General Mathematics Vol. 18, No. 4 (2010), 163–174
Common fixed point theorems for subcompatible
D-maps of integral type 1
H. Bouhadjera, A. Djoudi
Abstract
Some common fixed point theorems for two pairs of subcompatible
single and multivalued D-maps in metric spaces are obtained extending
some results of single-valued maps of Jungck and Rhoades [9].
2010 Mathematics Subject Classification: 47H10, 54H25.
Key words and phrases: Weakly compatible maps, Subcompatible maps,
Occasionally weakly compatible maps, Integral type, D-maps, Common fixed
point theorems.
1
Introduction
To generalize commuting maps, Sessa [10] introduced the notion of weakly
commuting maps.
Later on, Jungck generalized commuting and weakly commuting maps,
first to compatible maps [6] and then to weakly compatible maps [7].
And in 1998, the same author with Rhoades [8] extended the concept of
weakly compatible maps to the setting of single and multivalued maps by
giving the notion of subcompatible maps.
Recently in 2008, Al-Thagafi and Shahzad [2] introduced the concept of
occasionally weakly compatible maps (owc) which is a proper generalization
of nontrivial weakly compatible maps which do have a coincidence point.
1
Received 23 February, 2009
Accepted for publication (in revised form) 25 March, 2009
163
164
2
H. Bouhadjera, A. Djoudi
Preliminaries
Throughout this paper X stands for a metric with the metric d and B(X )
denotes the family of all nonempty, bounded subsets of X . Define for all A,
B in B(X )
δ(A, B) = sup{d(a, b) : a ∈ A, b ∈ B}.
If A = {a}, we write δ(A, B) = δ(a, B) and δ(A, B) = d(a, b) if A = {a} and
B = {b}. For all A, B, C in B(X ), the definition of δ yields the following
properties:
δ(A, B) = δ(B, A) ≥ 0,
δ(A, B) ≤ δ(A, C) + δ(C, B),
δ(A, A) = diamA,
δ(A, B) = 0 ⇔ A = B = {a}.
Definition 1 ([4]) A sequence {An } of nonempty subsets of X is said to be
convergent towards a subset A of X if,
(i) each point a of A is a limit of a convergent sequence {an }, where an ∈ An
for n ∈ N,
(ii) for arbitrary ² > 0, there is an integer m such that n > m, An ⊆ A² .
A² = {x ∈ X : ∃a ∈ A, a depending on x and d(x, a) < ²}. A is then said to
be the limit of the sequence {An }.
Lemma 1 ([4]) Let {An }, {Bn } be sequences in B(X ) converging respectively
to A and B in B(X ), then the sequence of numbers {δ(An , Bn )} converges to
δ(A, B).
Lemma 2 ([5]) Let {An } be a sequence in B(X ) and y be a point in X such
that δ(An , y) → 0. Then the sequence {An } converges to the set {y} in B(X ).
Definition 2 ([10]) Self-maps f and g of a metric space (X , d) are said to be
weakly commuting if, for all x ∈ X
d(f gx, gf x) ≤ d(gx, f x).
Definition 3 ([6]) Self-maps f and g of a metric space (X , d) are called compatible if
lim d(f gxn , gf xn ) = 0
n→∞
Common fixed point theorems for subcompatible...
165
whenever {xn } is a sequence in X such that lim f xn = lim gxn = t for some
n→∞
n→∞
t ∈ X.
Definition 4 ([7]) Two maps f , g : X → X are said to be weakly compatible
if they commute at their coincidence points.
Definition 5 ([8]) Maps f : X → X and F : X → B(X ) are said to be
subcompatible if they commute at coincidence points; that is,
{t ∈ X /F t = {f t}} ⊆ {t ∈ X /F f t = f F t}.
Definition 6 ([2]) Two self-maps f and g of a set X are owc if and only if
there is a point t ∈ X which is a coincidence point of f and g at which f and
g commute.
In their paper [3], Djoudi and Khemis gave the notion of D-maps which
extended the notion of property (E.A) given by Aamri and El Moutawakil [1].
Definition 7 ([3]) Maps f : X → X and F : X → B(X ) are said to be
D-maps iff there exists a sequence {xn } in X such that for some t ∈ X
lim f xn = t and lim F xn = {t}.
n→∞
n→∞
Our objective here is to prove some common fixed point theorems for two
pairs of subcompatible single and multivalued D-maps satisfying contractive
condition of integral type in metric spaces. These results extend the results of
Jungck and Rhoades [9].
