Gen. Math. Notes, Vol. 25, No. 2, December 2014, pp. 95-109
ISSN 2219-7184; Copyright © ICSRS Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
Quadruple Fixed Point Theorem for Four
Mappings
Anushri A. Aserkar1,2 and Manjusha P. Gandhi2
1
Rajiv Gandhi College of Engineering and Research, Nagpur, India
E-mail: aserkar_aaa@rediffmail.com
2
Yeshwantrao Chavan College of Engineering, Nagpur, India
E-mail: manjusha_g2@rediffmail.com
(Received: 3-8-14 / Accepted: 12-9-14)
Abstract
In this paper we have proved a unique common quadruple fixed point theorem
for four mappings satisfying w-compatible in partially ordered metric space with
two altering distance functions. An example has been given to validate the result.
Keywords: Quadruple fixed point, Compatible mapping, Partially ordered set,
Complete metric space.
1
Introduction
Fixed point theory has fascinated hundreds of researchers since 1922 with the
celebrated Banach’s fixed point theorem. This theorem provides a technique for
solving a variety of problems in mathematical sciences and engineering. This
study is a very active field of research at present.
T.G. Bhashkar et al. [13] introduced the concept of a coupled fixed point and
proved theorems in partially ordered complete metric spaces.
96
Anushri A. Aserkar et al.
V. Lakshmikantham et al. [14] proved coupled coincidence and coupled common
fixed point theorems for nonlinear mappings in partially ordered complete metric
spaces. Later, many results on coupled fixed point have been obtained [2, 3, 4, 10,
11, 12].
V. Berinde et al. [15] introduced the concept of a tripled fixed point.
B. Samet et al. [1] introduced fixed point of order N ≥ 3 for the first time. Very
recently, E. Karapınar [5] used the notion of quadruple fixed point and obtained
some quadruple fixed point theorems in partially ordered metric spaces. Many
researchers [6-9] were motivated and proved theorems on quadruple fixed points
with monotone property whereas in the present paper a unique common quadruple
fixed point theorem for four mappings without using the monotone property and
satisfying w-compatible condition in pairs has been proved.
2
Preliminaries
2.1
Quadruple Fixed Point
Let F: X×X×X×X →X. An element (x, y, z, w) is called a quadruple fixed point of
F if F(x, y, z, w) = x, F(y, z, w, x) = y, F(z, w, x, y) = z, F(w, x, y, z) = w.
2.2
Quadruple Coincidence Point
Let F: X×X×X×X →X and g: X →X. An element (x, y, z, w) is called a quadruple
coincidence point of F and g if F(x, y, z, w) = gx, F(y, z, w, x) = gy, F (z, w, x, y) =
gz, F (w, x, y, z) = gw.
2.3
Quadruple Common Fixed Point
Let F: X×X×X×X →X and g: X →X. An element (x, y, z, w) is called a quadruple
common fixed point of F and g if F(x, y, z, w) = gx = x, F(y, z, w, x) = gy = y, F(z,
w, x, y) = gz = z, F(w, x, y, z) = gw = w.
2.4
W-Compatible Mapping
F: X×X×X×X →X and g: X →X are called w-compatible if F(gx, gy, gz, gw) =
g(F(x, y, z, w)) whenever F(x, y, z, w) = gx, F(y, z, w, x) = gy, F(z, w, x, y) = gz,
F(w, x, y, z) = gw.
2.5
Alternating Distance Function
Let Φ denote all the functions ξ ∈ Φ such that ξ :[0, ∞) → [0, ∞) which satisfy
(i) ξ (t) = 0 if and only if t = 0,
Quadruple Fixed Point Theorem for Four...
97
(ii) ξ (t) is continuous and non-decreasing.
3
Main Theorem
X0
→≥
XL
:d
fn
,
ga
d
n
a
X
→
4
X
:
G
,
F
