DOI 10.1007/s10957-008-9495-y

**Growth Behavior of Two Classes of Merit Functions** **for Symmetric Cone Complementarity Problems**

**S.H. Pan· J.-S. Chen**

Published online: 14 January 2009

© Springer Science+Business Media, LLC 2009

**Abstract In the solution methods of the symmetric cone complementarity problem**
(SCCP), the squared norm of a complementarity function serves naturally as a merit
function for the problem itself or the equivalent system of equations reformulation.

In this paper, we study the growth behavior of two classes of such merit functions,
which are induced by the smooth EP complementarity functions and the smooth im-
plicit Lagrangian complementarity function, respectively. We show that, for the linear
symmetric cone complementarity problem (SCLCP), both the EP merit functions and
the implicit Lagrangian merit function are coercive if the underlying linear transfor-
*mation has the P -property; for the general SCCP, the EP merit functions are coercive*
*only if the underlying mapping has the uniform Jordan P -property, whereas the co-*
erciveness of the implicit Lagrangian merit function requires an additional condition
for the mapping, for example, the Lipschitz continuity or the assumption as in (45).

**Keywords Symmetric cone complementarity problem**· Jordan algebra · EP merit
functions· Implicit Lagrangian function · Coerciveness

Communicated by M. Fukushima.

The authors would like to thank the two anonymous referees for their helpful comments which improved the presentation of this paper greatly.

The research of J.-S. Chen was partially supported by National Science Council of Taiwan.

S.H. Pan

School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China e-mail:shhpan@scut.edu.cn

J.-S. Chen (

^{)}

Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan e-mail:jschen@math.ntnu.edu.tw

**1 Introduction**

Given a Euclidean Jordan algebra*A = (V, ◦, ·, ·), where ‘◦’ denotes the Jordan*
product andV is a finite-dimensional vector space over the real field R equipped
with the inner product*·, ·, let K be a symmetric cone in V and let F : V → V be*
a continuous mapping. The symmetric cone complementarity problem (SCCP) is to
*find ζ*∈ V such that

*ζ∈ K, F (ζ ) ∈ K, ζ, F (ζ ) = 0.* (1)
The model provides a simple unified framework for various existing complementar-
ity problems such as the nonlinear complementarity problem over nonnegative or-
thant cone (NCP), the second-order cone complementarity problem (SOCCP) and
the semidefinite complementarity problem (SDCP), and hence has extensive applica-
tions in engineering, economics, management science, and other fields; see [1–4] and
*references therein. When F (ζ )= L(ζ ) + b, L : V → V being a linear transformation*
*and b*∈ V, the SCCP becomes the linear complementarity problem over symmetric
cones (SCLCP),

*ζ∈ K, L(ζ ) + b ∈ K, ζ, L(ζ ) + b = 0.* (2)
Recently, there is much interest in the study of merit functions or complemen-
tarity functions associated with symmetric cones and the development of the merit
function approach or the smoothing method for solving the SCCP. For example, Liu,
Zhang and Wang [5] extended a class of merit functions proposed in [6] to the SCCP,
Kong, Tuncel and Xiu [7] studied the extension of the implicit Lagrangian function
proposed by Mangasarian and Solodov [8] to symmetric cones; Kong, Sun and Xiu
[9] proposed a regularized smoothing method by the natural residual complementar-
ity function associated with symmetric cones; and Huang and Ni [10] developed a
smoothing-type algorithm with the regularized CHKS smoothing function over the
symmetric cone.

*A mapping φ*: V × V → V is called a complementarity function associated with
the symmetric cone*K if the following equivalence holds:*

*φ (x, y)= 0 ⇐⇒ x ∈ K, y ∈ K, x, y = 0.* (3)
By Propositions III.4.4–4.5 and Theorem V.3.7 of [11], the Euclidean Jordan algebra
*V and the corresponding symmetric cone K can be written as*

V = V1× V2× · · · × V*m* and *K = K*^{1}*× K*^{2}*× · · · × K*^{m}*,* (4)
*where each (V**i**,◦, ·, ·) is a simple Euclidean Jordan algebra and K** ^{i}* is the sym-
metric cone inV

*i*

*. Moreover, for any x= (x*1

*, . . . , x*

_{m}*), y= (y*1

*, . . . , y*

_{m}*)*∈ V with

*x*

_{i}*, y*

*∈ V*

_{i}*i*,

*x◦ y = (x*1*◦ y*1*, . . . , x**m**◦ y**m**)* and *x, y = x*1*, y*1* + · · · + x**m**, y**m**.*

Therefore, the characterization (3) of complementarity function is equivalent to
*φ (x, y)= 0 ⇐⇒ x**i**∈ K*^{i}*, y*_{i}*∈ K*^{i}*,* *x**i**, y*_{i}* = 0 for all i = 1, 2, . . . , m. (5)*

*This means that, if φ is a complementarity function associated with the coneK,*
*then φ(x, y) for any x= (x*1*, . . . , x*_{m}*), y= (y*1*, . . . , y*_{m}*)∈ V with x**i**, y** _{i}*∈ V

*i*can be written as

*φ (x, y)*:=

*φ*^{(1)}*(x*_{1}*, y*_{1}*), φ*^{(2)}*(x*_{2}*, y*_{2}*), . . . , φ*^{(m)}*(x*_{m}*, y*_{m}*)*

*,*

*where φ** ^{(i)}*: V

*i*× V

*i*→ V

*i*is a complementarity function associated with

*K*

*, i.e.,*

^{i}*φ*

^{(i)}*(x*

_{i}*, y*

_{i}*)= 0 ⇐⇒ x*

*i*

*∈ K*

^{i}*, y*

_{i}*∈ K*

^{i}*,*

*x*

*i*

*, y*

_{i}*= 0.*(6) Consequently, the SCCP can be reformulated as the following system of equations:

*(ζ ):= φ(ζ, F (ζ )) =*

⎛

⎜⎝

*φ*^{(1)}*(ζ*1*, F*1*(ζ ))*
*...*
*φ*^{(m)}*(ζ*_{m}*, F*_{m}*(ζ ))*

⎞

⎟*⎠ = 0,*

*which naturally induces a merit function f*: V → R_{+}for the SCCP, defined as

*f (ζ ):= (1/2) (ζ ) *^{2}*= (1/2)*
*m*
*i=1*

* φ*^{(i)}*(ζ**i**, F**i**(ζ )) *^{2}*.*

In the rest of this paper, corresponding to the Cartesian structure ofV, we always
*write F= (F*1*, . . . , F*_{m}*)with F**i*: V → V*i**and ζ= (ζ*1*, . . . , ζ*_{m}*)with ζ**i*∈ V*i*.

