Zhu et al. Fixed Point Theory and Applications (2015) 2015:131
DOI 10.1186/s13663-015-0384-4
RESEARCH
Open Access
Common fixed point theorems for three
pairs of self-mappings satisfying the common
(E.A) property in Menger probabilistic
G-metric spaces
Chuanxi Zhu, Qiang Tu* and Zhaoqi Wu
*
Correspondence:
[email protected]
Department of Mathematics,
Nanchang University, Nanchang,
330031, P.R. China
Abstract
In this paper, we generalize the algebraic sum ⊕ of Fang. Based on this concept, we
prove some common fixed point theorems for three pairs of self-mappings satisfying
the common (E.A) property in Menger PGM-spaces. Finally, an example is given to
exemplify our main results.
MSC: Primary 47H10; secondary 46S10
Keywords: common fixed point; Menger PGM-space; common (E.A) property;
weakly compatible mappings
1 Introduction
As a generalization of a metric space, the concept of a probabilistic metric space has been
introduced by Menger [, ]. Fixed point theory in a probabilistic metric space is an important branch of probabilistic analysis, and many results on the existence of fixed points or
solutions of nonlinear equations under various types of conditions in Menger PM-spaces
have been extensively studied by many scholars (see e.g. [, ]). In , Mustafa and
Sims [] introduced the concept of a generalized metric space, based on the notion of
a generalized metric space, many authors obtained many fixed point theorems for mappings satisfying different contractive conditions in generalized metric spaces (see [–]).
Moreover, Zhou et al. [] defined the notion of a generalized probabilistic metric space
or a PGM-space as a generalization of a PM-space and a G-metric space. After that, Zhu
et al. [] obtained some fixed point theorems.
In , Aamri and Moutawakil [] defined a property for a pair of mappings, i.e., the
so-called property (E.A), which is a generalization of the concept of noncompatibility. In
, Fang and Gang [] defined the property (E.A) for two mappings in Menger PMspaces and studied the existence of and common fixed points in such spaces. Recently,
Wu et al. [] defined a property for two hybrid pairs of mappings satisfying the common
property (E.A) in Menger PM-spaces. Gu and Yin [] introduced the concept of common
(E.A) property and obtained some common fixed point theorems for three pairs of selfmappings satisfying the common (E.A) property in generalized metric spaces.
© 2015 Zhu et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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Zhu et al. Fixed Point Theory and Applications (2015) 2015:131
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The aim of this paper is to introduce the common (E.A) property in Menger PGMspaces, generalize the algebraic sum ⊕ in [], and study the common fixed point theorems
for three pairs of weakly compatible self-mappings under strict contractive conditions in
Menger PGM-spaces. Our results do not rely on any commuting or continuity condition
of the mappings.
2 Preliminaries
Throughout this paper, let R = (–∞, +∞), R+ = [, +∞), and Z+ be the set of all positive
integers.
A mapping F : R → R+ is called a distribution function if it is nondecreasing leftcontinuous with supt∈R F(t) =  and inft∈R F(t) = .
We shall denote by D the set of all distribution functions while H will always denote the
specific distribution function defined by
, t ≤ ,
H(t) =
, t > .
A mapping : [, ] × [, ] → [, ] is called a triangular norm (for short, a t-norm) if
the following conditions are satisfied:
() (a, ) = a;
() (a, b) = (b, a);
() a ≥ b, c ≥ d ⇒ (a, c) ≥ (b, d);
() (a, (b, c)) = ((a, b), c).
A typical example of a t-norm is m , where m (a, b) = min{a, b}, for each a, b ∈ [, ].
Definition . [] A Menger probabilistic G-metric space (for short, a PGM-space) is a
triple (X, G∗ , ), where X is a nonempty set, is a continuous t-norm, and G∗ is a mapping
