ABSTRACT
In this paper, we study a concept on the calm B-differentiability, a new kind of generalized differentiabilities for a given vector function introduced by Ye and Zhou in 2017, of the projector onto the
circular cone. Then, we discuss its applications in mathematical programming problems with circular cone complementarity constraints. Here, this problem can be considered to be a generalization
of mathematical programming problems with second-order cone complementarity constraints,
and thus it includes a large class of mathematical models in optimization theory. Consequently, the
obtained results for this problem are generalized, and then corresponding results for some special
mathematical problems can be implied from them directly. For more detailed information, we will
first prove the calmly B-differentiable property of the projector onto the circular cone. This result is
not easy to be shown by simply resorting to those of the projection operator onto the second-order
cone. By virtue of exploiting variational techniques, we next establish the exact formula for the regular (Fréchet) normal cone (this concept was proposed by Kruger and Mordukhovich in 1980) to
the circular cone complementarity set. Note that this set can be considered to be a generalization
of the second-order cone complementarity set. In finally, the exact formula for the regular (Fréchet)
normal cone to the circular cone complementarity set would be useful for us to study first-order
necessary optimality conditions for mathematical programming problems with circular cone complementarity constraints. Our obtained results in the paper are new, and they are generalized to
some existing ones in the literature.

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Science & Technology Development Journal, 23(4):727-736
Open Access Full Text Article Research Article
DongThap University
Correspondence
Vo Duc Thinh, Dong Thap University
Email: vdthinh@dthu.edu.vn
History
Received: 2020-07-30
Accepted: 2020-09-06
Published: 2020-10-10
DOI : 10.32508/stdj.v23i4.2426
Copyright
© VNU-HCM Press. This is an open-
access article distributed under the
terms of the Creative Commons
Attribution 4.0 International license.
On the calm b-differentiability of projector onto circular cone and
its applications
Vo Duc Thinh*
Use your smartphone to scan this
QR code and download this article
ABSTRACT
In this paper, we study a concept on the calm B-differentiability, a new kind of generalized differen-
tiabilities for a given vector function introduced by Ye and Zhou in 2017, of the projector onto the
circular cone. Then, we discuss its applications inmathematical programming problemswith circu-
lar cone complementarity constraints. Here, this problem can be considered to be a generalization
of mathematical programming problems with second-order cone complementarity constraints,
and thus it includes a large class ofmathematical models in optimization theory. Consequently, the
obtained results for this problem are generalized, and then corresponding results for some special
mathematical problems can be implied from them directly. For more detailed information, we will
first prove the calmly B-differentiable property of the projector onto the circular cone. This result is
not easy to be shownby simply resorting to those of the projection operator onto the second-order
cone. By virtue of exploiting variational techniques, we next establish the exact formula for the reg-
ular (Fréchet) normal cone (this concept was proposed by Kruger and Mordukhovich in 1980) to
the circular cone complementarity set. Note that this set can be considered to be a generalization
of the second-order cone complementarity set. In finally, the exact formula for the regular (Fréchet)
normal cone to the circular cone complementarity set would be useful for us to study first-order
necessary optimality conditions for mathematical programming problems with circular cone com-
plementarity constraints. Our obtained results in the paper are new, and they are generalized to
some existing ones in the literature.
Key words: calmly B-differentiable, circular cone, complementarity set, Fréchet normal cone,
optimality condition
INTRODUCTION
The second-order cone programming (SOCP) prob-
lem plays an important role in the optimization the-
ory and has attracted much attention from mathe-
maticians, see, e.g., 1–7. We refer the reader to1,2,4–7
and the references therein for some remarkable re-
sults on optimality conditions and stability analysis of
(SOCP).
Inspired by the second-order cone, many researchers
have investigated optimization and complementar-
ity problems where their constraints are involved in
second-order cones. It is called the second-order
cone complementarity problem (SOCCP), which in-
cludes a large class of optimization problems such
as quadratically constrained problems (see 8), the
second-order cone programming, and nonlinear
complementarity problem (see 9). In particular, re-
cent attention is paid to the second-order cone com-
plementarity set.
