International Journal of Mathematics and
Computer Science, 1(2006), no. 4, -
M
CS
A Generalization of a Fixed Point Theorem
of Kirk
Johnson O. Olaleru
Mathematics Department
University of Lagos
Lagos, Nigeria
e-mail: [email protected]
(Received July 4, 2006, Revised Nov. 8, 2006, Accepted Nov. 10, 2006)
Abstract
In this paper we prove the following theorem which is a generalization of Kirk’s fixed point theorem [7]:
Let X be a reflexive metrizable locally convex space and K a closed
convex bounded and nonempty subset of X with normal structure.
Let T : K → K be nonexpansive. Then T has a fixed point in K.
1
Introduction
Let (M, ρ) be a metric space. A mapping T : M → M is called a Lipschitz map
if there exist a number L ≥ 0 such that
ρ(T x, T y) ≤ Lρ(x, y)
for all x, y ∈ M . The mapping T is called strict contraction if L < 1, and
is called nonexpansive if L = 1. The importance of nonexpansive mappings
was outlined in 1980 by Bruck [4]. A fixed point theorem for a nonexpansive
map was proved by Kirk in 1965 [7] when the map is defined on a reflexive
Banach space. In this work we generalize his result to a reflexive metrizable
locally convex space in which the Tychonoff’s fixed point theorem becomes a
special case. Such locally convex spaces which are complete and metrizable
abound and are obviously generalizations of Banach spaces.
For example, the set of all real (or complex) valued indefinitely differentiable functions on the interval [a, b] becomes a metrizable locally convex
space under the topology defined by the seminorms pm (f ) = sup |f (m) (t)|,
a≤t≤b
Key words and phrases: Normal Structure, Fixed Points, Reflexive Locally
Convex Space.
AMS (MOS) Subject Classifications: 47H10, 46A03
c 2006, http://ijmcs.futureintech.net
ISSN 1814-0424 °
2
J. O. Olaleru
(m = 0, 1,. . . ). Also consider the set of all real (or complex) valued indefinitely differentiable functions on the interval (−∞, ∞). Under the topology of compact convergence for all the derivatives defined by the seminorms
pmn (f ) = sup |f (m) (t)|, (m = 0, 1,. . . ; n = 1, 2,. . . ), the set is a metriz−n≤t≤n
able locally convex space (see [9, p. 19]). In fact the duals of those spaces
of distributions under their appropriate strong topologies are also metrizable
locally convex spaces [9, p. 75]. Those spaces are also complete [9, p. 63].
A locally convex space (X, u) with topology u is a topological vector space
which has a local base of convex neighborhoods of zero.
Theorem S[9, Chap. 1, Theorem 4]. The topology of a metrizable locally
convex space (X, d) can always be defined by a metric d(x, y) = fc (x − y)
where
∞
X
fc (x) =
2−n min{pn (x), 1}.
n=1
It should be observed that if X is a normed linear space, then fc satisfies
the triangle inequality and will also be a norm. It is also easy to see that
fc (x) = 0 implies that x = 0 for any x ∈ X. It is also easy to prove that
fc (λx) ≤ fc (x) for any x whenever whenever 0 ≤ λ ≤ 1 and X is a metrizable
locally convex space. Henceforth fc will denote the function as defined above.
2
Normal Structure in Metrizable Locally Convex Spaces
We now introduce the concept of normal structure in metrizable locally convex spaces parallel to that of Banach spaces (compare with [1] or [2]). This
will be needed in our study of the existence of fixed points for nonexpansive
mappings.
Let C be a bounded subset of a metrizable locally convex space X. The
diameter d of C is defined by
d = sup{fc (x − y) : x, y ∈ C}
A Generalization of a Fixed Point Theorem of Kirk
3
A point xo ∈ X is called a diametrical point of C if
d = sup{fc (xo − x) : x ∈ C}
and a point xo ∈ X is called a nondiametrical point of C if
sup{fc (xo − x) : x ∈ C} < d
Let d(K) denote the diameter of K. Then a bounded subset K of X is said
to have normal structure if every non trivial convex subset C of K contains
at least one non diametral point, i.e., there exists xo ∈ X such that
sup{fc (xo − x) : x ∈ C} < sup{fc (x − y) : x, y ∈ C} = d(C)
The metrizable locally convex space X is said to have a normal structure if
every bounded convex subset of X has a normal structure.
We give an example which generalizes Brosdskii and Mil’man’s Theorem[2].
The proof is essentially the same as in [5,p. 158]. But we will include it for
completeness.
Proposition: Every compact convex subset of a metrizable locally convex
space X has normal structure.
