Journal of Optimization Theory and Applications, vol. 141, pp. 167-191, 2009

### Growth behavior of two classes of merit functions for symmetric cone complementarity problems

Shaohua Pan^{1}

School of Mathematical Sciences South China University of Technology

Guangzhou 510640, China E-mail: shhpan@scut.edu.cn

Jein-Shan Chen ^{2}
Department of Mathematics
National Taiwan Normal University

Taipei, Taiwan 11677 E-mail: jschen@math.ntnu.edu.tw

December 18, 2007 (revised March 29, 2008)

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 implicit La-
grangian 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 transformation has the
*P -property; for the general SCCP, the EP merit functions are coercive only if the un-*
*derlying mapping has the uniform Jordan P -property, whereas the coerciveness of the*
implicit Lagrangian merit function requires an additional condition for the mapping, for
example, the Lipschitz continuity or the assumption as in (46).

Key words. Symmetric cone complementarity problem, Jordan algebra, EP merit func- tions, implicit Lagrangian function, coerciveness.

1The author’s work is partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province.

2Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.

## 1 Introduction

*Given a Euclidean Jordan algebra A = (V, ◦, h·, ·i) with “◦” denoting the Jordan product*
and V being a finite-dimensional vector space over the real field R with the inner product
*h·, ·i. Let K be a symmetric cone in V and 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,* *hζ, F (ζ)i = 0.* (1)

The model provides a simple unified framework for various existing complementarity problems such as the nonlinear complementarity problem over nonnegative orthant cone (NCP), the second-order cone complementarity problem (SOCCP) and the semidefinite complementarity problem (SDCP), and hence has extensive applications in engineering, economics, management science, and other fields; see [1, 18, 26] and references therein.

*When F (ζ) = L(ζ) + b with 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,* *hζ, L(ζ) + bi = 0.* (2)
Recently, there is much interest in the study of merit functions or complementarity
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 [17] extended a class of merit functions proposed in [7] to the SCCP, Kong, Tun-
cel and Xiu [16] studied the extension of the implicit Lagrangian function proposed by
Mangasarian and Solodov [19] to symmetric cones; Kong, Sun and Xiu [14] proposed a
regularized smoothing method by the natural residual complementarity function associ-
ated with symmetric cones; and Huang and Ni [9] 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,* *hx, yi = 0.* (3)
By Propositions III.4.4–4.5 and Theorem V.3.7 of [5], the Euclidean Jordan algebra V
*and the corresponding symmetric cone K can be written as*

V = V_{1}*× V*_{2}*× · · · × V*_{m}*and K = K*^{1}*× K*^{2}*× · · · × K*^{m}*,* (4)
where each A*i* := (V*i**, ◦, h·, ·i) is a simple Euclidean Jordan algebra and K** ^{i}* is the sym-
metric cone in V

_{i}*. Moreover, for x = (x*

_{1}

*, . . . , x*

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

_{1}

*, . . . , y*

_{m}*) ∈ V with x*

_{i}*, y*

_{i}*∈ V*

*,*

_{i}*x ◦ y = (x*_{1}*◦ y*_{1}*, . . . , x*_{m}*◦ y*_{m}*) and hx, yi = hx*_{1}*, y*_{1}*i + · · · + hx*_{m}*, y*_{m}*i.*

Therefore, the characterization (3) of complementarity function is equivalent to

*φ(x, y) = 0 ⇐⇒ x*_{i}*∈ K*^{i}*, y*_{i}*∈ K*^{i}*, hx*_{i}*, y*_{i}*i = 0 for all i = 1, 2, . . . , m.* (5)
*This means that, if φ is a complementarity function associated with the cone K** ^{i}*, i.e.