*The merit function f is often involved in the design of the merit function methods*
or the equation reformulation methods for the SCCP. For these methods, the coer-
*civeness of f plays a crucial role in establishing the global convergence results. In*
this paper, we will study the growth behavior of two classes of such merit functions,
which respectively correspond to the EP-functions introduced by Evtushenko and
Purtov [12] and the implicit Lagrangian function by Mangasarian and Solodov [8].

The EP-functions over the symmetric cone *K were first introduced by Kong and*
Xiu [13], defined by

*φ*_{α}*(x, y):= −x ◦ y + (1/2α)*

*(x+ y)*_{−}2

*,* *0 < α≤ 1,* (7)

*φ*_{β}*(x, y):= −x ◦ y + (1/2β)*

*(x*_{−}*)*^{2}*+ (y*_{−}*)*^{2}

*,* *0 < β < 1,* (8)
*where (·)*_{−} denotes the minimum metric projection onto*−K. They showed that φ**α*

*and φ** _{β}* are continuously differentiable and strongly semismooth complementarity
functions associated with

*K. In addition, Kong, Tuncel and Xiu [*7] extended the implicit Lagrangian function to the symmetric cone

*K and studied its continuous*differentiability and strongly semismoothness. The function is defined as follows:

*φ*_{MS}*(x, y):= x ◦ y + (1/2α)*

*[(x − αy)*+]^{2}*− x*^{2}*+ [(y − αx)*+]^{2}*− y*^{2}
*,* (9)
*where α > 0 (= 1) is a fixed constant, and (·)*_{+}denotes the minimum metric projec-
tion on*K. Particularly, for the implicit Lagrangian merit function of the SCCP, they*
presented a mild stationary point condition and proved that it can provide a global
*error bound under the uniform Cartesian P -property and Lipschitz continuity of F .*

This paper is mainly concerned with the growth behavior of the merit functions induced by the above three types of smooth complementarity functions, that is,

*f*_{α}*(ζ ):= (1/2) φ**α**(ζ, F (ζ ))* ^{2}*= (1/2)*
*m*
*i*=1

* φ*_{α}^{(i)}*(ζ*_{i}*, F*_{i}*(ζ ))* ^{2}*,* (10)

*f*_{β}*(ζ ):= (1/2) φ**β**(ζ, F (ζ ))* ^{2}*= (1/2)*
*m*
*i*=1

* φ*_{β}^{(i)}*(ζ*_{i}*, F*_{i}*(ζ ))* ^{2}*,* (11)

*f*_{MS}*(ζ ):= (1/2) φ*_{MS}*(ζ, F (ζ )) *^{2}*= (1/2)*
*m*

*i=1*

* φ*^{(i)}_{MS}*(ζ**i**, F**i**(ζ )) *^{2}*,* (12)

*where φ*^{(i)}_{α}*, φ*^{(i)}

*β* *, φ*_{MS}* ^{(i)}*defined as in (7), (8), (9), respectively, are a complementarity
function associated with

*K*

*. Specifically, we show that for the SCLCP (2), the EP*

^{i}*merit functions f*

_{α}*and f*

_{β}*and the implicit Lagrangian function f*

_{MS}are coercive only

*if the linear transformation L has the P -property; for the general SCCP, f*

_{α}*and f*

*are*

_{β}*coercive if the mapping F has the uniform Jordan P -property, but the coerciveness of*

*f*

_{MS}

*needs an additional condition of F , for example, the Lipschitz continuity or the*assumption as in (45). WhenV = R

*and “◦” denotes the componentwise product of the vectors, the obtaining results precisely reduce to those of Theorems 2.1 and 2.3 in [14] and Theorem 4.1 in [15]. However, for the general Euclidean Jordan algebra even the Lorentz algebra, to the best of our knowledge, similar results have not been established for these merit functions.*

^{n}Throughout this paper, · represents the norm induced by the inner prod-
uct*·, ·, int(K) denotes the interior of the symmetric cone K, and (x*1*, . . . , x**m**)*∈
V1× · · · × V*m*is viewed as a column vector inV = V1× · · · × V*m**. For any x*∈ V,
*(x)*_{+}*and (x)*_{−}*denotes the metric projection of x ontoK and −K, respectively, i.e.,*
*(x)*_{+}:= argmin*y*∈K*{ x − y }.*

**2 Preliminaries**

This section recalls some concepts and materials of Euclidean Jordan algebras that will be used in the subsequent analysis. More detailed expositions of Euclidean Jor- dan algebras can be found in Koecher’s lecture notes [16] and the monograph by Faraut and Korányi [11]. Besides, one can find excellent summaries in [17–19].

*A Euclidean Jordan algebra is a triple (V, ◦, ·, ·*V*), where (V, ·, ·*V*)*is a finite-
dimensional inner product space over the real field*R and (x, y) → x ◦y : V×V → V*
is a bilinear mapping satisfying the following three conditions:

*(i) x◦ y = y ◦ x for all x, y ∈ V;*

*(ii) x◦ (x*^{2}*◦ y) = x*^{2}*◦ (x ◦ y) for all x, y ∈ V, where x*^{2}*:= x ◦ x;*

(iii) *x ◦ y, z*_{V}*= y, x ◦ z*_{V}*for all x, y, z*∈ V.

*We assume that there is an element e∈ V such that x ◦ e = x for all x ∈ V and*
*call e the unit element. Let ζ (x) be the degree of the minimal polynomial of x*∈ V,

*which can be equivalently defined as ζ (x):= min{k : {e, x, x*^{2}*, . . . , x** ^{k}*} are linearly

*dependent}. Since ζ (x)≤ dim(V) where dim(V) denotes the dimension of V, the*

*rank of (V, ◦) is well defined by q := max{ζ(x) : x ∈ V}. In a Euclidean Jordan*

*algebra (V, ◦, ·, ·*

_{V}

*), we denoteK := {x*

^{2}

*: x ∈ V} by the set of squares. From The-*orem III.2.1 of [11],

*K is a symmetric cone. This means that K is a self-dual closed*

*convex cone with nonempty interior int(K), and for any x, y ∈ int(K), there exists an*invertible linear transformation

