Directory UMM :Journals:Journal_of_mathematics:AUA:
ACTA UNIVERSITATIS APULENSIS
No 19/2009
A RELATED FIXED POINT THEOREM IN N COMPLETE METRIC
SPACES
Fayçal Merghadi, Abdelkrim Aliouche, Brian Fisher
Abstract. We prove a related fixed point theorem for n mappings in n complete
metric spaces via implicit relations. Our result generalizes Theorem 1.1 of [1].
2000 Mathematics Subject Classification: 47H10, 54H25.
1. Introduction
Recently, A. Aliouche and B. Fisher [1] proved the following related fixed point
theorem in two complete metric spaces.
Theorem 1.1 Let (X, d) and (Y, ρ) be complete metric spaces and let S, T be
mappings of Y into X and of X into Y respectively, satisfying the inequalities
f (ρ (T x, T Sy) , d (x, Sy) , ρ (y, T x) , ρ (y, T Sy)) ≤ 0 ,
g (d (Sy, ST x) , ρ (y, T x) , d (x, Sy) , d (x, ST x)) ≤ 0
for all x in X and y in Y , where f, g ∈ F . Then ST has a unique fixed point in
X and T S has a unique fixed point v in Y . Further, T u = v and Sv = u.
We denote by R+ the set of non-negative reals and by Φ the set of all functions
φ : R4+ → R such that
(i) φ is upper semi continuous in each coordinate variable,
(ii) if either φ (u, v, 0, u) ≤ 0 or φ (u, v, u, 0) ≤ 0 for all u, v ∈ R+ , then there
exists a real constant c, with 0 ≤ c < 1, such that u ≤ cv.
Example 1.2. φ (t1 , t2 , t3 , t4 ) = t1 − c max {t2 , t3 , t4 } , 0 ≤ c < 1.
Example 1.3. φ (t1 , t2 , t3 , t4 ) = t21 − c1 max t22 , t23 , t24 − c2 max {t1 t3 , t2 t4 } −
c3 t3 t4 , where c1 , c2 , c3 ∈ R+ and c1 + c2 < 1.
2. Main results
91
F. Merghadi, A. Aliouche, B. Fisher - A related fixed point Theorem in...
Theorem 2.1. Let (Xi , di ) be n complete metric spaces and let {Ai }i=n
i=1 be
n mappings such that Ai : Xi → Xi+1 for i = 1, . . . , n − 1 and An : Xn → X1 ,
satisfying the inequalities
d1 (An An−1 ...A2 x2 , An An−1 ...A1 x1 ) ,
≤0
d2 (x2 , A1 An An−1 ...A2 x2 ) ,
(2.1)
φ1
d1 (x1 , An An−1 ...A2 x2 ) , d1 (x1 , An An−1 ...A1 x1 )
for all x1 ∈ X1 and x2 ∈ X2 , in general
di (Ai−1 Ai−2 ...A1 An An−1 ...Ai+1 xi+1 , Ai−1 Ai−2 ...A1 An An−1 ...Ai xi ) ,
di+1 (xi+1 , Ai Ai−1 ..A1 An An−1 ...Ai+1 xi+1 ) ,
≤0
φi
di (xi , Ai−1 Ai−2 ..A1 An An−1 ...Ai+1 xi+1 ) ,
di (xi , Ai−1 Ai−2 ...A1 An An−1 ...Ai xi )
(2.i)
for all xi ∈ Xi , xi+1 ∈ Xi+1 and i = 2, . . . , n − 1, and
dn (An−1 An−2 ...A1 x1 , An−1 An−2 ...A1 An xn ) ,
d1 (x1 , An An−1 An−2 ...A1 x1 ) ,
≤0
(2.n)
φn
dn (xn , An−1 An−2 ...A1 x1 ) ,
dn (xn , An−1 An−2 ...A1 An xn )
for all x1 ∈ X1 , xn ∈ Xn , where φi ∈ Φ, for i = 1, . . . , n. Then
Ai−1 Ai−2 ...A1 An An−1 ...Ai
has a unique fixed point pi ∈ Xi for i = 1, . . . , n. Further, Ai pi = pi+1 for i =
1, . . . , n − 1 and An pn = p1 .
