Common fixed-point theorems and c-distance in ordered cone metric spaces
We present a generalization of several fixed and common fixed point theorems on c -distance in ordered cone metric spaces. In this way, we improve and generalize various results existing in the literature.
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Інститут математики НАН України
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irk-123456789-1657222020-02-17T01:26:06Z Common fixed-point theorems and c-distance in ordered cone metric spaces Rahimi, H. Soleimani Rad, G. Статті We present a generalization of several fixed and common fixed point theorems on c -distance in ordered cone metric spaces. In this way, we improve and generalize various results existing in the literature. Наведено узагальнення деяких теорем про нерухому точку та спільну нерухому точку для с-відстані в упорядкованих конічних метричних просторах. Таким чином, покращено та узагальнено різноманітні результати, що наведені в літературі. 2013 Article Common fixed-point theorems and c-distance in ordered cone metric spaces / H. Rahimi, G. Soleimani Rad // Український математичний журнал. — 2013. — Т. 65, № 12. — С. 1667–1680. — Бібліогр.: 26 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165722 515.12 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Rahimi, H. Soleimani Rad, G. Common fixed-point theorems and c-distance in ordered cone metric spaces Український математичний журнал |
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We present a generalization of several fixed and common fixed point theorems on c -distance in ordered cone metric spaces. In this way, we improve and generalize various results existing in the literature. |
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Article |
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Rahimi, H. Soleimani Rad, G. |
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Rahimi, H. Soleimani Rad, G. |
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Rahimi, H. |
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Common fixed-point theorems and c-distance in ordered cone metric spaces |
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Common fixed-point theorems and c-distance in ordered cone metric spaces |
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Common fixed-point theorems and c-distance in ordered cone metric spaces |
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Common fixed-point theorems and c-distance in ordered cone metric spaces |
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Common fixed-point theorems and c-distance in ordered cone metric spaces |
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common fixed-point theorems and c-distance in ordered cone metric spaces |
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Інститут математики НАН України |
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2013 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165722 |
| citation_txt |
Common fixed-point theorems and c-distance in ordered cone metric spaces / H. Rahimi, G. Soleimani Rad // Український математичний журнал. — 2013. — Т. 65, № 12. — С. 1667–1680. — Бібліогр.: 26 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT rahimih commonfixedpointtheoremsandcdistanceinorderedconemetricspaces AT soleimaniradg commonfixedpointtheoremsandcdistanceinorderedconemetricspaces |
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2025-07-14T19:41:47Z |
| last_indexed |
2025-07-14T19:41:47Z |
| _version_ |
1837652636650700800 |
| fulltext |
UDC 515.12
H. Rahimi (Islamic Azad. Univ., Central Tehran Branch, Iran),
G. Soleimani Rad (Young Researchers and Elite club, Islamic Azad. Univ., Central Tehran Branch, Iran)
COMMON FIXED POINT THEOREMS AND C-DISTANCE
IN ORDERED CONE METRIC SPACES
ТЕОРЕМИ ПРО СПIЛЬНУ НЕРУХОМУ ТОЧКУ ТА C-ВIДСТАНЬ
В УПОРЯДКОВАНИХ КОНIЧНИХ МЕТРИЧНИХ ПРОСТОРАХ
We present a generalization of several fixed and common fixed point theorems on the c-distance in ordered cone metric
spaces. In this way, we improve and generalize various results existing in the literature.
Наведено узагальнення деяких теорем про нерухому точку та спiльну нерухому точку для c-вiдстанi в упорядкованих
конiчних метричних просторах. Таким чином, покращено та узагальнено рiзноманiтнi результати, що наведенi в
лiтературi.
1. Introduction. Huang and Zhang [18] have introduced the concept of a cone metric space by
replacing the set of real numbers by an ordered Banach space and have showed some fixed point
theorems of contractive type mappings on cone metric spaces. Afterward, several fixed and common
fixed point results in cone metric spaces with related results have been introduced in [2, 4, 5, 8, 10,
14, 16, 17, 20] and the references contained therein. Also, the existence of fixed points in partially
ordered cone metric spaces has been studied in [6, 7, 24].
In 1996, Kada et al. [21] defined the concept of w-distance in complete metric spaces. Later,
many authors proved some fixed point theorems in complete metric spaces (see [3, 22]). Recently,
Saadati et al. [23] introduced a probabilistic version of the w-distance in a Menger probabilistic
metric space. In the sequel, Cho et al. [9] and Wang and Guo [26] defined a concept of the c-distance
in a cone metric space, which is a cone version of the w-distance of Kada et al. [21] and proved
some fixed point theorems in ordered cone metric spaces. Then Sintunavarat et al. [25] generalized
the Banach contraction theorem on c-distance of Cho et al. [9]. Also, Dordević et al. [12] proved
some fixed point and common fixed point theorems under c-distance for contractive mappings in
tvs-cone metric spaces.
The purpose of this work is to extend and generalize the main results of Cho et al. [9], Sintu-
navarat et al. [25], Huang and Zhang [18] on c-distance in ordered cone metric spaces.
2. Preliminaries.
Definition 2.1 (see [11, 18]). Let E be a real Banach space and let 0 denote the zero element in
E. A subset P of E is called a cone if the following conditions hold:
(C1) P is nonempty closed and P 6= {0};
(C2) a, b ∈ R, a, b ≥ 0 and x, y ∈ P imply that ax+ by ∈ P ;
(C3) if x ∈ P and −x ∈ P, then x = 0.
