Journal of Mathematical Analysis and Applications 231, 133]141 Ž1999.
Article ID jmaa.1998.6228, available online at http:rrwww.idealibrary.com on
An Estimate for Lipschitz Constants of
Metric Projections*
Chong Li
Department of Applied Mathematics, Southeast Uni¨ ersity, Nanjing 210096, P. R. China
Xinghua Wang
Department of Mathematics, Hangzhou Uni¨ ersity, Hangzhou 310028, P. R. China
and
Wenshan Yang
Department of Mathematics, Zhejiang Normal Uni¨ ersity, Jinhua 321004, P. R. China
Submitted by Joseph A. Ball
Received March 2, 1998
The main purpose of the paper is to give a global estimate for Lipschitz
constants of metric projections in a p-uniformly convex and q-uniformly smooth
Banach space. Q 1999 Academic Press
1. INTRODUCTION
Metric projection operators PV on convex closed sets V Žin the sense of
best approximations. are widely used in theoretical and applied areas of
mathematics, especially connected with problems of optimization and
approximation. As examples one can consider iterative-projective methods
for solving equations, variational inequalities and minimizations of functionals w1x, and methods of alternating projections for finding common
points of convex closed sets in Hilbert space w4, 5, 7x.
* Supported by National Natural Science Foundation of China.
133
0022-247Xr99 $30.00
Copyright Q 1999 by Academic Press
All rights of reproduction in any form reserved.
134
LI, WANG, AND YANG
It is well known w9x that in a Hilbert space X the following estimate
holds:
5 PV Ž x . y PV Ž y . 5 F 5 x y y 5 , for all x, y g X ,
Ž 1.
and many problems can be solved by applying the inequality Ž1.. Therefore, one natural method to solve problems in Banach spaces X is to
establish estimates analogous to Ž1.. This is in fact shown by some author’s
recent work; see, for example, Xu and Roach w11x. Alber and Notik w3x, and
Alber w2x.
However, applying their results to a Hilbert space X, we can only obtain
the following estimate:
5 PV Ž x . y PV Ž y . 5 F c 5 x y y 5 , for all x, y g X ,
Ž 2.
for some constant c ) 1. Comparing Ž2. with Ž1., it is natural to ask
whether there exists a common estimate for 5 PV Ž x . y PV Ž y .5 in a uniformly convex and uniformly smooth Banach space such that Ž1. holds for
a Hilbert space. The purpose of the present paper is to give such an
estimate for 5 PV Ž x . y PV Ž y .5 in a p-uniformly convex and q-uniformly
smooth Banach space.
2. PRELIMINARIES
Now let us recall the definition of the metric projection operator. Let V
be a convex closed subset in a Banach space X. For any x g X an element
x g V such that 5 x y x 5 s min m g V 5 x y m 5 is called a nearest point to x.
If for each x g X, it has a unique nearest point in V, then define PV Ž x . to
be the nearest point of x. It is known that PV is well defined in a
uniformly convex Banach space. The moduli of convexity and smoothness
of X are defined, respectively, by
d X Ž e . s inf 1 y
1
2
Ž x q y . : 5 x 5 s 5 y 5 , 5 x y y 5 s e 4 , 0 F e F 2,
and
r X Ž t . s sup 12 Ž 5 x q y 5 q 5 x y y 5 . y 1: 5 x 5 s 1, 5 y 5 s t 4 ,
t ) 0.
Let p G 2, 1 - q F 2, X is said to be p-uniformly convex Žresp. q-uniformly smooth. if there is a constant d ) 0 such that d X Ž e . G d e p Žresp.
r X Ž t . F dt q ..
135
LIPSCHITZ CONSTANTS OF PROJECTIONS
Let
Ž 1r2. 5 x 5 p q Ž 1r2. 5 y 5 p y 5 Ž x q y . r2 5 p
d p s inf
: 5 x y y5 ) 0 ,
5 Ž x y y . r2 5 p
½
½
5
5
Ž 1r2. 5 x 5 q q Ž 1r2. 5 y 5 q y 5 Ž x q y . r2 5 q
c q s sup
: 5 x y y5 ) 0 .
5 Ž x y y . r2 5 q
From w10x we have
PROPOSITION 1. Let p ) 1. Then a Banach space S is p-uniformly con¨ ex
if and only if d p ) 0.
PROPOSITION 2. Let q ) 1. Then a Banach space X is q-uniformly
smooth if and only if c q ) 0.
The following proposition is simple and well known.
PROPOSITION 3. Let q ) 1 and X be strictly con¨ ex. Then for any
x, y g X, f x g Jq Ž x ., f y g Jq Ž y ., there holds
² x y y, f x y f y : ) 0
where Jq Ž x . s x* g X * : ² x, x*: s 5 x 5 q, 5 x* 5 s 5 x 5 qy 14 .