Theorem 3.1: Let (X, ≤ ) be a partially ordered set and suppose there is a metric d
on X such that (X, d) is a complete metric space .Suppose
are such that F, G are
continuous. ξ , φ ∈Φ
such that
(i) ξ (d(F(x, y, z, w),G(p,q,r,s)
(
)
− φ ( max {d(fx,gp),d(fy,gq),d(fz,gr),d(fw,gs))})
≤ ξ max {d(fx,gp),d(fy,gq),d(fz,gr),d(fw,gs))}
+ L min ( d(fx,G(p,q,r,s)),d(gp, F(x, y,z, w),d(gp,G(p,q, r,s)) ) ...........(i)
for all x, y, z, w, p,q, r,s ∈ X.
(ii) F(X4 ) ⊂ g(X),G(X4 ) ⊂ f(X),
(iii) the pairs (F, f) and (G, g) are W - compatible .
Then F, G, f and g have a unique common coupled fixed point in X4 and also they
have a unique common fixed point in X.
Proof: Let x 0 , y 0 , z 0 , w 0 ∈ X .
∵F(X4 ) ⊂ g(X),G(X4 ) ⊂ f(X).
∴we may find x1, y1,z1,w1 so that F( x0 ,y0 ,z0,w0 ) = g(x1),F( y0 ,z0,w0 ,x0 ) = g(y1),
F( z0 ,w0 ,x0 ,y0 ) = g(z1),F( w0,x0 ,y0,z0 ) = g(w1).
Similarly wemayfind x2 , y2 ,z2 ,w2 so that G ( x1,y1,z1,w1 ) = f(x2 ),G ( y1,z1,w1,x1) = f(y2 ),
G ( z1,w1,x1,y1 ) = f(z2 ),G ( w1,x1,y1,z1 ) = f(w2 ).
Continuing in the same way we may form sequences
{x n } ,{y n } ,{z n },{w n } ,{a n },{b n } ,{c n } ,{d n } in X such that
(
(
)
)
(
(
a 2n = g(x2n+1) = F x 2n , y 2n ,z 2n ,w 2n ,b 2n = g(y2n+1) = F y 2n ,z 2n ,w 2n ,x 2n
c 2n = g(z2n+1) = F z 2n ,w 2n ,x 2n , y 2n ,d 2n = g(w 2n+1) = F w 2n ,x 2n , y 2n ,z 2n
)
)
98
Anushri A. Aserkar et al.
and
(
)
(
)
c 2n+1 = f(z2n+2 ) = G ( z 2n+1 , w 2n+1 , x 2n+1 , y 2n+1 ) ,
d 2n+1 = f(w 2n+2 ) = G ( w 2n+1 , x 2n+1 , y 2n+1 ,z 2n+1 )
a 2n+1 = f(x 2n+2 ) = G x 2n+1 , y 2n+1 ,z 2n+1 , w 2n+1 ,
b 2n+1 = f(y2n+2 ) = G y 2n+1 , z 2n+1 , w 2n+1 , x 2n+1
Putting x = x2n, y = y2n. z = z2n, w = w2n, p = x2n+1, q = y2n+1, r = z2n+1, s = w2n+1 in
(i) we get
ξ(d(a 2n ,a 2n+1)
(
)
- φ ( max {d(a 2n-1,a 2n ),d(b 2n-1,b 2n ),d(c2n-1,c2n ),d(d 2n-1,d 2n )})
≤ ξ max {d(a 2n-1,a 2n ),d(b 2n-1,b 2n ),d(c2n-1,c2n ),d(d 2n-1,d 2n )}
+ Lmin ( d(a 2n-1,a 2n+1),d(a 2n ,a 2n ),d(a 2n ,a 2n+1) )
∴ ξ(d(a 2n ,a 2n+1)
(
≤ ξ max {d(a 2n-1,a 2n ),d(b 2n-1,b 2n ),d(c2n-1,c2n ),d(d 2n-1,d 2n )}
(
)
)
- φ max {d(a 2n-1,a 2n ),d(b 2n-1,b 2n ),d(c2n-1,c2n ),d(d 2n-1,d 2n )} ....(ii)
Similarly if we consider,
ξ(d(F(y,z, w, x),G(q, r,s,p)
(
≤ ξ max {d(fy,gq),d(fz,gr),d(fw,gs),d(fx,gp))}
(
- φ max {d(fy,gq),d(fz,gr),d(fw,gs),d(fx,gp)}
)
)
+ Lmin ( d(fy,G(q, r,s,p)),d(gq, F(y, z, w, x)),d(gq,G(q,r,s, p)) )
We may prove that
ξ(d(b2n ,b2n+1)
(
≤ ξ max {d(b2n-1,b2n ),d(c2n-1,c2n ),d(d2n-1,d2n ),d(a 2n-1,a 2n )}
(
)
)
-φ max {d(b2n-1,b2n ),d(c2n-1,c2n ),d(d2n-1,d2n ),d(a 2n-1,a 2n )} ..........(iii)
Similarly we may prove that
ξ(d(c2n ,c2n+1)
(
≤ ξ max {d(c2n-1,c2n ),d(d 2n-1,d 2n ),d(a 2n-1,a 2n ),d(b2n-1,b 2n )}
(
)
)
- φ max {d(c2n-1,c2n ),d(d 2n-1,d 2n ),d(a 2n-1,a 2n ),d(b2n-1,b 2n )} ........(iv)
&
Quadruple Fixed Point Theorem for Four...
99
ξ(d(d2n ,d2n+1)
(
≤ ξ max {d(d 2n-1,d 2n ),d(a 2n-1,a 2n ),d(b2n-1,b2n ),d(c2n-1,c2n )}
(
)
)
-φ max{d(d 2n-1,d2n ),d(a 2n-1,a 2n ),d(b2n-1,b2n ),d(c2n-1,c2n )} ........(v)
Combining (ii), (iii), (iv), (v) we get
ξ(max {d(a 2n ,a 2n+1),d(b 2n ,b2n+1),d(c2n ,c2n+1),d(d 2n ,d 2n+1)}
(
)
-φ ( max {d(a 2n-1,a 2n ),d(b2n-1,b2n ),d(c2n-1,c2n ),d(d 2n-1,d 2n )}) ...(vi)
≤ ξ max {d(a 2n-1,a 2n ),d(b2n-1,b2n ),d(c2n-1,c2n ),d(d 2n-1,d 2n )}
∴ ξ(max {d(a 2n ,a 2n+1),d(b 2n ,b 2n+1),d(c2n ,c2n+1),d(d 2n ,d 2n+1)}
(