Let us now mention some existing results concern-
ing this set. In10, Liang et al. provided formula-
tions for Fréchet normal cone to the second-order
cone complementarity set. Unfortunately, the ob-
tained results were shown to be inexact in 11. In
that paper, Ye and Zhou gave exact formulas for the
proximal/regular (Fréchet)/limiting normal cone to
the second-order cone complementarity set by us-
ing the projection operator onto second-order cones
and the generalized differentiability called the calm
B-differentiability. Some first-order optimality con-
ditions formathematical programswith second-order
cone complementarity constraints were obtained in 12
and sufficient conditions for error bound property of
second-order cone complementarity problems were
established in13. To obtain these results, the au-
thors used the symmetric and self-dual property of the
second-order cone.
Recently, generalizations of second-order cones and
second-order cone complementarity sets have been
examined by many authors5,14–22. For example, au-
thors in 14,19–22 considered circular cones, which are
generalizations of second-order cones and are, in gen-
eral, nonsymmetric and non-self-dual cones. The
generalized differentiability of the projection operator
Cite this article : Thinh V D. On the calm b-differentiability of projector onto circular cone and its
applications. Sci. Tech. Dev. J.; 23(4):727-736.
727
Science & Technology Development Journal, 23(4):727-736
onto the circular cone was provided in 14,22. More-
over, the differentiability and calmness of vector-
valued functions associated with the circular cone
were also studied in19,23. In particular, authors in21
showed that the results of the projection operator onto
a circular cone could not be shown by simply resort-
ing to the results of the projection operator onto the
second-order cone, and hence, it is necessary to study
the results of circular cone directly.
To the best of our knowledge, there is no result on
the calmly B-differentiable property concerning the
circular cone and its extension. In this paper, in-
spired by11,13,22, we first study in Section 3, the calm
B-differentiability of the circular cone. We then pro-
vide in Section 4 the formula for the Fréchet normal
cone to a circular cone complementarity set, which
can be considered as a generalization of the second-
order cone complementarity set. This formula would
be useful for us to study optimality conditions for
mathematical programming problems with circular
cone complementarity constraints.
PRELIMINARIES
Throughout the paper, if not otherwise specified,
f(t)=o(t) (f(t)=O(t)) means f (t)jtj ! 0 (resp.,
f (t)
jtj
is uniformly bounded) as t ! 0, and ( f (x))+ :=
maxf f (x);0g; and ( f (x)) := minf f (x);0g:Br(x)
stands for the closed ball centered at x 2 Rn with ra-
dius r > 0. Given x;y 2 Rn; xTy stands for the scalar
product of x and y. For x := (x0;xr ) 2 RRn1;
we use the following notation
x? := fy 2 RnjxTy= 0g and
exr :={ xrjjxr jj if xr ̸= 0;
if otherwise:any unit vector e 2 Rn1
Let CRn be a nonempty subset, clC denotes its clo-
sure. The polar cone C◦ and the dual cone C⋆ of C
are
C◦ := fy 2 Rnjy⊤x 0;8x 2Cg and
C⋆ := fy 2 Rnjy⊤x 0;8x 2Cg
respectively.
The Fréchet normal cone to C at x 2 clC are defined
respectively by, see24,
bNC(x) :=
fx 2 Rn⟨x;x′ x⟩ o(∥ x′ x ∥);8x′ 2Cg:
Lemma 2.1 (24, Theorem 1.14) Let D={x | h(x)2 C}
and let Ñh(x) be surjective. Then
bND(x) = Ñh(x)T bNC(x):
Let f :Rn ! (¥; ¥] and x 2 Rn such that f
(
x
)
is
finite. The Fréchet subdifferential of f at x is defined
by, see [24, pages 89 and 90],
b¶ f (x) := fx 2 Rngj
limsup
x!x
⟨x;xx⟩ f (x)+ f (x)xx 0
The indicator function of a setC Rn is denoted by
dC(x) :=
{
0 if x ̸2C;
¥ otherwise:
It is known from [ 25, Proposition 1.18] that b¶dC(x) =bNC(x)
for any x 2C.