*φ(x**i**, y**i**) = 0 ⇐⇒ x**i* *∈ K*^{i}*, y**i* *∈ K*^{i}*, hx**i**, y**i**i = 0,* (6)
*then for any x = (x*_{1}*, . . . , x*_{m}*) ∈ V and y = (y*_{1}*, . . . , y*_{m}*) ∈ V with x*_{i}*, y*_{i}*∈ V** _{i}*,

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

*is exactly a complementarity function associated with the cone K. Consequently, the*
SCCP can be reformulated as the following system of equations:

*Φ(ζ) :=*

*φ(ζ*_{1}*, F*_{1}*(ζ))*
...

*φ(ζ*_{m}*, F*_{m}*(ζ))*

* = 0,* (7)

*which naturally induces a merit function f : V → R*_{+} for the SCCP, defined as
*f (ζ) :=* 1

2*kΦ(ζ)k*^{2} = 1
2

X*m*
*i=1*

*kφ(ζ*_{i}*, F*_{i}*(ζ))k*^{2}*.*

In the rest of this paper, corresponding to the Cartesian structure of V, 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 coerciveness
*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 [2] and the im-
plicit Lagrangian function by Mangasarian and Solodov [19]. The EP-functions over the
*symmetric cone K were first introduced by Kong and Xiu [15], defined by*

*φ**α**(x, y) := −x ◦ y +* 1

*2α[(x + y)**−*]^{2} *0 < α ≤ 1,* (8)
*φ**β**(x, y) := −x ◦ y +* 1

*2β*

£*(x**−*)^{2} *+ (y**−*)^{2}¤

*0 < β < 1* (9)
*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. Later, Kong, Tuncel and Xiu [16] 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}ª

*,* (10)

*where α > 0 (6= 1) is a fixed constant, and (·)*_{+} denotes the minimum metric projection
*on K. Particularly, for the implicit Lagrangian merit function for the SCCP, they pre-*
sented 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*kΦ**α**(ζ, F (ζ))k*^{2} = 1
2

X*m*
*i=1*

*kφ**α**(ζ**i**, F**i**(ζ))k*^{2}*,* (11)

*f**β**(ζ) :=* 1

2*kΦ**β**(ζ, F (ζ))k*^{2} = 1
2

X*m*
*i=1*

*kφ**β**(ζ**i**, F**i**(ζ))k*^{2}*,* (12)

*f*_{MS}*(ζ) :=* 1

2*kΦ*_{MS}*(ζ, F (ζ))k*^{2} = 1
2

X*m*
*i=1*

*kφ*_{MS}*(ζ*_{i}*, F*_{i}*(ζ))k*^{2}*.* (13)

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

_{β}*are 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 (46). When V = R*

*and*

^{n}*“◦” denotes the componentwise product of the vectors, the obtaining results precisely*
reduce to those of [24, Theorems 2.1 and 2.3] and [11, Theorem 4.1]. 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.

*Throughout this paper, k · k represents the norm induced by the inner product h·, ·i*
*and int(K) denotes the interior of the symmetric cone K. For a vector space V of finite*
*dimension, we write its dimension as dim(V). For any x ∈ V, (x)*_{+} *and (x)** _{−}* denotes the

*metric projection of x onto K and −K, respectively, i.e., (x)*+:= argmin

_{y∈K}*{kx − yk}.*

## 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 Jordan al- gebras can be found in Koecher’s lecture notes [13] and the monograph by Faraut and Kor´anyi [5]. Besides, one can find excellent summaries in the articles [21, 8, 22].

*A Euclidean Jordan algebra is a triple (V, ◦, h·, ·i*_{V}*), where (V, h·, ·i*_{V}) is a finite di-
*mensional inner product space over the real field R and (x, y) 7→ 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) hx ◦ y, zi*_{V} *= hy, x ◦ zi*_{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*ª