≤ ξ max {d(a 2n-1,a 2n ),d(b 2n-1,b 2n ),d(c2n-1,c2n ),d(d 2n-1,d 2n )}
)
∵ ξ(t) is an a non - decreasing sequence
∴ max {d(a 2n ,a 2n+1),d(b 2n ,b 2n+1),d(c 2n ,c 2n+1),d(d 2n ,d 2n+1)}
≤ max {d(a 2n-1,a 2n ),d(b2n-1,b2n ),d(c2n-1,c2n ),d(d 2n-1,d 2n )}
∴
{d(a 2n ,a 2n+1),d(b2n ,b2n+1),d(c2n ,c2n+1),d(d2n ,d 2n+1)}is a sequence of
non -increasing positive real numbers.So,it must converge to a positive real number
say δ > 0.
∴ lim {d(a 2n ,a 2n+1),d(b 2n ,b 2n+1),d(c2n ,c2n+1),d(d 2n ,d 2n+1)} = δ
n →∞
∴Taking lim on (vi) we get
n →∞
lim ξ(max {d(a 2n ,a 2n+1),d(b2n ,b2n+1),d(c2n ,c2n+1),d(d2n ,d2n+1)}
n →∞
≤ lim ξ max {d(a 2n-1,a 2n ),d(b2n-1,b2n ),d(c2n-1,c2n ),d(d 2n-1,d 2n )}
n →∞
- lim φ max {d(a 2n-1,a 2n ),d(b2n-1,b2n ),d(c2n-1,c2n ),d(d2n-1,d2n )}
n →∞
∴ξ(δ) ≤ ξ(δ)-φ (δ) ⇒ φ (δ) = 0
⇒δ =0
∴ lim {d(a 2n ,a 2n+1),d(b 2n ,b 2n+1),d(c2n ,c2n+1),d(d 2n ,d 2n+1)} = 0
n →∞
∴ Generalising we get,
lim {d(a n ,a n+1),d(b n ,b n+1),d(cn ,cn+1),d(d n ,d n+1)} = 0...................(vii)
n →∞
We will show that {a n } , {b n } , {c n } , {d n } are Cauchy sequences. Assume on the
contrary, that {a n } , {b n } , {c n } and {d n } are not Cauchy sequences,
(
(
consequently, lim d(a ,a ) ≠ 0 , lim d(b ,b ) ≠ 0 , lim d(c ,c ) ≠ 0
n →∞ n m
n →∞ n m
n →∞ n m
)
)
100
and lim d(d ,d ) ≠ 0 .
n m
n →∞
Anushri A. Aserkar et al.
{ k } and
Let there exists ∈> 0 for which we can find subsequence of integers m
{n k } such that n k > m k > k .
{d(a 2m k ,a 2n k ),d(b2m k ,b2n k ),d(c2m k ,c2n k ),d(d2m k ,d2n k )} ≥∈ and
{d(a 2m k ,a 2n k -1),d(b2m k ,b2n k -1),d(c2m k ,c2n k -1),d(d2m k ,d2n k -1)} <∈
By the triangle inequality, we have
∈≤ lim d(a 2m ,a 2n ) = lim d(a 2m ,a 2n
) + lim d(a
,a
)
k
k
k −1 k →∞ 2n k −1 2n k
k
k →∞
k →∞
∈≤ lim d(a 2m ,a 2n ) < lim d(a 2m ,a 2n
) ≤∈
−
1
k
k
k
k
k →∞
k →∞
∴ lim d(a 2m ,a 2n ) = lim d(a 2m ,a 2n
) =∈
−
1
k
k
k
k
k →∞
k →∞
Similarly we may that
lim d(b 2m ,b 2n ) = lim d(b 2m , b 2n −1 ) =∈
k
k
k
k
k →∞
k →∞
lim d(c 2m ,c 2n ) = lim d(c 2m ,c 2n −1 ) =∈
k
k
k
k
k →∞
k →∞
lim d(d 2m ,d 2n ) = lim d(d 2m ,d 2n −1 ) =∈
k
k
k
k
k →∞
k →∞
d(a 2m ,a 2n +1 ) ≤ d(a 2m ,a 2n ) + d(a 2m ,a 2n +1 )
k
k
k
k
k
k
d(a
,a
)
≤
d(a
,a
)
+
lim
lim
lim d(a 2m ,a 2n +1 )
2mk 2nk +1
2m k 2nk
k
k
k →∞
k →∞
k →∞
∴ lim d(a 2m ,a 2n +1 ) ≤∈
k
k
k →∞
d(a 2m -1 ,a 2n ) ≤ d(a 2m
,a
) + d(a 2m ,a 2n )
k -1 2mk
k
k
k
k
d(a
,a
)
≤
d(a
,a
)
+
lim
lim
lim d(a 2m ,a 2n )
2mk -1 2nk
2m k -1 2mk
k
k
k →∞
k →∞
k →∞
∴ lim d(a 2m -1 ,a 2n ) ≤∈
k
k
k →∞
d(a 2m -1 ,a 2n +1 ) ≤ d(a 2m
,a
) + d(a 2m ,a 2n ) + d(a 2n ,a 2n +1 )
k -1 2mk
k
k
k
k
k
k
lim d(a 2m -1 ,a 2n +1 )
k
k
k →∞
≤ lim d(a 2m
,a
) + lim d(a
,a
) + lim d(a
,a
)
k -1 2mk k →∞ 2mk 2nk k →∞ 2nk 2nk +1
k →∞
∴ lim d(a 2m -1 ,a 2n +1 ) ≤∈
k
k
k →∞
Quadruple Fixed Point Theorem for Four...
101
Similarly we may prove that
lim d(b 2m , b 2n +1 ) ≤∈, lim d(b 2m -1 ,b 2n ) ≤∈, lim d(b 2m -1 ,b 2n +1 ) ≤∈
k
k
k
k
k
k
k →∞
k →∞
k →∞
lim d(c 2m ,c 2n +1 ) ≤∈, lim d(c 2m -1 ,c 2n ) ≤∈, lim d(c 2m -1 ,c 2n +1 ) ≤∈
k
k
k
k
k
k
k →∞
k →∞
k →∞
lim d(d 2m ,d 2n +1 ) ≤∈, lim d(d 2m -1 ,d 2n ) ≤∈, lim d(d 2m -1 ,d 2n +1 ) ≤∈
k
k
k
k
k
k
k →∞
k →∞
k →∞
Putting x = x 2m , y = y2m , z = z2m , w = w 2m , p = x 2n +1,q = y2n +1,
k
k
k
k
k
k
r = z2n +1,s = w 2n +1 in (i), we get
k
k
ξ (d(F(x 2m , y2m ,z2m , w 2m ),G(x 2n +1, y2n +1,z2n +1, w 2n +1)
k
k
k
k
k
k
k
k

