International Journal of Analysis and Applications
ISSN 2291-8639
Volume 12, Number 2 (2016), 129-141
http://www.etamaths.com

C-CLASS FUNCTIONS ON SHORTER PROOFS OF SOME EVEN-TUPLED

COINCIDENCE THEOREMS IN ORDERED METRIC SPACES

ANUPAM SHARMA∗

Abstract. The purpose of this paper is to prove some even tupled coincidence theorems for mappings

with one variable in ordered complete metric spaces by using the concept of C-class functions. Our

results generalize and improve several results in the literature.

1. Introduction

Ran and Reurings [30] extended the Banach contraction principle on ordered metric spaces for
continuous monotone mappings with some applications to matrix equations. Thereafter Nieto and
López [25] modified Ran and Reurings’ fixed point theorem for an increasing mapping not necessarily
continuous by assuming an another hypothesis on the ordered metric space and proved some fixed point
theorems besides giving some applications to ordinary differential equations. In the same development,
Nieto and López [26] analogously proved a fixed point theorem for a decreasing mapping on ordered
metric space. In recent years, Nieto and López’s [25] fixed point theorems were extended and refined
by many authors ([1, 2, 7], [11]-[13], [18, 19, 24, 27]).

The idea of a coupled fixed point was introduced by Guo and Lakshmikantham [10] which was
well followed by Bhaskar and Lakshmikantham [5] where the authors introduced the notion of mixed
monotone property and proved some coupled fixed point theorems for weakly linear contractions en-
joying mixed monotone property in ordered complete metric spaces. In [23], Lakshmikantham and

Ćirić generalized these results for nonlinear contraction mappings by introducing the notion of coupled
coincidence point and mixed g-monotone property.

Recently, Berzig and Samet [6] extended and generalized some fixed point results to higher di-
mensions. However, they used permutations of variables and distinguished between the first and last
variables. Further, Roldán et al. [31] proved some existence and uniqueness theorems for nonlinear
mappings of any number of arguments, not necessarily permuted or ordered. For more details see
([20, 31, 32, 33]).

Recently, Imdad et al. [16] extended the idea of mixed g-monotone property to the mapping
F : Xn → X (where n is even natural number) and proved an even-tupled coincidence point theorem
for nonlinear contraction mappings satisfying mixed g-monotone property. Basically their results are
true for only even n but not for odd ones (for details see [14]-[17]). Very recently, Samet et al. [36]
have shown that the coupled (analogously n-tupled) fixed results can be more easily obtained by using
well known fixed point theorems on ordered metric spaces (see also [9, 28, 29]).

The concept of C-class functions was introduced by Ansari [3] which actually covers a large class
of contractive conditions. In this paper, we generalize the results of Sharma et al. [37] by using the
concept of C-class functions.

2. Preliminaries

With a view to make, our presentation self-contained, we collect some basic definitions and needed
results which will be used frequently in the text later.

2010 Mathematics Subject Classification. 47H10; 54H25.
Key words and phrases. partially ordered set; compatible mapping; mixed g-monotone property; n-tupled coincidence

point; C-class function.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

129



130 SHARMA

Definition 2.1. Let X be a non-empty set. A relation ‘ �’ on X is said to be a partial order if the
following properties are satisfied:
(i) reflexive: x � x for all x ∈ X,
(ii) anti-symmetric: x � y and y � x implies x = y,
(iii) transitive: x � y and y � z implies x � z for all x,y,z ∈ X.

A non-empty set X together with a partial order ‘ �’ is said to be an ordered set and we denote it
by (X,�).

Definition 2.2. Let (X,�) be an ordered set. Any two elements x and y are said to be comparable
elements in X if either x � y or y � x.

Definition 2.3. ([27]) A triplet (X,d,�) is called an ordered metric space if (X,d) is a metric space
and (X,�) is an ordered set. Moreover, if d is a complete metric on X, then we say that (X,d,�) is
an ordered complete metric space.

Recently, Kutbi et al. [22] introduced the concept of regular map.

Definition 2.4. ([22]) An ordered metric space (X,d,�) is said to be nondecreasing regular (resp.
nonincreasing regular) if it satisfies the following property:

if {xm} is a nondecreasing (resp. nonincreasing) sequence and xm → x, then xm � x (resp. x �
xm) ∀m ∈ N∪{0}.

Definition 2.5. ([22]) An ordered metric space (X,d,�) is said to be regular if it is both nonde-
creasing regular and nonincreasing regular.

Definition 2.6. Let (X,d,�) be an ordered metric space and g : X → X be a mapping. Then X is
said to be nondecreasing g-regular (resp. nonincreasing g-regular) if it satisfies the following property:

if {xm} is a nondecreasing (resp. nonincreasing) sequence and xm → x, then gxm � gx (resp.
gx � gxm) ∀m ∈ N∪{0}.

Definition 2.7. An ordered metric space (X,d,�) is said to be g-regular if it is both nondecreasing
g-regular and nonincreasing g-regular.

