# Vector Variational-Like Inequalities with Generalized Semimonotone Mappings

Authors: Suhel Ahmad Khan
Journal: International Journal of Analysis
 (2013)

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## Abstract

We introduce the concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity. We consider a class of generalized vector variationa-llike inequality problem involving generalized relaxed semimonotone mapping. By using Kakutani-Fan-Glicksberg’s fixed-point theorem, we prove the solvability for this class of vector variational-like inequality with relaxed monotonicity assumptions. The results presented in this paper generalize some known results for vector variational inequality in recent years.


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## Body

## 1. Introduction

Vector variational inequalities were initially introduced and considered by Giannessi [1] in a finite-dimensional Euclidean space in 1980. Due to its wide application the theory of the vector variational inequality is generalized in different directions and many existence results and algorithms for vector variational inequality problems have been established under various conditions; see for examples [2–7] and references therein.The concept monotonicity and the compactness operators are very useful in nonlinear functional analysis and its applications. In 1968, Browder [8] first combined the compactness and accretion of operators and posed the concept of a semiaccretive operator. Motivated by this idea, Chen [9] studied the concept of a semimonotone operator, which combines the compactness and monotonicity of an operator and posed it to the study of variational inequalities. Recently in 2003, Fang and Huang [10] introduced relaxed η-α-semimonotone mapping, a generalized concept of semimonotonicity, and they established several existence results for the variational-like inequality problem.In this paper, we pose two new concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity as well as two classes of generalized vector variational-like inequalities with generalized relaxed monotone mappings and generalized relaxed semimonotone mappings. We investigate the solvability of vector variational-like inequalities with generalized relaxed semimonotone mappings by means of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper generalize the results of Chen [9], Fang and Huang [10], Usman and Khan [11], and Zheng [7].


