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Inequalities in Banach spaces with applications
Nonlinear Annlysrs, Theory, Printed in Great Britain.
Methods
& Applicorrons,
Vol.
16, No.
12, pp. 1127-1138,
0
INEQUALITIES
0362-546X/91 $3.00+ .W.l 1991 Pergamon Press plc
IN BANACH SPACES WITH APPLICATIONS Xv
HONG-KUN Department
1991.
of Mathematics,
East China Institute of Chemical People’s Republic of China
(Received 21 May 1990; received in revised form
Technology,
Shanghai
200237,
10 October 1990; received for publication 6 November
1990)
Key words and phrases: Inequality, uniformly convex Banach space, uniformly smooth Banach space, uniformly convex functional, fixed point, uniformly Lipschitzian mapping, nearest point projection.
1. INTRODUCTION
IT IS well known identity:
that many problems
in a Hilbert
space H are solved by applying
IlAx + (1 - A)YI12= a412 + (1 - ~)llYl12 - A(1 - Mx
the following
- Yl12
(1.1)
for all x, y in H and 0 5 I 5 1. Therefore, one natural method to solve problems in a Banach space Xis to establish equalities and (usually) inequalities in Xanalogous to (1.1). This is in fact shown by some authors’ recent work, see for example, Reich [l], Poffald and Reich [2], Lim [3], Lim et af. [4], Prus and Smarzewski [5], Smarzewski [6], and Reich [21]. The purpose of this paper is to continue carrying out investigations in this line. More precisely, in Section 3, we establish some inequalities analogous to (1 .l) in either uniformly convex or uniformly smooth Banach spaces. As corollaries, we obtain new characteristics of p-uniform convexity and q-uniform smoothness of a Banach space X via the functionals I]. lip and II * llq respectively. We give in Section 4 applications of the inequalities obtained in Section 3 to fixed points of uniformly Lipschitzian mappings and to best approximation theory. 2. PRELIMINARIES
Throughout for its dual, R radius r. For x smoothness of &(E)
this paper, X will always stand for a real Banach space with norm II * /I and X* for the field of real numbers, and B, for the closed ball with center zero and in X and x* in X*, (x, x*> is the value of x* at x. The moduli of convexity and X are defined respectively by
= inf(l
-
11+(x + y))I : llxll = llyll = 1 and [Ix - y]I = E],
OIEI2,
and &Y(f) = SUPMIX
+ Yll + IIX - Yll) - 1 : llxll = 1, IIYII = 4,
t > 0.
X is said to be uniformly convex if dx(e) > 0 for all 0 < E I 2 and uniformly smooth if li~i px(t)/t = 0. Let p, q > 1 be real numbers. Then X is said to be p-uniformly convex (resp. q-uniformly smooth) if there is a constant c > 0 such that ax(e) 2 cY (resp. px(t) I ctq). Let f: X + i? := R U { + a~)be a proper functional and D a nonempty convex subset of X. Thenf is said to be convex on D if f(lx
+ (1 - AlY) 5 VW 1127
+ (1 - ~l_fCY)
HONG-KUN Xu
1128
for all 0 5 13 I 1 and x,y in D. f is said to be uniformly convex on D [7] if there exists a function ,u:R+ := [0, 00) --t R+ with p(t) = 0 if and only if t = 0 such that f(Ax + (1 - A)Y) 5 U(x)
+ (1 - J.)f(Y) - A(1 - ~Mllx
- Yll)
(2.1)
for all 0 5 A 5 1 and x, y in D. If the inequality (2.1) is valid for all x, y in D when A = 5, then fis said to be uniformly convex at center. Zalinescu [7, p. 3521 remarked that for a convex function f, f is uniformly convex on D if and only iffis uniformly convex at center on D. We now recall the conjugate function f *: X * + 17 of a convex function f is defined by f *(x*) = sup((x, x*> - f(x): x E X), It is well known that if f(x) = (l/p)llxllp(p > l), l/q + l/p = 1, and that for any constant c > 0, (cf)*(x*) subdifferential df: X + X* off is defined by
x* E x*. then f*(x*) = cf *(c-lx*).
= (l/q)llx*llq, where We also recall that the
af(x) = (x*E X* :f(y) 2 f(x) + (y - x, x*) for all y E X). On the other hand, with each p > 1 we can associate into X* defined as follows J,(x) In particular,
= {x* E X*: (x, x*)
J := J2 is called
the (generalized)
duality
map Jp from X
= llxll” and IIx*)l = Ilxllp-‘].
the normalized
duality
map
x1 f or x # 0 and that J,(x) is the subdifferential J,(x) = Il~ll~-~J(
on X. It is known now that of the functional (l/p) II - lip at
X.
Finally, we always denote A(1 - n)p + P(1 - A).
by
1
a
3. THE
number
in
[0, l]
and
by
W,(A)
the
function
INEQUALITIES
In this section we establish some inequalities which are analogous to (1.1) in either uniformly convex or uniformly smooth Banach spaces. Applications of these inequalities will be given in the next section. Let us first begin with the following. THEOREM 1. Let p > 1 be a fixed real number. Then the functional II* I/Pis uniformly convex on the whole Banach space X if and only if X is p-uniformly convex, i.e. there exists a constant c > 0 such that ax(e) 2 c. cp for all 0 < E 5 2.
Proof. Necessity. By definition P(t) := inf
i
of uniform
convexity
of II*IIP, we have for each t > 0
~llxllP + (1 - ~)IIYIIP - IlAx + (1 - aYIIp. Wp(4 0 < A < 1, x, y E X and IIx - yll = t
> 0, I
where W,(A) = A(1 - J_)p + Ap(l - A). Let p(O) = 0. It is easy to check ,~(ct) = cPp(t) for all c, t > 0. This leads to p(t) = p(l)tP for t > 0 and hence the following inequality
IlAx + (1 - ~)YllP 5 ~llxllp + (1 -411YllP - wpmllx
- YllP
(3.1)
Inequalities
in Banach
for all 0 5 A I 1 and x, y in X, where c = p(1) > 0. From 6x(&) I
(3.1) it easily follows
that
1 - (1 - 2-PC&P)1’P > p-‘2-pc&p
for all 0 < E I 2, which shows that X is p-uniformly Sufficiency. is uniformly
1129
spaces with applications
convex.
By Beauzamy [8, lemma 2, p. 3 lo], the p-uniform convexity of X implies that 11.lip convex at center on X and hence uniformly convex on X. This completes the proof.
COROLLARY 1. Let p > 1 be a given real number. Then the following statements about a Banach space X are equivalent. (i) X is p-uniformly convex. (ii) There is a constant c > 0 such that for every x, y in X, f, in J,(x) and& in J,(y), there holds the following inequality
llxllP+ P(YLL)
IIX + YllP 2 (iii) There is a constant holds the following
(i) * (ii). From P(Y,.L)
CllYIIP.
(3.2)
cr > 0 such that for every x, y in X, f, in J,(x) (X - Y9.G - fy) 2
Proof. O
+
(3.1)
and
and
fy in J,(y) there
CIIIX- YIIP.
the subdifferential
(3.3)
inequality,
5 (Ilx + lYllP - IlxllP)/~ = (IIU - Ah + w
it follows
that
for any
+ Y)IIP - llxllP)/~
5 IIX + YllP - IMP - ~-‘w,(441YIIp. Taking the limit as A + 0 yields (3.2). (ii) * (iii). For f, E J,(x) and& E J,(y),
we have by (3.2)
llYllP 2 lMIP + P(Y - x,fJ + cllx - YllP and
IlxllP 1 IIYIIP + P(X - Y&
+ cllx - YIIP.
