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Internat.
VOL.
J. Math.
4 (1994) 681686
& Math.
Sci.
17 NO.
681
FIXED POINT THEOREMS FOR A SUM OF TWO MAPPINGS
IN LOCALLY CONVEX SPACES
R VIJAYARAJU
Department of Mathematics
Anna University
Madras 600 025, India
(Received May 2, 1991 and in revised form March 12, 1992)
ABSTRACT.
theorem of Krasnoselskii for a sum of contraction and compact mappings in Banach spaces.
class of asymptotically nonexpansive mappings
mappings as well as the class of contraction mappings.
method some results concerning the existence of fixed points for a sum of nonexpansive and
continuous mappings and also a sum of asymptotically nonexpansive and continuous mappings in
locally convex spaces. These results extend a result of Cain and Nashed.
Cain and Nashed generalized to locally convex spaces a well known fixed point
The
includes properly
In this paper, we prove by using the same
theclass
of nonexpansive
KEY WORDS AND PHRASES.
uniformly asymptotically regular with respect to a map.
1991 AMS SUBJECT CLASSIFICATION CODES. 47H10, 54H25.
Asymptotically
nonexpansive and
continuous
mappings,
1.
INTRODUCTION.
Let K be a nonempty closed convex bounded subset of a Banach space x.
Krasnoselskii [6] proved first that a sum T+S of two mappings T and S has a fixed point in K,
when T:K.X is a contraction and S:K,X is compact (that is, a continuous mapping which maps
bounded sets into relatively compact sets) and satisfies the condition that
:,ye K.
Nashed and Wong [7] generalized Krasnoselskii’s theorem to sum T+S of a nonlinear
contraction mapping T of K into X (that is, [[TzTy[[ <([[zyl[) for all z,Ve K, where
real valued continuous function satisfying certain condition) and a compact mapping S of K into X.
Subsequently, Edmunds [4], Reinermann [8] extended Krasnoselskii’s theorem to a sum T+S of a
nonexpansive mapping T and a strongly continuous mapping S (that is, a continuous mapping from
the weak topology of x to the strong topology of x) when the underlying spaces x are Hilbert
spaces and uniformly convex Banach spaces respectively.
Krasnoselskii’s theorem was further extended by Cain and Nashed [2] to a sum T+S of a
contraction mapping T of a nonempty complete convex subset K of a locally convex space X into X
and a continuous mappings S of K into X.
Sehgal and Singh [9] generalized the above result of
Cain and Nashed [2] to a sum T+S of a nonlinear contraction mapping T of K into X and a
continuous mapping S of K into X. This result generalizes the result of Nashed and Wong [7].
The study of asymptotically nonexpansive mappings concerning the existence of fixed points
have become attractive to the authors working in nonlinear analysis, since the asymptotically
In 1955,
T;r.+Sy. K for all
is a
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682
P. VIJAYARAJU
nonexpansive mappings include nonexpansive as well as contraction mappings. Goebel and Kirk [5]
introduced the concept of asymptotically nonexpansive mappings in Banach spaces and proved a
theorem on the existence of fixed points for such mappings in uniformly convex Banach spaces.
The aim of this paper is to prove fixed point theorems for a sum of nonexpansive and
continuous mappings in locally convex spaces.
Throughout this paper, let X denote a Hausdorff
locally convex linear topological space with a family (Pa)ae Jof seminorms which defines the
topology on x, where J is any index set.
We recall the following definition.
DEFINITION 1.1. Let K be a nonempty subset of x.
(a) T is called a contraction
that
If T maps K into X, then
J, there is a real numberkcwith 0 _< ka<
[2] if for each c
such
p(Tx r) _< k,p(x )
(b) T is called a nonexpansive if ka
(c)
T is called an asymptotically nonexpansive [11] if
in (a).
for all x,vin K.
pa(Tnz
Tny) < knpcr(x
y)
for all x,Vin K,
for each a
It is assumed thatkn>
IVe introduce the following definition.
DEFINITION 1.2.
regular with respect to $if, for each a in d and r/> 0, there exists N
d and for n
2,
where{kn}is a sequence of real numbers such that lira
andkn>kn+
kn
1.
for n
1,2
If T and $ map K into X, then T is called a uniformly aymptotically
N(a,,/) such that
pa(TnxTn
Ix+ Sx) < r/
for all n > N and for all x in K.
EXAMPLE 1.3. Let X
We define a map T:K,X by Tx
Then T2x
T(1 + x)
R and K
[0,1].
+x for all x in K.
2+z. By induction, we prove that
Tnx
n+z.
