AJMS

1. www.ajms.com 1
ISSN 2581-3463
RESEARCH ARTICLE
Common Fixed Point Theorems for Non-linear Contractive Mapping in Hilbert
Space
M. Raji, F. Adegboye
Department of Mathematics, Confluence University of Science and Technology, Osara, Kogi State, Nigeria
Received: 25-02-2021; Revised: 18-03-2022; Accepted: 10-04-2022
ABSTRACT
In this manuscript, we prove the existence and uniqueness of a common fixed point for a continuous self-
mapping over a closed subset of Hilbert space satisfying non-linear rational type contraction condition.
Key words: Hilbert space, Fixed point, Closed subset, Completeness
Address for correspondence:
Muhammed Raji,
Email: rajimuhammed11@gmail.com
INTRODUCTION
Fixed point theory is an important tool used in
solving a variety of problems in control theory,
economic theory, global analysis, and non-
linear analysis. Several authors worked on the
applications, generalizations, and extensions of
the Banach contraction principle[1]
in different
directions by either weakening the hypothesis,
using different setups or considering different
mappings.
Banachcontractionprinciplehasbeenextendedand
generalized by several authors such as Kannan[2]
who investigated the extension of Banach fixed
point theorem by removing the completeness of
the space with different conditions. Chatterji[3]
considered various contractive conditions for
self-mappings in metric space. Dass and Gupta[5]
also investigated the rational type of contractions
to obtain a unique fixed point in complete metric
space. Fisher[10]
developed the approach of
Khann[2]
and proved analogous results involving
two mappings on a complete metric space.
PRELIMINARIES
The main aim of this paper is to prove the
existence and uniqueness of a common fixed point
for a continuous self-mapping T, some positive
integers r,s of a pair of continuous self-mapping
Tr
,Ts
. These results generalized and extend the
results of[2-13]
and[10]
here are the lists of some of
the results that motivated our results.
Theorem 2.1.[2]
proved that “If f is self-mapping
of a complete metric space X into itself satisfying
d(x,y)≤α[d(Tx,x)+d(Ty,y)], for all x,y∈X and
α∈[0,1/2), then f has a unique fixed point in X”.
Theorem 2.2.[10]
Prove the result with
d(x,y)≤α[d(Tx,x)+d(Ty,y)]+βd(x,y), for all x,y ∈ X
and α,β ∈[0,1/2), then f has a unique fixed point
in X”.
Theorem 2.3.[8]
Proved the common fixed point
theorems in Hilbert space with the following
condition.
Tx Ty
x Tx y Ty
x Tx y Ty
x y
− ≤
− + −
− + −










+ −
α β
2 2
for
all x,y ∈S,x ≠y, α ∈[0,1/2), β≥0 and 2α + β<1.
Theorem 2.4.[9]
Proved the fixed point theorems
in Hilbert space with the following condition.
Tx Ty
x Tx y Ty
x Tx y Ty
x Ty y Tx
x Ty y T
− ≤
− + −
− + −










+
− + −
− + −
α
β
2 2
2 2
x
x
x y










+ −
γ
for all x,y ∈S,x ≠y, 0≤ α,β < 1/2, 0 ≤ 𝛾 2α and 2β
+ 𝛾 < 1.
Theorem 2.5.[14]
Proved the unique fixed point
theorems in Hilbert space with the following
condition.

2. Raji and Adegboye: Common Fixed Point Theorems for Non-linear Contractive Mapping in Hilbert Space
AJMS/Apr-Jun-2022/Vol 6/Issue 2 2
Tx Ty a
x Tx y Ty x Ty y Tx
x Tx y Ty x Ty y Tx
− ≤
− + − + − + −
− + − + − + −








1
2 2 2 2



+
− + −
− + −










+
− + −
− + −
a
x Tx y Ty
x Tx y Ty
a
x Ty y Tx
x Ty y T
2
2 2
3
2 2
x
x
a x y










+ −
4
For all x,y ∈S and 4α1
,α2
,α3,
α4
≥ 0and 4α1
+ 2α2
+
2α3
+α4
1.
Theorem 2.6.[7]
Proved the unique common fixed
point theorems for two self-mappings T1
,T2
of closed
subset X in Hilbert space satisfying the inequality.
T x T y a
x T x y T y
x y
a
1 2 1
1 2
2
1
1
− ≤
− + −

 

+ −
+
y T y y T x
x y
− + −

 

+ −
+
2 1
1
1
a
x T y y T x
x y
3
2 1
1
1
− + −

 