Let F:Rn ) Rm be a set-valued mapping, the domain
and the graph of F are
domF := fx 2 RnjF(x) ̸=∅g;
gphF := f(x;y) 2 RnRmjy 2 F(x)g:
The Fréchet coderivative of F at (x, y) 2 gphF are re-
spectively defined by, see [24, Definition 1.32], for
each y2Rm,
bDF(x;y)(y)
:= fx 2 Rnj(x;y) 2 bNgphF (x;y)g:
When F(x) is single-valued, y can be omitted in the
above notations. Moreover, if F is continuously dif-
ferentiable, then for all y2Rm, we get
bDF(x)(y) = fÑF(x) yg :
The derivative in the directionh2Rn ofF at x is defined
by
F ′(x;y) := lim
t!0+
F(x+ th)F(x)
t
:
The circular cone is defined (cf.14,19–23) by
Kq :=
fx= (x0;xr) 2 RRmjx0 tanq jjxrjjg
(2.1)
with angle q 2 (0; p2 ). When q = p4 , it re-
duces to the second-order cone defined by Kq :=
fx= (x0;xr) 2 RRmjx0 jjxrjjg. In this case, the
set
Ω :=
{
(x;y)jx 2K ;y 2K ;xTy= 0} ; (2.2)
is called the second-order cone complementarity set. If
q ̸= p4 thenKq is a nonsymmetric and non-self-dual
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Science & Technology Development Journal, 23(4):727-736
cone. The boundary and the interior ofKq are given
respectively by
bdKq :=
fx= (x0;xr) 2 RRmjx0 tanq jjxrjjg ;
intKq :=Kqn(bdKq ):
The positive dual cone and the polar cone ofKq are
defined respectively by, see [ 20, Theorem 2.1],
Kq := fy= (y0;yr) 2 RRmjy0 cotq jjyrjjg ;
K q :=K q =
fy= (y0;yr) 2 RRmjy0 cotq jjyrjjg :
A relation between the boundary of Kq and that of
K q is established as follows.
Proposition 2.2 Let x 2 bdKqnf0g and
y 2 bdK q nf0g. Then, yT x= 0
if and only if x = k(y0cot2q ;yr ) with k =
x0
y0 tan
2 q (equivalently, y= k(x0 tan2 q ;xr) with k=
y0
x0 cot
2q :).
Proof. Let x 2 bdKqnf0g and y 2 bdK q nf0g.
“ If ”: Suppose that there exists k 2 R++ := (0;¥)
with x= k(y0cot2q ;yr ), then yT x= 0.
“ Only if ”: Let yT x = 0, then we get x0 tanq =∥ xr ∥
;y0 tan( p2 q) = jjyrjj and x0y0+yTr xr = 0:Thus, one
has
yTr xr =x0y0 = (jjxrjjcotq)(jjyrjjcot( p2 q))=jjxrjj : jjyrjj ;
which implies the existence ofk 2R++ such that xr =
kyr . Consequently, we obtain
x0 tanq = ky0 tan(
p
2
q);
i.e., x0 tan2 q = ky0. Hence, x = k(y0cot2q ;yr)
with k = x0y0 tan
2 q , and the proof is completed. □
We recall that for any given x := (x0;xr) 2RRm, it
can be decomposed by (see [ 20, Theorem 3.1])
x= l1(x) u1x +l2(x)u2x ;
where the spectral valuesl1(x);l2(x) and the spectral
vectors u1x ;u2x are defined respectively by
l1(x) := x0jjxrjjcotq ;
l2(x) := x0+ jjxrjj tanq ;
u1x :=
1
1+cot2 q
[
1 0
0 cotq
][
1
exr
]
;
u2x :=
1
1+tan2 q
[
1 0
0 tanq
][
1exr
]
:
The metric projection of x onto Kq , denoted by
PKq (x), is defined as follows
PKq (x) := argminz2Kq jjx zjj
= fz 2Kq j ∥ x z ∥∥ xu ∥;8u 2Kqg:
From22 and the convexity ofKq , we get thatPKq (x)
is a single-valued set and
PKq (x) = (l1(x)+)u
1
x+(l2(x)+)u2x: (2.3)
Moreover, [ 26, Proposition 2] states that, for all x 2
Rm+1,
x=PKq (x)+PK ◦q (x)
and ⟨PKq (x);PK ◦q (x)⟩= 0:
SinceP(Kq )(x);P(K ◦q )(x), one gets
PKq (x) =PK ◦q (x):
Let us define the circular cone complementarity set as
G := f(x;y)jx 2Kq ;y 2K q ;xTy= 0g; (2.4)
which is a generalized type of (2.2). Given (x,y)2G
and an arbitrary u 2Kq , it holds that
jj(x y)ujj2jj(x y) xjj2
= jj(xu) yjj2jjyjj2
= jjxujj22⟨xu;y⟩
= jjxujj22⟨x;y⟩+2⟨u;y⟩ 0;
which means that x = PKq (x y). Similarly, we get
that y 2PK q (y x).
