 Research
 Open Access
 Published:
On some recent coincidence and immediate consequences in partially ordered bmetric spaces
Fixed Point Theory and Applications volume 2015, Article number: 63 (2015)
Abstract
The aim of this article is to present, improve and generalize some recent coincidence and coupled coincidence point results from several papers in the framework of partially ordered bmetric spaces. Two examples are also provided to support the superiority of the obtained results.
Introduction and preliminaries
It is well known that the Banach contraction principle (see [1]) plays an important role in various fields of applied mathematical analysis and scientific applications, and it has been generalized and improved in many different directions. Some of such generalizations are obtained via rational metric spaces, such as ordered Banach spaces, partially ordered metric spaces, 2metric spaces, fuzzy metric spaces, probabilistic metric spaces, Gmetric spaces, cone metric spaces, cone Banach spaces, bmetric spaces or metric type spaces, etc. (see [2–27]). One of the most influential spaces is bmetric space, also called metric type space by some authors, introduced by Bakhtin (see [16]) in 1989. Since then, a large number of papers on fixed point results in the setting of bmetric spaces have appeared (see [17–27]).
The following definitions and results will be needed in what follows.
Definition 1.1
([28])
Let X be a (nonempty) set and \(s\geq1\) be a given real number. A function \(d:X\times X\rightarrow[0,\infty)\) is called a bmetric on X if, for all \(x,y,z\in X\), the following conditions hold:

(b1)
\(d ( x,y ) =0\) if and only if \(x=y\);

(b2)
\(d ( x,y ) =d (y,x )\);

(b3)
\(d ( x,z ) \leq s [ d ( x,y ) +d (y,z ) ]\).
In this case, the pair \((X,d)\) is called a bmetric space. If \((X,\preceq )\) is still a partially ordered set, then \((X,\preceq,d )\) is called a partially ordered bmetric space.
Otherwise, for more concepts such as bconvergence, bcompleteness, bCauchy sequence and bclosed set in bmetric spaces, we refer the reader to [20–29] and the references mentioned therein.
Definition 1.2
Let \(( X,\preceq ) \) be a partially ordered set and f, g, h be three selfmaps on X such that \(f(X)\cup g(X)\subseteq h(X)\). Then

(1)
elements \(x,y\in X\) are called comparable if \(x\preceq y\) or \(y\preceq x\) holds;

(2)
f is called monotone gnondecreasing w.r.t. ⪯ if \(gx\preceq gy\) implies \(fx\preceq fy\). In particular, f is called nondecreasing w.r.t. ⪯ if \(x\preceq y\) implies \(fx\preceq fy\);

(3)
the pair \(( f,g ) \) is said to be weakly increasing if \(fx\preceq gfx\) and \(gx\preceq fgx\) for all \(x\in X\);

(4)
the pair \(( f,g ) \) is said to be partially weakly increasing if \(fx\preceq gfx\) for all \(x\in X\);

(5)
f is said to be gweakly isotone increasing if \(fx\preceq gfx\preceq fgfx\) for all \(x\in X\);

(6)
the pair \((f,g )\) is said to be weakly increasing with respect to h if and only if for all \(x\in X\), \(fx\preceq gy\) for all \(y\in h^{1} ( fx ) \), and \(gx\preceq fy\) for all \(y\in h^{1} ( gx ) \);

(7)
the ordered pair \(( f,g ) \) is said to be partially weakly increasing with respect to h if \(fx\preceq gy\) for all \(y\in h^{1} ( fx ) \);

(8)
a partially ordered bmetric space \((X,\preceq,d )\) is said to be regular if the following conditions hold:

(i)
if a nondecreasing sequence \(x_{n}\rightarrow x\), then \(x_{n}\preceq x\) for all n,

(ii)
if a nonincreasing sequence \(y_{n}\rightarrow y\), then \(y_{n}\succeq y\) for all n;

(i)

(9)
the pair \((f,g)\) is said to be compatible if and only if \(\lim_{n\rightarrow\infty}d(fgx_{n},gfx_{n})=0\), whenever \(\{x_{n}\}\) is a sequence in X such that \(\lim_{n\rightarrow\infty}fx_{n}=\lim_{n\rightarrow\infty}gx_{n}=t\) for some \(t\in X\);