+ −
+
a
x y T x T y
x y
4
1 2
1
1
− + −

 

+ −
+
− + −

 

+ −
+
a
x y x T x
y T y
5
1
2
1
1
a
x T x x T y
y T y
a
x T y T x T y
x y
a
6
1 2
2
7
2 1 2
8
1
1
1
1
− + −

 

+ −
+
− + −

 

+ −
+
x y x T y
x y
a
x T x y T y x y
x T x x T y y T y x y
− + −

 

+ −
+
− + − + −
+ − − − −
+
1
1
1
2
9
1 2
1 2 2
a
x T y y T x
x T y y T x
a x T x y T y
a x T y y
10
2
2
1
2
2 1
11 1 2
12 2
− + −
− + −
+ − + −

 

+ − + −T
T x a x y
1 13

 
 + − .
For all x,y ∈ X andx ≠ y, where ai
(i=1,2,3,…,13) are
non-negative reals with 0 ≤ ∑8
i=1
ai
+3a9
+2∑12
i=10
ai
+a13
1.
Theorem 2.7.[6]
Proved the unique common fixed
point theorems for two self-mappings T1
T2
of
closed subset X in Hilbert space satisfying the
inequality.
T x T y
x T y y T x
x y
x y x T y
x y
1 2
2 2
2
1
2
2
2
2
2
2
1
1
1
1
− ≤
− + −




+ −
+
− + −




+ −
α
β +
+ − + −




+ −
γ
δ
y T x x T y
x y
1
2
2
2
2
for all x y S x y
, ,�
∈ ≠ where α,β,γ,δ and 4α + β +
4γ + δ 1.
Theorem 2.8.[6]
Proved the unique common fixed
point theorems for two self-mappings T1
,T2
and p,q
(positive integers) of closed subset X in Hilbert
space satisfying the inequality.
T x T y
x T y y T x
x y
x y x T y
p q
q p
q
1 2
2 2
2
1
2
2
2
2
2
1
1
1
− ≤
− + −






+ −
+
− + −



α
β




+ −
+ − + −






+ −
1
2 1
2
2
2
2
x y
y T x x T y
x y
p q
γ
δ
for all 𝑥,𝑦 ∈ 𝑆,𝑥≠𝑦, where 𝛼,𝛽,𝛾,𝛿 and 4𝛼 +𝛽+
4𝛾+𝛿 1.
MAIN RESULTS
Theorem 3.1. Let X be a closed subset of a Hilbert
space and T: X→X be a continuous self-mapping
satisfying the following inequality.
Tx Ty c
x Tx y Ty
x y
c
y Ty y Tx
x y
− ≤
− + −

 

+ −
+
− + −

 

+ −
+
1
2
1
1
1
1
c
x Ty y Tx
x y
c
x y Tx Ty
x y
3 4
1
1
1
1
− + −

 

+ −
+
− + −

 

+ −

3. Raji and Adegboye: Common Fixed Point Theorems for Non-linear Contractive Mapping in Hilbert Space
AJMS/Apr-Jun-2022/Vol 6/Issue 2 3
+
− + −

 

+ −
+
− + −

 

+ −
+
c
x y x Tx
y Ty
c
x Tx x Ty
y Ty
5 6
1
1
1
1
c
x Ty Tx Ty
x y
c
x y x Ty
x y
7 8
1
1
1
1
− + −

 

+ −
+
− + −

 

+ −
+
− + − + −
+ − − − −
c
x Tx y Ty x y
x Tx x Ty y Ty x y
9
1
+
− + −
− + −
+ − + −

 

+ − + −

 

c
x Ty y Tx
x Ty y Tx
c x Tx y Ty
c x Ty y Tx
10
2 2
11
12 +
+ −
c x y
13
for all x,y∈X and x≠y, where ci
(i=1,2,3,…,13) are
non-negative real numbers with 0≤ ci
i=
∑ 1
8
+ 3c9
+2
ci
i=
∑ 10
12
+ c13
1. Then, T has a unique fixed point
in X.
Proof. We construct a sequence {xn
} for an
arbitrary point x0
∈ X defined as follows
x2n+1
= Tx2n
, x2n+2
=Tx2n
+1, for 𝑛=1,2,3.,
We show that the sequence {xn
} is a Cauchy
sequence in X
x x Tx Tx
n n n n
2 1 2 2 2 1
+ −
− = −
≤
− + −

 

