I ntenat. J. kth. & Mah. Sci.
(1982)113-122
Vol. 5 No.
113
ON SOME FIXED POINT THEOREMS IN BANACH SPACES
D.V. PAl and P. VEERAMANI
Department of Mathematics, Indian Institute of Technology
Powai, Bombay- 400076, India
(Received March 28, 1980, and in revised form December 22, 1980)
ASTRACT.
In this paper, some fixed point theorems are proved for multi-mappings
as well as a pair of mappings.
Browder, Kanna,
iri
KEY WORDS AND PHRASES.
These extend certain known results due to Kirk,
and Rhoades.
Normal structure, Mi-mapping, Uiformly convex Banach
Space.
1980 MATi{EoIATICE SUBJECT CLASSIFICATION CODES.
i.
47H10, 54E15, 46A05.
INTRODUCTION.
A result of continuing interest in fixed point theory is one due to Kirk [6].
This states that a non-expansive self-mapping of bounded, closed and convex subset possessing normal structure in a reflexive Banach space has a fixed point.
The interest in this result has been further enhanced due to simultaneous and in-
dependent appearance of results of Browder [2] and GBhde [5] which are essentially
special cases of the result of Kirk.
Recently Kannan [6] and
iri6 [4]
have obtain-
ed results in basically the same spirit by suitably modifying the non-expansive
condition on the mapping and the condition of normal structure on the underlying
set.
In this paper we give a fixed point result for multi-mappings (Theorem 2.1)
and extend the results of Kannam [6] and Ciric [3] to a pair of mappings (Theorems
3.1 and 3.2).
[9]) for
This enables us to establish convergence of Ishikawa iterates (cf.
a pair of mappings.
D.V. PAl AND P. VEERAMANI
114
A FIXED POINT THEOREM FOR FULTI-IAPPINGS.
2.
For x
Let K be a closed, bounded and convex subset of a Banach space X.
lx-kll
let (x;K) denote sup
X,
K} and let (K) genote the diameter of K.
k
K is called a non-diametral point of K if (x’K) <
Recall that a point x
.
(K)
and that K is said to have normal structure whenever given any closed bounded convex subset C of K with more than one point, there exists a non-diametral x e C.
It is well-knocn (cf. [A]) that a compact convex subset of an arbitrary Banach
space and a closed, bounded and convex subset of a uniformly convex Banach space
With K as before, let r(K) denote the radius of
have normal structure.
K
K
n,n-empry
(K
weakly ,ompa,-t
onvex
stb.qet
K
2
Let 2
K
K
then
is n,nempty
denote the collection of all non-empty
let ;.(A,B denet sup
a-b,
a
A, b
B}
Let K be a nonempty weakly ompat cenvc subset of the Banach
..I.
Assume K has normal structure.
space X.
8ana’h =pa. X,
,f
t K and, turtht-rmre tI K has normat structure, then
subsets of K and, tot A,B
Fheor,’m
denote the Chebyshev centre of
,,,nvex ubset
(K) (whenever 6(K) > 0).
<
C
C
It is well known (cf. Opial [8]) that if K is
(x,K) }.
r(K)
K
x e
I,.ed
x e K} and let K
f(x K)
inf
2
Let T:K
K
be a mapping satisfying:
for each closed convex subset F of K invariant under T, there exists some
a(F), 0
a(F)
<
< i, such that
5(Tx,Ty)
for each x, y
(x,F), (F) 5(F)
< max
F.
Then T has a fixed point x
O
satisfying Tx
{x }.
O
O
We imitate in parts the proof of Kirk’s theorem.
Proof.
Let
denote the
collection of non-empty closed convex subsets C of K that are left invariant by
T(i.e., TC
c
C, where TC
u[Tc
compactness of K, we can apply
Zorn’s
By weak
lemma to get a minimal element M.
It suf-
").
Suppose that M contains more than one ele-
fices to show that M is a singleton.
ment.
Order
by set-inclusion.
c e C
By the definition of normal structure there exists x
sup
Hence g(x ,M) < a
O
I(M)
lx-YI[
O
(M)
Y e
)I
for some
5(x ,M)<
O
0 <a
1
5(M),
< 1.
O
M such that
115
FIXED POINT THEOREMS IN BANACH SPACES
If 6(Tx, Ty) S 6(x,M) for all x, y
{x
M, let M6
E M:
(x,:.[) -<
i
6(M)}.
Otherwise, by hypothesis there exists (M), 0 < (M) < i, such that
6(Tx, Ty)
B
Let
As
< e
max
Xo M6,
uous, M
6
Let x
6(M) for some x, y
{a, a I)
and
M6
M6 is nonempty.
M.
(x
6(x,M)
M:
Evidently, M
B6(M)).
_<
6 is convex.
Since x
is closed.
