Hacettepe Journal of Mathematics and Statistics
Volume 39 (2) (2010), 151 – 158
A COMMON FIXED POINT THEOREM FOR
WEAKLY COMPATIBLE MAPPINGS IN
SYMMETRIC SPACES SATISFYING AN
INTEGRAL TYPE CONTRACTIVE
CONDITION
Rakesh Tiwari∗†, S. K. Shrivastava‡ and V. K. Pathak§
Received 28 : 08 : 2008 : Accepted 03 : 07 : 2009
Abstract
In this note a common fixed point theorem for weakly compatible mappings satisfying a contractive condition of integral type and the common
(E.A) property is established in symmetric spaces. This theorem generalizes and improves results of M. Aamri and D. El Moutawakil (Common fixed points under contractive conditions in symmetric spaces,
Applied Mathematics E-notes 3, 156–162, 2003; Some new common
fixed point theorems under strict contractive conditions, J. Math. Anal.
Appl. 270, 181–188, 2002) and Abdelkrim Aliouche (A common fixed
point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type, J. Math. Anal. Appl.
322 (2), 796–802, 2006).
Keywords: Common fixed points, Weakly compatible mappings, Symmetric spaces,
Common (E.A) property.
2000 AMS Classification: Primary: 47 H 10. Secondary: 54 H 25.
∗
Department Of Mathematics, Govt. V. Y. T. PG. Autonomous College, Durg (C.G.), 491001
India. E-mail: [email protected]
†
Corresponding Author.
‡
Department of Mathematics, D. D. U. University, Gorakhpur (U.P.), 273009 India.
E-mail: [email protected]
§
Department Of Mathematics, Govt. PG. College, Dhamtari (C.G.), 493773 India.
E-mail: [email protected]
152
R. Tiwari, S. K. Shrivastava, V. K. Pathak
1. Introduction
A symmetric on a set X is a nonnegative real valued function d on X × X such that
(i) d(x, y) = 0 if and only if x = y,
(ii) d(x, y) = d(y, x).
Hicks and Rhoades[4] established some common fixed point theorems in symmetric spaces
using the fact that some of the properties of metrics are not required in the proofs of
certain metric theorems.
Let d be a symmetric on a set X and for r > 0 and any x ∈ X, let B(x, r) = {y ∈ X :
d(x, y) < r}. A topology t(d) on X is given by U ∈ t(d) if and only if for each x ∈ U ,
B(x, r) ⊂ U for some r > 0. A symmetric d is a semi-metric if for each x ∈ X and each
r > 0, B(x, r) is a neighborhood of x in the topology t(d). Note that limn→∞ d(xn , x) = 0
if and only if xn → x in the topology t(d).
The following two axioms appeared in Wilson[9] for a symmetric space (X, d).
(W.3) Given {xn }, x and y in X,
lim d(xn , x) = 0 and lim d(xn , y) = 0 imply x = y.
n→∞
n→∞
(W.4) Given {xn }, {yn } and x in X,
lim d(xn , x) = 0 and lim d(xn , yn ) = 0 imply that lim d(yn , x) = 0.
n→∞
n→∞
n→∞
Commuting, weakly commuting, compatible and weakly compatible mappings have been
frequently used to prove existence theorems in common fixed point theory. Recall that,
Jungck and Rhoades [6] defined S and T to be weakly compatible if they commute at
there coincidence points, i.e. if Su = T u for some u ∈ X, then ST u = T Su.
Many examples in the literature illustrate that commuting implies weakly commuting
implies compatible implies weakly compatible maps, but the converse need not be true
(see [5] and [8]).
Amri and Moutawakil [2] have established some common fixed point theorems under
strict contractive conditions on a metric space for mappings satisfying the property (E.A).
Again recall that the pair (S, T ) satisfies the property (E.A) if there exists a sequence
{xn } in X such that
lim Sxn = lim T xn = t for some t ∈ X.
n→∞
n→∞
Most recently, W. Liu et al. [7] defined a common (E.A) property for pair of mappings
as follows :
1.1. Definition. Let A, B, S, T : X → X. The pairs (A, S) and (B, T ) satisfy the
common (E.A) property if there exist two sequences {xn } and {yn } in X such that
lim Axn = lim Sxn = lim Byn = lim T yn = t ∈ X.
n→∞
n→∞
n→∞
n→∞
If B = A and S = T in the above, we obtain the definition of the property (E.A).