*T : V → V such that T (K) = K.*

A Euclidean Jordan algebra is said to be simple if it is not the direct sum of two
Euclidean Jordan algebras. By Propositions III.4.4–III.4.5 and Theorem V.3.7 of [11],
any Euclidean Jordan algebra is, in a unique way, a direct sum of simple Euclidean
Jordan algebras. Moreover, the symmetric cone in a given Euclidean Jordan alge-
bra is, in a unique way, a direct sum of symmetric cones in the constituent simple
Euclidean Jordan algebras. Here are two popular examples of simple Euclidean Jor-
dan algebras. One is the algebraS^{n}*of n× n real symmetric matrices with the inner*
product*X, Y *_{S}^{n}*:= Tr(XY ) and the Jordan product X ◦ Y := (XY + Y X)/2, where*
*Tr(X) is the trace of X and XY is the usual matrix multiplication of X and Y . In*
this case, the unit element is the identity matrix inS* ^{n}* and the cone

*K is the set of*all positive semidefinite matrices. The other is the Lorentz algebra L

*, also called the quadratic forms algebra, withV = R*

^{n}*,*

^{n}*·, ·*

_{V}being the usual inner product inR

*and the Jordan product defined by*

^{n}*x◦ y := (x, y*_{R}^{n}*, x*_{1}*y*_{2}*+ y*1*x*_{2}*) ,* (13)
*for any x= (x*1*, x*2*), y= (y*1*, y*2*)*∈ R × R^{n}^{−1}. Under this case, the unit element
*e= (1, 0, . . . , 0) ∈ R** ^{n}*, and the associate cone, called the Lorentz cone (or the second-
order cone), is given by

*K := {x = (x*1

*, x*2

*)*∈ R × R

^{n}^{−1}

*: x*2

*≤ x*1}.

*Recall that an element c∈ V is said to be idempotent if c*^{2}*= c. Two idempotents c*
*and d are said to be orthogonal if c◦ d = 0. We say that {c*1*, c*2*, . . . , c**k*} is a complete
system of orthogonal idempotents if

*c*^{2}_{j}*= c**j**,* *c*_{j}*◦ c**i**= 0 if j = i, j, i = 1, 2, . . . , k, and*
*k*
*j*=1

*c*_{j}*= e.*

A nonzero idempotent is said to be primitive if it cannot be written as the sum of two other nonzero idempotents. We call a complete system of orthogonal primitive idempotents a Jordan frame. Then, we have the following spectral decomposition theorem (see Theorem III.1.2 in [11]).

**Theorem 2.1 Suppose that**A = (V, ◦, ·, ·_{V}*)is a Euclidean Jordan algebra with*
*rank q. Then, for each x∈ V, there exist a Jordan frame {c*1*, c*2*, . . . , c*_{q}*} and real*
*numbers λ*1*(x), λ*2*(x), . . . , λ*_{q}*(x)such that x*=*q*

*j*=1*λ*_{j}*(x)c*_{j}*. The numbers λ**j**(x)*
*(counting multiplicities), which are uniquely determined by x, are called the eigen-*
*values of x.*

*In the sequel, we denote by λ*max*(x)and λ*min*(x)*the maximum eigenvalue and the
*minimum eigenvalue of x respectively and by tr(x)*:=*q*

*j*=1*λ*_{j}*(x)the trace of x.*

By Proposition III.1.5 of [11], a Jordan algebra*A = (V, ◦) over R with a unit*
*element e∈ V is Euclidean if and only if the symmetric bilinear form tr(x ◦ y) is*
positive definite. Therefore, we may define an inner product*·, · on V by*

*x, y := tr(x ◦ y), ∀x, y ∈ V.*

Let* · be the norm on V induced by the inner product ·, ·, namely,*

* x :=*

*x, x =*

_{q}

*j*=1

*λ*^{2}_{j}*(x)*

*1/2*

*,* *∀x ∈ V.*

Then, by the Schwartz inequality, it is easy to verify that

* x ◦ y ≤ x · y , ∀x, y ∈ V.* (14)
*For a given x∈ V, we define the linear operator L : V → V by*

*L(x)y := x ◦ y, for every y ∈ V.*

Since the inner product*·, · is associative by the associativity of tr(·) (see Propo-*
sition II.4.3 of [11]), i.e., for all x, y, z*∈ V, it holds that x, y ◦ z = y, x ◦ z, the*
linear operator*L(x) for each x ∈ V is symmetric with respect to ·, · in the sense*
that

*L(x)y, z = y, L(x)z, ∀y, z ∈ V.*

*We say that elements x and y operator commute ifL(x) and L(y) commute, i.e.,*
*L(x)L(y) = L(y)L(x).*

*Let ϕ*: R → R be a real-valued function. Then, it is natural to define a vector-
valued function associated with the Euclidean Jordan algebra*A = (V, ◦, ·, ·) by*

*ϕ*_{V}*(x):= ϕ(λ*1*(x))c*1*+ ϕ(λ*2*(x))c*2*+ · · · + ϕ(λ**q**(x))c**q**,* (15)
*where x∈ V has the spectral decomposition x =**q*

*j*=1*λ**j**(x)c**j**. The function ϕ*_{V} is
also called the Löwner operator in [19] and shown to inherit many properties from
*ϕ. Especially, when ϕ(t) is chosen as max{0, t} and min{0, t} for t ∈ R, respectively,*
*ϕ*_{V} becomes the metric projection operator onto*K and −K,*

*(x)*_{+}:=

*q*
*j*=1

max

*0, λ**j**(x)*

*c*_{j}*,* *(x)*_{−}:=

*q*
*j*=1

min

*0, λ**j**(x)*

*c*_{j}*.* (16)

*It is easy to verify that x= (x)*_{+}*+ (x)*_{−},*|x| = (x)*_{+}*− (x)*_{−}and* x *^{2}*= (x)*_{+} ^{2}+
* (x)*_{−} ^{2}.

An important part in the theory of Euclidean Jordan algebras is the Peirce decom- position theorem which is stated as follows (see Theorem IV.2.1 of [11]).