-
6(x,M)
is contin-
M6
6(Tx, Ty)
{(x,M),e 6(M)}
< max
6(M) for y e M.
< 8
Hence T(M) is contained in a closed ball of arbitrary centre in Tx and radius
8(M).
By the minimality of M, if m e Tx, then M
of centre m and radius 8(M)), whence m
(M6)
M
and
c
U( m
T(M)
c
8(M))
M.
< 86(M) < (M) which contradicts the minimality of M.
(the closed ball
But
Thus M is a singleton
and this completes the proof.
Corollary 2.2.
space X.
Let K be a nonempty weakly compact convex subset of the Banach
Assume K has normal structure.
Let T be a mapping of K into itself which
for each closed convex subset F of K invariant under T there exists
satisfies:
some (F), 0 < e(F) < i, such that
max
for each x, y
F.
Corollary 2.3.
space X.
6(x,F), e(F)
Then T has a fixed point.
Let K be a nonempty weakly compact convex subset of the Banach
Assume K has normal structure.
satisfies:
Let T be a mapping of K into itself which
for each closed convex subset F of K invariant under T there exists
some e(F), 0 < e(F) < i, such that
max
for each x, y
Remark.
[2].
F.
lx-yll, r(F),
a (F)
Then T has a fixed point.
The preceeding results generalize the results of Kirk [7] and Browder
D.V. PAl AND P. VEERAMANI
116
3.
COMMON FIXED POINTS OF MAPPINGS.
Let K be a weakly compact convex subset of the Banach space X.
Theorem 3.1.
Let T I, T be two mappings of K into itself satisfying:
2
lrlx r2Yll
(i)
<_
(I Ix-rlxl I+I IY-T2Yl I)/2,
(I
[+I lY-rlxl I)/3,
(I !x-Yl I+l Ix-rlxl +:Iy-Tmyl I)/3}
max
Ix-TroY
for each x, y e K,
(2)
TIC
(3)
either
C if and only if
lZ-TlZll
Iz-T2zll
zSPc
z sp C
or
C for each closed subset C of K;
T2C
_< (C)/2,
<
(C)/2
holds for each closed convex subset C of K invariant under T
Then there exists a unique common fixed point of T
Proof.
Let
and T
and T
2.
2.
the family of all non-empty closed convex subsets of K,
denote
each of which is mapped into itself by T
Zorn’s lemma,
weak compactness of K and
I
I
I
and T
by
Ordering
2.
set-inclusion, by
we obtain a minimal element F of K.
With-
out loss of generality, assume that
lz-rmzll
zSPF
Since (F)/2 < r(f)
Let x e F
(y e F).
T2(F)
This gives that
we obtain using (i) that
c
by hypotheses (2)
c
This gives
whence
hypothesis
then F
(x,F)
C
TI(F U)
(TIX,F) r(F),
F
(2) T2(F c)
(F)/2.
U(TIX
r(F))
IITlX-r2Yll
U, whence
T2(F
n U)
r(F).
<
F n U and
By the minimality of F, we obtain F
F n U.
TlX e Ft.
Therefore,
Fc
Tl(Fc)
c
U.
and by
We now show that if F contains more than one element,
C
is a proper subset of F.
r(F) for each x
Assume the contrary that F
F, we obtain (F)
r(F)
C
<_
max {3
(F)/4, ((F) + (F))/3,
((F) + (F) + (F)/2)/3}
56 (F)/6.
Since
(x,F), (x e F).
from (I), we get
lrlx r2Yll
F.
Again
117
FIXED POINT THEOREMS IN BANACH SPACES
(TIX,F)
The same argument as before yields
<_ 5(F)/6
(F), which is a contradiction.
<
Consequently, if F contains more than one element, then F
c
Hence F contains exactly
But this in view of above contradicts the minimality of F.
one element, say, xO, whence
Y0
e K such that
x
TlX0
YO T2Yo"
TlY 0
0
T2xO.
is a proper subset of F.
Assume there exists another element
Then using (I), we obtain
ITlX0 T2YoII <_2 If TlX 0 T2Y011
x
whence
0
TlX0 T2Y 0 Y0"
Let K be a weakly compact convex subset of the Banach space X.
THEOREM 3.2.
Assume K has normal structure.
(i)
lTlX TmYll
Let T I, T be mappings of K into itself satisfying:
2
TlXll + IY-TmYl I)/2,
(I Ix-rmYll + I!Y-TlXl I)/2,
(I Ix-yl I+I IX-TlXl + Iy-Tmyl I)/3}
<_ max {(I
x
for each x,y e K,
c C if and only if
(2)
TI C
(3)
either
sup
z e D
lz
or
sup
z e D
lz
c C for each closed convex subset C of K,
T2C
TlZll
T2zll
<_ r(D),
<_ r(D)
holds for each closed convex subset D of K invariant under T
Then there exists a unique common fixed point of T
PROOF.
