EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 5, 2017, 1023-1034
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Some Coincidence and Fixed Point Results in Partially
Ordered Complete Generalized D∗-Metric Spaces

Alaa. M. F. AL. Jumaili

Department of Mathematics, University of Anbar, College of Education for Pure Sciences,Iraq

Abstract. In the present paper, several coincidence fixed point theorems established for mappings
satisfying contractive conditions related to a non-Decreasing φ-maps in partially ordered complete
generalized D∗-metric spaces where the cone is not necessarily normal which is the main result of
our article.

2010 Mathematics Subject Classifications: primary 47H10; secondary 54H25

Key Words and Phrases: Coincidence point (C.P), D∗-Metric Spaces, generalized D∗-Metric
Spaces

1. Introduction

The fixed point theorems in metric spaces are playing a fundamental role to construct
methods in mathematical sciences. So the metric fixed point theorems (F.P.Ths.) has
been researched extensively in the past two decades. The concept of cone metric spaces is
a generalization of metric spaces (M.SPS). The Banach fixed point theorems [5] provides
a technique for solving variety problems in mathematical science and engineering. In the
literature there are several generalizations of the Banach’s contraction principle, for some
of these generalizations of the Banach’s fixed point theorems and various contractive def-
initions that have been employed; we refer the readers to [1,6-8,11-14,17,21], and other
references listed in the reference section of this article.
Recently,(F.P.Th.) has developed rapidly in partially ordered metric spaces such as [17,
18], Ran and Reurings [23] and [22] studied several new facts for contractions in partially
ordered metric spaces. The authors in [15] generalized the conception of (M.SPS), substi-
tute the R by an ordered Banach spaces (B.S) and defined cone-Metric spaces.
B.C.Dhage, [9] in 1992, defined D-Metric spaces as a generalization of (M.SPS) and he
proved the existence of unique fixed point of a self-map satisfying a contractive condition
in complete and bounded D-Metric spaces. In 2007, S. Shaban, etal [24] have been estab-
lished the meaning of D∗-Metric spaces which as a probable modification of the definition
of (D-Metric) established via the author in [9], and proved several basic properties in

Email addresses: alaa−mf1970@yahoo.com (Alaa. M. F. AL. Jumaili)

http://www.ejpam.com 1023 c© 2017 EJPAM All rights reserved.



A. M. AL. Jumaili / Eur. J. Pure Appl. Math, 10 (5) (2017), 1023-1034 1024

D∗-Metric spaces. Afterwards, many authors [25,26,16] proved several (F.P.Ths.) in these
spaces. Fixed point problems have as well been considered partially ordered D∗-M.SPS, in
[4] Alaa.M. AL. Jumaili and Xiao-Song Yang, They used the meaning of D∗-Metric spaces
presented a new notion of the ∇∗-distance on a complete D∗-Metric spaces and estab-
lished several (F.P.Ths.) in partially ordered D∗-metric spaces. Recently, the authors in
[2] extension the concept of D∗-Metric spaces by changing R by a Real-(B.S) in D∗-Metric
spaces, they established several (F.P.Th.) under certain contractive conditions.
The motivation of this article is to study several coincidence (F.P.Ths.) for functions
satisfying contractive conditions concerning to a non-decreasing φ-maps [3,10] partially
ordered complete generalized D∗-M.SPS, where the cone is not necessarily normal.

2. Preliminaries

Assume E is real-(B.S) and P is proper sub set of E. P is called an order cone (O.C)
if:
a) P is closed, P 6= φ and P 6= {0},
b) ax+ by ∈ P ∀ x, y ∈ P and a, b ∈ R+,
c) x ∈ P and −x ∈ P implies x = 0.
For an (O.C) P ⊂ E, we define a partial ordering 4 on E with respect to P via x 4 y
iff y − x ∈ P . we shall using x ≺ y to indicate that x 4 y but x 6= y, while x � y for
y − x ∈ intP , where intP refer to the interior of P.
The (O.C) P is called normal if ∃ a number K > 0 (s.t) ∀x, y ∈ E, 0 ≤ x ≤ y ⇒,

‖x‖ ≤ K‖y‖ ..................................(2.1)

Or equivalently,

inf{‖x+ y‖ : x, y ∈ P and ‖x‖ = ‖y‖ = 1} > 0 ...................................(2.2)

We name the positive element K which satisfying (2.1) normal constant of P.
From (2.2) we can deduce that P is non-normal iff ∃ sequences {xs}, {ys} ∈ P (s.t),

0 ≤ {xs} ≤ {xs}+ {ys}, lims→∞({xs}+ {ys}) = 0, but lims→∞{xs} 6= 0.