∗
from X × X × X into D (Gx,y,z
denotes the value of G∗ at the point (x, y, z)) satisfying the
following conditions:
∗
(t) =  for all x, y, z ∈ X and t >  if and only if x = y = z;
(PGM-) Gx,y,z
∗
∗
(t) ≥ Gx,y,z
(t) for all x, y, z ∈ X with z = y and t > ;
(PGM-) Gx,x,y
∗
∗
∗
(t) = · · · (symmetry in all three variables);
(PGM-) Gx,y,z (t) = Gx,z,y (t) = Gy,x,z
∗
∗
∗
(t)) for all x, y, z, a ∈ X and s, t ≥ .
(PGM-) Gx,y,z (t + s) ≥ (Gx,a,a (s), Ga,y,z
Example . [] Let (X, G) be a G-metric space, where G(x, y, z) = |x – y| + |y – z| + |z – x|.
t
∗
(t) = t+G(x,y,z)
for all x, y, z ∈ X. Then (X, G∗ , m ) is a Menger PGM-space.
Define Gx,y,z
Definition . [] Let (X, G∗ , ) be a Menger PGM-space and x be any point in X. For
any >  and δ with  < δ < , and (, δ)-neighborhood of x is the set of all points y in X
∗
for which Gx∗ ,y,y () >  – δ and Gy,x
() >  – δ. We write
 ,x
∗
() >  – δ ,
Nx (, δ) = y ∈ X : Gx∗ ,y,y () >  – δ, Gy,x
 ,x
which means that Nx (, δ) is the set of all points y in X for which the probability of the
distance from x to y being less than is greater than  – δ.
Definition . [] Let (X, G∗ , ) be a PGM-space, and {xn } is a sequence in X.
Zhu et al. Fixed Point Theory and Applications (2015) 2015:131
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() {xn } is said to be convergent to a point x ∈ X (write xn → x), if for any >  and
 < δ < , there exists a positive integer M,δ such that xn ∈ Nx (, δ) whenever
n > M,δ ;
() {xn } is called a Cauchy sequence, if for any >  and  < δ < , there exists a positive
integer M,δ such that Gx∗n ,xm ,xl () >  – δ whenever n, m, l > M,δ ;
() (X, G∗ , ) is said to be complete, if every Cauchy sequence in X converges to a point
in X.
Remark . Let (X, G∗ , ) be a Menger PGM-space, {xn } is a sequence in X. Then the
following are equivalent:
() {xn } is convergent to a point x ∈ X;
() Gx∗n ,xn ,x (t) →  as n → ∞, for all t > ;
() Gx∗n ,x,x (t) →  as n → ∞, for all t > .
We can analogously prove the following lemma as in Menger PM-spaces.
Lemma . Let (X, G∗ , ) be a Menger PGM-space with a continuous t-norm, {xn }, {yn },
and {zn } be sequences in X and x, y, z ∈ X, if {xn } → x, {yn } → y, and {zn } → z as n → ∞.
Then
∗
() lim infn→∞ Gx∗n ,yn ,zn (t) ≥ Gx,y,z
(t) for all t > ;
∗
∗
() Gx,y,z (t + o) ≥ lim supn→∞ Gxn ,yn ,zn (t) for all t > .
Particularly, if t is a continuous point of Gx,y,z (·), then limn→∞ Gxn ,yn ,zn (t ) = Gx,y,z (t ).
Lemma . [] Let (X, G∗ , ) be a Menger PGM-space. For each λ ∈ (, ], define a function Gλ∗ by
∗
(t) >  – λ ,
Gλ∗ (x, y, z) = inf t ≥  : Gx,y,z
t
for x, y, z ∈ X, then
∗
(t) >  – λ;
() Gλ∗ (x, y, z) < t if and only if Gx,y,z
∗
() Gλ (x, y, z) =  for all λ ∈ (, ] if and only if x = y = z;
() Gλ∗ (x, y, z) = Gλ∗ (y, x, z) = Gλ∗ (y, z, x) = · · · ;
() if = m , then for every λ ∈ (, ], Gλ∗ (x, y, z) ≤ Gλ∗ (x, a, a) + Gλ∗ (a, y, z).
Definition . [] Let f and g be self-mappings of a set X. If w = fx = gx for some x in X,
then x is called a coincidence point of f and g, and w is called point of coincidence of f
and g.
Definition . Let S and T be two self-mappings of a Menger PGM-space (X, G∗ , ).
S and T are said to be weakly compatible (or coincidentally commuting) if they commute
at their coincidence points, i.e., if Tu = Su for some u ∈ X implies that TSu = STu.
Definition . [] Let (X, d) be a G-metric space and A, B, S, and T four self-mappings
on X. The pairs (A, S) and (B, T) are said to satisfy the common (E.A) property if there exist
two sequences {xn } and {yn } in X such that limn→∞ Axn = limn→∞ Sxn = limn→∞ Byn =
limn→∞ Tyn = t for some t ∈ X.
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Definition . Let (X, G∗ , ) be a Menger PGM-space and A, B, S, and T four selfmappings on X. The pairs (A, S) and (B, T) are said to satisfy the common (E.A) property if there exist two sequences xn and yn in X such that limn→∞ Axn = limn→∞ Sxn =