We define a map S:KX by Sz
Therefore
Tnx Tn lx+Sx
REMARK 1.4. T is uniformly asymptotically regular with respect to the zero operator means
that T is uniformly asymptotically regular [11]. The following example shown in [11] that uniform
asymptotic regularity is stronger than asymptotic regularity.
Let X
for all x in K.
0. Hence T is uniformly asymptotically regular with respect to S.
eP, < p < oo and K denote the closed unit ball in x. Define a map T:K by
Tx
(2,3
for all z
(t[i,2,3
K.
2.
MAIN RESULTS.
We state the following TychonoWs theorem and Banach’s contraction principle which will be
used to prove our theorems 2.1 and 2.2.
THEOREM A [10].
Let K be a nonempty compact convex subset of X.
mapping of K into itself, then T has a fixed point in K.
If T is continuous
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FIXED POINT THEOREMS IN LOCALLY CONVEX SPACES
683
THEOREM B [2].
contraction mapping of K into itself, then T has a unique fixed point u in K and Tnx.,u for all r in
Let K be a nonempty sequentially complete subset of X.
If T is a
K.
The following theorem is an extension of Theorem 3.1 of Cain and Nashed [2] for a sum of
contraction and continuous mappings to a sum of certain type of asymptotically nonexpansive
mapping T and continuous mapping S in locally convex spaces X by assuming the conditions that T
is uniformly asymptotically regular with respect to S and Tnx+Sy E K for all r,y in K and n
This result is new even in the case of normed linear spaces.
THEOREM
2.1
Let K be a nonempty compact convex subset of X.
asymptotically nonexpansive selfmapping of K.
Suppose that T is uniformly asymptotically regular selfmapping of K with respect to the mapping
K for all z,y e K and n
1,2
PROOF. For each fixed y in K, we define a map Hnfrom K to K by
1,2
Let T be an
Let S be a continuous mapping of K into X.
S and that Tnz+Sy_
Then T+S has a fixed point in K.
Hn(z
an(Tnz+ SV) for all z e K.
where an
it follows that
(1 1/n)/knand {kn}is an in Definition 1.1(c).
Since T is asymptotically nonexpansive,
pa(Hn(a) Hn(b))
anPa(Tna Tnb)
< (11/n)t,a(ab)
for all a,b in K.
Hence Hnis a contraction on K. By Theorem B, Hnhas a unique fixed point, say, Lny in K.
Therefore
Lny
Hn(LnY
an(Tn(Lny) + Sy).
Let u, v. K be arbitrary. Then we have
Therefore
pct(Lnu
Lnv< anpa(Tn(Lnu)
Tn(Lnv)) + anpa(Su
Sv)
_< (1
1/n)pa(Lnu
Lnv 4 anpa(Su
Sv)
(2.2)
Tnzn
Tn
lzn+ Szn,Oas n.oo.
From (2.4) and (2.5)we obtain
znTnlznOas
(2.5)
pct(LnuLnv< nanpa(Su Sv).
Since S is continuous, Lnis continuous.
Using Tychonoffs Theorem A, we see that Lnhas a fixed point, say,rnin K. Therefore
xn
Lnxn
an(Tnr,n+ S:n).
(2.3)
Hence znTnr,nSXn
Tn+Sy
K for all ,y E K.
Since T is uniform!y asymptotically regular with respect to S, it follows that
(an
1)(Tnn + SZn)O as noo,
sincean,l as ncx and K is bounded and
(2.4)
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684
P. VIJAYARAJU
Now
po(Zn(T + S)zn)< po(Zn(Tn+ S)n) + po,((Tn+ S)Zrn(T + S)zn)
< po(Zn(Tn+ S)Zn)+ klPo(Tnlzn
Zn).
Using (2.4) and (2.6)in (2.7) we get
zn
(T + S)zn,Oas n,.
Since K is compact, there exists a subnet(z/)of the sequence {zn} such that
u
for some u
K.
Since T and S are continuous, it follows that
(I(T +S)(fl))

(I(T + S))(u)
and by (2.8) we get
(z.7)
(T +S)(,)
0.
Since x is Hausdorff, it follows that (I(T + S))(u) =0. Hence T+S has a fixed point
For nonexpansive mapping T, the condition that T is uniformly asymptotically regular with
respect to the map S is not needed in the following theorem.
Theorem 3.1 of Cain and Nashed [2] for a sum of contraction and continuous mappings in locally
convex spaces.
THEOREM 2.2. Let K be a nonempty compact convex subset of X. Let T be a nonexpansive
mapping of K into X and S be a continuous mapping of K into X such that Tz+ St fi K for all
K. Then T+S has a fixed point in K.
PROOF. For each fixedtin K, we define amap//nfrom K to K by
This theorem is an extension of
t
Hn(.