In this paper, E stands for a real-(B.S), P is a cone in E with intP 6= {0} (such cones are
called solid) and 4 is a partial-ordering (P-O) with respect to P, where the cone is not
necessarily normal unless otherwise stated.

Now, recall several basic definitions and results of generalizedD∗-Metric spaces, and for
more details on D∗-Metric spaces and generalized D∗-Metric spaces, we refer the authors
for review [24] and [2] respectively.

Definition 1. [2] Let X be a non empty set. A generalized D∗-M.SP on a set X is a
function, D∗: X ×X ×X → E, that satisfies the following conditions ∀x, y, z, a ∈ X:

A. D∗(x, y, z) ≥ 0,



A. M. AL. Jumaili / Eur. J. Pure Appl. Math, 10 (5) (2017), 1023-1034 1025

B. D∗(x, y, z) = 0 ⇔ x = y = z,
C. D∗(x, y, z) = D∗(p{x, y, z}),(symmetry) where p is a permutation function,
D. D∗(x, y, z) ≤ D∗(x, y, a) +D∗(a, z, z) .

In that case D∗ is called a generalized D∗-Metric (D∗-Cone metric) and (X,D∗) is
called a generalized D∗-M.SP (D∗-Cone metric space).

Remark 1. It is obvious that the concept of a generalized D∗-M.SP (D∗-Cone metric
space) is more general than that of D∗-M.SP or Cone metric space. If E = R and P =
[0,+∞) after that a generalized D∗-M.SP becomes D∗-M.SP.

Example 1. Suppose E = R2, P = {(x, y) ∈ E : x, y ≥ 0}, X = R, defined a function,
D∗: X3 → E via :
D∗(x, y, z) = (|x− y|+ |y− z|+ |x− z|, α(|x− y|+ |y− z|+ |x− z|)) 3 α ≥ 0 is a constant
(see[2]). In that case (X,D∗) is a generalized D∗-M.SP (D∗-Cone metric space). over the
normal cone P.

Example 2. let E = C1
R[0, 1] with ‖u‖ = ‖u‖∞ + ‖u′‖∞ and P = {u ∈ E : u(t) ≥ 0 on

[0, 1]} (see, e.g., [27]). Let X = [0,+∞) and,
d(x, y) = |x− y|, g(x, y, z) = d(x, y) + d(y, z) + d(z, x) ∀x, y, z ∈ X, defined a function D∗:
X3 → P via D∗ (x, y, z) = g (x, y, z)u where u ∈ P is fixed. In that case (X,D∗) is a
generalized D∗-M.SP over the non-normal cone P.

Lemma 1. [2] Assume (X,D∗) is a generalized D∗-M.SP, in that case ∀x, y ∈ X, obtain
D∗(x, x, y) equal to D∗(x, y, y).

Remark 2. For the case of non-normal cones, the following remarks holed and useful in
the sequel for elements u, v, w ∈ P :
(R1)− If u ≤ v and v � w, in that case u� w.
(R2)− If u� v and v ≤ w, in that case u� w.
(R3)− If 0 ≤ u� d ∀ d ∈ intP, in that case u = 0.

Definition 2. [2] suppose (X,D∗) is a generalized D∗-M.SP in that case:
a) A sequence{xs} in X is called Cauchy sequence if ∀ d belong to E with 0 � d, there
exist H (s.t) ∀ r, s, l ≥ H, D∗(xr, xs, xl)� d.
b) If each Cauchy sequence is convergent in X, in that case X is called complete generalized
D∗-Metric.
c) A sequence{xs} → x ∈ X, if ∀ d ∈ E with 0 � d there exist H (s.t) ∀ r, s ≥ H,
D∗(xr, xs, x)� d, and x is the limit point of {xs} with indicate via xs → x, as (s→∞).

Proposition 1. [2] Assume (X,D∗) is a generalized D∗-M.SP in X, If xs → x, in that
case {xs} is a Cauchy sequence.

