I ntrn. J. Math. & Math. Sci.
(1978)57-62
57
Vol.
A FIXED POINT THEOREM FOR A NONLINEAR TYPE CONTRACTION
S. A. HUSAIN and V. M. SEHGAL
Department of Mathematics
University of Wyoming
Laramie, Wyoming 82071
(Received January 31, 1978)
ABSTRACT.
A well-known result of Boyd and Wong [i] on nonlinear contractions is
extended.
Several other known results are obtained as special cases.
INTRODUCTION.
In this paper, we extend a well-known result of Boyd and Wong [i] and obtain
as consequences several other known results (see
[2], [3], [4], [5]).
+
Throughout this paper, let (X,d) be a complete metric space, R the nonnegatlve
(tl,t2,t3,t4,t5):(R+)
reals and
S
/
R+ a function which is (a) continuous from
right in each coordinate variable (b) nondecreasing in
fies the inequality (c) (t,s,s,as,bs) <
{a,b} c (0,i,2}
with a
+
2.
b
Max{t,s}
if
t2, t3, t4, t5,
Max{t,s}
and satis-
0 where
Note that (c) implies that (t,t,t,t,t) < t for
any t > 0.
2.
MAIN RESULTS.
The following is the main result of this paper.
THEOREM i.
Let f,g:X
/
X be two comutatlve mappings such that
(1)
fX c X,
(ii)
g
(ill)
d(fx,fy) <_ (d(gx,gy),
for each x, y
is continuous,
X.
d(fx, gx),
d(fy,gy),
Then, there exists a unique u
d(fx,gy),
X with fu
gu
d(fy,gx)),
u.
S. A. HUSAIN & V. M. SEHGAL
58
We first prove the following lemma which simplifies the proof of the above
theorem.
LEMMA.
fv
Under the conditions of Theorem i, if there exists a v
X with fu
gv, then there exists a unique u
PROOF.
g(w)
Suppose
t
d(fv,fw)
> 0.
Thus fv
a contradiction.
(2.1)
f (w)
f(v)
implies
Then it follows by (iii) that
(t,0,0,t,t)
t <
u.
X
We show that for any w
f(w)
gu
X such that
<
(t,t,t,t,t)
Now, since fw
fw.
< t,
gw, therefore, f(fw)
g(fw) and
consequently by (2.1)
f(w)
f(w), then fu
Thus, if we set u
g(fw).
f(fw)
gu
u.
The uniqueness of u now follows from
(2 .).
PROOF OF THEOREM i.
{yn
in X as follows.
Yo
gx I.
Set
Yl
exists by
(i)
a
Let x be an arbitrary point in X.
O
Let
fxl"
Xn+I
By (i) there exists a x
I
fXo.
Yo
Thus
if
e X such
Construct a sequence
Yo Yl "’Yn are
that
Yn gXn+l"
X such that
obtained with
Let
Yn+l
Yn
fXn+l"
fx
n’ there
Thus, for
each n e I (nonnegative Integers),
Yn
{yn }
d(Yn,Yn+l )"
We shall show that
n
I,
dn
fXn
dn+I
X.
is a Cauchy sequence in
For this, let for each
Then by (i) and (b),
dn+I d(fxn+l,fxn+2)
Now, if for some n e I,
(2.2)
gXn+l"
dn+I > dn,
_< @(d n,dn,dn+l,0,dn+
then by (b) and (c)
<_ (dn, dn+ I, dn+l, 0,2dn+I)
<
dn+l,
dn+l)"
(2.3)
NONLINEAR TYPE CONTRACTION
I,
Thus for each n
a contradiction.
dn+I
_< dn,
that is
59
{dn }
is a nonincreasing
+
sequence of nonnegative reals and consequently there exists a d e R such that
{d
/
n
d.
0, for otherwise by (2.3) and (c),
Clearly d
d < (d,d,d,0,2d) < d,
Thus,
a contradiction.
-
{yn
Suppose, now that
d
/
n
(2.4)
0.
is not a Cauchy sequence.
