorname{gph}S_{i}|_{\Omega}\big{)}\cap[T(x_{ik};\Omega)\times
Y]$

for such $k$ and $i$. Since $\\{x_{1k}^{*}\\}$ is bounded along a subsequence,
there exists $x_{0}^{*}\in X$ such that
$x_{1k}^{*}\rightharpoonup^{*}x_{0}^{*}$ and also
$x_{2k}^{*}\rightharpoonup^{*}-x_{0}^{*}$ as $k\to\infty$ without loss of
generality. Thus

$x_{0}^{*}\in
D_{\Omega}^{c*}S_{1}(\bar{x}|\bar{y}_{1})(0)\cap(-D_{\Omega}^{c*}S_{2}(\bar{x}|\bar{y}_{2}))(0),$

which implies by the qualification condition (6.15) that $x_{0}^{*}=0$. For
definiteness, suppose that $S_{1}$ is PSNC relative to $\Omega$ at
$(\bar{x},\bar{y}_{1})$. Then $x_{1k}^{*}\to 0$ that contradicts the condition
$\max\\{\lambda_{k},\|(x_{1k}^{*},y_{1k}^{*})\|_{X\times Y}\\}=1$ due to
$\lambda_{k}\downarrow 0$ and $y_{1k}^{*}\to 0$ as $k\to\infty$.

Hence we can put $\lambda_{k}=1$ without loss of generality. Taking into
account that $x_{k}^{*}\rightharpoonup^{*}x^{*}$ and $y_{k}^{*}\to y^{*}$, it
follows from (6.19) that $y_{ik}^{*}\to y^{*}$ and
$x_{ik}^{*}\rightharpoonup^{*}x_{i}^{*}$, $i=1,2$, for some $x_{i}^{*}$ with
$x^{*}=x_{1}^{*}+x_{2}^{*}$. Then we only need to show that $x_{i}^{*}\in
D_{\Omega}^{c*}S_{i}(\bar{x}|\bar{y}_{i})(y^{*})$ for $i=1,2$. To proceed,
observe that if $(x_{ik}^{*},-y_{ik}^{*})\to(0,0)$ as $k\to\infty$, then
$(x_{i}^{*},y^{*})=(0,0)$, which immediately gives us (6.16). In the other
case, there exists $M>0$ such that $\|(x_{ik}^{*},-y_{ik}^{*})\|_{X\times
Y}>M$ for all $k\in\mathbb{N}$, and then we get

$(x_{ik}^{*},-y_{ik}^{*})\in\widehat{N}_{(L+1)\varepsilon_{k}/M}^{c}\big{(}(x_{k},y_{ik});\operatorname{gph}S_{i}|_{\Omega}\big{)}\cap[T(x_{k};\Omega)\times
Y]$

for all $k\in\mathbb{N}$ and $i=1,2$. This yields $x_{i}^{*}\in
D_{\Omega}^{c*}S_{i}(\bar{x}|\bar{y}_{i})(y^{*})$ for $i=1,2$, which verifies
(6.16) and thus completes the proof of the theorem. ∎

Finally, the pointbased characterizations of the relative Lipschitz-like
property from Section 4 lead us to the following corollary ensuring the
fulfillment of the qualification and relative PSNC conditions of Theorem 6.5.

######  Corollary 6.6.

In the general setting of Theorem 6.5, assume that either $S_{1}$ is relative
Lipschitz-like around $(\bar{x},\bar{y}_{1})$, or $S_{2}$ is relative
Lipschitz-like around $(\bar{x},\bar{y}_{2})$. Then both sum rule inclusions
(6.16) and (6.17) are satisfied.

###### Proof.

Suppose for definiteness that $S_{1}$ is relative Lipschitz-like around
$(\bar{x},\bar{y}_{1})$. Then Theorem 4.5 tells us that
$D^{c*}_{\Omega}S_{1}(\bar{x}|\bar{y}_{1})(0)=\\{0\\}$, and hence the
qualification condition (6.15) holds. This coderivative criterion also yields
the fulfillment of the relative PSNC property from Definition 4.4, and thus
the claimed statement is verified. ∎

##  7 Concluding Remarks

This paper introduces and develops new notions of relative generalized normals
and corresponding coderivatives for set-valued mappings in Banach spaces and
use them for deriving necessary conditions, sufficient conditions, and
characterizations of the relative Lipschitz-like property of general
multifunctions. Moreover, it is shown that the limiting conic contingent
coderivatives in the obtained characterizations enjoy pointbased calculus
rules, which justify their broad usage in applications. To achieve such a
calculus, we develop a powerful technique of variational analysis resulting in
the novel relative extremal principle for systems of sets and the
corresponding fuzzy intersection rule.

The major goals of our future research revolve around applications of the
newly developed results to various problems of optimization and equilibria.
From the technical viewpoint, we plan to relax the convexity assumption on the
constraint set in the stability results and calculus rules. Furthermore, we
intend to generalize the Hilbert space requirement in the fuzzy intersection
and pointbased sum rules derived in Section 6.

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