International Journal of Analysis and Applications
ISSN 2291-8639
Volume 4, Number 2 (2014), 87-99
http://www.etamaths.com

QUADRUPLE FIXED POINT OF MULTIVALUED NONLINEAR

CONTRACTION MAPPINGS

ANIMESH GUPTA1∗, R.N. YADAVA2, S.S. RAJPUT3

Abstract. The notion of Quadruple fixed point is introduced by Karapinar

E. [6]. Samet and Vetro [12] established some coupled fixed point theorems for

multivalued non linear contraction mapping in partially ordered metric spaces.
In this paper, we obtain existence of quadrupled fixed point of multivalued non

linear contraction mappings in framework work of partially ordered metric

spaces. Also, we give an example.

1. Introduction and Preliminary

Let (X,d) be a metric space. We denote by CB(X) the collection of non- empty
closed bounded subsets of X. For A,B ∈ CB(X) and x ∈ X, suppose that

D(x,A) = inf
a∈A

d(x,a)

H(A,B) = max{sup
a∈A

D(a,B), sup
b∈B

D(b,A)}.

Such mapping H is called a Housdorff metric on CB(X) induced by d.

Definition 1. An element x ∈ X is said to be a fixed point of a multivalued mapping
T : X → CB(X) iff x ∈ Tx.

In 1969, Nadlar [8] extended the famous Banach contraction principle from sin-
gle valued mapping to multivalued mapping and proved the following fixed point
theorem for the multivalued contraction which state as follows,

Theorem 2. Let (X,d) be a complete metric space and let T be a mapping from
X into CB(X). Assume that there exists c ∈ [0, 1) such that H(Tx,Ty) ≤ cd(x,y)
for all x,y ∈ X. Then T has a fixed point.

The existence of fixed points for various multi valued contraction mappings has
been studied by many authors under different conditions. In 1989, Mizoguchi and
Takahashi [7] proved the following interesting fixed point theorem for a weak con-
traction.

Theorem 3. Let (X,d) be a complete metric space and let T be a mapping for-
m X into CB(X). Assume that there exists c ∈ [0, 1) such that H(Tx,Ty) ≤
α(d(x,y))d(x,y) for all x,y ∈ X, where α is a function from [0,∞) into [0, 1),

2010 Mathematics Subject Classification. 47H10, 54H25, 46J10, 46J15.
Key words and phrases. Quadrupled Coincidence Point, Quadrupled Fixed Point, Mixed

Monotone, Mixed g- monotone , Nonlinear contraction.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

87



88 GUPTA,YADAVA AND RAJPUT

satisfying the condition lim sups→t+ α(s) < 1 for all t ∈ [0,∞). Then T has a fixed
point.

Several authors studies the problem of existence of fixed point of multivalued
mappings satisfying different contractive conditions (see e.g., [1, 2, 3, 4, 5, 7, 10,
11]). The theory of multivalued mapping has application in control theory, convex
optimization, differential equations, and economics.

Existence of fixed points in ordered metric spaces has been initiated in 2004 by
Ran and Reurings [10] further studied by Nieto and Rodriguez - Lopez [9]. Samet
and Vetro [12] introduced the notion of fixed point of N order in case of single-valued
mappings. In particular for N = 4 (Quadruple case) i.e., Let (X,�) be partially
ordered set and (X,d) be a complete metric space. We consider the following partial
order on the product space X4 = X ×X ×X ×X:

(u,v,r,t) � (x,y,z,w) iff x � u, y � v, z � r, t � w,(1.1)

where (u,v,r,t), (x,y,z,w) ∈ X4.
Regarding this partial order Karapinar [6] give the following definitions,

Definition 4. Let (X,�) be partially ordered set and F : X4 → X. We say that F
has the mixed monotone property if F(x,y,z,w) is monotone non decreasing in x
and z and it is monotone non increasing in y and w, that is, for any x,y,z,w ∈ X

x1,x2 ∈ X, x1 � x2 =⇒ F(x1,y,z,w) � F(x2,y,z,w)
y1,y2 ∈ X, y1 � y2 =⇒ F(x,y2,z,w) � F(x,y1,z,w)
z1,z2 ∈ X, z1 � z2 =⇒ F(x,y,z1,w) � F(x,y,z2,w)

w1,w2 ∈ X, w1 � w2 =⇒ F(x,y,z,w2) � F(x,y,z,w1).(1.2)