The above observation allows us to obtain a relation
between the complementarity set G, and the projec-
tion ontoKq as follows.
Proposition 2.3 Let G be as in (2.4). Then, we get
[(x;y) 2 G]() [x 2PKq (x y)]
() [y 2PK q (y x)]() [y 2PK ◦q (x y)]:
By Proposition 2.3, G can be expressed by
G= f(x;y)j(x y;x) 2 gphPKq g:
Let f :Rm+1Rm+1 ! Rm+1Rm+1 be defined by
f (x;y) := (x y;x) for all (x;y) 2 Rm+1Rm+1, we
can check that f is continuously differentiable and
Ñ f (x;y) =
[
Im+1 Im+1
Im+1 0
]
;
where Im+1 is the unit matrix of the degree m+1, has
full rank. It follows from [ 27, Exercise 6.7] thatbNG(x;y) =
fÑ f (x;y)(x;y)j (x;y) 2 bNgphPKq ( f (x;y))g
= fx+ y;x)j (x;y) 2 bNgphPKq )(x y;x)g
= f(u;v)j v 2 bDPKq (x y)(u v)g:
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Science & Technology Development Journal, 23(4):727-736
From the above discussion, one obtains the following
result, which plays an important role in computing the
Fréchet normal cone to complementarity set.
Proposition 2.4 Let G be as in (2.4) and (x,y)2G.
Then, we getbNG(x;y)
= f(u;v)j v 2 bDPKq (x y)(u v)g: (2.5)
CALMB-DIFFERENTIABILITY OF THE
PROJECTMAPPINGONTO A
CIRCULAR CONE
In this section, we first show that the projection oper-
ator PKq is calmly B-differentiable at any x 2 Rm+1.
Then, we provide a characterization for a proximal
normal vector of G.
Definition 3.1 (11) The function F : Rn ! Rm is
called calmly B-differentiable at x if, for all h suffi-
ciently close to 0, we get
F(x+h)F(x)F ′(x;h) = O(∥ h ∥2 ):
Theorem 3.2The projection mapping PKq is calmly
B-differentiable for any x 2 Rm+1.
Proof.Given an arbitrary x 2 Rm+1, it is enough to
show that, for h sufficiently close to 0,
PKq (x+h)PKq (x)P
′
Kq
(x;h)
= 0= O(∥ h ∥2): (3.1)
We consider the following cases.
Case 1: x 2 intKq . Then, we have PKq (x) =
x, PKq (x + h) = x + h. Moreover, it follows
from the definition of the directional derivative that
P′Kq (x;h) = h. So, (3.1) is fulfilled.
Case 2: x 2 intK . We get, in this case that
PKq (x) = 0, PKq (x+h) = 0. On the other hand, by
the definition ofP′Kq (x;h), one hasP
′
Kq
(x;h) = 0, so
(3.1) holds.