For our main results we need the following:
Let Ψ be the set of all continuous maps ψ : R+ → R such that
(ψ1 ) : for all u, v in R+ , if
(ψa ) : ψ(u, v, v, u, u + v, 0) ≤ 0 or
(ψb ) : ψ(u, v, u, v, 0, u + v) ≤ 0
we have u ≤ v
(ψ2 ) : ϕ(u, u, 0, 0, u, u) > 0 for all u > 0,
next, let Φ be the set of all maps ϕ : R+ → R+ such that ϕ is LebesgueR²
integrable which is summable nonnegative and satisfies 0 ϕ(t)dt > 0 for each
² > 0,
and let F be the set of all continuous maps z : R+ → R+ such that z(t) = 0
iff t = 0.
166
3
H. Bouhadjera, A. Djoudi
Main results
Theorem 1 Let (X , d) be a metric space and let f , g : X → X ; F , G : X →
B(X ) be single and multivalued maps, respectively. Suppose that
(1) f and g are surjective,
ÃZ
(2)
Z
δ(F x,Gy)
ψ
ϕ(t)dt,
0
Z
Z
d(f x,gy)
0
Z
δ(gy,Gy)
ϕ(t)dt ,
0
Z
δ(f x,Gy)
ϕ(t)dt,
0
δ(f x,F x)
ϕ(t)dt,
ϕ(t)dt,
0
!
δ(gy,F x)
ϕ(t)dt
≤0
0
for all x, y in X , where ψ ∈ Ψ and ϕ ∈ Φ. If either
(3) f and F are subcompatible D-maps; g and G are subcompatible, or
(30 ) g and G are subcompatible D-maps; f and F are subcompatible.
Then, f , g, F and G have a unique common fixed point t ∈ X such that
F t = Gt = {f t} = {gt} = {t}.
Proof. Suppose that f and F are D-maps, then, there exists a sequence {xn }
in X such that lim f xn = t and lim F xn = {t} for some t ∈ X . By vertue of
n→∞
n→∞
condition (1) there are two points u and v in X such that t = f u = gv.
We show that Gv = {gv}. Indeed, by inequality (2) we have
ÃZ
Z
δ(F xn ,Gv)
ψ
ϕ(t)dt,
0
Z
Z
d(f xn ,gv)
0
Z
δ(gv,Gv)
ϕ(t)dt ,
0
Z
δ(f xn ,Gv)
ϕ(t)dt,
0
δ(f xn ,F xn )
ϕ(t)dt,
!
δ(gv,F xn )
ϕ(t)dt,
ϕ(t)dt
0
≤ 0.
0
Since ψ is continuous, we get at infinity
ÃZ
ψ
Z
δ(gv,Gv)
ϕ(t)dt, 0, 0,
0
Z
δ(gv,Gv)
ϕ(t)dt,
0
δ(gv,Gv)
!
ϕ(t)dt, 0
≤0
0
R δ(gv,Gv)
which from (ψa ), gives 0
ϕ(t)dt ≤ 0, and hence δ(gv, Gv) = 0, which
implies that Gv = {gv} = {t}. Since the pair (g, G) is subcompatible, then,
Ggv = gGv; i.e., Gt = {gt}.
Common fixed point theorems for subcompatible...
167
We claim that Gt = {gt} = {t}. If not, then condition (2) implies that
ÃZ
Z d(f xn ,gt)
Z δ(f xn ,F xn )
δ(F xn ,Gt)
ϕ(t)dt,
ϕ(t)dt,
ϕ(t)dt ,
ψ
0
Z
0
Z
δ(gt,Gt)
0
Z
δ(f xn ,Gt)
ϕ(t)dt,
ϕ(t)dt,
0
ϕ(t)dt
0
At infinity we get
ÃZ
Z
d(t,gt)
ψ
ϕ(t)dt,
0
!
δ(gt,F xn )
≤ 0.
0
Z
d(t,gt)
Z
d(t,gt)
ϕ(t)dt, 0, 0,
ϕ(t)dt,
0
0
!
d(gt,t)
ϕ(t)dt
≤0
0
R d(t,gt)
which contradicts (ψ2 ). Thus, 0
ϕ(t)dt = 0, which implies that {gt} =
{t} = Gt.