Definition 5. An element (x,y,z,w) ∈ X4 is called a quadruple fixed point of
F : X4 → X 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.(1.3)

For a metric space (X,d) the function ρ : X4 → [0,∞), given by

ρ((x,y,z,w), (u,v,r,t)) = d(x,u) + d(y,v) + d(z,r) + d(w,t)(1.4)

forms a metric space on X4, that is, (X4,ρ) is a metric induced by (X,d).

2. Quadruple Fixed Point Result for Multivalued Mappings

First we introduced the following concepts.

Definition 6. An element (x,y,z,w) ∈ X4 is called a Quadruple fixed point of
F : X4 → CL(X) if

x ∈ F(x,y,z,w), y ∈ F(y,z,w,x),
z ∈ F(z,w,x,y), w ∈ F(w,x,y,z)(2.1)

Definition 7. A mapping f : X4 → R is called lower semi continuous if, for the
sequences {xn},{yn},{zn},{wn} in X and (x,y,z,w) ∈ X4, one has



QUADRUPLE FIXED POINT 89

lim
n→∞

({xn},{yn},{zn},{wn}) = (x,y,z,w) =⇒ f(x,y,z,w)

� lim
n→∞

inf{{xn},{yn},{zn},{wn}}(2.2)

Let (X,d) be a metric space endowed with the partial order � and T : X → X.
Define the set Ψ ⊂ X4 by,

Ψ = {(x,y,z,w) ∈ X4 : T(x) � T(y) � T(z) � T(w)}(2.3)

Definition 8. A mapping F : X4 → X is said to have a Ψ- property if,

(x,y,z,w) ∈ Ψ =⇒ F(x,y,z,w) ×F(y,z,w,x) ×F(z,w,x,y) ×F(w,x,y,z) ⊂ Ψ.(2.4)

We give some examples to illustrate Definition 8.

Example 9. Let X = R be endowed with the usual order � and T : X → X.
Define F : X4 → CL(X) by,

F(x,y,z,w) = {x}(2.5)

Obviously F has the Ψ- property.

Example 10. Let X = R+ be endowed with the usual order ≤ and T : X → X be
defined by Tx = exp(x). Define F : X4 → CL(X) by,

F(x,y,z,w) = {x + w} ∀x,y,z,w ∈ R+(2.6)

We have Ψ = {(x,y,z,w) ∈ X4,exp(x) � exp(y) � exp(z) � exp(w)}. More-
over, F has the Ψ− property.

Now, we prove the following theorem.

Theorem 11. Let (X,d) be a complete metric space endowed with a partial order �
and Ψ 6= φ that is there exists (x0,y0,z0,w0) ∈ Psi. Suppose that F : X4 → CL(X)
has a Ψ− property such that f : X4 → [0,∞) given by for all x,y,z,w ∈ X,

f(x,y,z,w) = D(x,F(x,y,z,w)) + D(y,F(y,z,w,y))

+ D(z,F(z,w,x,y)) + D(w,F(w,x,y,z))(2.7)

is lower semi continuous and there exists a function φ : [0,∞) → [M, 1), 0 <
M < 1 satisfying

lim
r→s+

sup φ(r) < 1 for each s ∈ [0,∞)(2.8)

if for any (x,y,z,w) ∈ Ψ there exist u ∈ F(x,y,z,w),v ∈ F(y,z,w,x),r ∈
F(z,w,x,y), t ∈ F(w,x,y,z) with

√
φ(f(x,y,z,w))[d(x,u) + d(y,v) + d(z,r) + d(w,t)] ≤ f(x,y,z,w)(2.9)

such that

f(u,v,r,t) ≤ φ(f(x,y,z,w))[d(x,u) + d(y,v) + d(z,r) + d(w,t)](2.10)

then F has a quadruple fixed point.



