# Presic Type Fixed Point Theorem for Four Maps in Metric Spaces

Authors: K. P. R. Rao; Sk. Sadik; S. Manro
Journal: Journal of Mathematics
 (2016)

---

## Abstract

We obtained a Presic type fixed point theorem for two pairs of jointly2
k-weakly compatible maps in metric spaces. We also have given an example to illustrate our main theorem.


---

## Body

## 1. Introduction and Preliminaries

Banach [1], in 1932, proved the following theorem, popularly known as Banach contraction principle.Theorem 1.
Let(
X
,
d
) be a complete metric space and let f be a self-mapping on X satisfying d
(
f
x
,
f
y
)
≤
k
d
(
x
,
y
) for all x
,
y
∈
X, where 0
≤
k
<
1. Then f has a unique fixed point in X.There are number of generalizations of such principle (see [2–12]). One such generalization is given by Presic [13] in 1965.Letf
:
X
k
→
X, where k
≥
1 is a positive integer. A point x
∗
∈
X is called a fixed point of f if x
∗
=
f
(
x
∗
,
x
∗
,
…
,
x
∗
). Consider the k-order nonlinear difference equation(1)
x
n
+
1
=
f
x
n
-
k
+
1
,
x
n
-
k
+
2
,
…
,
x
n
for
n
=
k
-
1
,
k
,
k
+
1
,
…with the initial values x
0
,
x
1
,
x
2
,
…
,
x
k
-
1
∈
X.Equation (1) can be studied by means of fixed point theory in view of the fact that x
∈
X is a solution of (1) if and only if x is a fixed point of f. One of the most important results in this direction is obtained by Presic [13] in the following way.Theorem 2 (see [13]).
Let(
X
,
d
) be a complete metric space, k be a positive integer, and f
:
X
k
→
X. Suppose that(2)
d
f
x
1
,
x
2
,
…
,
x
k
,
f
x
2
,
x
3
,
…
,
x
k
+
1
≤
∑
i
=
1
k
q
i
d
x
i
,
x
i
+
1holds for all x
1
,
x
2
,
…
,
x
k
,
x
k
+
1
∈
X, where q
i
≥
0 and ∑
i
=
1
k
‍
q
i
∈
[
0,1
). Then f has a unique fixed point x
∗. Moreover, for any arbitrary points x
1
,
x
2
,
…
,
x
k
+
1 in X, the sequence {
x
n
} defined by x
n
+
k
=
f
(
x
n
,
x
n
+
1
,
…
,
x
n
+
k
-
1
), for all n
∈
N converges to x
∗.Later Ciric and Presic [14] generalized the above theorem as follows.Theorem 3 (see [14]).
Let(
X
,
d
) be a complete metric space, k be a positive integer, and f
:
X
k
→
X. Suppose that (3)
d
f
x
1
,
x
2
,
…
,
x
k
,
f
x
2
,
x
3
,
…
,
x
k
+
1
≤
λ
max
⁡
d
x
i
,
x
i
+
1
:
1
≤
i
≤
kholds for all x
1
,
x
2
,
…
,
x
k
,
x
k
+
1 in X, where λ
∈
[
0,1
). Then f has a fixed point x
∗
∈
X. Moreover, for any arbitrary points x
1
,
x
2
,
…
,
x
k
+
1 in X, the sequence {
x
n
} defined by x
n
+
k
=
f
(
x
n
,
x
n
+
1
,
…
,
x
n
+
k
-
1
), for all n
∈
N converges to x
∗. Moreover, if d
(
f
(
u
,
u
,
…
,
u
)
,
f
(
v
,
v
,
…
,
v
)
)
<
d
(
u
,
v
) holds for all u
,
v
∈
X with u
≠
v, then x
∗ is the unique fixed point of f.Recently Rao et al. [2, 3] obtained some Presic type theorems for two and three maps in metric spaces. Now we give the following definition of [2, 3].Definition 4.
LetX be a nonempty set and T
:
X
2
k
→
X and f
:
X
→
X. The pair (
f
,
T
) is said to be 2
k-weakly compatible if f
(
T
(
x
,
x
,
…
,
x
,
x
)
)
=
T
(
f
x
,
f
x
,
…
,
f
x
,
f
x
) whenever x
∈
X such that f
x
=
T
(
x
,
x
,
…
,
x
,
x
).Using this definition, Rao et al. [2] proved the following.Theorem 5 (see [2]).
Let(
X
,
d
) be a metric space, k be a positive integer, and S
,
T
:
X
2
k
→
X, f
:
X
→
X be mappings satisfying (1.4.1)
d
(
S
(
x
1
,
x
2
,
…
,
x
2
k
)
,
T
(
x
2
,
x
3
,
…
,
x
2
k
+
1
)
)
≤
λ
max
⁡
{
d
(
f
x
i
,
f
x
i
+
1
)
:
1
≤
i
≤
2
k
} for all x
1
,
x
2
,
…
,
x
2
k
,
x
2
k
+
1 in X,
(1.4.2)
d
(
T
(
y
1
,
y
2
,
…
,
y
2
k
)
,
S
(
y
2
,
y
3
,
…
,
y
2
k
+
1
)
)
≤
λ
max
⁡
{
d
(
f
y
i
,
f
y
i
+
1
)
:
1
≤
i
≤
2
k
} for all y
1
,
y
2
,
…
,
y
2
k
,
y
2
k
+
1 in X, where 0
≤
λ
<
1,
(1.4.3)
d
(
S
(
u
,
…
,
u
)
,
T
(
v
,
…
,
v
)
)
<
d
(
f
u
,
f
v
), for all u
,
v
∈
X with u
≠
v.
(1.4.4)
Suppose thatf
(
X
) is complete and either (
f
,
S
) or (
f
,
T
) is a 2
k-weakly compatible pair.Then there exists a unique point p
∈
X such that f
p
=
p
=
S
(
p
,
…
,
p
)
=
T
(
p
,
…
,
p
).Now we give a definition of jointly2
k-weakly compatible pairs.Definition 6.
LetX be a nonempty set, k be a positive integer, S
,
T
:
X
2
k
→
X, and f
,
g
:
X
→
X. The pairs (
f
,
S
) and (
g
,
T
) are said to be jointly 2
k-weakly compatible pairs if f
(
S
(
x
,
x
,
…
,
x
)
)
=
S
(
f
x
,
f
x
,
…
,
f
x
)
) and g
(
T
(
x
,
x
,
…
,
x
)
)
=
T
(
g
x
,
g
x
,
…
,
g
x
)
) whenever there exists x
∈
X such that f
x
=
S
(
x
,
x
,
…
,
x
) and g
x
=
T
(
x
,
x
,
…
,
x
).In this paper, we obtain a Presic type theorem for four mappings satisfying a slight different contractive condition in metric spaces. We also give an example and two corollaries to our main theorem.Now we prove our main result.


