Volume 2007, Article ID 84146,10pages doi:10.1155/2007/84146
Research Article
Some Characteristic Quantities Associated with Homogeneous
P
-Type and
M
-Type Functions
Ya-Ping Fang, Nan-Jing Huang, and Yeol Je Cho
Received 22 November 2006; Accepted 20 March 2007
Recommended by Patricia J. Y. Wong
Several characteristic quantities associated with homogeneousP-type andM-type func-tions are introduced and studied in this paper. Further, the concepts ofP-property and
M-property for a couple of functions are introduced and some quantities for a pair of ho-mogeneous functions havingP-property andM-property are obtained, respectively. As an application, a bound for the solution of the homogeneous complementarity problem with aP-type function is derived.
Copyright © 2007 Ya-Ping Fang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
For any givenP-matrix [1](see also [2–4]),M∈Rn×n, Mathias and Pang [5] introduced
a quantityα(M) by
α(M)= min
x∞=11max≤i≤nxi(Mx)i. (1.1)
In terms of α(M), a bound for the solution of the linear complementarity problem LCP(M,q) (see [2–4]) with aP-matrixMis established in [5]. Recently, Xiu and Zhang [6] further gave some new properties of α(M) and introduced a new quantity β(M), which is defined by
β(M)= max
x∞=1
max
Moreover, Xiu and Zhang [6] introduced a fundamental quantityα{A,B}associated with a pair{A,B}havingv-columnP-property (see [2–4,7]) by
α{A,B} = min
x∞=11max≤i≤n(Ax)i(Bx)i, (1.3)
whereA,B∈Rm×n. They developed some characteristic quantities ofα{A,B}. By means
of these quantities, Xiu and Zhang [6] established global error bounds for the vertical and horizontal linear complementarity problems.
Motivated by these works, in this paper, we introduce the concepts ofP-type andM -type functions and give several quantities for homogeneousP-type andM-type func-tions. Furthermore, we give the concepts ofP-property andM-property for a couple of functions, and obtain some quantities for homogeneous continuous pair withP-property andM-property, respectively. As an application, a bound of the solution to the homoge-neous complementarity problem with aP-type function is obtained.
2. Characteristic quantities forP-Type andM-Type functions
LetT:Rn→Rnbe a function. We say thatTis positively homogeneous with degreeθ >0 ifT(λx)=λθT(x) for allx∈Rnandλ >0. DefineᏴby
Ᏼ=T|T:Rn−→Rnis continuous and positively homogeneous. (2.1)
GivenT∈Ᏼ, define
T =max
x=1
T(x)=sup x=0
T(x)
xθ , (2.2)
whereθ >0 is the positively homogeneous degree ofTand · is a norm onRn.
Theorem2.1. LetT,S:Rn→Rnbe two positively homogeneous functions with degreesθ
andρ, respectively. Then the following conclusions hold:
(i)T(x) ≤ T · xθ;
(ii)if the inverseT−1inᏴexists, thenT−1is positively homogeneous with degree1/θ;
(iii)T·S ≤ T · Sθ.
Proof. (i) This follows directly from (2.2).
(ii) SinceT−1∈Ᏼ, we suppose the degree ofT−1isθ. It follows that
T−1·T(λx)=λx=T−1λθT(x)=λθθ
T−1·T(x)=λθθ
x. (2.3)
(iii) It is easy to see thatT·Sis positively homogeneous with degreeθρ. By (2.2),
T·S =sup
x=0 T
S(x) xθρ ≤sup
x=0
T ·S(x)θ xθρ
≤sup
x=0
T · Sθ· xθρ
xθρ = T · Sθ.
(2.4)
This completes the proof.
LetT:Rn→Rnbe a function. Recall thatTis aP-function (see [3,4]) if
max
1≤i≤n
xi−yiT(x)−T(y)i>0 (2.5)
for allx=y.
We now introduce the concepts ofM-type andP-type functions as follows.
Definition 2.2. LetT:Rn→Rnbe a function.Tis said to be (i)M-type if
min
1≤i≤nxi
T(x)i>0, ∀x=0; (2.6)
(ii)P-type if
max
1≤i≤nxi
T(x)i>0, ∀x=0. (2.7)
Note that aP-functionTwithT(0)=0 isP-type and a functionT:Rn→RnisM-type
ifT(0)=0 andTi:Rn→Ris strictly monotone for eachi, whereTi(x)=[T(x)]i.