Now we present an example of the above definition as follows:
A Common Fixed Point Theorem
153
1.2. Example. Let A, B, S and T be self maps on X = [0, 1], with the usual metric
d(x, y) = |x − y|, defined by:
(
1 − x2 when x ∈ 0, 12 ,
Ax =
1
when x ∈ 21 , 1 ,
Bx = 1 − x, for all x ∈ X,
(
1 − 2x when x ∈ 0, 12 ,
1 Sx =
1
when x ∈ 2 , 1 ,
T x = 1 − x3 , for all x ∈ X.
Let {xn } and {yn } be the sequences defined by xn =
1
n+1
and yn =
1
.
n2 +1
Then
lim Axn = lim Sxn = lim Byn = lim T yn = 1 ∈ X.
n→∞
n→∞
n→∞
n→∞
Thus the common (E.A) property is satisfied.
2. Preliminaries
In the sequel, we need a function φ : R+ → R+ satisfying the condition 0 < φ(t) < t
for each t > 0.
2.1. Definition. [3, Definition 4] Let (X, d) be a symmetric space. We say that (X, d)
satisfies property (HE) if given {xn }, {yn } and x in X,
lim d(xn , x) = 0 and lim d(yn , x) = 0 imply lim d(xn , yn ) = 0.
n→∞
n→∞
n→∞
Recently Abdelkrim Aliouche [3] proved a common fixed point theorem for self mappings in a symmetric space under a contractive condition of integrals and satisfying a
new property introduced recently in[2].
The main object of this paper is to prove a common fixed point theorem for weakly
compatible mappings in the setting of symmetric spaces satisfying an integral type contractive condition and the common (E.A) property. In this process, we are dropping two
conditions, namely the properties (H.E) and (W.4). Thus, this Theorem generalizes and
improves the results of Aamri et al. [1, 2] and Abdelkrim Aliouche [3].
3. Main results
Now we present our main result.
3.1. Theorem. Let d be a symmetric for X which satisfies (W.3 ). Let A, B, S and T
be self mappings of X such that
(3.1)
(3.2)
Z
d(Ax,By)
0
A(X) ⊂ T (X) and B(X) ⊂ S(X),
Z aL(x,y)+(1−a)M (x,y)
ϕ(t) dt ≤ φ
ϕ(t) dt ,
0
for all x, y ∈ X, where ϕ : R+ → R+ is a Lebesgue-integrable mapping which is summable,
non-negative and such that
Z ǫ
(3.3)
ϕ(t)dt > 0, for all ǫ > 0,
0
154
R. Tiwari, S. K. Shrivastava, V. K. Pathak
where,
L(x, y) = max d(Sx, T y), d(Sx, By), d(By, T y)},
M (x, y) = max{d2 (Sx, T y), d(Sx, By)d(By, T y), d(Sx, T y)d(Sx, By),
d(Sx, T y)d(By, T y), d2 (By, T y)
1/2
and 0 ≤ a ≤ 1. Suppose that (A, S) and (B, T ) satisfy the common (E.A) property and
are weakly compatible. If one of the subspaces AX, BX, SX and T X of X is complete,
then A, B, S and T have a unique common fixed point in X.
Proof. Since (A, S) and (B, T ) satisfies the common (E.A) property, there exist two
sequences {xn } and {yn } in X such that
lim d(Bxn , z) = lim d(T xn , z) = lim d(Ayn , z) = lim d(Syn , z) = 0
n→∞
n→∞
n→∞
n→∞
for some z ∈ X.
Now suppose that SX is a complete subspace of X. Then z = Su for some u ∈ X.