+ −
+
− +
− −
−
− −
c
x Tx x Tx
x x
c
x Tx x
n n n n
n n
n n
1
2 2 2 1 2 1
2 2 1
2
2 1 2 1
1
1
1 2
2 1 2
2 2 1
1
n n
n n
Tx
x x
−
−
−

 

+ −
+
− + −

 

+ −
− −
−
c
x Tx x Tx
x x
n n n n
n n
3
2 2 1 2 1 2
2 2 1
1
1
+
− + −

 

+ −
+
− + −
− −
−
−
c
x x Tx Tx
x x
c
x x x
n n n n
n n
n n n
4
2 2 1 2 2 1
2 2 1
5
2 2 1 2
1
1
1 T
Tx
x Tx
n
n n
2
2 1 2 1
1

 

+ −
+
− −
c
x Tx x Tx
x Tx
n n n n
n n
6
2 2 2 2 1
2 1 2 1
1
1
− + −

 

+ −
−
− −
+
− + −

 

+ −
+
− +
− −
−
−
c
x Tx Tx Tx
x x
c
x x x
n n n n
n n
n n n
7
2 2 1 2 2 1
2 2 1
8
2 2 1 2
1
1
1 −
−

 

+ −
−
−
Tx
x x
n
n n
2 1
2 2 1
1
+
− + −
+ −
+ − −
− −
−
−
−
c
x Tx x Tx
x x
x Tx x Tx
x
n n n n
n n
n n n n
n
9
2 2 2 1 2 1
2 2 1
2 2 2 2 1
2
1
1
1 2 1 2 2 1
− −
− −
Tx x x
n n n
+
− + −
− + −
− −
− −
c
x Tx x Tx
x Tx x Tx
n n n n
n n n n
10
2 2 1
2
2 1 2
2
2 2 1 2 1 2
+ − + −

 

+ − + −

 

− −
− −
c x Tx x Tx
c x Tx x Tx
n n n n
n n n n
11 2 2 2 1 2 1
12 2 2 1 2 1 2 +
+ − −
c x x
n n
13 2 2 1
This implies
x x a n Tx Tx
n n n n
2 1 2 2 2 1
+ −
− = ( ) − ,
where
a n
B c c c c c
x x
B c c c c c c
n n
( )=
+ + + + +
( )
−
+ − − − − − −
−
1 2 9 10 11 12
2 2 1
2 1 2 4 5 9
2
1 1
10 11 12
2 2 1
− −
( )
− −
c c
x x
n n
,
1 2 4 5 8 9 10 11 12 13
2
B c c c c c c c c c
= + + + + + + + + and
B2
=1-c1
-c6
-c9
-c10
-c11
-c12 S
Clearly, δ=a (n)1,∀n=1,2,3,.., we get
x x x x
n n n n
+ −
− = −
1 1
 ,
Recursively, we have
x x x x n
n n
n
+ − = − ≥
1 1 0 1
 , ,
Takingn→∞, we have x x
n n
+ − →
1 0.
Hence, {xn
} is a Cauchy sequence in X and so it has
a limit λ in X. Since the sequences {x2n+1
}={Tx2n
}
and {x2n+2
}={Tx2n+1
}are subsequences of {xn
}, and
also these subsequences have the same limit λ in X.
We now show that λ is a common fixed point of T
Now consider the following inequality

4. Raji and Adegboye: Common Fixed Point Theorems for Non-linear Contractive Mapping in Hilbert Space
AJMS/Apr-Jun-2022/Vol 6/Issue 2 4
    
  
− = − + − = −
( )
+ −
( )≤ − + −
+ + +
+ +
T x x T x
x T x T Tx
n n n
n n n
2 2 2 2 2 2
2 2 2 2 2 +
+1
≤
− + −

 

+ −
+
− + −
+ +
+
+ + +
c
T x Tx
x
c
x Tx x T
n n
n
n n n
1
2 1 2 1
2 1
2
2 1 2 1 2 1
1
1
1
 





 

+ − +
1 2 1
x n
c
Tx x T
x
c
x T Tx
n n
n
n n
3
2 1 2 1
2 1
4
2 1 2 1
1
1
1
 

 
− + −

 

+ −
+
− + −

 

+ +
+
+ +
1
1 2 1
+ − +
 x n
+
− + −

 

+ −
+
− + −

 

+
+
+ +
+
c
x T
x Tx
c
T Tx
x
n
n n
n
5
2 1
2 1 2 1
6
2 1
1
1
1
1
  
  
2
2 1 2 1
n n
Tx
+ +
−
+
− + −

 