Summing the two inequalities above, we obtain (3.3) with cr = 2c/p. (iii) * (i). We first show that (3.2) holds. For x, y E X and 0 I t I 1, let g(t) = /Ix + ty]lp. Then, since g(t + h) - g(t) 2 hp(y, f,) for any ft in J,(x + ty) and h > 0, it follows that g!+(t) = lim (g(t + h) - g(t))/h 2 p(y, f,) and hence h-O+
11~
+
~11’ - llxllp = g(l)
- g(O) 2
’ g:(t) I’0
dt
1
(v,ft> dt
2P i; 0
1 =
P(Y,fx)
+ P
_fi - fx> dt I 0
1 1 P(Y,~,>
+ PC,I~Y~~~ s 0
=
P(Y,fx)
+ C,IIYllP,
tP-’ dt
1130
HONG-KUNXu
that is (3.2) holds with c = cr. Now for any x, y E X and A E (0, l), by (3.2) we have Il~llP 2 IlAx + (1 - 4ulP
+ (1 - A)P(X - Y,fxx+(l-x~y)
+ C,lPllY - xllP
and
IIYIIP 2 IlAx + (1 - AIYIIP + her Multiplying obtain
the two inequalities
MIP
above
by A and
+ (1 - ~)IIYIIP 2 IlAx + (1 -
1 - I, respectively,
]I* lip is uniformly
The dual versions THEOREM 1’. Let uniformly smooth
of theorem
convex
on X. The proof
1 and corollary
and then summing,
is complete.
1 are the following.
and X be a smooth Banach if and only if there exists a constant c > 0 such that
IIAX + (1 - aYllq 2 ~llxllq + (1 - ~)llYl14 1, where
we
- n)l]x - y]IP.
q > 1 be a fixed real number
for all x, y in X and 0 I 1 I
- YIIP*
~>YIIP + c,~p(~)llx - YllP
2 IlAx + (1 - A)Yll” + 2-pc,l(l This shows that
4%
- X,fL+(l-h)y) + c,(l -
space. Then X is
JYq(4cllx - Yl14
q-
(3.1’)
W,(A) = A(1 - A)’ + Aq(l - A).
COROLLARY 1’. let q > 1 be a given real number and X be a smooth Banach space. Then the following statements are equivalent. (i)’ X is q-uniformly smooth. (ii)’ There is a constant c > 0 such that for every x, y in X there holds the following inequality
IIX + Yl145 llxllq + dY, J,(x)) + 41Yl14. (iii)’ There inequality
is a constant
cr > 0 such that
for every x,y
in X there
holds
the following
(x - Y, J,(x) - J,(Y)> 5 CIIIX - Yl14. To prove theorem
1’ and corollary
l’, we need a result of Zalinescu
LEMMA 1 [7, theorem 2.21. Let X be a reflexive Banach space,f: g: X -+ R a functional. Then the following are equivalent.
f(Y) Lf(X)
(a)
+ (Y - x,x*)
[7].
X -+ R a convex functional
and
+ ‘dllx - Yll)
is valid for every x, y in X and x* in af(x). (b)
f*(y*)
sf*(x*)
+ (x,y*
- x*> + g*(lly* - x*ll)
is valid for every x in X and x*, y* in X* with x* E af(x), functionals off, g respectively. We now turn to proofs
of theorem
1’ and corollary
1’.
where
f *, g* are the conjugate
Inequalities
in Banach
1131
spaces with applications
Proof of theorem 1’. Suppose X is q-uniformly smooth. Then X* is p-uniformly p = q(q - 1))’ is the conjugate number of q); hence by corollary 1, we have + 4Y*IIp
lb* + y*P 2 IIx*IIP+ P(Y*,fx*>
for every x*, y* in X* and f,* in J,(x*). Since the conjugate functional u~dllxllq(x E n we get from the above inequality and lemma 1 that
IIX + YIP 5 IlxllQ+ 4(Y, J,(x)) for every x, y in X. In particular,
llxl145 IL
convex (here
(l/p)llx*IIp(x*
E X*) is
+ cl-qllYllq
for x, y in X and 0 I A I
1, we have
+ (1 - J.1Yl14+ q(l - A)(x - y, J,(Ax + (1 - I)y)) + d( 1 - A>“/x - yp
and
IIYIP 5
IIAX+ (1 - AlYll” + qA(y - x, J, + dhqItx- Yl14
where d = clmq. Summing
these two inequalities
yields
IlAx + (1 - ~lYl142 ~llxllq + (1 - ~)llYl14 - ~,Wllx
- Yl14
for all x, y in X and 0 I A I 1. This verifies the inequality (3.1)‘. Suppose conversely that (3,l)’ is valid. Then we have for x, Y in X and t in (0, 1) that
(Ilx + tyl14- llxllqvt = llu - f)X + tcx + Y)l14- Ilxll”>/t 2 IIX + YllQ - llxllq - ~,W-‘cllYl14. It follows
that 4(Y1 J,(x))
= ;i:_
(Ilx + trl14 - llxllq)~t
2 IIX + Yl14 - llxllq - cllYllq, that is, we have
IIX + Yl145 llxllq + 4(Y, J,(x)) + 41Yl14 for x, y in X. It follows
from lemma
1 that
I/x* + y*IP 2 IIx*IIP + p(y*,f,*> for x*, y* in X* and f,* in J,(x*). hence X is q-uniformly smooth.
+ cl-pllY*llp
This shows by corollary
1 that X* isp-uniformly
convex and
Proof of corollary 1’. The equivalence of(i)’ and (ii)’ is in fact proved. So, it remains to show the equivalence of (ii)’ and (iii)’ and this can be done similarly to the proof of that of (ii) and (iii) of corollary 1. We now apply the above results to Lp spaces. Assume tion of the equation (p - 2)F’
+ (p - l)F
1 < p < + COand tp is the unique
- 1 = 0,
Let cp = (1 + t,P-l)(l + tp)lmp. Since Lp is p-uniformly p > 2, whereas 2-uniformly convex and p-uniformly following.
solu-
o
convex and 2-uniformly smooth if smooth if 1 < p I 2, we have the
1132
HONG-KUN Xu
COROLLARY 2. (i) If 2 < p < + co, then we have for all x, y in L!’ and 0 I A I 1 5 ~llxlIP + (1 - ~)lIYIP - ~p(mpllx
IlAx + (1 - M”
- YIP.
1>Ilx-
IlAx + (1 - A)Yl12 2 ~11xl12+ (1 - ~)IlYl12 - A(1 - AMP IIX + YllP 2 llxllP + P(Y, J,(x)>
(3.4) Yl12.
+ CpllYllP.
(3.4)’ (3.5)
IIX + Yl12 5 llxl12 + %YP J(x)) + (P - l)llYl12.
(3.5)’
- J,(x))
2 2P-1cpllx - YIIP.
(3.6)
(x - Y,J(X) - J(Y)) 5 (P - 1)Ilx - YI12.
(3.6)’
(x - y, J,(x)
(ii)If1
- A(1 - m
IlAx + (1 - J)Yl14 2 ~llxllq + (1 - ~)IlYl14 -
- l)llx - YI12.
~qW*llx -
Yl14.
IIX + Yl12 2 llxl12 + %Y, J(x)) + CqllYl12. IIX + Yl14 5 llxllq + 4(Y, J,(x))
- J,(Y)>
5
(3.7)’ (3.8) (3.8)’
+ cqlIY114.
l)llx - Yl12. w’cqllx- Yl14.
