Int. Journal of Math. Analysis, Vol. 5, 2011, no. 16, 757 - 763
Common Fixed Point Theorems for Self Maps of
a Generalized Metric Space Satisfying
A-Contraction Type Condition
M. Akram1, A. A. Zafar1 and A. A. Siddiqui2
Department of Mathematics, GC University
Lahore 54000, Pakistan
dr.makram@gcu.edu.pk, azharali@gcu.edu.pk
1
2
Department of Mathematics, King Saud University
P. O. Box 2455-5, Riyadh-11451, Saudi Arabia
asiddiqui@ksu.edu.sa
Abstract
We introduced a general class of contraction maps on a metric space, called Acontractions (that includes the contractions originally studied by R. Kannan, M. S.
Khan at el, R. Bianchini, and S. Reich), and extended some common fixed point
theorems on M. S. Khan’s contractions to general self maps of a metric
space satisfying certain A-contraction type condition; in this paper, we prove
exact analogues of these results in the setting of generalized metric spaces
(defined originally by A. Branciari).
Mathematics Subject Classification: 47H10, 54H25
Keywords: generalized metric space, self maps, fixed point, A-contraction.
1 Introduction
A general class of metric spaces was introduced by A. Branciari [5]. Various
fixed point theorems for self mappings on such generalized metric spaces have
been obtained (see [1, 5, 7], for instance). The present authors introduced the
class of A-contractions [3] that includes classes of contractions originally studied
758
M. Akram, A. A. Zafar and A. A. Siddiqui
by R. Kannan [8], M. S. Khan at el [9, 10], R. Bianchini [4] and S. Reich [12] (see
[3], for details). In [2], we extended some common fixed point theorems (appeared
in [6,11]) from M.S. Khan’s contractions to general self maps of a metric space
satisfying certain A-contraction type condition, namely, the condition A (see
below); in the sequel, a self map of a generalized metric space satisfying the
condition A is called an A-contraction. We shall prove exact analogues of our
above mentioned results for A-contractions of a generalized metric space.
2 Generalized metric spaces (gms)
We begin with the following definition of generalized metric space as introduced
originally by Branciari [5]. Throughout the sequel, N and R+ will symbolize the
set of all natural numbers and the set of all non-negative real numbers,
respectively.
Definition: Let X be a nonempty set, d: X2 → R+ a mapping such that for all x, y
E X and for all distinct points z, w E X-{x,y}, each of them different from x, y
one has
(i)
d(x, y) = 0 if and only if x = y,
(ii)
d(x ,y) = d (y, x),
(iii)
d(x, y) ≤ d(x, z) + d(z, w) + d(w, y),
then we say that (X, d) is a generalized metric space (or gms ).
Following example shows that there exists a generalized metric which is not a
metric:
Example: Let (X, d)be an infinite metric space andx, z E X be distinct elements
which are fixed throughout the discussion. Define d* : X → R by
2
d *( x , z) = d(x, z);
1
d w* (y, )= d x ( y, ) = d z ( y, )= d x( ,z), ∀ y E X x z
*
*
\{ , };
3
1
d y *w
=
d
y
x
=
d
y
z
=
(,)
( , ) d x( ,z), ∀ y E X\ x ,z ;
* ( , )
*
3
{
}
d*(y,y)=0, ∀ yE X.
We show that d* satisfies all the axioms of the generalized metric and hence d* is
a generalized metric on X . Indeed,
⎛ 1 1 1⎞
d(x,z)=⎜++⎟d(x,z),
gives
⎝333⎠
1
1
1
1
d *y( w
, ) = d x z( <, ) d x z (+ , d) x z + (d, x) z , ∀ ( y, ), w E X \
3
3
3
3
{x,z}
Common fixed point theorems
759
so that
d* (y,w) < d* (y,x)+ 31 d(x,z)+ 31 d(x, z) ,∀ y,w e X \ { x, z} .
*
Hence, d (y,w) < d (y, x) + d (x, z) + d (z,w), ∀ y,w e X \{ x,z } .
*
*
*
If p,q e X \{ x, z} then d* (y,w) < d (y, p)+ d (p, q) + d (q,w) .
