Internat. J. Math. & Math. Sci.
VOL. II NO. 3 (1988) 527-534
527
ON SEQUENCES OF CONTRACTIVE MAPPINGS AND THEIR FIXED POINTS
M. IMDAD
Aligarh Muslim University
Department of Mathematics
Aligarh-202001, India
M.S. KHAN
King Abdul Aziz University
Department of Mathematics
Faculty of Science
P.O. BOX 9028
Jeddah, Saudi Arabia
S. SESSA
Universita d[ Napoli
Facolt di Architettura
Istituto di Matematica
Via Monteoliveto, 3
80134 Napoli, Italy
(Received October 12, 1984 and in revised form February 9, 1987)
By using a condition of Reich, we establish two fixed point theorems
concerning sequences of contractive mappings and their fixed points. A suitable
example is also given.
ABSTRACT.
KEY WORDS AND PHRASES: Complete metric space, Fixed point, Sequence of mappings.
1980 AMS SUBJECT CLASSIFICATION CODE. 47H10, 54H25.
i.
INTRODUCTIONS.
Throughout this paper, (X,d) denotes a complete metric space and T stands
It is well known that each of the following conditions
for a mapping of X into itself.
ensure the existence and uniqueness of a fixed point of T:
(A) (Banach).
There exists a number k, 0 & k < I, such that for each x,y in X,
d(Tx, Ty)k. d(x,y)
(B) (Rakotch [I]).
There exists a monotonically decreasing function g: (o,)[0,i)
such that for each x,y in X,xy,
d(Tx,Ty)g(d(x,y) d(x,y).
(C) (Reich [2]).
in X,xy,
There exist nonnegative numbers a,b
d(Tx,Ty)
such that for each x,y
a.d(x,y) +b. [d(x,Tx)+d(y,Ty].
M. IMDAD, M.S. KHAN and S. SESSA
528
(D)
(Reich [3]).
There exist monotonically decreasing functions a,b:
such that for each x,y in X,
(0,)[0,I)
xy,
d(Tx,Ty)<-a(d(x,y) )-d(x,y)+b(d(x,y) )- [d(x,Tx)+d(y ,Ty) ],
I.
where for any t>O,
a(t)+2b(t)
(1.2)
i.
(E) (Reich [4]). There exist functions a,b: (0,)+[0,i) such that (I.I) holds for
each x,y in X, xy, satisfying (1.2) and
limsup [a(r)+2b(r)] < i.
r+t +
(F) (Hardy and Rogers [5]). There exist monotonically decreasing functions a,b,c:
(0,)[0,I) such that for each x,y in X, xy,
d(Tx,Ty)&a(d(x,y).d(x,y)+b(d(x,y)).[d(x,Tx)+d(y,Ty)]
+c(d(x,y))’[d(x,Ty)+d(y,Tx)],
(1.3)
where for any t > 0,
a(t)+2b(t)+2c(t)
i.
(1.4)
It is not hard to show that, adopting the same proof of [4], that T has a unique
fixed point if
(G) There
exist functions a,b,c:
(0,)[0,i) such that (1.3) holds for each x,y
in
X, xy, satisfying (1.4) and
limpsup [a(r)+2b(r)+2c(r)] < I.
rt +
Evidently (A) implies (B) and (C), (B) and (C) imply (D), (D) imply (E) and (F),
(E) and (F) imply (G). Suitable examples can be found in Rhoades [6] to illustrate
some of the above implications. In the sequel, N stands for the set of natural
numbers.
The following result was established in [5] and [6].
THEOREM I. Let T heN, be mappings of (X,d) into itself satisfying condition
n
(F) with the same functions a,b,c and with fixed points z
Suppose that a mapping
n
T of X into itself can be defined pointwise by T(x)=limn Tn(X) for any x in X.
Then T has a unique fixed point z and
z=limn zN.
Theorem
generalizes an analogous result of Bonsall [7], Theorem 6 of [2] and
Theorem 4 of [3] established for mappings T satisfying conditions (A),(C) and (D),
n
respectively. Results due to Chatterjea [8] and Singh [9], Kannan type mappings,
are also included in Theorem i.
The proof of Theorem
consists essentially in the fact that the sequence
is regular, i.e. it possesses a limit
to Theorem
z(say).
{Zn}
It appears that a result corresponding
for mappings satisfying condition (G) does not exist in print in the
SEQUENCES OF CONTRACTIVE MAPPINGS AND THEIR FIXED POINTS
529
In this note, we wish to study this problem.
literature of fixed point theory.
Other
results and open problems related to the stability of fixed point of mappings can
be found in Rhoades [6], Nadler [i0], Rus [II] and Singh [12].
RESULTS.
2.
We first state the following result more general than Theorem I.
THEOREM 2. Let T ,neN, be mappings of (X,d) into itself satisfying condition
n
(G) with the same functions a,b,c and with fixed points z
Suppose that a mapping
n
T of X into itself can be defined pointwlse by T(x)= lim
T (x) for any x in X
n
{Zn
and the sequence
PROOF.
satisfies the inequality
[0,)
X
X
Since the metric d:
n
Then T has a unique fixed point z and
is regular.
z=limn_=Zn.
is continuous, the limit mapping T
By condition (G), T has a unique fixed point z.