(x - Y, J(x) - J(Y)) 2 (4 (x - Y, J,(x)
(3.7)
(3.9) (3.9)’
Proof. Inequality (3.4) is due essentially to Lim [3]. The proof of Smarzewski [6] is false and the correct proof can be found in Lim [9]. Inequality (3.7) is contained in [4]. Proofs of implications (3.4) * (3.5) * (3.6) and (3.7) * (3.8) * (3.9) are the same as those in corollaries 1 and 1’ and therefore omitted here. It remains to show (3.4)‘-(3.9)‘. Since all these inequalities can be proved by the dual method, we only prove the inequality (3.8)’ (any other proof is similar). To this end, we let p = q(q - 1))’ be the conjugate number of q and let f(x) = p-lIIxJlp. Then f*(x*) = q-‘llx*11*. Noticing J,(x) = df(x), we see that (3.5) is equivalent to the following inequality: in Lp. for all x, y f(x + Y) 2 f(x) + (Y, J,(x)) + Cpf(Y)
Since (cf)*(x*)
= cf *(c-lx*)
IIX+ Yl14 5
for any constant c > 0, it follows from lemma 1 that
llxllq + dY, J,(x))
+ c~-qllYllq
for all x, y
in Lq,
where cp is given as in (3.5). It thus remains only to show cd-” = c4. One easily sees that t, is the solution of the equation (q - 2)F’
if and only if t,“(p-l)
+ (q - l)F2
- 1 = 0,
o
is the solution of the equation (p - 2)F’
+ (p - 1y-2
- 1 = 0,
o
This implies by the uniqueness of the solutions of the two equations above that t~‘(p-‘) From this, it follows that 1-q CP
=
(1
+
t;-l)l-4/(1
This completes the proof.
+
tP)(P-‘)(‘-4)
=
(1 + t,)/(l
+ t,y
= (1 + fq4_‘)/(1 + t,y-’
= tp. = cq.
Inequalities
in Banach
1133
spaces with applications
Remark 1. All constants appeared in the inequalities of corollary 2 (e.g. the cP in (3.4)-(3.6) p - 1 in (3.4)‘-(3.6)‘, etc.) are the best possible (cf. Lim [9] and Lim et al. [4]). Remark 2. In [lo, Chapter 3, p. 911, it is asked whether such that for all x, Y in Lp, we have when p > 2
there are constants
and
cr , c2, CT;,c; > 0
(x - Y, J(x) - J(Y))
2 c1IIx - Yl?
(3.10)
- J,(Y)>
5 c,llx - YIP
(3.11)
and (x - Y1 J,(x) and when
1< p I 2
(x - Y, J(x) - J(Y)) 5 41x - Yl12
(3.10)’
2 c;llx - YIIP.
(3.11)’
and (x - Y, J,(x) Our corollaries
- J,(Y))
1 and 1’ show that no such constants
exist.
We have seen from theorem 1 that the inequality (3.1) (or its equivalent version for every x, y in X iff X is p-uniformly convex; however, when x, y are restricted sets of X, we have the following characteristic of uniform convexity of X.
(3.2)) holds in bounded
THEOREM 2. Let p > 1 and r > 0 be two fixed real numbers. Then a Banach space X is uniformly convex if and only if there is a continuous, strictly increasing and convex function g: R+ + R+, g(0) = 0, such that
llnx + (1 - 4Yll” 5 J.w for all x, Y in B, and 0 I ,I I
- Yll)
(3.12)
1.
Proof. Necessity. Since X is uniformly on B, and hence P(t) .= inf
+ (1 - ~)IIYIIP - ~,m(llx
convex,
by [7, theorem
4.11, ]I* lip is uniformly
convex
JllxllP + (1 - ~)llYllP - IlAx + (1 - 4Yll”. W,(A) x,yEB,withI/x-Yll=tandO<,I
>0
for all 0 < t 5 2r.
1 Set p(O) = 0. The uniform convexity of X implies that ,Uis continuous and strictly increasing on [0, 2r]. Take g = CO(D), i.e. g is the maximal convex function majorized by ,u. Then g satisfies (3.12); for details see Zalinescu [7, Appendix]. Sufficiency.
(3.12) clearly implies dx(&) 1 1 - (1 - (2r)-pg(re))1’p
which shows that X is uniformly The following
convex
is the dual version
> (2r)-pp-1g(re)
and the proof
of theorem
2.
is complete.
> 0,
1134
HONG-KUN Xu
THEOREM 2’. Let q > 1 and r > 0 be two fixed real numbers. Then a smooth Banach space X is uniformly smooth if and only if there exists a continuous, strictly increasing and convex function g*: Ri + R+, g*(O) = 0, such that
4IIYIP - Wq@)g*(llx - A)
llnx + (1 - J.)Yl14 2 ~llxl14 + (1 forallx,yinB,andOIAI
1.
COROLLARY3. let p > 1 and r > 0 be two fixed real numbers the following are equivalent. (i) X is uniformly convex. (ii) There is a continuous, strictly increasing and convex such that
IIX+ YllP2 llxllP+ P(Y,L)
for every x, y in B, and f, in Jp (x). (iii) There is a continuous, strictly such that
increasing
(x - r,f, for every x, y in B, , f, in Jp (x) and
+
space. Then
function
g: R+ + R+, g(0) = 0,
gdlvll)
(3.13)
and convex
- fy) 1
and X be a Banach
function
g: R+ -+ Rf,
g(0) = 0,
dllx - A)
(3.14)
fy in Jp (y).
COROLLARY2’. Let q > 1 and r > 0 be two fixed real numbers and X be a smooth Banach space. Then the following are equivalent. (i)’ X is uniformly smooth. (ii)’ There is a continuous, strictly increasing and convex function g*: Rf -+ R+, g*(O) = 0, such that
IIX+ YIP5 llxl14 + 4(Y,J,(d) + s*(llull>
for all x, y in B,. (iii)’ There is a continuous, such that
strictly increasing (x - Y, J,(x)
and convex function
- J,(Y)> 5
g*: R+ + R+, g*(O) = 0,
g*cllx- A>
for all x, y in B,. 4. APPLICATIONS In this section we give applications of the established inequalities in the previous section. We first prove an existence theorem for fixed points of uniformly Lipschitzian mappings in Banach spaces. Let K be a nonempty closed convex subset of a Banach space X and CY> 0 a constant. A mapping T: K + K is said to be uniformly ol-Lipschitzian if IIT”x - T”_Y/~5 &
- ~11
for each integer n L 1 and all x,y in K. This class of mappings has been studied by many authors (see [3-5, 11-141). Our theorem 3 below seems new so far. To prove it, we recall that the normal structure coefficient N(X) of X is defined (cf. Bynum [20]) by N(X) = inf( diam(K)/r, K a bounded
(K) : convex
subset of X consisting
of more than one point},
Inequalities
in Banach
spaces with applications
= sup1 /Ix - y/I : x, y in K] is the diameter
where diam(K)
1135
of K and
rK(K) = inf sup IIx - yl] XEKYEK is the Chebyshev radius of K relative to itself. X is said to have uniformly normal structure if N(X) > 1. It is known that a uniformly convex Banach has uniformly normal structure and for a Hilbert space H, N(H) = 2l”. THEOREM 3. Let p > 1 and let X be a p-uniformly convex Banach space, K a nonempty closed convex subset of X, T: K + K a uniformly cz-Lipschitzian mapping. Suppose that there is an x, in K for which (T”x,) is bounded and that (Y < [t(l + (1 + ~cJV)“~)]“~, where N is the normal structure coefficient of X and c is the constant given in inequality (3.1). Then T has a fixed point, i.e. there is a z in K such that T(z) = z. To prove the theorem,
we need the following
LEMMA 2. Let K, X be as in theorem unique point z in K such that
lemma.
3 and let lx,] be a bounded
sequence.
Then there exists a
lim sup ]]x, - zllP I lim sup Ilx, - x]lp - cllx - zllp n+OO n-m for every x in K, where c is the constant
given in (3.1).
Proof. Let r(x) = lim sup [lx, - xljP, x E X. By theorem 1, r(x) is uniformly convex on X and n-co therefore, there is a unique point z in K (called the asymptotic center of the sequence lx,) in K) such that r(z) = inf r(x). It follows from inequality (3.1) that XCK
r(z + A(x - z)) 5 k(x)
+ (1 - A)r(z) - W,(A)cllx - zllp
for x in K and 0 5 A I 1. Noticing r(z + A(x - z)) 2 r(z) for x in K and 0 I A I that 0 I lim (r(z + A(x - z)) - r(z))/A I r(x) - r(z) - cllx - zIIp,
1, we derive
X-O+
and the desired
inequality
follows.