## 2. Main Result

Theorem 7.
Let(
X
,
d
) be a complete metric space, k be a positive integer, and S
,
T
:
X
2
k
→
X and f
,
g
:
X
→
X be mappings satisfying(2.1.1)
S
(
X
2
k
)
⊆
g
(
X
), T
(
X
2
k
)
⊆
f
(
X
),
(2.1.2)
d
(
S
(
x
1
,
x
2
,
…
,
x
2
k
-
1
,
x
2
k
)
,
T
(
y
1
,
y
2
,
…
,
y
2
k
-
1
,
y
2
k
)
)
≤
λ
m
a
x
{
d
(
g
x
1
,
f
y
1
)
,
d
(
f
x
2
,
g
y
2
)
,
d
(
g
x
3
,
f
y
3
), d
(
f
x
4
,
g
y
4
)
,
…
,
d
(
g
x
2
k
-
1
,
f
y
2
k
-
1
)
,
d
(
f
x
2
k
,
g
y
2
k
)
}
for allx
1
,
x
2
,
…
,
x
2
k
,
y
1
,
y
2
,
…
,
y
2
k
∈
X and 0
<
λ
<
1,
(2.1.3)
(
f
,
S
) and (
g
,
T
) are jointly 2
k-weakly compatible pairs.
(2.1.4)
Supposez