limn→∞ Byn = limn→∞ Tyn = t for some t ∈ X.
Definition . [] Let F , F ∈ D . The algebraic sum F ⊕ F of F and F is defined by
(F ⊕ F )(t) = sup min F (t ), F (t )
(.)
t +t =t
for all t ∈ R.
As a generalization, we give the following definition.
Definition . Let F , F , F ∈ D . The algebraic sum F ⊕F ⊕F of F , F , and F is defined
by
(F ⊕ F ⊕ F )(t) =
sup
min F (t ), F (t ), F (t )
(.)
t +t +t =t
for all t ∈ R.
Remark . Let F (t) = H(t), then (.) and (.) are equivalent.
Definition . [] Let φ : R+ → R+ be a function and φ n (t) be the nth iteration of φ(t),
(i) φ is nondecreasing;
(ii) φ is upper semi-continuous from the right;
∞ n
(iii)
n= φ (t) < +∞ for all t > .
We define the class of functions φ : R+ → R+ satisfying conditions (i), (ii), and (iii).
Lemma . Let (X, G∗ , ) be a Menger PGM-space and x, y, z ∈ X. If there exists φ ∈ ,
such that
∗
∗
φ(t) + o ≥ Gx,y,z
(t),
Gx,y,z
(.)
for all t > . Then x = y = z.
Proof Let λ ∈ (, ] and we put a = Gλ∗ (x, y, z). Since φ(·) is upper semi-continuous from
the right at the point a, for given > , there exists s > a such that φ(s) < φ(a) + ε. By
∗
Lemma ., s > Gλ∗ (x, y, y) implies that Gx,y,z
(s) >  – λ. So, it follows from (.) that
∗
∗
∗
φ(s) + ≥ Gx,y,z
φ(s) + o ≥ Gx,y,z
(s) >  – λ,
Gx,y,z
which implies that Gλ∗ (x, y, z) < φ(s) + < φ(a) + . By the arbitrariness of , we get a =
Gλ∗ (x, y, z) ≤ φ(a), thus a = , i.e., Gλ∗ (x, y, z) = . By () of Lemma ., we conclude that
x = y = z.
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3 Main results
In this section, we will establish some new common fixed point theorems in Menger PGMspaces.
Theorem . Let (X, G∗ , ) be a Menger PGM-space. Suppose the self-mappings f , g, h,
R, S, and T : X → X satisfy the following conditions:
∗
∗
∗
∗
∗
φ(t) ≥ min GRx,Sy,Tz
(t), Gfx,Rx,Rx
(t), Ggy,Sy,Sy
(t), Ghz,Tz,Tz
(t),
Gfx,gy,hz
∗
∗
∗
Gfx,Sy,Tz ⊕ GRx,gy,Tz
(t),
⊕ GRx,Sy,hz
∗
∗
∗
(t)
Gfx,gy,Tz ⊕ Gfx,Sy,hz
⊕ GRx,gy,hz
(.)
for all x, y, and z ∈ X, t > , where φ ∈ . If one of the following conditions is satisfied, then
the pairs (f , R), (g, S), and (h, T) have a common fixed point of coincidence in X:
(i) the subspace Rx is closed in X, fx ⊆ Sx, gx ⊆ Tx, and the two pairs of (f , R) and (g, S)
satisfy the common (E.A) property;
(ii) the subspace Sx is closed in X, gx ⊆ Tx, hx ⊆ Rx, and the two pairs of (g, S) and
(h, T) satisfy the common (E.A) property;
(iii) the subspace Tx is closed in X, fx ⊆ Sx, hx ⊆ Rx, and the two pairs of (f , R) and
(h, T) satisfy the common (E.A) property.
Moreover, if the pairs (f , R), (g, S), and (h, T) are weakly compatible, then f , g, h, R, S, and
T have a unique common fixed point in X.
Proof First, we suppose that the subspace Rx is closed in X, fx ⊆ Sx, gx ⊆ Tx, and the
two pairs of (f , R) and (g, S) satisfy the common (E.A) property. Then by Definition . we
know that there exist two sequences {xn } and {yn } in X such that
lim fxn = lim Rxn = lim gyn = lim Syn = t,
n→∞
n→∞
n→∞
n→∞
for some t ∈ X. Since gx ⊆ Tx, there exists a sequence {zn } in X such that gyn = Tzn . Hence
limn→∞ Tzn = a. Next, we will show limn→∞ hzn = a. In fact, if limn→∞ hzn = z = a, then
from (.) we can get
∗
∗
∗
∗
∗
Gfx
(t), Gfx
(t), Ggy
(t), Ghz
(t),
φ(t) ≥ min GRx
n ,gyn ,hzn
n ,Syn ,Tzn
n ,Rxn ,Rxn
n ,Syn ,Syn
n ,Tzn ,Tzn
∗
∗
∗
(t),
⊕ GRx
Gfxn ,Syn ,Tzn ⊕ GRx
n ,gyn ,Tzn
n ,Syn ,hzn
∗
∗
∗
⊕ GRx
(t) .
Gfxn ,gyn ,Tzn ⊕ Gfx
n ,Syn ,hzn
n ,gyn ,hzn
On letting n → ∞, and by () of Lemma ., we can obtain
∗
∗
φ(t) + o ≥ lim sup Gfx
φ(t)
Ga,a,z
n ,gyn ,hzn
n→∞
∗
≥ min , , , Gz,a,a
(t),
∗
∗
∗
(t),
⊕ GRx
⊕ GRx
lim Gfx
n ,Syn ,Tzn
n ,gyn ,Tzn
n ,Syn ,hzn
n→∞
∗
∗
∗
(t)
.
lim Gfx
⊕
G
⊕
G
,gy
,Tz
,Sy
,hz
,gy
,hz
fx
Rx
n n
n
n n n
n n n
n→∞
(.)
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In addition, by Definition ., it is easy to verify that