d(fx 2m ,gx 2n +1),d(fy 2m ,gy 2n +1),d(fz 2m ,gz 2n +1),  



k
k
k
k
k
k
≤ ξ  max 


d(fw 2m k ,gw 2n k +1))
 



d(fx 2m ,gx 2n +1),d(fy 2m ,gy 2n +1),d(fz 2m ,gz 2n +1),  



k
k
k
k
k
k
− φ  max 

d(fw 2m ,gw 2n +1))


 
k
k




 d(fx 2m ,G(x 2n +1, y2n +1,z 2n +1, w 2n +1)), 
k
k
k
k
k


+ Lmin  d(gx 2n +1, F(x 2m , y2m ,z2m , w 2m ),

k
k
k
k
k


 d(gx

,G(x
,
y
,
z
,
w
))

2n k +1
2n k +1 2n k +1 2n k +1 2n k +1 


d(a 2m -1,a 2n ),d(b2m -1,b2n ),d(c2m -1,c2n ), 


k
k
k
k
k
k 
ξ(d(a2m ,a 2n +1) ≤ ξ  max 

k
k
d(d2m -1,d2n )


 
k
k



{
}

-φ  max d(a 2m -1,a 2n ),d(b2m -1,b2n ),d(c2m -1,c2n ),d(d2m -1,d2n ) 
k
k
k
k
k
k
k
k 