=

f
p

=

g
p for some p
∈
X whenever there exists a sequence {
y
2
k
+
n
}
n
=
1
∞ in X such that  l
i
m
n
→
∞
y
2
k
+
n
=
z
∈
X.
Thenz is the unique point in X such that z
=
f
z
=
g
z
=
s
(
z
,
z
,
…
,
z
,
z
)
=
T
(
z
,
z
,
…
,
z
,
z
).Proof.
Supposex
1
,
x
2
,
…
,
x
2
k are arbitrary points in X. From (2.1.1), define(4)
y
2
k
+
2
n
-
1
=
S
x
2
n
-
1
,
x
2
n
,
…
,
x
2
k
+
2
n
-
2
=
g
x
2
k
+
2
n
-
1
y
2
k
+
2
n
=
T
x
2
n
,
x
2
n
+
1
,
…
,
x
2
k
+
2
n
-
1
=
f
x
2
k
+
2
n
for
n
=
1,2
,
…
.Let(5)
α
2
n
=
d
f
x
2
n
,
g
x
2
n
+
1
,
α
2
n
-
1
=
d
g
x
2
n
-
1
,
f
x
2
n
,
n
=
1,2
,
…
.Let θ
=
λ
1
/
2
k and μ
=
m
a
x
{
α
1
/
θ
,
α
2
/
θ
2
,
…
,
α
2
k
/
θ
2
k
}.
Then0
<
θ
<
1 and by selection of μ we have(6)
α
n
≤
μ
θ
n
,
for
n
=
1,2
,
…
,
2
k
.
(7)
α
2
k
+
1
=
d
g
x
2
k
+
1
,
f
x
2
k
+
2
=
d
S
x
1
,
x
2
,
x
3
,
x
4
,
…
,
x
2
k
-
1
,
x
2
k
,
T
x
2
,
x
3
,
x
4
,
…
,
x
2
k
,
x
2
k
+
1
≤
λ
max
⁡
d
g
x
1
,
f
x
2
,
d
f
x
2
,
g
x
3
,
d
g
x
3
,
f
x
4
,
d
f
x
4
,
g
x
5
,
…
,
d
g
x
2
k
-
1
,
f
x
2
k
,
d
f
x
2
k
,
g
x
2
k
+
1
from
2.1
.
2
=
λ
max
⁡
α
1
,
α
2
,
α
3
,
α
4
,
…
,
α
2
k
-
1
,
α
2
k
≤
λ
max
⁡
μ
θ
,
μ
θ
2
,
μ
θ
3
,
μ
θ
4
,
…
,
μ
θ
2
k
-
1
,
μ
θ
2
k
from
6
=
λ
μ
θ
=
θ
2
k
μ
θ
=
μ
θ
2
k
+
1
,
(8)
α
2
k
+
2
=
d
f
x
2
k
+
2
,
g
x
2
k
+
3
=
d
S
x
3
,
x
4
,
x
5
,
x
6
,
…
,
x
2
k
+
1
,
x
2
k
+
2
,
T
x
2
,
x
3
,
x
4
,
…
,
x
2
k
,
x
2
k
+
1
≤
λ
max
⁡
d
g
x
3
,
f
x
2
,
d
f
x
4
,
g
x
3
,
d
g
x
5
,
f
x
4
,
d
f
x
6
,
g
x
5
,
…
,
d
g
x
2
k
+
1
,
f
x
2
k
,
d
f
x
2
k
+
2
,
g
x
2
k
+
1
from
2.1
.
2
=
λ
max
⁡
α
2
,
α
3
,
α
4
,
α
5
,
…
,
α
2
k
,
α
2
k
+
1
≤
λ
max
⁡
μ
θ
2
,
μ
θ
3
,
μ
θ
4
,
μ
θ
5
,
…
,
μ
θ
2
k
,
μ
θ
2
k
+
1
from
6
,
7
=
λ
μ
θ
2
=
θ
2
k
μ
θ
2
=
μ
θ
2
k
+
2
.Continuing in this way, we get(9)
α
n
≤
μ
θ
n
for
n
=
1,2
,
…
.Now consider(10)
d
y
2
k
+
2
n
-
1
,
y
2
k
+
2
n
=