Given a cone P ⊂ E, we define a partial ordering � with respect to P by x � y ⇐⇒ y−x ∈ P.
We write x ≺ y if x � y and x 6= y. Also, we write x� y if and only if y − x ∈ intP, where int P
is interior of P. If intP 6= ∅, the cone P is called solid. The cone P is called normal if there exists
a number k > 0 such that, for all x, y ∈ E,
0 � x � y =⇒ ‖x‖ ≤ k‖y‖.
c© H. RAHIMI, G. SOLEIMANI RAD, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12 1667
1668 H. RAHIMI, G. SOLEIMANI RAD
The least positive number satisfying the above is called the normal constant of P.
Definition 2.2 (see [18]). Let X be a nonempty set and E be a real Banach space equipped with
the partial ordering � with respect to the cone P ⊂ E. Suppose that a mapping d : X × X → E
satisfies the following conditions:
(CM1) 0 � d(x, y) for all x, y ∈ X and d(x, y) = 0 if and only if x = y;
(CM2) d(x, y) = d(y, x) for all x, y ∈ X;
(CM3) d(x, y) � d(x, z) + d(z, y) for all x, y, z ∈ X.
Then d is called a cone metric on X and (X, d) is called a cone metric space.
Definition 2.3 (see [18]). Let (X, d) be a cone metric space, let {xn} be a sequence in X and
x ∈ X.
(1) {xn} is said to be convergent to x if, for any c ∈ E with 0� c, there exists n0 ≥ 1 such that
d(xn, x)� c for all n > n0 and we write limn→∞ d(xn, x) = 0.
(2) {xn} is called a Cauchy sequence if, for any c ∈ E with 0� c, there exists n0 ≥ 1 such that
d(xn, xm)� c for all m,n > n0 and we write limn,m→∞ d(xn, xm) = 0.
(3) If every Cauchy sequence in X is convergent, then X is called a complete cone metric space.
Lemma 2.1 (see [18]). Let (X, d) be a cone metric space and P be a normal cone with normal
constant k. Also, let {xn} and {yn} be sequences in X and x, y ∈ X. Then the following hold:
(1) {xn} converges to x if and only if d(xn, x)→ 0 as n→∞.
(2) If {xn} converges to x and {xn} converges to y, then x = y.
(3) If {xn} converges to x, then {xn} is a Cauchy sequence.
(4) If xn → x and yn → y as n→∞, then d(xn, yn)→ d(x, y) as n→∞.
(5) {xn} is a Cauchy sequence if and only if d(xn, xm)→ 0 as n,m→∞.
Lemma 2.2 (see [6, 19]). Let E be a real Banach space with a cone P in E. Then, for all
u, v, w, c ∈ E, the following hold:
(1) If u � v and v � w, then u� w.
(2) If 0 � u� c for all c ∈ int P, then u = 0.
(3) If u � λu where u ∈ P and 0 < λ < 1, then u = 0.
(4) Let xn → 0 in E, 0 � xn and 0� c. Then there exists a positive integer n0 such that xn � c
for each n > n0.
(5) If 0 � u � v and k is a nonnegative real number, then 0 � ku � kv.
(6) If 0 � un � vn for all n ≥ 1 and un → u, vn → v as n→∞, then 0 � u � v.
Definition 2.4 (see [9, 26]). Let (X, d) be a cone metric space. A mapping q : X ×X → E is
called a c-distance on X if the following are satisfied:
(CD1) 0 � q(x, y) for all x, y ∈ X;
(CD2) q(x, z) � q(x, y) + q(y, z) for all x, y, z ∈ X;
(CD3) for all n ≥ 1 and x ∈ X, if q(x, yn) � u for some u = ux, then q(x, y) � u whenever
{yn} is a sequence in X converging to a point y ∈ X;
(CD4) for all c ∈ E with 0 � c, there exists e ∈ E with 0 � e such that q(z, x) � e and
q(z, y)� e imply d(x, y)� c.
Remark 2.1 (see [9]). Each w-distance q in a metric space (X, d) is a c-distance with E = R+
and P = [0,∞). But the converse does not hold. Thus the c-distance is a generalization of the
w-distance.
Example 2.1 (see [9, 26]). (1) Let E = C1
R[0, 1] with the norm ‖x‖ = ‖x‖∞ + ‖x′‖∞ and
consider the cone P = {x ∈ E : x(t) ≥ 0 on [0, 1]}. Also, let X = [0,∞) and define a mapping
d : X ×X → E by d(x, y) = |x− y|ψ for all x, y ∈ X, where ψ : [0, 1] → R such that ψ(t) = 2t.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
COMMON FIXED POINT THEOREMS AND C-DISTANCE IN ORDERED CONE METRIC SPACES 1669
Then (X, d) is a cone metric space. Define a mapping q : X ×X → E by q(x, y) = (x+ y)ψ for all
x, y ∈ X. Then q is c-distance.
(2) Let (X, d) be a cone metric space and P be a normal cone. Put q(x, y) = d(w, y) for all
x, y ∈ X, where w ∈ X is a fixed point. Then q is a c-distance.
(3) Let (X, d) be a cone metric space and P be a normal cone. Define q(x, y) = d(x, y) for all
x, y ∈ X. Then q is a c-distance.
(4) Let E = R, P = {x ∈ E : x ≥ 0} and X = [0,∞). Define a mapping d : X × X → E
by d(x, y) = |x − y| for all x, y ∈ X. Then (X, d) is a cone metric space. Define a mapping
q : X ×X → E by q(x, y) = y for all x, y ∈ X. Then q is a c-distance.