(10)
the pair \((f,g)\) is said to be weakly compatible if f and g commute at their coincidence points (i.e., \(fgx=gfx\), whenever \(fx=gx\)).
Fixed point results in partially ordered metric spaces were firstly obtained by Ran and Reurings (see [30]) and then by Nieto and López (see [31, 32]). Subsequently, many authors presented numerous interesting and significant results in ordered metric and ordered bmetric spaces (see [3, 28, 29, 33–38]).
Throughout this paper, we introduce the denotations Ψ, ϒ, Φ, Θ as follows.
Let Ψ be the family of all functions \(\psi:[0,\infty )\rightarrow[0,\infty)\) satisfying the following conditions:

(a1)
ψ is continuous,

(a2)
ψ is nondecreasing,

(a3)
\(\psi ( 0 ) =0<\psi ( t ) \) for every \(t>0\).
In this case, ψ is said to be an altering distance function.
Let ϒ be the family of all functions \(\varphi:[0,\infty )\rightarrow[0,\infty)\) satisfying the following conditions:

(b1)
φ is right continuous,

(b2)
φ is nondecreasing,

(b3)
\(\varphi(t)< t\) for every \(t>0\).
Let Φ be the family of all functions \(\phi:[0,\infty )\rightarrow[0,\infty)\) satisfying the following conditions:

(c1)
ϕ is lower semicontinuous,

(c2)
\(\phi(t)=0\) if and only if \(t=0\).
Let Θ be the family of all continuous functions \(\theta :[0,\infty)\rightarrow[0,\infty)\) with \(\theta ( t ) =0\) if and only if \(t=0\).
In [29] authors introduced and proved two theorems as follows.
Let \((X,\preceq,d)\) be an ordered bmetric space with \(s>1\), and let \(f,g,R,S:X\rightarrow X\) be four mappings. For all \(x,y\in X\), set
Theorem 1.3
([29])
Let \((X,\preceq,d)\) be a partially ordered complete bmetric space with \(s>1\). Let \(f,g,R,S:X\rightarrow X\) be four mappings such that \(f ( X ) \subseteq R ( X ) \) and \(g ( X ) \subseteq S ( X ) \). Suppose that for every two comparable elements \(Sx,Ry\in X\), we have
where \(\psi,\varphi:[0,\infty)\rightarrow[0,\infty)\) are altering distance functions. Let f, g, R and S be continuous, the pairs \(( f,S ) \) and \(( g,R ) \) be compatible and the pairs \(( f,g )\) and \(( g,f ) \) be partially weakly increasing with respect to R and S, respectively. Then the pairs \(( f,S ) \) and \(( g,R ) \) have a coincidence point z in X. Moreover, if Rz and Sz are comparable, then z is a coincidence point of f, g, R and S.
Theorem 1.4
([29])
Let \((X,\preceq,d)\) be a regular partially ordered complete bmetric space with \(s>1\), \(f,g,R,S:X\rightarrow X\) be four mappings such that \(f ( X ) \subseteq R ( X ) \) and \(g ( X ) \subseteq S ( X ) \) and \(R ( X ) \) and \(S ( X ) \) are bclosed subsets of X. Suppose that for every two comparable elements \(Sx,Ry\in X\), we have
where \(\psi,\varphi:[0,\infty)\rightarrow[0,\infty)\) are altering distance functions. Then the pairs \(( f,S ) \) and \(( g,R ) \) have a coincidence point z in X provided that the pairs \(( f,S ) \) and \(( g,R ) \) are weakly compatible and the pairs \(( f,g ) \) and \(( g,f )\) are partially weakly increasing with respect to R and S, respectively. Moreover, if Rz and Sz are comparable, then z is a coincidence point of f, g, R and S.
Similarly, in [35] authors introduced and proved the following results.
Let \((X,\preceq,d)\) be a partially ordered bmetric space with \(s>1\) and \(T:X\rightarrow X\) and \(g:X\rightarrow X\) be two mappings. For all \(x,y\in X\), put
and
The mapping T is called an almost generalized \((\psi,\varphi ,L )\)contractive mapping with respect to g for some \(\psi\in \Psi\), \(\varphi\in\Upsilon\), and \(L\geq0\) if
for all \(x,y\in X\) with \(gx\preceq gy\).
Theorem 1.5
([35])
Suppose that \((X,\preceq,d)\) is a partially ordered complete bmetric space with \(s>1\). Let \(T:X\rightarrow X\) be an almost generalized \(( \psi,\varphi,L ) \)contractive mapping with respect to \(g:X\rightarrow X\), and T and g be continuous such that T is a monotone gnondecreasing mapping, commutative with g and \(T ( X ) \subseteq g ( X ) \). If there exists \(x_{0}\in X \) such that \(gx_{0}\preceq Tx_{0}\), then T and g have a coincidence point in X.
Theorem 1.6
([35])
Suppose that \((X,\preceq,d)\) is a partially ordered complete bmetric space with \(s>1\). Let \(T:X\rightarrow X\) be an almost generalized \(( \psi,\varphi,L ) \)contractive mapping with respect to \(g:X\rightarrow X\), T be a monotone gnondecreasing mapping and \(T ( X ) \subseteq g ( X ) \). Also suppose that if \(\{ gx_{n} \} \subset X\) is a nondecreasing sequence with \(gx_{n}\rightarrow gz\) in gX, then \(gx_{n}\preceq gz\), \(gz\preceq g ( gz ) \) for all n hold. Also suppose that gX is bclosed. If there exists \(x_{0}\in X\) such that \(gx_{0}\preceq Tx_{0}\), then T and g have a coincidence point. Further, if T and g commute at their coincidence points, then T and g have a common fixed point.
In [34] authors repeated some wellknown notions and proved the following new results.
Let \(( X,\preceq )\) be a partially ordered set and \(F:X\times X\rightarrow X\) and \(g:X\rightarrow X\) be two mappings. Then