d
S
x
2
n
-
1
,
x
2
n
,
x
2
n
+
1
,
x
2
n
+
2
,
…
,
x
2
k
+
2
n
-
3
,
x
2
k
+
2
n
-
2
,
T
x
2
n
,
x
2
n
+
1
,
x
2
n
+
2
,
x
2
n
+
3
,
…
,
x
2
k
+
2
n
-
2
,
x
2
k
+
2
n
-
1
≤
λ
max
⁡
d
g
x
2
n
-
1
,
f
x
2
n
,
d
f
x
2
n
,
g
x
2
n
+
1
,
d
g
x
2
n
+
1
,
f
x
2
n
+
2
,
d
f
x
2
n
+
2
,
g
x
2
n
+
3
,
…
,
d
g
x
2
k
+
2
n
-
3
,
f
x
2
k
+
2
n
-
2
,
d
f
x
2
k
+
2
n
-
2
,
g
x
2
k
+
2
n
-
1
=
λ
max
⁡
α
2
n
-
1
,
α
2
n
,
α
2
n
+
1
,
α
2
n
+
2
,
…
,
α
2
k
+
2
n
-
3
,
α
2
k
+
2
n
-
2
≤
λ
max
⁡
μ
θ
2
n
-
1
,
μ
θ
2
n
,
μ
θ
2
n
+
1
,
μ
θ
2
n
+
2
,
…
,
μ
θ
2
k
+
2
n
-
3
,
μ
θ
2
k
+
2
n
-
2
from
9
=
λ
μ
θ
2
n
-
1
=
θ
2
k
μ
θ
2
n
-
1
=
μ
θ
2
k
+
2
n
-
1
.
Also(11)
d
y
2
k
+
2
n
,
y
2
k
+
2
n
+
1
=
d
S
x
2
n
+
1
,
x
2
n
+
2
,
x
2
n
+
3
,
x
2
n
+
4
,
…
,
x
2
k
+
2
n
-
1
,
x
2
k
+
2
n
,
T
x
2
n
,
x
2
n
+
1
,
x
2
n
+
2
,
x
2
n
+
3
,
…
,
x
2
k
+
2
n
-
2
,
x
2
k
+
2
n
-
1
≤
λ
max
⁡
d
g
x
2
n
+
1
,
f
x
2
n
,
d
f
x
2
n
+
2
,
g
x
2
n
+
1
,
d
g
x
2
n
+
3
,
f
x
2
n
+
2
,
d
f
x
2
n
+
4
,
g
x
2
n
+
3
,
…
,
d
g
x
2
k
+
2
n
-
1
,
f
x
2
k
+
2
n
-
2
,
d
f
x
2
k
+
2
n
,
g
x
2
k
+
2
n
-
1
=
λ
max
⁡
α
2
n
,
α
2
n
+
1
,
α
2
n
+
2
,
α
2
n
+
3
,
…
,
α
2
k
+
2
n
-
2
,
α
2
k
+
2
n
-
1
≤
λ
max
⁡
μ
θ
2
n
,
μ
θ
2
n
+
1
,
μ
θ
2
n
+
2
,
μ
θ
2
n
+
3
,
…
,
μ
θ
2
k
+
2
n
-
2
,
μ
θ
2
k
+
2
n
-
1
from
9
=
λ
μ
θ
2
n
=
θ
2
k
μ
θ
2
n
=
μ
θ
2
k
+
2
n
.Thus from (10) and (11) we have(12)
d
y
2
k
+
n
,
y
2
k
+
n
+
1
≤
μ
θ
2
k
+
n
for
n
=
1,2
,
3
,
…
.Now from m
>
n consider(13)
d
y
2
k
+
n
,
y
2
k
+
m
≤
d
y
2
k
+
n
,
y
2
k
+
n
+
1
+
d
y
2
k
+
n
+
1
,
y
2
k
+
n
+
2
+
d
y
2
k
+
n
+
2
,
y
2
k
+
n
+
3
+
⋯
+
d
y
2
k
+
m
-
1
,
y
2
k
+
m
≤
μ
θ
2
k
+
n
+
μ
θ
2
k
+
n
+
1
+
μ
θ
2
k
+
n
+
2
+
⋯
+
μ
θ
2
k
+
m
-
1
from
12
≤
μ
θ
2
k
θ
n
1
-
θ
⟶
0
as
n
⟶
∞
,
m
⟶
∞
.Hence {
y
2
k
+
n
} is a Cauchy sequence in X. Since X is complete there exists z