Remark 2.2 (see [9, 26]). From (2) and (4) in Example 2.1, we have two important results:
(1) For any c-distance q, q(x, y) = 0 is not necessarily equivalent to x = y for all x, y ∈ X.
(2) For any c-distance q, q(x, y) = q(y, x) does not necessarily hold for all x, y ∈ X.
Lemma 2.3 (see [9, 25, 26]). Let (X, d) be a cone metric space and q be a c-distance on X.
Also, let {xn} and {yn} be sequences in X and x, y, z ∈ X. Suppose that {un} and {vn} are two
sequences in P converging to 0. Then the following hold:
(1) If q(xn, y) � un and q(xn, z) � vn for n ≥ 1, then y = z.
(2) If q(xn, yn) � un and q(xn, z) � vn for each n ≥ 1, then {yn} converges to z.
(3) If q(xn, xm) � un for all m > n, then {xn} is a Cauchy sequence in X.
(4) If q(y, xn) � un for each n ≥ 1, then {xn} is a Cauchy sequence in X.
Definition 2.5 (see [6, 9]). Let (X,v) be a partially ordered set. Two mappings f, g : X → X
are said to be weakly increasing if fx v gfx and gx v fgx hold for all x ∈ X.
3. Main results. Our first result is the following theorem of Hardy – Rogers type (see [15]) for
any c-distance in a cone metric space without normality condition of cone.
Theorem 3.1. Let (X,v) be a partially ordered set and (X, d) be a complete cone metric space.
Suppose that there exist mappings αi : X → [0, 1) such that the following condition hold:
αi(fx) ≤ αi(x)
for all x ∈ X and i = 1, 2, . . . , 5. Also, let q be a c-distance on X and f : X → X be a continuous
and nondecreasing mapping with respect to v satisfying the following conditions:
q(fx, fy) � α1(x)q(x, y) + α2(x)q(x, fx) + α3(x)q(y, fy) + α4(x)q(x, fy) + α5(x)q(y, fx),
(3.1)
q(fy, fx) � α1(x)q(y, x) + α2(x)q(fx, x) + α3(x)q(fy, y) + α4(x)q(fy, x) + α5(x)q(fx, y)
(3.2)
for all comparable x, y ∈ X such that
(α1 + α2 + α3 + 2α4 + 2α5)(x) < 1. (3.3)
If there exists x0 ∈ X such that x0 v fx0, then f has a fixed point. Moreover, if fz = z, then
q(z, z) = 0.
Proof. If fx0 = x0, then x0 is a fixed point of f and the proof is finished. Now, suppose that
fx0 6= x0. Since f is nondecreasing with respect to v and x0 v fx0, we obtain by induction that
x0 v fx0 v f2x0 v . . . v fnx0 v fn+1x0 v . . . ,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
1670 H. RAHIMI, G. SOLEIMANI RAD
where xn = fxn−1 = fnx0. Now, setting x = xn and y = xn−1 in (3.1), we have
q(xn+1, xn) = q(fxn, fxn−1) �
� α1(xn)q(xn, xn−1) + α2(xn)q(xn, fxn) + α3(xn)q(xn−1, fxn−1)+
+α4(xn)q(xn, fxn−1) + α5(xn)q(xn−1, fxn) =
= α1(fxn−1)q(xn, xn−1) + α2(fxn−1)q(xn, xn+1) + α3(fxn−1)q(xn−1, xn)+
+α4(fxn−1)q(xn, xn) + α5(fxn−1)q(xn−1, xn+1) �
� α1(xn−2)q(xn, xn−1) + α2(xn−2)q(xn, xn+1) + α3(xn−2)q(xn−1, xn)+
+α4(xn−2)[q(xn, xn+1) + q(xn+1, xn)] + α5(xn−2)[q(xn−1, xn) + q(xn, xn+1)] � . . .
. . . � α1(x0)q(xn, xn−1) + (α2 + α4 + α5)(x0)q(xn, xn+1+
+(α3 + α5)(x0)q(xn−1, xn) + α4(x0)q(xn+1, xn). (3.4)
Similarly, setting x = xn and y = xn−1 in (3.2), we get
q(xn, xn+1) � α1(x0)q(xn−1, xn) + (α2 + α4 + α5)q(xn+1, xn)+
+α4(x0)q(xn, xn+1) + (α3 + α5)(x0)q(xn, xn−1). (3.5)
Thus, adding up (3.4) and (3.5), we obtain
q(xn+1, xn) + q(xn, xn+1) � (α1 + α3 + α5)(x0)[q(xn, xn−1) + q(xn−1, xn)]+
+(α2 + 2α4 + α5)(x0)[q(xn+1, xn) + q(xn, xn+1)].
Set vn = q(xn+1, xn) + q(xn, xn+1) and then we have
vn � (α1 + α3 + α5)(x0)vn−1 + (α2 + 2α4 + α5)(x0)vn.
Thus we get vn � λvn−1, where λ =
(α1 + α3 + α5)(x0)
1− (α2 + 2α4 + α5)(x0)
< 1 by (3.4). By repeating the
procedure, we obtain vn � λnv0 for all n ≥ 1. Thus it follows that
q(xn, xn+1) � vn � λn[q(x1, x0) + q(x0, x1)]. (3.6)
Let m > n, then it follows from (3.6) and λ < 1 that
q(xn, xm) � q(xn, xn+1) + q(xn+1, xn+2) + · · ·+ q(xm−1, xm) �
� (λn + λn+1 + · · ·+ λm−1)[q(x1, x0) + q(x0, x1)] �
� λn
1− λ
[q(x1, x0) + q(x0, x1)].