In order to give the main theorem we need a lemma about the constants
d p and c q .
LEMMA 1. Let X be a Banach space and n be a positi¨ e integer. Then for
any x, y g X the following hold
ž
1y
1
2
n
/
xq
1
2
n
y PF 1y
ž
1
2
n
/
5 x5 p q
ny1
y dp
ž
1
2
n
/
xq
1
2
n
q
y
G 1y
ž
1
2
n
/
2p
1
5 y5 p
ny1yi
2
5 x5q q
ny1
y cq
2n
½ ž /ž / 5
Ý
is0
1y
i
1
1
1
i
1
2n
1
2q
2
p
Ž 3.
2
5 y5q
ny1yi
½Ýž /ž / 5
is0
xyy
xyy
2
q
. Ž 4.
136
LI, WANG, AND YANG
Proof. We will prove formula Ž3. by induction while formula Ž4. can be
proved similarly. From the definition of d p , we have that for any x, y g X,
1
2
xq
1
2
p
F
y
1
5 x5 p q
2
1
2
xyy
5 y 5 p y dp
p
.
2
This means that Ž3. holds for n s 1. Now assume that Ž3. holds for n s k.
Let us prove that Ž3. holds for n s k q 1. In fact, for any x, y g X,
ž
1y
1
2 kq 1
s
F
F
1
2
1
/
xq
xq
1
2 kq1
ž
2
5 x5 p q
p
1
y
1
1y
1
2
/
k
xq
1
1y
1
2
5 x5 p q
2
1
1y
2
ky1
1
y dp
2
s 1y
is0
1
2 kq1
1
2
2k
1
2
i
p
5 x5 p q
k
2k
xq
ž /
ž /
½Ýž /ž /
ž
/
2
1
y
p
1
2k
5 x5 p q
1
ky1yi
2
5
1
2k
xyy
ž /
2k
p
2
5 y5 p
xyy
2
1
2
p
1
y dp
y
5 y 5 p y dp
kq1
p
y dp
k
1
ž /
½Ýž / ž /
k
2
i
1
1
2p
is0
xyy
p
p
2
kyi
2
5
xyy
2
p
.
Hence Ž3. holds for n s k q 1 and the proof is complete.
PROPOSITION 4. Let q ) 1 and X be q-uniformly smooth. Then for any
x, y g X the following inequality holds
5 x q y 5 q F 5 x 5 q q q² y, Jq Ž x . : q
cq
2
qy1
y1
5 y 5 q.
Proof. From Ž4., it follows that for any n ) 1
5 x q Ž 1r2 n . y 5 q y 5 x 5 q
1r2 n
s
Ž 1 y Ž 1r2 n . . x q Ž 1r2 n . Ž x q y .
G 5 x q y5q y 5 x5q y
q
y 5 x5 p
1r2 n
c q Ž 1 y 2 ny n q .
2 qy 1 y 1
5 y 5 q.
Ž 5.
137
LIPSCHITZ CONSTANTS OF PROJECTIONS
Thus we have that
q² y, Jq Ž x . : s lim
5 x q Ž 1r2 n . y 5 q y 5 x 5 q
1r2 n
nª`
G 5 x q y5q y 5 x5q y
cq
2
qy 1
y1
5 y 5 q.
This proves the proposition.
3. MAIN THEOREM
Now we are ready to prove the main theorem
THEOREM 1. Let X be a p-uniformly con¨ ex and q-uniformly smooth
Banach space. Let V be a closed con¨ ex subset in X. Then for any x, y g X
there holds
5 PV x y PV y 5 F
pc q Ž 2 py 1 y 1 .
qd p Ž 2 qy1 y 1 .
c pyq 5 x y y 5 q r p
where c s max5 x y PV y 5, 5 y y PV x 54 .
Proof. Let
Px s PV x,
Py s PV y,
and
x n s Px q 2 n Ž x y Px .
Then Px n s Px for any positive integer n. It follows that
5 x n y Px 5 p F 5 x n y Py 5 p F 5 x y Py q Ž 2 n y 1 . Ž x y Px . 5 p .
Dividing the inequality by 2 n and using Lemma 1, we have
1
1
2
2
5 x y Px 5 p F
n
5 x y Py 5 p y d p
n
ny1
1
i
1
ny1yi
½Ýž /ž / 5
is0
2p
2
Multiplying the inequality by 2 n and letting n ª `, we obtain
dp
2
py 1
y1
5 Px y Py 5 p F 5 x y Py 5 p y 5 x y Px 5 p .
Px y Py
2
p
.
138
LI, WANG, AND YANG
Using Cauchy mean-valued theorem we obtain
dp
2
y1
py 1
5 Px y Py 5 p F
p
q
c pyq Ž 5 x y Py 5 q y 5 x y Px 5 q . .