. Since
*ζ(x) ≤ dim(V), the rank of (V, ◦) is well defined by q := max{ζ(x) : x ∈ V}. In a*
*Euclidean Jordan algebra A = (V, ◦, h·, ·i*_{V}*), we denote K := {x*^{2} *: x ∈ V} by the set*
*of squares. From [5, Theorem III.2.1], 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 and III.4.5 and Theorem V.3.7 of
[5], 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 algebra 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 Jordan algebras. One is the
algebra S^{n}*of n × n real symmetric matrices with the inner product hX, Y i*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 I in S*^{n}*and the cone K is the set of all positive semidefinite matrices. The*
other is the Lorentz algebra L* ^{n}*, also called the quadratic forms algebra, with V = R

*,*

^{n}*h·, ·i*

_{V}being the usual inner product in R

*and the Jordan product defined by*

^{n}*x ◦ y := (hx, yi*_{R}^{n}*, x*_{1}*y*_{2}*+ y*_{1}*x*_{2})* ^{T}* (14)

*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),*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}*: kx*2*k*2 *≤ x*1

ª*.*

*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. One says 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 6= i, j, i = 1, 2, . . . , k,* and P_{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.

*Theorem 2.1 [5, Theorem III.1.2] Suppose that A = (V, ◦, h·, ·i*_{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 =* P_{q}

*j=1**λ*_{j}*(x)c*_{j}*.*

*The numbers λ**j**(x) (counting multiplicities), which are uniquely determined by x, are*
*called the eigenvalues of x. In the sequel, we write the maximum eigenvalue and the*
*minimum eigenvalue of x as λ*_{max}*(x) and λ*_{min}*(x), respectively. Furthermore, the trace of*
*x, denoted by tr(x), is defined as tr(x) :=*P_{q}

*j=1**λ*_{j}*(x).*

*By [5, Proposition III.1.5], 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 h·, ·i on V by*
*hx, yi := tr(x ◦ y), ∀ x, y ∈ V.*

*Let k · k be the norm on V induced by the inner product h·, ·i, namely,*
*kxk :=* p

*hx, xi =*³P_{q}

*j=1**λ*^{2}_{j}*(x)*

´_{1/2}

*,* *∀ x ∈ V.*

Then, from equation (12) of [20], it follows that

*kx ◦ yk ≤ kxk · kyk,* *∀ x, y ∈ V.* (15)

*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 h·, ·i is associative by the associativity of tr(·) (see [5, Proposition*
*II.4.3]), i.e., for all x, y, z ∈ V, it holds that hx, y ◦ zi = hy, x ◦ zi, the linear operator*
*L(x) for each x ∈ V is symmetric with respect to h·, ·i in the sense that*

*hL(x)y, zi = hy, L(x)zi,* *∀ y, z ∈ V.*

*We say that elements x and y operator commute if L(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, ◦, h·, ·i) by*

*ϕ*_{V}*(x) := ϕ(λ*_{1}*(x))c*_{1}*+ ϕ(λ*_{2}*(x))c*_{2}*+ · · · + ϕ(λ*_{q}*(x))c*_{q}*,* (16)
*where x ∈ V has the spectral decomposition x =* P_{q}

*j=1**λ*_{j}*(x)c*_{j}*. The function ϕ*_{V} is also
*called L¨owner operator in [22] 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)*_{+} :=

X*q*
*j=1*

max©

*0, λ*_{j}*(x)*ª

*c*_{j}*and (x)** _{−}* :=

X*q*
*j=1*

min©

*0, λ*_{j}*(x)*ª

*c*_{j}*.* (17)

*It is easy to verify that x = (x)*+*+ (x)**−**, |x| = (x)*+*− (x)**−**and kxk*^{2} *= k(x)*+*k*^{2}*+ k(x)**−**k*^{2}.
An important part in the theory of Euclidean Jordan algebras is the Peirce decom-
position theorem which is stated as follows.