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COMMON FIXED POINT THEOREMS AND C-DISTANCE IN ORDERED CONE METRIC SPACES 1671
Lemma 2.3 implies that {xn} is a Cauchy sequence in X. Since X is complete, there exists a point
x′ ∈ X such that xn → x′ as n → ∞. The continuity of f implies that xn+1 = fxn → fx′ as
n → ∞ and, since the limit of a sequence is unique, we get that fx′ = x′. Thus x′ is a fixed point
of f.
Now, suppose that fz = z. Then, by using (3.1), we have
q(z, z) = q(fz, fz) �
� α1(z)q(z, z) + α2(z)q(z, fz) + α3(z)q(z, fz) + α4(z)q(z, fz) + α5(z)q(z, fz) �
� (α1 + α2 + α3 + α4 + α5)(z)q(z, z).
Since (α1 + α2 + α3 + α4 + α5)(z) < (α1 + α2 + α3 + 2α4 + 2α5)(z) < 1, we get that q(z, z) = 0
by Lemma 2.2.
Theorem 3.1 is proved.
Corollary 3.1 ([25], Theorem 3.1). Let (X,v) be a partially ordered set and (X, d) be a com-
plete cone metric space. Suppose that there exist mappings αi : X → [0, 1) such that the following
condition hold:
αi(fx) ≤ αi(x)
for all x ∈ X and i = 1, 2, 3. Also, let q be a c-distance on X and f : X → X be a continuous and
nondecreasing mapping with respect to v satisfying the following condition:
q(fx, fy) � α1(x)q(x, y) + α2(x)q(x, fx) + α3(x)q(y, fy)
for all x, y ∈ X with y v x such that
(α1 + α2 + α3)(x) < 1.
If there exists x0 ∈ X such that x0 v fx0, then f has a fixed point. Moreover, if fz = z, then
q(z, z) = 0.
Theorem 3.2. Let (X,v) be a partially ordered set, (X, d) be a complete cone metric space
and q be a c-distance on X. Suppose that there exists a continuous and nondecreasing mapping
f : X → X with respect to v such that the following conditions hold:
q(fx, fy) � α1q(x, y) + α2q(x, fx) + α3q(y, fy) + α4q(x, fy) + α5q(y, fx),
q(fy, fx) � α1q(y, x) + α2q(fx, x) + α3q(fy, y) + α4q(fy, x) + α5q(fx, y)
for all comparable x, y ∈ X, where αi are nonnegative coefficients for i = 1, 2, . . . , 5 with
α1 + α2 + α3 + 2(α4 + α5) < 1.
If there exists x0 ∈ X such that x0 v fx0, then f has a fixed point. Moreover, if fz = z, then
q(z, z) = 0.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
1672 H. RAHIMI, G. SOLEIMANI RAD
Proof. We can prove this result by applying Theorem 3.1 with αi(x) = αi for i = 1, 2, . . . , 5.
Corollary 3.2 ([9], Theorem 3.1). Let (X,v) be a partially ordered set, (X, d) be a complete cone
metric space and q be a c-distance on X. Suppose that there exists a continuous and nondecreasing
mapping f : X → X with respect to v such that the following condition hold:
q(fx, fy) � α1q(x, y) + α2q(x, fx) + α3q(y, fy)
for all x, y ∈ X with y v x, where αi are nonnegative coefficients for i = 1, 2, 3 with
α1 + α2 + α3 < 1.
If there exists x0 ∈ X such that x0 v fx0, then f has a fixed point. Moreover, if fz = z, then
q(z, z) = 0.
Our second result is the following theorem of Hardy – Rogers type (see [15]) for any c-distance
in a cone metric space with a normal cone.
Theorem 3.3. Let (X,v) be a partially ordered set, P be a normal cone and (X, d) be a com-
plete cone metric space. Suppose that there exist mappings αi : X → [0, 1) such that the following
condition hold:
αi(fx) ≤ αi(x)
for all x ∈ X and i = 1, 2, . . . , 5. Also, let q be a c-distance on X and f : X → X be a nondecreas-
ing mapping with respect to v satisfying the following conditions:
q(fx, fy) � α1(x)q(x, y) + α2(x)q(x, fx) + α3(x)q(y, fy) + α4(x)q(x, fy) + α5(x)q(y, fx),
q(fy, fx) � α1(x)q(y, x) + α2(x)q(fx, x) + α3(x)q(fy, y) + α4(x)q(fy, x) + α5(x)q(fx, y)
for all comparable x, y ∈ X such that
(α1 + α2 + α3 + 2α4 + 2α5)(x) < 1.
If there exists x0 ∈ X such that x0 v fx0 and inf{‖q(x, y)‖ + ‖q(x, fx)‖ : x ∈ X} > 0 for all
y ∈ X with y 6= fy, then f has a fixed point. Moreover, if fz = z, then q(z, z) = 0.