Notice that, on setting g = I (identity mapping on X), Definitions 2.6 and 2.7 reduce to Definitions
2.4 and 2.5 respectively.

Throughout the paper, n stands for a general even natural number. Let us denote by Xn the
product space X ×X × . . .×X of n identical copies of X.

Definition 2.8. ([16]) Let (X,�) be an ordered set and F : Xn → X and g : X → X two mappings.
Then F is said to have the mixed g-monotone property if F is g-nondecreasing in its odd position
arguments and g-nonincreasing in its even position arguments, that is, for x1,x2,x3, ...,xn ∈ X, if
for all x11,x

1
2 ∈ X, gx11 � gx12 ⇒ F(x11,x2,x3, ...,xn) � F(x12,x2,x3, ...,xn)

for all x21,x
2
2 ∈ X, gx21 � gx22 ⇒ F(x1,x22,x3, ...,xn) � F(x1,x21,x3, ...,xn)

for all x31,x
3
2 ∈ X, gx31 � gx32 ⇒ F(x1,x2,x31, ...,xn) � F(x1,x2,x32, ...,xn)

...
for all xn1 ,x

n
2 ∈ X, gxn1 � gxn2 ⇒ F(x1,x2,x3, ...,xn2 ) � F(x1,x2,x3, ...,xn1 ).

For g = I (identity mapping), Definition 2.8 reduces to mixed monotone property (for details see
[16]).



C-CLASS FUNCTIONS ON SHORTER PROOFS OF SOME EVEN-TUPLED COINCIDENCE THEOREMS 131

Definition 2.9. ([34]) An element (x1,x2, ...,xn) ∈ Xn is called an n-tupled fixed point of the
mapping F : Xn → X if 



F(x1,x2,x3, ...,xn) = x1

F(x2,x3, ...,xn,x1) = x2

F(x3, ...,xn,x1,x2) = x3

...

F(xn,x1,x2, ...,xn−1) = xn.

Definition 2.10. ([16]) An element (x1,x2, ...,xn) ∈ Xn is called an n-tupled coincidence point of
mappings F : Xn → X and g : X → X if



F(x1,x2,x3, ...,xn) = g(x1)

F(x2,x3, ...,xn,x1) = g(x2)

F(x3, ...,xn,x1,x2) = g(x3)
...

F(xn,x1,x2, ...,xn−1) = g(xn).

Remark 2.1. For n = 2, Definitions 2.9 and 2.10 yield the definitions of coupled fixed point and
coupled coincidence point respectively while on the other hand, for n = 4 these definitions yield the
definitions of quadrupled fixed point and quadrupled coincidence point respectively.

Definition 2.11. An element (x1,x2, ...,xn) ∈ Xn is called an n-tupled common fixed point of
mappings F : Xn → X and g : X → X if



F(x1,x2,x3, ...,xn) = g(x1) = x1

F(x2,x3, ...,xn,x1) = g(x2) = x2

F(x3, ...,xn,x1,x2) = g(x3) = x3

...

F(xn,x1,x2, ...,xn−1) = g(xn) = xn.

Definition 2.12. ([14]) Let X be a non-empty set. Then the mappings F : Xn → X and g : X → X
are said to be compatible if



lim
m→∞

d(g(F(x1m,x
2
m, ...,x

n
m)),F(gx

1
m,gx

2
m, ...,gx

n
m)) = 0

lim
m→∞

d(g(F(x2m, ...,x
n
m,x

1
m)),F(gx

2
m, ...,gx

n
m,gx

1
m)) = 0

...

lim
m→∞

d(g(F(xnm,x
1
m, ...,x

n−1
m )),F(gx

n
m,gx

1
m, ...,gx

n−1
m )) = 0,

where {x1m},{x2m}, ...,{xnm} are sequences in X such that


lim
m→∞

F(x1m,x
2
m, ...,x

n
m) = lim

m→∞
g(x1m) = x

1

lim
m→∞

F(x2m, ...,x
n
m,x

1
m) = lim

m→∞
g(x2m) = x

2

...

lim
m→∞

F(xnm,x
1
m, ...,x

n−1
m ) = lim

m→∞
g(xnm) = x

n,

for some x1,x2, ...,xn ∈ X are satisfied.

The following families of control functions are indicated in Choudhury et al. [8].

(1) = := {ζ : [0,∞) → [0,∞) : ζ is continuous and ζ(t) = 0 if and only if t = 0}
(2) Ω := {ϕ : [0,∞) → [0,∞) : ϕ is continuous and monotone nondecreasing and ϕ(t) = 0 if and

only if t = 0}
(3) =u := {ζ : [0,∞) → [0,∞) : ζ is continuous and ζ(t) > 0 , t > 0 and ζ(0) ≥ 0}.



132 SHARMA

Notice that members of Ω are called altering distance functions (cf. [21]).

Ansari [3] introduced the concept of C-class functions which covers a large class of contractive
conditions (see Example 2.1 (1),(2),(9),(15)).