∗
∗
∗
(t)
lim Gfx
⊕ GRx
⊕ GRx
n ,Syn ,Tzn
n ,gyn ,Tzn
n ,Syn ,hzn
n→∞
∗
∗
∗
(t), GRx
(t), GRx
(t)
≥ lim min Gfx
n ,Syn ,Tzn
n ,gyn ,Tzn
n ,Syn ,hzn
n→∞
∗
∗
∗
∗
(t), Ga,a,a
(t), Ga,a,z
(t) = Ga,a,z
(t).
≥ min Ga,a,a
(.)
Similarly, we also have
∗
∗
∗
∗
lim Gfx
(t) ≥ Ga,a,z
⊕ Gfx
⊕ GRx
(t).
n ,gyn ,Tzn
n ,Syn ,hzn
n ,gyn ,hzn
n→∞
Then (.) is
∗
∗
Ga,a,z
φ(t) + o ≥ Ga,a,z
(t)
for all t > . By Lemma ., we have a = z. So, limn→∞ hzn = a.
Since Rx is a closed subset of X and limn→∞ Rxn = a, there exists p ∈ X such that a = Rp,
we claim that fp = a. Suppose not, then by using (.), we obtain
∗
∗
∗
∗
∗
Gfp,gy
φ(t) ≥ min GRp,Sy
(t), Gfp,Rp,Rp
(t), Ggy
(t), Ghz
(t),
n ,hzn
n ,Tzn
n ,Syn ,Syn
n ,Tzn ,Tzn
∗
∗
∗
Gfp,Syn ,Tzn ⊕ GRp,gy
(t),
⊕ GRp,Sy
n ,Tzn
n ,hzn
∗
∗
∗
(t) .
Gfp,gyn ,Tzn ⊕ Gfp,Sy
⊕ GRp,gy
n ,hzn
n ,hzn
Taking n → ∞ on the two sides of the above inequality, similar to (.), we get
∗
∗
∗
∗
∗
φ(t) + o ≥ min , Gfp,a,a
(t),
(t), , , lim Gfp,Sy
⊕ GRp,gy
⊕ GRp,Sy
Gfp,a,a
n ,Tzn
n ,Tzn
n ,hzn
n→∞
∗
∗
∗
(t)
⊕ Gfp,Sy
⊕ GRp,gy
lim Gfp,gy
n ,Tzn
n ,hzn
n ,hzn
n→∞
∗
∗
∗
∗
≥ min , Gfp,a,a
(t), , , Gfp,a,a
(t), Gfp,a,a
(t) = Gfp,a,a
(t).
By Lemma ., we have fp = a = Rp. Hence, p is the coincidence point of the pair (f , R).
By condition fx ⊆ Sx and fp = a, there exists u ∈ X such that a = Su. Now we claim that
gu = a. In fact, if gu = a, then from (.), we have
∗
∗
∗
∗
∗
φ(t) ≥ min GRp,Su,Tz
Gfp,gu,hz
(t), Gfp,Rp,Rp
(t), Ggu,Su,Su
(t), Ghz
(t),
n
n
n ,Tzn ,Tzn
∗
∗
∗
Gfp,Su,Tzn ⊕ GRp,gu,Tz
(t),
⊕ GRp,Su,hz
n
n
∗
∗
∗
Gfp,gu,Tzn ⊕ Gfp,Su,hz
(t) .
⊕ GRp,gu,hz
n
n
Letting n → ∞ on the two sides of the above inequality, we get
∗
∗
∗
∗
∗
(t), , lim Gfp,Su,Tz
⊕ GRp,gu,Tz
⊕ GRp,Su,hz
Ga,gu,a
φ(t) + o ≥ min , , Ggu,a,a
(t),
n
n
n
n→∞
∗
∗
∗
(t)
⊕ Gfp,Su,hz
⊕ GRp,gu,hz
lim Gfp,gu,Tz
n
n
n
n→∞
∗
∗
∗
∗
≥ min , , Ggu,a,a
(t), , Ga,gu,a
(t), Ga,gu,a
(t) = Ga,gu,a
(t).
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By Lemma ., we can also obtain gu = a, and so u is the coincidence point of the pair
(g, S).
Since gX ⊆ TX, there exists v ∈ X such that a = Tv. We claim that hv = a. If not, from
(.), we have
∗
∗
Gfp,gu,hv
φ(t) + o ≥ Gfp,gu,hv
φ(t)
∗
∗
∗
∗
≥ min GRp,Su,Tv
(t), Gfp,Rp,Rp
(t), Ggu,Su,Su
(t), Ghv,Tv,Tv
(t),
∗
∗
∗
Gfp,Su,Tv ⊕ GRp,gu,Tv
(t),
⊕ GRp,Su,hv
∗
∗
∗
Gfp,gu,Tv ⊕ Gfp,Su,hv
(t)
⊕ GRp,gu,hv
∗
∗
∗
∗
≥ min , , , Ghv,a,a
(t),
(t), Ga,a,a
⊕ Ga,a,a
⊕ Ga,a,hv
∗
∗
∗
Ga,a,a ⊕ Ga,a,hv
(t)
⊕ Ga,a,hv
∗
∗
∗
∗
≥ min , , , Ghv,a,a
(t), Ga,a,hv
(t), Ga,a,hv
(t) = Ga,a,hv
(t).
By Lemma ., we have hv = a = Tv, so v is the coincidence point of the pair (h, T).
Therefore, in all the above cases, we obtain fp = Rp = a, gu = Su = hv = Tv = a. Now, weak
compatibility of the pairs (f , R), (g, S), and (h, T) give fa = Ra, ga = Sa, and ha = Ta.
Next, we show that fa = a. In fact, if fa = a, then from (.) we have
∗
∗
∗
∗
∗
Gfa,a,a
φ(t) + o ≥ min GRa,Su,Tv
(t), Gfa,Ra,Ra
(t), Ggu,Su,Su
(t), Ghv,Tv,Tv
(t),
∗
∗
∗
(t),
Gfa,Su,Tv ⊕ GRa,gu,Tv
⊕ GRa,Su,hv
∗
∗
∗
Gfa,gu,Tv ⊕ Gfa,Su,hv
(t)
⊕ GRa,gu,hv
∗
∗
∗
∗
≥ min GRa,a,a
(t),
(t), , , , Gfa,a,a
⊕ GRa,a,a
⊕ GRa,a,a
∗
∗
∗
⊕ GRa,a,a
Gfa,a,a ⊕ Gfa,a,hv
(t)
∗
∗
∗
∗
(t), Gfa,a,a
(t), Gfa,a,a
(t) = Gfa,a,a
(t).
≥ min , , , Gfa,a,a
From Lemma . we know fa = a and so fa = Ra = a. Similarly, it can be show that ga =
Sa = a and ha = Ta = a, so we get fa = ga = ha = Ra = Sa = Ta = a, which means that a is a
common fixed point of f , g, h, R, S, and T.
Next, we will show the uniqueness. Actually, suppose that w ∈ X, w = a is another common fixed point of f , g, h, R, S, and T. Then by (.), we have
∗
∗
∗
∗
∗
Gw,a,a
φ(t) + o ≥ min GRw,Sa,Ta
(t), Gfw,Rw,Rw
(t), Gga,Sa,Sa
(t), Gha,Ta,Ta
(t),
∗
∗
∗
Gfw,Sa,Ta ⊕ GRw,ga,Ta
(t),
⊕ GRw,Sa,ha
∗
∗
∗
Gfw,ga,Ta ⊕ Gfw,Sa,ha
(t)
⊕ GRw,ga,ha
∗
∗
∗
∗
≥ min Gw,a,a
(t),
(t), , , , Gfa,a,a
⊕ Gw,a,a
⊕ Gw,a,a
∗
∗
∗
Gw,a,a ⊕ Gw,a,hv
(t)
⊕ Gw,a,a
∗
∗
∗
∗