(1)
an element \(( x,y ) \in X\times X\) is called a coupled coincidence point of F and g if
$$ F ( x,y ) =gx, \qquad F ( y,x ) =gy. $$ 
(2)
F and g are commutative if for all \(x,y\in X\),
$$ F ( gx,gy ) =g \bigl( F ( x,y ) \bigr) . $$ 
(3)
F is said to have the mixed gmonotone property if F is nondecreasing gmonotone in its first argument and is nonincreasing gmonotone in its second argument, that is, for any \(x,y\in X\),
$$ x_{1},x_{2}\in X,\quad gx_{1}\preceq gx_{2}\quad\Rightarrow\quad F ( x_{1},y ) \preceq F ( x_{2},y ) $$and
$$ y_{1},y_{2}\in X,\quad gy_{1}\preceq gy_{2}\quad\Rightarrow\quad F ( x,y_{1} ) \succeq F ( x,y_{2} ) . $$In particular, if g is an identity mapping, then F is said to have the mixed monotone property.
Let \((X,\preceq,d )\) be a partially ordered bmetric space with \(s>1\), and let \(T:X\times X\rightarrow X\) and \(g:X\rightarrow X\) be two mappings. Set
and
Let \(\psi\in\Psi\), \(\phi\in\Phi\) and \(\theta\in\Theta\). The mapping T is called an almost generalized \(( \psi,\phi,\theta )\)contractive mapping with respect to g if there exists \(L\geq0\) such that
for all \(x,y,u,v\in X\) with \(gx\preceq gu\) and \(gy\succeq gv\).
Theorem 1.7
([34])
Suppose that \((X,\preceq,d)\) is a partially ordered complete bmetric space with \(s>1\). Let \(T:X\times X\rightarrow X\) be an almost generalized \(( \psi,\phi,\theta )\)contractive mapping with respect to \(g:X\rightarrow X\), and T and g be continuous such that T has the mixed gmonotone property and commutes with g. Also, suppose that \(T ( X\times X ) \subseteq g ( X ) \). If there exists \(( x_{0},y_{0} ) \in X\times X\) such that \(gx_{0}\preceq T ( x_{0},y_{0} ) \) and \(gy_{0}\succeq T ( y_{0},x_{0} ) \), then T and g have a coupled coincidence point in X.
It needs emphasizing that the following crucial lemma is utilized again and again in proving of all main results from [29, 34] and [35].
Lemma 1.8
([33])
Let \((X,d)\) be a bmetric space with \(s\geq1\) and suppose that \(\{x_{n}\}\) and \(\{y_{n}\}\) are bconvergent to x and y, respectively. Then
In particular, if \(x=y\), then \(\lim_{n\rightarrow\infty }d ( x_{n},y_{n} ) =0\).
Moreover, for each \(z\in X\), we have
Main results
In this section, we improve and generalize coincidence and coupled coincidence point results of Theorems 1.31.7 in several directions without utilizing Lemma 1.8 in the proofs.
Theorem 2.1
Let \((X,\preceq,d)\) be a partially ordered complete bmetric space with \(s>1\). Let \(f,g,R,S:X\rightarrow X\) be four mappings such that \(f ( X ) \subseteq R ( X ) \) and \(g ( X ) \subseteq S ( X ) \). Suppose that for every two comparable elements \(Sx,Ry\in X\), we have