**Theorem 2.2 Let**A = (V, ◦, ·, ·) be a Euclidean Jordan algebra with rank q and*let{c*1*, c*2*, . . . , c**q**} be a Jordan frame in V. For i, j ∈ {1, 2, . . . , q}, define*

V*ii*:=

*x∈ V : x ◦ c**i**= x*

*,* V*ij*:=

*x∈ V : x ◦ c**i**= (1/2)x = x ◦ c**j*

*,* *i= j.*

*Then, the spaceV is the orthogonal direct sum of subspaces V**ij* *(i≤ j). Further-*
*more,*

(a) V*ij*◦ V*ij*⊆ V*ii*+ V*jj*;
(b) V*ij*◦ V*j k*⊆ V*ik**if i= k;*

(c) V*ij*◦ V*kl**= {0} if {i, j} ∩ {k, l} = ∅.*

*To close this section, we recall the concepts of the P -property and the uniform*
*Jordan P -property for a linear transformation and a nonlinear mapping.*

**Definition 2.1 A linear transformation L**: V → V is said to have the P -property if*ζ* *and L(ζ ) operator commute*

*ζ◦ L(ζ ) ∈ −K* *⇒ x = 0.*

* Definition 2.2 A mapping F= (F*1

*, . . . , F*

*m*

*)with F*

*i*: V → V

*i*is said to have:

*(i) the uniform Cartesian P -property if there is a positive scalar ρ such that, for any*
*ζ, ξ∈ V, there is an index ν ∈ {1, 2, . . . , m} such that*

*ζ**ν**− ξ**ν**, F*_{ν}*(ζ )− F**ν**(ξ ) ≥ ρ ζ − ξ *^{2}*,*

*(ii) the uniform Jordan P -property if there is a positive scalar ρ such that, for any*
*ζ, ξ∈ V, there is an index ν ∈ {1, 2, . . . , m} such that*

*λ*_{max}*[(ζ**ν**− ξ**ν**)◦ (F**ν**(ζ )− F**ν**(ξ ))]≥ ρ ζ − ξ *^{2}*.*

Unless otherwise stated, in the subsequent analysis, we assume that A =
*(V, ◦, ·, ·) is a simple Euclidean Jordan algebra of rank q and dim(V) = n.*

**3 Coerciveness of f**_{α}**and f**_{β}

*In this section, we study the conditions under which the EP merit functions f*_{α}*and f** _{β}*
are coercive. For this purpose, we need Lemma 3.1 of [13], which is stated as follows.

* Lemma 3.1 For a given Jordan frame{c*1

*, c*

_{2}

*, . . . , c*

_{q}*}, if z ∈ V can be written as*

*z*=
*q*
*i*=1

*z*_{i}*c** _{i}*+

1*≤i<j≤q*

*z*_{ij}*,*

*with z**i**∈ R for i = 1, 2, . . . , q and z**ij*∈ V*ij* *for 1≤ i < j ≤ q, then*

*z*_{+}=
*q*
*i*=1

*s**i**c**i*+

1*≤i<j≤q*

*s**ij**,* *z*_{−}=
*q*
*i*=1

*w**i**c**i*+

1*≤i<j≤q*

*w**ij**,*

*where s**i* *≥ (z**i**)*_{+}*≥ 0, 0 ≥ (z**i**)*_{−}*≥ w**i* *with s**i* *+ w**i* *= z**i* *for i* *= 1, . . . , q, and*
*s*_{ij}*, w** _{ij}*∈ V

*ij*

*with s*

*ij*

*+ w*

*ij*

*= z*

*ij*

*for 1≤ i < j ≤ q.*

The following lemma summarizes some important inequalities involved in the
*maximum eigenvalue and the minimum eigenvalue for any x*∈ V. Since their proofs
can be found in Lemma 14 of [17] and Proposition 2.1 of [20], we here omit them.

**Lemma 3.2 For any x, y**∈ V, the following inequalities always hold:

*(a) λ*min*(x) c *^{2}*≤ x, c ≤ λ*max*(x) c *^{2}*for any nonzero idempotent c;*

(b) *|λ*max*(x+ y) − λ*max*(x)| ≤ y and |λ*min*(x+ y) − λ*min*(x)| ≤ y ;*
*(c) λ*max*(x+ y) ≤ λ*max*(x)+ λ*max*(y)and λ*min*(x+ y) ≥ λ*min*(x)+ λ*min*(y).*

Using Lemmas 3.1–3.2, we may establish a lower bound for * φ**α**(x, y)* and
* φ**β**(x, y)* .

**Lemma 3.3 Let φ**_{α}*and φ*_{β}*be given by (7) and (8), respectively. Then, for any*
*x, y*∈ V,

* φ**α**(x, y)* ≥

*(2α− α*^{2}*)/(2α)*

max

*[(λ*min*(x))*_{−}]^{2}*,* *[(λ*min*(y))*_{−}]^{2}
*,* (17)
* φ**β**(x, y)* ≥

*(1− β*^{2}*)/(2β)*

max

*[(λ*min*(x))*_{−}]^{2}*,[(λ*min*(y))*_{−}]^{2}

*.* (18)
*Proof Suppose that x has the spectral decomposition x*=*q*

*i*=1*x**i**c**i* *with x**i*∈ R and
*{c*1*, c*_{2}*, . . . , c** _{q}*} being a Jordan frame. From Theorem2.2, y∈ V can be expressed by

*y*=
*q*
*i*=1

*y*_{i}*c** _{i}*+

1*≤i<j≤q*

*y*_{ij}*,* (19)

*where y**i**∈ R for i = 1, 2, . . . , q and y**ij*∈ V*ij**. Therefore, for any l∈ {1, 2, . . . , q},*

*c**l**, x◦ y = c**l**◦ x, y =*

*x*_{l}*c*_{l}*,*

*q*
*i*=1

*y*_{i}*c** _{i}*+

1*≤i<j≤q*

*y*_{ij}

*= x**l*

*c*_{l}*,*

*q*
*i*=1

*y*_{i}*c*_{i}

*+ x**l*

*c*_{l}*,*

1*≤i<j≤q*

*y*_{ij}

*= x**l**y**l**,* (20)

where the last equality is due to the fact that*c**l**,*

1*≤i<j≤q**y** _{ij}* = 0 by the orthogo-

*nality of V*

*ij*

*(i≤ j).*

We next prove the inequality (17). From (19) and the spectral decomposition of x,

*x+ y =*
*q*
*i*=1

*(x*_{i}*+ y**i**)c** _{i}*+

1*≤i<j≤q*

*y*_{ij}*,*

which together with Lemma3.1implies that

*(x+ y)*_{−}=
*q*
*i*=1

*u*_{i}*c** _{i}*+

1*≤i<j≤q*

*u*_{ij}*,*

*where u**i**≤ (x**i**+ y**i**)*_{−}*≤ 0 for i = 1, 2, . . . , q and u**ij*∈ V*ij*. By this, we can compute