Next, we show that F u = {f u} = {t}. Suppose not. Then inequality (2) gives
ÃZ
Z
Z
δ(F u,Gt)
ψ
d(f u,gt)
ϕ(t)dt,
0
Z
δ(f u,F u)
ϕ(t)dt,
0
Z
δ(gt,Gt)
ϕ(t)dt ,
0
Z
δ(f u,Gt)
ϕ(t)dt,
!
δ(gt,F u)
ϕ(t)dt,
0
ϕ(t)dt
0
≤ 0;
0
that is,
ÃZ
Z
δ(F u,t)
ψ
ϕ(t)dt, 0,
0
Z
δ(t,F u)
!
δ(t,F u)
ϕ(t)dt, 0, 0,
ϕ(t)dt
0
≤0
0
R δ(F u,t)
which implies by (ψb ) that 0
ϕ(t)dt ≤ 0 and hence F u = {t} = {f u}.
Since f and F are subcompatible, then, F f u = f F u; i.e., F t = {f t}.
Then, the use of (2) gives
ÃZ
Z
Z
δ(F t,Gt)
ψ
d(f t,gt)
ϕ(t)dt,
0
Z
0
Z
δ(gt,Gt)
0
Z
ϕ(t)dt,
ϕ(t)dt, 0, 0,
0
ϕ(t)dt
≤ 0;
0
Z
d(f t,t)
!
δ(gt,F t)
ϕ(t)dt,
0
Z
d(f t,t)
ϕ(t)dt ,
0
δ(f t,Gt)
ϕ(t)dt,
0
i.e.,
ÃZ
ψ
δ(f t,F t)
ϕ(t)dt,
Z
d(f t,t)
ϕ(t)dt,
0
d(t,f t)
!
ϕ(t)dt
0
≤0
168
H. Bouhadjera, A. Djoudi
contradicts (ψ2 ). Hence, {f t} = {t} = F t. Therefore t is a common fixed
point of maps f , g, F and G.
Now, suppose that there exists another common fixed point t0 such that t0 6= t.
Then, using inequality (2) we obtain
ÃZ
Z
Z
0
0
δ(F t,Gt )
ψ
d(f t,gt )
ϕ(t)dt,
0
Z
0
Z
δ(gt0 ,Gt0 )
ÃZ
Z
Z
d(t,t0 )
ϕ(t)dt,
0
ϕ(t)dt
0
Z
d(t,t0 )
!
δ(gt0 ,F t)
ϕ(t)dt,
0
=ψ
ϕ(t)dt ,
0
δ(f t,Gt0 )
ϕ(t)dt,
0
δ(f t,F t)
ϕ(t)dt,
ϕ(t)dt, 0, 0,
0
Z
d(t,t0 )
ϕ(t)dt,
0
!
d(t,t0 )
ϕ(t)dt
0
≤0
which contradicts (ψ2 ). Thus, t0 = t.
The proof is similar by replacing (3) with (30 ).
If we let in Theorem 1, f = g and F = G, then, we get the next corollary.
Corollary 1 Let (X , d) be a metric space and let f : X → X ; F : X → B(X )
be a single and a multivalued map, respectively. If
(1) f is surjective,
ÃZ
(2)
Z
δ(F x,F y)
ψ
0
Z
Z
δ(f y,F y)
ϕ(t)dt ,
0
Z
δ(f x,F y)
ϕ(t)dt,
0
δ(f x,F x)
ϕ(t)dt,
0
ϕ(t)dt,
0
Z
d(f x,f y)
ϕ(t)dt,
!
δ(f y,F x)
ϕ(t)dt
≤0
0
for all x, y in X , where ψ ∈ Ψ and ϕ ∈ Φ,
(3) f and F are subcompatible D-maps.
Then, f and F have a unique common fixed point t ∈ X such that F t = {f t} =
{t}.
Now, if we put in Theorem 1, f = g, then, we obtain the following result.
Corollary 2 Let (X , d) be a metric space and let f : X → X ; F , G : X →
B(X ) be maps satisfying the conditions
(1) f is surjective,
Common fixed point theorems for subcompatible...
ÃZ
(2)
Z
δ(F x,Gy)
ψ
Z
d(f x,f y)
ϕ(t)dt,
ϕ(t)dt ,
0
Z
δ(f y,Gy)
δ(f x,F x)
ϕ(t)dt,
0
Z
169
0
Z
δ(f x,Gy)
ϕ(t)dt,
ϕ(t)dt,
0
0
!