Definition 2.13. ([3]) A continuous function F : [0,∞)2 → R is called a C-function if F is
continuous and satisfies the following:
(1) F(s,t) ≤ s;
(2) F(s,t) = s implies that either s = 0 or t = 0 for all s,t ∈ [0,∞).

An extra condition on F is that F(0, 0) = 0 could be imposed in some cases if required. The letter
C denotes the class of all C-functions.

Example 2.1. ([3]) Define F : [0,∞)2 → R by
(1) F(s,t) = s− t, F(s,t) = s ⇒ t = 0;
(2) F(s,t) = ms, 0<m<1, F(s,t) = s ⇒ s = 0;
(3) F(s,t) = s

(1+t)r
; r ∈ (0,∞), F(s,t) = s ⇒ s = 0 or t = 0;

(4) F(s,t) = log(t + as)/(1 + t), a > 1, F(s,t) = s ⇒ s = 0 or t = 0;
(5) F(s,t) = ln(1 + as)/2, a > e, F(s, 1) = s ⇒ s = 0;
(6) F(s,t) = (s + l)(1/(1+t)

r) − l, l > 1,r ∈ (0,∞), F(s,t) = s ⇒ t = 0;
(7) F(s,t) = s logt+a a, a > 1, F(s,t) = s ⇒ s = 0 or t = 0;
(8) F(s,t) = s− ( 1+s

2+s
)( t

1+t
), F(s,t) = s ⇒ t = 0;

(9) F(s,t) = sβ(s), β : [0,∞) → [0, 1), and is continuous, F(s,t) = s ⇒ s = 0;
(10) F(s,t) = s− t

k+t
, F(s,t) = s ⇒ t = 0;

(11) F(s,t) = s − ϕ(s), F(s,t) = s ⇒ s = 0, where ϕ : [0,∞) → [0,∞) is a continuous function
such that ϕ(t) = 0 ⇔ t = 0;

(12) F(s,t) = sh(s,t), F(s,t) = s ⇒ s = 0, where h : [0,∞) × [0,∞) → [0,∞) is a continuous
function such that h(s,t) < 1 for all t,s > 0;

(13) F(s,t) = s− ( 2+t
1+t

)t, F(s,t) = s ⇒ t = 0;
(14) F(s,t) = n

√
ln(1 + sn), F(s,t) = s ⇒ s = 0;

(15) F(s,t) = φ(s),F(s,t) = s ⇒ s = 0, where φ : [0,∞) → [0,∞) is an upper semi-continuous
function such that φ(0) = 0, and φ(t) < t for t > 0,

(16) F(s,t) = s
(1+s)r

; r ∈ (0,∞), F(s,t) = s ⇒ s = 0 ;
(17) F(s,t) = ϑ(s); ϑ : R+×R+ → R is a generalized Mizoguchi-Takahashi type function, F(s,t) = s

⇒ s = 0;
(18) F(s,t) = s

Γ(1/2)

∫∞
0

e−x√
x+t

dx, where Γ is the Euler Gamma function;

for all s,t ∈ [0,∞). Then F is an element of C.

3. Main results

(A) Let (X,�) be an ordered set. Define the following partial order v on the product space Xn, for
U = (x1,x2, . . . ,xn), V = (y1,y2, . . . ,yn) ∈ Xn

U v V ⇔ x1 � y1, y2 � x2, x3 � y3, . . . ,yn � xn.

(B) Let (X,d) be a metric space. Define the following metric D̃ on the product space Xn, for U =
(x1,x2, . . . ,xn), V = (y1,y2, . . . ,yn) ∈ Xn,

D̃(U,V ) = max
1≤i≤n

d(xi,yi).

The proof of the following lemmas are immediately. We note the same idea here, but in the case of
coupled and tripled fixed point theorems, we have been first used in ([4], [28], [35]).

Lemma 3.1. Let (X,d,�) be an ordered complete metric space. Then (Xn,D̃,v) is an ordered
complete metric space.



C-CLASS FUNCTIONS ON SHORTER PROOFS OF SOME EVEN-TUPLED COINCIDENCE THEOREMS 133

Lemma 3.2. Let (X,d,�) be an ordered metric space and F : Xn → X and g : X → X be two
mappings. Define mappings TF : X

n → Xn and Tg : Xn → Xn by

TF (x
1,x2, . . . ,xn) = (F(x1,x2, . . . ,xn),F(x2, . . . ,xn,x1), . . . ,F(xn,x1, . . . ,xn−1))

and Tg(x
1,x2, . . . ,xn) = (gx1,gx2, . . . ,gxn). Then the following hold:

(1) If F has the mixed g-monotone property, then TF is monotone Tg-nondecreasing with respect to
v .
(2) If F and g are compatible, then TF and Tg are compatible.
(3) If g is continuous, then Tg is continuous.
(4) If F is continuous, then TF is continuous.

(5) If (X,d,�) is g-regular, then (Xn,D̃,v) is nondecreasing g-regular.
(6) A point (x1,x2, . . . ,xn) ∈ Xn is an n-tupled coincidence point of F and g iff (x1,x2, . . . ,xn) is a
coincidence point of TF and Tg.