∈
X such that y
2
k
+
n
→
z as n
→
∞. From (2.1.4), there exists u
∈
X such that(14)
z
=
f
u
=
g
u
.Now consider(15)
d
S
u
,
u
,
…
,
u
,
u
,
y
2
k
+
2
n
=
d
S
u
,
u
,
…
,
u
,
u
,
T
x
2
n
,
x
2
n
+
1
,
…
,
x
2
k
+
2
n
-
2
,
x
2
k
+
2
n
-
3
≤
λ
max
⁡
d
g
p
,
f
x
2
n
,
d
f
p
,
g
x
2
n
+
1
,
d
g
p
,
f
x
2
n
+
2
,
d
f
p
,
g
x
2
n
+
3
,
…
,
d
g
p
,
f
x
2
k
+
2
n
-
2
,
d
f
p
,
g
x
2
k
+
2
n
-
1
.Letting n
→
∞ and using (14) d
(
S
(
u
,
u
,
…
,
u
,
u
)
,
f
u
)
≤
0 so that(16)
f
u
=
S
u
,
u
,
…
,
u
,
u
.Similarly we have(17)
f
u
=
g
u
=
T
u
,
u
,
…
,
u
,
u
.Since (
f
,
S
) and (
g
,
T
) are jointly 2
k-weakly compatible pairs, we have(18)
f
z
=
f
f
u
=
f
S
u
,
u
,
…
,
u
,
u
=
S
f
u
,
f
u
,
…
,
f
u
,
f
u
=
S
z
,
z
,
…
,
z
,
z
,
(19)
g
z
=
T
z
,
z
,
…
,
z
,
z
(20)
d
f
z
,
z
=
d
f
z
,
g
u
=
d
S
z
,
z
,
…
,
z
,
z
,
T
u
,
u
,
…
,
u
,
u
from
18
,
17
≤
λ
max
⁡
d
g
z
,
f
u
,
d
f
z
,
g
u
,
d
g
z
,
f
u
,
d
f
z
,
g
u
,
…
,
d
g
z
,
f
u
,
d
f
z
,
g
u
=
λ
max
⁡
d
g
z
,
z
,
d
f
z
,
z
from
14
.Similarly, we have(21)
d
g
z
,
z
≤
λ
max
⁡
d
z
,
f
z
,
d
z
,
g
z
.From (20) and (21) we have(22)
max
⁡
d
z
,
f
z
,
d
z
,
g
z
≤
λ
max
⁡
d
z
,
f
z
,
d
z
,
g
zwhich in turn yields that(23)
f
z
=
z
,
g
z
=
z
.From (18), (19), and (23), we have(24)
z
=
f
z
=
g
z
=
S
z
,
z
,
…
,
z
,
z
=
T
z
,
z
,
…
,
z
,
z
.Suppose there exists z
′
∈
X such that (25)
z
′
=
f
z
′
=
g
z
′
=
S
z
′
,
z
′
,
…
,
z
′
,
z
′
=
T
z
′
,
z
′
,
…
,
z
′
,
z
′
.Then from (2.1.2) we have(26)
d
z
,
z
′
=
d
S
z
,
z
,
…
,
z
,
z
,
T
z
′
,
z
′
,
…
,
z
′
,
z
′
≤
λ
max
⁡
d
g
z
,
f
z
′
,
d
f
z
,
g
z
′
,
…
,
d
g
z
,
f
z
′
,
d
f
z
,
g
z
′
=
λ
d
z
,
z
′
.This implies that z
′
=
z.
Thusz is the unique point in X satisfying (24).Now we give an example to illustrate our main Theorem7.Example 8.
LetX
=
[
0,1
], k
=
1, and d
(
x
,
y
)