δ(f y,F x)
ϕ(t)dt
≤0
0
for all x, y in X , where ψ ∈ Ψ and ϕ ∈ Φ. If either
(3) f and F are subcompatible D-maps; f and G are subcompatible, or
(30 ) f and G are subcompatible D-maps; f and F are subcompatible.
Then, f , F and G have a unique common fixed point t ∈ X such that F t =
Gt = {f t} = {t}.
Using recurrence on n, we obtain the following result.
Theorem 2 Let (X , d) be a metric space and let f , g : X → X ; Fn : X →
B(X ), n = 1, 2, . . . be maps such that
(1) f and g are surjective,
ÃZ
(2)
Z
δ(Fn x,Fn+1 y)
ψ
ϕ(t)dt,
0
Z
Z
d(f x,gy)
0
Z
δ(gy,Fn+1 y)
δ(f x,Fn x)
ϕ(t)dt,
Z
δ(f x,Fn+1 y)
ϕ(t)dt,
ϕ(t)dt ,
0
ϕ(t)dt,
0
!
δ(gy,Fn x)
ϕ(t)dt
0
≤0
0
for all x, y in X , where ψ ∈ Ψ and ϕ ∈ Φ. If either
(3) f and Fn are subcompatible D-maps; g and Fn+1 are subcompatible, or
(30 ) g and Fn+1 are subcompatible D-maps; f and Fn are subcompatible.
Then, there exists a unique point t ∈ X such that Fn t = {f t} = {gt} = {t}.
Now, we prove our second main theorem.
Theorem 3 Let (X , d) be a metric space and let f , g : X → X ; F , G : X →
B(X ) be single and multivalued maps, respectively. Suppose that
(a) F (X ) ⊆ g(X ) and G(X ) ⊆ f (X ),
ÃZ
(b)
Z
z(δ(F x,Gy))
ψ
0
Z
Z
z(δ(gy,Gy))
ϕ(t)dt ,
0
Z
z(δ(f x,Gy))
ϕ(t)dt,
0
z(δ(f x,F x))
ϕ(t)dt,
0
ϕ(t)dt,
0
Z
z(d(f x,gy))
ϕ(t)dt,
z(δ(gy,F x))
!
ϕ(t)dt
0
≤0
170
H. Bouhadjera, A. Djoudi
for all x, y in X , where ψ ∈ Ψ, ϕ ∈ Φ and z ∈ F. If either
(c) f and F are subcompatible D-maps; g and G are subcompatible and F (X )
is closed, or
(c0 ) g and G are subcompatible D-maps; f and F are subcompatible and G(X )
is closed.
Then, f , g, F and G have a unique common fixed point t ∈ X such that
F t = Gt = {f t} = {gt} = {t}.
Proof. Suppose that g and G are D-maps, then, there is a sequence {yn } in
X such that lim gyn = t and lim Gyn = {t} for some t ∈ X . Since G(X ) is
n→∞
n→∞
closed and G(X ) ⊆ f (X ), then, there exists a point u ∈ X such that f u = t.
First, we claim that F u = {f u} = {t}. If not, then, from (b),
ÃZ
Z
Z
z(δ(F u,Gyn ))
z(d(f u,gyn ))
ψ
ϕ(t)dt,
0
Z
z(δ(f u,F u))
ϕ(t)dt,
ϕ(t)dt ,
0
Z
z(δ(gyn ,Gyn ))
0
Z
z(δ(f u,Gyn ))
ϕ(t)dt,
ϕ(t)dt,
0
0
ϕ(t)dt
≤ 0.
0
Since ψ and z are continuous, at infinity we get
ÃZ
Z z(δ(f u,F u))
Z
z(δ(F u,f u))
ψ
ϕ(t)dt, 0,
ϕ(t)dt, 0, 0,
0
!
z(δ(gyn ,F u))
0
!
z(δ(f u,F u))
ϕ(t)dt
≤0
0
R z(δ(F u,f u))
which from (ψb ) gives 0
ϕ(t)dt ≤ 0 and therefore z(δ(F u, f u)) = 0
which implies that F u = {f u} = {t}. Since f and F are subcompatible, then,
F f u = f F u; i.e., F t = {f t}.