n
n o
n
o
n
o
o
(r)
(i)
(2)
(1)
, . . . , xr
, . . . , xr
, xr
Proof. Let xr
be sequences
r∈N
r∈N
r∈N
r∈N
in X1 , X2 , . . . , Xi , . . . , Xn , respectively.
n o
(1)
(i)
Now let x0 be an arbitrary point in X1 . We define the sequences xr
r∈N
i = 1, . . . , n by
(1)
r
x(1)
r = (An An−1 ...A1 ) x0 ,
(1)
r
x(i)
r = Ai−1 Ai−2 ...A1 (An An−1 ...A1 ) x0
for i = 2, . . . , n.
92
for
F. Merghadi, A. Aliouche, B. Fisher - A related fixed point Theorem in...
(1)
(1)
For n = 1, 2, . . . , we assume that xn 6= xn+1 . Applying the inequality (2.1) for
(1)
(1)
x2 = A1 (An An−1 ...A1 )r−1 x0 , x1 = (An An−1 ...A1 )r x0 we get
(1)
(1)
d1 (An An−1 ...A1 )r x0 , (An An−1 ...A1 )r+1 x0 ,
(1)
r−1 (1)
x0 , A1 (An An−1 ...A1 )r x0 ,
d2 A1 (An An−1 ...A1 )
φ1
(1)
(1)
d1 (An An−1 ...A1 )r x0 , (An An−1 ...A1 )r x0 ,
r (1)
r+1 (1)
d1 (An An−1 ...A1 ) x0 , (An An−1 ...A1 )
x0
(1) (1)
(2)
(2)
d1 xr , xr+1 , d2 xr−1 , xr ,
≤ 0.
= φ1
(1) (1)
0,
d1 xr , xr+1
From the implicit relation we have
(1)
(2)
(2)
d1 x(1)
,
r , xr+1 ≤ cd2 xr−1 , xr
(3.1)
for r = 1, 2, . . . .
(1)
Applying the inequality (2.i) for xi+1 = Ai ...A1 (An ...A1 )r−1 x0 and xi =
(1)
Ai−1 ...A1 (An ...A1 )r x0 , we obtain
(1)
(1)
di Ai−1 ...A1 (An ...A1 )r x0 , Ai−1 (An ...A1 )r+1 x0 ,
(1)
r−1 (1)
x0 , Ai ...A1 (An ...A1 )r x0 ,
di+1 Ai ...A1 (An ...A1 )
φi
r (1)
r (1)
di Ai−1 ...A1 (An ...A1 ) x0 , Ai−1 ...A1 (An ...A1 ) x0 ,
(1)
(1)
di Ai−1 ...A1 (Ai−1 ...A1 )r x0 , Ai−1 ...A1 (An ...A1 )r+1 x0
(i) (i)
(i+1) (i+1)
di xr , xr+1 , di+1 xr−1 , xr
,
≤0
= φi
(i+1) (i+1)
0,
di+1 xr
, xr+1
and so
(i)
(i+1) (i+1)
di x(i)
,
x
≤
cd
,
x
,
x
i+1
r
r
r+1
r−1
(3.i)
for i = 2, . . . , n − 1 and r = 1, 2, . . . .