(1.3).
We
claim that
inf
neN
d(Zn,Z)
0,
otherwise assume Z > 0. By observing that d(z ,z)>-Z>0 and hence z z for any neN,
n
n
.we deduce using (1.3) and the triangular property of the metric d,
d(Zn, z)
d(TnZn,Tz)
<_-
d(TnZn,Tnz)
+
d(TnZ,Tz)
<=
a.d(Zn,Z)+b.d(Z,TnZ)
+
c-[d(Zn,Z)+d(Z,TnZ)+d(Z,TnZn)]
+
d(TnZ,Tz)
(a+2c).d(Zn,Z)+(l+b+c).d(TnZ,Z)
for any neN, where a=a(d(z
n’ z) and similarly for b and
c.
Thus
d(z ,z)
n
2d(T z,z)
n
=<
(2 I)
1
for any neN.
If we denote with
<.
(a+2c)
{Zk(n)
a subsequence of
d(Zk(n),Z) <+
l/n,
d(Zk(n),Z)
0.
{Zn}
such that
we obtain that
lim
n-
=Z>
(2.2)
Following Reich [4], we oserve that the assumptions about the functions a,b,c of
(G) imply the existence of two functions h,k: (0,)(0,) for which, given
t>0, there exists an h(t)>0 such that 0r-t<h(t) implies
condition
a(r)+2b(r)+2c(r)k(t)<l.
By (2.2), let pen such that
0Sd(Zk(n),Z)-Z <h(Z)
(2.3)
M. IMDAD, M.S. KHAN and S. SESSA
530
for any
n>.p.
From (2.3), it follows that
a(d(Zk(n),Z))+2c(d(Zk(n),Z))<.k(Z)<
for any
n>p.
On the other hand, since the sequence
we can find an integer
(l-k(Z))/2
2d(Tz,Tk(n)(Z))
l-[a(d(Zk(n), z))+2c(d(zk(n), z))
Thus Z=0 and therefore the sequence
a contradiction.
{Zk(n)
converges to z.
Since
is regular, it has limit z and this concludes the proof.
THEOREM 3.
Zn.
converges to z=T(z),
By (2. I), then we have for any n>max {p,q},
n>q.
Z<d(Zk(n),Z)<
{zn
{Tk(n)(Z)
qeN such that
d(Tz,Tk(n)(Z))<
for any
1
Let T
n
be mappings of
(X,d)
into itself with at least one fixed point
If {Tn} converges uniformly to a mapping T of X into itself satisfying condition
(G) and if the sequence {z
n
is regular, then
z=limnZn
where z is the unique fixed
point of T.
PROOF.
We have that
d(z n
z)=d(Zn,TZ)<=d( zn TZn)+d(TZn,TZ).
<_-d( Zn, Tzn)+a. d( Zn, z )+b. d( Zn, Tz
n
+c.[d(Zn,Z)+d(z, Zn)+d(Zn TZn)]
=(a+2c).d(z n’ z) + (l+b+c)-d(Zn,TZ n)
for any neN, where
a=a(d(Zn,Z))
d(zn ,z)<
and similarly for b and c.
Thus
2d(Zn ’TZn)
1
(a+2c)
for any neN and proceeding as in the proof of Theorem 2, we get the thesis.
REMARK i.
It is evident that there certainly exists a subsequence of {z
n
converging to z, even if {z n is not regular. It is not yet known if the regularity
of
{Zn}
is a necessary condition in Theorems 2 and 3.
REMARK 2.
Theorem 3 generalizes Theorem 5 of Ray [13], which in turn extends
Theorem 9 of Reich [3].
REMARK 3.
a result
Following Ray [13] and Fraser and Nadler [14], one can establish
analogous to Theorem i0 of Reich [3] by using condition (G).
SEQUENCES OF CONTRACTIVE MAPPINGS AND THEIR FIXED POINTS
3.
531
AN EXAMPLE.
In order to illustrate the degree of generality of Theorem 2 over Theorem i,
we furnish an example which shows that there exist mappings T of X into itself satlsn
fying condition (G) but no condition (F).
EXAMPLE.
Let X=[0,1] be equipped with metric d defined as follows,
Ix-yl
f x,y
x+y
if one of x,ygN
[0,1],
d(x,y)
{I}
Then (X,d) is a complete metric space because it is isometric to a closed
subspace of the space of absolutely sunnable sequences.
For further details, see
Boyd and Wong [15].
Now we define T
n
XX nN
Tl(X)=T2(x)=0
by setting
for any x in X and for
nZ3,
x-(n-l)x2/(2n-3)
if x e[0,1],
x-I
if x e N
T (X)
n
{I}.
Further, T does not satisfy condition (F) for n.>3, otherwise we should have for y=0
n
and x=t e(0,1],
t
(n-1)t 2
(n-i)t 2
2n-3
2n-3
---.< a(t).t+b(t)
(n-i)t 2
+c(t)" [t
2n
+ t]
3
(n-l)t 2
.< [a(t)+2c(t)].t+ [b(t)-c(t)]
<
2n
Since
b(t)<I/2 for any t
3
> 0, we obtain
(n-l)t
I
2b(t) +
< 1
2n- 3
for any t e (0,I].