where \(\varepsilon>1\) is a constant and \(M_{s} ( x,y ) \) is given by (1.1). Let f, g, R and S be continuous, the pairs \(( f,S ) \) and \((g,R ) \) be compatible and the pairs \(( f,g )\) and \(( g,f ) \) be partially weakly increasing with respect to R and S, respectively. Then the pairs \(( f,S )\) and \(( g,R ) \) have a coincidence point z in X. Moreover, if Rz and Sz are comparable, then z is a coincidence point of f, g, R and S.
Proof
Let \(x_{0}\) be an arbitrary point of X. Similar to [29], we construct a sequence \(\{ z_{n} \} \) in X such that \(z_{2n+1}=fx_{2n}=Rx_{2n+1}\) and \(z_{2n+2}=gx_{2n+1}=Sx_{2n+2}\) for all \(n\geq0\). Since the pairs \(( f,g )\) and \(( g,f ) \) are partially weakly increasing with respect to R and S, respectively, it follows that \(z_{n}\preceq z_{n+1}\) for all \(n\geq1\). We complete the proof only in two steps.
Step I. We prove that
for all \(n\geq1\), where \(\lambda\in[0,\frac{1}{s})\).
We first assume that \(z_{n}\neq z_{n+1}\) for all \(n\geq1\). Since \(Sx_{2n}=z_{2n}\) and \(Rx_{2n1}=z_{2n1}\) are comparable, then (2.1) means that
If \(d ( z_{2n1},z_{2n} ) \leq d ( z_{2n},z_{2n+1} ) \), then (2.3) becomes
which gives a contradiction (because \(s^{ \varepsilon}>1\)). Thus
Again, since \(Sx_{2n}=z_{2n}\) and \(Rx_{2n+1}=z_{2n+1}\) are comparable, then (2.1) implies that
If \(d ( z_{2n},z_{2n+1} ) \leq d ( z_{2n+1},z_{2n+2} ) \), then (2.5) becomes
which gives a contradiction (because \(s^{ \varepsilon}>1\)). So
Now, combining (2.4) and (2.6), we get that (2.2), where \(\lambda =\frac{1}{s^{ \varepsilon}}\in[0,\frac{1}{s})\).
Assume now that \(z_{n_{0}}=z_{n_{0}+1}\) for some \(n_{0}\). If \(n_{0}=2k1\), then \(z_{2k1}=z_{2k}\) gives that \(z_{2k}=z_{2k+1}\). Indeed, since \(Sx_{2k}=z_{2k}\) and \(Rx_{2k1}=z_{2k1}\) are comparable, then by (2.3) we have that
which establishes that \(d(z_{2k},z_{2k+1})=0\), that is, \(z_{2k}=z_{2k+1}\). If \(n_{0}=2k\), then \(z_{2k}=z_{2k+1}\) gives that \(z_{2k+1}=z_{2k+2}\). Actually, since \(Sx_{2k}=z_{2k}\) and \(Rx_{2k+1}=z_{2k+1}\) are comparable, then by (2.5) we have that
which implies that \(d(z_{2k+1},z_{2k+2})=0\), that is, \(z_{2k+1}=z_{2k+2}\). Consequently, the sequence \(\{ z_{n} \} \) in both cases becomes constant for \(n\geq n_{0}\) and hence (2.2) holds.
Step II. We show that f, g, R and S have a coincidence point.
Making the most of (2.2) and Lemma 3.1 of [18], we obtain that \(\{ z_{n} \}\) is a bCauchy sequence. Since \((X,d)\) is bcomplete, then there exists \(z\in X\) such that \(z_{n}\) bconverges to z. Accordingly,
Note that \(( f,S ) \) is compatible, that is, \(\lim_{n\rightarrow\infty}d ( Sfx_{2n},fSx_{2n} ) =0\). Otherwise, by the continuity of f and S, it is valid that \(Sfx_{2n}\rightarrow Sz\) and \(fSx_{2n}\rightarrow fz\), as \(n\rightarrow\infty\). Now, we have that