(1)
Now applying the inequality (2.n) for xn = An−1 ...A1 (An ...A1 )r x0 and x1 =
93
F. Merghadi, A. Aliouche, B. Fisher - A related fixed point Theorem in...
(1)
(An ...A1 )r x0 , we have
(1)
(1)
dn An−1 ...A1 (An ...A1 )r x0 , An−1 ...A1 (An ...A1 )r+1 x0 ,
(1)
(1)
d1 (An ...A1 )r x0 , (An ...A1 )r+1 x0 ,
φn
(1)
(1)
r
r
dn An−1 ...A1 (An ...A1 ) x0 , An−1 ...A1 (An ...A1 ) x0 ,
(1)
(1)
dn An−1 ...A1 (An ...A1 )r x0 , An−1 ...A1 (An ...A1 )r x0
(n) (n)
(1)
(1)
dn xn , xn+1 , d1 xn−1 , xn ,
≤0
= φn
(n) (n)
0,
dn xn , xn+1
and so
(n)
(1)
(1)
dn x(n)
,
x
≤
cd
,
x
,
x
1
r
r+1
r−1 r
(3.n)
for r = 1, 2, . . . .
It now follows from (3.1), (3.i) and (3.n) that for large enough n
(i)
(i+1) (i+1)
,
x
di x(i)
≤
cd
x
,
x
i+1
r
r
r+1
r−1
≤ ...
(n)
(n)
≤ cn−i dn xr+i−n , xr+i−n+1
(1)
(1)
≤ cn−i+1 d1 xr+i−n−1 , xr+i−n
≤ ...
(1)
(1)
≤ c2n−i+1 d1 xr+i−2n−1 , xr+i−2n
≤ ...
(1)
(1)
≤ cmn−i+1 d1 xr+i−mn−1 , xr+i−mn
n
o
(1) (1)
(n) (n)
≤ cmn max d1 x1 , x2 , . . . , dn x1 , x2
.
n o
(i)
Since c < 1, it follows that xr
is Cauchy sequences in Xi with a limit pi in Xi
for i = 1, 2, . . . , n.
To prove that pi is a fixed point of Ai−1 ...A1 An ...Ai pi for i = 2, . . . , n − 1,
suppose that Ai−1 ...A1 An ...Ai pi 6= pi . Using the inequality (2.i) for xi = pi and
(i+1)
xi+1 = xr
, we obtain
(i)
di xr+1 , Ai−1 ...A1 An ...Ai pi ,
(i+1) (i+1)
(i)
φi
, xr+1 , di pi , xr , ≤ 0.
di+1 xr
di (pi , Ai−1 ...A1 An ...Ai pi )
94
F. Merghadi, A. Aliouche, B. Fisher - A related fixed point Theorem in...
Letting r → ∞, we have
di (pi , Ai−1 ...A1 An ...Ai pi ) , 0, 0,
≤ 0.
φi
di (pi , Ai−1 ...A1 An ...Ai pi )
It follows from (ii) that pi = Ai−1 ...A1 An ...Ai pi for i = 2, . . . , n − 1.
(1)
For the case i = 1, we use (2.1) with x1 = p1 and x2 = A1 (An ...A1 )r−1 x0 ,
giving
(1)
d1 xr , An ...A1 p1 ,
(2) (2)
(1)
φ1
d2 xr , xr+1 , d1 p1 , xr , ≤ 0.
d1 (p1 , An ...A1 p1 )
Letting r → ∞, we have
φi
d1 (p1 , An ...A1 p1 ) , 0, 0,
d1 (p1 , An ...A1 p1 )
≤ 0.
It follows from (ii) that p1 = An ...A1 p1 .
(1)
Finally, if i = n, using the inequality (2.n) for xn = pn and x1 = xn we get
(n)
dn xr+1 , An−1 ...A1 An pn ,
(1) (1)
(n)
φn
d1 xr , xr+1 , dn pn , xr+1 , ≤ 0.
dn (pn , An−1 ...A1 An pn )
Letting r → ∞, we have
dn (pn , An−1 ...A1 An pn ) , 0, 0,
φn
≤0
dn (pn , An−1 ...A1 An pn )
and by (ii), pn = An−1 ...A1 An pn .
To prove the uniqueness, suppose that Ai−1 ...A1 An ...Ai has a second fixed point
zi 6= pi in Xi .