90 GUPTA,YADAVA AND RAJPUT

Proof. By our assumption, φ(f(x,y,z,w)) < 1 for each (x,y,z,w) ∈ X4. Hence
, for any (x,y,z,w) ∈ X4, there exist u ∈ F(x,y,z,w),v ∈ F(y,z,w,x),r ∈
F(z,w,x,y), t ∈ F(w,x,y,z) satisfying

√
φ(f(x,y,z,w))d(x,u) � D(x,F(x,y,z,w))√
φ(f(x,y,z,w))d(y,v) � D(y,F(y,z,w,x))√
φ(f(x,y,z,w))d(z,r) � D(z,F(z,w,x,y))√
φ(f(x,y,z,w))d(w,t) � D(w,F(w,x,y,z)).(2.11)

Let (x0,y0,z0,w0) be an arbitrary point in Ψ. From (2.8) and (2.9) we can
choose x1 ∈ F(x0,y0,z0,w0),y1 ∈ F(y0,z0,w0,x0),z1 ∈ F(z0,w0,x0,y0),w1 ∈
F(w0,x0,y0,z0) satisfying

√
φ(f(x0,y0,z0,w0))[d(x0,x1) + d(y0,y1) + d(z0,z1) + d(w0,w1)] � f(x0,y0,z0,w0)

(2.12)

such that

f(x1,y1,z1,w1) � φ(f(x0,y0,z0,w0))[d(x0,x1) + d(y0,y1) + d(z0,z1) + d(w0,w1)]
(2.13)

By 2.12 and 2.13, we obtain

f(x1,y1,z1,w1) � φ(f(x0,y0,z0,w0))[d(x0,x1) + d(y0,y1) + d(z0,z1) + d(w0,w1)]
�

√
φ(f(x0,y0,z0,w0)) (φ(f(x0,y0,z0,w0))

[d(x0,x1) + d(y0,y1) + d(z0,z1) + d(w0,w1)])

f(x1,y1,z1,w1) �
√
φ(f(x0,y0,z0,w0))f(x0,y0,z0,w0).

Since F has a Ψ− property and (x0,y0,z0,w0) ∈ Ψ, so we have

F(x0,y0,z0,w0) ×F(y0,z0,w0,x0) ×F(z0,w0,x0,y0) ×F(w0,x0,y0,z0) ⊂ Ψ
(2.14)

which implies that (x1,y1,z1,w1) ∈ Ψ.

Again by 2.9 and 2.10 we can choose ,

x2 ∈ F(x1,y1,z1,w1),y2 ∈ F(y1,z1,w1,x1),z2 ∈ F(z1,w1,x1,y1),w2 ∈ F(w1,x1,y1,z1)

satisfying

√
φ(f(x1,y1,z1,w1))[d(x1,x2) + d(y1,y2) + d(z1,z2) + d(w1,w2)] � f(x1,y1,z1,w1)

(2.15)

such that



QUADRUPLE FIXED POINT 91

f(x1,y1,z1,w1) � φ(f(x1,y1,z1,w1))[d(x1,x2) + d(y1,y2) + d(z1,z2) + d(w1,w2)]
(2.16)

Thus we have

f(x1,y1,z1,w1) �
√
φ(f(x1,y1,z1,w1))f(x1,y1,z1,w1)

(2.17)

which implies that (x2,y2,z2,w2) ∈ Ψ.

Continuing this process, we can choose sequences {xn},{yn},{zn},{wn}, in X
such that for each n ∈ N with (xn,yn,zn,wn) ∈ Ψ.

Now xn+1 ∈ F(xn,yn,zn,wn), yn+1 ∈ F(yn,zn,wn,xn), zn+1 ∈ F(zn,wn,xn,yn),
wn+1 ∈ F(wn,xn,yn,zn) satisfying

√
φ(f(xn,yn,zn,wn))[d(xn,xn+1) +

d(yn,yn+1) + d(zn,zn+1) + d(wn,wn+1)] � f(xn,yn,zn,wn)
(2.18)

such that

f(xn+1,yn+1,zn+1,wn+1) � φ(f(xn,yn,zn,wn)[d(xn,xn+1)
+d(yn,yn+1) + d(zn,zn+1) + d(wn,wn+1)].