=
x
-
y. Define S
(
x
,
y
)
=
3
x
2
+
2
y
/
72, T
(
x
,
y
)
=
2
x
+
3
y
2
/
72, f
x
=
x
/
6, and g
x
=
x
2
/
4 for all x
,
y
∈
X.
Then, for allx
1
,
x
2
,
y
1
,
y
2
∈
X, we have(27)
d
S
x
1
,
x
2
,
T
y
1
,
y
2
=
3
x
1
2
+
2
x
2
72
-
2
y
1
+
3
y
2
2
72
=
1
72
3
x
1
2
-
2
y
1
+
2
x
2
-
3
y
2
2
≤
1
72
3
x
1
2
-
2
y
1
+
2
x
2
-
3
y
2
2
≤
1
36
max
⁡
3
x
1
2
-
2
y
1
,
2
x
2
-
3
y
2
2
=
1
3
max
⁡
d
g
x
1
,
f
y
1
,
d
f
x
2
,
g
y
2
.Thus the condition (2.1.2) of Theorem 7 is satisfied. One can easily verify the remaining conditions of Theorem 7. Clearly, 0 is the unique point in X such that 0
=
f
0
=
g
0
=
S
(
0,0
,
…
,
0,0
)
=
T
(
0,0
,
…
,
0,0
).Corollary 9.
Let(
X
,
d
) be a metric space and S
,
T
:
X
2
k
→
X and f
:
X
→
X be mappings satisfying (2.3.1)
S
(
X
2
k
)
⊆
f
(
X
), T
(
X
2
k
)
⊆
f
(
X
),
(2.3.2)
d
(
S
(
x
1
,
x
2
,
…
,
x
2
k
-
1
,
x
2
k
)
,
T
(
y
1
,
y
2
,
…
,
y
2
k
-
1
,
y
2
k
)
)
≤
λ
max
⁡
d
(
f
x
i
,
f
y
i
)
/
1
≤
i
≤
2
k for all x
1
,
x
2
,
…
,
x
2
k
,
y
1
,
y
2
,
…
,
y
2
k
∈
X and 0
<
λ
<
1,
(2.3.3)
f
(
X
) is a complete subspace of X,
(2.3.4)
the pairs(
f
,
S
) or (
f
,
T
) is 2
k-weakly compatible.Then there exists a unique u
∈
X such that u
=
f
u
=
S
(
u
,
u
,
…
,
u
,
u
)
=
T
(
u
,
u
,
…
,
u
,
u
).Corollary 10.
Let(
X
,
d
) be a complete metric space and S
,
T
:
X
2
k
→
X be mappings satisfying (2.4.1)
d
(
S
(
x
1
,
x
2
,
…
,
x
2
k
-
1
,
x
2
k
)
,
T
(
y
1
,
y
2
,
…
,
y
2
k
-
1
,
y
2
k
)
)
≤
λ
max
⁡
d
(
x
i
,
y
i
)
/
1
≤
i
≤
2
k for all x
1
,
x
2
,
…
,
x
2
k
,
y
1
,
y
2
,
…
,
y
2
k
∈
X and 0
<
λ
<
1.Then there exists a unique u
∈
X such that u
=
S
(
u
,
u
,
…
,
u
)
=
T
(
u
,
u
,
…
,
u
).Remark 11.
Corollaries9 and 10 are slight variant of theorems of [2, 3].



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*Source: 2121906-2016-11-22.xml*