Proof of theorem 3. Since (T”x,) is bounded (and hence (T”x} is bounded for any x in K), by lemma 2, we can inductively construct a sequence (x,),~ 1 in K as follows: for each integer center of the sequence [T”x,) in K. Let m 2 0, x,+~ is the asymptotic r,
= lim sup IIT*x, n+m
Then by Lim [ 15, theorem
- x,,
1II
and
R,
= sup ]]x, - T”x,)). n21
11, we have
r, I N-’ diam(( T”x,),
Z 1) I N-‘aR,,
m = 1,2, . . . .
1136
HONG-KUNXv
where N is the normal structure coefficient of X. For each fixed m L 1 and all n > k L 1, we have from (3.1) Il%l+, + (1 - W-%+,
- ~“XX
+
cw,~4Il&n+,- h?l+#
5 4I&+1 - T”x,lIP + (1 - A)]]T”x, - TkX,+# 5
an,l - T”x,ljP + (1
- A)(wpIIx,+I - T”-kX,II?
Taking the limit as n + 00 leads to r; + cW,(A)l]xm+t - TkX,+#
I (A + (1 - A)&)/$.
It then follows that %+I 5
(1 - A)@ - 1) ,.P cW,V)
5
m
(I
-
‘)@’
-
I)
N-PUP
. RP
ItI*
cW,(4
Letting I + 1, we conclude that l)l’pR,
R m+l I (crP(d - l)N-Pc-
where A = (cY~((Y” - l)N-pc-l)l’p
=: A * R,,
m = 1,2, . . . .
< 1 by assumption of the theorem. Since
II&+, - x,,,ll5 r,,,
+ R, I 2R, I ... I 2A”-‘RI,
it follows that lx,) is Cauchy. Let z = lim x,. Then we have m+m
llz - Tzll 5 llz - m?JI+ Il&?l - RAI + IIR?I - Ml I (1 + a)llx, - z]] + R,,, + 0,
asm + co,
and hence Tz = z. The proof is complete. Recently, Prus [16] calculated that N(Lp) = min(21’P, 21’q), where q = p(p - I)-’ is the conjugate number of p. Therefore, we have the following result. COROLLARY 4. Let K be a closed convex subset of Lp(l < p < 00) and T: K -, K a uniformly a-Lipschitzian mapping such that {T”x,) is bounded for some x0 in K. Suppose (Y< (3 + +(l + 4(p - 1)2(p-1)‘p)1’2)1’2 if 1 < p I 2 and (Y< (i + *(I + 8~~)~‘~)~‘~if p > 2 (here cp is as in corollary 2). Then T has a fixed point. Remark
3. For generalizations author [14].
of theorem 3 and corollary 4 to semigroups see the present
Next, we give estimates of moduli of continuity of nearest projections in Banach spaces. It is well known that if Mis a nonempty closed convex subset of a uniformly convex Banach space X, then for every x in X, there is a unique element m in M (called the nearest point of x in M) such that for all y in M. llx - mll 5 Ilx - AI
Inequalities in Banach spaces with applications
1137
Let P,(x) = m. Then we define a mapping PM.- X + A4 which is called the nearest point (or metric) projection of X onto M. If X is assumed in addition to be smooth, then m E M is the nearest point of x E X in M if and only if (m - y, J(x - m)) L 0
for all y in M.
(4.1)
We now know (cf. [17]) that if X is uniformly convex, PM is continuous. However, unfortunately, one is unable to give an estimate of the modulus of continuity of PM. Recently, Bjornestal [18] and Abatzoglou [19] discussed locally this problem in a Banach space which is both uniformly convex and uniformly smooth. The following theorem improves to some extent upon theirs. THEOREM 4. Let X be a Banach space which is both uniformly convex and uniformly smooth, Ma nonempty closed convex subset of X, P: X + A4 the nearest point projection of X onto M, and Q = 1 - P (I is the identity operator on X). Then for each constant r > 0, there exists a strictly increasing continuous function p: R+ + R+, p(0) = 0, such that
llpx -
PYll 5
~u(llX - Yll)
and
IlQx - QyIi 5 ~(llx - ~11)
for all x, y in B, . Proof. Since P is bounded on B,, by corollaries 3 and 3’, there are strictly increasing continuous functions g, g*: R+ + R+, g(0) = g*(O) = 0, such that (r’ is chosen so large that x-y,xPx,PxPYEB,, wheneverx,yEB,)
IIX + Yl122
llxl12 + 2(Y,
J(x)) + gdlrll)
for x, y in B,, ,
IIX + Yl125 llxl12+ 3Y7 JO) + g*(llrll)
for x, y in B,, .
(4.2) (4.3)
Combining (4.2) and (4.3), we obtain for all x,y in B,, [Ix - Py112 = [1(x - Px) + (Px - Py)II2 L IIX - Pxl12 + 2(Px - Py, J(x - Px)) + g(IIPx - Py(l) 2 I/x - pxl12 + g(llPx - PYIO.
Similarly, we have IIY - pxl12 2 IIY - PYl12 +
gm -
PYII).
It follows that %llPx
- PYII) 5
IIX - PYl12 + IIY - pAI2 - Ilx - pxl12- IIY - PYl12.
Using (4.3), we also have as above IIX - PYl12 = II@ - Y> + (Y - PY)l12 5 IIY - PYl12 + ax - Y, J(Y - PY)> + g*(llx - Yll) and Ily - Pxl12I
IIX- Pxl12 + ZY - x, Ax - Px)> + g*(lly - XII).
(4.4)
1138
HONG-KUN Xu
This, together
with (4.4), implies
g(lPx
- WI)
5 (x - Y, J(Y - 0)
- J(x - Px)> + g*cllx - yll)
=
JQx) + (Px -
QY, JQY -
+ (PY -
Px, 4x
dQx
-
-
Py, J(y -
Px)> + g*(llx -
Py))
yl()
- t2.d) + g*(llx - _dl),
and thus g(l/Px Now set ,D = g -lg*.
- Pub
+
Then the desired
g(llQx - Qd) 5 g*dlx - ~4).
conclusions:
Ilpx- &II 5 Pc(llX - YII) follows.
The proof
Acknowledgement-1 manuscript.
IlQx - Qd 5 &
and
- rib
is complete. am extremely
grateful
to the referee
for his careful
reading
and good
suggestions
for this
REFERENCES zeros of accretive sets in Banach spaces, Nonlinear Analysis 2, 1. REICH S., An iterative procedure for constructing 85-92 (1978). 2. POFFALD E. I. & REICH S., An incomplete Cauchy problem, J. m&h. Analysis Applic. 113, 514-543 (1986). 3. LIM T. C., Fixed point theorems for uniformly Lipschitzian mappings in Lp spaces, Nonlinear Analysis 7, 555-563
(1983). to fixed point theory and approximation 4. LIM T. C., Xu H. K. & Xu Z. B., An Lp inequality and its application theory, J. upprox. Theory (to appear). and centers in uniformly convex spaces, J. math. 5. PRUS B. & SMARZEWSKIR., Strongly unique best approximation
Analysis Applic. 121, lo-21 (1987). 6. SMARZEWSKI R., Strongly unique minimization of functionals in Banach spaces with applications to theory of approximations and fixed points, J. math. Analysis Applic. 115, 155-172 (1986). 7. ZALINESCU C., On uniformly convex functions, J. math. Analysis Applic. 95, 344-374 (1983). 8. BEAUZAMYB., Introduction to Banach Spaces and Their Geometry. North-Holland, Amsterdam (1985). 9. LIM T. C., On some Lp inequalities in best approximation theory (to appear). 10. You Z. Y., Nonlinear Analysis. Xi’an Jiaotong University Press, Xi’an, Shaaxi Province, People’s Republic of China (1990). mappings in spaces with uniformly normal struc11. CASINI E. & MALUTA E., Fixed points of uniformly Lipschitzian ture, Nonlinear Analysis 9, 103-108 (1985). whose iterates have uniform Lipschitz con12. GOEBEL K. & KIRK W. A., A fixed point theorem for transformations stant, Studia math. 47, 135-140 (1973). 13. LIFSCHITZ E. A., Fixed point theorems for operators in strongly convex spaces, Voronez Gos. Univ. Trudy Math. Fak. 16, 23-28 (1975). (In Russian.) 14. Xv H. K., Fixed point theorems for uniformly Lipschitzian semigroups in uniformly convex spaces, J. math.