The following lemma is crucial for our main result.

Lemma 3.3. Let (X,d,�) be an ordered complete metric space and f and g be two self-mappings on
X. Suppose that the following conditions are satisfied:
(i) f(X) ⊆ g(X),
(ii) f is monotone g-nondecreasing,
(iii) f and g are compatible,
(iv) g is continuous,
(v) either f is continuous or X is nondecreasing g-regular,
(vi) there exists x0 ∈ X such that g(x0) � f(x0),
(vii) there exist ϕ ∈ Ω and ζ ∈=u and F ∈C such that for all x,y ∈ X,

ϕ(d(f(x),f(y))) ≤F(ϕ(d(g(x),g(y))),ζ(d(g(x),g(y)))), with g(x) � g(y). (3.1)

Then f and g have a coincidence point.

Proof. In view of assumption (vi), if g(x0) = f(x0), then x0 is a coincidence point of f and g and
hence proof is finished. On the other hand if g(x0) 6= f(x0), then we have g(x0) ≺ f(x0). So according
to assumption (i), that is, f(X) ⊆ g(X), we can choose x1 ∈ X such that g(x1) = f(x0). Again from
f(X) ⊆ g(X), we can choose x2 ∈ X such that g(x2) = f(x1). Continuing this process, we define a
sequence {xm}⊂ X of joint iterates such that

g(xm+1) = f(xm) ∀m ∈ N∪{0}. (3.2)

Now, we assert that {g(xm)} is a non-decreasing sequence, that is

g(xm) � g(xm+1) ∀m ∈ N∪{0}. (3.3)

We prove this fact by mathematical induction. On using (3.2) for m = 0 and assumption (vi), we have

g(x0) � f(x0) = g(x1).

Thus, (3.3) holds for m = 0. Suppose that (3.3) holds for m = r > 0, that is,

g(xr) � g(xr+1). (3.4)

Then we have to show that (3.3) holds for m = r + 1. To accomplish this we use (3.2), (3.4) and
assumption (ii) so that

g(xr+1) = f(xr) � f(xr+1) = g(xr+2).
Thus, by induction, (3.3) holds for all m ∈ N∪{0}.
If g(xm) = g(xm+1) for some m ∈ N, then by using (3.2), we have g(xm) = f(xm), that is, xm is a
coincidence point of f and g and hence proof is finished. On the other hand if g(xm) 6= g(xm+1) for
each m ∈ N∪{0}, we can define a sequence

δm := d(g(xm),g(xm+1)), m ∈ N∪{0}. (3.5)



134 SHARMA

On using (3.2), (3.3), (3.5) and assumption (vii), we obtain

ϕ(δm+1) = ϕ(d(g(xm+1),g(xm+2))) = ϕ(d(f(xm),f(xm+1)))

≤ F(ϕ(d(g(xm),g(xm+1))),ζ(d(g(xm),g(xm+1))))
= F(ϕ(δm),ζ(δm)) ≤ ϕ(δm). (3.6)

On using the property of ϕ, we have ϕ(δm+1) ≤ ϕ(δm), which implies that δm+1 ≤ δm. Therefore {δm}
is a monotone decreasing sequence of nonnegative real numbers. Hence there exists δ ≥ 0 such that
δm → δ as m → ∞. Taking limit as m → ∞ in (3.6) and using the continuities of ϕ and ζ, we have
ϕ(δ) ≤F(ϕ(δ),ζ(δ)), so ϕ(δ) = 0, or ,ζ(δ) = 0, therefore δ = 0 , which is a contradiction . Therefore

lim
m→∞

δm = lim
m→∞

d(g(xm),g(xm+1)) = 0. (3.7)

Now, we show that {g(xm)} is a Cauchy sequence. On contrary suppose that {g(xm)} is not a Cauchy
sequence. Then there exists an � > 0 and sequences of positive integers {m(k)} and {t(k)} such that
for all positive integers k, t(k) > m(k) > k, such that

ηk = d(g(xm(k)),g(xt(k))) ≥ �, and d(g(xm(k)),g(xt(k)−1)) < �.
Now,

� ≤ ηk = d(g(xm(k)),g(xt(k)))
≤ d(g(xm(k)),g(xt(k)−1)) + d(g(xt(k)−1),g(xt(k)))
< � + δt(k)−1

that is,
� ≤ ηk < � + δt(k)−1.

Letting k →∞ in above inequality and using (3.7), we get
lim
k→∞

ηk = �. (3.8)

Again,

ηk+1 = d(g(xm(k)+1),g(xt(k)+1))

≤ d(g(xm(k)+1),g(xm(k))) + d(g(xm(k)),g(xt(k))) + d(g(xt(k)),g(xt(k)+1))
< δm(k)+1 + ηk + δt(k)+1

⇒ ηk+1 < δm(k)+1 + ηk + δt(k)+1.