Taking the limit as \(n\rightarrow\infty\) in (2.7), we get \(\frac{1}{s}d ( Sz,fz ) \leq0\), i.e., \(fz=Sz\).
Following an argument similar to that mentioned above, we obtain \(gz=Rz\). Now that Sz and Rz are comparable, hence by (2.1) it is obvious that
which establishes that \(fz=gz\) (because \(s^{ \varepsilon}>1\)). Therefore, \(fz=gz=Sz=Rz\). □
Theorem 2.2
Let \((X,\preceq,d)\) be a regular partially ordered complete bmetric space with \(s>1\), \(f,g,R,S:X\rightarrow X\) be four mappings such that \(f ( X ) \subseteq R ( X ) \) and \(g ( X ) \subseteq S ( X ) \) and \(R ( X ) \) and \(S ( X ) \) are bclosed subsets of X. Suppose that for every two comparable elements \(Sx,Ry\in X\), we have
where \(\varepsilon>1\) is a constant and \(M_{s} ( x,y ) \) is given by (1.1). Then the pairs \(( f,S ) \) and \(( g,R ) \) have a coincidence point z in X provided that the pairs \(( f,S ) \) and \(( g,R ) \) are weakly compatible and the pairs \(( f,g ) \) and \(( g,f ) \) are partially weakly increasing with respect to R and S, respectively. Moreover, if Rz and Sz are comparable, then z is a coincidence point of f, g, R and S.
Proof
Similar to the proof of Theorem 2.1, we can construct the sequence \(\{z_{n}\}\) and obtain that there exists \(z\in X\) such that \(z_{n}\rightarrow z\) as \(n\rightarrow\infty\). Since \(R ( X ) \) and \(S (X )\) are bclosed, \(\{ z_{2n+1} \} \subseteq R ( X )\) and \(\{ z_{2n+2} \} \subseteq S ( X )\), then there exist \(u,v\in X\) such that \(z=Ru\) and \(z=Sv\). That is,
We now prove that z is a coincidence point of f and S.
By using \(Rx_{2n+1}\rightarrow Sv\) (\(n\rightarrow\infty\)) and the regularity of \((X,\preceq,d)\), it follows that \(Rx_{2n+1}\preceq Sv\). As a consequence, by (2.8) we have that
where
By virtue of the triangle inequality and (2.9), it may be verified that
Taking the limit as \(n\rightarrow\infty\) in (2.11) together with (2.10), we arrive at
As a result, \(Sv=fv\) (because \(s^{ \varepsilon}>s>1\)). Hence, \(z=Sv=fv\). Next by the compatibility of f and S, we claim that \(fz=fSv=Sfv=Sz\). That is to say, z is a coincidence point of f and S.
Similarly, it can be shown that z is a coincidence point of g and R. The remainder is the same as the proof of Theorem 2.1 and therefore we omit it. □
Corollary 2.3
Let \((X,\preceq,d)\) be a partially ordered complete bmetric space with \(s>1\). Let \(f,g:X\rightarrow X\) be two mappings. Suppose that for every comparable elements \(x,y\in X\),
where
Then the pair \(( f,g ) \) has a common fixed point z in X provided that the pair \(( f,g ) \) is weakly increasing and either,