Analysis Applic. 152, 391-398 (1990). 15. LIM T. C., On the normal
structure
coefficient
and the bounded
sequence
coefficient,
Proc. Am. math. Sot. 88,
262-264 (1983). 16. PRUS B., On Bynum’s fixed point theorem, Atti Semin. mat. fis. Univ. Modena (to appear). 17. GEOBEL K. & REICH S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984). 18. BJORNESTALB. O., Local Lipschitz continuity of the metric projection operator, in Approximation Theory, Bunuch Center Publ. No. 4, pp. 43-53. PWN, Warsaw (1979). 19. ABATZOCLOU T., Continuity of metric projection in uniformly convex and uniformly smooth Banach spaces, J.
approx. Theory 39, 299-307 (1983). 20. BYNUM W. L., Normal structure coefficients for Banach spaces, Pucif. J. math. 86, 427-436 (1980). 21. REICH S., Constructive techniques for accretive and monotone operators, in Applied Nonlinear Analysis, pp. 335-345.
Academic
Press,
New York (1979).
Methods
& Applicorrons,
Vol.
16, No.
12, pp. 1127-1138,
0
INEQUALITIES
0362-546X/91 $3.00+ .W.l 1991 Pergamon Press plc
IN BANACH SPACES WITH APPLICATIONS Xv
HONG-KUN Department
1991.
of Mathematics,
East China Institute of Chemical People’s Republic of China
(Received 21 May 1990; received in revised form
Technology,
Shanghai
200237,
10 October 1990; received for publication 6 November
1990)
Key words and phrases: Inequality, uniformly convex Banach space, uniformly smooth Banach space, uniformly convex functional, fixed point, uniformly Lipschitzian mapping, nearest point projection.
1. INTRODUCTION
IT IS well known identity:
that many problems
in a Hilbert
space H are solved by applying
IlAx + (1 - A)YI12= a412 + (1 - ~)llYl12 - A(1 - Mx
the following
- Yl12
(1.1)
for all x, y in H and 0 5 I 5 1. Therefore, one natural method to solve problems in a Banach space Xis to establish equalities and (usually) inequalities in Xanalogous to (1.1). This is in fact shown by some authors’ recent work, see for example, Reich [l], Poffald and Reich [2], Lim [3], Lim et af. [4], Prus and Smarzewski [5], Smarzewski [6], and Reich [21]. The purpose of this paper is to continue carrying out investigations in this line. More precisely, in Section 3, we establish some inequalities analogous to (1 .l) in either uniformly convex or uniformly smooth Banach spaces. As corollaries, we obtain new characteristics of p-uniform convexity and q-uniform smoothness of a Banach space X via the functionals I]. lip and II * llq respectively. We give in Section 4 applications of the inequalities obtained in Section 3 to fixed points of uniformly Lipschitzian mappings and to best approximation theory. 2. PRELIMINARIES
Throughout for its dual, R radius r. For x smoothness of &(E)
this paper, X will always stand for a real Banach space with norm II * /I and X* for the field of real numbers, and B, for the closed ball with center zero and in X and x* in X*, (x, x*> is the value of x* at x. The moduli of convexity and X are defined respectively by
= inf(l
-
11+(x + y))I : llxll = llyll = 1 and [Ix - y]I = E],
OIEI2,
and &Y(f) = SUPMIX
+ Yll + IIX - Yll) - 1 : llxll = 1, IIYII = 4,
t > 0.
X is said to be uniformly convex if dx(e) > 0 for all 0 < E I 2 and uniformly smooth if li~i px(t)/t = 0. Let p, q > 1 be real numbers. Then X is said to be p-uniformly convex (resp. q-uniformly smooth) if there is a constant c > 0 such that ax(e) 2 cY (resp. px(t) I ctq). Let f: X + i? := R U { + a~)be a proper functional and D a nonempty convex subset of X. Thenf is said to be convex on D if f(lx
+ (1 - AlY) 5 VW 1127
+ (1 - ~l_fCY)
HONG-KUN Xu
1128
for all 0 5 13 I 1 and x,y in D. f is said to be uniformly convex on D [7] if there exists a function ,u:R+ := [0, 00) --t R+ with p(t) = 0 if and only if t = 0 such that f(Ax + (1 - A)Y) 5 U(x)
+ (1 - J.)f(Y) - A(1 - ~Mllx
- Yll)
(2.1)
for all 0 5 A 5 1 and x, y in D. If the inequality (2.1) is valid for all x, y in D when A = 5, then fis said to be uniformly convex at center. Zalinescu [7, p. 3521 remarked that for a convex function f, f is uniformly convex on D if and only iffis uniformly convex at center on D. We now recall the conjugate function f *: X * + 17 of a convex function f is defined by f *(x*) = sup((x, x*> - f(x): x E X), It is well known that if f(x) = (l/p)llxllp(p > l), l/q + l/p = 1, and that for any constant c > 0, (cf)*(x*) subdifferential df: X + X* off is defined by
x* E x*. then f*(x*) = cf *(c-lx*).
= (l/q)llx*llq, where We also recall that the
af(x) = (x*E X* :f(y) 2 f(x) + (y - x, x*) for all y E X). On the other hand, with each p > 1 we can associate into X* defined as follows J,(x) In particular,
= {x* E X*: (x, x*)
J := J2 is called
the (generalized)
duality
map Jp from X
= llxll” and IIx*)l = Ilxllp-‘].
the normalized
duality
map
x1 f or x # 0 and that J,(x) is the subdifferential J,(x) = Il~ll~-~J(
on X. It is known now that of the functional (l/p) II - lip at
X.
Finally, we always denote A(1 - n)p + P(1 - A).
by
1
a
3. THE
number
in
[0, l]
and
by
W,(A)
the
function
INEQUALITIES
In this section we establish some inequalities which are analogous to (1.1) in either uniformly convex or uniformly smooth Banach spaces. Applications of these inequalities will be given in the next section. Let us first begin with the following. THEOREM 1. Let p > 1 be a fixed real number. Then the functional II* I/Pis uniformly convex on the whole Banach space X if and only if X is p-uniformly convex, i.e. there exists a constant c > 0 such that ax(e) 2 c. cp for all 0 < E 5 2.
Proof. Necessity. By definition P(t) := inf
i
of uniform
convexity
of II*IIP, we have for each t > 0
~llxllP + (1 - ~)IIYIIP - IlAx + (1 - aYIIp. Wp(4 0 < A < 1, x, y E X and IIx - yll = t
> 0, I
where W,(A) = A(1 - J_)p + Ap(l - A). Let p(O) = 0. It is easy to check ,~(ct) = cPp(t) for all c, t > 0. This leads to p(t) = p(l)tP for t > 0 and hence the following inequality
IlAx + (1 - ~)YllP 5 ~llxllp + (1 -411YllP - wpmllx
- YllP
(3.1)
Inequalities
in Banach
for all 0 5 A I 1 and x, y in X, where c = p(1) > 0. From 6x(&) I
(3.1) it easily follows
that
1 - (1 - 2-PC&P)1’P > p-‘2-pc&p
for all 0 < E I 2, which shows that X is p-uniformly Sufficiency. is uniformly
1129
spaces with applications
convex.
By Beauzamy [8, lemma 2, p. 3 lo], the p-uniform convexity of X implies that 11.lip convex at center on X and hence uniformly convex on X. This completes the proof.