## 2. Preliminaries

Throughout the paper unless otherwise specified, letX and Y be two real Banach spaces, K⊂X be a nonempty closed and convex subset of X.  P is said to be a closed convex and pointed cone with its apex at the origin, if the following conditions hold: (i)
λP⊂P, for all λ>0,(ii)
P+P⊂P,(iii)
P∩(-P)={0}.The partial order≤P in Y, induced by the pointed cone P, is defined by declaring x≤Py if and only if y-x∈P for all x,y in Y. An ordered Banach space is a pair (Y,P) with the partial order induced by P. The weak order ≤intP in an ordered Banach space (Y,P) with intP≠∅ is defined as x≤intPy if and only if y-x∈intP for all x,y in Y, where intP denotes the interior of P. Let L(X,Y) denote the space of all continuous linear mappings from X into Y.  A set-valued mapping P:K→2Y be such that for each x∈K,P(x) is a proper, closed, convex cone with intP(x)≠∅ and let P-=⋂x∈KP(x).First, we give the concept of generalized relaxed monotone mapping. In order to do so, the definition of a vector monotone mapping is needed, which was posed by Chen [9].Definition 1 (see [9]).
LetT:K→L(X,Y) be a mapping, K⊂X be a nonempty, closed and convex subset in X. Let P:K→2Y be such that for each x∈K,P(x) is a proper, closed, convex cone with intP(x)≠∅.T is said to be P--monotone on K if and only if it satisfies the following condition:
(1)〈T(y)-T(x),y-x〉≥P-0,∀x,y∈K,
where P-=⋂x∈KP(x).We now give the concept of generalized relaxed monotone mapping.Definition 2.
LetP:K→2Y be such that for each x∈K,P(x) is a proper, closed, convex cone with intP(x)≠∅ and a mapping η:K×K→X. A mapping T:K→L(X,Y) is said to be generalized relaxed monotone, if
(2)〈Ty-Tx,η(y,x)〉-α(x,y)≥P-0,∀x,y∈K,
where α:X×X→Y is a mapping such that limt→0+(α(x,ty+(1-t)x)/t)=0.Remark 3.
(i) Ifα(x,y)=α(x-y), where α:X→ℝ with α(λz)=λpα(z) for λ>0,p>1, then inclusion (2)
(3)〈Ty-Tx,η(y,x)〉-α(x,y)≥P-0,∀x,y∈K
reduces to
(4)〈Ty-Tx,η(y,x)〉-α(x-y)∈P-,∀x,y∈K.
Also if P(x)=ℝ+, for all x∈K, then P-=⋂x∈KP(x)=ℝ+. As a result the above inclusion reduces to
(5)〈Ty-Tx,η(y,x)〉-α(x-y)≥0,∀x,y∈K,⇒〈Ty-Tx,η(y,x)〉≥α(x-y),∀x,y∈K,
then T is said to be relaxed η-α-monotone, introduced and studied by Fang and Huang [10].
(ii) In the above inclusion (5), if we take η(y,x)=y-x, for all x,y∈K, then it reduces to
(6)〈Ty-Tx,y-x〉≥α(x-y),∀x,y∈K,
then T is said to be α-relaxed monotone.
(iii) In the above inclusion (6), if we take α(z)=k∥z∥p, where k>0 is constant, then it reduces to
(7)〈Ty-Tx,y-x〉≥k∥y-x∥p,∀x,y∈K.
ThenT is called p-monotone, see for example [12–14].We recall the following concepts and results which are needed in the sequel.Definition 4.
A mappingf:K→Y is said to be (i)
P--convex, if f(tx+(1-t)y)≤P-tf(x)+(1-t)f(y),  for all x,y∈K,t∈[0,1];(ii)
P--concave, if -f is P--convex.Definition 5.
A mappingη:K×K→X is said to be affine in the first argument if for any xi∈K and λi≥0,(1≤i≤n), with ∑i=1nλi=1 and any y∈K,
(8)η(∑i=1n‍λixi,y)=∑i=1n‍λiη(xi,y).Lemma 6 (see [3]).
Let(Y,P) be an ordered Banach space with a proper, closed, convex and solid cone P(x) then for all x,y,z∈Y, one has (i)
z≰intP(x)x and x≥P(x)y⇒z≰intP(x)y,(ii)
z≱intP(x)x and x≤P(x)y⇒z≱intP(x)y.Lemma 7 (see [15]).
LetK be a subset of a topological vector space X and let F:K→2X be a KKM mapping. If for each x∈K, F(x) is closed and for at least one x∈K, F(x) is compact, then
(9)⋂x∈K‍F(x)≠∅.Definition 8.
A mappingT:K→L(X,Y) is said to be η-hemicontinuous, if for any x,y∈K, the mapping  t→〈Tx+t(y-x),η(y,x)〉 is continuous at 0+.Now, we have the following Minty’s type Lemma.Lemma 9.