(a)
f and g are continuous, or

(b)
\(( X,d,\preceq ) \) is regular.
Proof
Taking \(R=S=I_{X}\) (an identity mapping on X) in Theorems 2.1 and 2.2, the desired result holds. □
Remark 2.4
Compared with Theorem 2.1, Theorem 2.2 omits the assumption of continuity of f, g, R and S, and replaces the compatibility of the pairs \(( f,S ) \) and \(( g,R ) \) by the weak compatibility of the pairs.
Remark 2.5
Theorem 2.1 and Theorem 2.2 greatly generalize Theorem 1.3 and Theorem 1.4, respectively. In fact, condition (2.1) or (2.8) is much wider than condition (1.2) or (1.3). On the one hand, we delete the functions ψ and φ. On the other hand, our condition is much more general because \(\varepsilon>1\) is arbitrary. In addition, the proofs of Theorem 2.1 and Theorem 2.2 are shorter than those of Theorem 1.3 and Theorem 1.4 because we never use Lemma 1.8, but Theorem 1.3 and Theorem 1.4 are strongly dependent on this lemma.
Definition 2.6
Let \((X,\preceq,d)\) be a partially ordered bmetric space with \(s>1\). The mapping \(T:X\rightarrow X\) is called an almost generalized \(( \psi,L ) \)contractive mapping with respect to \(g:X\rightarrow X\) for some \(\psi\in\Psi\), \(\varepsilon>1\) and \(L\geq0\) if
for all \(x,y\in X\) with \(gx\preceq gy\), where \(M(x,y)\) and \(N(x,y)\) are given by (1.4) and (1.5), respectively.
Theorem 2.7
Suppose that \((X,\preceq,d)\) is a partially ordered complete bmetric space with \(s>1\). Let \(T:X\rightarrow X\) be an almost generalized \((\psi ,L ) \)contractive mapping with respect to \(g:X\rightarrow X\), and T and g be continuous such that T is a monotone gnondecreasing mapping, compatible with g and \(T ( X ) \subseteq g (X )\). If there exists \(x_{0}\in X\) such that \(gx_{0}\preceq Tx_{0}\), then T and g have a coincidence point in X.
Proof
Similar to [35], we construct sequences \(\{ x_{n} \} \) and \(\{ y_{n} \} \) in X such that
for which
The same as in [35] we can assume that \(y_{n}\neq y_{n+1}\) for all \(n\geq0\). For this case we shall show that
for all \(n\geq1\), where \(\lambda\in[0,\frac{1}{s})\).
Indeed, by (2.12)(2.14), we have that
where
and
Hence, by (2.16)(2.18), we arrive at
If \(d ( y_{n},y_{n+1} )\geq d ( y_{n1},y_{n} ) >0\) for some \(n\in\Bbb{N}\), then by (2.19) we get that
or equivalently,
This is a contradiction. Thus from (2.19) it follows that
that is, (2.15) holds, where \(\lambda=\frac{1}{s^{ \varepsilon }}\in[0,\frac{1}{s})\).
Now combining (2.15) and Lemma 3.1 of [18], we claim that \(\{ y_{n} \} = \{ Tx_{n} \} = \{ gx_{n+1} \} \) is a bCauchy sequence. Since \(( X,d ) \) is bcomplete, then there exists \(x\in X \) such that
Thus by the compatibility of T and g, we get that
By the continuity of T and g, it may be verified that
Hence by the triangle inequality together with (2.20) and (2.21), it ensures us that
Therefore, we obtain that \(Tx=gx\), that is, x is a coincidence point of T and g. □
In the following theorem we omit the assumption of continuity of T and g.
Theorem 2.8
Suppose that \((X,\preceq,d)\) is a partially ordered complete bmetric space with \(s>1\). Let \(T:X\rightarrow X\) be an almost generalized \(( \psi ,L ) \)contractive mapping with respect to \(g:X\rightarrow X\), T be a monotone gnondecreasing mapping and \(T ( X ) \subseteq g ( X ) \). Also suppose that if \(\{ gx_{n} \} \subset X\) is a nondecreasing sequence with \(gx_{n}\rightarrow gz\) in gX, then \(gx_{n}\preceq gz\), \(gz\preceq g ( gz ) \) for all n hold. Also suppose that gX is bclosed. If there exists \(x_{0}\in X\) such that \(gx_{0}\preceq Tx_{0}\), then T and g have a coincidence point. Further, if T and g commute at their coincidence points, then T and g have a common fixed point.
Proof
By the proof of Theorem 2.7, we can show that \(\{ y_{n} \}= \{Tx_{n} \} = \{ gx_{n+1} \} \) is a bCauchy sequence. Since gX is bclosed, then there exists \(x\in X\) such that
We shall show that x is a coincidence point of T and g. As a matter of fact, owing to \(gx_{n}\preceq gx\) for all n, by (2.12) it should be noticed that
where
and
Taking the limit as \(n\rightarrow\infty\) from (2.22) together with (2.23) and (2.24), we deduce that
or equivalently,
However, note that
then (2.25) and (2.26) lead to a contradiction if \(gx\neq Tx\). In other words, \(gx=Tx\).
Set \(y=gx=Tx\). Now that T and g commute at x, it follows that \(Ty=T ( gx )=g ( Tx ) =gy\). Since \(gx\preceq g ( gx ) =gy\), then by (2.12) and \(gx=Tx\) and \(gy=Ty\), we demonstrate that
or equivalently,
This is a contradiction if \(Tx\neq Ty\). Hence, we claim that \(Tx=Ty=y\). Therefore, \(Ty=gy=y\). That is to say, y is a common fixed point of T and g. □
Corollary 2.9
Let \((X,\preceq,d)\) be a partially ordered complete bmetric space with \(s>1\) and \(T:X\rightarrow X\) be a nondecreasing mapping. Suppose that there exist \(\psi\in\Psi\) and \(L\geq0\) such that
where
and
for all \(x,y\in X\) with \(x\preceq y\). Also suppose that either