≥ min , , , Gw,a,a
(t), Gw,a,a
(t), Gw,a,a
(t) = Gw,a,a
(t).
By Lemma . we have a = w, a contradiction, so, f , g, h, R, S, and T have a unique common
fixed point.
Zhu et al. Fixed Point Theory and Applications (2015) 2015:131
Page 8 of 14
Finally, if condition (ii) or (iii) holds, then the argument is similar to the above, so we
omit it. This completes the proof of Theorem ..
Taking φ(t) = λt, λ ∈ (, ), then we can obtain the following results.
Corollary . Let (X, G∗ , ) be a Menger PGM-space. Suppose the self-mappings f , g, h,
R, S, and T : X → X satisfy the following conditions:
∗
∗
∗
∗
∗
Gfx,gy,hz
(λt) ≥ min GRx,Sy,Tz
(t), Gfx,Rx,Rx
(t), Ggy,Sy,Sy
(t), Ghz,Tz,Tz
(t),
∗
∗
∗
(t),
⊕ GRx,Sy,hz
Gfx,Sy,Tz ⊕ GRx,gy,Tz
∗
∗
∗
(t)
⊕ GRx,gy,hz
Gfx,gy,Tz ⊕ Gfx,Sy,hz
for all x, y, and z ∈ X, where λ ∈ (, ). If one of the following conditions is satisfied, then the
pairs (f , R), (g, S), and (h, T) have a common fixed point of coincidence in X:
(i) the subspace Rx is closed in X, fx ⊆ Sx, gx ⊆ Tx, and the two pairs of (f , R) and (g, S)
satisfy the common (E.A) property;
(ii) the subspace Sx is closed in X, gx ⊆ Tx, hx ⊆ Rx, and the two pairs of (g, S) and
(h, T) satisfy the common (E.A) property;
(iii) the subspace Tx is closed in X, fx ⊆ Sx, hx ⊆ Rx, and the two pairs of (f , R) and
(h, T) satisfy the common (E.A) property.
Moreover, if the pairs (f , R), (g, S), and (h, T) are weakly compatible, the f , g, h, R, S, and
T have a unique common fixed point in X.
Theorem . Let (X, G∗ , ) be a Menger PGM-space. Suppose self-mappings f , g, h, R, S,
and T : X → X satisfying the following conditions:
∗
(t) ≥ ψ m(x, y, z, t)
Gfx,gy,hz
(.)
for all x, y, and z ∈ X, where
∗
∗
∗
∗
(t), Gfx,Rx,Rx
(t), Ggy,Sy,Sy
(t), Ghz,Tz,Tz
(t),
m(x, y, z, t) = min GRx,Sy,Tz
∗
∗
∗
(t),
⊕ GRx,Sy,hz
Gfx,Sy,Tz ⊕ GRx,gy,Tz
∗
∗
∗
⊕ GRx,gy,hz
Gfx,gy,Tz ⊕ Gfx,Sy,hz
(t) ,
ψ is continuous and ψ(t) > t for all t > . If one of the following conditions is satisfied, then
the pairs (f , R), (g, S), and (h, T) have a common fixed point of coincidence in X:
(i) the subspace Rx is closed in X, fx ⊆ Sx, gx ⊆ Tx, and the two pairs of (f , R) and (g, S)
satisfy the common (E.A) property;
(ii) the subspace Sx is closed in X, gx ⊆ Tx, hx ⊆ Rx, and the two pairs of (g, S) and
(h, T) satisfy the common (E.A) property;
(iii) the subspace Tx is closed in X, fx ⊆ Sx, hx ⊆ Rx, and the two pairs of (f , R) and
(h, T) satisfy the common (E.A) property.
Moreover, if the pairs (f , R), (g, S), and (h, T) are weakly compatible, the f , g, h, R, S, and
T have a unique common fixed point in X.
Zhu et al. Fixed Point Theory and Applications (2015) 2015:131
Page 9 of 14
Proof First, we suppose that condition (i) is satisfied. Then there exist two sequences {xn }
and {yn } in X such that
lim fxn = lim Rxn = lim gyn = lim Syn = t,
n→∞
n→∞
n→∞
n→∞
for some t ∈ X.
Since gx ⊆ Tx, there exists a sequence {zn } in X such that gyn = Tzn . Hence limn→∞ Tzn =
a. We claim that limn→∞ hzn = a. In fact, if limn→∞ hzn = z = a, it is not difficult to prove
that there exists t >  such that
∗
∗
(t ) > Ga,a,z
(t ).
ψ Ga,a,z
(.)
∗
∗
∗
If not, we have Ga,a,z
(t) ≥ ψ(Ga,a,z
(t)) > Ga,a,z
(t) for all t > , which is a contradiction. Then
by (.), there exists t >  such that
∗
Gfx
(t ) ≥ ψ m(xn , yn , zn , t ) ,
n ,gyn ,hzn 
(.)
where
ψ m(xn , yn , zn , t )
∗
∗
∗
∗
= ψ min GRx
(t ), Gfx
(t ), Ggy
(t ), Ghz
(t ),
n ,Syn ,Tzn 
n ,Rxn ,Rxn 
n ,Syn ,Syn 
n ,Tzn ,Tzn 
∗
∗
∗
⊕ GRx
(t ),
Gfxn ,Syn ,Tzn ⊕ GRx
n ,gyn ,Tzn
n ,Syn ,hzn
∗
∗
∗
(t ) .
⊕ GRx
Gfxn ,gyn ,Tzn ⊕ Gfx
n ,Syn ,hzn
n ,gyn ,hzn
(.)
Letting n → ∞ in (.) and by the property of ψ, we can obtain
lim ψ m(xn , yn , zn , t )
∗
∗
∗
∗
(t ), Gfx
(t ), Ggy
(t ), Ghz
(t ),
= ψ lim min GRx
n ,Syn ,Tzn 
n ,Rxn ,Rxn 
n ,Syn ,Syn 
n ,Tzn ,Tzn 
n→∞
n→∞
∗
∗
⊕ GRx
⊕ GRx
(t ),
n ,gyn ,Tzn
n ,Syn ,hzn
∗
∗
∗
Gfxn ,gyn ,Tzn ⊕ Gfx
(t
⊕
G
)
.

