Download 24. ON SOME FIXED POINT RESULTS IN d p -COMPLETE HAUSDORFF SPACES Author: V. Naga Raju

Int Jr. of Mathematical Sciences & Applications
Vol.4, No.1, January-June 2014
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ISSN No: 2230-9888
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ON SOME FIXED POINT RESULTS IN dp-COMPLETE HAUSDORFF SPACES
V. Naga Raju
Department of Mathematics, College of Engineering
Osmania University,Hyderabad - 500007, Andhra Pradesh,India
Email : [email protected]
Abstract
In this paper we define
d p - complete topological spaces for p  2 . A d 2 - complete
topological space is a generalized d-complete topological space introduced by Troy L. Hicks.
Some fixed point theorems for self-maps of
d p - complete Hausdorff topological spaces
are established which generalize the results of Troy L. Hicks.
Keywords:
d p - complete topological spaces, d-complete topological spaces, orbitally lower
semi continuous and orbitally continuous maps.
AMS (2000) Mathematics classification: 47H10, 54H25
1 . Introduction
Troy L. Hicks [5] has introduced d-complete topological spaces, attributing the basic ideas of these spaces to
Kasahara ([8], [9]) and Iseki [7] as follows:
1.1
Definition: A topological space (X, t) is said to be d- complete if there is a mapping d : X  X  [0, ) such that
(i) d ( x, y)  0  x  y and (ii)
 xn  is a sequence in X such that

 d (x
n 1
n
, x n 1 ) is convergent implies that
 xn  converges in (X, t).
In this paper we introduce d 2 - complete topological spaces as a generalization of
d-complete topological spaces. In fact , we define
non-empty set X, let
1.2
X
p
be its p-fold cartesian product.
Definition: A topological space (X, t) is said to be
that (i)
d p - complete topological spaces for any integer p  2 . For a
d p - complete if there is a mapping d p : X p  [0, ) such
d p ( x1 , x 2 , . . . , x p )  0  x1  x 2  . . .  x p and (ii)  xn  is a sequence in X with
lim d p ( xn , xn1 , xn2 , . . . , xn p 1 )  0 implies that  xn  converges to some point in (X, t). A
n
d p - complete topological space is denoted by (X, t, d p )
1.3
Remark: The function d in the Definition 1.1 and the function d 2 (the case p = 2) in Definition 1.2 are both
defined on
XX
and satisfy condition (i) of the definitions which are identical. Since the convergence of an

infinite series

n 1
n
of real numbers implies that lim  n  0 , but not conversely; it follows that every dn 
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complete topological space is d 2 - complete, but not conversely. Therefore the class of d 2 - complete topological
spaces is wider than the class of d-complete spaces and hence a separate study of fixed point theorems of selfmaps on d 2 - complete topological spaces is meaningful.
.
2 . Preliminaries
Let X be a non-empty set. A mapping
p non-negative on X provided
2.1
d p : X p  [0, ) is called a
d p ( x1 , x 2 , . . . , x p )  0  x1  x 2  . . .  x p .
Definition: Suppose (X, t) is a topological space and
d p is a p non- negative on X. A sequence  xn  in X is
d p - Cauchy sequence if d p ( x n , x n 1 , . . . , x n  p 1 )  0 as n   .
said to be a
In view of Definition 2.1, a topological space (X, t) is
p-non- negative
d p - complete if there is a
d p on X such that every d p - Cauchy sequence in X converges to some point in (X, t).
If T is a self map of a non-empty set X and x  X , then the orbit of x, OT (x) is given by


OT ( x)  x, Tx , T 2 x, . . . . If T is a self map of a topological space X, then a mapping G : X  [0, ) is said to
be T-orbitally lower semi-continuous (resp. T-orbitally continuous) at
some
x X
with x n
x*  X
if
 xn  is a sequence in OT (x) for
 x * as n   then G( x*)  lim inf G( xn ) ( resp. G( x*)  lim G( xn ) ). A self map T
n 
of topological space X is said to be w-continuous at
x X
if
n
x n  x as n   implies Tx n  Tx as n   .
d p is a p non-negative on a non-empty set X, and T : X  X then we write, for simplicity of
If
notation, that
(2.2)
G p ( x) : d p ( x, Tx, T 2 x, . . . , T p1 x) for x  X
Clearly we have
(2.3)
G p ( x )  0 if and only if x is a fixed point of T .
(2.4)
H p ( x, y)  d p ( x, y, Ty, T 2 y, . . . , T p2 y) for x, y X
(2.5)
E p ( x, y )  d p ( x, y, y, . . . , y ) for x, y X
Clearly
(2.6)
3.1
H p ( x, Tx )  G p ( x) and E p ( x, x )  0 .
3 . Main results
Theorem: Suppose (X, t, dp) is a dp- complete Hausdorff topological space and T: X X is a
w-continuous map such that Gp (Tx)  k (Gp(x)) for all x  X, where k : [0,)  [0,) is a
non-decreasing function with k(0) = 0. Then T has a fixed point iff there is an x  X such that
lim kn(Gp(x))=0. In this case lim Tnx=z and Tz=z.
n
n
Proof :
Suppose T has a fixed point, say z, so that z = Tz =…..= T p-1 z and hence Gp(z) = 0 which gives kn(Gp(
z )) = 0 for all n  1.
 lim kn(Gp(z))=0.
n
Conversely, suppose that lim kn(Gp(x))=0 for some x  X.
n
Let xn=
Tnx
for n  0 with x0 = x. Then
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d p ( xn , xn1 ,....,xn p 1 )  d p (T n x, T n1 x,....,T n p 1 x)
 G p (T n x)
 