For any givenP-type and positively homogeneous functionT with degreeθ >0, we defineα(T) andβ(T) by
α(T)= min
x∞=11max≤i≤nxi
T(x)i=inf
x=0
max1≤i≤nxiT(x)i
xθ+1
∞ , (2.8)
β(T)= max
x∞=1
max
1≤i≤nxi(T(x))i=supx=0
max1≤i≤nxi(T(x))i
xθ+1
∞ , (2.9)
wherex∞=max1≤i≤n{|xi|}. In addition, ifT isM-type, we can further defineα(T) andβ(T) by
α(T)= max x∞=1
min
1≤i≤nxi(T(x))i=supx=0
min1≤i≤nxiT(x)i
xθ+1
∞ , (2.10)
β(T)= min x∞=11min≤i≤nxi
T(x)i=inf
x=0
min1≤i≤nxi(T(x))i
xθ+1
∞ . (2.11)
Remarks 2.3. The definitions ofα(T),β(T) associated with aP-type positively homoge-neous functionTgeneralize the definitions ofα(M),β(M) associated with aP-matrix in [5,6], respectively.
By (2.8)–(2.11), we can obtain the following proposition.
Proposition2.4. Let T:Rn→Rnbe a positively homogeneous function with degreeθ. Then the following conclusions hold:
(i)ifTisP-type, then
α(T)xθ+1
∞ ≤1max≤i≤nxi
T(x)i≤β(T)xθ+1
∞ ; (2.12)
(ii)ifTisM-type, then
β(T)xθ+1
∞ ≤1min≤i≤nxi
T(x)i≤α(T)xθ+1
∞ , (2.13)
β(T)≤α(T)≤α(T)≤β(T). (2.14)
Theorem2.5. LetT:Rn→Rnbe aP-type and positively homogeneous function with
de-greeθand have inverseT−1inᏴ. Then the following conclusions hold:
(a)β(T)≤ T∞;
(b)α(T)≤1/T−1θ
∞;
(c) 1/T−1θ+1
∞ ≤β(T)/β(T−1),α(T)/α(T−1)≤ T1+1∞ /θ.
Proof. For any nonzerox∈Rn, we know
xiT(x)i≤ x∞· T(x)∞≤ T∞· x∞θ+1, i=1, 2,...,n. (2.15)
By (2.9), we obtainβ(T)≤ T∞. Hence (a) is true.
From (2.2) andTheorem 2.1,
T−1
∞=sup
x=0
T−1(x)
∞ x1/θ
∞ =supy=0
y∞
T(y)1/θ
∞ =sup
y=0
y1+θ
∞
1/θ T(y)
∞· y∞1/θ
. (2.16)
SinceT(y)∞· y∞≥max1≤i≤nyi(T(y))i, we have
T−1
∞≤sup
y=0
y1+θ
∞
max1≤i≤nyiT(y)i 1/θ
=
sup
y=0
y1+θ
∞
max1≤i≤nyiT(y)i 1/θ
=
1
α(T)
1/θ (2.17)
and so
α(T)≤ 1
T−1θ
∞
. (2.18)
From (2.8), (2.9), andTheorem 2.1, we know
βT−1=sup x=0
max1≤i≤nxiT−1(x)i
x1+1∞ /θ =sup y=0
max1≤i≤nyiT(y)i T(y)1+1/θ
∞
≥sup
y=0
max1≤i≤nyiT(y)i
T1+1/θ
∞ · y1+∞θ
= β(T) T1+1/θ
∞ ,
αT−1=inf x=0
max1≤i≤nxiT−1(x)i
x1+1/θ
∞ =infy=0
max1≤i≤nyiT(y)i T(y)1+1/θ
∞
≥inf
y=0
max1≤i≤nyiT(y)i T1+1/θ
∞ · y1+∞θ
= α(T) T1+1/θ
∞ ,
(2.19)
which yields the second inequality in (c). By the same arguments, we can prove
β(T)≥β(T−1) T−11+θ
∞
, α(T)≥ α
T−1 T−11+θ
∞
, (2.20)
which yields the first inequality in (c). This completes the proof.
Similarly, we can obtain the following results.
Theorem2.6. LetT:Rn→Rnbe anM-type and positively homogeneous function with
degreeθand have inverseT−1inᏴ. Then
(i)β(T)≤1/T−1θ
∞;
(ii) 1/T−1θ+1
∞ ≤β(T)/β(T−1),α(T)/α(T−1)≤ T1+1∞ /θ.