Letting k →∞ in above inequality and using (3.7) and (3.8), we get
lim
k→∞

ηk+1 = �. (3.9)

Since t(k) > m(k), hence by (3.3), we get g(xm(k)) ≤ g(xt(k)). Therefore, owing to (3.1) and assumption
(vii), we get

ϕ(ηk+1) = ϕ(d(g(xm(k)+1),g(xt(k)+1))) = ϕ(d(f(xm(k)),f(xt(k))))

≤ F(ϕ(d(g(xm(k)),g(xt(k)))),ζ(d(g(xm(k)),g(xt(k)))))
= F(ϕ(ηk),ζ(ηk))

that is,
ϕ(ηk+1) ≤F(ϕ(ηk),ζ(ηk)).

Letting k →∞ in above inequality and using (3.8), (3.9) and continuities of ϕ and ζ, we get
ϕ(�) ≤F(ϕ(�),ζ(�))

so ϕ(�) = 0, or ,ζ(�) = 0 thus � = 0 which is a contradiction . Therefore the sequence {g(xm)} is
Cauchy. From the completeness of X, there exists x ∈ X such that

lim
m→∞

f(xm) = lim
m→∞

g(xm) = x. (3.10)

Since F and g are compatible, we have from (3.10),

lim
m→∞

d(f(gxm),g(fxm)) = 0. (3.11)



C-CLASS FUNCTIONS ON SHORTER PROOFS OF SOME EVEN-TUPLED COINCIDENCE THEOREMS 135

Now, we use assumption (v). Firstly, we assume that f is continuous. Then for all m ∈ N ∪{0}, we
have

d(g(x),f(gxm)) ≤ d(g(x),g(fxm)) + d(g(fxm),f(gxm)).
Taking k → ∞ in above inequality and using (3.10), (3.11) and continuities of f and g, we get
d(g(x),f(x)) = 0, that is, g(x) = f(x). Hence the element x ∈ X is a coincidence point of f and g.
Next, we suppose that X is nondecreasing g-regular. From (3.3) and (3.10), we get

g(gxm) � g(x). (3.12)

Since f and g are compatible and g is continuous by (3.10) and (3.11), we have

lim
m→∞

g(gxm) = g(x) = lim
m→∞

g(fxm) = lim
m→∞

f(gxm). (3.13)

Now, using triangle inequality, we have

d(f(x),g(x)) ≤ d(f(x),g(gxm+1)) + d(g(gxm+1),g(x))
= d(f(x),g(fxm)) + d(g(gxm+1),g(x)).

Taking k →∞ in above inequality and using (3.13), we have

d(f(x),g(x)) ≤ lim
m→∞

d(f(x),g(fxm)) + lim
m→∞

d(g(gxm+1),g(x))

= lim
m→∞

d(f(x),f(gxm)).

Since ϕ is continuous and monotone nondecreasing, from the above inequality we have

ϕ(d(f(x),g(x))) ≤ ϕ( lim
m→∞

d(f(x),f(gxm)))

= lim
m→∞

ϕ(d(f(x),f(gxm))).

By (3.12) and assumption (vii), we get

ϕ(d(f(x),g(x))) ≤ lim
m→∞

ϕ(d(f(x),f(gxm)))

≤ lim
m→∞

F(ϕd(g(x),g(gxm))),ζ(d(g(x),g(gxm))))

= F( lim
m→∞

ϕ(d(g(x),g(gxm))), lim
m→∞

ζ(d(g(x),g(gxm))))

= F(ϕ(d(f(x),g(x))),ζ(d(f(x),g(x))).

so ϕ(d(f(x),g(x))) = 0, or ,ζ(d(f(x),g(x)) = 0 ,which implies that d(f(x),g(x)) = 0, that is, g(x) =
f(x). Hence x ∈ X is a coincidence point of f and g.

Lemma 3.4. In addition to the hypotheses of Lemma 3.3, suppose that for real x,y ∈ X there exists,
z ∈ X such that f(z) is comparable to f(x) and f(y). Then f and g have a unique common fixed point.

Proof. The set of coincidence points of f and g is non-empty due to Lemma 3.3. Assume now, x
and y are two coincidence points of f and g, that is,

f(x) = g(x) and f(y) = g(y).

Now we will show that g(x) = g(y). By assumption, there exists z ∈ X such that f(z) is comparable
to f(x) and f(y). Put z10 = z and choose z1 ∈ X such that g(z1) = f(z0). Further define sequence
{g(zm)} such that g(zm+1) = f(zm). Further set x0 = x and y0 = y. In the same way, define the
sequences {g(xm)} and {g(ym)}. Then it is easy to show that

g(xm+1) = f(xm) and g(ym+1) = f(ym).

Since f(x) = g(x1) = g(x) and f(z) = g(z1) are comparable, we have

g(x) � g(z1).

It is easy to show that g(x) and g(zm) are comparable, that is, for all m ∈ N,

g(x) � g(zm).



136 SHARMA

Thus from (3.1) we have

ϕ(d(g(x),g(zm+1))) = ϕ(d(f(x),f(zm)))
≤ F(ϕ(d(g(x),g(zm))),ζ(d(g(x),g(zm)))).