(2.19)

Hence, we obtain

f(xn+1,yn+1,zn+1,wn+1) �
√
φ(f(xn,yn,zn,wn)f(xn,yn,zn,wn)(2.20)

with

(xn+1,yn+1,zn+1,wn+1) ∈ Ψ.(2.21)

We claim that f(xn,yn,zn,wn) → 0 as n →∞.
If f(xn,yn,zn,wn) = 0 for some n ∈ N, then

D(xn,F(xn,yn,zn,wn)) = 0 implies that

xn ∈ F(xn,yn,zn,wn) = F(xn,yn,zn,wn).

Analogously,

D(yn,F(yn,zn,wn,xn)) = 0 implies that

yn ∈ F(yn,zn,wn,xn) = F(yn,zn,wn,xn) ,

D(zn,F(zn,wn,xn,yn)) = 0 implies that

zn ∈ F(zn,wn,xn,yn) = F(zn,wn,xn,yn) ,



92 GUPTA,YADAVA AND RAJPUT

D(wn,F(wn,xn,yn,zn)) = 0 implies that

wn ∈ F(wn,xn,yn,zn) = F(wn,xn,yn,zn).

Hence (xn,yn,zn,wn) become a quadruple fixed point of F for such n and the
result follows. Suppose that f(xn,yn,zn,wn) > 0 for all n ∈ N.
Using 2.20 and φ(t) < 1, we conclude that {F(xn,yn,zn,wn)} is decreasing se-
quence of positive real numbers. Thus, there exists a δ ≥ 0 such that

lim
n→∞

f(xn,yn,zn,wn) = δ(2.22)

We will show that δ = 0. Assume on contrary that δ > 0. Let n → ∞ in 2.20
and by assumption 2.8 we obtain

δ � lim
f(xn,yn,zn,wn)→δ+

sup
√
φ(f(xn,yn,zn,wn))δ < δ,(2.23)

a contradiction, Hence

lim
n→∞

f(xn,yn,zn,wn) = 0
+(2.24)

Now, we prove that sequences {xn},{yn},{zn},{wn} in X are Cauchy sequences
in (X,d). Assume that

α = lim f(xn,yn,zn,wn) → 0+ sup
√
φ(f(xn,yn,zn,wn)).(2.25)

By 2.8 we conclude that α < 1. Let k be a real number such that α < k < 1.
Thus there exists n0 ∈ N such that

√
φ(f(xn,yn,zn,wn)) � k for each n ≥ n0.(2.26)

Using 2.20 we obtain

f(xn+1,yn+1,zn+1,wn+1) � kf(xn,yn,zn,wn) for each n ≥ n0.(2.27)

By mathematical induction,

f(xn+1,yn+1,zn+1,wn+1) � kn+1−n0f(xn0,yn0,zn0,wn0 ) for each n ≥ n0.
(2.28)

Since φ(t) ≥ M < 0 for all t ≥ 0 so 2.18 and 2.28 gives that

[d(xn,xn+1) + d(yn,yn+1) + d(zn,zn+1) + d(wn,wn+1)] �
kn−n0
√
M

(xn0,yn0,zn0,wn0 )

(2.29)

for each n ≥ n0, which yields that the sequences {xn},{yn},{zn},{wn} in X
are Cauchy sequences in (X,d). Since X is complete then there exists (a,b,c,d) ∈
X4 such that



QUADRUPLE FIXED POINT 93

lim
n→∞

xn = a, lim
n→∞

yn = b,

lim
n→∞

zn = c, lim
n→∞

wn = d.(2.30)

Finally we show that (a,b,c,d) ∈ X4 is quadruple fixed point of F . As f is lower
semi continuous 2.24 implies that