Proposition 2. [2] Assume (X,D∗) is a generalized D∗-M.SP, and P is normal cone with
normal constant K. Suppose {xs} in X, in that case {xs} converges to x⇔ D∗(xr, xs, x)→
0, as (r, s→∞).



A. M. AL. Jumaili / Eur. J. Pure Appl. Math, 10 (5) (2017), 1023-1034 1026

Proposition 3. [2] Assume (X,D∗) is a generalized D∗-M.SP, and P is normal cone.
Suppose {xs} in X and x ∈ X. in that case the following equivalent:
a) {xs} is D∗-convergent to x;
b) D∗(xs, xs, x) convergent to 0, when (s→∞);
c) D∗(xs, x, x) convergent to 0, when (s→∞).

Definition 3. Suppose X 6= φ, in that case (X,D∗,4) is said to be an ordered generalized
D∗-M.SP if the following hold:
a) (X,D∗) is a generalized D∗-M.SP,
b) (X,4) is a partially ordered set (P.O.S).

Recall that if (X,4) is a (P.O.S), in that case x, y ∈ X are said to be comparable [19]
when x 4 y or y 4 x satisfies.
Also Nashine and Samet, in [20] introduced the following concept:
Let X 6= φ and let £ : X → X, ∀x ∈ X, we denoted via £−1(x) the sub set of X given
through £−1(x) := {u ∈ X : £u = x}.

Definition 4. [19] Assume (X,4) is a (P.O.S) and let T,G,£ : X → X be given mappings
(s.t) TX ⊆ £X and GX ⊆ £X. Describe that G and T are weakly increasing with respect
to £ if for each x ∈ X, we obtain:
Tx 4 Gy,∀ y ∈ £−1(Tx) and Gx 4 Ty,∀ y ∈ £−1(Gx). If T = G, we say that T is weakly
increasing with related to £.

Remark 3. If £ : X → X is the identity mapping (£x = x ∀x ∈ X), in that case G
and T are weakly increasing with respect to £⇔ G and T are weakly increasing mappings
[19], i.e., Tx 4 G(Tx) and Gx 4 T (Gx) hold ∀x ∈ X.

Definition 5. Let (X,4) be an ordered generalized D∗-M.SP, X is called regular [19] if
the next condition holds:
if {zs} is a non-Decreasing sequence in (X,4) (s.t) zs → z ∈ X, when (s → ∞), in that
case zs 4 z, for each s belong to N.

3. Coincidence Fixed Point Theorems in Partially Ordered Complete
Generalized D∗-M.SPS

In this section, we establish several coincidence (F.P.Ths.) in partially ordered com-
plete generalized D∗-M.SPS. We start with the following definition (φ-maps).

Definition 6. [3, 10]. Let P be (O.C). A non-Decreasing function φ : P → P is called an
φ-maps if:
a) φ(0) = 0 and 0 < φ(w) < w for w ∈ P\{0},
b) w ∈ intP implies w − φ(w) ∈ intP,
c) If w ∈ P\{0} and d ∈ intP , So ∃ s0 ∈ N (s.t) φs(w)� d for each s ≥ s0.

Example 3 (3). (a) - If P is an arbitrary cone in (B.S) E and δ ∈ (0, 1), in that case
φ : P → P , defined by φ(w) = δw for w ∈ P , is a φ-maps.



A. M. AL. Jumaili / Eur. J. Pure Appl. Math, 10 (5) (2017), 1023-1034 1027

(b) - let Ψ : [0,+∞) → [0,+∞) be any real valued φ-map and let P be a cone in (B.S) E
and δ ∈ (0, 1), be fixed. In that case the function φδ : P → P defined by: φδ(w) = Ψ(δ)w,
is a φ-maps. Examples of this type are of particular interest in the case when the cone P
is non-normal. (See Example 2), one can take,

E = C1
R[0, 1], P = {x ∈ E : x(t) ≥ 0 on [0, 1]} as well Ψ(δ) =

δ

1 + δ
; δ ∈ (0, 1).

The following theorem is our first main results.