Let Rm = d(g(x),g(zm+1)). Then

ϕ(Rm) ≤F(ϕ(Rm−1),ζ(Rm−1)). (3.14)
Using the property of ϕ, we have ϕ(Rm) ≤ ϕ(Rm−1), which implies that Rm ≤ Rm−1 (by the property
of ϕ). Therefore {Rm} is a monotone decreasing sequence of nonnegative real numbers. Hence there
exists r ≥ 0 such that Rm → r as m → ∞. Taking the limit as m → ∞ in (3.14) and using the
continuities of ϕ and ζ, we have ϕ(r) ≤ F(ϕ(r),ζ(r)),so ϕ(r) = 0, ,ζ(r) = 0 thus r = 0 which is a
contradiction . Therefore Rm → 0 as m →∞, that is,

lim
m→∞

d(g(x),g(zm+1)) = 0.

Similarly we can prove that
lim

m→∞
d(g(y),g(zm+1)) = 0.

Therefore by triangle inequality

d(g(x),g(y)) ≤ d(g(x),g(zm+1)) + d(g(zm+1),g(y)) → 0 as m →∞.
Hence

g(x) = g(y). (3.15)

Since g(x) = f(x) and f and g are compatible, we have gg(x) = f(gx). Write g(x) = a, then we have

g(a) = f(a). (3.16)

Thus a is the coincidence point of f and g. Then owing to (3.15) with y = a, it follows that g(x) = g(a),
that is,

g(a) = a. (3.17)

Using (3.16) and (3.17), we have a = g(a) = f(a). Thus a is the common fixed point of f and g. To
prove the uniqueness, assume that b is another common fixed point of f and g. Then by (3.15), we
have

b = g(b) = g(a) = a.

This completes the proof of Lemma.

Theorem 3.1. Let (X,d,�) be an ordered complete metric space and F : Xn → X and g : X → X
be two mappings. Suppose that the following conditions are satisfied:
(i) F(Xn) ⊆ g(X),
(ii) F and g are compatible,
(iii) F has the mixed g-monotone property,
(iv) g is continuous,
(v) either F is continuous or X is g-regular,
(vi) there exist x10,x

2
0,x

3
0, ...,x

n
0 ∈ X such that



gx10 � F(x10,x20,x30, ...,xn0 )
F(x20,x

3
0, ...,x

n
0 ,x

1
0) � gx20

gx30 � F(x30, ...,xn0 ,x10,x20)
...

F(xn0 ,x
1
0,x

2
0, ...,x

n−1
0 ) � gx

n
0 ,

(3.18)

(vii) there exist ϕ ∈ Ω and ζ ∈=u and F a C-function such that
ϕ(d(FU,FV )) ≤F(ϕ( max d(gxi,gyi)),ζ(max d(gxi,gyi)))

for all U = (x1,x2, ...,xn), V = (y1,y2, ...,yn) ∈ Xn with gy1 � gx1,gx2 � gy2,gy3 � gx3, . . . ,gxn �
gyn.

Then F and g have an n-tupled coincidence point.



C-CLASS FUNCTIONS ON SHORTER PROOFS OF SOME EVEN-TUPLED COINCIDENCE THEOREMS 137

Proof. Consider the product space Y = Xn equipped with the metric D̃ (given by (B)) and the
partial order v (given by (A)). Then by Lemma 3.1, (Y,D̃,v) is an ordered complete metric space.
Also F and g induce mappings TF : Y → Y and Tg : Y → Y (defined in Lemma 3.2). Clearly,
• (i) implies that TF (Y ) ⊆ Tg(Y ),
• (ii) implies that TF is monotone Tg-nondecreasing (by item (1) of Lemma 3.2),
• (iii) implies that TF and Tg are compatible (by item (2) of Lemma 3.2),
• (iv) implies that Tg is continuous (by item (3) of Lemma 3.2),
• (v) implies that either TF is continuous (by item (4) of Lemma 3.2) or (Y,D̃,v) is nondecreasing
g-regular (by item (5) of Lemma 3.2),
• (vi) is equivalent to the condition: there exists U0 = (x10,x20, . . . ,xn0 ) ∈ Y such that Tg(U0) ⊆ TF (U0).

Now, in view of (vii), for given U,V ∈ Y such that Tg(U) v Tg(V ) implies that

(gx1,gx2, . . . ,gxn) v (gy1,gy2, . . . ,gyn).