Proof. If fx0 = x0, then x0 is a fixed point of f and the proof is finished. Now, suppose that
fx0 6= x0. As in the proof of Theorem 3.1, we have
x0 v fx0 v f2x0 v . . . v fnx0 v fn+1x0 v . . . ,
where xn = fxn−1 = fnx0. Moreover, {xn} converges to a point x′ ∈ X and
q(xn, xm) � λn
1− λ
[q(x1, x0) + q(x0, x1)]
for all positive numbers with m > n ≥ 1, where λ =
(α1 + α3 + α5)(x0)
1− (α2 + 2α4 + α5)(x0)
< 1. By (CD3), it
follows that
q(xn, x
′) � λn
1− λ
[q(x1, x0) + q(x0, x1)]
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COMMON FIXED POINT THEOREMS AND C-DISTANCE IN ORDERED CONE METRIC SPACES 1673
for all n ≥ 1. Since P is a normal cone with normal constant k, we get
‖q(xn, xm)‖ ≤ k( λn
1− λ
)‖q(x1, x0) + q(x0, x1)‖
for all m > n ≥ 1. In particular, we obtain
‖q(xn, xn+1)‖ ≤ k
(
λn
1− λ
)
‖q(x1, x0) + q(x0, x1)‖ (3.7)
for all n ≥ 1. Also, we get
‖q(xn, x′)‖ ≤ k
(
λn
1− λ
)
‖q(x1, x0) + q(x0, x1)‖ (3.8)
for all n ≥ 1. Suppose that x′ 6= fx′. Then, by the hypothesis, (3.7) and (3.8), we have
0 < inf{‖q(x, x′)‖+ ‖q(x, fx)‖ : x ∈ X} ≤
≤ inf{‖q(xn, x′)‖+ ‖q(xn, fxn)‖ : n ≥ 1} =
= inf{‖q(xn, x′)‖+ ‖q(xn, xn+1)‖ : n ≥ 1} ≤
≤ inf
{
k
(
λn
1− λ
)
‖q(x1, x0) + q(x0, x1)‖+ k
(
λn
1− λ
)
‖q(x1, x0) + q(x0, x1)‖ : n ≥ 1
}
= 0
which is a contradiction. Hence x′ = fx′.
Moreover, suppose that fz = z. Then, we have q(z, z) = 0 by the final part of the proof of
Theorem 3.1.
Theorem 3.3 is proved.
Corollary 3.3 ([25], Theorem 3.2). Let (X,v) be a partially ordered set, P be a normal cone
and (X, d) be a complete cone metric space. Suppose that there exist mappings αi : X → [0, 1) such
that the following condition hold:
αi(fx) ≤ αi(x)
for all x ∈ X and i = 1, 2, 3. Also, let q be a c-distance on X and f : X → X be a nondecreasing
mapping with respect to v satisfying the following condition:
q(fx, fy) � α1(x)q(x, y) + α2(x)q(x, fx) + α3(x)q(y, fy)
for all x, y ∈ X with y v x such that
(α1 + α2 + α3)(x) < 1.
If there exists x0 ∈ X such that x0 v fx0 and inf{‖q(x, y)‖ + ‖q(x, fx)‖ : x ∈ X} > 0 for all
y ∈ X with y 6= fy, then f has a fixed point. Moreover, if fz = z, then q(z, z) = 0.
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1674 H. RAHIMI, G. SOLEIMANI RAD
Theorem 3.4. Let (X,v) be a partially ordered set, P be a normal cone, (X, d) be a complete
cone metric space and q be a c-distance on X. Suppose that there exists a nondecreasing mapping
f : X → X with respect to v such that the following conditions hold:
q(fx, fy) � α1q(x, y) + α2q(x, fx) + α3q(y, fy) + α4q(x, fy) + α5q(y, fx),
q(fy, fx) � α1q(y, x) + α2q(fx, x) + α3q(fy, y) + α4q(fy, x) + α5q(fx, y)
for all comparable x, y ∈ X, where αi are nonnegative coefficients for i = 1, 2, . . . , 5 with
α1 + α2 + α3 + 2(α4 + α5) < 1.
If there exists x0 ∈ X such that x0 v fx0 and inf{‖q(x, y)‖ + ‖q(x, fx)‖ : x ∈ X} > 0 for all
y ∈ X with y 6= fy, then f has a fixed point. Moreover, if fz = z, then q(z, z) = 0.
Proof. We can prove this result by applying Theorem 3.3 with αi(x) = αi for i = 1, 2, . . . , 5.
Corollary 3.4 ([9], Theorem 3.2). Let (X,v) be a partially ordered set, P be a normal cone,
(X, d) be a complete cone metric space and q be a c-distance on X. Suppose that there exists a
nondecreasing mapping f : X → X with respect to v such that the following condition hold:
q(fx, fy) � α1q(x, y) + α2q(x, fx) + α3q(y, fy)
for all x, y ∈ X with y v x, where αi are nonnegative coefficients for i = 1, 2, 3 with
α1 + α2 + α3 < 1.
If there exists x0 ∈ X such that x0 v fx0 and inf{‖q(x, y)‖ + ‖q(x, fx)‖ : x ∈ X} > 0 for all
y ∈ X with y 6= fy, then f has a fixed point. Moreover, if fz = z, then q(z, z) = 0.
Our third result include two mappings and the existence of their common fixed point for any
c-distance in a cone metric space without the normality condition of the cone.
Theorem 3.5. Let (X,v) be a partially ordered set and (X, d) be a complete cone metric space.