Similarly, we also have
dp
2
y1
py 1
5 Px y Py 5 p F
p
c pyq Ž 5 y y Px 5 q y 5 y y Py 5 q . .
q
It follows that
2
dp
2
py 1
y1
5 Px y Py 5 p F
p
q
c pyq Ž 5 x y Py 5 q y 5 x y Px 5 q
q5 y y Px 5 q y 5 y y Py 5 q . .
Note that Ž5. implies
5 x y Py 5 q y 5 y y Py 5 q F q² x y y, Jq Ž y y Py . : q
5 y y Px 5 q y 5 x y Px 5 q F q² y y x, Jq Ž x y Px . : q
cq
2
qy1
2
qy1
y1
cq
y1
5 x y y5 q,
5 x y y 5 q.
Hence
2
dp
2
5 Px y Py 5 p
y1
p
F c py q Ž q² x y y, Jq Ž y y Py . : q q² y y x, Jq Ž x y Px . : .
q
py 1
q2
cq
2
qy 1
y1
5 x y y 5 q.
From the characterization of a best approximation it follows that
² Px y Py, Jq Ž y y Py . : F 0,
² Py y Px, Jq Ž x y Px . : F 0.
This, together with Proposition 3, gives
2
dp
2
5 Px y Py 5 p
y1
p
cq
5 x y y5q
F 2 c pyq q² Qx y Qy, Jq Ž Qy . y Jq Ž Qx . : q qy1
q
2
y1
py 1
ž
F2
p
q
c pyq
cq
2
qy 1
y1
5 x y y5q
/
139
LIPSCHITZ CONSTANTS OF PROJECTIONS
where Qx s x y Px. That is,
5 Px y Py 5 F
pc q Ž 2 py 1 y 1 .
qd p Ž 2 qy 1 y 1 .
c pyq 5 x y y 5 q r p .
The proof is complete.
If X is a Hilbert space then p s q s 2 and d p s c q s 1. Therefore we
have
COROLLARY 1. Let X be a Hilbert space and V be a con¨ ex closed subset
in X. Then the following holds
PV Ž x . y PV Ž y . F 5 x y y 5 ,
for all x, y g X .
COROLLARY 2. Let X be the L r Ž m .-space and V be a con¨ ex closed
subset in X. Then the following holds
¡r Ž r y 1 . c
PV Ž x . y PV Ž y . F
~
ry2
Ž 2 ry1 y 1 .
2
2 c 2yr
¢r Ž r y 1 . Ž 2
ry1
y 1.
5 x y y 5 2r r ,
5 x y y 5 r r2 ,
rG2
r F 2.
Proof. We divide two cases, r G 2 and r F 2, to prove the corollary.
Ži. If r G 2, then p s r, q s 2, that is, L r Ž m . is r-uniformly convex
and 2-uniformly smooth. It follows from w6x that d p s d r s 1, c q s c 2 F
p y 1. Thus the result follows from Theorem 1.
Žii. If r F 2, then p s 2, q s r, that is, L r Ž m . is 2-uniformly convex
and r-uniformly smooth. From w8x we have d p s d 2 G r y 1. Now let us
show that c q s c r s 1.
For y1 F t - 1, let
f Ž t. s
1r2 q Ž 1r2 . < t < r y Ž Ž 1 q t . r2 .
Ž Ž 1 y t . r2.
r
r
.
Then the derivative of f Ž t . for t / 0 is
f 9Ž t . s
r Ž < t < ry1 s Ž t . y 2 Ž Ž 1 q t . r2 .
4 Ž Ž 1 y t . r2 .
rq1
ry1
q 1.
140
LI, WANG, AND YANG
where sŽ t . s tr< t < for t / 0 and sŽ0. s 0. Write
g Ž t . s < t < ry1 s Ž t . y 2
ž
1qt
2
ry1
/
q 1.
Then the derivative of g Ž t . for t / 0 is
g 9 Ž t . s Ž r y 1. Ž < t <
ry2
y
ž
1qt
2
ry2
/
.
Thus
g 9Ž t . ) 0
; t g Ž y1r3, 0 . j Ž 0, 1 .
g 9Ž t . - 0
; t g Ž y1, y1r3. .
From g Ž1. s g Žy1. s 0, g Ž0. - 0 it follows that g Ž t . F 0 for all t g
wy1, 1x. This implies that f 9Ž t . F 0 for all t g wy1, 0. j Ž0, 1. so that
sup f Ž t . : y1 F t - 1 4 s f Ž y1 . s 1.
Thus we have
sup
½
Ž 1r2. < x < r q Ž 1r2. < y < r y < Ž x q y . r2 < r
: x / y s 1.
< Ž x y y . r2 < r
5
This implies that c q s c r s 1 and the proof is complete.
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