+Lmin  d(a2m -1,a2n +1),d(a2n ,a2m ),d(a2n ,a2n +1) 
k
k
k
k
k
k 

Taking lim , weget
n→∞
ξ(∈) ≤ ξ(∈)-φ (∈)
∴φ(∈) = 0
∴∈= 0,which is not possible
{an }is a cauchy sequence.
Similarly we may prove that {bn },{cn },{dn } arecauchysequences.
As (X, d) is a complete metric space. So,
a 2n+1 = f(x 2n+2 ),b 2n+1 = f(y2n+2 ), c 2n+1 = f(z2n+2 ),d 2n+1 = f(w 2n+2 )
converge to some α,β, γ,θ in X.
102
Anushri A. Aserkar et al.
Hence there exists x, y,z, w in X,such that α = fx , β = fy , γ = fz , θ = fw.
{
}{
}{ }{
}
Also the subsequences a 2n , b 2n , c 2n , d 2n converge to α.β, γ,θ
ξ(d(F(x, y, z, w),G(x 2n+1, y2n+1,z2n+1, w 2n+1)
(
≤ ξ max {d(fx,gx 2n+1),d(fy,gy 2n+1),d(fz,gz 2n+1),d(fw,gw 2n+1))}
(
- φ max {d(fx,gx 2n+1),d(fy,gy 2n+1),d(fz,gz 2n+1),d(fw,gw 2n+1))}
)
)
 d(fx,G(x 2n+1, y2n+1, z 2n+1, w 2n+1)),d(gx 2n+1,F(x, y, z, w)), 

 d(gx

2n+1,G(x 2n+1, y2n+1,z 2n+1, w 2n+1))


+ Lmin 
ξ(d(F(x, y, z, w),a 2n+1)
(
≤ ξ max {d(α,a 2n ),d(β,b 2n ),d(γ,c2n ),d(θ,d 2n )}
(
- φ max {d(α,a 2n ),d(β,b 2n ),d(γ,c2n ),d(θ,d 2n )}
)
)
+ Lmin ( d(α,a 2n+1),d(a 2n ,F(x, y,z, w)),d(a 2n ,a 2n+1) )
Taking lim to both sides, we get
n→∞
∴ ξ(d(F(x, y,z, w), α) ≤ 0
∴ F(x, y,z, w) = α
∴ F(x, y,z, w) = fx = α
∵ (F,f) are w -compatible.
∴ fα = f(F(x, y, z, w)) = F(fx,fy,fz,fw) = F(α, β, γ, θ)
Similarly we may prove that
fβ = F(β, γ, θ, α)
fγ = F(γ, θ, α, β)
fθ = F(θ, α, β, γ)
Putting x = α, y = β,z = γ, w = θ we get
ξ(d(F(α,β, γ,θ),G(x 2n+1, y2n+1,z2n+1, w 2n+1)
(
≤ ξ max {d(fα,gx 2n+1),d(fβ,gy 2n+1),d(fγ,gz 2n+1),d(fθ,gw 2n+1))}
(
- φ max {d(fα,gx 2n+1),d(fβ,gy 2n+1),d(fγ,gz 2n+1),d(fθ,gw 2n+1))}
)
 d(fα,G(x 2n+1, y2n+1,z 2n+1, w 2n+1)),d(gx 2n+1,F(α,β, γ,θ)), 

 d(gx

2n+1,G(x 2n+1, y2n+1, z2n+1, w 2n+1))