Rx
,Sy
,hz
,gy
,hz
n n n
n n n
∗
Gfx
n ,Syn ,Tzn
(.)
As the proof of Theorem ., we know
∗
∗
∗
∗
(t ) ≥ Ga,a,z
⊕ GRx
⊕ GRx
(t ),
lim Gfx
n ,Syn ,Tzn
n ,gyn ,Tzn
n ,Syn ,hzn
n→∞
∗
∗
∗
∗
(t ) ≥ Ga,a,z
⊕ Gfx
⊕ GRx
(t ).
lim Gfx
n ,gyn ,Tzn
n ,Syn ,hzn
n ,gyn ,hzn
n→∞
Then (.) is
∗
∗
∗
(t ), Gz,a,a
(t ), Gz,a,a
(t )
lim ψ m(xn , yn , zn , t ) ≥ ψ min , , , Gz,a,a
n→∞
∗
(t ) .
= ψ Gz,a,a
(.)
Zhu et al. Fixed Point Theory and Applications (2015) 2015:131
Page 10 of 14
Without loss of generality, we assume that t in (.) is a continuous point of Ga,a,z (·). By
the left-continuity of the distribution function and the continuity of ψ, there exists δ > 
such that
∗
∗
(t) > Ga,a,z
(t),
ψ Ga,a,z
for all t ∈ (t – δ, t ]. Since Ga,a,z (·) is nondecreasing, the set of all discontinuous points of
Ga,a,z (·) is a countable set at most. Thus, when t is a discontinuous point of GTa,Ta,Sa (·),
we can choose a continuous point t of GTa,Ta,Sa (·) in (t – δ, t ] to replace t .
∗
∗
(t ) ≥ ψ{Ga,a,z
(t )}, which contradicts (.). Then
Let n → ∞ in (.), then we have Ga,a,z
a = z, limn→∞ hzn = a.
Since Rx is a closed subset of X and limn→∞ Rxn = a, there exists p in X such that a = Rp,
we claim that fp = a. Suppose not, then by using (.), we obtain
∗
∗
∗
∗
∗
(t) ≥ ψ min GRp,Sy
(t), Gfp,Rp,Rp
(t), Ggy
(t), Ghz
(t),
Gfp,gy
n ,hzn
n ,Tzn
n ,Syn ,Syn
n ,Tzn ,Tzn
∗
∗
∗
(t),
⊕ GRp,Sy
Gfp,Syn ,Tzn ⊕ GRp,gy
n ,Tzn
n ,hzn
∗
∗
∗
(t) .
Gfp,gyn ,Tzn ⊕ Gfp,Sy
⊕ GRp,gy
n ,hzn
n ,hzn
Similarly, we can get fp = Rp = a. Hence, p is the coincidence point of the pair (f , R).
By the condition fx ⊆ Sx and fp = a, there exists u ∈ X such that a = Su. Now we claim
that gu = a. In fact, if gu = a, then from (.), we have
∗
∗
∗
∗
∗
Gfp,gu,hz
(t) ≥ ψ min GRp,Su,Tz
(t), Gfp,Rp,Rp
(t), Ggu,Su,Su
(t), Ghz
(t),
n
n
n ,Tzn ,Tzn
∗
∗
∗
Gfp,Su,Tzn ⊕ GRp,gu,Tz
(t),
⊕ GRp,Su,hz
n
n
∗
∗
∗
(t) ;
Gfp,gu,Tzn ⊕ Gfp,Su,hz
⊕ GRp,gu,hz
n
n
in the same way, we can also obtain gu = a, and so u is the coincidence point of the pair
(g, S).
Since gX ⊂ TX, there exists v ∈ X such that a = Tv. We claim that hv = a. If not, from
(.) and the property of ψ, we have
∗
∗
∗
∗
∗
Gfp,gu,hv
(t) ≥ ψ min GRp,Su,Tv
(t), Gfp,Rp,Rp
(t), Ggu,Su,Su
(t), Ghv,Tv,Tv
(t),
∗
∗
∗
(t),
⊕ GRp,Su,hv
Gfp,Su,Tv ⊕ GRp,gu,Tv
∗
∗
∗
(t)
Gfp,gu,Tv ⊕ Gfp,Su,hv
⊕ GRp,gu,hv
∗
∗
∗
∗
(t),
(t), Ga,a,a
⊕ Ga,a,a
⊕ Ga,a,hv
= ψ min , , , Ghv,a,a
∗
∗
∗
⊕ Ga,a,hv
(t)
Ga,a,a ⊕ Ga,a,hv
∗
∗
∗
≥ ψ min , , , Ghv,a,a
(t), Ga,a,hv
(t), Ga,a,hv
(t)
∗
∗
(t) > Ga,a,hv
(t),
= ψ Ga,a,hv
a contradiction. Hence hv = Tv = a, and so v is the coincidence point of the pair (h, T).
Therefore, in all the above cases, we obtain fp = Rp = a, gu = Su = a, and hv = Tv = a.
Now, the weak compatibility of the pairs (f , R), (g, S), and (h, T) give fa = Ra, ga = Sa, and
ha = Ta.
Zhu et al. Fixed Point Theory and Applications (2015) 2015:131
Page 11 of 14