Theorem2.7. LetT,S:Rn→Rnbe two positively homogeneous functions with the same degreeθ. Then the following conclusions hold:
(1)if bothTandSareP-type, thenβ(T+S)≤β(T) +β(S);
(2)ifTisP-type andSisM-type, thenα(T+S)≥α(T),β(T+S)≥β(T);
(3)if bothTandSareM-type, then
β(T+S)≥β(T) +β(S), α(T+S)≥max{α(T),α(S)},
β(T+S)≥max{β(T),β(S)}. (2.21)
Proof. The facts directly follow from the definitions ofα,β,α,β, and simple arguments.
Remarks 2.8. Theorems2.5–2.7generalize partly Theorems2.1and2.5of Xiu and Zhang [6].
3. Extensions
Definition 3.1. LetT,S:Rn→Rnbe two functions. Say that{T,S}has
(1)P-property if for any nonzerox∈Rn,
max
1≤i≤n(T(x))i(S(x))i>0; (3.1)
(ii)M-property if for any nonzerox∈Rn,
min
1≤i≤n(T(x))i(S(x))i>0. (3.2)
LetT,S∈Ᏼwith positively homogeneous degreesθ andρ, respectively, and{T,S} haveP-property. Defineα{T,S}andβ{T,S}as follows:
α{T,S} = min
x∞=11max≤i≤n
T(x)i(S(x))i=inf x=0
max1≤i≤n(T(x))i(S(x))i
xθ∞+ρ
, (3.3)
β{T,S} = max
x∞=11max≤i≤n
T(x)iS(x)i=sup
x=0
max1≤i≤n(T(x))i(S(x))i
xθ∞+ρ .
(3.4)
Remarks 3.2. The definitions ofα{T,S},β{T,S}associated with a positively homoge-neous function pair {T,S} having P-property generalize the definitions of α{M,N},
β{M,N}associated with a matrix pair havingv-columnP-property in [2,6,7]. In addition, if{T,S}hasM-property, we can defineα{T,S}andβ{T,S}by
α{T,S} = max x∞=11min≤i≤n
T(x)iS(x)i=sup
x=0
min1≤i≤n(T(x))i(S(x))i
xθ∞+ρ
, (3.5)
β{T,S} = min x∞=11min≤i≤n
T(x)iS(x)i=inf
x=0
min1≤i≤n(T(x))i(S(x))i
xθ∞+ρ .
(3.6)
By the definitions ofα{T,S},β{T,S},α{T,S}, andβ{T,S}, we can obtain the follow-ing proposition.
Proposition3.3. LetT,S:Rn→Rnbe two positively homogeneous functions with degrees θandρ, respectively. Then the following conclusions hold:
(i)if{T,S}hasP-property, then
α{T,S}xθ∞+ρ≤max
1≤i≤n
T(x)iS(x)i≤β{T,S}xθ∞+ρ; (3.7)
(ii)if{T,S}hasM-property, then
β{T,S}xθ+ρ
∞ ≤min
1≤i≤n
T(x)iS(x)i≤α{T,S}xθ+ρ
∞ , (3.8)
β{T,S} ≤α{T,S} ≤α{T,S} ≤β{T,S}. (3.9)
Note that, ifT−1exists, then the condition that{T,S}hasP-property (M-property) is equivalent to the condition thatST−1isP-type (M-type).
Theorem3.4. LetT,S:Rn→Rnbe two positively homogeneous functions with degreesθ
the following conclusions hold:
(a)β{T,S} ≤ T∞· S∞;
(b)α{T,S} ≤ S∞/T−1θ
∞;
(c) 1/T−1θ+ρ
∞ ≤β{T,S}/β(ST−1),α{T,S}/α(ST−1)≤ T1+(∞ ρ/θ).
Proof. (a) For any nonzerox∈Rn, it follows from (i) ofTheorem 2.1that
T(x)iS(x)i≤ T(x)∞· S(x)∞≤ T∞· S∞· x∞θ+ρ, i=1, 2,...,n. (3.10)
By (3.4),
β{T,S} ≤ T∞· S∞. (3.11)
(b) From (2.2) and (i) ofTheorem 2.1,
T−1
∞=sup
x=0
T−1(x)
∞ x1/θ
∞ =supy=0
y∞
T(y)1/θ
∞ =sup
y=0
yθ
∞·S(y)∞
1/θ T(y)
∞·S(y)∞
1/θ. (3.12)
SinceT(y)∞· y∞≥max1≤i≤nyi(T(y))iandS(y)∞≤ S∞· yρ∞, we have
T−1
∞≤sup
y=0
S
∞· yρ∞+θ
max1≤i≤nT(y)iS(y)i 1/θ
= S1/θ
∞ ·
sup
y=0
yρ∞+θ
max1≤i≤nT(y)i
S(y)i
1/θ
=
S
∞ α{T,S}
1/θ
.