LetK⊂X be a nonempty, closed, and convex subset of X. Let P:K→2Y be such that for each x∈K,P(x) is a proper, closed, convex cone with intP(x)≠∅. Let f:K×K→Y be  P--convex in the first argument with the condition f(x,x)=0, for all x∈K. Suppose the following conditions hold: (i)
η:K×K→X is an affine in the first argument with the condition η(x,y)+η(y,x)=0, for all x,y∈K;(ii)
the set-valued mappingW:K→2Y defined as W(x)=Y∖{-intP(x)} for all x∈K is weakly upper semicontinuous;(iii)
T:K→L(X,Y) is η-hemicontinuous and generalized relaxed monotone mapping.
Then the following two problems are equivalent:(A)
x∈K,〈Tx,η(y,x)〉+f(y,x)≰intP(x)0, for  all y∈K,(B)
x∈K,〈Ty,η(y,x)〉+f(y,x)-α(x,y)≰intP(x)0, for  all y∈K.Proof.
Following the lines of proof given by Chen [9], one can easily prove.Theorem 10.
LetX be real reflexive Banach space, and Y be a Banach space. Let K⊂X be a nonempty, bounded, closed, and convex subset of X. Let P:K→2Y be such that for each x∈K,P(x) is a proper, closed, convex cone with intP(x)≠∅. Let f:K×K→Y be P--convex and upper semicontinuous in the first and second arguments, respectively, with the condition f(x,x)=0 for all x∈K. Let conditions (i)–(iii) of Lemma 9 be satisfied and suppose the following conditions hold: (i)
η:K×K→X is lower semicontinuous in the second argument;(ii)
α:X×X→Y is weakly lower semicontinuous and P--convex in the first and second arguments, respectively. Then there exist x∈K, such that
(10)〈Tx,η(y,x)〉+f(y,x)≰intP(x)0,∀y∈K.Proof.
Define two set-valued mappingsF,G:K→2K as follows:
(11)F(y)={{≰intP(x)}x∈K:〈Tx,η(y,x)〉+f(y,x)≰intP(x)0},∀y∈K,G(y)={{≰intP(x)}x∈K:〈Ty,η(y,x)〉+f(y,x)-α(x,y)≰intP(x)0},∀y∈K.
Then F(y) and G(y) are nonempty since y∈G(y)∩F(y). We claim that F is a KKM mapping. If this is not true, then there exist a finite set {y1,…,yn}⊂K and ti≥0,i=1,…,n with ∑i=1nti=1 such that y=∑i=1ntiyi∉⋃i=1nF(yi). Now by the definition of F, we have
(12)〈Ty,η(yi,y)〉+f(yi,y)≤intP(y)0,i=1,…,n.
Now we have
(13)0=〈Ty,η(y,y)〉+f(y,y)=〈Ty,η(∑i=1n‍tiyi,y)〉+f(∑i=1n‍tiyi,y)=∑i=1n‍ti[〈Ty,η(yi,y)〉+f(yi,y)]≤intP(y)0,
which leads to a contradiction since P is a proper cone. Thus our claim is verified. So F is a KKM mapping.
Now we proveF(y)⊂G(y) for every y∈K. Indeed let x∈F(y). Then, we have
(14)〈Tx,η(y,x)〉+f(y,x)≰intP(x)0.
Since T is generalized relaxed monotone, we have
(15)〈Ty,η(y,x)〉+f(y,x)≥P-〈Tx,η(y,x)〉+f(y,x)+α(x,y).
From Lemma 6, we have
(16)〈Ty,η(y,x)〉+f(y,x)-α(x,y)≰intP(x)0,
that is, x∈G(y), which follows F(y)⊂G(y), for each y∈K and so G is also a KKM mapping. Now we claim that for each y∈K,G(y)⊂K is closed in the weak topology of X.
Indeed supposex-∈G(y)¯w, the weak closure of G(y). Since X is reflexive, there is a sequence {xn} in G(y) such that {xn} converges weakly to x-∈K. Then
(17)〈Ty,η(y,xn)〉+f(y,xn)-α(xn,y)≰intP(xn)0,
which implies that
(18)〈Ty,η(y,xn)〉+f(y,xn)-α(xn,y)∈Y∖{-intP(xn)}.
Since η(y,·) and f(y,·) are lower and upper semicontinuous and α(·,y) and Y∖{-intP(xn)} are weakly lower semicontinuous, therefore
(19)〈Ty,η(y,x-)〉+f(y,x-)-α(x-,y)∈Y∖{-intP(x-)}.
Thus we get
(20)〈Ty,η(y,x-)〉+f(y,x-)-α(x-,y)≰intP(x-)0
and so x-∈G(y). This shows that G(y) is weakly closed for each y∈K. Our claim is then verified. Since X is reflexive and K⊂X is nonempty, bounded, closed, and convex, K is a weakly compact subset of X and so G(y) is also weakly compact. According to Lemma 7,
(21)⋂y∈K‍G(y)≠∅.
This implies that there exists x∈K such that
(22)〈Ty,η(y,x)〉+f(y,x)-α(x,y)≰intP(x)0,∀y∈K.
Therefore by Minty’s type Lemma 9, we conclude that there exist x∈K such that
(23)〈Tx,η(y,x)〉+f(y,x)≰intP(x)0,∀y∈K.
This completes the proof.