It follows that for odd i,

(gxi,gxi+1, . . . ,gxn,gx1, . . . ,gxi−1) v (gyi,gyi+1, . . . ,gyn,gy1, . . . ,gyi−1), (3.19)

and for even i,

(gyi,gyi+1, . . . ,gyn,gy1, . . . ,gyi−1) v (gxi,gxi+1, . . . ,gxn,gx1, . . . ,gxi−1). (3.20)

If i is odd, then by using (3.19) and (vii), we get

d(F(xi,xi+1, . . . ,xn,x1,x2, . . . ,xi−1),F(yi,yi+1, . . . ,yn,y1,y2, . . . ,yi−1))

≤ ϕ(max{d(gxi,gyi),d(gxi+1,gyi+1), . . . ,d(gxn,gyn),d(gx1,gy1),

F(d(gx2,gy2), . . . ,d(gxi−1,gyi−1)}),ζ(max{d(gxi,gyi),d(gxi+1,gyi+1), . . . ,

d(gxn,gyn),d(gx1,gy1),d(gx2,gy2), . . . ,d(gxi−1,gyi−1)}))

= F(ϕ( max
1≤i≤n

d(gxi,gyi)),ζ( max
1≤i≤n

d(gxi,gyi))).

If i is even, then by using (3.20) and (vii), we get

d(F(xi,xi+1, . . . ,xn,x1,x2, . . . ,xi−1),F(yi,yi+1, . . . ,yn,y1,y2, . . . ,yi−1))

= d(F(yi,yi+1, . . . ,yn,y1,y2, . . . ,yi−1),F(xi,xi+1, . . . ,xn,x1,x2, . . . ,xi−1))

≤ ϕ(max{d(gyi,gxi),d(gyi+1,gxi+1), . . . ,d(gyn,gxn),d(gy1,gx1),

d(gy2,gx2), . . . ,d(gyi−1,gxi−1)}) − ζ(max{d(gyi,gxi),d(gyi+1,gxi+1), . . . ,

d(gyn,gxn),d(gy1,gx1),d(gy2,gx2), . . . ,d(gyi−1,gxi−1)})

= ϕ( max
1≤i≤n

d(gxi,gyi)) − ζ( max
1≤i≤n

d(gxi,gyi)).

Hence, in both the cases, for each i (1 ≤ i ≤ n), we have

d(F(xi,xi+1, . . . ,xn,x1,x2, . . . ,xi−1),F(yi,yi+1, . . . ,yn,y1,y2, . . . ,yi−1))

≤ ϕ( max
1≤i≤n

d(gxi,gyi)) − ζ( max
1≤i≤n

d(gxi,gyi)). (3.21)



138 SHARMA

Hence by using (3.21), we have

D̃(TF (U),TF (V ))

= max
1≤i≤n

d(F(xi,xi+1, . . . ,xn,x1,x2, . . . ,xi−1),F(yi,yi+1, . . . ,yn,y1,y2, . . . ,yi−1))

≤F( max
1≤i≤n

[ϕ( max
1≤i≤n

d(gxi,gyi)),ζ( max
1≤i≤n

d(gxi,gyi))])

= F(ϕ( max
1≤i≤n

d(gxi,gyi)),ζ( max
1≤i≤n

d(gxi,gyi)))

= F(ϕ(D̃(Tg(U),Tg(V )),ζ(D̃(Tg(U),Tg(V ))).

Thus all conditions of Lemma 3.3 are satisfied for ordered complete metric space (Y,D̃,v) and map-
pings TF : Y → Y and Tg : Y → Y. Therefore TF and Tg have a coincidence point in Y = Xn.
According to item (6) of Lemma 3.2, the mappings F and g have an n-tupled coincidence point.

Corollary 3.1. ([37]) Let (X,d,�) be an ordered complete metric space and F : Xn → X and
g : X → X be two mappings. Suppose that the following conditions are satisfied:
(i) F(Xn) ⊆ g(X),
(ii) F and g are compatible,
(iii) F has the mixed g-monotone property,
(iv) g is continuous,
(v) either F is continuous or X is g-regular,
(vi) there exist x10,x

2
0,x

3
0, ...,x

n
0 ∈ X such that (3.18) holds,

(vii) there exist ϕ ∈ Ω and ζ ∈= such that

ϕ(d(FU,FV )) ≤ ϕ( max
1≤i≤n

d(gxi,gyi)) − ζ( max
1≤i≤n

d(gxi,gyi)),

for all U = (x1,x2, ...,xn), V = (y1,y2, ...,yn) ∈ Xn with gy1 � gx1,gx2 � gy2,gy3 � gx3, . . . ,gxn �
gyn.

Then F and g have an n-tupled coincidence point.

Proof. It is sufficient to take F(s,t) = s− t in Theorem 3.1.

Corollary 3.2. Corollary 3.1 remains true if condition (vii) is replaced by the following:
(vii)’ there exist ϕ ∈ Ω such that

ϕ(d(FU,FV )) ≤ kϕ( max d(gxi,gyi)), 0 < k < 1,
for all U = (x1,x2, ...,xn), V = (y1,y2, ...,yn) ∈ Xn with gy1 � gx1,gx2 � gy2,gy3 � gx3, . . . ,gxn �
gyn.

Proof. It is sufficient to take F(s,t) = ks, 0 < k < 1 in Theorem 3.1.