Suppose that f t 6= t, then, from inequality (b),
ÃZ
Z z(d(f t,gyn ))
Z z(δ(f t,F t))
z(δ(F t,Gyn ))
ψ
ϕ(t)dt,
ϕ(t)dt,
ϕ(t)dt ,
0
Z
0
Z
z(δ(gyn ,Gyn ))
0
Z
z(δ(f t,Gyn ))
ϕ(t)dt,
0
ϕ(t)dt,
0
At infinity we obtain
ÃZ
ψ
Z
z(d(f t,t))
ϕ(t)dt, 0, 0,
0
Z
z(d(f t,t))
ϕ(t)dt,
0
z(d(f t,t))
ϕ(t)dt,
z(d(t,f t))
!
ϕ(t)dt
0
!
ϕ(t)dt
0
0
Z
z(δ(gyn ,F t))
≤0
≤ 0.
Common fixed point theorems for subcompatible...
171
R z(d(f t,t))
which contradicts (ψ2 ). Therefore 0
ϕ(t)dt = 0 which implies that
z(d(f t, t)) = 0; i.e., {f t} = {t} = F t.
Since F (X ) ⊆ g(X ), there exists an element v ∈ X such that gv = t. We claim
that Gv = {gv} = {t}. If not, then, using condition (b) we have
ÃZ
Z
Z
z(δ(F t,Gv))
ψ
z(d(f t,gv))
ϕ(t)dt,
0
Z
ϕ(t)dt ,
0
Z
z(δ(gv,Gv))
0
Z
z(δ(f t,Gv))
ϕ(t)dt,
0
z(δ(f t,F t))
ϕ(t)dt,
ϕ(t)dt,
ϕ(t)dt
0
ÃZ
0
Z
z(δ(t,Gv))
=ψ
!
z(δ(gv,F t))
ϕ(t)dt, 0, 0,
0
Z
z(δ(t,Gv))
ϕ(t)dt,
0
!
z(δ(t,Gv))
ϕ(t)dt, 0
≤0
0
R z(δ(t,Gv))
which from (ψa ) gives 0
ϕ(t)dt = 0 and hence z(δ(t, Gv)) = 0 which
implies that Gv = {t} = {gv}. Since the pair (G, g) is subcompatible, then,
Ggv = gGv; i.e., Gt = {gt}.
Suppose that gt 6= t. Then, by (b) we have
ÃZ
Z
Z
z(δ(F t,Gt))
ψ
z(d(f t,gt))
z(δ(f t,F t))
ϕ(t)dt,
0
Z
ϕ(t)dt,
0
Z
z(δ(gt,Gt))
ÃZ
ϕ(t)dt,
0
ϕ(t)dt, 0, 0,
0
Z
ϕ(t)dt
z(d(t,gt))
ϕ(t)dt,
0
!
0
Z
z(d(t,gt))
z(d(t,gt))
z(δ(gt,F t))
ϕ(t)dt,
0
=ψ
Z
Z
z(δ(f t,Gt))
ϕ(t)dt,
0
ϕ(t)dt ,
0
!
z(d(gt,t))
ϕ(t)dt
≤0
0
R z(d(t,gt))
contradicts (ψ2 ). Therefore 0
ϕ(t)dt = 0 which implies that z(d(t, gt)) =
0; i.e., {gt} = {t} = Gt, and t is a common fixed point of f , g, F and G.
The uniqueness of the common fixed point follows easily from condition (b).
The proof is thus completed.
The proof is similar by replacing (c0 ) with (c).
Corollary 3 Let (X , d) be a metric space and let f : X → X ; F : X → B(X )
be a single and a multivalued map, respectively. Suppose that
(a) F (X ) ⊆ f (X ),
172
H. Bouhadjera, A. Djoudi
ÃZ
(b)
Z
z(δ(F x,F y))
ψ
Z
z(d(f x,f y))
ϕ(t)dt,
0
Z
ϕ(t)dt ,
0
Z
z(δ(f y,F y))
z(δ(f x,F x))
ϕ(t)dt,
0
Z
z(δ(f x,F y))
ϕ(t)dt,
ϕ(t)dt,
0
0
!
z(δ(f y,F x))
ϕ(t)dt
≤0
0
for all x, y in X , where ψ ∈ Ψ, ϕ ∈ Φ and z ∈ F. If f and F are subcompatible
D-maps and F (X ) is closed, then, f and F have a unique common fixed point
t ∈ X such that F t = {f t} = {t}.