Theorem 1. suppose (X,4) is (P.O.S) with assume (X,D∗) is a generalized D∗-M.SP
and P is (O.C) with normal cone K.
let £, T : X → X be two maps (s.t),
D∗(Tx, Ty, Tz) ≤ φ(D∗(£x,£y,£z)) ........................................................... (3.1)
And assume the following:
a) T is weakly increasing with respect to £,
b) £X is a complete sub-space of X,
c) X is regular.
For each x, y, z ∈ X with £z 4 £y 4 £x where φ is a φ-map. In that case £ and T have
a coincidence point.

Proof. Assume that a point x0 ∈ X is arbitrary. By definition (4) we have TX ⊆ £X,
thus we can construct a sequence {xs} in X via: £xs+1 = Txs, ∀s ∈ N0.
Since T is weakly increasing with respect to £ and x1 ∈ £−1(Tx0) and x2 ∈ £−1(Tx1),
in that case we get: £x1 4 £x2 4 £x3 4 ......... 4 £xs 4 £xs+1 4 .........

Now establish that {£xs} is a Cauchy sequence in (£(X), D∗). We will discuss two
cases:
(a)- There exists s ∈ N (s.t) £xs = £xs+1. Using the considered contractive condition,
get:
Txs = Txs+1, that is, £xs+1 = £xs+2. Therefore we have £xr = £xs, ∀ r ≥ s ⇒ {£xs}
is a Cauchy sequence in (£(X), D∗).
(b) - The successive conditions of a sequence {£xs} are different. From the above inequal-
ity (3.1), we obtain:
D∗(£xs,£xs,£xs+1)
≤ D∗(Txs−1, Txs−1, Txs)
≤ φ(D∗(£xs−1,£xs−1,£xs))
≤ φ2(D∗(£xs−2,£xs−2,£xs−1)).......... ≤ φs(D∗(£x0,£x0,£x1)).
Fix d, 0� d. By means of the characteristic (c) of definition (6), ∃ s0 ∈ N (s.t),
φs(D∗(£x0,£x0,£x1))� d ∀ s ≥ s0.
According the Remark (2−R1), we get:
D∗(£xs,£xs,£xs+1)� d ∀ s ≥ s0. In the same method choose a natural number s1 ∈ N
(s.t): D∗(£xr,£xr,£xr+1) < d− φ(d) ∀ r ≥ s1 ............................................(3.2)



A. M. AL. Jumaili / Eur. J. Pure Appl. Math, 10 (5) (2017), 1023-1034 1028

Now via induction on r we will prove the following claim:
D∗(£xs,£xs,£xr)� d ∀ r > s ≥ s1 .............................................................(3.3)

By using the inequality (3.2) and the truth that d− φ(d) < d, we have the inequality
(3.3) holds for r = s+ 1. Assume that the inequality (3.3) holds for r = h.
For r = h+ 1 and by using Remark (2) we obtain:
D∗(£xs,£xs,£xh+1)
≤ D∗(£xs,£xs,£xs+1) +D∗(£xs+1,£xs+1,£xh+1)
� d− φ(d) + φ(D∗(£xs,£xs,£xh))
� d− φ(d) + φ(d) = d.
By induction on r, we conclude that the inequality (3.3) holds ∀ r > s ≥ s1. Now the part
(D) of definition (1) implies that,
D∗(xr, xs, xl) ≤ D∗(xr, xr, xs) +D∗(xs, xl, xl)� 2d holds for r, s, l ≥ s1.
Therefore, {£xs} is (D∗−Cauchysequence) in (£(X), D∗) which is complete by assump-
tion. Thus, ∃ u = £v and z ∈ X (s.t),
lims→∞£xs = u = £z......................................................................(3.4)
Since X is regular and {£xs} is a non-Decreasing sequence, we get from (3.4) that
£xs 4 £z ∀ s ∈ N. Assume that £xs 6= £z. Fix d, 0� d, choose, s ∈ N (s.t),

D∗(£xs,£z,£z)�
d

2
and D∗(£xs+1,£z,£z)�

d

2
.

Therefore, we can apply the considered contractive condition to obtain:
D∗(Tz, Tz,£z)
≤ D∗(Tz, Tz, Txs) +D∗(Txs,£z,£z)
≤ φ(D∗(£xs,£z,£z)) +D∗(£xs+1,£z,£z) (Using (3.1))
< D∗(£xs,£z,£z) +D∗(£xs+1,£z,£z)

� d

2
+
d

d
= d.