0 � f(a,b,c,d)
= D(a,F(a,b,c,d)) + D(b,F(b,c,d,a))

+ D(c,F(c,d,a,b)) + D(d,F(d,a,b,c))

� lim
n→∞

inf f(xn,yn,zn,wn) = δ.(2.31)

Hence,

D(a,F(a,b,c,d)) = D(b,F(b,c,d,a)) = 0

D(c,F(c,d,a,b)) = D(d,F(d,a,b,c)) = 0

gives that (a,b,c,d) is a quadruple fixed point of F.
�

Theorem 12. Let (X,d) be a complete metric space endowed with a partial order �
and Ψ 6= φ that is there exists (x0,y0,z0,w0) ∈ Psi. Suppose that F : X4 → CL(X)
has a Ψ− property such that f : X4 → [0,∞) given by

f(x,y,z,w) = D(x,F(x,y,z,w)) + D(y,F(y,z,w,y))

+ D(z,F(z,w,x,y)) + D(w,F(w,x,y,z))(2.32)

for all x,y,z,w ∈ X and f is lower semi continuous and there exists a function
φ : [0,∞) → [M, 1), 0 < M < 1, satisfying

lim
r→s+

sup φ(r) < 1 for each s ∈ [0,∞)(2.33)

if for any (x,y,z,w) ∈ Ψ there exist u ∈ F(x,y,z,w),v ∈ F(y,z,w,x),r ∈
F(z,w,x,y), t ∈ F(w,x,y,z) with

√
φ(∆)∆ � D(x,F(x,y,z,w)) + D(y,F(y,z,w,y))

+ D(z,F(z,w,x,y)) + D(w,F(w,x,y,z))(2.34)

such that

D(u,F(u,v,r,t)) + D(v,F(v,r,t,u)) + D(r,F(r,t,u,v)) + D(t,F(t,u,v,r)))

� φ(∆)∆
(2.35)

where ∆ = ∆((x,y,z,w), (u,v,r,t)) = [d(x,u) + d(y,v) + d(z,r) + d(w,t)] then F
has a quadruple fixed point.



94 GUPTA,YADAVA AND RAJPUT

Proof. By replacing φ(f(x,y,z,w)) with [d(x,u) + d(y,v) + d(z,r) + d(w,t)] in the
proof of Theorem 11 we obtain sequences {xn},{yn},{zn},{wn}, in X such that
for each n ∈ N with, (xn,yn,zn,wn) ∈ Ψ

xn+1 ∈ F(xn,yn,zn,wn), yn+1 ∈ F(yn,zn,wn,xn)
zn+1 ∈ F(zn,wn,xn,yn), wn+1 ∈ F(wn,xn,yn,zn)(2.36)

such that

√
φ(∆n)∆n � D(xn,F(xn,yn,zn,wn)) + D(yn,F(yn,zn,wn,yn))

+ D(zn,F(zn,wn,xn,yn)) + D(wn,F(wn,xn,yn,zn))(2.37)

D(xn+1,F(xn+1,yn+1,zn+1,wn+1)) + D(yn+1,F(yn+1,zn+1,wn+1,yn+1))

+ D(zn+1,F(zn+1,wn+1,xn+1,yn+1))

+ D(wn+1,F(wn+1,xn+1,yn+1,zn+1))

�
√
φ(∆n)(D(xn,F(xn,yn,zn,wn))

+ D(yn,F(yn,zn,wn,yn))

+ D(zn,F(zn,wn,xn,yn))

+ D(wn,F(wn,xn,yn,zn))).