*Theorem 2.2 [5, Theorem IV.2.1] Let A = (V, ◦, h·, ·i) be a Euclidean Jordan algebra*
*with rank q and {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*ª

*, V** _{ij}* :=©

*x ∈ V : x ◦ c** _{i}* = 1

2*x = x ◦ c** _{j}*ª

*, i 6= j.*

*Then the space V is the orthogonal direct sum of subspaces V*_{ij}*(i ≤ j). Furthermore,*
(a) V_{ij}*◦ V*_{ij}*⊆ V** _{ii}*+ V

_{jj}*;*

(b) V_{ij}*◦ V*_{jk}*⊆ V*_{ik}*if i 6= 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*

¾

*⇒ ζ = 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*

*hζ*_{ν}*− ξ*_{ν}*, F*_{ν}*(ζ) − F*_{ν}*(ξ)i ≥ ρkζ − ξk*^{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))] ≥ ρkζ − ξk*^{2}*.*

*Unless otherwise stated, in the subsequent analysis, we assume that A = (V, ◦, h·, ·i)*
*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 under what conditions the EP merit functions f*_{α}*and f** _{β}* are
coercive. For this purpose, we first present several technical lemmas.

*Lemma 3.1 [15, Lemma 3.1] For a given Jordan frame {c*_{1}*, c*_{2}*, . . . , c*_{q}*}, if z ∈ V can be*
*written as*

*z =*
X*q*

*i=1*

*z**i**c**i*+ X

*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*_{+} =
X*q*

*i=1*

*s*_{i}*c** _{i}* + X

*1≤i<j≤q*

*s*_{ij}*, z** _{−}* =
X

*q*

*i=1*

*w*_{i}*c** _{i}* + X

*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 maxi-
*mum eigenvalue and the minimum eigenvalue for any x ∈ V. Since their proofs can be*
found in [21, Lemma 14] and [23, Proposition 2.1], we here omit them.

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

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

(b) µ

*λ*_{max}*(x + y) − λ*_{max}*(x)*

¶

*≤ kyk and*
µ

*λ*_{min}*(x) − λ*_{min}*(x + y)*

¶

*≤ kyk;*

*(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 kφ*_{α}*(x, y)k and kφ*_{β}*(x, y)k.*

*Lemma 3.3 Let φ*_{α}*and φ*_{β}*be given by (8) and (9), respectively. Then, for any x, y ∈ V,*
*kφ*_{α}*(x, y)k ≥* *2α − α*^{2}

*2α* max
n

*[(λ*_{min}*(x))** _{−}*]

^{2}

*, [(λ*

_{min}

*(y))*

*]*

_{−}^{2}o

*,* (18)

*kφ*_{β}*(x, y)k ≥* *1 − β*^{2}
*2β* max

n

*[(λ*_{min}*(x))** _{−}*]

^{2}

*, [(λ*

_{min}

*(y))*

*]*

_{−}^{2}o

*.* (19)

*Proof. Suppose that x has the spectral decomposition x =*P_{q}

*i=1**x*_{i}*c*_{i}*with x*_{i}*∈ R and*
*{c*1*, c*2*, . . . , c**q**} being a Jordan frame. From Theorem 2.2, y ∈ V can be expressed by*

*y =*
X*q*

*i=1*

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

*1≤i<j≤q*

*y*_{ij}*,* (20)

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

*hc*_{l}*, x ◦ yi = hc*_{l}*◦ x, yi =*

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

X*q*
*i=1*

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

*1≤i<j≤q*

*y** _{ij}*
+

*= x*_{l}

*
*c*_{l}*,*

X*q*
*i=1*

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

*+ x*_{l}

*

*c*_{l}*,* X

*1≤i<j≤q*

*y** _{ij}*
+

*= x**l**y**l**,* (21)

*where the last equality is since hc*_{l}*,*P

*1≤i<j≤q**y** _{ij}*®

*= 0 by the orthogonality of V*_{ij}*(i ≤ j).*

*We next prove the inequality (18). From (20) and the spectral decomposition of x,*

*x + y =*
X*q*

*i=1*

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

*1≤i<j≤q*

*y*_{ij}*.*

which together with Lemma 3.1 implies that

*(x + y)** _{−}* =
X

*q*

*i=1*

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

*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
D

*c*_{l}*, [(x + y)** _{−}*]

^{2}E

=

*
*c*_{l}*◦*

Ã * _{q}*
X

*i=1*

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

*1≤i<j≤q*

*u*_{ij}

!