+ −
+
− + −

 
+ +
+
+ +
c
Tx T Tx
x
c
x Tx
n n
n
n n
7
2 1 2 1
2 1
8
2 1 2 1
1
1
1
 

  

+ − +
1 2 1
 x n
+
− + − + −
+ − − − −
+ + +
+ + +
c
T x Tx x
T Tx x Tx x
n n n
n n n
9
2 1 2 1 2 1
2 1 2 1 2 1 2
1
  
    n
n
n n
n n
c
Tx x T
Tx x T
+
+ +
+ +
+
− + −
− + −
1
10
2 1
2
2 1
2
2 1 2 1
 
 
+ − + −

 

+ − + −

 
 + −
+ +
+ +
c T x Tx
c Tx x T c x
n n
n n
11 2 1 2 1
12 2 1 2 1 13 2
 
   n
n+1 .
Letting n→∞, we get ‖λ-Tλ‖ ≤
(c1
+c6
+c9
+c10
+c11
+c12
)‖λ-Tλ‖, since
c1
+c6
+c9
+c10
+c11
+c12
1, hence λ = Tλ.
Similarly, from the hypothesis, we get λ = Tλ by
considering the following
   
− = −
( )+ −
( )
+ +
T x x T
n n
2 1 2 1 .
We now show that λ is a unique fixed point of T.
Suppose that t (λ≠t) is also a common fixed point
of T.
Then, by the hypothesis, we get
 
 



− = − ≤
− + −

 

+ −
+
− + −

 

+ −
+
t T Tt c
T t Tt
t
c
t Tt t T
t
c
1
2 3
1
1
1
1
 

 

  
− + −

 

+ −
+
− + −

 

+ −
+
− + −

 

Tt t T
t
c
t T Tt
t
c
t T
1
1
1
1
1
1
4
5
+
+ −
t Tt
+
− + −

 

+ −
+
− + −

 

+ −
+
− + −
c
T Tt
t Tt
c
Tt T Tt
t
c
t Tt
6 7
8
1
1
1
1
1
    

 


 

+ −
1  t
+
− + − + −
+ − − − −
+
− + −
− + −
c
T t Tt t
T Tt t Tt t
c
Tt t T
Tt t T
9
10
2 2
1
  
   
 
 
+ − + −

 
 + − + −

 

+ −
c T t Tt c Tt t T
c t
11 12
13
   
 .
Thus,

 
− ≤ + + + + + + + +
( )
− −
t c c c c c c c c c
t t
3 4 5 7 8 9 10 12 13
2
a contradiction. Hence λ=t (common fixed point λ
is unique in X).
Theorem 3.2. Let X be a closed subset of a Hilbert
space and T: X→X be a continuous self mapping
satisfying
T x T y c
x T x y T y
x y
c
y T y y T x
x y
r s
r s
s r
− ≤
− + −




+ −
+
− + −




+ −
+
1
2
1
1
1
1
c
x T y y T x
x y
c
x y T x T y
x y
s r
r s
3
4
1
1
1
1
− + −




+ −
+
− + −




+ −

5. Raji and Adegboye: Common Fixed Point Theorems for Non-linear Contractive Mapping in Hilbert Space
AJMS/Apr-Jun-2022/Vol 6/Issue 2 5
+
− + −




+ −
+
− + −




+ −
+
−
c
x y x T x
y T y
c
x T x x T y
y T y
c
x T y
r
s
r s
s
s
5 6
7
1
1
1
1
1
1
1
1
1
8
+ −




+ −
+
− + −




+ −
T x T y
x y
c
x y x T y
x y
r s s
+
− + − + −
+ − − − −
+
− + −
−
c
x T x y T y x y
x T x x T y y T y x y
c
x T y y T x
x T
r s
r s s
s r
9
10
2 2
1
s
s r
y y T x
+ −
+ − + −




+ − + −



 + −
c x T x y T y
c x T y y T x c x y
r s
s r
11
12 13
for all x, y∈ X and x≠y, where ci
(i=1,2,3,…,13)
are non-negative real numbers with
0 3 2 1
1
8
9
10
12
13
≤ + + +
= =
∑ ∑
i
i
i
i
c c c c and r,s are two
positive integers. Then T has a unique fixed point
in X.
Proof. From theorem 3.1.Tr
and Ts
have a unique
common fixed point λ∈X, so that Tr
λ=λ and Ts
λ=λ.
From Tr
(Tλ)=T(Tr
λ)=Tλ, it follows that Tλ is a
fixed point of Tr
. But λ is a unique fixed point of
Tr
. Therefore Tλ=λ.
Similarly,wecangetTλ=λfromTs
(Tλ)=T(Ts
λ)=Tλ.
Hence λ is a unique fixed point of T.
Now, we show uniqueness. Let t be another fixed
point of T, so that Tt=t. then from the hypothesis,
we have
 