Suppose that there exist mappings αi : X → [0, 1) such that the following conditions hold:
αi(fx) ≤ αi(x), αi(gx) ≤ αi(x)
for all x ∈ X and i = 1, 2, . . . , 5. Also, let q be a c-distance on X and f, g : X → X be two
continuous and weakly increasing mappings with respect to v satisfying the following conditions:
q(fx, gy) � α1(x)q(x, y) + α2(x)q(x, fx) + α3(x)q(y, gy) + α4(x)q(x, gy) + α5(x)q(y, fx),
(3.9)
q(gy, fx) � α1(x)q(y, x) + α2(x)q(fx, x) + α3(x)q(gy, y) + α4(x)q(gy, x) + α5(x)q(fx, y)
(3.10)
for all comparable x, y ∈ X such that
(α1 + α2 + α3 + 2α4 + 2α5)(x) < 1. (3.11)
Then f and g have a common fixed point. Moreover, if fz = gz = z, then q(z, z) = 0.
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COMMON FIXED POINT THEOREMS AND C-DISTANCE IN ORDERED CONE METRIC SPACES 1675
Proof. Let x0 be an arbitrary point in X. We construct the sequence {xn} in X as follows:
x2n+1 = fx2n , x2n+2 = gx2n+1
for all n ≥ 0. Since f and g are weakly increasing mappings, there exist x1, x2, x3 ∈ X such that
x1 = fx0 v gfx0 = gx1 = x2 , x2 = gx1 v fgx1 = fx2 = x3.
Continuing in this manner, it follows that there exist x2n+1 ∈ X and x2n+2 ∈ X such that
x2n+1 = fx2n v gfx2n = gx2n+1 = x2n+2,
x2n+2 = gx2n+1 v fgx2n+1 = fx2n+2 = x2n+3
for all n ≥ 0. Thus x1 v x2 v · · · v xn v xn+1 v . . . for all n ≥ 1, that is, {xn} is a nondecreasing
sequence. Since x2n v x2n+1 for all n ≥ 1, by using (3.9) for x = x2n and y = x2n+1, we have
q(x2n+1, x2n+2) = q(fx2n, gx2n+1) �
� α1(x2n)q(x2n, x2n+1) + α2(x2n)q(x2n, fx2n) + α3(x2n)q(x2n+1, gx2n+1)+
+α4(x2n)q(x2n, gx2n+1) + α5(x2n)q(x2n+1, fx2n) =
= (α1 + α2)(gx2n−1)q(x2n, x2n+1) + α3(gx2n−1)q(x2n+1, x2n+2)+
+α4(gx2n−1)q(x2n, x2n+2) + α5(gx2n−1)q(x2n+1, x2n+1) �
� (α1 + α2)(x2n−1)q(x2n, x2n+1) + α3(x2n−1)q(x2n+1, x2n+2)+
+α4(x2n−1)[q(x2n, x2n+1) + q(x2n+1, x2n+2)]+
+α5(x2n−1)[q(x2n+1, x2n+2) + q(x2n+2, x2n+1)] =
= (α1 + α2 + α4)(fx2n−2)q(x2n, x2n+1) + α5(fx2n−2)q(x2n+2, x2n+1)+
+(α3 + α4 + α5)(fx2n−2)q(x2n+1, x2n+2) � · · ·
· · · � (α1 + α2 + α4)(x0)q(x2n, x2n+1)+
+(α3 + α4 + α5)(x0)q(x2n+1, x2n+2) + α5(x0)q(x2n+2, x2n+1).
Similarly, by using (3.10) for x = x2n and y = x2n+1, we get
q(x2n+2, x2n+1) � (α1 + α2 + α4)(x0)q(x2n+1, x2n) + α5(x0)q(x2n+1, x2n+2)+
+(α3 + α4 + α5)(x0)q(x2n+2, x2n+1).
Thus, adding up two previous relations, we obtain
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1676 H. RAHIMI, G. SOLEIMANI RAD
q(x2n+2, x2n+1) + q(x2n+1, x2n+2) � (α1 + α2 + α4)(x0)[q(x2n+1, x2n) + q(x2n, x2n+1)]+
+(α3 + α4 + 2α5)(x0)[q(x2n+2, x2n+1) + q(x2n+1, x2n+2)].
Setting vn = q(x2n+1, x2n)+q(x2n, x2n+1) and un = q(x2n+2, x2n+1)+q(x2n+1, x2n+2), it follows
that
un � (α1 + α2 + α4)(x0)vn + (α3 + α4 + 2α5)(x0)un.
Thus we have
un � λvn, (3.12)
where λ =
(α1 + α2 + α4)(x0)
1− (α3 + α4 + 2α5)(x0)
∈ [0, 1) by (3.11). By a similar procedure, starting with
x = x2n+2 and y = x2n+1, we have
vn+1 � λun. (3.13)
From (3.12) and (3.13), we get that
vn+1 � λ2vn , un � λ2un−1
for all n ≥ 1. Therefore, {un} and {vn} are two sequences converging to 0. Also, we obtain
q(x2n, x2n+1) � vn and q(x2n+1, x2n+2) � un and so q(xn, xn+1) � vn + un.
On the other hand, it is easy to show that, if {un} and {vn} are two sequences in E converging
to 0, then {un + vn} is a sequence converging to 0 (see [9, 12]). Lemma 2.3 implies that {xn} is
a Cauchy sequence in X. Since X is complete, there exists a point x′ ∈ X such that xn → x′ as
n → ∞. The continuity of f and g implies that x2n+1 = fx2n → fx′ and x2n+2 = gx2n+1 → gx′
as n → ∞. Since the limit of a sequence is unique, we get fx′ = x′ and gx′ = x′. Thus x′ is a
common fixed point of f and g.