1
(n-l)t
2
2n- 3
This implies, for n>=3, that
i
t
<
i
3
2
(n
2n
l)-t<
I
2b(t)
3
b(t).<2b(t)<t for any t e (0,i]. Then, since b
(0,), we should have b(t)=0 for any t>0.
i.e.
3
(n-1)t 2
2b(t)].t+b(t)"
[1
2n
is monotonically decreasing in
532
M. IMDAD, M.S. KHAN and S. SESSA
Therefore, for each x,y in X, xy, and n3, the condition (F) reduces to
d( TnX, TnY)<fa(d(x, y) -d(x, y)+c( d(x, y) ). d(x, TnY)+d(y,
TnX
].
Now for x=0 and y=qeN- {I}, we deduce that
q-l<-a(q)-q+c(q) [q-l+q]<a(q).q+2c(q).q.
This implies, since a and c are monotonically decreasing functions, that
q-i
<
a(1) + 2c(i).
q
As q+, we obtain Ia(i)+2c(1)<i, a contradiction which shows that the condition
(F) is not satisfied by Tn for n3.
c(t)
On the other hand, for any neN the condition (G) holds if we choose b(t)
0 for any t>0 and a(t)=l-t/2
obviously satisfied by T
and T
For n3 and x,y in [0,I],
2.
d(TnX,TnY)=Ix-y
The condition (G) is
0<tl, a(t)=l-i/t if t>l.
if
n
[I
we get
(x+y)]
3
2n
l-l- (x+y)] <.lx-yl-[1
<]x-yl" [1
xy,
2
__1 Ix-yl]
2
c(d(x,y))*d(x,y).
Furthermore, if one of x,y lies in N
d(TnX,TnY) TnX+Tny
(x+y)-
{I} with xy and n3, then we have
"< x+y
1
1/(x+y)
+ c(d(x,y))-d(x,y).
We now define T(x)=x
x2/2
if 0<x<=l, T(x)--x-I if x is in N
z =z=O are the unique fixed points of T and T
n
n
lim T
n(X)
{i}.
Of course,
respectively and we have
T(x)
nfor any x in X.
{z n}
Thus the conclusion of Theorem 2 holds good since the sequence
converges to z.
The idea of this example appears in [15].
ACKNOWLEDGEMENT.
The authors thank an anonymous referee for some useful suggestions.
SEQUENCES OF CONTRACTIVE MAPPINGS AND THEIR FIXED POINTS
533
REFERENCES
1.
RAKOTCH, E. A Note on Contractive Mappings, Proc. Amer. Math. Soc. 1__3 (1962),
459-465.
2.
REICH, S.
Some Remarks Concerning Contraction Mappings, Canad. Math. Bull., 1__4
3.
(1971), 121-124.
REICH, S. Kannan’s Fixed
4.
REICH, S.
5.
HARDY, G.E. and ROGERS, T.D.
6.
RHOADES, B.E. A Comparison of Various Definitions of Contractive Mappings,
Trans. Amer. Math. Soc., 226 (1977), 257-290.
7.
BONSALL, F.F. Lectures on Some Fixed Point Theorems of Functional Analysis,
Tata Institute of Fundamental Research, Bombay, 1962.
8.
CHATTERJEA, S.K.
Point Theorem,
Bol....Uq.
at.
Ital.,
(4) (1971), 1-11.
(4)
Fixed Points of Contractive Functions, Boll. Un. Mat. Ital., 5
(1972), 26-42.
A Generalization of a fixed point theorem of Reich,
Canad. Math. Bull., 16 (2) (1973), 201-206.
Fixed Point Theorems,
Comptes Rend. Acad. Bulgare Sci., 2__5
(1972), 727-730.
9.
SlNGH, S.P.
Some Results on Fixed Point Theorems, Riv. Mat. Univ. Parma,
i_7(2)
(1969), 61-64.
I0.
NADLER Jr., S.B.
Sequences of Contractions and Fixed Points, Pacific. J. Math., 2_7
(1968), 579-585.
II.
RUS, I.A.
12.
SlNGH, S.P.
Results and Problems in the Metrical Fixed Point Theory, Math. Rev.
Anal. Numer. Theor. Approximation, 2__I(44) 2 (1979), 189-194.
On Sequence of Contraction Mappings, Riv. Mat. Univ. Parma,
1__1(2)
(1970), 227-231.
On Ciric’s Fixed Point Theorem, Fund. Math., 94 (1977), 221-229.
13.
RAY, B.K.
14.
FRASER Jr., R.B. and NADLER, Jr., S.B. Sequences of Contractive Maps and Fixed
Points, Pacific J. Math., 51 (1969), 659-667.
15.
BOYD, D.W. and WONG, J.S.W.
20
(1969), 458-464.
On Nonlinear Contractions, Proc. Amer. Math. Soc.,