Corollary 3.3. Corollary 3.1 remains true if condition (vii) is replaced by the following:
(vii)” there exist ϕ ∈ Ω and β : [0,∞) → [0, 1) which is semi-continuous such that

ϕ(d(FU,FV )) ≤ ϕ( max d(gxi,gyi))β(ϕ( max d(gxi,gyi)))
for all U = (x1,x2, ...,xn), V = (y1,y2, ...,yn) ∈ Xn with gy1 � gx1,gx2 � gy2,gy3 � gx3, . . . ,gxn �
gyn.

Proof. It is sufficient to take F(s,t) = sβ(s) (where β : [0,∞) → [0, 1) and semi-continuous) in
Theorem 3.1.

Corollary 3.4. Corollary 3.1 remains true if condition (vii) is replaced by the following:
(vii)”’ there exist ϕ ∈ Ω and φ : [0,∞) → [0,∞) which is an upper semi-continuous function such that



C-CLASS FUNCTIONS ON SHORTER PROOFS OF SOME EVEN-TUPLED COINCIDENCE THEOREMS 139

φ(0) = 0 and φ(t) < t for t > 0 such that

ϕ(d(FU,FV )) ≤ φ(ϕ( max d(gxi,gyi)))

for all U = (x1,x2, ...,xn), V = (y1,y2, ...,yn) ∈ Xn with gy1 � gx1,gx2 � gy2,gy3 � gx3, . . . ,gxn �
gyn.

Proof. It is sufficient to take F(s,t) = φ(s) (where φ : [0,∞) → [0,∞) is an upper semi-continuous
function such that φ(0) = 0 and φ(t) < t for t > 0) in Theorem 3.1.

Corollary 3.5. Let (X,d,�) be an ordered complete metric space and F : Xn → X be a mapping.
Suppose that the following conditions are satisfied:
(i) F has the mixed monotone property,
(ii) either F is continuous or X is regular,
(iii) there exist x10,x

2
0,x

3
0, ...,x

n
0 ∈ X such that



x10 � F(x10,x20,x30, ...,xn0 )
F(x20,x

3
0, ...,x

n
0 ,x

1
0) � x20

x30 � F(x30, ...,xn0 ,x10,x20)
...

F(xn0 ,x
1
0,x

2
0, ...,x

n−1
0 ) � x

n
0 ,

(iv) there exist ϕ ∈ Ω and ζ ∈=u and F a C-function such that

ϕ(d(FU,FV )) ≤F(ϕ( max d(xi,yi)),ζ(max d(xi,yi)))

for all U = (x1,x2, ...,xn), V = (y1,y2, ...,yn) ∈ Xn with x1 � y1,y2 � x2,x3 � y3, . . . ,yn � xn.

Then F has an n-tupled fixed point.

Proof. It is sufficient to take g = I (identity mapping) in Theorem 3.1.

Corollary 3.6. Corollary 3.5 remains true if condition (iv) is replaced by the following:
(iv)’ there exists ζ ∈=u such that

d(FU,FV ) ≤F( max
1≤i≤n

d(xi,yi),ζ( max
1≤i≤n

d(xi,yi))),

for all U = (x1,x2, ...,xn), V = (y1,y2, ...,yn) ∈ Xn with x1 � y1,y2 � x2,x3 � y3, . . . ,yn � xn.

Proof. It is sufficient to take ϕ and g to be identity mappings in Theorem 3.1.

Corollary 3.7. Corollary 3.1 remains true if condition (iv) is replaced by the following:
(iv)” there exists k ∈ (0, 1) such that

d(FU,FV ) ≤ k max
1≤i≤n

d(xi,yi),

for all U = (x1,x2, ...,xn), V = (y1,y2, ...,yn) ∈ Xn with x1 � y1,y2 � x2,x3 � y3, . . . ,yn � xn.

Proof. It is sufficient to take ϕ and g to be identity mappings and ζ(t) = (1 − k)t, k ∈ (0, 1) in
Theorem 3.1.

Now we shall prove the uniqueness of n-tupled fixed point.

Theorem 3.2. In addition to the hypotheses of Theorem 3.1, suppose that for real (x1,x2, ...,xn) and
(y1,y2, ...,yn) ∈ Xn there exists, (z1,z2, ...,zn) ∈ Xn such that (F(z1,z2, ...,zn),F(z2, ...,zn,z1), ...,F(zn,
z1, ...,zn−1)) is comparable to (F(x1,x2, ...,xn), F(x2, ...,xn,x1), ...,F(xn,x1, ...,xn−1)) and (F(y1,y2, ...,
yn),F(y2, ...,yn,y1), ...,F(yn,y1, ...,yn−1)). Then F and g have a unique n-tupled common fixed point.



140 SHARMA

Proof. Set U = (x1,x2, . . . ,xn), V = (y1,y2, . . . ,yn) and W = (z1,z2, . . . ,zn). Then we have

TF (W) v TF (U) or TF (U) v TF (W)

and

TF (W) v TF (V ) or TF (V ) v TF (W).
Hence by using Lemma 3.4, TF and Tg have a unique n-tupled common fixed point.

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Department of Mathematics and Statistics,Indian Institute of Technology, Kanpur 208 016, India

∗Corresponding author: annusharma241@gmail.com