## 3. Main Results

Throughout this section, letX be real reflexive Banach space, Y a Banach space. Let K⊂X be a nonempty, bounded, closed, and convex subset of X. Let P:K→2Y be such that for each x∈K,P(x) is a proper, closed, convex cone with intP(x)≠∅ and let P-=⋂x∈KP(x).Some nonlinear mapping consisting with two variables, may be monotone with respect to the first variable and compact with respect to the second one. However, we cannot always expect for them to be monotone or compact with respect to the two variables simultaneously. Keeping this complexity in mind we are interested in the so called semimonotone mapping. We now give the concept of a generalized relaxed semimonotone mapping.Definition 11.
LetK⊂X be a nonempty, closed, and convex subset of X. Let η:K×K→X be a mapping and let α:X×X→Y be a bifunction such that limt→0+(α(x,ty+(1-t)x)/t)=0 and f:K×K→Y. A mapping A:K×K→L(X,Y) is said to be generalized relaxed semimonotone mapping, if the following conditions hold: (i)
for everyu∈K,A(u,·) is a generalized relaxed monotone mapping, that is,
(24)〈A(u,y)-A(u,x),η(y,x)〉-α(x,y)≥P-0,∀x,y∈K;(ii)
for everyy∈K,A(·,y) is completely continuous, that is, when un→wu,A(un,y)→A(u,y) (by the norm of operators), where →w denotes the weak convergence.Remark 12.
WhenY=ℝ,P(x)=ℝ+, and η(y,x)=y-x, for  all x,y∈K and α≡0, then this is exactly the concept which was introduced by Chen [9].Example 13.
LetX=Y=ℝ2,K=[0,1]×[0,1]. Let P:K→2Y be defined by
(25)P(x)={(y1,y2)∈Y∣y1≥0,y2≥0},x∈K.
So
(26)P-=⋂x∈K‍P(x)={(y1,y2)∈Y∣y1≥0,y2≥0}.
Let T:K→L(X,Y) be defined by
(27)T(x)=(x1x2x1+1x2+1),∀x=(x1,x2)∈K.
Also η:K×K→X, α:X×X→Y are define by η(y,x)=y-x, for  all x,y∈K and α(x,y)=(x-y)2, for  all x,y∈X, respectively.
Let us supposeA:K×K→L(X,Y) is defined by
(28)A(x,y)=(x1+y1x2+y2y1y2),∀x=(x1,x2),y=(y1,y2)∈K.
First, we show that A is a generalized relaxed monotone mapping. Indeed, for each u=(u1,u2),v=(v1,v2)∈K, it follows that
(29)〈A(u,y)-A(v,y),η(u-v)〉-α(v,u)=(u1-v1u2-v2u1-v1u2-v2)(u1-v1u2-v2)-((v1-u1)2(v2-u2)2)=((u2-v2)2(u1-v1)2)≥P-0.
The norm ofA is defined as ∥A∥=|x+y|+|y|, for  all x=(x1,x2), and y=(y1,y2)∈K.
Now, for fixedv∈K, if un∈K,u∈K, and un→wu, it is easy to prove that
(30)∥A(un,v)-A(u,v)∥→0.
Hence, for every v∈K,A(·,v) is completely continuous.
Therefore, the mapping defined as above is a generalized relaxed semimonotone mapping.Now we will pose the main problem of our study. In this paper, we investigate the following generalized vector variational-like inequality problem (for short, GVVLIP) is to find a vectoru∈K satisfying
(31)〈A(u,u),η(v,u)〉+f(v,u)≰intP(u)0,∀v∈K,
where A:K×K→L(X,Y) is a nonlinear mapping and η:K×K→X, f:K×K→Y are the two vector-valued bimappings.The GVVLIP (31) includes many variational inequality problems as special cases.Some special cases of GVVLIP (31) are as follows. (I)
Ifη(v,u)=v-u and f≡0, then the GVVLIP (31) reduces to the following variational inequality problem of finding u∈K such that
(32)〈A(u,u),v-u〉≰intP(u)0,∀v∈K,
which was introduced and studied by Zheng [7](II)
IfY=ℝ,P(u)=[0,+∞), for  all u∈K. Also if η(v,u)=v-u,f≡0 and L(X,Y) is the dual space X* of X, then the GVVLIP (31) reduces to the following variational inequality problem of finding u∈K such that
(33)〈A(u,u),v-u〉≥0,∀v∈K,