Using the inequality (2.i) for xi+1 = Ai zi and xi = pi we get
d1 (Ai ...A1 An ...Ai zi , Ai−1 ...A1 An ...Ai pi ) ,
di+1 (Ai zi , Ai ...A1 An ...Ai+1 zi )
φ1
di (pi , Ai−1 ...A1 An ...Ai+1 zi ) , di (pi , Ai−1 ...A1 An ...Ai pi )
= φ1 (di (zi , pi ) , 0, di (pi , zi ) , 0) ≤ 0
which implies that zi = pi , proving the uniqueness of pi for i = 2, . . . , n − 1. The
uniqueness of p1 in X1 and pn in Xn follow similarly.
95
F. Merghadi, A. Aliouche, B. Fisher - A related fixed point Theorem in...
We finally note that
Ai pi = Ai ...A1 An ...Ai+1 (Ai pi ),
so that Ai pi is a fixed point of Ai ...A1 An ...Ai+1 . Since the fixed point is unique, it
follows that Ai pi = pi+1 for i = 1, . . . , n − 1. It follows similarly that An pn = p1 .
This completes the proof of the theorem.
Example 2.2. Let (Xi , d) for i = 1, . . . , n be n complete metric spaces where
Xi = {xi : i − 1 ≤ xi ≤ i} for i = 1, . . . , n and d is the usual metric for the real
numbers. Define Ai : Xi → Xi+1 for i = 1, . . . , n − 1 and An : Xn → X1 by
5/4 if 0 ≤ x1 < 1/2,
A1 x1 =
3/2 if 1/2 ≤ x1 ≤ 1,
i + 1/4 if i − 1 ≤ xi < i − 3/4,
Ai xi =
i + 1/2 if i − 3/4 ≤ xi ≤ i
for i = 2, . . . , n − 1,
An xn =
3/4 if n − 1 ≤ xn < n − 3/4,
1 if n − 3/4 ≤ xn ≤ n
and φ1 = φ2 = ...φn = φ ∈ Φ such that φ (t1 , t2 , t3 , t4 ) = t1 − c max {t2 , t3 , t4 } and
0≤c
No 19/2009
A RELATED FIXED POINT THEOREM IN N COMPLETE METRIC
SPACES
Fayçal Merghadi, Abdelkrim Aliouche, Brian Fisher
Abstract. We prove a related fixed point theorem for n mappings in n complete
metric spaces via implicit relations. Our result generalizes Theorem 1.1 of [1].
2000 Mathematics Subject Classification: 47H10, 54H25.
1. Introduction
Recently, A. Aliouche and B. Fisher [1] proved the following related fixed point
theorem in two complete metric spaces.
Theorem 1.1 Let (X, d) and (Y, ρ) be complete metric spaces and let S, T be
mappings of Y into X and of X into Y respectively, satisfying the inequalities
f (ρ (T x, T Sy) , d (x, Sy) , ρ (y, T x) , ρ (y, T Sy)) ≤ 0 ,
g (d (Sy, ST x) , ρ (y, T x) , d (x, Sy) , d (x, ST x)) ≤ 0
for all x in X and y in Y , where f, g ∈ F . Then ST has a unique fixed point in
X and T S has a unique fixed point v in Y . Further, T u = v and Sv = u.
We denote by R+ the set of non-negative reals and by Φ the set of all functions
φ : R4+ → R such that
(i) φ is upper semi continuous in each coordinate variable,
(ii) if either φ (u, v, 0, u) ≤ 0 or φ (u, v, u, 0) ≤ 0 for all u, v ∈ R+ , then there
exists a real constant c, with 0 ≤ c < 1, such that u ≤ cv.
Example 1.2. φ (t1 , t2 , t3 , t4 ) = t1 − c max {t2 , t3 , t4 } , 0 ≤ c < 1.
Example 1.3. φ (t1 , t2 , t3 , t4 ) = t21 − c1 max t22 , t23 , t24 − c2 max {t1 t3 , t2 t4 } −
c3 t3 t4 , where c1 , c2 , c3 ∈ R+ and c1 + c2 < 1.