*, (x + y)** _{−}*
+

=

*
*u*_{l}*c** _{l}*+

Ã

*c*_{l}*◦* X

*1≤i<j≤q*

*u*_{ij}

!
*,*

X*q*
*i=1*

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

*1≤i<j≤q*

*u** _{ij}*
+

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

*

*c*_{l}*,* X

*1≤i<j≤q*

*u** _{ij}*
+

+

* X

*1≤i<j≤q*

*u*_{ij}*, c*_{l}*◦*
X*q*

*i=1*

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

+

*

*c*_{l}*◦* X

*1≤i<j≤q*

*u*_{ij}*,* X

*1≤i<j≤q*

*u** _{ij}*
+

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

*
*c**l**,*

Ã X

*1≤i<j≤q*

*u**ij*

!_{2}+

*,* *∀ l = 1, 2, . . . , q,* (22)

*where the last equality is since hc*_{l}*,*P

*1≤i<j≤q**u** _{ij}*®

*= 0 by the orthogonality of V*_{ij}*(i ≤ j).*

Now, using equations (21)–(22), we obtain that
*hc*_{l}*, −φ*_{α}*(x, y)i =*

¿

*c*_{l}*, x ◦ y −* 1

*2α[(x + y)** _{−}*]

^{2}À

*= x**l**y**l**−* 1
*2α*

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

*
*c**l**,*

Ã X

*1≤i<j≤q*

*u**ij*

!_{2}+

*≤ x*_{l}*y*_{l}*−* 1

*2α[(x*_{l}*+ y** _{l}*)

*]*

_{−}^{2}

*, ∀ l = 1, 2, . . . , q,*(23) where the inequality is due to the following facts

*u*_{l}*≤ (x*_{l}*+ y** _{l}*)

_{−}*≤ 0 and*

*
*c*_{l}*,*

Ã X

*1≤i<j≤q*

*u*_{ij}

!_{2}+

*≥ 0.*

On the other hand, from Lemma 3.2 (a) we have that

*hc*_{l}*, −φ*_{α}*(x, y)i ≥ λ*_{min}*(−φ*_{α}*(x, y))kc*_{l}*k*^{2} *= λ*_{min}*(−φ*_{α}*(x, y)),* *∀ l = 1, 2, . . . , q.* (24)
Thus, combining (23) with (24), 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 particularly have that*

*2αλ*min*(−φ**α**(x, y)) ≤ 2αλ*min*(x)y**ν* *− [(λ*min*(x) + y**ν*)*−*]^{2}*.* (25)
*We next proceed the proof by the two cases: λ*min*(x) ≤ 0 and λ*min*(x) > 0.*

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

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

*]*

_{−}^{2}

*≤ −(2α − α*

^{2}

*)[(λ*

_{min}

*(x))*

*]*

_{−}^{2}

*,*(26) which, together with (25), immediately implies that

*kφ**α**(x, y)k ≥ |λ*min*(−φ**α**(x, y))| ≥* *(2α − α*^{2})

*2α* *[(λ*min*(x))**−*]^{2}*.* (27)
*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}; and otherwise we will 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. For this case, the inequality (27) clearly holds.*

*Summing up the above discussions, the inequality (27) holds for any x, y ∈ V. In view*
*of the symmetry of x and y in φ*_{α}*(x, y), we also have that*

*kφ*_{α}*(x, y)k ≥* *(2α − α*^{2})

*2α* *[(λ*_{min}*(y))** _{−}*]

^{2}

*for any x, y ∈ V. Thus, the proof of the inequality (18) is completed.*

*We next prove the inequality (19). By the spectral decomposition of x, we have that*
*(x** _{−}*)

^{2}=P

_{q}*i=1**[(x** _{i}*)