which was introduced and investigated by Chen [9]. He obtained some existence results and discussed their applications in partial differential equations of divergence form.We recall the following fixed-point theorem, by Zeidle [16], which will play an important role in establishing our existing results for GVVLIP (31).Theorem 14 (see [16]).
The set-valued mappingF:M→2M has a fixed point, if the following conditions are satisfied: (1)
M is compact, convex, and nonempty set in locally convex space;(2)
F(x) is convex, closed, and nonempty for every x∈M;(3)
F is upper semicontinuous on M.Now, we have the following existence results for GVVLIP (31) involving a generalized relaxed semimonotone mapping in reflexive Banach spaces.Theorem 15.
LetX be real reflexive Banach space, and let Y be a Banach space. Let K⊂X be a nonempty, bounded, closed, and convex subset of X. Let P:K→2Y be such that for each x∈K,P(x) is a proper, closed, convex cone with intP(x)≠∅. Let A:K×K→L(X,Y) be a generalized relaxed semimonotone mapping. Suppose the following conditions hold: (i)
f:K×K→Y is P--convex with f(x,x)=0, for  all x∈K and upper semicontinuous in the second argument;(ii)
η:K×K→X is an affine mapping with η(x,y)+η(y,x)=0, for  all x,y∈K and lower semicontinuous in the second argument;(iii)
The set-valued mappingW:K→2Y defined as W(x)=Y∖{-intP(x)}, for  all x∈K, is weakly upper semicontinuous and concave;(iv)
α:X×X→Y is P--concave and weakly lower semicontinuous in the second and first argument, respectively;(v)
For each fixedv∈K, A(v,·):K×K→L(X,Y) is continuous on each finite dimensional subspace of X. Then there exists a u0∈K such that
(34)〈A(u0,u0),η(v,u0)〉+f(v,u0)≰intP(u0)0,∀v∈K.Proof.
LetF be a finite dimensional subspace of X and KF=K⋂‍F≠∅. For each v∈KF, we consider the following generalized vector variational-like inequality problem. Find u0∈KF such that
(35)〈A(v,u0),η(u,u0)〉+f(u,u0)≰intP(u0)0,∀u∈KF.
Since KF⊂F is bounded, closed, and convex, A(v,·) is continuous on KF and generalized relaxed monotone for each fixed v∈K. From Theorem 10, we know our problem has solution u0∈KF.
Define a set-valued mappingT:KF→2KF as follows:
(36)F(v)={{≰intP(w)}w∈KF:〈A(v,w),η(u,w)〉+f(u,w)≰intP(w)0,∀u∈KF}.
It follows from Minty’s type Lemma 9, that for each fixed v∈KF(37){{≰intP(w)}w∈KF:〈A(v,w),η(u,w)〉+f(u,w)≰intP(w)0,∀u∈KF}={{≰intP(w)}w∈KF:〈A(v,u),η(u,w)〉+f(u,w)-α(w,u)≰intP(w)0,∀u∈KF}.
Now we will use the fixed-point theorem to verify the existence of the solution of the problem in a finite dimensional. Since F is of finite dimensional, hence KF is compact. First, we claim that
(38)F(v)={{≰intP(w)}w∈KF:〈A(v,u),η(u,w)〉+f(u,w)-α(w,u)∈Y∖{-intP(w)}=W(w){≰intP(w)}}
is convex. Indeed, let w1,w2∈F(v) and m,n≥0, such that m+n=1(39)m[〈A(v,u),η(u,w1)〉+f(u,w1)-α(w1,u)]=mW(w1),n[〈A(v,u),η(u,w2)〉+f(u,w2)-α(w2,u)]=mW(w2).
Since η(u,·) is affine and f(u,·),α(·,u) are P--convex, then from preceding two inclusions we have
(40)〈A(v,u),η(u,mw1+nw2)〉+f(u,mw1+nw2)-α((mw1+nw2),u)∈mW(w1)+nW(w2).
Since W is concave, we have mw1+nw2∈F(v), that is, F(v) is convex and our claim is then verified. Now, we claim that F(v) is closed. Let wj∈F(v) such that wj→w, then
(41)〈A(v,u),η(u,wj)〉+f(u,wj)-α(wj,u)∈Y∖{-intP(wj)},〈A(v,u),η(u,wj)〉+f(u,wj)-α(wj,u)∈W(wj).