Since d ∈ intP, via Remark (2−R1) it follows that D∗(Tz, Tz,£z) = 0 such that Tz = £z.
Thus z is a coincidence point for £ and T.

Next, remember the following case see [19] to explain the validity of Theorem (1).

Example 4. Assume that (X,D∗) is a generalized D∗-M.SP. Consider Example (2), with
the reverse order: x 4 y ⇔ x ≥ y. Define a maps T : X ×X → X and £ : X ×X → X

as follows: Tx = 2x and £x = 3x and a φ-map define by φ(w) =
w

2
, w ∈ P. Hence

all the conditions of Theorem (1) are satisfied; particularly one can be able to reduce
condition (3.1) to: 2(|x − y| + |y − z| + |z − x|)u ≥ 3

2(|x − y| + |y − z| + |z − x|)u,
and holds ∀ x, y, z ∈ [0,+∞). As well, T is weakly increasing with respect to £, since
£y = Tx ⇒ 3y = 2x ⇒ y = 2x

3 , which mean implies Tx = 2x ≥ 2y = Ty, it mean
Tx 4 Ty. Clear that, 0 a coincidence point of £ and T.

Corollary 1. Assume (X,4) is (P.O.S) with suppose that (X,D∗) is a generalized D∗-
M.SP and P is (O.C). Let £, T : X → X be two non-Decreasing maps. Assume that ∀
x, y, z ∈ X with z 4 y 4 x ∃ some h ∈ [0, 1) (s.t): D∗(Tx, Ty, Tz) ≤ hD∗(£x,£y,£z)
holds. Assume the following:



A. M. AL. Jumaili / Eur. J. Pure Appl. Math, 10 (5) (2017), 1023-1034 1029

a) T is weakly increasing with respect to £,
b) £X is a complete sub-space of X,
c) X is regular.
In that case £ and T have a coincidence point.

Proof. The proof is directly consequence from Theorem (1) when taking φ(w) = hw.

Remark 4. if the mapping £ : X → X is identity, we get the following (F.P ) result.

Corollary 2. Assume (X,4) is (P.O.S) and suppose that (X,D∗) is a complete gener-
alized D∗-M.SP and P is (O.C). Let T : X → X be a mapping (s.t) D∗(Tx, Ty, Tz) ≤
φ(D∗(x, y, z)) holds ∀ x, y, z ∈ X with z 4 y 4 x where φ is a φ-map. Assume the follow-
ing:
a) Tx 4 T (Tx) ∀ x ∈ X,
b) X is regular.
In that case T has a (F.P ).

Next, our result is the following generalization of Theorem (1).

Theorem 2. Assume (X,4) is (P.O.S) and suppose that (X,D∗) is a complete general-
ized D∗-M.SP and P is (O.C), and Let £, T : X → X be a non-Decreasing maps. Assume
that ∀ x, y, z ∈ X with £z 4 £y 4 £x there exists,
Φ(x, y, z) ∈ {D∗(£x,£y,£z), D∗(£x,£x, Tx), D∗(£y,£y, Ty), D∗(Tx,£y,£z)} (s.t):
D∗(Tx, Ty, Tz) ≤ φ(Φ(x, y, z)) where φ is a φ-map. Assume the following:
a) T is weakly increasing with respect to £,
b) X is regular.
In that case £ and T have a coincidence point.

Proof. Assume that a point x0 ∈ X is arbitrary. By definition (4) we have, TX ⊆ £X,
thus construct a sequence {xs} in X defined via:
£xs+1 = Txs, for each s ∈ N . since x1 ∈ £−1(Tx0) and x2 ∈ £−1(Tx1), and via T is
weakly increasing with respect to £, we get that:
£x1 = Tx0 4 Tx1 = £x2 4 Tx2 = £x3. Continuing this process, we obtain that:
£x1 4 £x2 4 £x3 4 ......... 4 £xs 4 £xs+1 4 .........
Assume that ∃ s0 ∈ {1, 2, 3, .....} (s.t) Φ(xs0 , xs0 , xs0−1) = 0 thus it is clear that £xs0−1 =
£xs0 = Txs0−1 therefore we are completed.
Next we can assume Φ(xs, xs, xs−1) > 0 ∀ s ≥ 1.
Assume £xs−1 6= £xs ∀ s ∈ N therefore for s ∈ N we obtain:
D∗(£xs,£xs,£xs+1) = D∗(Txs−1, Txs−1, Txs) ≤ φ(Φ(xs−1, xs−1, xs)) where,
Φ(xs−1, xs−1, xs) ∈ {D∗(£xs−1,£xs−1,£xs), D∗(£xs−1,£xs−1, Txs−1),
D∗(£xs,£xs, Txs), D