+ Lmin 
)
Quadruple Fixed Point Theorem for Four...
103
(
)
- φ ( max {d(fα,a 2n ),d(fβ,b 2n ),d(fγ,c2n ),d(fθ,d 2n )})
∴ ξ(d(fα ,a 2n+1) ≤ ξ max {d(fα,a 2n ),d(fβ,b 2n ),d(fγ,c2n ),d(fθ,d 2n )}
(
+ Lmin ( d(fα,a 2n+1),d(a 2n , F(α,β, γ,θ)),d(a 2n ,a 2n+1) )
)
ξ(d(fα, α)) ≤ ξ max {d(fα,α),d(fβ,β),d(fγ, γ),d(fθ,θ)} -
φ ( max {d(fα,α),d(fβ,β),d(fγ, γ),d(fθ,θ)})
(
)
(
)
ξ(d(fβ, β)) ≤ ξ max {d(fβ,β),d(fγ, γ),d(fθ,θ),d(fα,α)} -
φ ( max {d(fβ,β),d(fγ, γ),d(fθ,θ),d(fα,α)})
ξ(d(fθ , θ ) ) ≤ ξ max {d(fθ,θ),d(fα,α),d(fβ,β),d(fγ,γ)} -
φ ( max {d(fθ,θ),d(fα,α),d(fβ,β),d(fγ,γ)})
(
)
φ ( max {d(fγ,γ),d(fθ,θ),d(fα,α),d(fβ,β)})
ξ(d(fγ , γ ) ) ≤ ξ max {d(fγ,γ),d(fθ,θ),d(fα,α),d(fβ,β)} ∴ ξ(max {d(fα,α),d(fβ,β),d(fγ,γ),d(fθ,θ)}) ≤
(
)
- φ ( max {d(fα,α),d(fβ,β),d(fγ,γ),d(fθ,θ)})
∴φ ( max {d(fα ,α ),d(fβ , β ),d(fγ ,γ ),d(fθ ,θ )}) = 0
ξ max {d(fα,α),d(fβ,β),d(fγ,γ),d(fθ,θ)}
max {d(fα,α),d(fβ,β),d(fγ,γ),d(fθ,θ)} = 0
⇒ d(fα,α) = 0 ,d(fβ,β) = 0 ,d(fγ, γ) = 0 & d(fθ,θ) = 0
∴ fα = α, fβ = β ,fγ = γ ,fθ = θ
∴ F(α,β, γ,θ) = fα = α
F(β, γ,θ,α,) = fβ = β
F(γ,θ,α,β) = fγ = γ F(θ,α,β, γ) = fθ = θ
∵ F(X4 ) ⊂ g(X), there exists υ, ν,σ, τ such that g(υ) = F(α,β, γ,θ) = fα = α
g(ν) = F(β, γ,θ,α,) = fβ = β g(σ) = F(γ,θ,α,β) = fγ = γ g(τ ) = F(θ,α,β, γ) = fθ = θ
∴ξ (d ( g(υ),G(υ, ν,σ, τ)) = ξ (d(F(α,β, γ,θ),G(υ, ν,σ, τ))
(
)
− φ ( max {d(fα ,gυ),d(fβ ,gν),d(fγ ,gσ),d(fθ ,gτ))})
≤ ξ max {d(fα ,gυ),d(fβ ,gν),d(fγ ,gσ),d(fθ ,gτ))}
+ Lmin ( d(fα ,G(υ, ν,σ, τ)),d(gυ,F(α,β, γ,θ),d(gυ,G(υ, ν,σ, τ)) )
∴ξ (d ( g(υ),G(υ, ν,σ, τ)) ≤ 0 ⇒ d ( g(υ),G(υ, ν,σ, τ) = 0
∴g(υ) = G(υ, ν,σ, τ)
Similarly we may prove that
g(ν) = G(ν,σ, τ, υ), g(σ) = G(σ, τ, υ, ν), g(τ) = G(τ, υ, ν,σ)
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Anushri A. Aserkar et al.
∵ (G,g) are W compatible
∴ gα = gg(υ) = gG(υ, ν,σ, τ) = G(g(υ),g(ν),g(σ), g(τ)) = G(α,β, γ,θ)
Similarly we may prove that
gγ = G(γ,θ,α,β),
gθ = G(θ,α,β, γ)
gβ = G(β, γ,θ,α),
Putting x = x2n, y = y2n. z = z2n, w = w2n, p = α , q = β , r = γ , s = θ in (i) we get
ξ(d(F(x 2n , y2n ,z2n , w 2n ),G(α,β, γ,θ)
(
)
− φ ( max {d(fx 2n ,gα),d(fy 2n ,gβ),d(fz 2n ,gγ),d(fw 2n ,gθ))})
≤ ξ max {d(fx 2n ,gα),d(fy 2n ,gβ),d(fz 2n ,gγ),d(fw 2n ,gθ))}
+ Lmin ( d(fx 2n ,G(α,β, γ,θ)),d(gα,F(x 2n , y2n , z2n , w 2n ),d(gα,G(α,β, γ,θ)) )
Taking lim , we get
n →∞
ξ(d(α,gα)) ≤ ξ max {d(α,gα),d(β,gβ),d(γ,gγ),d(θ,gθ))}
(
)
- φ ( max {d(α,gα),d(β,gβ),d(γ,gγ),d(θ,gθ))})
+ Lmin ( d(α,gα),d(gα, α),d(gα,gα) )
(
∴ ξ(d(α,gα)) ≤ ξ max {d(α,gα),d(β,gβ),d(γ,gγ),d(θ,gθ))}
(
- φ max {d(α,gα),d(β,gβ),d(γ,gγ),d(θ,gθ))}
)
Similarly we may prove that
(
)
- φ ( max {d(β,gβ),d(γ,gγ),d(θ,gθ),d(α,gα)})
ξ(d(γ,gγ)) ≤ ξ ( max {d(γ,gγ),d(θ,gθ),d(α,gα),d(β,gβ)})
- φ ( max {d(γ,gγ),d(θ,gθ),d(α,gα),d(β,gβ)})
ξ(d(θ, gθ)) ≤ ξ ( max {d(θ,gθ),d(α,gα),d(β,gβ),d(γ,gγ)})
- φ ( max {d(θ,gθ),d(α,gα),d(β,gβ),d(γ,gγ)})
∴ ξ ( max {d(α,gα),d(β,gβ),d(γ,gγ),d(θ,gθ))})