*c*_{l}*,[(x + y)*_{−}]^{2}

=

*c** _{l}*◦

_{q}

*i*=1

*u*_{i}*c** _{i}*+

1*≤i<j≤q*

*u*_{ij}

*, (x+ y)*_{−}

=

*u**l**c**l*+

*c**l*◦

1*≤i<j≤q*

*u**ij*

*,*

*q*
*i=1*

*u**i**c**i*+

1*≤i<j≤q*

*u**ij*

*= u*^{2}_{l}*+ u**l*

*c*_{l}*,*

1*≤i<j≤q*

*u*_{ij}

+

1*≤i<j≤q*

*u*_{ij}*, c** _{l}*◦

*q*

*i*=1

*u*_{i}*c*_{i}

+

*c** _{l}*◦

1*≤i<j≤q*

*u*_{ij}*,*

1*≤i<j≤q*

*u*_{ij}

*= u*^{2}* _{l}* +

*c*_{l}*,*

1*≤i<j≤q*

*u*_{ij}

2

*,* *∀l = 1, 2, . . . , q,* (21)

where the last equality is due to the fact that*c**l**,*

1*≤i<j≤q**u** _{ij}* = 0 by the orthogo-

*nality of V*

*ij*

*(i≤ j). Now, using (20)–(21), we obtain that*

*c**l**,−φ**α**(x, y)* =

*c*_{l}*, x◦ y − (1/2α)*

*(x+ y)*_{−}2

*= x**l**y*_{l}*− (1/2α)*

*u*^{2}* _{l}* +

*c*_{l}*,*

1*≤i<j≤q*

*u*_{ij}

2

*≤ x**l**y*_{l}*− (1/2α)*

*(x*_{l}*+ y**l**)*_{−}2

*,* *∀l = 1, 2, . . . , q,* (22)
where the inequality is due to the following facts

*u*_{l}*≤ (x**l**+ y**l**)*_{−}≤ 0 and

*c*_{l}*,*

1*≤i<j≤q*

*u*_{ij}

2

*≥ 0.*

On the other hand, from Lemma3.2(a) we have that

*c**l**,−φ**α**(x, y) ≥ λ*min*(−φ**α**(x, y)) c**l* ^{2}*= λ*min*(−φ**α**(x, y)),* *∀l = 1, 2, . . . , q.*

(23) Thus, combining (22) with (23), it follows that

*2αλ*min*(−φ**α**(x, y))≤ 2αx**l**y*_{l}*− [(x**l**+ y**l**)*_{−}]^{2}*,* *∀l = 1, 2, . . . , q.*

*Let λ*min*(x)= x**ν* *with ν∈ {1, 2, . . . , q}. Then, we have particularly that*

*2αλ*min*(−φ**α**(x, y))≤ 2αλ*min*(x)y*_{ν}*− [(λ*min*(x)+ y**ν**)*_{−}]^{2}*.* (24)
*We next proceed to the proof for two cases: λ*min*(x)≤ 0 and λ*min*(x) >*0.

*Case (i): λ*min*(x)*≤ 0. Under this case, we prove the following inequality:

*2αλ*min*(x)y*_{ν}*− [(λ*min*(x)+ y**ν**)*_{−}]^{2}*≤ −(2α − α*^{2}*)[(λ*min*(x))*_{−}]^{2}*,* (25)
which, together with (24), implies immediately

* φ**α**(x, y)* ≥!!*λ*min*(−φ**α**(x, y))*!! ≥

*(2α− α*^{2}*)/(2α)*

*[(λ*min*(x))*_{−}]^{2}*.* (26)

*In fact, if λ*min*(x)+ y**ν*≥ 0, then we can deduce that

*2αλ*min*(x)y*_{ν}*− [(λ*min*(x)+ y**ν**)*_{−}]^{2}*= 2α(λ*min*(x))*_{−}*(y*_{ν}*)*_{+}

*≤ −(2α − α*^{2}*)[(λ*min*(x))*_{−}]^{2};
otherwise, we have that

*2αλ*min*(x)y*_{ν}*− [(λ*min*(x)+ y**ν**)*_{−}]^{2}*= 2αλ*min*(x)y*_{ν}*− [(λ*min*(x)+ y**ν**)*]^{2}

*≤ −(2α − α*^{2}*)[λ*min*(x)]*^{2}

*= −(2α − α*^{2}*)[(λ*min*(x))*_{−}]^{2}*.*
*Case (ii): λ*min*(x) >*0. Under this case, the inequality (26) clearly holds.

Summing up the above discussions, the inequality (26) holds for any x, y∈ V. In
*view of the symmetry of x and y in φ*_{α}*(x, y), we also have that*

* φ**α**(x, y)* ≥

*(2α− α*^{2}*)/(2α)*

*[(λ*min*(y))*_{−}]^{2}*,*

*for any x, y*∈ V. Thus, the proof of the inequality (17) is completed.

We next prove the inequality (18). By the spectral decomposition of x, we have
*that (x*_{−}*)*^{2}=*q*

*i*=1*[(x**i**)*_{−}]^{2}*c** _{i}*, which in turn implies

*c**l**, (x*_{−}*)*^{2}* = [(x**l**)*_{−}]^{2}*,* *∀l = 1, 2, . . . , q.* (27)
In addition, from Lemma3.1*and the expression of y given by (19), it follows that*

*y*_{−}=
*q*
*i*=1

*v*_{i}*c** _{i}*+

1*≤i<j≤q*

*v*_{ij}*,*

*where v**i* *≤ (y**i**)*_{−}*≤ 0 for i = 1, 2, . . . , q and v**ij* ∈ V*ij*. By the same arguments
as (21),

*c**l**, (y*_{−}*)*^{2}* = v*_{l}^{2}+

*c*_{l}*,*

1*≤i<j≤q*

*v*_{ij}

2

*,* *∀l = 1, 2, . . . , q.* (28)

Now, from (20), (27) and (28), it follows that

*c**l**,−φ**β**(x, y)* =

*c*_{l}*, x◦ y − (1/2β)*

*(x*_{−}*)*^{2}*+ (y*_{−}*)*^{2}

*= x**l**y*_{l}*− (1/2β)*

*((x*_{l}*)*_{−}*)*^{2}*+ v*_{l}^{2}+

*c*_{l}*,*

1*≤i<j≤q*

*v*_{ij}

2

*≤ x**l**y*_{l}*− (1/2β)*

*((x*_{l}*)*_{−}*)*^{2}*+ (v**l**)*^{2}

*≤ x**l**y**l**− (1/2β)*

*((x**l**)*_{−}*)*^{2}*+ ((y**l**)*_{−}*)*^{2}

*,* *∀l = 1, 2, . . . , q,*
where the first inequality is due to the nonnegativity of*c**l**, (*

1*≤i<j≤q**v*_{ij}*)*^{2} and the
*second one is due to the fact that v**l**≤ (y**l**)*_{−}≤ 0. On the other hand, by Lemma3.2(a),