2. Main results
91
F. Merghadi, A. Aliouche, B. Fisher - A related fixed point Theorem in...
Theorem 2.1. Let (Xi , di ) be n complete metric spaces and let {Ai }i=n
i=1 be
n mappings such that Ai : Xi → Xi+1 for i = 1, . . . , n − 1 and An : Xn → X1 ,
satisfying the inequalities
d1 (An An−1 ...A2 x2 , An An−1 ...A1 x1 ) ,
≤0
d2 (x2 , A1 An An−1 ...A2 x2 ) ,
(2.1)
φ1
d1 (x1 , An An−1 ...A2 x2 ) , d1 (x1 , An An−1 ...A1 x1 )
for all x1 ∈ X1 and x2 ∈ X2 , in general
di (Ai−1 Ai−2 ...A1 An An−1 ...Ai+1 xi+1 , Ai−1 Ai−2 ...A1 An An−1 ...Ai xi ) ,
di+1 (xi+1 , Ai Ai−1 ..A1 An An−1 ...Ai+1 xi+1 ) ,
≤0
φi
di (xi , Ai−1 Ai−2 ..A1 An An−1 ...Ai+1 xi+1 ) ,
di (xi , Ai−1 Ai−2 ...A1 An An−1 ...Ai xi )
(2.i)
for all xi ∈ Xi , xi+1 ∈ Xi+1 and i = 2, . . . , n − 1, and
dn (An−1 An−2 ...A1 x1 , An−1 An−2 ...A1 An xn ) ,
d1 (x1 , An An−1 An−2 ...A1 x1 ) ,
≤0
(2.n)
φn
dn (xn , An−1 An−2 ...A1 x1 ) ,
dn (xn , An−1 An−2 ...A1 An xn )
for all x1 ∈ X1 , xn ∈ Xn , where φi ∈ Φ, for i = 1, . . . , n. Then
Ai−1 Ai−2 ...A1 An An−1 ...Ai
has a unique fixed point pi ∈ Xi for i = 1, . . . , n. Further, Ai pi = pi+1 for i =
1, . . . , n − 1 and An pn = p1 .
n
n o
n
o
n
o
o
(r)
(i)
(2)
(1)
, . . . , xr
, . . . , xr
, xr
Proof. Let xr
be sequences
r∈N
r∈N
r∈N
r∈N
in X1 , X2 , . . . , Xi , . . . , Xn , respectively.
n o
(1)
(i)
Now let x0 be an arbitrary point in X1 . We define the sequences xr
r∈N
i = 1, . . . , n by
(1)
r
x(1)
r = (An An−1 ...A1 ) x0 ,
(1)
r
x(i)
r = Ai−1 Ai−2 ...A1 (An An−1 ...A1 ) x0
for i = 2, . . . , n.
92
for
F. Merghadi, A. Aliouche, B. Fisher - A related fixed point Theorem in...
(1)
(1)
For n = 1, 2, . . . , we assume that xn 6= xn+1 . Applying the inequality (2.1) for
(1)
(1)
x2 = A1 (An An−1 ...A1 )r−1 x0 , x1 = (An An−1 ...A1 )r x0 we get
(1)
(1)
d1 (An An−1 ...A1 )r x0 , (An An−1 ...A1 )r+1 x0 ,
(1)
r−1 (1)
x0 , A1 (An An−1 ...A1 )r x0 ,
d2 A1 (An An−1 ...A1 )
φ1
(1)
(1)
d1 (An An−1 ...A1 )r x0 , (An An−1 ...A1 )r x0 ,
r (1)
r+1 (1)
d1 (An An−1 ...A1 ) x0 , (An An−1 ...A1 )
x0
(1) (1)
(2)
(2)
d1 xr , xr+1 , d2 xr−1 , xr ,
≤ 0.