≤ ξ ( max {d(α,gα),d(β,gβ),d(γ,gγ),d(θ,gθ))})
- φ ( max {d(α,gα),d(β,gβ),d(γ,gγ),d(θ,gθ))})
∴φ ( max {d(α,gα),d(β,gβ),d(γ,gγ),d(θ,gθ))}) = 0
ξ(d(β,gβ)) ≤ ξ max {d(β,gβ),d(γ,gγ),d(θ,gθ),d(α,gα))}
⇒ max {d(α,gα),d(β,gβ),d(γ,gγ),d(θ,gθ))} = 0
)
Quadruple Fixed Point Theorem for Four...
∴ gα = α, gβ = β,
gα = G(α,β, γ,θ) = α
gγ = γ,
105
gθ = θ
Similarly we may prove that
gβ = G(β, γ,θ,α) = β,
gγ = G(γ,θ,α,β) = γ,
gθ = G(θ,α,β, γ) = θ
∴ F(α,β, γ,θ) = fα = gα = G(α,β,γ,θ) = α
F(β, γ,θ,α,) = fβ = gβ = G(β, γ,θ,α) = β
F(γ,θ,α,β) = fγ = gγ = G(γ,θ,α,β) = γ,
F(θ,α,β, γ) = fθ = gθ = G(θ,α,β, γ) = θ
Thus (α,β, γ,θ) is a quadruple fixed point of F,G,f,g
Uniqueness: Let if possible there are two fixed points say
(α,β, γ,θ) and (α*,β*, γ*,θ*) of F,G,f,g
∴ F(α,β, γ,θ) = fα = gα = G(α,β, γ,θ) = α
F(β, γ,θ,α,) = fβ = gβ = G(β, γ,θ,α) = β
F(γ,θ,α,β) = fγ = gγ = G(γ,θ,α,β) = γ,
F(θ,α,β, γ) = fθ = gθ = G(θ,α,β, γ) = θ
and
F(α*,β*, γ*,θ*) = fα* = gα* = G(α*,β*, γ*,θ*) = α*
F(β*, γ*,θ*,α*) = fβ* = gβ* = G(β*, γ*,θ*,α*) = β*
F(γ*,θ*,α*,β*) = fγ* = gγ* = G(γ*,θ*,α*,β*) = γ*,
F(θ*,α*,β*, γ*) = fθ* = gθ* = G(θ*,α*,β*, γ*) = θ*
Putting x = α, y = β,z = γ, w = θ and p = α*,q = β*,r = γ*,s = θ*
ξ(d(F(α,β, γ,θ),G(α*,β*, γ*,θ*)
(
≤ ξ max {d(fα,gα*),d(fβ,gβ*),d(fγ,gγ*),d(fθ,gθ*)}
(
- φ max {d(fα,gα*),d(fβ,gβ*),d(fγ,gγ*),d(fθ,gθ*)}
)
)
+ Lmin ( d(fα,G(α*,β*, γ*,θ*)),d(gα*, F(α, β, γ,θ)),d(gα*,G(α*,β*, γ*,θ*)) )
(
)
- φ ( max {d(α,α*),d(β,β*),d(γ,γ*),d(θ,θ*)})
ξ(d(α,α*)) ≤ ξ max {d(α,α*),d(β,β*),d(γ,γ*),d(θ,θ*)}
+ Lmin ( d(α,α*),d(α*,α),d(α*, α*) )
(
)
- φ ( max {d(α,α*),d(β,β*),d(γ,γ*),d(θ,θ*)})
ξ(d(α,α*)) ≤ ξ max {d(α,α*),d(β,β*),d(γ,γ*),d(θ,θ*)}
Similarly we may prove that
(
)
- φ ( max {d(β,β*),d(γ,γ*),d(θ,θ*),d(α,α*)})
ξ(d(β,β*)) ≤ ξ max {d(β,β*),d(γ,γ*),d(θ,θ*),d(α,α*)}
106
Anushri A. Aserkar et al.
(
)
- φ ( max {d(γ,γ*),d(θ,θ*),d(α,α*),d(β,β*)})
ξ(d(θ,θ*)) ≤ ξ ( max {d(θ,θ*),d(α,α*),d(β,β*),d(γ,γ*)})
- φ ( max {d(θ,θ*),d(α,α*),d(β,β*),d(γ,γ*)})
∴ξ ( max {d(α,α*),d(β,β*),d(γ,γ*),d(θ,θ*)})
≤ ξ ( max {d(α,α*),d(β,β*),d(γ,γ*),d(θ,θ*)})
- φ ( max {d(α,α*),d(β,β*),d(γ,γ*),d(θ,θ*)})
∴φ ( max {d(α,α*),d(β,β*),d(γ,γ*),d(θ,θ*)}) = 0
ξ(d(γ, γ*)) ≤ ξ max {d(γ,γ*),d(θ,θ*),d(α,α*),d(β,β*)}
∴ max {d(α,α*),d(β,β*),d(γ,γ*),d(θ,θ*)} = 0
⇒ d(α,α*) = d(β,β*) = d(γ, γ*) = d(θ,θ*) = 0
∴ α = α*, β = β*, γ = γ*, θ = θ*
Thus (α,β, γ,θ) is a unique common quadruple fixed point of F,G,f,g .
Corollary 3.2: Let (X, ≤ ) be a partially ordered set and suppose there is a metric
d on X such that (X,d) is a complete metric space .Suppose
F,G : X4 → X and g,f : X → X are such that F,G are continuous. k ∈[0,1)
(i) d(F(x, y,z, w),G(p,q,r,s)
(
≤ k max {d(fx,gp),d(fy,gq),d(fz,gr),d(fw,gs))}
for all x, y, z, w, p,q, r,s ∈ X.
(ii) F(X4 ) ⊂ g(X),G(X4 ) ⊂ f(X),
)
(iii) the pairs (F, f) and (G, g) are W - compatible .
Then F, G, f and g have a unique common coupled fixed point in X4 and also they