Proof . We prove by contradiction. Assume K does not have a normal
structure. The method of proof is to construct a sequence in K which does
not have a convergent subsequence.
Since K does not have a normal structure, there exists a nontrivial convex
subset C of K such that all points of C are diametral. Without loss of
generality we can assume that C is closed, thus compact. Let d > 0 denote
the diameter of C. We shall construct a sequence {xi }∞
i=1 in C such that
fc (xi − xj ) = d; i, j = 1, 2, ..., i 6= j. To do this we choose x1 ∈ C arbitrary
and assume that x2 , x3 , ..., xn have already been chosen such that fc (xi −xj ) =
d; i, j = 1, 2, ....n. By the convexity of C,
1
(x1 + x2 + ... + xn ) ∈ C
n
and so it is a diametral point by assumption. By compactness, sup{fc (x−y) :
x, y ∈ C} is achieved. So, there exists xn+1 ∈ C such that
fc (xn+1 −
x1 + x2 + .... + xn
)=d
n
4
J. O. Olaleru
Hence,
d =
≤
1
fc ((xn+1 − x1 ) + (xn+1 − x2 ) + ...
n P
n
1
1
j=1 fc (xn+1 − xj ) ≤ n .nd = d
n
+ (xn+1 − xn ))
Consequently, fc (xn+1 − xj ) = d for j = 1,2,... This implies that the sequence
{xn }∞
n=1 has no convergent subsequence, contradicting the compactness of K
and completing the proof.
Corollary 1 (Brosdskii and Mil’man [2]): Every compact convex subset of a
Banach space X has normal structure.
The following example shows that a complete metrizable locally convex
space may not have a normal structure even if it is reflexive. Consider the
set D(R) of all indefinitely differentiable functions on R under the topology
determined by the seminorms
pmn (f ) = sup |f m (t)| (m = 0, 1, ...; n = 1, 2, ...)
−n≤t≤n
This space is a complete metrizable locally convex space and it is not normable
[9,p. 236]. It is a Montel space and thus reflexive [9, p. 75]. Consider the
subset K of D(R) defined by
K = {f ∈ D(R) : f (0) = 0, f (1) = 1, 0 ≤ f (t) ≤ 1 ∀t ∈ [0, 1]}.
Then K is clearly bounded and convex. It does not have a normal structure.
This can easily be shown. For since [0, 1] is compact, the locally convex
topology as defined above is the same as the ”sup norm” and for any point
fo ∈ K,
sup{fc (fo − f ), f ∈ K} = sup{kfo − f k, f ∈ K} = 1 = diameter of K.
Hence each point of K is diametral.
3
The Main Theorem
The following results will be needed for our theorem.
Lemma 1 [9, Proposition 5, p. 72]: Let X be a Hausdorff locally convex
space with dual X 0 . Then X is reflexive if and only if every bounded set
A Generalization of a Fixed Point Theorem of Kirk
5
in X is contained in a weakly compact set and every bounded set in X 0 is
equicontinuous.
Just like in a normed linear space, every closed set in the weak topology
of a locally convex space is also a closed set in the strong topology. However,
the converse is false. We now show a condition which makes the converse to
be true. This will generalize Theorem 6.17 of Chidume [5] as we use a similar
approach.
Lemma 2: Let X be a locally convex space and K ⊂ X be convex and closed
in the strong topology, then K is closed in the weak topology.
Proof . By an application of Hahn Banach Extension Theorem which is true
for locally convex space [8, &20, 7(1)], the proof is essentially the same as
the well known case when X is assumed to a normed linear space (e.g.see [5,
Theorem 6.17]).
The following well known characterization of compactness will also be
used.
Lemma 3: Let (X, d) be a metric space. Then (X, d) is compact if and only
if every collection of closed nonempty subsets of X with finite intersection
property has a nonempty intersection.
We now proceed to our main result. The proof is a slight modification of
the one used by Kirk as used by Chidume [5] when X is a Banach space.
Theorem: Let X be a reflexive metrizable locally convex space and K a
closed convex bounded and nonempty subset of X with normal structure. Let
T : K → K be nonexpansive. Then T has a fixed point in K.
Proof . Let = denote the collection of all closed convex bounded and nonempty subsets of X, each of which is mapped into itself by T , and ordered
by inclusion. We claim that every chain in = has a lower bound. For let Z
be an arbitrary chain in = and let
K ∗ = ∩{Kα , Kα ∈ Z}.