∗(Txs−1, Txs−1,£xs)} = {D∗(£xs−1,£xs−1,£xs),
D∗(£xs−1,£xs−1,£xs), D

∗(£xs,£xs,£xs+1), D
∗(£xs,£xs,£xs)}

= {D∗(£xs−1,£xs−1,£xs), D∗(£xs,£xs,£xs+1), 0}.



A. M. AL. Jumaili / Eur. J. Pure Appl. Math, 10 (5) (2017), 1023-1034 1030

I. If Φ(xs−1, xs−1, xs) = D∗(£xs,£xs,£xs+1) therefore, D∗(£xs,£xs,£xs+1) ≤ φ(D∗(£xs,£xs,£xs+1)),
and via the characteristic of φ-map we get:
D∗(£xs,£xs,£xs+1) < D∗(£xs,£xs,£xs+1) which is impossible.

II. If Φ(xs−1, xs−1, xs) = 0, hence D∗(£xs,£xs,£xs+1) ≤ φ(0) < 0 which is a contra-
diction. So, Φ(xs−1, xs−1, xs) = D∗(£xs−1,£xs−1,£xs), thus
D∗(£xs,£xs,£xs+1) ≤ Φ(D∗(£xs−1,£xs−1,£xs)) after that for s ∈ N , we obtain:
D∗(£xs,£xs,£xs+1)
= D∗(Txs−1, Txs−1, Txs)
≤ φ(D∗(£xs−1,£xs−1,£xs))
≤ φ2(D∗(£xs−2,£xs−2,£xs−1)).......... ≤ φs(D∗(£x0,£x0,£x1))

We can show that {£xs} is a Cauchy sequence by similar method to that in the evi-
dence of Theorem (1). Since X is D∗-complete, so {£xs} is convergent to a point u in X.
Now we explain that £u = Tu.
Since {£xs} non-Decreasing sequence and £xs → u, therefore by regularity of X we have
£xs 4 u ∀ s. if £xs = u for some u, hence, by construction we obtain, £xs+1 = u and u
is (F.P ). So we presume that £xs 6= u, thus for s ∈ N we obtain:
D∗(£u, Tu, Tu)
≤ D∗(£u,£xs,£xs) +D∗(£xs, Tu, Tu)
= D∗(£u,£xs,£xs) +D∗(Txs−1, Tu, Tu)
≤ D∗(£u,£xs,£xs) + φ(Φ(xs−1, u, u)) where,
Φ(xs−1, u, u) ∈ {D∗(£xs−1,£u,£u), D∗(£xs−1,£xs−1, Txs−1),
D∗(£xs−1,£xs−1, Txs−1), D

∗(Txs−1,£xs−1,£u)} =
{D∗(£xs−1,£u,£u), D∗(£xs−1,£xs−1,£xs), D

∗(£xs,£xs−1,£u)}

Fix d, 0� d, and choose N1 ∈ N (s.t),

D∗(£u,£xs,£xs)�
d

2
and D∗(£xs−1,£u,£u)� d

2
,∀ s ≥ N1.

We can discuss three cases as following:

A. If Φ(xs−1, u, u) = D∗(£xs−1,£u,£u), therefore we have:
D∗(£u, Tu, Tu)
≤ D∗(£u,£xs,£xs) + φ(D∗(£xs−1,£u,£u))
< D∗(£u,£xs,£xs) +D∗(£xs−1,£u,£u)

� d

2
+
d

2
= d.

B. If Φ(xs−1, u, u) = D∗(£xs−1,£xs−1,£xs), after that we obtain:
D∗(£u, Tu, Tu)
≤ D∗(£u,£xs,£xs) + φ(D∗(£xs−1,£xs−1,£xs))
< D∗(£u,£xs,£xs) +D∗(£xs−1,£xs−1,£u)

� d

2
+
d

2
= d.