Since f(y,·)+α(·,y) is upper semicontinuous, also A(v,u)∈L(X,Y) and W is upper semicontinuous; therefore
(42)〈A(v,u),η(u,wj)〉+f(u,wj)-α(wj,u)→〈A(v,u),η(u,w)〉+f(u,w)-α(w,u)∈W(w).
This implies w∈F(v); hence F(v) is closed. Next, we claim that F is upper semicontinuous. Let vj→v,wj→F(vj), and wj→w, then we have
(43)〈A(vj,u),η(u,wj)〉+f(u,wj)-α(wj,u)∈W(wj).
From the complete continuity of A(·,u) and the lower semicontinuity of η(u,·), we have
(44)〈A(vj,u),η(u,wj)〉→〈A(v,u),η(u,w)〉
which implies that
(45)〈A(v,u),η(u,w)〉+f(u,w)-α(w,u)∈W(w),
that is, w∈F(v); thus our claim is then verified. Hence F is upper semicontinuous. By Fan-Glicksberg fixed-point theorem, there exists a v0∈F(v0) that is, there exists a v0∈KF such that
(46)〈A(v0,v0),η(u,v0)〉+f(u,v0)≰intP(v0)0,∀u∈KF.
Now we generalize this result to whole space.
LetΩ≡{F⊂X:Fbefinitedimensional,F⋂‍K≠∅} and let ΓF≡{w∈K:〈A(w,u),η(u,w)〉+f(u,w)-α(w,u)≰intP(w)0,forallu∈KF},F∈Ω.
From the above we know thatforallF∈Ω,ΓF≠∅. Let ΓFw¯ denote the weak closure of ΓF. For any Fi∈Ω,i=1,…,n, we know that Γ⋃i=1nFi⊆⋂i=1nΓFi⊆⋂i=1nΓFi¯w. Therefore, ⋂i=1nΓFi¯w≠∅. Since K is weakly compact, from the finite intersection property, we have ⋂x∈ΩΓF¯w≠∅. Let w0∈⋂x∈ΩΓF¯w, we see that
(47)〈A(w0,w0),η(u,w0)〉+f(u,w0)≰intP(w0)0.
Indeed, for each u∈K, let F∈Ω, such that u∈KF, w0∈KF. From w0∈ΓF¯w, there exists wj∈ΓF, that is,
(48)〈A(wj,u),η(u,wj)〉+f(u,wj)-α(wj,u)∈Y∖{-intP(wj)}∈W(wj),
such that wj→ww0, from the complete continuity of A(·,v), upper semicontinuity of f(u,·), and lower semicontinuity of α(·,u), we have
(49)〈A(w0,u),η(u,w0)〉+f(u,w0)-α(w0,u)∈W(w0).
From Lemma 9, we have
(50)〈A(w0,w0),η(u,w0)〉+f(u,w0)≰intP(w0)0,∀u∈K.
This completes the proof.If the boundedness ofK is dropped off, then we have the following theorem under certain coercivity condition.Theorem 16.
LetX be real reflexive Banach space and Y be a Banach space. Let 0∈K⊂X be a nonempty, closed, and convex subset of X. Let P:K→2Y be such that for each x∈K,P(x) is a proper, closed, convex cone with intP(x)≠∅. Let A:K×K→L(X,Y) be a generalized relaxed semimonotone mapping. Let conditions (i)–(v) of Theorem 15 be satisfied and
(51)lim∥u∥→∞〈A(u,u),η(u,0)〉+f(0,u)≥intP(u)0
then there exist u∈K such that
(52)〈A(u,u),η(v,u)〉+f(v,u)≰intP(u)0,∀v∈K.Proof.
For eachr>0, let B[0,r] denote the closed ball in the Banach space X with center 0 and radius r. By Theorem 15, for each r∈ℕ, there exists ur∈B[0,r]∩K such that
(53)〈A(ur,ur),η(v,ur)〉+f(v,ur)≰intP(ur)0,∀v∈B[0,r]∩K.
Since 0∈K, we have
(54)〈A(ur,ur),η(0,ur)〉+f(0,ur)≰intP(ur)0.
From condition (vi), we know that {ur} is bounded. So, we may suppose that ur converges to u as r→∞. It follows from Lemma 9, that
(55)〈A(ur,v),η(v,ur)〉+f(v,ur)-α(ur,v)≰intP(ur)0,∀v∈K∩Br.
Since A is completely continuous, η, α are lower semicontinuous, and f,W are upper semicontinuous. Therefore, it follows that
(56)〈A(u,v),η(v,u)〉+f(v,u)-α(u,v)≰intP(u)0∀v∈K∩Br.
Again from Lemma 9, we get
(57)〈A(u,u),η(v,u)〉+f(v,u)≰intP(u)0,∀v∈K.
This completes the proof.Remark 17.
Theorems15 and 16 improve and generalize Theorems 3.1 and 3.2 of Fang and Huang [10] and Theorem 2.1 to Theorem 2.6 of Chen [9].



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*Source: 762380-2013-01-14.xml*