COROLLARY 1. Let p > 1 be a given real number. Then the following statements about a Banach space X are equivalent. (i) X is p-uniformly convex. (ii) There is a constant c > 0 such that for every x, y in X, f, in J,(x) and& in J,(y), there holds the following inequality
llxllP+ P(YLL)
IIX + YllP 2 (iii) There is a constant holds the following
(i) * (ii). From P(Y,.L)
CllYIIP.
(3.2)
cr > 0 such that for every x, y in X, f, in J,(x) (X - Y9.G - fy) 2
Proof. O
+
(3.1)
and
and
fy in J,(y) there
CIIIX- YIIP.
the subdifferential
(3.3)
inequality,
5 (Ilx + lYllP - IlxllP)/~ = (IIU - Ah + w
it follows
that
for any
+ Y)IIP - llxllP)/~
5 IIX + YllP - IMP - ~-‘w,(441YIIp. Taking the limit as A + 0 yields (3.2). (ii) * (iii). For f, E J,(x) and& E J,(y),
we have by (3.2)
llYllP 2 lMIP + P(Y - x,fJ + cllx - YllP and
IlxllP 1 IIYIIP + P(X - Y&
+ cllx - YIIP.
Summing the two inequalities above, we obtain (3.3) with cr = 2c/p. (iii) * (i). We first show that (3.2) holds. For x, y E X and 0 I t I 1, let g(t) = /Ix + ty]lp. Then, since g(t + h) - g(t) 2 hp(y, f,) for any ft in J,(x + ty) and h > 0, it follows that g!+(t) = lim (g(t + h) - g(t))/h 2 p(y, f,) and hence h-O+
11~
+
~11’ - llxllp = g(l)
- g(O) 2
’ g:(t) I’0
dt
1
(v,ft> dt
2P i; 0
1 =
P(Y,fx)
+ P
_fi - fx> dt I 0
1 1 P(Y,~,>
+ PC,I~Y~~~ s 0
=
P(Y,fx)
+ C,IIYllP,
tP-’ dt
1130
HONG-KUNXu
that is (3.2) holds with c = cr. Now for any x, y E X and A E (0, l), by (3.2) we have Il~llP 2 IlAx + (1 - 4ulP
+ (1 - A)P(X - Y,fxx+(l-x~y)
+ C,lPllY - xllP
and
IIYIIP 2 IlAx + (1 - AIYIIP + her Multiplying obtain
the two inequalities
MIP
above
by A and
+ (1 - ~)IIYIIP 2 IlAx + (1 -
1 - I, respectively,
]I* lip is uniformly
The dual versions THEOREM 1’. Let uniformly smooth
of theorem
convex
on X. The proof
1 and corollary
and then summing,
is complete.
1 are the following.
and X be a smooth Banach if and only if there exists a constant c > 0 such that
IIAX + (1 - aYllq 2 ~llxllq + (1 - ~)llYl14 1, where
we
- n)l]x - y]IP.
q > 1 be a fixed real number
for all x, y in X and 0 I 1 I
- YIIP*
~>YIIP + c,~p(~)llx - YllP
2 IlAx + (1 - A)Yll” + 2-pc,l(l This shows that
4%
- X,fL+(l-h)y) + c,(l -
space. Then X is
JYq(4cllx - Yl14
q-
(3.1’)
W,(A) = A(1 - A)’ + Aq(l - A).
COROLLARY 1’. let q > 1 be a given real number and X be a smooth Banach space. Then the following statements are equivalent. (i)’ X is q-uniformly smooth. (ii)’ There is a constant c > 0 such that for every x, y in X there holds the following inequality
IIX + Yl145 llxllq + dY, J,(x)) + 41Yl14. (iii)’ There inequality
is a constant
cr > 0 such that
for every x,y
in X there
holds
the following
(x - Y, J,(x) - J,(Y)> 5 CIIIX - Yl14. To prove theorem
1’ and corollary
l’, we need a result of Zalinescu
LEMMA 1 [7, theorem 2.21. Let X be a reflexive Banach space,f: g: X -+ R a functional. Then the following are equivalent.
f(Y) Lf(X)
(a)
+ (Y - x,x*)
[7].
X -+ R a convex functional
and
+ ‘dllx - Yll)
is valid for every x, y in X and x* in af(x). (b)
f*(y*)
sf*(x*)
+ (x,y*
- x*> + g*(lly* - x*ll)
is valid for every x in X and x*, y* in X* with x* E af(x), functionals off, g respectively. We now turn to proofs
of theorem
1’ and corollary
1’.
where
f *, g* are the conjugate
Inequalities
in Banach
1131
spaces with applications
Proof of theorem 1’. Suppose X is q-uniformly smooth. Then X* is p-uniformly p = q(q - 1))’ is the conjugate number of q); hence by corollary 1, we have + 4Y*IIp
lb* + y*P 2 IIx*IIP+ P(Y*,fx*>
for every x*, y* in X* and f,* in J,(x*). Since the conjugate functional u~dllxllq(x E n we get from the above inequality and lemma 1 that
IIX + YIP 5 IlxllQ+ 4(Y, J,(x)) for every x, y in X. In particular,
llxl145 IL
convex (here
(l/p)llx*IIp(x*
E X*) is
+ cl-qllYllq
for x, y in X and 0 I A I
1, we have
+ (1 - J.1Yl14+ q(l - A)(x - y, J,(Ax + (1 - I)y)) + d( 1 - A>“/x - yp
and
IIYIP 5
IIAX+ (1 - AlYll” + qA(y - x, J,
where d = clmq. Summing
these two inequalities
yields
IlAx + (1 - ~lYl142 ~llxllq + (1 - ~)llYl14 - ~,Wllx
- Yl14
for all x, y in X and 0 I A I 1. This verifies the inequality (3.1)‘. Suppose conversely that (3,l)’ is valid. Then we have for x, Y in X and t in (0, 1) that
(Ilx + tyl14- llxllqvt = llu - f)X + tcx + Y)l14- Ilxll”>/t 2 IIX + YllQ - llxllq - ~,W-‘cllYl14. It follows
that 4(Y1 J,(x))
= ;i:_
(Ilx + trl14 - llxllq)~t
2 IIX + Yl14 - llxllq - cllYllq, that is, we have
IIX + Yl145 llxllq + 4(Y, J,(x)) + 41Yl14 for x, y in X. It follows
from lemma
1 that
I/x* + y*IP 2 IIx*IIP + p(y*,f,*> for x*, y* in X* and f,* in J,(x*). hence X is q-uniformly smooth.
+ cl-pllY*llp
This shows by corollary
1 that X* isp-uniformly
convex and
Proof of corollary 1’. The equivalence of(i)’ and (ii)’ is in fact proved. So, it remains to show the equivalence of (ii)’ and (iii)’ and this can be done similarly to the proof of that of (ii) and (iii) of corollary 1. We now apply the above results to Lp spaces. Assume tion of the equation (p - 2)F’
+ (p - l)F
1 < p < + COand tp is the unique
- 1 = 0,
Let cp = (1 + t,P-l)(l + tp)lmp. Since Lp is p-uniformly p > 2, whereas 2-uniformly convex and p-uniformly following.
solu-
o
convex and 2-uniformly smooth if smooth if 1 < p I 2, we have the
1132
HONG-KUN Xu
COROLLARY 2. (i) If 2 < p < + co, then we have for all x, y in L!’ and 0 I A I 1 5 ~llxlIP + (1 - ~)lIYIP - ~p(mpllx
IlAx + (1 - M”
- YIP.
1>Ilx-
IlAx + (1 - A)Yl12 2 ~11xl12+ (1 - ~)IlYl12 - A(1 - AMP IIX + YllP 2 llxllP + P(Y, J,(x)>
(3.4) Yl12.
+ CpllYllP.
(3.4)’ (3.5)
IIX + Yl12 5 llxl12 + %YP J(x)) + (P - l)llYl12.
(3.5)’
- J,(x))
2 2P-1cpllx - YIIP.
(3.6)
(x - Y,J(X) - J(Y)) 5 (P - 1)Ilx - YI12.