Clearly, Kα is closed, convex and bounded and hence contained in a weakly
compact set of X by Lemma 1. By Lemma 2, Kα is weakly closed and
6
J. O. Olaleru
hence weakly compact. Then = has finite intersection property and so K ∗ ,
is nonempty by Lemma 3. K ∗ is a lower bound for Z and hence by Zorn’s
Lemma, = has a minimal element which we shall denote K 1 . Since K 1 is
closed, convex, bounded and nonempty, the closed convex hull coT (K 1 ) of
K 1 is K 1 .
It remains to show that K 1 consists of a single point. This we do by contradiction. Suppose on the contrary it consists of more than one element. Then
the d(K 1 ) > 0. Since K has normal structure, there exists x1 ∈ K 1 such
that K 1 ⊆ U (x1 , d1 ) for some d1 < d(K 1 ) where U (x1 , d1 ) is a neighborhood
of x1 with radius d1 . Define
K 2 = {x ∈ K 1 : K 1 ⊂ U (x, d1 )}
= K 1 ∩ (∩t∈K 1 U (t, d1 ))
K 2 is a closed convex subset of K 1 . Clearly, K 2 6= K 1 . Let x2 ∈ K 2 be
arbitrary, then, for an arbitrary x ∈ K 1 ,
fc (T x2 − T x) ≤ fc (x2 − x) ≤ d1
Hence
T (K 1 ) ⊆ U (T x2 , d1 )
since x ∈ K 1 is arbitrary. Also
coT (K 1 ) ⊆ U (T x2 , d1 ) and coT (K 1 ) = K 1 and so K 1 ⊆ U (T x2 , d1 )
which implies that T x2 ∈ K 2 . Thus K 1 is a closed convex nonempty bounded
proper subset of K 2 which is invariant under T . This contradicts the minimality of K 1 in =. Hence K 1 is a singleton. This completes the proof.
If K is compact, then our theorem becomes a special case of Tychonov’s
fixed point theorem. However we do not know if the theorem is valid without
the metrizability of X.
Corollary 2 [7]: Let X be a reflexive Banach space and let K be a nonempty closed bounded and convex subset of X with normal structure. Let
T : K → K be nonexpansive. Then T has a fixed point in K.
Since a uniformly convex space is both reflexive and has a normal structure, the following result follows.
A Generalization of a Fixed Point Theorem of Kirk
7
Corollary 3 (F. E. Browder[2]; Gohde[6]): Let X be a uniformly convex
space and let K be a nonempty closed bounded and convex subset of X with
normal structure. Let T : K → K be nonexpansive. Then T has a fixed point
in K.
Remarks
1. It is still an open problem whether the Theorem is true for Banach spaces
(not even locally convex spaces) if the assumption of normal structure is not
there. However, Belluce and Kirk in [1] proved that if the assumption of normal structure is replaced with the concept of diminishing orbital diameter of
f, (see [1]), the Theorem is true. It is still an open question whether that
result can be extended from Banach space to locally convex space.
2. In the Corollary 2 above, Kirk was able to prove that the result still holds
if the condition that K is bounded is replaced by the requirement that the
sequence {φn (p)} be bounded for some p ∈ K. We conjecture that this is
also true for our theorem although it has not been proved. We do not know
whether there are other conditions that may substitute for the boundedness
of K.
Acknowledgement
The author is grateful to the referee for useful comments and suggestions.
References
[1] L. P. Belluce and W. A. Kirk, Fixed point theorem for certain classes of
nonexpansive maps, Proc. Amer. Math. Soc. 20 (1969), 141-146.
[2] M. S. Brosdskii and D. P. Mil’man, On the center of a convex set, Dokl.
Akad. Nauk. SSSR, 5 (1948), 837-840.
[3] F. E. Browder, Nonexpansive nonlinear operators in a Banach space,
Proc. Nat. Acad. Sci., 54 (1965), 1041-1044.
[4] R. E. Bruck, Asymptotic Behaviour of Nonexpansive Mappings, Contemporary Mathematics, 18, Fixed Points and Nonexpansive Mappings,
(R. C. Sine, Editor), AMS, Providence, RI, 1980.
[5] C. E. Chidume, Applicable Functional Analysis, International Centre
for Theoretical Physics, Italy, 1996.
8
J. O. Olaleru
[6] D. Gohde, Zum prin zip der kontraktiven abblidung, Math. Nachr. 30
(1965), 251-258.
[7] W. A. Kirk, A fixed point Theorem for mappings which do not increase
distance, Amer. Math. Monthly 72 (1965), 1004-1006.
[8] G. Kothe, Topological Vector Spaces 1, Springer-Verlag, 1969.
[9] A. P. Robertson, W. J. Robertson, Topological vector spaces, Cambridge
University Press, 1980.