Corollary 4 Let (X , d) be a metric space and let f : X → X ; F , G : X →
B(X ) be maps. If
(a) F (X ) ⊆ f (X ) and G(X ) ⊆ f (X ),
ÃZ
(b)
Z
z(δ(F x,Gy))
ψ
Z
z(d(f x,f y))
ϕ(t)dt,
0
Z
ϕ(t)dt ,
0
Z
z(δ(f y,Gy))
0
Z
z(δ(f x,Gy))
ϕ(t)dt,
0
z(δ(f x,F x))
ϕ(t)dt,
ϕ(t)dt,
0
!
z(δ(f y,F x))
ϕ(t)dt
≤0
0
for all x, y in X , where ψ ∈ Ψ, ϕ ∈ Φ and z ∈ F. If either
(c) f and F are subcompatible D-maps; f and G are subcompatible and F (X )
is closed, or
(c0 ) f and G are subcompatible D-maps; f and F are subcompatible and G(X )
is closed.
Then, there is a unique point t ∈ X such that F t = Gt = {f t} = {t}.
By recurrence on n, we get the next result.
Theorem 4 Let (X , d) be a metric space and let f , g : X → X ; Fn : X →
B(X ) be single and multivalued maps, respectively. Suppose that
(a) Fn (X ) ⊆ g(X ) and Fn+1 (X ) ⊆ f (X ),
ÃZ
(b)
Z
z(δ(Fn x,Fn+1 y))
ψ
0
Z
Z
z(δ(gy,Fn+1 y))
ϕ(t)dt ,
0
Z
z(δ(f x,Fn+1 y))
ϕ(t)dt,
0
z(δ(f x,Fn x))
ϕ(t)dt,
0
ϕ(t)dt,
0
Z
z(d(f x,gy))
ϕ(t)dt,
!
z(δ(gy,Fn x))
ϕ(t)dt
0
≤0
Common fixed point theorems for subcompatible...
173
for all x, y in X , where ψ ∈ Ψ, ϕ ∈ Φ, z ∈ F and n ∈ N∗ = {1, 2, . . .}. If
either
(c) f and Fn are subcompatible D-maps; g and Fn+1 are subcompatible and
Fn (X ) is closed, or
(c0 ) g and Fn+1 are subcompatible D-maps; f and Fn are subcompatible and
Fn+1 (X ) is closed.
Then, there exists a unique point t in X such that Fn t = {f t} = {gt} = {t}.
References
[1] M. Aamri and D. El Moutawakil, Some new common fixed point theorems
under strict contractive conditions, J. Math. Anal. Appl., 270(1), 2002,
181-188.
[2] M.A. Al-Thagafi and N. Shahzad, Generalized I-nonexpansive selfmaps
and invariant approximations, Acta Math. Sin. (Engl. Ser.), 24(5), 2008,
867-876.
[3] A. Djoudi and R. Khemis, Fixed points for set and single valued maps
without continuity, Demonstratio Mathematica Vol., XXXVIII, no. 3,
2005, 739-751.
[4] B. Fisher, Common fixed points of mappings and set-valued mappings,
Rostock. Math. Kolloq., no. 18, 1981, 69-77.
[5] B. Fisher and S. Sessa, Two common fixed point theorems for weakly
commuting mappings, Period. Math. Hungar., 20(3), 207-218.
[6] G. Jungck, Compatible mappings and common fixed points, Internat. J.
Math. Math. Sci., 9(4), 1986, 771-779.
[7] G. Jungck, Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J. Math. Sci., 4(2), 1996, 199-215.
[8] G. Jungck and B.E. Rhoades, Fixed points for set valued functions without
continuity, Indian J. Pure Appl. Math., 29(3), 1998, 227-238.
[9] G. Jungck and B.E. Rhoades, Fixed point theorems for occasionally weakly
compatible mappings, Fixed Point Theory, 7(2), 2006, 287-296.
174
H. Bouhadjera, A. Djoudi
[10] S. Sessa, On a weak commutativity condition in fixed point considerations,
Publ. Inst. Math. (Beograd) (N.S.), 32(46), 1982, 149-153.
Hakima Bouhadjera
Laboratoire de Mathématiques Appliquées
Faculté des Sciences
Université Badji Mokhtar d’Annaba, B.P. 12, 23000, Annaba, Algérie
e-mail: b hakima2000@yahoo.fr
Ahcène Djoudi
Laboratoire de Mathématiques Appliquées
Faculté des Sciences
Université Badji Mokhtar d’Annaba, B.P. 12, 23000, Annaba, Algérie
e-mail: adjoudi@yahoo.com