(3.6)’
(x - y, J,(x)
(ii)If1
- A(1 - m
IlAx + (1 - J)Yl14 2 ~llxllq + (1 - ~)IlYl14 -
- l)llx - YI12.
~qW*llx -
Yl14.
IIX + Yl12 2 llxl12 + %Y, J(x)) + CqllYl12. IIX + Yl14 5 llxllq + 4(Y, J,(x))
- J,(Y)>
5
(3.7)’ (3.8) (3.8)’
+ cqlIY114.
l)llx - Yl12. w’cqllx- Yl14.
(x - Y, J(x) - J(Y)) 2 (4 (x - Y, J,(x)
(3.7)
(3.9) (3.9)’
Proof. Inequality (3.4) is due essentially to Lim [3]. The proof of Smarzewski [6] is false and the correct proof can be found in Lim [9]. Inequality (3.7) is contained in [4]. Proofs of implications (3.4) * (3.5) * (3.6) and (3.7) * (3.8) * (3.9) are the same as those in corollaries 1 and 1’ and therefore omitted here. It remains to show (3.4)‘-(3.9)‘. Since all these inequalities can be proved by the dual method, we only prove the inequality (3.8)’ (any other proof is similar). To this end, we let p = q(q - 1))’ be the conjugate number of q and let f(x) = p-lIIxJlp. Then f*(x*) = q-‘llx*11*. Noticing J,(x) = df(x), we see that (3.5) is equivalent to the following inequality: in Lp. for all x, y f(x + Y) 2 f(x) + (Y, J,(x)) + Cpf(Y)
Since (cf)*(x*)
= cf *(c-lx*)
IIX+ Yl14 5
for any constant c > 0, it follows from lemma 1 that
llxllq + dY, J,(x))
+ c~-qllYllq
for all x, y
in Lq,
where cp is given as in (3.5). It thus remains only to show cd-” = c4. One easily sees that t, is the solution of the equation (q - 2)F’
if and only if t,“(p-l)
+ (q - l)F2
- 1 = 0,
o
is the solution of the equation (p - 2)F’
+ (p - 1y-2
- 1 = 0,
o
This implies by the uniqueness of the solutions of the two equations above that t~‘(p-‘) From this, it follows that 1-q CP
=
(1
+
t;-l)l-4/(1
This completes the proof.
+
tP)(P-‘)(‘-4)
=
(1 + t,)/(l
+ t,y
= (1 + fq4_‘)/(1 + t,y-’
= tp. = cq.
Inequalities
in Banach
1133
spaces with applications
Remark 1. All constants appeared in the inequalities of corollary 2 (e.g. the cP in (3.4)-(3.6) p - 1 in (3.4)‘-(3.6)‘, etc.) are the best possible (cf. Lim [9] and Lim et al. [4]). Remark 2. In [lo, Chapter 3, p. 911, it is asked whether such that for all x, Y in Lp, we have when p > 2
there are constants
and
cr , c2, CT;,c; > 0
(x - Y, J(x) - J(Y))
2 c1IIx - Yl?
(3.10)
- J,(Y)>
5 c,llx - YIP
(3.11)
and (x - Y1 J,(x) and when
1< p I 2
(x - Y, J(x) - J(Y)) 5 41x - Yl12
(3.10)’
2 c;llx - YIIP.
(3.11)’
and (x - Y, J,(x) Our corollaries
- J,(Y))
1 and 1’ show that no such constants
exist.
We have seen from theorem 1 that the inequality (3.1) (or its equivalent version for every x, y in X iff X is p-uniformly convex; however, when x, y are restricted sets of X, we have the following characteristic of uniform convexity of X.
(3.2)) holds in bounded
THEOREM 2. Let p > 1 and r > 0 be two fixed real numbers. Then a Banach space X is uniformly convex if and only if there is a continuous, strictly increasing and convex function g: R+ + R+, g(0) = 0, such that
llnx + (1 - 4Yll” 5 J.w for all x, Y in B, and 0 I ,I I
- Yll)
(3.12)
1.
Proof. Necessity. Since X is uniformly on B, and hence P(t) .= inf
+ (1 - ~)IIYIIP - ~,m(llx
convex,
by [7, theorem
4.11, ]I* lip is uniformly
convex
JllxllP + (1 - ~)llYllP - IlAx + (1 - 4Yll”. W,(A) x,yEB,withI/x-Yll=tandO<,I
>0
for all 0 < t 5 2r.
1 Set p(O) = 0. The uniform convexity of X implies that ,Uis continuous and strictly increasing on [0, 2r]. Take g = CO(D), i.e. g is the maximal convex function majorized by ,u. Then g satisfies (3.12); for details see Zalinescu [7, Appendix]. Sufficiency.
(3.12) clearly implies dx(&) 1 1 - (1 - (2r)-pg(re))1’p
which shows that X is uniformly The following
convex
is the dual version
> (2r)-pp-1g(re)
and the proof
of theorem
2.
is complete.
> 0,
1134
HONG-KUN Xu
THEOREM 2’. Let q > 1 and r > 0 be two fixed real numbers. Then a smooth Banach space X is uniformly smooth if and only if there exists a continuous, strictly increasing and convex function g*: Ri + R+, g*(O) = 0, such that
4IIYIP - Wq@)g*(llx - A)
llnx + (1 - J.)Yl14 2 ~llxl14 + (1 forallx,yinB,andOIAI
1.
COROLLARY3. let p > 1 and r > 0 be two fixed real numbers the following are equivalent. (i) X is uniformly convex. (ii) There is a continuous, strictly increasing and convex such that
IIX+ YllP2 llxllP+ P(Y,L)
for every x, y in B, and f, in Jp (x). (iii) There is a continuous, strictly such that
increasing
(x - r,f, for every x, y in B, , f, in Jp (x) and
+
space. Then
function
g: R+ + R+, g(0) = 0,
gdlvll)
(3.13)
and convex
- fy) 1
and X be a Banach
function
g: R+ -+ Rf,
g(0) = 0,
dllx - A)
(3.14)
fy in Jp (y).
COROLLARY2’. Let q > 1 and r > 0 be two fixed real numbers and X be a smooth Banach space. Then the following are equivalent. (i)’ X is uniformly smooth. (ii)’ There is a continuous, strictly increasing and convex function g*: Rf -+ R+, g*(O) = 0, such that
IIX+ YIP5 llxl14 + 4(Y,J,(d) + s*(llull>
for all x, y in B,. (iii)’ There is a continuous, such that
strictly increasing (x - Y, J,(x)
and convex function
- J,(Y)> 5
g*: R+ + R+, g*(O) = 0,
g*cllx- A>
for all x, y in B,. 4. APPLICATIONS In this section we give applications of the established inequalities in the previous section. We first prove an existence theorem for fixed points of uniformly Lipschitzian mappings in Banach spaces. Let K be a nonempty closed convex subset of a Banach space X and CY> 0 a constant. A mapping T: K + K is said to be uniformly ol-Lipschitzian if IIT”x - T”_Y/~5 &
- ~11
for each integer n L 1 and all x,y in K. This class of mappings has been studied by many authors (see [3-5, 11-141). Our theorem 3 below seems new so far. To prove it, we recall that the normal structure coefficient N(X) of X is defined (cf. Bynum [20]) by N(X) = inf( diam(K)/r, K a bounded
(K) : convex
subset of X consisting
of more than one point},
Inequalities
in Banach
spaces with applications
= sup1 /Ix - y/I : x, y in K] is the diameter
where diam(K)
1135
of K and
rK(K) = inf sup IIx - yl] XEKYEK is the Chebyshev radius of K relative to itself. X is said to have uniformly normal structure if N(X) > 1. It is known that a uniformly convex Banach has uniformly normal structure and for a Hilbert space H, N(H) = 2l”. THEOREM 3. Let p > 1 and let X be a p-uniformly convex Banach space, K a nonempty closed convex subset of X, T: K + K a uniformly cz-Lipschitzian mapping. Suppose that there is an x, in K for which (T”x,) is bounded and that (Y < [t(l + (1 + ~cJV)“~)]“~, where N is the normal structure coefficient of X and c is the constant given in inequality (3.1). Then T has a fixed point, i.e. there is a z in K such that T(z) = z. To prove the theorem,
we need the following
LEMMA 2. Let K, X be as in theorem unique point z in K such that
lemma.