Case 3: x 2 bdKqnf0g. It implies that l1(x) = 0 and
l2(x)> 0. By (2.3) and Lemma 3.2(b) in22, we have
PKq (x) = x;P
′
Kq
(x;h)
= h (1+ cot2 q)((ux1)Th)_ux1;
PKq (x+h) = ((x0+h0 ∥ xr+hr ∥ cotq)+)
11+cot2 q
[
1
(xrhr)cotq
jjxr+hr jj
]
+((x0+h0+ ∥ xr+hr ∥ tanq)+)
11+tan2 q
[
1
(xr+hr) tanq
jjxr+hr jj
]
: (3:2)
Let ePh = (ePh0; ePhr ) 2 Rm+1 be defined byePh =PKq (x+h)PKq (x)P′Kq (x;h) (3.3)
we now show computations for ePh0 and ePhr . By (3.2)
and (3.3), ePh0 is given by Figure 1.
By the expression of ∥ xr ∥, for all hr sufficiently close
to 0, we get ∥ xr+hr ∥=∥ xr ∥+⟨exr;hr⟩+O(∥ hr ∥2),
which implies that
x0+h0 ∥ xr+hr ∥ cotq
= h0⟨exr;hr⟩cotq +O(∥ hr ∥2): (3.5)
It follows from (3.4), (3.5), and the Lipschitz property
of the function ()_ with modulus 1 that
j (x0+h0 ∥ xr+hr ∥ cotq)_
+(h0⟨exr;hr⟩cotq)_j O(∥ h_r ∥2): (3.6)
On the other hand, we also get equation 3.7 in Figure 2
Using the Taylor expression of the function x∥x∥ , one
has
xr+hr
jjxr+hrjj
= exr+ 1jjxjj (exr;exTr )hr+O(jjhrjj2) : (3.8)
Thus, (3.7) is given by Figure 3
Moreover, from (3.5) and (3.6), we obtain
ePhr = (exr;exTr )hr+O(∥ hr ∥2)∥ xr ∥ (cotq + tanq)
((h0⟨xr;hr⟩cotq +O(∥ hr ∥2))_)
+
exr
cotq + tanq
(∥ hr ∥2)):
(3.10)
It is necessary to show that ∥ ePhr ∥ O(∥ h ∥2). In-
deed, it follows from (3.10) that
∥ ePhr ∥ ∥ (exr;exTr )hr+O(∥ hr ∥2) ∥∥xr∥(cotq+tanq) (
∥ h ∥
p
1+ cot2 q +O(∥ h ∥2 )
)
+O(∥ h ∥2 ) = O(∥ h ∥2 ):
Combining to (3.6), we get ∥ ePh ∥=O(∥ h ∥2 ), which
means that (3.1) holds.
Case 4: x2bdK q nf0g. It is obvious thatPKq (x)=
0. Then, for each h 2 Rm+1 sufficiently close to 0,
we have l1(x) < 0;l2(x) = 0 and l1(x+ h) = (x0+
h0) ∥ xr+hr ∥ cotq < 0:. It follows from (2.3) and
Lemma 3.2 in22 that
PKq (x+h) = ((x0+h0+ tanq jjxr+hrjj)+)u2x+h
P′Kq (x;h) = (1+ tan
2 q)(⟨u2x ;h⟩)+u2x ;
where
u2x =
1
1+ tan2 q
[
1exr tanq
]
;
u2x+h =
1
1+tan2 q
[
1
xr+hr
jjxr+hr jj tanq
]
= u2x +
1
1+tan2 q
[
0
1
jjxr jj (exrexTr )hr+O(jjhrjj2)
]
730
Science & Technology Development Journal, 23(4):727-736
Figure 1: Equation 3.4
Figure 2: Equation 3.7
Figure 3: Equation 3.9
Thus, one gets
ePh = (x0+h0+ tanq jjxr+hrjj)+u2x+h
(1+ tan2 q)(⟨u2x ;h⟩)+u2x
= (h0+ ⟨exr tanq ;hr⟩+O(jjhrjj2))+(
u2x +
1
1+tan2 q
[
0
1
jjxr jj (exrexTr )hr+O(jjhrjj2)
])
(h0+ ⟨exr tanq ;hr⟩)+u2x (3:11)
with ePh :=PKq (x+h)PKq (x)P′Kq (x;h). Since
the function ()+ is Lipschitz with modulus 1, from
(3.11), we obtainePh O(jjhrjj2)+ u2x
+jj(h0+ ⟨exr tanq⟩+O(jjhrjj2))+(
1
1+tan2 q
)[ 0
1
jjxr jj (exrexTr )hr+O(jjhrjj2)
]
jj
O(jjhrjj2)+(jjhjj
p
1+ tan2 q +O(jjhrjj2)
(
1
1+tan2 q
√(
1
jjxr jj2 exr;hr)2+O(jjhrjj4)
)
= O(jjhjj2):
Note that the last inequality holds by ∥hr∥∥ h∥.