Next, we show that fa = a. In fact, if fa = a, then from (.) we have
∗
∗
∗
∗
∗
(t) ≥ ψ min GRa,Su,Tv
(t), Gfa,Ra,Ra
(t), Ggu,Su,Su
(t), Ghv,Tv,Tv
(t),
Gfa,a,a
∗
∗
∗
∗
∗
∗
⊕ GRa,Su,hv
⊕ Gfa,Su,hv
⊕ GRa,gu,hv
(t), Gfa,gu,Tv
(t)
Gfa,Su,Tv ⊕ GRa,gu,Tv
∗
∗
∗
∗
= ψ min GRa,a,a
(t),
(t), , , , Gfa,a,a
⊕ GRa,a,a
⊕ GRa,a,a
∗
∗
∗
Gfa,a,a ⊕ Gfa,a,hv
(t)
⊕ GRa,a,a
∗
∗
∗
≥ ψ min , , , Gfa,a,a
(t), Gfa,a,a
(t), Gfa,a,a
(t)
∗
∗
= ψ Gfa,a,a
(t) > Gfa,a,a
(t),
which is a contradiction, hence fa = a and so fa = Ra = a. Similarly, it can be shown that
ga = Sa = a and ha = Ta = a, so we get fa = ga = ha = Ra = Sa = Ta = a, which means that
a is a common fixed point of f , g, h, R, S, and T.
Next, we will show the uniqueness. Actually, suppose that w ∈ X, w = a is another common fixed point of f , g, h, R, S, and T. Then by (.), we have
∗
∗
∗
∗
∗
Gw,a,a
(t) ≥ ψ min GRw,Sa,Ta
(t), Gfw,Rw,Rw
(t), Gga,Sa,Sa
(t), Gha,Ta,Ta
(t),
∗
∗
∗
(t),
Gfw,Sa,Ta ⊕ GRw,ga,Ta
⊕ GRw,Sa,ha
∗
∗
∗
⊕ GRw,ga,ha
(t)
Gfw,ga,Ta ⊕ Gfw,Sa,ha
∗
∗
∗
∗
≥ ψ min Gw,a,a
(t),
(t), , , , Gfa,a,a
⊕ Gw,a,a
⊕ Gw,a,a
∗
∗
∗
Gw,a,a ⊕ Gw,a,hv
(t)
⊕ Gw,a,a
∗
∗
∗
≥ ψ min , , , Gw,a,a
(t), Gw,a,a
(t), Gw,a,a
(t)
∗
∗
= ψ Gw,a,a
(t) > Gw,a,a
(t),
which is a contradiction, so f , g, h, R, S, and T have a unique common fixed point.
Finally, if condition (ii) or (iii) holds, then the argument is similar to the above, so we
omit it. This completes the proof of Theorem ..
Taking ψ(t) = ρt, ρ ∈ (, +∞), then we can obtain the following results.
Corollary . Let (X, G∗ , ) be a Menger PGM-space. Suppose the self-mappings f , g, h,
R, S, and T : X → X satisfy the following conditions:
∗
∗
∗
∗
∗
Gfx,gy,hz
(t) ≥ ρ min GRx,Sy,Tz
(t), Gfx,Rx,Rx
(t), Ggy,Sy,Sy
(t), Ghz,Tz,Tz
(t),
∗
∗
∗
(t),
Gfx,Sy,Tz ⊕ GRx,gy,Tz
⊕ GRx,Sy,hz
∗
∗
∗
Gfx,gy,Tz ⊕ Gfx,Sy,hz
(t) ,
⊕ GRx,gy,hz
for all x, y, and z ∈ X, where ρ ∈ (, +∞). If one of the following conditions is satisfied, then
the pairs (f , R), (g, S), and (h, T) have a common fixed point of coincidence in X:
(i) the subspace Rx is closed in X, fx ⊆ Sx, gx ⊆ Tx, and the two pairs of (f , R) and (g, S)
satisfy the common (E.A) property;
Zhu et al. Fixed Point Theory and Applications (2015) 2015:131
Page 12 of 14
(ii) the subspace Sx is closed in X, gx ⊆ Tx, hx ⊆ Rx, and the two pairs of (g, S) and
(h, T) satisfy the common (E.A) property;
(iii) the subspace Tx is closed in X, fx ⊆ Sx, hx ⊆ Rx, and the two pairs of (f , R) and
(h, T) satisfy the common (E.A) property.
Moreover, if the pairs (f , R), (g, S), and (h, T) are weakly compatible, the f , g, h, R, S, and
T have a unique common fixed point in X.
4 An application
In this section, we will provide an example to exemplify the validity of the main result.
t
∗
Example . Let X = [, ], Gx,y,z
(t) = t+|x–y|+|y–z|+|z–x|
, from Example ., we know
∗
(X, G , ) is a PGM-space. We define the mappings f , g, h, R, S, and T by
,
fx = 
,