*]*

_{−}^{2}

*c*

*, which in turn implies that*

_{i}*hc*_{l}*, (x** _{−}*)

^{2}

*i = [(x*

*)*

_{l}*]*

_{−}^{2}

*,*

*∀ l = 1, 2, . . . , q.*(28)

*In addition, from Lemma 3.1 and the expression of y given by (20), it follows that*

*y** _{−}*=
X

*q*

*i=1*

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

*1≤i<j≤q*

*v*_{ij}*,*

*where v*_{i}*≤ (y** _{i}*)

_{−}*≤ 0 for i = 1, . . . , q and v*

_{ij}*∈ V*

*. By the same arguments as (22),*

_{ij}*hc*_{l}*, (y** _{−}*)

^{2}

*i = v*

^{2}

*+*

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

Ã X

*1≤i<j≤q*

*v*_{ij}

!_{2}+

*,* *∀ l = 1, 2, . . . , q.* (29)

Now, from equations (21), (28) and (29), it follows that
*hc*_{l}*, −φ*_{β}*(x, y)i =*

¿

*c*_{l}*, x ◦ y −* 1
*2β*

£*(x** _{−}*)

^{2}

*+ (y*

*)*

_{−}^{2}¤À

*= x*_{l}*y*_{l}*−* 1
*2β*

*((x** _{l}*)

*)*

_{−}^{2}

*+ v*

_{l}^{2}+

*
*c*_{l}*,*

Ã X

*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 hc*_{l}*, (*P

*1≤i<j≤q**v** _{ij}*)

^{2}

*i, and the*

*second one is since v*

*l*

*≤ (y*

*l*)

*−*

*≤ 0. On the other hand, by Lemma 3.2 (a),*

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

Combining the last two inequalities immediately leads 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, . . . , 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}¤

= *− [β(λ*_{min}*(x))*_{−}*− (y** _{ν}*)

*]*

_{−}^{2}

*− (1 − β*

^{2}

*)[(λ*

_{min}

*(x))*

*]*

_{−}^{2}

*2β*

*≤ −(1 − β*^{2})

*2β* *[(λ*_{min}*(x))** _{−}*]

^{2}

*,*which in turn implies that

*kφ*_{β}*(x, y)k ≥ |λ*_{min}*(−φ*_{β}*(x, y))| ≥* *(1 − β*^{2})

*2β* *[(λ*_{min}*(x))** _{−}*]

^{2}

*.*(30)

*If λ*

_{min}

*(x) = x*

_{ν}*> 0, then the inequality (30) is obvious. Thus, (30) holds for any x, y ∈ V.*

*In view of the symmetry of x and y in φ**β**(x, y), we also have*
*kφ*_{β}*(x, y)k ≥ |λ*_{min}*(−φ*_{β}*(x, y))| ≥* *(1 − β*^{2})

*2β* *[(λ*_{min}*(y))** _{−}*]

^{2}

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

*2*

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 (8) and (9), respectively. Let {x*^{k}*} ⊂ V*
*and {y*^{k}*} ⊂ V be the 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 kx*^{k}*◦ y*^{k}*k → +∞.*

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

Proof. If Case (i) is satisfied, then 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}ª

*, {((x** ^{k}*)

*)*

_{−}^{2}

*} and {((y*

*)*

^{k}*)*

_{−}^{2}

*} are bounded. Since*

*kx*

^{k}*◦ y*

^{k}*k → +∞, we must have λ*min

*(x*

^{k}*◦ y*

^{k}*) → −∞ or λ*max

*(x*

^{k}*◦ y*

^{k}*) → +∞.*

*If λ*_{min}*(x*^{k}*◦ y*^{k}*) → −∞ as k → ∞, then by Lemma 3.2(b) there hold 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 k((x*^{k}*+y** ^{k}*)