Suppose that z ∈ X is another point satisfying fz = gz = z. Then (3.9) implies that
q(z, z) = q(fz, gz) �
� α1(z)q(z, z) + α2(z)q(z, fz) + α3(z)q(z, gz) + α4(z)q(z, gz) + α5(z)q(z, fz) �
� (α1 + α2 + α3 + α4 + α5)(z)q(z, z).
Since (α1+α2+α3+α4+α5)(z) < (α1+α2+α3+2α4+2α5)(z) and (α1+α2+α3+2α4+2α5)(z) <
1 for all z ∈ X, by (3.9), we get q(z, z) = 0 by Lemma 2.2.
Theorem 3.5 is proved.
Corollary 3.5 ([25], Theorem 3.3). Let (X,v) be a partially ordered set and (X, d) be a com-
plete cone metric space. Suppose that there exist mappings αi : X → [0, 1) such that the following
conditions hold:
αi(fx) ≤ αi(x), αi(gx) ≤ αi(x)
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COMMON FIXED POINT THEOREMS AND C-DISTANCE IN ORDERED CONE METRIC SPACES 1677
for all x ∈ X and i = 1, 2, 3. Also, let q be a c-distance on X and f, g : X → X be two continuous
and weakly increasing mappings with respect to v satisfying the following conditions:
q(fx, gy) � α1(x)q(x, y) + α2(x)q(x, fx) + α3(x)q(y, gy),
q(gx, fy) � α1(x)q(x, y) + α2(x)q(x, gx) + α3(x)q(y, fy)
for all x, y ∈ X with y v x such that
(α1 + α2 + α3)(x) < 1.
Then f and g have a common fixed point. Moreover, if fz = gz = z, then q(z, z) = 0.
Theorem 3.6. Let (X,v) be a partially ordered set, (X, d) be a complete cone metric space
and q be a c-distance on X. Suppose that there exist two continuous and weakly increasing mappings
f, g : X → X with respect to v such that the following conditions hold:
q(fx, gy) � α1q(x, y) + α2q(x, fx) + α3q(y, gy) + α4q(x, gy) + α5q(y, fx),
q(gy, fx) � α1q(y, x) + α2q(fx, x) + α3q(gy, y) + α4q(gy, x) + α5q(fx, y)
for all comparable x, y ∈ X, where αi are nonnegative coefficients for i = 1, 2, . . . , 5 with
α1 + α2 + α3 + 2(α4 + α5) < 1.
Then f and g have a common fixed point. Moreover, if fz = gz = z, then q(z, z) = 0.
Proof. We can prove this result by applying Theorem 3.5 with αi(x) = αi for i = 1, 2, . . . , 5.
Corollary 3.6 ([9], Theorem 3.3). Let (X,v) be a partially ordered set, (X, d) be a complete
cone metric space and q be a c-distance on X. Suppose that there exist two continuous and weakly
increasing mappings f, g : X → X with respect to v such that the following conditions hold:
q(fx, gy) � α1q(x, y) + α2q(x, fx) + α3q(y, gy),
q(gx, fy) � α1q(x, y) + α2q(x, gx) + α3q(y, fy)
for all comparable x, y ∈ X, where αi are nonnegative coefficients for i = 1, 2, 3 with
α1 + α2 + α3 < 1.
Then f and g have a common fixed point. Moreover, if fz = gz = z, then q(z, z) = 0.
Our next result include two mappings and the existence of their common fixed point for any
c-distance in a cone metric space with the normal cone.
Theorem 3.7. Let (X,v) be a partially ordered set, P be a normal cone and (X, d) be a com-
plete cone metric space. Suppose that there exist mappings αi : X → [0, 1) such that the following
conditions hold:
αi(fx) ≤ αi(x), αi(gx) ≤ αi(x)
for all x ∈ X and i = 1, 2, . . . , 5. Also, let q be a c-distance on X and f, g : X → X be two weakly
increasing mappings with respect to v satisfying the following conditions:
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1678 H. RAHIMI, G. SOLEIMANI RAD
q(fx, gy) � α1(x)q(x, y) + α2(x)q(x, fx) + α3(x)q(y, gy) + α4(x)q(x, gy) + α5(x)q(y, fx),
q(gy, fx) � α1(x)q(y, x) + α2(x)q(fx, x) + α3(x)q(gy, y) + α4(x)q(gy, x) + α5(x)q(fx, y)
for all comparable x, y ∈ X such that
(α1 + α2 + α3 + 2α4 + 2α5)(x) < 1.
If inf{‖q(x, y)‖ + ‖q(x, fx)‖ : x ∈ X} > 0 and inf{‖q(x, y)‖ + ‖q(x, gx)‖ : x ∈ X} > 0 for all
y ∈ X with y 6= fy and y 6= gy, respectively, then f and g have a common fixed point. Moreover, if
fz = gz = z, then q(z, z) = 0.
Proof. The proof is similar to Theorem 3.3. One can prove this theorem by using the proof of
Theorems 3.3 and 3.6.
Corollary 3.7 ([25], Theorem 3.4). Let (X,v) be a partially ordered set, P be a normal cone
and (X, d) be a complete cone metric space. Suppose that there exist mappings αi : X → [0, 1) such
that the following conditions hold:
αi(fx) ≤ αi(x), αi(gx) ≤ αi(x)
for all x ∈ X and i = 1, 2, 3. Also, let q be a c-distance on X and f, g : X → X be two weakly
increasing mappings with respect to v satisfying the following conditions:
q(fx, gy) � α1(x)q(x, y) + α2(x)q(x, fx) + α3(x)q(y, gy),
q(gx, fy) � α1(x)q(x, y) + α2(x)q(x, gx) + α3(x)q(y, fy)
for all comparable x, y ∈ X such that
(α1 + α2 + α3)(x) < 1.