3 and let lx,] be a bounded
sequence.
Then there exists a
lim sup ]]x, - zllP I lim sup Ilx, - x]lp - cllx - zllp n+OO n-m for every x in K, where c is the constant
given in (3.1).
Proof. Let r(x) = lim sup [lx, - xljP, x E X. By theorem 1, r(x) is uniformly convex on X and n-co therefore, there is a unique point z in K (called the asymptotic center of the sequence lx,) in K) such that r(z) = inf r(x). It follows from inequality (3.1) that XCK
r(z + A(x - z)) 5 k(x)
+ (1 - A)r(z) - W,(A)cllx - zllp
for x in K and 0 5 A I 1. Noticing r(z + A(x - z)) 2 r(z) for x in K and 0 I A I that 0 I lim (r(z + A(x - z)) - r(z))/A I r(x) - r(z) - cllx - zIIp,
1, we derive
X-O+
and the desired
inequality
follows.
Proof of theorem 3. Since (T”x,) is bounded (and hence (T”x} is bounded for any x in K), by lemma 2, we can inductively construct a sequence (x,),~ 1 in K as follows: for each integer center of the sequence [T”x,) in K. Let m 2 0, x,+~ is the asymptotic r,
= lim sup IIT*x, n+m
Then by Lim [ 15, theorem
- x,,
1II
and
R,
= sup ]]x, - T”x,)). n21
11, we have
r, I N-’ diam(( T”x,),
Z 1) I N-‘aR,,
m = 1,2, . . . .
1136
HONG-KUNXv
where N is the normal structure coefficient of X. For each fixed m L 1 and all n > k L 1, we have from (3.1) Il%l+, + (1 - W-%+,
- ~“XX
+
cw,~4Il&n+,- h?l+#
5 4I&+1 - T”x,lIP + (1 - A)]]T”x, - TkX,+# 5
an,l - T”x,ljP + (1
- A)(wpIIx,+I - T”-kX,II?
Taking the limit as n + 00 leads to r; + cW,(A)l]xm+t - TkX,+#
I (A + (1 - A)&)/$.
It then follows that %+I 5
(1 - A)@ - 1) ,.P cW,V)
5
m
(I
-
‘)@’
-
I)
N-PUP
. RP
ItI*
cW,(4
Letting I + 1, we conclude that l)l’pR,
R m+l I (crP(d - l)N-Pc-
where A = (cY~((Y” - l)N-pc-l)l’p
=: A * R,,
m = 1,2, . . . .
< 1 by assumption of the theorem. Since
II&+, - x,,,ll5 r,,,
+ R, I 2R, I ... I 2A”-‘RI,
it follows that lx,) is Cauchy. Let z = lim x,. Then we have m+m
llz - Tzll 5 llz - m?JI+ Il&?l - RAI + IIR?I - Ml I (1 + a)llx, - z]] + R,,, + 0,
asm + co,
and hence Tz = z. The proof is complete. Recently, Prus [16] calculated that N(Lp) = min(21’P, 21’q), where q = p(p - I)-’ is the conjugate number of p. Therefore, we have the following result. COROLLARY 4. Let K be a closed convex subset of Lp(l < p < 00) and T: K -, K a uniformly a-Lipschitzian mapping such that {T”x,) is bounded for some x0 in K. Suppose (Y< (3 + +(l + 4(p - 1)2(p-1)‘p)1’2)1’2 if 1 < p I 2 and (Y< (i + *(I + 8~~)~‘~)~‘~if p > 2 (here cp is as in corollary 2). Then T has a fixed point. Remark
3. For generalizations author [14].
of theorem 3 and corollary 4 to semigroups see the present
Next, we give estimates of moduli of continuity of nearest projections in Banach spaces. It is well known that if Mis a nonempty closed convex subset of a uniformly convex Banach space X, then for every x in X, there is a unique element m in M (called the nearest point of x in M) such that for all y in M. llx - mll 5 Ilx - AI
Inequalities in Banach spaces with applications
1137
Let P,(x) = m. Then we define a mapping PM.- X + A4 which is called the nearest point (or metric) projection of X onto M. If X is assumed in addition to be smooth, then m E M is the nearest point of x E X in M if and only if (m - y, J(x - m)) L 0
for all y in M.
(4.1)
We now know (cf. [17]) that if X is uniformly convex, PM is continuous. However, unfortunately, one is unable to give an estimate of the modulus of continuity of PM. Recently, Bjornestal [18] and Abatzoglou [19] discussed locally this problem in a Banach space which is both uniformly convex and uniformly smooth. The following theorem improves to some extent upon theirs. THEOREM 4. Let X be a Banach space which is both uniformly convex and uniformly smooth, Ma nonempty closed convex subset of X, P: X + A4 the nearest point projection of X onto M, and Q = 1 - P (I is the identity operator on X). Then for each constant r > 0, there exists a strictly increasing continuous function p: R+ + R+, p(0) = 0, such that
llpx -
PYll 5
~u(llX - Yll)
and
IlQx - QyIi 5 ~(llx - ~11)
for all x, y in B, . Proof. Since P is bounded on B,, by corollaries 3 and 3’, there are strictly increasing continuous functions g, g*: R+ + R+, g(0) = g*(O) = 0, such that (r’ is chosen so large that x-y,xPx,PxPYEB,, wheneverx,yEB,)
IIX + Yl122
llxl12 + 2(Y,
J(x)) + gdlrll)
for x, y in B,, ,
IIX + Yl125 llxl12+ 3Y7 JO) + g*(llrll)
for x, y in B,, .
(4.2) (4.3)
Combining (4.2) and (4.3), we obtain for all x,y in B,, [Ix - Py112 = [1(x - Px) + (Px - Py)II2 L IIX - Pxl12 + 2(Px - Py, J(x - Px)) + g(IIPx - Py(l) 2 I/x - pxl12 + g(llPx - PYIO.
Similarly, we have IIY - pxl12 2 IIY - PYl12 +
gm -
PYII).
It follows that %llPx
- PYII) 5
IIX - PYl12 + IIY - pAI2 - Ilx - pxl12- IIY - PYl12.
Using (4.3), we also have as above IIX - PYl12 = II@ - Y> + (Y - PY)l12 5 IIY - PYl12 + ax - Y, J(Y - PY)> + g*(llx - Yll) and Ily - Pxl12I
IIX- Pxl12 + ZY - x, Ax - Px)> + g*(lly - XII).
(4.4)
1138
HONG-KUN Xu
This, together
with (4.4), implies
g(lPx
- WI)
5 (x - Y, J(Y - 0)
- J(x - Px)> + g*cllx - yll)
=
JQx) + (Px -
QY, JQY -
+ (PY -
Px, 4x
dQx
-
-
Py, J(y -
Px)> + g*(llx -
Py))
yl()
- t2.d) + g*(llx - _dl),
and thus g(l/Px Now set ,D = g -lg*.
- Pub
+
Then the desired
g(llQx - Qd) 5 g*dlx - ~4).
conclusions:
Ilpx- &II 5 Pc(llX - YII) follows.
The proof
Acknowledgement-1 manuscript.
IlQx - Qd 5 &
and
- rib
is complete. am extremely
grateful
to the referee
for his careful
reading
and good
suggestions
for this
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approx. Theory 39, 299-307 (1983). 20. BYNUM W. L., Normal structure coefficients for Banach spaces, Pucif. J. math. 86, 427-436 (1980). 21. REICH S., Constructive techniques for accretive and monotone operators, in Applied Nonlinear Analysis, pp. 335-345.
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