Let
be
and T
I
Without loss of generality, assume that sup
z e F
Then using (i) we obtain
lTlX T2Yll
<_ r(F).
This gives exactly as in Theorem 3.1 that T
normal structure, one has (F
c
<
contradicts the minimality of F.
the unique common fixed point of T
REMARK.
(i)
and T
2.
2.
Exactly as in Theorem 3.1.,
as in Theorem 3.1.
minimal element F.
Let x eF c
I
has a
lz-T2zll
<_ r(F).
(y e F)
l(F)c c
F
c
and
T2(Fc)CFc
Since K has
(F) if K contains more than one element, which
Thus F contains Drecisely one element, which is
I
and T
2
as in Theorem 3.1.
One can replace condition (i) of Theorem 3.2 by
lrlx T2Yll
<_max {I Ix-Yll, (I
IX-TlXll + IY-TmYlI)/2
(I Ix-T2Yl + IY-TlXl I)/3, ([ Ix-Yl I+I IX-TlXl I+I IY-T2Yl I)/3}
D.V. PAl AND P. VEERAMANI
118
and T
This also yields the existence of a common fixed point of T
2.
However,
it need not be unique.
Let K be a weakly compact convex subset of the Banach space X.
THEOREM 3.3.
Let T I, T 2 be mappings of K into itself satisfying
Assume K has normal structure.
(2) and (3) of the preceding theorem and,
lTlX T2Yll < max {I Ix-Yll, lX-TlXll, Ix-TIYlI, lx-T2xll, Ix-T2YlI}"
(i)
Then there exists a common fixed point of T
I and T 2.
The proof of the above theorem is similar to that of Theorem 3.2 and hence it
is omitted.
4.
ISHIKAWA ITERATION FOR COMMON FIXED POINTS.
A uniformly convex Banach space is reflexive.
A bounded, closed and convex
subset of a uniformly convex Banach space is therefore weakly compact;
it
Hence Theorems 2.1, 3.2 and 3.3 can be particularized to
has normal structure.
Rhoades [9] has extended a result of
such a setting.
als%
iri
(cf. [3], Theorem 2)
to a wider class of transformations by using Ishikawa iterative scheme.
With a
suitable modification of arguments, this extends to a pair of mappings of the type
as in Theorem 3.2.
Let K be a non-empty closed bounded and convex subset of a uni-
THEOREM 4.1.
formly convex Banach space X.
x
(5)
Yn
(6)
Xn+ I
-
{e }, {B
where
n
(ii)
n
an(l
n
n
0
of iterates be defined by
e K
8n)Xn
(i
(i -a )x
n
n
+
(i)
and
(iii)
1
8n T 1
+ an
satisfy
unique common fixed point of T
PROOF.
T 2 be mappings of K into itself satisfying
Let the sequence {x
n
(i), (2) and (3) of Theorem 3.2.
(4)
TI,
Let
and T
0 <
x
n
n
lim n
0
n > 0
T2 Yn
n
_>
<
Bn
B
< 1 for all n,
< I
Then {x
2.
The existence of the unique common fixed point of T
from Theorem 3.2.
converges to the
n
Let the unique common fixed point be v.
lTlXn -vl[ <
]Ixn
vll
1
and T
From (1)
2
results
119
FIXED POINT THEOPdMS IN BANACH SPACES
and
Following exactly the same lines as in the proof of Theorem i of [9] we obtain
subsequences
x
Ynk,
of
n
k
(7)
x
Yn’
respectively such that
n
lim
k
II xnk
T2
Ynk II
lim
IIxnk
T
xnkll
0
we show that
(8)
I
0
It would be sufficient, with (7), to show that
limk
lTlXnk- T2Ynkll
O.
For any integer n, from
lTlxn
T2Yn II
<
(I Ixn
TlXnll
T2Ynll
< (2
Bn) II
flYn T2Y nil)/2
+
we ob rain
lTlxn
(9)
T2Ynll/(l 8n).
x
n
It follows from
lTlxn T2Ynl] < (I Ixn T2Ynll
flYn TlXn[ I)/3
+
that
T2Yn II _<
lTlXn
(i0)
(2
6n) Ixn T2Yn ]I/
(2 +
8n
From
lTlXn T2Ynl
(I
<
Ixn Ynll
+
fixn
Tlxnl I+I IYn-T2Ynl I/3
we obtain
lTxn T2Yn If _<
(ii)
From (9)
lxn T2Yn II
6n
(ii) we obtain
T2Ynl
_<
211Xn T2Ynll
Therefore,
liT iXn
k
and (7)
/ (i
implies
T2 ynk II
<_
2
IIxnk
limk ITlXn T2Ynkll
T
2Ynk
0,
k
whence
lim
k
llxn TlXnk II
Now let us prove that this implies that
0
/ (I
6n
D.V. PAl AND P. VEERAMANI
120
ko
This follows easily from
T
x
T
I Xnk
2
Xnk
xnk TlXnk II + max{(l Ixnk
<
TlXnkl I+l Ixnk-T2Xnl I)/2,
T2Xnkll lx TlXnklI)/3
(!1 %1I+ Ilx TlXn !I+ Ii xnk
+
x
x
x
k
0 as k +
which tends to
.