(3.13)
This implies that
α{T,S} ≤S∞ T−1θ
∞
. (3.14)
(c) It follows from (2.9) that
βST−1=sup x=0
max1≤i≤nxiST−1(x)i
x1+∞ρ/θ
=sup
y=0
max1≤i≤nT(y)i
S(y)i
T(y)1+ρ/θ
∞
≥sup
y=0
max1≤i≤nT(y)iS(y)i
T1+∞ρ/θ· yρ∞+θ
= β{T,S} T1+∞ρ/θ .
(3.15)
By (3.4) andTheorem 2.1,
T−1(y)
∞≤T−1∞· y1∞/θ,
β{T,S} =sup
x=0
max1≤i≤nT(x)iS(x)i
xρ∞+θ
=sup
y=0
max1≤i≤nyiST−1(y)i T−1(y)ρ+θ
∞
It follows that
β{T,S} ≥sup
y=0
max1≤i≤nyiST−1(y)i T−1θ+ρ
∞ · y
1+ρ/θ
∞
= β
ST−1 T−1θ+ρ
∞
. (3.17)
Hence
1
T−1θ+ρ
∞
≤ β{T,S} βST−1≤ T
1+ρ/θ
∞ . (3.18)
By similar arguments, we can prove that
1
T−1θ+ρ
∞
≤ α{T,S} αST−1≤ T
1+ρ/θ
∞ . (3.19)
This completes the proof.
Remarks 3.5. Theorem 3.4generalizes and improvesTheorem 2.7of Xiu and Zhang [6].
Similarly, we can obtain the following result.
Theorem3.6. LetT,S:Rn→Rnbe two positively homogeneous functions with degreesθ
andρ, respectively. Suppose that{T,S}hasM-property andThas inverseT−1inᏴ. Then the following conclusions hold:
(1)β{T,S} ≤ S∞/T−1θ
∞;
(2) 1/T−1θ+ρ
∞ ≤α{T,S}/α(ST−1),β{T,S}/β(ST−1)≤ T1+(∞ ρ/θ).
4. An application
In this section, we give a bound for the solution of the homogeneous complementarity problem, denoted by HCP(T,q), which consists of findingx∈Rnsuch that
x≥0, T(x) +q≥0, xTT(x) +q=0, (4.1)
whereT:Rn→Rnis aP-type and positively homogenous function andq∈Rn.
Theorem4.1. LetT:Rn→Rnbe aP-type and positively homogeneous function with
de-greeθ. Suppose thatT has inverseT−1 inᏴandx is the unique solution ofHCP(T,q). Then
αT−1θ(−q)
+∞≤ xθ∞≤
(−q)+
∞
α(T) , (4.2)
where(−q)+denotes the nonnegative part of−q.
Proof. Ifx=0, then (−q)+=0. The conclusion holds trivially. In the sequel we always
suppose that x=0, equivalently, q is not nonnegative. Since x solves HCP(T,q), by
Proposition 2.4, one has
α(T)xθ+1
∞ ≤1max≤i≤nxiT(x)i=1max≤i≤nxi(−q)i
≤max
1≤i≤nxi
(−q)+
i≤ x∞·(−q)+∞.
This implies that
xθ∞≤(−q)+∞
α(T) . (4.4)
It follows from (2.12) that
αT−1y1+1/θ
∞ ≤1max≤i≤nyiT−1(y)i. (4.5)
Thus we have
αT−1T(x)1+1/θ
∞ ≤1max≤i≤nxi
T(x)i. (4.6)
SinceT(x)≥ −q, we know that|T(x)| ≥(T(x))+≥(−q)+and so T(x)
∞≥(−q)+∞. (4.7)
By (4.6), (4.7), and the fact thatxi(T(x) +q)i=0, we know
αT−1(−q)
+1+1∞ /θ≤α
T−1T(x)1+1/θ
∞ ≤1max≤i≤nxi
T(x)i
=max
1≤i≤nxi(−q)i≤ x∞·(−q)+∞.
(4.8)
Hence
αT−1θ(−q)
+∞≤ xθ∞. (4.9)
This completes the proof.
Acknowledgments
The authors thank the referees for their helpful comments and suggestions leading to the improvements of this paper. This work was supported by the National Natural Sci-ence Foundation of China (10671135), the Specialized Research Fund for the Doctoral Program of Higher Education (20060610005) and the Korea Research Foundation Grant (KRF-2004-041-C00033).
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Ya-Ping Fang: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Email address:[email protected]
Nan-Jing Huang: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Email address:[email protected]
Yeol Je Cho: Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Korea