*
*
*
Thus, in each case, we have
d (y,w) < d (y, p)+ d (p, q) + d (q,w)
*
*
*
*⇒
d* (x, z) < d* (x, p)+ d* (p, q) + d* (q, z),∀ p,qe X
This shows that d*is a generalized metric on X . However, d* is not a
metric on X since d* does not satisfy the triangle inequality because
2
1
1
d (x, z) > d (x, z) = d(x,z) + d(x,z)
3
3
3
*
*
d* (x, z) > d (x,w) + d (w,z), we X \{ x, z} .
Remarks: (i) If (X, d) is an infinite metric space, then this admits infinitely many
generalized metrics on X . From the above example, observe that x, z is an
arbitrary pair of distinct elements which admits d* from d depending on the choice
of x, z . Denoting the generalized metric d* corresponding to the pair x, z by d x,z ,
we obtain infinitely many generalized metrics d x,z on the infinite set X , which are
not metrics on X .
(ii) As in the usual metric space settings, a generalized metric space is a
topological space with respect to the basis given byβ = { B(x, r) : xe X, r e R} ,
where B(x, r) = { y e X : d(x, y) < r} is open ball centered at x .
Definition: Let (X,d) be a gms. A sequence {xn} in X is said to to be a Cauchy
sequence if for any 6 > 0 there exists n6 in N such that for all m, n 6 N with n < n6,
one has d(xn , xn+m) < 6. The space (X, d) is called complete if every Cauchy
sequence in X is convergent.
3 A-contractions
By A, we would mean the set of all functions α: R 3 →R + satisfying the
+
conditions:
(i) α is continuous on the set R3+ of all triplets of non-negative reals (with respect
to the Euclidean metric on R3);
(ii) a< kb for some ke[0,1) whenever a < α(a,b,b) or a < α(b,a,b) or a<α(b,b,a) for
all a, b.
Next, we introduce the following A-contraction type condition for a pair of self
maps of a generalized metric space:
760
Common
fixed point theorems
M. Akram, A. A. Zafar and A. A. Siddiqui
761
Definition: Two self maps S and T on a generalized metric space X are said to
satisfy condition (A) if
d(Sx,TSy)≤α(d(x,Sx),d(Sy,TSy),d(x,Sy)) for all x,y EX, for some α∈ A. ... (A)
4 Some common fixed point theorems
Following theorem is a generalization of [3; Theorem 1] in the setting of gms;
recall that a point x∈ X is said to be a fixed point of the self-map T of X if x = Tx.
Theorem 1: Let S and T be two self maps defined on a complete generalized
metric space X satisfying the condition (A). Then S and T have a unique common
fixed point.
Proof: Let x0EX. Define a sequence {xn} in X by x2n+1 = Sx2n, x2n = Tx2n -1 for
n=1,2,... By condition (A),
d(x2n+1,x2n+2) = d(Sx2n,TSx2n) ≤ α(d(x2n,Sx2n),d(Sx2n,TSx2n),d(x2n,Sx2n))
= α(d(x2n,x2n+1),d(x2n+1,x2n+2),d(x2n,x2n+1))
≤ kd(x2n,x2n+1) ≤ k(kd(x2n-1,x2n)) ≤ k2d(x2n-1,x2n)
with kE [0,1). In general, we have d(xn, xn+1)≤kn d(xo,x1) for all n.
Suppose that xm ≠ xn for different m, n E N and consider
d(xn,xn+3)≤d(xn,xn+1)+d(xn+1,xn+2)+d(xn+2,xn+3)≤knd(xo,x1)+kn+1d(xo,x1)+kn+2d(xo,x)
= kn(1+k+k2)d(xo,x1).
Now, for m > 2 and using the supposition that xp ≠ xr for p, r E N and p ≠ r, for
p = n, r = n + m we have d(xn,xn+m) ≤ d(xn,xn+1) + d(xn+1,xn+2)+--- +d(xn+m-1,xn+m) ≤ kn
d(x0,x1) + kn+1 d(x0,x1) +--- + kn+m-1 d(x0,x1) ≤ kn(1+k+k2+ --- +km-1) d(x0,x1), where,
kE[0,1). Thus, {xn} is a Cauchy sequence in X. Since X is complete gms, xn→x for
some x in X. We claim the point x is the required common fixed point of S and T. To
justify our claim, we proceed as follows: observe that
d(Sx,xn) = d(Sx,TSx2n-2) ≤ α(d(Sx,x) , d(TSx2n-2,Sx2n-2) , d(x, Sx2n-2))
= α(d(Sx,x), d(x2n,x2n-1), d(x, x2n-1)).