Consequently, we have
lim d(Ayn , Su) = lim d(Bxn , Su) = lim d(T xn , Su) = lim d(Syn , Su) = 0.
n→∞
n→∞
n→∞
n→∞
If Au 6= z, using (3.2) we get
(3.4)
Z
d(Au,Bxn )
ϕ(t) dt ≤ φ
0
<
Z
Z
aL(u,xn )+(1−a)M (u,xn )
ϕ(t) dt
0
aL(u,xn )+(1−a)M (u,xn )
ϕ(t) dt,
0
where
L(u, xn ) = max{d(Su, T xn ), d(Su, Bxn ), d(Bxn , T xn )}, and
M (u, xn ) = max d2 (Su, T xn ), d(Su, Bxn )d(Bxn , T xn ), d(Su, T xn )d(Su, Bxn ),
1/2
d(Su, T xn )d(Bxn , T xn ), d2 (Bxn , T xn )
.
Taking n → ∞, we get L(u, xn ) = 0, and M (u, xn ) = 0 respectively. Now (3.4) becomes
limn→∞
Z
d(Au,Bxn )
ϕ(t) dt = 0,
0
and (3.3) implies that limn→∞ d(Au, Bxn ) = 0. By (W.3), we have z = Au = Su.
Since AX ⊂ T X, there exists v ∈ X such that z = Au = T v.
If Bv 6= z, using (3.2) we have
(3.5)
Z
d(z,Bv)
ϕ(t) dt =
0
Z
d(Au,Bv)
ϕ(t) dt
0
≤φ
Z
aL(u,v)+(1−a)M (u,v)
0
ϕ(t) dt ,
A Common Fixed Point Theorem
155
1/2
from which we get L(u, v) = d(z, Bv), and M (u, v) = d2 (z, Bv)}
, respectively. Now
(3.5) becomes
Z
d(Au,Bv)
ϕ(t)dt ≤ φ
0
=φ
<
Z
Z
ad(z,Bv)+(1−a)d(z,Bv)
0
Z
d(z,Bv)
0
d(z,Bv)
ϕ(t) dt ,
ϕ(t) dt ,
ϕ(t)dt,
0
which is a contradiction. Hence
Z
d(z,Bv)
ϕ(t) dt = 0,
0
and (3.3) implies that z = Bv = T v.
The pair (A, S) is weakly compatible, so ASu = SAu whenever Au = Su, which
implies Az = Sz.
Let us show that z is a common fixed point of A, B, S and T .
If z 6= Az, using (3.2) we get
(3.6)
Z
d(z,Az)
ϕ(t) dt =
0
Z
d(Az,Bv)
ϕ(t) dt
0
≤φ
Z
aL(z,v)+(1−a)M (z,v)
0
ϕ(t) dt ,
1/2
so L(z, v) = d(z, Az), and M (z, v) = d2 (z, Az)}
respectively. Now (3.6) becomes
Z
d(Az,z)
ϕ(t) dt ≤ φ
0
<
Z
Z
ad(Az,z)+(1−a)d(Az,z)
ϕ(t) dt ,
0
d(z,Az)
ϕ(t) dt,
0
which is a contradiction. Therefore
Z
d(z,Az)
ϕ(t) dt = 0,
0
and (3.3) implies that z = Az = Sz.
Similarly, the weak compatibility of B and T implies BT v = T Bv, i.e. Bz = T z. If
z 6= Bz, by using (3.2) and (3.3) a similar calculation to the above yields z = Bz = T z.
Thus z is a common fixed point of A, B, S and T .
When T X is assumed to be complete subspace of X, then the proof is similar. On
the other hand the cases in which AX or BX is a complete subspace of X are similar to
the cases in which T X or SX is complete, respectively, by (3.1).
156
R. Tiwari, S. K. Shrivastava, V. K. Pathak
For the uniqueness of the common fixed point z, let w 6= z be another common fixed
point of A, B, S and T . Then using (3.2), we obtain
Z d(z,w)
Z d(Az,Bw)
ϕ(t)dt =
ϕ(t) dt
0
0
≤φ
=φ
≤φ
<
Z
Z
Z
Z
aL(z,w)+(1−a)M (z,w)
ϕ(t) dt
0
ad(z,w)+(1−a)d(z,w)
ϕ(t) dt
0
d(z,w)
ϕ(t)dt
0
d(z,w)
ϕ(t) dt,
0
which is a contradiction. Therefore
R d(z,w)
0
ϕ(t) dt = 0, and (3.3) implies that z = w.
3.2. Remark. If we take a = 1 in Theorem 3.1 we obtain an improved and more general
version of Aliouche [3, Theorem 1].