since
IlXnk TlXnk II +o
Also
II T i Xnk
T
i
Xn II <_ II T I Xnk
T2
Xnk II
T2Xnk !!)/3}.
llT2Xn- rlXn.l
+
From (i) of Theorem 3.2,
r2Xnkl max([IIXn TlXnl lxnk
]TlXn
[II Xn
[I
+
<
T
Z
2
Xnk II
+
llx
T
nk
I
T x
2%
1{]/2
Xn
IXn. x%ii II Xn TlXn II I1=% T2x%] 1]/3
+
+
If
I! TlXn%
3
!1
T2Xnkll
<
Ill Xn T2Xn !1
k
TlXn T2Xnkl] < IIxn
+
lXnk T2x%ll + fiT
T x
i
n
II
+
IIxnk
+
TlXn 111/3,
IIT.xn
x
2 n
k
TlXn II
T
Xn II+ II xnk
then
T2Xnk il
which imp lies
()
II T I Xn
T2
Xnk II <_II xn
i
T
2
Xnk II.
If
II T i Xn
-T
2
Xnk ll![llxn Z -=nk ll+ll xn
Ti
Xn II+ II xnk
it follows, in a similar manner, that (II) holds.
is satisfied.
T2
Xnk II]/3,
Therefore, in all cases, (II)
FIXED POINT THEOREMS IN BANACH SPACES
121
Therefore,
II T i Xnk
T1
Xn II
<
Irlxnk-
.
which tends to 0 as k-
converges, say, to u.
x
n
k
II
+
Therefore
II xnk
T
{TlXnk }
2
Xnk II +II xn
TlXn l+llh-T2Xnk II,
is a Cauchy sequence and hence it
Consequently
lim
lim
xnk
TlXnk
u.
Also,
lu-T 2ull
lU-Xnkll Ixnk-TIxnkll +fITIxnk-T2II IU-Xnk
+
{(I
IXnk T].Xnkl + I1, T2ul I)/2,
(I
I(xnk T2ull + lu TlXnkl I)/3
+
<
<
Xnk IXnk
max
Taking the limit as k
Now,
-
(llxn
=o,
-u
!1
we obtain
+
II
T
nk
lu
lxnk II +
T2ull
O.
Ilu- T2ull)/:}
Therefore, u
T2u
Ilu- T].ul <_ I1- T2uli + II T2u- Tzull
<_ mx {(llu :pli + lu Teul I)/2
(I lu T2il + lu
(llu- !1 + I1-
lu
This implies
ACKNOWLEDGEMENT.
manuscript.
T].ull
0
Therefore,
TlUl I)/3
Tlull + Iiu
u
Tlu
Thanks are due to the referees for a critical reading of the
In particular, the present improved version of Theorem 4.1 is due
to the suggestions of one of the referees who pointed our attention to
reference [9].
REFERENCES
i.
BONSALL, F.F.
2.
BROWDER, F.E. Non-expansive nonlinear operators in a Banach space.
Nat. Acad. Sci., 54 (1965), 1041-1043.
Lectures on some fixed point theorems of functional analysis,
Tata Institute of Fundamental Research, Bombay, India, 1962.
Proc.
122
3.
D.V. PAl AND P. VEERAMANI
61R16,
Lj.B.
On fixed point theorems in Banach spaces, Publ. Inst. Math.,
1__9
(33), (1975), 43-50.
4.
DIESTEL, J.
Geometry of Banach spaces, Lecture notes, No. 485, Sp_ringer-Verlag,
Berlin, 1975.
5.
GHDE,
6.
KANNAN, R.
7.
KIRK, W.A. A fixed point theorem for mappings which do not increase distances,
Amer. Math. Monthly, 72 (1965), 1004-1006.
8.
OPIAL, Z.
9.
RHOADES, B.E. Some fixed point theorems in Banach spaces, Math. Seminar Notes,
5 (1977), 69-74.
D. Zum Prinzip der Kontraktiven Abbildung, Math. Nach., 30 (1965),
251-258.
Some results on fixed points
III, Fund. Math., 70 (1971), 169-177.
Nonexpansive and monotone mappings in Banach spaces, Lecture notes
from January, 1967 lectures given at Center for Dynamical Systems at Brown
University.