When n → ∞, d(Sx,x) ≤ α(d(Sx,x) d(x,x) ,d( x,x)) ≤ 0 gives Sx = x.
Similarly, one can show that Tx=x. Thus x is the common fixed point of S and T.
For uniqueness, let y be another fixed point of S and T. Then TSy = y so that
d(x,y) = d(Sx,TSy) ≤ α (d(Sx,x),d(Sy,TSy),d(x,Sy)) = α(d(x,x),d(y,y),d(x,y)) ≤ 0.
Hence, d(x,y) = 0 giving in turn the uniqueness. □
Corollary 1: Let S and T be two self maps of complete generalized metric space
X satisfying the condition:
d(Sx,TSy) ≤ h(d(x,Sx) + d(Sy,TSy))
... (K)
for all x, y in X and 0≤ h <1/2. Then S and T have a unique common fixed point.
Proof: We construct α: R3 → R+ by α(u, v, w) = h(v + w) where 0 ≤ h <1/2. Then
α∈ A. By taking u = d(x, y), v = d(x, Sx) and w = d(Sy, TSy) in the above
construction of α and exploiting the above condition (K) of metric d in relation to
S and T, we get d(Sx, TSy) ≤ α (d(x, Sy), d(x, Sx), d(Sy, TSy)). Thus the required
result follows from Theorem 1. □
Corollary 2: Let S and T be two self maps of complete generalized metric space
X satisfying the following condition:
d(Sx,TSy) ≤ h d(x, Sx)d(Sy , TSy)
... (M)
for all x, y in X and 0~ h < 1. Then S and T have a unique common fixed point.
Proof: If we define α: R3 —* R+ by α(u,v,w)=h(vw)1/2 where 0~h<1. So that αEA.
Now, by taking u = d(x, Sy), v = d(x, Sx) and w = d(Sy, TSy) in definition of α
and using (M) we get, d(Sx, TSy) ≤ α (d(x, Sy), d(x, Sx), d(Sy, TSy)). Thus by
Theorem1, we obtain the required conclusion. □
Corollary 3: Let S and T be two self maps of complete generalized metric space
X satisfying the following condition:
d(Sx,TSy)≤βmax{d(Sx,x)+d(TSy,Sy),d(Sx,x)+d(x,Sy),d(TSy,Sy)+d(x,Sy)}... (O)
for all x,y in X and some βE [0,1/2). Then S and T have a unique common fixed
point.
Proof: We construct α: R3 —* R+ by α(u,v,w) = β max(v+w, u+v, u+w) where βE
[0,1/2) so that αEA. Taking u = d(x, y), v = d(x, Sx) and w = d(Sy, TSy) in the
definition of α and using condition (O), we get
d(Sx, TSy) ≤ α (d(x, y), d(x, Sx), d(Sy, TSy)). Thus by Theorem1, S and T have a
unique common fixed point. □
Corollary 4: Let S and T be two self maps of complete generalized metric space
X satisfying the following condition:
d(Sx,TSy)≤βmax{ d(Sx, x), d(TSy, Sy)}
... (B)
for all x,y in X and βE [0,1). Then S and T have a unique common fixed point.
Proof: By constructing α: R3 —* R+ by α (u, v, w) = β max(v, w) where βE [0,1)
and taking u, v and w as in the Corollary 3, we see that the condition (A) satisfied.
Thus, S and T have a unique common fixed point by Theorem 1. □
Corollary 5: Let S and T be two self maps of complete generalized metric space
X such that there exist non-negative real numbers a, b, c with a + b + c ≤ 1
satisfying the following condition:
d(Sx, TSy) ≤ ad(x, Sy) + bd(Sx, x) + cd(TSy, Sy)
... (R)
for all x,y in X. Then S and T have a unique common fixed point.
Proof: Define α: R3 —* R+ by α (u, v, w) = au + bv + cw, where a, b and c are
non-negative real numbers with a + b + c ≤ 1. so that αEA. By taking suitable u, v
and w and using condition (R), one can easily get the condition (A) satisfied,
which in turn leads to the required conclusion by Theorem 1. □
The following result on sequences of self-maps generalizes [3; Theorem 2]:
762
M. Akram, A. A. Zafar and A. A. Siddiqui
Theorem 2: Let {Fn} and {Gn} be sequence of self maps in complete generalized
metric space X satisfying condition A for each m, n. Then {Fn} and {Gn} have a
unique common fixed point.