An(T+Sy) for all
K,
where{An}is a sequence of real numbers with 0 <An<
Proceeding as in the above theorem, we obtain a sequence {n} in K such that
Since K is compact and{Xn} C K, there exists a subset(x#)of the sequence {n}such that
and,nlas nc.
/
for some : in K.
Thereforez/
T+S has a fixed point
The following example shows that the above theorem cannot be deduced from Theorem 2.1.
EXAMPLE 2.3.
Let X=space (s), the space of all sequences of complex number whose
topology is defined by the family of seminormsPndefined by
,/(Tz/+ Sz/).
Since T and S are continuous, it follows that
(T+ S)z.
Hence
in K.
pn(X)
rna[
for
(1,2
X and n
1,2,....
LetK={,X:Ijl<lforj=l,2
Then K is compact [3, Problem 47, p. 346]. Also K is convex.
We define a map T from K to K by Tz
We define a map S from K to K by Su
If a,b
K, then we have
}.
(3/4) : for all z
(1/4) ufor all
K. Then T is nonexpansive.
K. Then S is continuous.
Pn(Ta+ Sb) < (3[4)pn(a) + (l[4)Pn(b).
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FIXED POINT THEOREMS IN LOCALLY CONVEX SPACES
685
Therefore Ta+Sb_
Suppose that
K for all a,b E K.
)E K. Then we have
(1,0
T(el)
(3/4,0,0
), Tml(el)
((3/4)m1,0,0
Tin(el)
((3/4)m,0,0
andS(el)
(1/4,0
Therefore
Pn(Tm(el)_Tml(el) +S(el))=1(3/4)m_(3/4)m + 1/4[
1(1/4)(1(3/4)m1)1,1/4 as moo.
Hence T is not uniformly asymptotically regular with respect to S.
The following example shows that if the condition Tx+Sy in K for all x, y
dropped, then the conclusion of theorem fails.
EXAMPLE 2.4. Let XR and K
We define a map T from K to K by Tx=x/2 for all zeK.
nonexpansive. We define a map S from K to K by Sy
Suppose that
3/4,b
K. Then
Ta+ Sbl
If u is a fixed point of T+S in K, then u
has no fixed point in K.
To prove of the following Theorems 2.5 and 2.6, we need the following extension of Tychonoff’s
Theorem A.
THEOREM C [1, p. 169]. Let K be a nonempty closed convex subset of a locally convex space
X.
K, then T has a fixed point in K.
In Theorems 2.5 and 2.6, the compactness of the set K of Theorems 2.1 and 2.2 is replaced by
a weaker condition that the set K is a complete and bounded set, but the mappings T and S are
required to satisfy additional conditions that S(K) is contained in some compact subsets of K and
(I
T
S)(K) is closed.
THEOREM 2.5.
Let K be a nonempty complete bounded convex subset of X.
asymptotically nonexpansive selfmapping of K. Suppose that S is a continuous mapping of K into
X such that S(K) is contained in some compact subset M of K.
uniformly asymptotically regular with respect to S and that TnX+Sy in K for all x,y e K and
1,2,.... If (I
T
S)(K)is closed, then T+S has a fixed point in K.
PROOF.
Define a map Unas in the proof of Theorem 2.1. Proceeding as in Theorem 2.1, K
and Lnsatisfy all hypotheses of Theorem C, where Lnis as in the proof of Theorem 2.1.
Theorem C, Lnhas a fixed point, say, xnin K.
that 0
(I TS)(K). Hence the proof is complete.
THEOREM 2.6.
Let K be a nonempty complete bounded convex subset of X.
nonexpansive mapping of K into X. Suppose that S is a continuous mapping of K into X such that
S(K) is contained in a compact subset M of K and Tz +Sy. K for all z,V
closed, then T+S has a fixed point in K.
PROOF.
Define a map Hnas in the proof of Theorem 2.2.
Theorem 2.2 and using Theorem C instead of Tychonoff’s Theorem A, we obtain a sequence {zn} in
K such that zn
An(Tzn+ Szn).
Since,nl as n*o and K is bounded, it follows that(ITS)znOas no.
Since (ITS)(K) is closed, it follows that 0
(IT$)(K).
K of Theorem 2.2 is
[0,1].
Then T is a contraction and hence
for all y E K. Then S is continuous.
11/8 <
Tu+Su
K. Therefore Ta+SbfK for some a,b. K.
(u/2) +
and therefore u
2 $ K. Hence T+S
If T is a continuous mapping of K into itself such that T(K) is contained in a compact subset of
Let T be an
Assume further that T is a
n
By
Since (ITS)(K) is closed, it follows from (2.8)
Let T be a
K.
If (ITS)(K) is
Proceeding as in the proof of