*c**l**,−φ*_{β}*(x, y) ≥ λ*min*(−φ*_{β}*(x, y)) c**l* ^{2}*= λ*min*(−φ*_{β}*(x, y)),* *∀l = 1, 2, . . . , q.*

Combining the last two inequalities leads immediately to
*λ*min*(−φ**β**(x, y))≤ x**l**y*_{l}*− (1/2β)*

*((x*_{l}*)*_{−}*)*^{2}*+ ((y**l**)*_{−}*)*^{2}

*,* *∀l = 1, 2, . . . , q.*

*Let λ*min*(x)= x**ν* *with ν∈ {1, 2, . . . , q}, and suppose that λ*min*(x)*≤ 0. Then,
*λ*_{min}*(−φ**β**(x, y))≤ λ*min*(x)y*_{ν}*− (1/2β)*

*((λ*_{min}*(x))*_{−}*)*^{2}*+ ((y**ν**)*_{−}*)*^{2}

*≤ [(λ*min*(x))*_{−}*][(y**ν**)*_{−}*] − (1/2β)*

*((λ*min*(x))*_{−}*)*^{2}*+ ((y**ν**)*_{−}*)*^{2}

*= −(1/2β)*

*β(λ*min*(x))*_{−}*− (y**ν**)*_{−}2

*+ (1 − β*^{2}*)[(λ*min*(x))*_{−}]^{2}

≤ −

*(1− β*^{2}*)/(2β)*

*[(λ*min*(x))*_{−}]^{2}*,*

which in turn implies

* φ**β**(x, y)* ≥!!λ_{min}*(−φ**β**(x, y))*!! ≥

*(1− β*^{2}*)/(2β)*

*[(λ*min*(x))*_{−}]^{2}*.* (29)

*If λ*min*(x)= x**ν**>*0, then the inequality (29) is obvious. Thus, (29) holds for any
*x, y∈ V. In view of the symmetry of x and y in φ**β**(x, y), we also have*

* φ**β**(x, y) ≥ |λ*min*(−φ**β**(x, y))*| ≥

*(1− β*^{2}*)/(2β)*

*[(λ*min*(y))*_{−}]^{2}

*for any x, y*∈ V. Consequently, the desired result follows.

The following proposition characterizes an important property for the smooth EP
*complementarity functions φ*_{α}*and φ** _{β}* under a unified framework.

**Proposition 3.1 Let φ**_{α}*and φ*_{β}*be given as in (7) and (8), respectively. Let{x** ^{k}*} ⊂ V

*and{y*

^{k}*} ⊂ V be sequences satisfying one of the following conditions:*

*(i) either λ*min*(x*^{k}*)→ −∞ or λ*min*(y*^{k}*)*→ −∞;

*(ii) λ*min*(x*^{k}*), λ*_{min}*(y*^{k}*) >−∞, λ*max*(x*^{k}*), λ*_{max}*(y*^{k}*)→ +∞ and x*^{k}*◦ y** ^{k}* → +∞.

*Then, φ**α**(x*^{k}*, y*^{k}*) → +∞ and φ**β**(x*^{k}*, y*^{k}*)* → +∞.

*Proof Under Case (i) the assertion is direct by Lemma*3.3. In what follows, we
will prove the assertion under Case (ii). Notice that, in this case, the sequences
*{x*^{k}*}, {y*^{k}*} and {x*^{k}*+ y*^{k}*} are all bounded below since λ*min*(x*^{k}*), λ*min*(y*^{k}*) >*−∞

*and λ*min*(x*^{k}*+ y*^{k}*)≥ λ*min*(x*^{k}*)+ λ*min*(y*^{k}*) >−∞. Therefore, the sequences {[(x** ^{k}*+

*y*

^{k}*)*

_{−}]

^{2}

*}, {((x*

^{k}*)*

_{−}

*)*

^{2}

*} and {((y*

^{k}*)*

_{−}

*)*

^{2}

*} are bounded. In addition, we also have λ*min

*(x*

*◦*

^{k}*y*

^{k}*)→ −∞ or λ*max

*(x*

^{k}*◦ y*

^{k}*)→ +∞, since x*

^{k}*◦ y*

*→ +∞.*

^{k}*If λ*min*(x*^{k}*◦ y*^{k}*)→ −∞ as k → ∞, then by Lemma*3.2(c) there holds that
*λ*min*(−φ**α**(x, y))= λ*min

*(x*^{k}*◦ y*^{k}*)− (1/2α)((x*^{k}*+ y*^{k}*)*_{−}*)*^{2}

*≤ λ*min*(x*^{k}*◦ y*^{k}*)+ (1/2α)*""*"((x*^{k}*+ y*^{k}*)*_{−}*)*^{2}""*" ,*
*λ*min*(−φ**β**(x, y))= λ*min

*(x*^{k}*◦ y*^{k}*)− (1/2β)*

*((x*^{k}*)*_{−}*)*^{2}*+ ((y*^{k}*)*_{−}*)*^{2}

*≤ λ*min*(x*^{k}*◦ y*^{k}*)+ (1/2β)*""*"((x*^{k}*)*_{−}*)*^{2}*+ ((y*^{k}*)*_{−}*)*^{2}""*" ,*

which, together with the boundedness of* ((x*^{k}*+y*^{k}*)*_{−}*)*^{2}* and ((x*^{k}*)*_{−}*)*^{2}*+((y*^{k}*)*_{−}*)*^{2} ,
*implies that λ*min*(−φ**α**(x*^{k}*, y*^{k}*))→ −∞ and λ*min*(−φ**β**(x*^{k}*, y*^{k}*))*→ −∞. Since

* φ**α**(x*^{k}*, y*^{k}*) ≥ |λ*min*(−φ**α**(x, y))| and φ**β**(x*^{k}*, y*^{k}*) ≥ |λ*min*(−φ**β**(x, y))|,*
we obtain immediately that* φ**α**(x*^{k}*, y*^{k}*) → +∞ and φ**β**(x*^{k}*, y*^{k}*)* → +∞.