have a unique common fixed point in X.
Proof: Substituting ξ(t) = t and φ (t) = (1- k) t and L=0, in the main theorem we get
the proof.
Corollary 3.3: Let (X, ≤ ) be a partially ordered set and suppose there is a metric
d on X such that (X, d) is a complete metric space .Suppose
F, G : X4 → X and g, f : X → X are such that F, G are continuous.
k
( d(fx,gp)+d(fy,gq)+d(fz,gr)+d(fw,gs) )
4
for all x, y,z, w,p,q,r,s ∈ X.
(i) d(F(x, y,z, w),G(p,q,r,s)) ≤
Quadruple Fixed Point Theorem for Four...
107
(ii) F(X4 ) ⊂ g(X),G(X4 ) ⊂ f(X),
(iii) the pairs (F, f) and (G, g) are W - compatible .
Then F, G, f and g have a unique common coupled fixed point in X4 and also they
have a unique common fixed point in X.
Proof: We know that
1
{d(fx,gp)+d(fy,gq)+d(fz,gr)+d(fw,gs)} ≤ max {d(fx,gp),d(fy,gq),d(fz,gr),d(fw,gs)}
4
So we get the proof from Corollary 3.1.
Corollary 3.4: Let (X, ≤ ) be a partially ordered set and suppose there is a metric
d on X such that (X, d) is a complete metric space .Suppose
F, G : X4 → X and g, f : X → X are such that F, G are continuous. φ ∈Φ
(i) d(F(x, y,z, w),G(p,q, r,s)
≤ max {d(fx,gp),d(fy,gq),d(fz,gr),d(fw,gs))} −
φ ( max {d(fx,gp),d(fy,gq),d(fz,gr),d(fw,gs))})
for all x, y, z, w, p,q, r,s ∈ X.
(ii) F(X4 ) ⊂ g(X),G(X4 ) ⊂ f(X),
(iii) the pairs (F, f) and (G, g) are W - compatible .
Then F, G, f and g have a unique common coupled fixed point in X4 and also they
have a unique common fixed point in X.
Proof: Substituting ξ(t) = t and L=0, in the main theorem we get the proof.
Corollary 3.5: Let (X, ≤ ) be a partially ordered set and suppose there is a metric
d on X such that (X,d) is a complete metric space .Suppose
F : X 4 → X and f : X → X are such that F,G are continuous. k is a constant such
that k ∈ [0,1)
(i) d(F(x, y,z, w),F(p,q, r,s) ≤ k max {d(fx,fp),d(fy,fq),d(fz,fr),d(fw,fs))}
for all x, y, z, w, p,q, r,s ∈ X.
(ii) F(X4 ) ⊂ f(X),
(iii) the pairs (F, f) are W -compatible .
Then F, f has a unique common coupled fixed point in X4 and also they have a
unique common fixed point in X.
108
Anushri A. Aserkar et al.
Proof: Substituting F = G and f = g in Corollary 3.1, we get the proof.
Example 3.6: Let X= [0, 1] and d(x, y) = x − y , (X,d) is a complete metric space
F(x, y,z, w) = max {x,y,z,w}
1
ξ(t) = t
5
fx = 4x
1
G(x, y, z, w) = max {x,y,z,w}
2
1
6
gx = 3x
φ (t) = t
Thus F, G : X4 → X and g, f : X → X and F, G are
continuous. ξ , φ ∈Φ and L ≥ 0 such that
F(X4 ) ⊂ g(X),G(X4 ) ⊂ f(X),
and the pairs (F, f) and (G, g) are W - compatible .
Thus all the conditions of the theorem are satisfied .The unique fixed point is (0, 0,
0, 0).
Conclusion
A unique common quadruple fixed point theorem by introducing a new
contractive condition for two altering distance functions and four maps satisfying
w-compatible condition in pairs has been proved without using monotone
property.
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