Consequently, (3.1) is implied.
Case 5: x=0. Then, for all h 2 Rm+1, we get
l1(x) = l2(x) = 0 and
PKq (x) = 0;PKq (x+h) =PKq (h);
P′Kq (x) =PKq (h):
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Science & Technology Development Journal, 23(4):727-736
Thus, one has
jjPKq (x+h)PKq (x)P
′
Kq
(x;h)jj= 0;
which means that (3.1) holds.
Case 6: x 2 Rm+1n(Kq [ (K q )). Since
Rm+1n(Kq [ (K q )) is open, for h sufficiently
close to 0, one has x+ h 2 Rm+1n(Kq [ (K q )).
Moreover, we can check that l1(x)< 0; l1(x+h)< 0
and l2(x) > 0; l2(x+ h) > 0. Thus, it follows from
(2.3) and [ 22, Lemma 3.2(a) and (3.6)] that
PKq (x) = (x0+ tanq jjxrjj)u2x ;
PKq (x+h) = (x0+h0+ tanq jjxr+hrjj)u2x+h
P′Kq (x;h) =
1
tanq+cotq[
cotq exTrexr x0+jjxr jj tanqjjxr jj I x0jjxr jjexrexTr
][
h0
hr
]
:
By directly computations, we get
jjxr+hrjj= jjxrjj+ ⟨exr;hr⟩+O(jjhrjj2) ;
xr+hr
jjxr+hr jj =
xr
jjxr jj
+ 1jjxrjj
(
IexrexTr )hr+O(jjhrjj2) ;
and
P′Kq (x;h) =[ 1
1+tan2 q (h0+ tanq⟨exr;hr⟩)
1
cotq+tanq
(
h0exr+ x0+jjxr jj tanqjjxr jj hr)+B
]
B= x0jjxr jj ⟨exr;hr⟩exr:
By letting ePh :=(
1+ tan2 q
)(
PKq (x+h)PKq (x)P
′
Kq
(x;h)
)
,
then one has
ePh0 = O(jjhrjj2)= O(jjhjj2) ; (3.12)
O
(
jjhrjj2
)
(x0+ tanq jjxrjjexr tanq)
h0 tanqexr x0 tanq + jjxrjj tan2 qjjxrjj hr
+
x0 tanq
jjxrjj ⟨exr;hr⟩exr
= h0⟨exr;hr⟩+O(jjhrjj2)= O(jjhjj2) ;
(3.13)
where the last equation holds by the fact that
jjh0⟨exr ;hr⟩jj
jjhjj2
jh0j:jjxrjj:jjhrjj
jjhjj2
jjxr jj:jjhjj
2
2jjhjj2 =
jjxr jj
2 :
Thus, (3.1) is fulfilled. □
APPLICATION
In this section, we first establish the formulation for
the Fréchet normal cone to the circular cone comple-
mentarity set G.
Theorem 4.1 Let G be defined as in (2.4) and (x;y) 2
G. Then, we get Figure 4
Proof . We consider the following cases.