*)*

_{−}^{2}

*k and k((x*

*)*

^{k}*)*

_{−}^{2}

*+((y*

*)*

^{k}*)*

_{−}^{2}

*k implies*

*that λ*

_{min}

*(−φ*

_{α}*(x*

^{k}*, y*

^{k}*)) → −∞ and λ*

_{min}

*(−φ*

_{β}*(x*

^{k}*, y*

^{k}*)) → −∞. Since*

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

*−*)

^{2}

*k and k((x*

*)*

^{k}*−*)

^{2}

*+ ((y*

*)*

^{k}*−*)

^{2}

*k, implies that*

*λ*

_{max}

*(−φ*

_{α}*(x*

^{k}*, y*

^{k}*)) → +∞ and λ*

_{max}

*(−φ*

_{β}*(x*

^{k}*, y*

^{k}*)) → +∞. Noting that*

*kφ*_{α}*(x*^{k}*, y*^{k}*)k ≥ |λ*_{max}*(−φ*_{α}*(x*^{k}*, y*^{k}*))| and kφ*_{β}*(x*^{k}*, y*^{k}*)k ≥ |λ*_{max}*(−φ*_{β}*(x*^{k}*, y*^{k}*))|,*
*we readily obtain that kφ**α**(x*^{k}*, y*^{k}*)k → +∞ and kφ**β**(x*^{k}*, y*^{k}*)k → +∞.* *2*

When V = R^{n}*with “◦” being the componentwise product of the vectors, the condi-*
*tion kx*^{k}*◦ y*^{k}*k → +∞ automatically holds if λ*_{max}*(x*^{k}*), λ*_{max}*(y*^{k}*) → +∞, and Proposition*
3.1 reduces to the result of [10, Lemma 2.5] for the NCPs. But 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}*, ◦, h·, ·i*

_{R}

*) introduced in Section*

^{n}*2. Assume that n = 3 and take the sequences {x*

^{k}*} and {y*

^{k}*} as follows:*

*x** ^{k}* =

*k*
*k*
0

* and 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*
*kx*^{k}*◦y*^{k}*k 9 +∞. For such {x*^{k}*} and {y*^{k}*}, by computation we have that kφ**α**(x*^{k}*, y*^{k}*)k = 0*
*and kφ*_{β}*(x*^{k}*, y*^{k}*)k = 0, i.e. the conclusion of Proposition 3.1 does not hold.*

In addition, in the subsequent analysis, we also use the continuity of Jordan product stated by the following lemma. Since the proof can be found in [9] or [20], 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, ◦, h·, ·i) is a general Euclidean Jordan algebra. We first consider the SCLCP*case.

*Theorem 3.1 Let f*_{α}*and f*_{β}*be given by (11) and (13), 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 kζ*^{k}*k → +∞. We only need to prove that*
*f**α**(ζ*^{k}*) → +∞ and f**β**(ζ*^{k}*) → +∞.* (31)
*By passing to a subsequence if necessary, we assume that ζ*^{k}*/kζ*^{k}*k → ¯ζ, and consequently*
*(L(ζ*^{k}*) + b)/kζ*^{k}*k → L(¯ζ). If λ*_{min}*(ζ*^{k}*) → −∞, then from Proposition 3.1 it follows that*
*kφ*_{α}*(ζ*^{k}*, L(ζ*^{k}*) + b)k, kφ*_{β}*(ζ*^{k}*, L(ζ*^{k}*) + b)k → +∞, which in turn implies (31).*

*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*

*kζ*^{k}*k* *= 0 ∈ K.*

*Since {ζ*^{k}*} is bounded below and λ*max*(ζ*^{k}*) → +∞ by kζ*^{k}*k → +∞, there is an element*
*d ∈ V such that (ζ*¯ ^{k}*− ¯d)/kζ*^{k}*− ¯dk ∈ K for each k. Noting that K is closed, and we have*