(2.38)

where

∆n = ∆((xn,yn,zn,wn)(xn+1,yn+1,zn+1,wn+1))(2.39)

= d(xn,xn+1) + d(yn,yn+1) + d(zn,zn+1) + d(wn,wn+1).(2.40)

Again following arguments similar to those given in proof of Theorem 11 we
deduce that

D(xn,F(xn,yn,zn,wn)) + D(yn,F(yn,zn,wn,yn))

+ D(zn,F(zn,wn,xn,yn)) + D(wn,F(wn,xn,yn,zn)).(2.41)

is a decreasing sequence of real numbers. Thus, there exists a δ > 0 such that

lim
n→∞

(D(xn,F(xn,yn,zn,wn)) + D(yn,F(yn,zn,wn,yn))

+ D(zn,F(zn,wn,xn,yn)) + D(wn,F(wn,xn,yn,zn))) = δ.(2.42)

Now we need to proof that {∆n} admits a subsequence converging to certain η+
for some η ≥ 0. Since φ(t) ≤ M > 0, using 2.37 we obtain

δn �
1
√
a

(D(xn,F(xn,yn,zn,wn)) + D(yn,F(yn,zn,wn,yn))

+ D(zn,F(zn,wn,xn,yn)) + D(wn,F(wn,xn,yn,zn)))(2.43)

from 2.42 and 2.43 it is clear that the sequence



QUADRUPLE FIXED POINT 95

(D(xn,F(xn,yn,zn,wn)) + D(yn,F(yn,zn,wn,yn))

+ D(zn,F(zn,wn,xn,yn)) + D(wn,F(wn,xn,yn,zn)))(2.44)

is bounded. Therefore, there is some θ ≥ 0 such that

lim
n→∞

inf ∆n = θ(2.45)

from 2.36 we have

xn+1 ∈ F(xn,yn,zn,wn),yn+1 ∈ F(yn,zn,wn,xn),

zn+1 ∈ F(zn,wn,xn,yn),wn+1 ∈ F(wn,xn,yn,zn),

∆n � D(xn,F(xn,yn,zn,wn)) + D(yn,F(yn,zn,wn,yn))
+ D(zn,F(zn,wn,xn,yn)) + D(wn,F(wn,xn,yn,zn)).(2.46)

By comparing 2.42 to 2.45 we get that θ ≥ δ. Now, we shall show that θ = δ.
If δ = 0, by 2.42 and 2.43 we get θ = lim infn→∞+ ∆n = 0 and consequently
θ = δ = 0. Suppose that δ > 0. Assume on contrary that θ > δ. From 2.42 and
2.45 there is a positive integer n0 such that

D(xn,F(xn,yn,zn,wn)) + D(yn,F(yn,zn,wn,yn))

+ D(zn,F(zn,wn,xn,yn)) + D(wn,F(wn,xn,yn,zn))

� δ +
θ − δ

4
(2.47)

δ −
θ −δ

4
� ∆n(2.48)

for all n ≥ n0. We combine 2.37, 2.47 and 2.48 to obtain√
φ((∆n)

(
δ −

θ −δ
4

)
�

√
φ((∆n)∆n

� D(xn,F(xn,yn,zn,wn)) + D(yn,F(yn,zn,wn,yn))
+ D(zn,F(zn,wn,xn,yn)) + D(wn,F(wn,xn,yn,zn))

� δ +
θ −δ

4
(2.49)

for all n ≥ n0. It follows that

√
φ((∆n) �

θ + 3δ

3θ + δ
∀n ≥ n0.(2.50)

By 2.38 and 2.50 we have



96 GUPTA,YADAVA AND RAJPUT

D(xn+1,F(xn+1,yn+1,zn+1,wn+1)) + D(yn+1,F(yn+1,zn+1,wn+1,yn+1))

+ D(zn+1,F(zn+1,wn+1,xn+1,yn+1))

+ D(wn+1,F(wn+1,xn+1,yn+1,zn+1))

≤ hD(xn,F(xn,yn,zn,wn))
+ D(yn,F(yn,zn,wn,yn))

+ D(zn,F(zn,wn,xn,yn))

+ D(wn,F(wn,xn,yn,zn)).