,
hx = 
,

⎧
⎪
⎨,
Sx =  ,
⎪
⎩
,

,
gx = 
,

,
Rx = 
,

⎧
⎪
⎨,
Tx =  ,
⎪
⎩
,

x ∈ [,  ],
x ∈ (  , ],
x ∈ [,  ],
x ∈ (  , ],
x ∈ [,  ),
x =  ,
x ∈ (  , ],
x ∈ [,  ],
x ∈ (  , ],
x ∈ [,  ],
x ∈ (  , ],
x ∈ [,  ],
x =  ,
x ∈ (  , ].
Noting that f , g, h, R, S, and T are discontinuous mappings, RX is closed in X. From the
definition of f , g, h, R, S, and T, we have fx ⊆ Sx, gx ⊆ Tx; let xn = n +  , yn = n +  , then
the pairs (f , R) and (g, S) satisfy the common (E.A) property. Thus, the condition (i) in
Theorem . is satisfied. It is not difficult to find that (f , R), (g, S), and (h, T) are weakly

(t). Next we will show that (.) is also satisfied.
compatible. Let φ(t) = 
∗
∗
To prove (.), we just need to show Gfx,gy,hz
(φ(t)) ≥ GRx,Sy,Tz
(t); we discuss the following
cases.
∗
(φ(t)) = , then (.) is obviously satisfied.
Case (). For x, y, z ∈ [,  ], we have Gfx,gy,hz

Case (). For x, y, z ∈ (  , ], we have
∗
φ(t) =
Gfx,gy,hz
t
t+


≥
t
∗
= GRx,Sy,Tz
(t).

t + 
∗
Case (). For x, y ∈ [,  ], z ∈ (  , ], it is not difficult to find that GRx,Sy,Tz
(t) =
y ∈ [,

)

nor y =

.

On the other hand,
∗
Gfx,gy,hz
(φ(t)) = t+t 

t
t+ 
, neither
, we have
∗
∗
Gfx,gy,hz
φ(t) ≥ GRx,Sy,Tz
(t).
Case (). For x, z ∈ [,  ], y ∈ (  , ], similar to Case (), we have
∗
Gfx,gy,hz
φ(t) =
t
t+


≥
t
t+


∗
= GRx,Sy,Tz
(t).
∗
Case (). For y, z ∈ [,  ], x ∈ (  , ], we have Gfx,gy,hz
(φ(t)) =
into two subcases.
t
t+ 
. Next we divide the study
Zhu et al. Fixed Point Theory and Applications (2015) 2015:131
Page 13 of 14
∗
(a) If y = z =  , x ∈ (  , ], we have GRx,Sy,Tz
(t) =
t
t+ 

, then
∗
∗
φ(t) ≥ GRx,Sy,Tz
(t).
Gfx,gy,hz
(b) If y =


∗
or z =  , x ∈ (  , ], we have GRx,Sy,Tz
(t) =
t
t+ 
, then
∗
∗
Gfx,gy,hz
(t)
φ(t) ≥ GRx,Sy,Tz
is also satisfied.
Case (). For x ∈ [,  ], y, z ∈ (  , ], we have
∗
φ(t) =
Gfx,gy,hz
t
t+


≥
t
t+


∗
= GRx,Sy,Tz
(t).
∗
(φ(t)) =
Case (). For y ∈ [,  ], x, z ∈ (  , ], we have Gfx,gy,hz
into two subcases.
∗
(a) If y ∈ [,  ), x, z ∈ (  , ], we have GRx,Sy,Tz
(t) =
t
t+ 
t
t+ 
. Next we divide the study
t
t+ 

. Next we divide the study
, then
∗
∗
Gfx,gy,hz
φ(t) ≥ GRx,Sy,Tz
(t).
∗
(t) =
(b) If y =  , x, z ∈ (  , ], we have GRx,Sy,Tz
t
t+ 

, then
∗
∗
φ(t) ≥ GRx,Sy,Tz
(t).
Gfx,gy,hz
∗
(φ(t)) =
Case (). For z ∈ [,  ], x, y ∈ (  , ], we have Gfx,gy,hz
into two subcases.
∗
(a) If z ∈ [,  ), x, y ∈ (  , ], we have GRx,Sy,Tz
(t) =
t
t+ 
, then
∗
∗
(t).
Gfx,gy,hz
φ(t) ≥ GRx,Sy,Tz
∗
(b) If z =  , x, y ∈ (  , ], we have GRx,Sy,Tz
(t) =
t
t+ 
, then
∗
∗
Gfx,gy,hz
φ(t) ≥ GRx,Sy,Tz
(t).
Then in all the above cases, f , g, h, R, S, and T satisfy the conditions (.) and (i) of
Theorem .. So, f , g, h, R, S, and T have a unique common fixed point in [, ]. In fact, 
is the unique common fixed point of f , g, h, R, S, and T.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the editor and the referees for their constructive comments and suggestions. The
research was supported by the National Natural Science Foundation of China (11361042, 11326099, 11461045, 11071108)
and the Provincial Natural Science Foundation of Jiangxi, China (20132BAB201001, 20142BAB211016, 2010GZS0147).
Received: 3 May 2015 Accepted: 13 July 2015
Zhu et al. Fixed Point Theory and Applications (2015) 2015:131
Page 14 of 14
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