*k→∞*lim

*ζ*^{k}*− ¯d*

*kζ*^{k}*− ¯dk* = lim

*k→∞*

*(ζ*^{k}*− ¯d)/kζ*^{k}*k*

°*°ζ*^{k}*/kζ*^{k}*k − ¯d/kζ*^{k}*k*°

° =
*ζ*¯

*k¯ζk* = ¯*ζ ∈ K.*

Thus, ¯*ζ ∈ K, L(¯ζ) ∈ K and ¯ζ ◦ L(¯ζ) = 0. From Proposition 6 of [8], 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 k¯ζk = 1. Hence, the sequence {L(ζ*^{k}*) + b}*

*is unbounded. Without loss of generality, assume that kL(ζ*^{k}*) + bk → +∞.*

*If λ*min*(L(ζ*^{k}*) + b) → −∞, then using Proposition 3.1 yields the desired result of (31).*

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

*kζ*^{k}*k* *◦* *L(ζ*^{k}*) + b*

*kζ*^{k}*k* *9 0.* (32)

Suppose that (32) does not hold, then from Lemma 3.4 it follows that
*ζ ◦ L(¯*¯ *ζ) = lim*

*k→+∞*

*ζ*^{k}*− d*

*kζ*^{k}*k* *◦* *L(ζ*^{k}*) + b − d*

*kζ*^{k}*k* *= 0 ∀d ∈ V.* (33)

*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*

*kζ*^{k}*k* *∈ K and* *L(ζ*^{k}*) + b − ˜d*

*kζ*^{k}*k* *∈ K,* *∀ k.*

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

*ζ ∈ K and L(¯*¯ *ζ) ∈ K.* (34)

From (33) and (34) and Proposition 6 of [8], it follows that ¯*ζ and L(¯ζ) operator commute.*

*Using the P -property of L and noting that ¯ζ ◦ L(¯ζ) = 0 ∈ −K, we then obtain ¯ζ =*
*0, which clearly contradicts k¯ζk = 1. Therefore, (32) holds. Since kζ*^{k}*k → +∞, we*
*have kζ*^{k}*◦ (L(ζ*^{k}*) + b)k → +∞. Combining with λ*_{min}*(ζ*^{k}*), λ*_{min}*(L(ζ*^{k}*) + b) > −∞ and*
*kζ*^{k}*k, kL(ζ*^{k}*) + bk → +∞, it follows that the sequences {ζ*^{k}*} and {L(ζ*^{k}*) + b} satisfy*
condition (ii) of Proposition 3.1. This means that the result of (31) holds. *2*

*Theorem 3.2 Let f*_{α}*and f*_{β}*be defined as in (11) and (13), respectively. If the mapping*
*F has the uniform Jordan P -property, then f*_{α}*and f*_{β}*are coercive.*

Proof. The proof technique is similar to that of [11, Theorem 4.1]. For completeness,
*we include it. Let {ζ*^{k}*} be a sequence such that kζ*^{k}*k → +∞. Corresponding to the*
*Cartesian structure of V, we write ζ*^{k}*= (ζ*_{1}^{k}*, . . . , ζ*_{m}^{k}*) with ζ*_{i}^{k}*∈ V**i* *for each k. Define*

*J :=*©

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

*Clearly, the set J 6= ∅ since {ζ*^{k}*} is unbounded. Let {ξ*^{k}*} be a bounded sequence with*
*ξ*^{k}*= (ξ*_{1}^{k}*, . . . , ξ*_{m}^{k}*) and ξ*_{i}^{k}*∈ V*_{i}*for i = 1, 2, . . . , m, where ξ*_{i}^{k}*for each k is defined as follows:*

*ξ*_{i}* ^{k}* =

½ 0 *if i ∈ J;*

*ζ*_{i}^{k}*otherwise,* *i = 1, 2, . . . , m.*

*Since F has the uniform Jordan P -property, there is a constant ρ > 0 such that*
*ρkζ*^{k}*− ξ*^{k}*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}*))¤

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

*≤ kζ*_{ν}^{k}*kkF*_{ν}*(ζ*^{k}*) − F*_{ν}*(ξ*^{k}*)k,* (35)