Proof: Let x0∈ X. Define a sequence {x n} in X by x2n+1 = Fn+1x2n and x2n = Gnx2n1 for n = 1, 2, 3 ... The case for which xn= x2n+1 is trivial. Next, suppose x2n ≠ x2n+1
for some n. Then we have
d(x2n+1,x2n)=d(Fn+1x2n,Gnx2n-1)≤α(d(Fn+1x2n,x2n),d(Gnx2n-1,x2n-1),d(x2n,x2n-1)
= α(d(x2n+1,x2n) , d(x2n,x2n-1) , d(x2n,x2n-1))
≤ k d(x2n, x2n-1) ≤ k(kd(x2n-1, x2n-2)).
n
In general, d(x n+1,xn) ≤ k d(x1,x o) for some k ∈ [0,1).
Next, consider d(xn,xn+3) ≤ d(xn,xn+1) +d(xn+1,xn+2)+d(xn+2,xn+3)
≤ kn d(xo,x1)+kn+1 d(xo,x1)+kn+2 d(xo,x1)
= kn(1+k+k2)d(xo,x1)
Now, for m > 2 and using the supposition that x p ≠ x r for p, r ∈ N and p ≠ r,
p = n, r = n + m we have
d(xn,xn+m) ≤ d(xn,xn+1) + d(xn+1,xn+2) +--- +d(xn+m-1,xn+m)
≤ kn d(x0,x1) + kn+1 d(x0,x1) +--- + kn+m-1 d(x0,x1)
≤ kn(1+k+k2+ --- +km-1) d(x0,x1), where k∈ [0,1).
Taking limit as n →∞ , d(x n , x n+m ) → 0. This shows that {x n } is Chauchy
sequence in complete gms X and hence converges to some x in X.
Next, for each m, we have that
d(x,Fmx)≤d(x,x2n-1)+d(x2n-1,x2n)+d(x2n,Fmx)= d(x,x2n)+ d(x2n-1,x2n)+d(Gnx2n-1,Fmx)
= d(x,x2n)+ d(x2n-1,x2n)+α(d(Gnx2n-1,x2n-1),d(Fmx,x),d(x2n-1,x))
= d(x,x2n)+ d(x2n-1,x2n)+α(d(x2n,x2n-1),d(Fmx,x),d(x2n-1,x))
and so by letting n→∞, we get
d(x,Fmx) ≤ d(x,x)+ d(x,x)+α(d(x,x),d(Fmx,x),d(x,x))
≤ 0+ 0+α(0,d(Fmx,x) ,0) ≤ 0.
This gives d(x, Fmx ) = 0 so that x = Fmx for all m. Similarly x = Gmx for all m.
Thus x is a common fixed point of F n and G n for all n. The uniqueness of
common fixed point x can be seen easily. □
The results given below are immediate consequences of Theorem 2:
Corollary 6: Let {Fn} and {Gn} be sequence of self maps in complete gms X
satisfying d(Fnx, Gny) ≤ h(d(Fnx, x) + d(Gny, y)) for all x, y in X and 0 ≤ h < 2
1 . Then Fn and Gn have a unique common fixed point.
Corollary 7: Let {Fn} and {Gn} be sequence of self maps in complete gms X
satisfying d(Fnx, Gny) ≤ h d(x,Fnx)d(y,Gny) for all x, y in X and 0≤ h < 1.
Then Fn and Gn have a unique common fixed point.
Common fixed point theorems
763
Corollary 8: Let {Fn} and {Gn} be sequence of self maps in complete gms X
satisfying d(Fnx,Gny)<3max{d(Fnx,x)+d(Gny,y),d(Fnx,x)+d(x,y),d(Gny,y)+d(x,y)}
for all x,y in X and some 3∈ [0,1/2). Then Fn and Gn have unique common fixed
point.
Corollary 9: Let {Fn} and {Gn} be sequence of self maps in complete gms X
satisfying d(Fnx, Gny) < 3 max{d(Fnx, x), d(Gny, y))for all x,y in X and some
3∈ [0,1). Then Fn and Gn have a unique common fixed point.
Corollary 10: Let {Fn} and {Gn} be sequence of self maps in complete gms X
such that there exists non-negative real numbers a, b, c with a + b + c < 1
satisfying d(Fnx, Gny) < ad(Fnx, x) + bd(Gny, y) + cd(x, y) for all x, y in X. Then
Fn and Gn have a unique common fixed point.
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Received: September, 2010