 k (G p (T n1 x))  k 2 G p T n2 x  ........ k n G p x   0 as n   which gives
(xn) is a dp- Cauchy sequence in X and hence converges, say to z. That is,
lim Tnx=z exists. Since
n
x n  z as n   and since T is w- continuous, we get that Tx n  Tz as n   . That is,
x n 1  Tz as n   which gives x n  Tz as n   .
Since X is Hausdorff, we get Tz =z.
3.2 Remark: It may be noted that in view of Remark 1.3, the result proved by Troy L. Hicks ([5], Theorem 2,pp.437) is
a particular case of Theorem 3.1 .
3.3 Theorem : Suppose (X,t,dp) is a dp-complete Hausdorff topological space and T : X  X such that
there is an x  X with Gp(Ty)  k (Gp(y)) for all y  OT(x), where k : [0,)  [0,) is a nondecreasing function with k (0) = 0. Suppose kn(Gp(x))  0 as n  .Then
a) lim Tnx = z exists and
n
b) Tz = z iff Gp(x) is T - orbitally lower semi-continuous at z.
Proof :
(a) Consider dp (Tnx, Tn+1x,……, Tn+p-1x)
= Gp (Tn x)  k (Gp(Tn-1x))  k2 (Gp(Tn-2 x))……  kn (Gp(x))  0 as n   by hypothesis, which
n
gives that (T x) is a dp-Cauchy sequence in X and hence converges, say, to z.
(b) Suppose Tz = z and (Tn x) is a sequence in OT(x) with lim Tnx=z.
n
Then Gp(z) = 0 ≤ lim inf
n 
Gp(Tnx)
which gives the T-orbitally lower semi-continuity of Gp(x) at z.
Conversely, suppose that Gp(x) is T-orbitally lower semi-continuous at z.
By proof of part (a), (Tnx) is a dp-Cauchy sequence and hence 0 =
lim Gp(Tnx).
n
This together with the T-orbitally lower semi-continuity of Gp(x) at z imply Tz=z by Theorem 3.1.
3.4 Remark:The result of Troy L. Hicks ([5], Theorem 3, pp.437, 438) is a particular case of Theorem 3.3.
We now present some consequences of Theorems 3.1 and 3.3.
3.5 Corollary: Suppose (X, t, dp) is a dp-complete Hausdorff topological space, T : X  X is
w-continuous and satisfying Hp(Tx, Ty)   Hp (x, y) for all x, yX, where 0 <  < 1. Then T has
a unique fixed point z and z = lim Tnx for any x  X.
n
Proof : Taking k(t) = t, where 0 <  < 1.
Then kn (Gp(x)) = n Gp(x) and therefore, by Theorem 3.1, we have xn  z as n  , where Tz = z.
Suppose z and z*are two fixed points and taking x = z, y = z* in the inequality, we get
Hp(Tz, Tz*)   Hp(z, z*)  Hp(z, z*)   Hp(z, z*), which gives Hp(z, z*) = 0. That is, z = z*, proving
the uniqueness.
3.6 Remark: The result proved by Troy L. Hicks ([5], Corollary 2,pp.438) is a particular case
of Corolloary 3.5.
3.7 Corollary: Suppose (X, t, dp) is a dp-complete Hausdorff topological space, T : XX and
Hp(Tx, Ty)  (Hp(x,y)) Hp(x,y) for all x, y  X, where  : [0, )  [0,1) is non-decreasing.
If xn = Tny, then
a)
lim xn= z exists and
n
b) Tz = z iff Gp(x) is T-orbitally lower semi-continuous at z.
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Proof :Taking k(t) = t (t).
We find by induction that kn(t)  t ((t))n and the inequality of the theorem gives
Hp(Tx, Ty)  k(Hp(x,y)) for all x, y  X.
Now, taking y = Tx in the above inequality, we get
Hp(Tx, T2x)  k(Hp(x, Tx) for all x  X
 Gp (Tx)  k(Gp(x)) for all x  X.
Since (t) < 1, kn(Gp(x))  0 as n   and therefore the theorem follows from Theorem 3.3.
3.8 Remark : Note that, the result of Troy L. Hicks ([5], pp.438) is a particular case
of Corollary 3.7.
References:
[1]
J. Achari, Results on fixed point theorems, Maths. Vesnik 2 (15) (30) (1978), 219-221.
[2]
Ciric, B. Ljubomir, A certain class of maps and fixed point theorems, Publ. L’Inst. Math. (Beograd) 20 (1976), 73-77.
[3]
B. Fisher, Fixed point and constant mappings on metric spaces, Rend. Accad. Lincei 61(1976), 329 – 332.
[4]
K.M. Ghosh, An extension of contractive mappings, JASSY 23(1977), 39-42.
[5]
Troy. L. Hicks, Fixed point theorems for d-complete topological spaces I, Internet. J. Math & Math. Sci. 15 (1992), 435440.
[6]
Troy. L. Hicks and B.E. Rhoades, Fixed point theorems for d-complete topological spaces II, Math. Japonica 37, No.
5(1992), 847-853.
[7]
K. Iseki, An approach to fixed point theorems, Math. Seminar Notes, 3(1975), 193-202.
[8]
S. Kasahara, On some generalizations of the Banach contraction theorem, Math. Seminar Notes, 3(1975), 161-169.
[9]
S. Kasahara, Some fixed point and coincidence theorems in L-Spaces, Math. Seminar Notes, 3(1975), 181-187.
[10]
M.S. Khan, Some fixed point theorems IV, Bull. Math. Roumanie 24 (1980), 43-47.
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