A. M. AL. Jumaili / Eur. J. Pure Appl. Math, 10 (5) (2017), 1023-1034 1031

C. If Φ(xs−1, u, u) = D∗(£xs,£xs−1,£u), thus we get:
D∗(£u, Tu, Tu)
≤ D∗(£u,£xs,£xs) + φ(D∗(£xs,£xs−1,£u))
< D∗(£u,£xs,£xs) +D∗(£xs,£xs−1,£u)
≤ D∗(£u,£xs,£xs) +D∗(£xs,£xs,£xs−1) +D∗(£xs−1,£u,£u)� d.
Whenever s ∈ N , Therefore in all above cases we have:
D∗(£u, Tu, Tu)� d for arbitrary d ∈ intP. According the Remark (2−R3),
it follows that D∗(£u, Tu, Tu) = 0, which implies that £u = Tu.
Thus we conclude that u is a coincidence point for £ and T.

Corollary 3. Assume (X,4) is (P.O.S) and suppose that (X,D∗) is a complete general-
ized D∗-M.SP and P is (O.C). Let £, T : X → X be non-Decreasing mappings. Assume
that for some h ∈ [0, 1), and ∀ x, y, z ∈ X with Fz 4 £y 4 £x, ∃
Φ(x, y, z) ∈ {D∗(£x,£y,£z), D∗(£x,£x, Tx), D∗(£y,£y, Ty), D∗(Tx,£y,£z)}
(s.t): D∗(Tx, Ty, Tz) ≤ hΦ(x, y, z). We assume the following:
a) T is weakly increasing with respect to £,
b) X is regular.
In that case £ and T have a coincidence point.

Proof. The proof is direct result from Theorem (2).

Remark 5. if the mapping £ : X → X is identity, so from Theorem (2) we get easily the
following (F.P ) result.

Corollary 4. Assume (X,4) is (P.O.S) and suppose that (X,D∗) is a complete general-
ized D∗-M.SP and P is (O.C). Let £, T : X → X be non-Decreasing mappings, (s.t):
D∗(Tx, Ty, Tz) ≤ φ(Φ(x, y, z)) where
Φ(x, y, z) ∈ {D∗(x, y, z), D∗(x, x, Tx), D∗(y, y, Ty), D∗(Tx, y, z)} and ∀ x, y, z ∈ X with
z 4 y 4 x, where φ is a φ-map. Assume the following:
a) Tx 4 T (Tx) ∀ x ∈ X,
b) X is regular.
In that case T has (F.P ).

Next, we present a sufficient condition for the unique-ness of the point of coincidence
in the following result.

Theorem 3. Assume that (X,4) is a totally ordered set and (X,D∗) is a complete gen-
eralized D∗-M.SP, P is (O.C). Let £, T : X → X be non-Decreasing mappings. Assume
that ∀ x, y, z ∈ X with £z 4 £y 4 £x ∃,
Φ(x, y, z) ∈ {D∗(£x,£y,£z), D∗(£x,£x, Tx), D∗(£y,£y, Ty), D∗(Tx,£y,£z)}
(s.t): D∗(Tx, Ty, Tz) ≤ φ(Φ(x, y, z)) where φ is a φ-map. Assume the following:
a) T is weakly increasing with respect to £,
b) X is regular.
In that case £ and T have a unique coincidence point.



REFERENCES 1032

Proof. Assume that £ and T have two points of coincidence, u and w, (s.t) £u = Tu
and £w = Tw,£u 6= £w. As (X,4) is a totally ordered set and u,w ∈ X, assume that
u ≺ w. by means of the contractive condition we have that:
D∗(£u,£w,£w) = D∗(Tu,£w,£w) ≤ φ(Φ(u,w,w)) holds for some,
Φ(u,w,w) ∈ {D∗(£u,£w,£w), D∗(£u,£u, Tu), D∗(£u,£u, Tu), D∗(Tu,£u,£w)}
= {0, D∗(£u,£w,£w)}
By means of characteristic of φ-mapping we get a contradiction. Therefore £u = £w.
Thus £ and T have a unique point of coincidence £u = Tu.

Acknowledgements

I would like to express my sincere gratitude to the referees for their valuable suggestions
and comments which improved the paper.

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