(2.51)

where h = θ+3δ
3θ+δ

. Since θ > δ > 0, therefore h < 1, so proceeding by induction
and combining the above inequalities, it follows that

δ � D(xn0+k0,F(xn0+k0,yn0+k0,zn0+k0,wn0+k0 ))
+ D(yn0+k0,F(yn0+k0,zn0+k0,wn0+k0,yn0+k0 ))

+ D(zn0+k0,F(zn0+k0,wn0+k0,xn0+k0,yn0+k0 ))

+ D(wn0+k0,F(wn0+k0,xn0+k0,yn0+k0,zn0+k0 ))

� hk0 [D(xn0,F(xn0,yn0,zn0,wn0 ))
+ D(yn0,F(yn0,zn0,wn0,yn0 ))

+ D(zn0,F(zn0,wn0,xn0,yn0 ))

+ D(wn0,F(wn0,xn0,yn0,zn0 ))]δ.

(2.52)

for a positive integer k0. Then, we obtain a contradiction, so we must have θ = δ.
Now, we shall show that θ = 0. Since

θ = δ � D(xn,F(xn,yn,zn,wn)) + D(yn,F(yn,zn,wn,yn))
+ D(zn,F(zn,wn,xn,yn)) + D(wn,F(wn,xn,yn,zn))∆n(2.53)

then we rewrite 2.45 as

(2.54) lim
n→∞+

inf ∆n = θ
+.

Hence, there exists a subsequence {∆nk} of {∆n} such that limk→∞+inf∆nk =
θ+.

By 2.33 we have

lim
∆nk→∞

+
sup

√
φ(∆nk ) < 1.(2.55)

From 2.38 we obtain



QUADRUPLE FIXED POINT 97

D(xnk+1,F(xnk+1,ynk+1,znk+1,wnk+1)) + D(ynk+1,F(ynk+1,znk+1,wnk+1,ynk+1))

+D(znk+1,F(znk+1,wnk+1,xnk+1,ynk+1)) + D(wnk+1,F(wnk+1,xnk+1,ynk+1,znk+1))

�
√
φ(∆nk )[D(xnk,F(xnk,ynk,znk,wnk ))

+ D(ynk,F(ynk,znk,wnk,ynk ))

+ D(znk,F(znk,wnk,xnk,ynk ))

+ D(wnk,F(wnk,xnk,ynk,znk ))].

(2.56)

Taking the limit as k →∞ and using 2.42 we have

δ = lim
k→∞+

{sup[D(xnk+1,F(xnk+1,ynk+1,znk+1,wnk+1))

+ D(ynk+1,F(ynk+1,znk+1,wnk+1,ynk+1))

+ D(znk+1,F(znk+1,wnk+1,xnk+1,ynk+1))

+ D(wnk+1,F(wnk+1,xnk+1,ynk+1,znk+1))]}

� lim
k→∞+

sup
[√

φ(∆nk )
]

(2.57)

lim
k→∞+

{sup[D(xnk,F(xnk,ynk,znk,wnk )) + D(ynk,F(ynk,znk,wnk,ynk ))

+ D(znk,F(znk,wnk,xnk,ynk ))

+ D(wnk,F(wnk,xnk,ynk,znk ))]

�
(

lim
k→∞+

sup
√
φ(∆nk )

)
δ.(2.58)

Assume that δ > 0, then from 2.57 we get that

1 � lim
k→∞+

sup
√
φ(∆nk )(2.59)

a contradiction with respect to 2.55 so δ = 0.
Now, from 2.38 and 2.42 we have

α = lim
∆n→0

sup
√
φ(∆n) < 1(2.60)

The rest of the proof is similar to the proof of the Theorem 11 so it is omitted. �

We improve and corrected the example of Samet and Vetro [12].

3. Examples

Example 13. Let X = [0, 2], and let d : X × X → [0,∞) be the usual metric.
Suppose that T(x) = M for all x ∈ [0, 2] where M is a constant in [0,2], and
F : X4 → CL(X) is defined for all x,y,z,w ∈ X as follows

F(x,y,z,w) =

{
x2

4
if x ∈

[
0, 15

32

)
∪
(

15
32
, 2
]

{15
96
, 1

5
} if x = 15

32



98 GUPTA,YADAVA AND RAJPUT

Oviously, F has the Ψ- property. Set φ : [0,∞) → [0,∞)