If inf{‖q(x, y)‖ + ‖q(x, fx)‖ : x ∈ X} > 0 and inf{‖q(x, y)‖ + ‖q(x, gx)‖ : x ∈ X} > 0 for all
y ∈ X with y 6= fy and y 6= gy, respectively, then f and g have a common fixed point. Moreover, if
fz = gz = z, then q(z, z) = 0.
Theorem 3.8. Let (X,v) be a partially ordered set, P be a normal cone, (X, d) be a complete
cone metric space and q be a c-distance on X. Suppose that there exist two weakly increasing
mappings f, g : X → X with respect to v such that the following conditions hold:
q(fx, gy) � α1q(x, y) + α2q(x, fx) + α3q(y, gy) + α4q(x, gy) + α5q(y, fx),
q(gy, fx) � α1q(y, x) + α2q(x, gx) + α3q(y, fy) + α4q(gy, x) + α5q(fx, y)
for all comparable x, y ∈ X, where αi are nonnegative coefficients for i = 1, 2, . . . , 5 with
α1 + α2 + α3 + 2(α4 + α5) < 1.
If inf{‖q(x, y)‖ + ‖q(x, fx)‖ : x ∈ X} > 0 and inf{‖q(x, y)‖ + ‖q(x, gx)‖ : x ∈ X} > 0 for
all y ∈ X with y 6= fy and with y 6= gy, respectively, then f and g have a common fixed point.
Moreover, if fz = gz = z, then q(z, z) = 0.
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COMMON FIXED POINT THEOREMS AND C-DISTANCE IN ORDERED CONE METRIC SPACES 1679
Proof. We can prove this result by applying Theorem 3.7 with αi(x) = αi for i = 1, 2, . . . , 5.
Corollary 3.8 ([9], Theorem 3.4). Let (X,v) be a partially ordered set, P be a normal cone,
(X, d) be a complete cone metric space and q be a c-distance on X. Suppose that there exist two
weakly increasing mappings f, g : X → X with respect to v such that the following conditions hold:
q(fx, gy) � α1q(x, y) + α2q(x, fx) + α3q(y, gy),
q(gx, fy) � α1q(x, y) + α2q(x, gx) + α3q(y, fy)
for all comparable x, y ∈ X, where αi are nonnegative coefficients for i = 1, 2, 3 with
α1 + α2 + α3 < 1.
If inf{‖q(x, y)‖ + ‖q(x, fx)‖ : x ∈ X} > 0 and inf{‖q(x, y)‖ + ‖q(x, gx)‖ : x ∈ X} > 0 for all
y ∈ X with y 6= fy and y 6= gy, respectively, then f and g have a common fixed point. Moreover, if
fz = gz = z, then q(z, z) = 0.
Example 3.1. Let E = R and P = {x ∈ E : x ≥ 0}. Let X = [0, 1] and define a mapping
d : X ×X → E by d(x, y) = |x− y| for all x, y ∈ X. Then (X, d) is a cone metric space. Define a
function q : X ×X → E by q(x, y) = d(x, y) for all x, y ∈ X. Then q is a c-distance (by Example
2.1). Let an order relation v be defined by x v y ⇐⇒ x ≤ y. Also, let a mapping f : X → X
be defined by f(x) =
x2
4
for all x ∈ X. Define the mappings α1(x) =
x+ 1
4
, α4(x) =
x
8
and
α2 = α3 = α5 = 0 for all x ∈ X. Observe that:
(1) α1(fx) =
1
4
(
x2
4
+ 1
)
≤ 1
4
(
x2 + 1
)
≤ x+ 1
4
= α(x) for all x ∈ X.
(2) α4(fx) =
x2
32
≤ x2
8
≤ x
8
= α4(x) for all x ∈ X.
(3) αi(fx) = 0 ≤ 0 = αi(x) for all x ∈ X and i = 2, 3, 5.
(4) (α1 + α2 + α3 + 2α4 + 2α5)(x) =
x+ 1
4
+
2x
8
=
2x+ 1
4
< 1 for all x ∈ X.
(5) For all comparable x, y ∈ X, we get
q(fx, fy) =
∣∣∣x2
4
− y2
4
∣∣∣ ≤ |x+ y||x− y|
4
=
(x+ y
4
)
|x− y| ≤
(x+ 1
4
)
|x− y| ≤
≤ α1(x)q(x, y) + α2(x)q(x, fx) + α3(x)q(y, fy)+
+α4(x)q(x, fy) + α5(x)q(y, fx).
(6) Similarly, we have
q(fy, fx) ≤ α1(x)q(y, x) + α2(x)q(fx, x) + α3(x)q(fy, y)+
+α4(x)q(fy, x) + α5(x)q(fx, y)
for all comparable x, y ∈ X.
Moreover, f is a nondecreasing and continuous mapping with respect to v . Hence all the
conditions of Theorem 3.1 are satisfied. Thus f has a fixed point x = 0 and q(0, 0) = 0.
Remark 3.1. There exist many examples on fixed point results under c-distance in cone metric
spaces (see, for example, [9, 12, 25, 26]). Also, most of the examples in [1, 6, 24] can be easily
translated into the c-distance on ordered cone metric spaces with q(x, y) = d(x, y).
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1680 H. RAHIMI, G. SOLEIMANI RAD
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Received 11.05.12,
after revision — 19.04.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 12
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