*If λ*max*(x*^{k}*◦ y*^{k}*)→ +∞ as k → ∞, from Lemma*3.2(c) it then follows that
*λ*_{max}*(−φ*_{α}*(x, y))= λ*max

*(x*^{k}*◦ y*^{k}*)− (1/2α)((x*^{k}*+ y*^{k}*)*_{−}*)*^{2}

*≥ λ*max*(x*^{k}*◦ y*^{k}*)− (1/2α)*""*"((x*^{k}*+ y*^{k}*)*_{−}*)*^{2}""*" ,*
*λ*_{max}*(−φ**β**(x, y))= λ*max

*(x*^{k}*◦ y*^{k}*)− (1/2β)*

*((x*^{k}*)*_{−}*)*^{2}*+ ((y*^{k}*)*_{−}*)*^{2}

*≥ λ*max*(x*^{k}*◦ y*^{k}*)− (1/2β)*""*"((x*^{k}*)*_{−}*)*^{2}*+ ((y*^{k}*)*_{−}*)*^{2}""*" ,*

which, by the boundedness of* ((x*^{k}*+ y*^{k}*)*_{−}*)*^{2}* and ((x*^{k}*)*_{−}*)*^{2}*+ ((y*^{k}*)*_{−}*)*^{2} , implies
*that λ*max*(−φ**α**(x*^{k}*, y*^{k}*))→ +∞ and λ*max*(−φ**β**(x*^{k}*, y*^{k}*))*→ +∞. Noting that

* φ**α**(x*^{k}*, y*^{k}*) ≥ |λ*max*(−φ**α**(x*^{k}*, y*^{k}*))| and φ**β**(x*^{k}*, y*^{k}*) ≥ |λ*max*(−φ**β**(x*^{k}*, y*^{k}*))|,*

we obtain readily that* φ**α**(x*^{k}*, y*^{k}*) → +∞ and φ**β**(x*^{k}*, y*^{k}*)* → +∞.
WhenV = R* ^{n}*, ‘◦’ being the componentwise product of the vectors, x

^{k}*◦ y*

*→*

^{k}*+∞ automatically holds if λ*max

*(x*

^{k}*), λ*

_{max}

*(y*

^{k}*)*→ +∞, and Proposition3.1reduces to the result of Lemma 2.5 in [21] for the NCPs. However, for the general Euclidean Jordan algebra, this condition is necessary as illustrated by the following example.

*Example 3.1 Consider the Lorentz algebra L*^{n}*= (R*^{n}*,◦, ·, ·*_{R}^{n}*)* introduced in
Sect.2. Assume that n*= 3 and take the sequences {x*^{k}*} and {y** ^{k}*} as follows:

*x** ^{k}*=

⎛

⎝*k*
*k*
0

⎞

*⎠ ,* *y** ^{k}*=

⎛

⎝ *k*

*−k*
0

⎞

*⎠ , for each k.*

*It is easy to verify that λ*min*(x*^{k}*)= 0, λ*min*(y*^{k}*)= 0, λ*max*(x*^{k}*), λ*_{max}*(y*^{k}*)*→ +∞,
but * x*^{k}*◦ y*^{k}* +∞. For such {x*^{k}*} and {y** ^{k}*}, by computation we have that

*φ*

*α*

*(x*

^{k}*, y*

^{k}*) = 0 and φ*

*β*

*(x*

^{k}*, y*

^{k}*)*= 0, i.e. the conclusion of Proposition3.1does not hold.

In the subsequent analysis, we use often the continuity of the Jordan product stated by the following lemma. Since the proof can be found in [10], we omit it.

**Lemma 3.4 Let**{x^{k}*} and {y*^{k}*} be the sequences such that x*^{k}*→ ¯x and y*^{k}*→ ¯y when*
*k→ ∞. Then, we have that x*^{k}*◦ y*^{k}*→ ¯x ◦ ¯y.*

*Now, we are in a position to establish the coerciveness of f*_{α}*and f** _{β}*. Assume that

*A = (V, ◦, ·, ·) is a general Euclidean Jordan algebra. We consider first the SCLCP*case.

**Theorem 3.1 Let f**_{α}*and f*_{β}*be given by (10) and (12), respectively. If F (ζ )*=
*L(ζ )+ b, with the linear transformation L having the P -property, then f**α* *and f*_{β}*are coercive.*

*Proof Let{ζ*^{k}*} be a sequence such that ζ** ^{k}* → +∞. We need only to prove that

*f*

_{α}*(ζ*

^{k}*)→ +∞,*

*f*

_{β}*(ζ*

^{k}*)→ +∞.*(30)

*By passing to a subsequence if necessary, we assume that ζ*

^{k}*/ ζ*

^{k}*→ ¯ζ, and conse-*

*quently (L(ζ*

^{k}*)+ b)/ ζ*

^{k}*→ L(¯ζ). If λ*min

*(ζ*

^{k}*)*→ −∞, then from Proposition3.1 it follows that

*φ*

*α*

*(ζ*

^{k}*, L(ζ*

^{k}*)+ b) , φ*

*β*

*(ζ*

^{k}*, L(ζ*

^{k}*)+ b) → +∞, which in turn im-*plies (30).

Now, assume that*{ζ*^{k}*} is bounded below. We argue that the sequence {L(ζ*^{k}*)+ b}*

is unbounded by contradiction. Suppose that*{L(ζ*^{k}*)+ b} is bounded. Then,*
*L( ¯ζ )*= lim

*k*→∞

*(L(ζ*^{k}*)+ b)/ ζ*^{k}

*= 0 ∈ K.*

Since *{ζ*^{k}*} is bounded below and λ*max*(ζ*^{k}*)→ +∞ by ζ** ^{k}* → +∞, there is an
element ¯

*d∈ V such that (ζ*

^{k}*− ¯d)/ ζ*

^{k}*− ¯d ∈ K for each k. Noting that K is closed,*we have

*k*lim→∞*(ζ*^{k}*− ¯d)/ ζ*^{k}*− ¯d = ¯ζ/ ¯ζ = ¯ζ ∈ K.*

*Thus, ¯ζ* *∈ K, L(¯ζ) ∈ K and ¯ζ ◦ L(¯ζ) = 0. From Proposition 6 of [18], it follows*
*that ¯ζ and L( ¯ζ ) operator commute. This, together with ¯ζ◦ L(¯ζ) = 0 ∈ −K and the*
*P-property of L, implies that ¯ζ= 0, yielding a contradiction to ¯ζ = 1. Hence, the*
sequence*{L(ζ*^{k}*)+b} is unbounded. Without loss of generality, assume that L(ζ*^{k}*)+*

*b* → +∞.