φ(s) =

{
11
12
s if s ∈

[
0, 2

3

]
10
16

if s ∈
(

2
3
,∞
)

Consider a function

f(x,y,z,w) =




A if x,y,z,w ∈
[
0, 15

32

)
∪
(

15
32
, 2
]

B if x,y,z ∈
[
0, 15

32

)
∪
(

15
32
, 2
]
with w = 15

32
C if x,y ∈

[
0, 15

32

)
∪
(

15
32
, 2
]
with z = w = 15

32
D if x ∈

[
0, 15

32

)
∪
(

5
32
, 2
]
with y = z = w = 15

32
E if x = y = z = w = 15

32

where, A = x+y+z+w−1
4
(x2 +y2 +z2 +w2), B = x+y+z−1

4
(x2 +y2 +z2)+ 43

160
,

C = x + y − 1
4
(x2 + y2) + 86

160
, D = x− 1

4
(x2) + 129

160
, E = 172

160

which is lower semicontinuous. Thus for all x,y,z,w ∈ X with x,y,z,w 6= 5
32

,

there exists u ∈ F(x,y,z,w) = x
2

4
, v ∈ F(y,z,w,x) = y

2

4
, r ∈ F(z,w,x,y) = z

2

4
,

t ∈ F(w,x,y,z) = w
2

4
such that

D(u,F(u,v,r,t)) + D(v,F(v,r,t,u)) + D(r,F(r,t,u,v)) + D(t,F(t,u,v,r))

=
x2

4
−
x4

16
+
y2

4
−
y4

16
+
z2

4
−
z4

16
+
w2

4
−
w4

16

=
1

4

[(
x +

x2

4

)(
x−

x2

4

)
+

(
y +

y2

4

)(
y −

y2

4

)
+

(
z +

z2

4

)(
z −

z2

4

)
+

(
w +

w2

4

)(
w −

w2

4

)]
≤

1

4

[(
x +

x2

4

)
d(x,u) +

(
y +

y2

4

)
d(y,v) +

(
z +

z2

4

)
d(z,r) +

(
w +

w2

4

)
d(w,t)

]
�

1

4
max{

(
x +

x2

4

)
,

(
y +

y2

4

)
,

(
z +

z2

4

)
,

(
w +

w2

4

)
}

d(x,u) + d(y,v) + d(z,r) + d(w,t)

�
10

12
max{

(
x−

x2

4

)
,

(
y −

y2

4

)
,

(
z −

z2

4

)
,

(
w −

w2

4

)
}

d(x,u) + d(y,v) + d(z,r) + d(w,t)

� φ(d(x,u) + d(y,v) + d(z,r) + d(w,t))[d(x,u) + d(y,v) + d(z,r) + d(w,t)]

Hence for all x,y,z,w ∈ X with x,y,z,w 6= 15
32

, the conditions 2.9 and 2.10
are satisfied. Analogously, one can easy show that conditions 2.9 and 2.10 are
satisfied for the cases x,y,z ∈

[
0, 15

32

)
∪
(

15
32
, 2
]

with w = 15
32

and x ∈
[
0, 15

32

)
∪(

15
32
, 2
]
with y = z = w = 15

32
. For the last case, that is x = y = z = w = 15

36
, we

assume that u = v = r = t = 15
96
, it follows that,

[d(x,u) + d(y,v) + d(z,r) + d(w,t)] =
5

4
>

2

3



QUADRUPLE FIXED POINT 99

As a consequence, we conclude that all the conditions of Theorem 2.7 are satisfied
and F admits a quadruple fixed point i.e. (0, 0, 0, 0).

4. ACKNOWLEDGEMENT

The authors thank the referees for their careful reading of the manuscript and
for their suggestions.

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1Department of Applied Mathematics, Vidhyapeeth Institute of Science & Technol-

ogy, Near SOS Balgram, Bhopal (M.P.) India

2Patel Group of Institutions, Ratibad, Bhopal (M.P.), India

3Department of Mathematics, Govt. P.G. Collage, Gadarwara Dist. Narsingpur
(M.P.), India

∗Corresponding author