= φ1
(1) (1)
0,
d1 xr , xr+1
From the implicit relation we have
(1)
(2)
(2)
d1 x(1)
,
r , xr+1 ≤ cd2 xr−1 , xr
(3.1)
for r = 1, 2, . . . .
(1)
Applying the inequality (2.i) for xi+1 = Ai ...A1 (An ...A1 )r−1 x0 and xi =
(1)
Ai−1 ...A1 (An ...A1 )r x0 , we obtain
(1)
(1)
di Ai−1 ...A1 (An ...A1 )r x0 , Ai−1 (An ...A1 )r+1 x0 ,
(1)
r−1 (1)
x0 , Ai ...A1 (An ...A1 )r x0 ,
di+1 Ai ...A1 (An ...A1 )
φi
r (1)
r (1)
di Ai−1 ...A1 (An ...A1 ) x0 , Ai−1 ...A1 (An ...A1 ) x0 ,
(1)
(1)
di Ai−1 ...A1 (Ai−1 ...A1 )r x0 , Ai−1 ...A1 (An ...A1 )r+1 x0
(i) (i)
(i+1) (i+1)
di xr , xr+1 , di+1 xr−1 , xr
,
≤0
= φi
(i+1) (i+1)
0,
di+1 xr
, xr+1
and so
(i)
(i+1) (i+1)
di x(i)
,
x
≤
cd
,
x
,
x
i+1
r
r
r+1
r−1
(3.i)
for i = 2, . . . , n − 1 and r = 1, 2, . . . .
(1)
Now applying the inequality (2.n) for xn = An−1 ...A1 (An ...A1 )r x0 and x1 =
93
F. Merghadi, A. Aliouche, B. Fisher - A related fixed point Theorem in...
(1)
(An ...A1 )r x0 , we have
(1)
(1)
dn An−1 ...A1 (An ...A1 )r x0 , An−1 ...A1 (An ...A1 )r+1 x0 ,
(1)
(1)
d1 (An ...A1 )r x0 , (An ...A1 )r+1 x0 ,
φn
(1)
(1)
r
r
dn An−1 ...A1 (An ...A1 ) x0 , An−1 ...A1 (An ...A1 ) x0 ,
(1)
(1)
dn An−1 ...A1 (An ...A1 )r x0 , An−1 ...A1 (An ...A1 )r x0
(n) (n)
(1)
(1)
dn xn , xn+1 , d1 xn−1 , xn ,
≤0
= φn
(n) (n)
0,
dn xn , xn+1
and so
(n)
(1)
(1)
dn x(n)
,
x
≤
cd
,
x
,
x
1
r
r+1
r−1 r
(3.n)
for r = 1, 2, . . . .
It now follows from (3.1), (3.i) and (3.n) that for large enough n
(i)
(i+1) (i+1)
,
x
di x(i)
≤
cd
x
,
x
i+1
r
r
r+1
r−1
≤ ...
(n)
(n)
≤ cn−i dn xr+i−n , xr+i−n+1
(1)
(1)
≤ cn−i+1 d1 xr+i−n−1 , xr+i−n
≤ ...
(1)
(1)
≤ c2n−i+1 d1 xr+i−2n−1 , xr+i−2n
≤ ...
(1)
(1)
≤ cmn−i+1 d1 xr+i−mn−1 , xr+i−mn
n
o
(1) (1)
(n) (n)
≤ cmn max d1 x1 , x2 , . . . , dn x1 , x2
.
n o
(i)
Since c < 1, it follows that xr
is Cauchy sequences in Xi with a limit pi in Xi
for i = 1, 2, . . . , n.
To prove that pi is a fixed point of Ai−1 ...A1 An ...Ai pi for i = 2, . . . , n − 1,
suppose that Ai−1 ...A1 An ...Ai pi 6= pi . Using the inequality (2.i) for xi = pi and
(i+1)
xi+1 = xr
, we obtain
(i)
di xr+1 , Ai−1 ...A1 An ...Ai pi ,
(i+1) (i+1)
(i)
φi
, xr+1 , di pi , xr , ≤ 0.
di+1 xr
di (pi , Ai−1 ...A1 An ...Ai pi )
94
F. Merghadi, A. Aliouche, B. Fisher - A related fixed point Theorem in...
Letting r → ∞, we have
di (pi , Ai−1 ...A1 An ...Ai pi ) , 0, 0,
≤ 0.