*If λ*min*(L(ζ*^{k}*)+ b) → −∞, then using Proposition* 3.1 yields the desired re-
sult (30). We next assume that the sequence*{L(ζ*^{k}*)+ b} is bounded below. We prove*
that

*(ζ*^{k}*/ ζ*^{k}* ) ◦*

*(L(ζ*^{k}*)+ b)/ ζ*^{k}

* 0.* (31)

Suppose that (31) does not hold; then, from Lemma3.4, it follows that

*¯ζ ◦ L(¯ζ) = lim*

*k*→+∞

*(ζ*^{k}*− d)/ ζ*^{k}

◦

*(L(ζ*^{k}*)+ b − d)/ ζ*^{k}

*= 0 ∀d ∈ V. (32)*

Since*{ζ*^{k}*} and {L(ζ*^{k}*)+ b} are bounded below and λ*max*(ζ*^{k}*), λ*_{max}*(L(ζ*^{k}*)+ b) →*
*+∞, there is an element ˜d such that ζ*^{k}*− ˜d ∈ K and L(ζ*^{k}*)+ b − ˜d ∈ K for each k.*

Therefore,

*(ζ*^{k}*− ˜d)/ ζ*^{k}

*∈ K,*

*(L(ζ*^{k}*)+ b − ˜d)/ ζ*^{k}

*∈ K, ∀k.*

Noting that*K is closed, ¯ζ = lim**k*→∞*(ζ*^{k}*− ˜d)/ ζ*^{k}* and L(¯ζ)=lim**k*→∞*[(L(ζ*^{k}*)*+
*b− ˜d)/ ζ** ^{k}* ], we have

*¯ζ ∈ K,* *L( ¯ζ )∈ K.* (33)

From (32) and (33) and Proposition 6 of [18], it follows that ¯ζ and L( ¯ζ ) operator com-
*mute. Using the P -property of L and noting that ¯ζ◦ L(¯ζ) = 0 ∈ −K, we then obtain*

*¯ζ = 0, which clearly contradicts ¯ζ = 1. Therefore, (*31) holds. Since* ζ** ^{k}* → +∞,
we have

*ζ*

^{k}*◦ (L(ζ*

^{k}*)+ b) → +∞. Combining with λ*min

*(ζ*

^{k}*), λ*min

*(L(ζ*

^{k}*)+ b) >*

*−∞ and ζ*^{k}* , L(ζ*^{k}*)+ b → +∞, it follows that the sequences {ζ*^{k}*} and {L(ζ*^{k}*)*+
*b*} satisfy condition (ii) of Proposition3.1. This means that the result (30) holds.
**Theorem 3.2 Let f**_{α}*and f*_{β}*be defined as in (10) and (12), respectively. If the map-*
*ping F has the uniform Jordan P -property, then f*_{α}*and f*_{β}*are coercive.*

*Proof The proof technique is similar to that of Theorem 4.1 in [15]. For complete-*
ness, we include it. Let*{ζ*^{k}*} be a sequence such that ζ** ^{k}* → +∞. Corresponding to
the Cartesian structure of

*V, let ζ*

^{k}*= (ζ*

_{1}

^{k}*, . . . , ζ*

_{m}

^{k}*)with ζ*

_{i}*∈ V*

^{k}*i*

*for each k. Define*

*J*:=

*i∈ {1, 2, . . . , m} | {ζ*_{i}* ^{k}*} is unbounded

*.*

*Clearly, the set J= ∅, since {ζ*^{k}*} is unbounded. Let {ξ** ^{k}*} be a bounded sequence with

*ξ*

^{k}*= (ξ*

_{1}

^{k}*, . . . , ξ*

_{m}

^{k}*)and ξ*

_{i}*∈ V*

^{k}*i*

*for i= 1, 2, . . . , m, where ξ*

_{i}

^{k}*for each k is defined as*follows:

*ξ*_{i}* ^{k}*=

#*0,* *if i∈ J,*
*ζ*_{i}^{k}*,* *otherwise,*

*with i= 1, 2, . . . , m. Since F has the uniform Jordan P -property, there is a constant*
*ρ >*0 such that

*ρ ζ*^{k}*− ξ*^{k}^{2}≤ max

*i**=1,...,m**λ*_{max}

*(ζ*_{i}^{k}*− ξ*_{i}^{k}*)◦ (F**i**(ζ*^{k}*)− F**i**(ξ*^{k}*))*

*= λ*max

*ζ*_{ν}^{k}*◦ (F**ν**(ζ*^{k}*)− F**ν**(ξ*^{k}*))*

*≤ ζ*_{ν}^{k}*◦ (F**ν**(ζ*^{k}*)− F**ν**(ξ*^{k}*)) *

*≤ ζ*_{ν}^{k}* F**ν**(ζ*^{k}*)− F**ν**(ξ*^{k}*) ,* (34)
*where ν is an index from{1, 2, . . . , m} for which the maximum is attained and the*
last inequality is due to (14). Clearly, ν*∈ J by the definition of {ξ** ^{k}*}; consequently,

*{ζ*

*ν*

*} is unbounded. Without loss of generality, we assume that*

^{k}* ζ**ν*^{k}* → +∞.* (35)

Since

* ζ*^{k}*− ξ*^{k}^{2}*≥ ζ*_{ν}^{k}*− ξ*_{ν}^{k}^{2}*= ζ*_{ν}^{k}^{2}*,* *for each k,* (36)
dividing both sides of (34) by* ζ*_{ν}* ^{k}* then yields that

*ρ ζ*_{ν}^{k}* ≤ F**ν**(ζ*^{k}*)− F**ν**(ξ*^{k}*) ≤ F**ν**(ζ*^{k}*) + F**ν**(ξ*^{k}*) .*

Notice that *{F (ξ*^{k}*)} is bounded, since the mapping F is continuous and {ξ** ^{k}*} is
bounded. Hence, the last inequality implies immediately

* F**ν**(ζ*^{k}*) → +∞.* (37)

In addition, we can verify by contradiction that

* ζ*_{ν}^{k}*◦ F**ν**(ζ*^{k}*) → +∞.* (38)

In fact, if*{ ζ*_{ν}^{k}*◦ F**ν**(ζ*^{k}*)* } is bounded, then on the one hand we have

*k*lim→∞* ζ**ν*^{k}*◦ (F**ν**(ζ*^{k}*)− F**ν**(ξ*^{k}*)) / ζ**ν*^{k}^{2}*= 0,*
but on the other hand, the inequality (36) implies

*k*→+∞lim *ρ ζ*^{k}*− ξ*^{k}^{2}*/ ζ**ν*^{k}^{2}*≥ ρ > 0,*

which clearly contradicts the third inequality in (34). Thus, from (35), (37), (38),
the sequences*{ζ*_{ν}^{k}*} and {F**ν**(ζ*^{k}*)*} satisfy the conditions of Proposition3.1. Therefore,