 



− = − ≤
− + −




+ −
+
− + −




+ −
t T T t c
T t T t
t
c
t T t t T
r s
r r
s r
1
2
1
1
1
1 t
t
+
c
T t t T
t
c
t T T t
t
c
t T
s r r s
3 4
5
1
1
1
1
1
 

 

 
− + −




+ −
+
− + −




+ −
+
− + − r
r
s
t T t





+ −
1
+
− + −




+ −
+
− + −




+ −
+
−
c
T T t
t T t
c
T t T T t
t
c
t
r s
s
s r s
6
7 8
1
1
1
1
  
 

 1
1
1
1
9
+ −




+ −
+
− + − + −
+ − − − −


  
   
T t
t
c
T t T t t
T T t t T t t
s
r s
r s s
+
− + −
− + −
+ − + −




+ − +
c
T t t T
T t t T
c T t T t
c T t t
s r
s r
r s
s
10
2 2
11
12
 
 
 
 −
−



 + −
T c t
r
 
13 .
Therefore, we have

 
− ≤ + + + + + + + +
( )
− −
t c c c c c c c c c
t t
3 4 5 7 8 9 10 12 13
2
Implies λ=t, since c3
+c4
+c5
+c7
+c8
+c9
+c10
+2c12
+c13
1.
Hence, λ is a unique common fixed point of
Tin X.
Example 3.3. Let T: [0,1]→[0,1] be a mapping
defined by Tx=x3
/6, for all x∈[0,1]with usual norm
‖x-y‖=|x-y|, for all x∈[0,1].
Proof. From theorem 3.1, we have
Tx Ty
x y
c
x
x
y
y
x y
c
y
y
y
x
− = − ≤
− + −






+ −
+
− + −




3 3
1
3 3
2
3 3
6 6
6
1
6
1
6
1
6 


+ −
+
1 x y
c
x
y
y
x
x y
c
x y
x y
x y
c
x y
3
3 3
4
3 3
5
6
1
6
1
1
6 6
1
1
− + −






+ −
+
− + −






+ −
+
− +
+ −






+ −
x
x
y
y
3
3
6
1
6

6. Raji and Adegboye: Common Fixed Point Theorems for Non-linear Contractive Mapping in Hilbert Space
AJMS/Apr-Jun-2022/Vol 6/Issue 2 6
+
− + −






+ −
+
− + −






+ −
+
c
x
x
x
y
y
y
c
x
y x y
x y
6
3 3
3
7
3 3 3
6
1
6
1
6
6
1
6 6
1
c
x
y
y
x
x y
c
x y
x y
x y
c
x y
3
3 3
4
3 3
5
6
1
6
1
1
6 6
1
1
− + −






+ −
+
− + −






+ −
+
− +
+ −






+ −
x
x
y
y
3
3
6
1
6
+
− + −






+ −
+
− + −






+ −
+
c
x
x
x
y
y
y
c
x
y x y
x y
6
3 3
3
7
3 3 3
6
1
6
1
6
6
1
6 6
1
c
x y x
y
x y
c
x
x
y
y
x y
x
x
x
y
y
y
8
3
9
3 3
3 3 3
1
6
1
6 6
1
6 6 6
− + −






+ −
+
− + − + −
+ − − − x
x y
−
+
− + −
− + −
+ − + −






+ −
c
x
y
y
x
x
y
y
x
c x
x
y
y
c x
y
10
3 2 3 2
3 3 11
3 3
12
3
6 6
6 6
6 6
6
+
+ −





 + −
y
x
c x y
3
13
6
Continuewiththeprocedureintheproofoftheorem
3.1, we get that 0 is the only fixed point of T.
CONCLUSION
In this paper, we proved the existence and
uniqueness of a common fixed point for a
continuous self mapping,T some positive integers
r,s of a pair of continuous self mapping ,
r s
T T
of Hilbert space. These results generalized and
extend the results of some literatures.
ACKNOWLEDGMENT
The authors would like to extend their sincere thanks
to all that have contributed in one way or the other
for the actualization of the review of this work.
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