Then there exists a E > 0 such
that for each k e I, there exist integers n(k), re(k) with k < n(k) < m(k) satisylng
d(Yn(k),Ym(k))
>
E.
Let m(k) be the least integer greater than n(k) such (2.4) holds.
that for each k e I,
E < E
k
<_
d(Yn(k),Ym(k)_l )
d(Yn(k),Ym(k)_l)
+
Hence, it follows by (2.4) that as k
However, for each k
E
Therefore, as k
+
k
_< E.
Consequently, for each k e I,
d(Ym(k)_l,Ym(k))
/
This implies
,
/
<_ E + d k.
(2.5)
E.
I,
<_
dn(k)
<_
2d
k
+
+
d(fXn(k)+ffXm(k)+l) + dm(k),
Ek, dk, dk, Ek+dk, Ek+dk),
,
E < (E,0,0,E,E) < E,
contradicting the existence of E > 0.
Thus,
Consequently, there is a v e X such that
fx
n
{yn }
gxn+l
/
{yn }
/
v.
We show that for this v,
d(fv,gv)
0.
is a Cauchy sequence in X.
v, that is
(2 6)
S. A. HUSAIN & V. M. SEHGAL
60
> 0.
Suppose
Now by (ii) and (2.6) we have,
f gx
n
gfxn
gv and
/
g2xn
+
gv.
Also, it follows by (b) and (iii) that,
d(f(gXn),fv) <_
Therefore, as n
+
(d(g2Xn,gV) d(fgXn,g2Xn)
,
a
+
d(gv,g2Xn)).
d(gv, fv) <_ (0,0,a,0,a)
<
a,
gv and hence by the above lemma, there is a unique
Thus fv
u s X satisfying fu
d(fgXn,gV)
the above inequality yields that
a
a contradiction.
,
gu
u.
In the special case when g is taken to be the identity map of x in Theorem i,
we have
COROLLARY i.
Let f:X
/
X satisfy either of the following conditions: for all
x,y e X,
(A).
d(fx, fy) <_ (d(x,y), d(x,fx), d(y,fy), d(x,fy), d(y,fx)).
(B).
d(fx, fy) <_ a(d(x,fx) + d(y,fy)) + B(d(x, fy) + d(y,fx)) + (d(x,y))
where a
(t)
<
+
_> 0, 8 _> 0 and :R
/
(l-2a-2)t if t > 0.
PROOF.
+
R
is a right continuous function satisfying
Then f has a unique fixed point in X.
The conclusion is an obvious consequence of Theorem i if (A) holds.
In case of condition (B), let
’(R+) 5
+
(tl,t2,t3,t4,t5)
then
satisfies conditions
+
R
be defined by
(tl)
(a), (b) and (c).
+ a(t 2 +
t
3)
+ B(t 4 +
t5).
Thus the conclusion again follows
by Theorem I.
It may be remarked that if a
known result of Boyd and Wong [i].
results of Hardy and Rogers
B
0 in (B) then Corollary 1 yields a wellIf (t)
[2], Kannan [3],
results are special cases of Theorem i.
at, then Corollary I yields certain
Reich
[4], Sehgal [5].
All these
NONLINEAR TYPE CONTRACTION
61
REFERENCES
i.
Boyd, D. W. and S. W. Wong.
20 (1969) 458-464.
2.
Hardy, G. E. and T. D. Rogers. A Generalization of a Fixed Point Theorem of
Reich, Canad. Math. Bull. 16 (2) (1973) 201-206.
3.
Kannan, R.
4.
Reich, S.
5.
Sehgal, V. M. Some Fixed and Common Fixed Point Theorems in Metric Spaces,
Canad. Math. Bull. 16 (1974) 257-259.
On Nonlinear Contractions, Proc. Amer. Math Soc.
Some Remarks on Fixed Points, Bull. Cal. Math. Soc. 60 (1968) 71-76.
Kannan’s
KEY WORDS AND PHRASES.
Fixed Point Theorems, Bull. U.M.I. 4 (1971) i-ii.
Nonlinea type coction, Fixed point theorems.
AMS(MOS) SUBJECT CLASSIFICATIONS (1970).
47H10o