For ϕ(t) = 1 in Theorem 3.1, we obtain the following corollary.
3.3. Corollary. Let d be a symmetric for X which satisfies (W3 ). Let A, B, S and T
be self mappings of X such that
(3.7)
(3.8)
A(X) ⊂ T (X) and B(X) ⊂ S(X),
d(Ax, By) ≤ φ aL(x, y) + (1 − a)M (x, y)
for all x, y ∈ X, where
L(x, y) = max{d(Sx, T y), d(Sx, By), d(By, T y)},
M (x, y) = max d2 (Sx, T y), d(Sx, By)d(By, T y), d(Sx, T y)d(Sx, By),
d(Sx, T y)d(By, T y), d2 (By, T y)
1/2
and 0 ≤ a ≤ 1. Suppose that (A, S) and (B, T ) satisfies the common (E.A) property and
are weakly compatible. If one of the subspaces AX, BX, SX or T X of X is complete,
then A, B, S and T have a unique common fixed point of X.
3.4. Remark. If we take a = 1 in Corollary 3.3, we obtain an improved version of [1,
Theorem 2.2].
Again, for B = A and T = S in Theorem 3.1, we get the following corollary.
3.5. Corollary. Let d be a symmetric on X which satisfies (W.3 ). Let A and S be self
mappings of X such that
(3.9)
(3.10)
Z
d(Ax,Ay)
0
A(X) ⊂ S(X),
Z aL(x,y)+(1−a)M (x,y)
ϕ(t) dt ≤ φ
ϕ(t) dt ,
0
for all x, y ∈ X, where ϕ : R+ → R+ is a Lebesgue-integrable mapping which is summable,
non-negative and satisfies (3.3),
L(x, y) = max{d(Sx, Sy), d(Sx, Ay), d(Ay, Sy)},
M (x, y) = max{d2 (Sx, Sy), d(Sx, Ay)d(Ay, Sy), d(Sx, Sy)d(Sx, Ay),
1/2
d(Sx, Sy)d(Ay, Sy), d2 (Ay, Sy)}
,
A Common Fixed Point Theorem
157
and 0 ≤ a ≤ 1. Suppose that (A, S) satisfies the property (E.A) and is weakly compatible.
If one of the subspaces AX or SX of X is complete, then A and S have a unique common
fixed point in X.
Again for ϕ(t) = 1 in Corollary 3.5, we obtain the following corollary.
3.6. Corollary. Let A and B be self mappings of X such that
(3.11)
(3.12)
A(X) ⊂ S(X),
d(Ax, Ay) ≤ φ aL(x, y) + (1 − a)M (x, y)
for all x, y ∈ X, where
L(x, y) = max{d(Sx, Sy), d(Sx, Ay), d(Ay, Sy)},
M (x, y) = max d2 (Sx, Sy), d(Sx, Ay)d(Ay, Sy), d(Sx, Sy)d(Sx, Ay),
d(Sx, Sy)d(Ay, Sy), d2 (Ay, Sy)
1/2
and 0 ≤ a ≤ 1. Suppose that (A, S) satisfies the property (E.A) and is weakly compatible.
If one of the subspaces AX or SX of X is complete, then A and S have a unique common
fixed point in X.
3.7. Remark. If we take a = 1 in Corollary 3.5, we obtain a generalized and improved
form of [1, Theorem 2.1].
Since the non-compatibility of two mappings in a symmetric space implies that they
satisfy the common (E.A) property, we get the following corollary.
3.8. Corollary. Let d be a symmetric for X which satisfies (W.3 ). Let A, B, S and T
be self mappings of X satisfying (3.1) and (3.2) for all x, y ∈ X, where ϕ : R+ → R+
is a Lebesgue-integrable mapping which is summable, non-negative and satisfies (3.3).
Suppose that (A, S) and (B, T ) are weakly compatible but noncompatible. If one of the
subspaces AX, BX, SX or T X of X is complete, then A, B, S and T have a unique
common fixed point in X.
3.9. Remark. If we take ϕ(t) = 1 in Corollary 3.8, we obtain a generalized and improved
form of [2, Theorem 2].
Acknowledgement The authors are very thankful to Prof. H. K. Pathak for his valuable
suggestions regarding this paper.
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