φi
di (pi , Ai−1 ...A1 An ...Ai pi )
It follows from (ii) that pi = Ai−1 ...A1 An ...Ai pi for i = 2, . . . , n − 1.
(1)
For the case i = 1, we use (2.1) with x1 = p1 and x2 = A1 (An ...A1 )r−1 x0 ,
giving
(1)
d1 xr , An ...A1 p1 ,
(2) (2)
(1)
φ1
d2 xr , xr+1 , d1 p1 , xr , ≤ 0.
d1 (p1 , An ...A1 p1 )
Letting r → ∞, we have
φi
d1 (p1 , An ...A1 p1 ) , 0, 0,
d1 (p1 , An ...A1 p1 )
≤ 0.
It follows from (ii) that p1 = An ...A1 p1 .
(1)
Finally, if i = n, using the inequality (2.n) for xn = pn and x1 = xn we get
(n)
dn xr+1 , An−1 ...A1 An pn ,
(1) (1)
(n)
φn
d1 xr , xr+1 , dn pn , xr+1 , ≤ 0.
dn (pn , An−1 ...A1 An pn )
Letting r → ∞, we have
dn (pn , An−1 ...A1 An pn ) , 0, 0,
φn
≤0
dn (pn , An−1 ...A1 An pn )
and by (ii), pn = An−1 ...A1 An pn .
To prove the uniqueness, suppose that Ai−1 ...A1 An ...Ai has a second fixed point
zi 6= pi in Xi .
Using the inequality (2.i) for xi+1 = Ai zi and xi = pi we get
d1 (Ai ...A1 An ...Ai zi , Ai−1 ...A1 An ...Ai pi ) ,
di+1 (Ai zi , Ai ...A1 An ...Ai+1 zi )
φ1
di (pi , Ai−1 ...A1 An ...Ai+1 zi ) , di (pi , Ai−1 ...A1 An ...Ai pi )
= φ1 (di (zi , pi ) , 0, di (pi , zi ) , 0) ≤ 0
which implies that zi = pi , proving the uniqueness of pi for i = 2, . . . , n − 1. The
uniqueness of p1 in X1 and pn in Xn follow similarly.
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F. Merghadi, A. Aliouche, B. Fisher - A related fixed point Theorem in...
We finally note that
Ai pi = Ai ...A1 An ...Ai+1 (Ai pi ),
so that Ai pi is a fixed point of Ai ...A1 An ...Ai+1 . Since the fixed point is unique, it
follows that Ai pi = pi+1 for i = 1, . . . , n − 1. It follows similarly that An pn = p1 .
This completes the proof of the theorem.
Example 2.2. Let (Xi , d) for i = 1, . . . , n be n complete metric spaces where
Xi = {xi : i − 1 ≤ xi ≤ i} for i = 1, . . . , n and d is the usual metric for the real
numbers. Define Ai : Xi → Xi+1 for i = 1, . . . , n − 1 and An : Xn → X1 by
5/4 if 0 ≤ x1 < 1/2,
A1 x1 =
3/2 if 1/2 ≤ x1 ≤ 1,
i + 1/4 if i − 1 ≤ xi < i − 3/4,
Ai xi =
i + 1/2 if i − 3/4 ≤ xi ≤ i
for i = 2, . . . , n − 1,
An xn =
3/4 if n − 1 ≤ xn < n − 3/4,
1 if n − 3/4 ≤ xn ≤ n
and φ1 = φ2 = ...φn = φ ∈ Φ such that φ (t1 , t2 , t3 , t4 ) = t1 − c max {t2 , t3 , t4 } and
0≤c