Case 1: x=0 and y 2 int K q . Then, we get x-y=-y,
which implies that
l1(x y) =y0jjyrjjcotq < 0;
l2(x y) =y0+ jjyrjj tanq :
Since y 2 int K q , we get y0 tan
( p
2 q
)
>
jjyrjj ; i:e:; y0 > jjyrjj tanq . Consequently, one
has
l2(x y) =y0+ jjyrjj tanq < 0:
It follows from [22, Lemma 3.1(a) and (3.6)] that
¶B(PKq )(x y) =
{
ÑPKq (x y)
}
= 0:
By [22, Theorem 3.5(a)], we have
bDPKq (x y)(y) = {ÑPKq (x y)y}= 0:
Consequently, from (2.5), one obtains
bNG(x;y) = {(u;v) ju 2 Rm+1;v= 0} :
Case 2: x 2 intKq and y=0. Then we get x-y=x, so
l1(x y) = x0jjxrjjcotq < 0;
l2(x y) = x0+ jjxrjj tanq :
It follows from [22, Lemma 3.1(a), (3.6) andTheorem
3.5(a)] that
¶B(PKq )(x y) =
{
ÑPKq (x y)
}
= I;bDPKq (x y)(y) = {ÑPKq (x y)y}= y:
Therefore, by (2.5), we have
bNG(x;y) = {(u;v) ju 2 Rm+1;v 2 Rm+1} :
Case 3: x 2 bdKqnf0g,y 2 bdK q nf0g and y⊤x =
0. We get from Lemma 2.2 that y = k(x0 tan2 q ;xr)
with k = y0x0 cot
2q > 0, which implies that
x y= (x0;xr) k
(
x0 tan2 q ;xr
)
=
(
(1 k tan2 q)x0;(1+ k)xr
)
:
Moreover, we have
l1(x y)
= (1 k tan2 q)x0 (1+ k) jjxrjjcotq
=k(1 tan2 q)x0 < 0
(4.1)
732
Science & Technology Development Journal, 23(4):727-736
Figure 4: Theorem 4.1
and
l2(x y)
= (1 k tan2 q)x0+(1+ k) jjxrjjcotq
= (1 tan2 q)x0 > 0:
(4.2)
By (3.6) in 22, we get
ÑPKq (x y) = 1tanq+cotq[
cotq exTrexr 1+tan2 qtanq(1+k) I 1k tan2 q(1+k) tanq exrexTr
]
:
On the other hand, it follows from [22, Theorem 3.5]
and (2.5) that
bNG(x;y)
=
{
(u;v) j v 2 bDPKq (x y)(u v)}
=
{
(u;v) j v 2 ÑPKq (x y)(u v)
}
:
(4.4)
Let (u;v) 2 bNG(x;y) and x′ 2 bd Kqnf0g, x′ 2
bdK q nf0gwith y′= k(x
′
0 tan
2 q ;x′r). Then y′Tx′=
0, which implies that (x’,y’ )2G. Consequently, one
has
⟨(u;v);(x′;y′ )(x;y)⟩
jj(x′;y′ )(x;y)jj =
⟨u;x′x⟩+⟨v;y′y⟩
jj(x′x;y′y)jj
=
⟨u;x′x⟩+⟨kv;(x′0 tan2 q ;x
′
r)(x0 tan2 q ;xr)⟩
jj(x′x;k(x′0 tan2 q ;xr)k(0tan2 q ;xr ))jj
⟨u+k(v0 tan2 q ;vr );x′x⟩p
1+k2jjx′xjj : (4:5)
Since (u;v)2 bNG(x;y), passing to the limit in (4.5), we
get
limsup
x′
bdKq nf0g!x
⟨u+k(v0 tan2 q ;vr );x′x⟩
jjx′xjj
limsup
(x′;y′)
G!(x;y)
p
1+k2(⟨(u;v);(x′;y′ )(x;y)⟩)
jj(x′;y′ )(x;y)jj 0
which implies that
u+ k(v0 tan
2 q ;vr ) 2 bNbdKq nf0g(x) (4.6.)
By x2 bdKqnf