(a)
\(( X,d,\preceq ) \) is regular, or

(b)
T is continuous.
If there exists \(x_{0}\in X\) such that \(x_{0}\preceq Tx_{0}\), then T has a fixed point in X.
Remark 2.10
Theorems 2.7 and 2.8 improve and generalize Theorems 1.5 and 1.6 in many ways. First, Theorems 2.7 and 2.8 delete the function φ in (1.6), since due to the proofs of Theorem 1.5 and Theorem 1.6, it is superfluous based on the fact that \(\varphi ( t ) \leq t\) for each \(t\in[0,\infty)\). Second, condition (2.12) is wider than (1.6) because the constant \(\varepsilon>1\) is optional. Third, the compatible condition of Theorem 2.7 is weaker than the commutative condition of Theorem 1.5. This is because if T and g are commutative, then \(Tgx_{n}=gTx_{n}\). This is natural that \(\lim_{n\rightarrow\infty}d(Tgx_{n}, gTx_{n})=0\). That is to say, the pair \((T,g)\) is compatible. However, the converse is not true. Otherwise, the proofs of Theorems 2.7 and 2.8 are shorter than the ones of Theorems 1.5 and 1.6 since they do not utilize Lemma 1.8, but Theorems 1.5 and 1.6 rely on this lemma entirely.
Definition 2.11
Let \((X,\preceq,d)\) be a partially ordered bmetric space with \(s>1\), \(\psi\in\Psi\) and \(\theta\in \Theta\). The mapping \(T:X\times X\rightarrow X\) is called an almost generalized \(( \psi,\theta ) \)contractive mapping with respect to \(g:X\rightarrow X\) if there exists \(L\geq0\) such that
for all \(x,y,u,v\in X\) with \(gx\preceq gu\) and \(gy\succeq gv\), where \(M_{s,T,g} ( x,y,u,v ) \) and \(N_{T,g} ( x,y,u,v ) \) are given by (1.7) and (1.8), respectively.
Theorem 2.12
Suppose that \((X,\preceq,d)\) is a partially ordered complete bmetric space with \(s>1\). Let \(T:X\times X\rightarrow X\) be an almost generalized \(( \psi ,\theta )\)contractive mapping with respect to \(g:X\rightarrow X\), and T and g be continuous such that T has the mixed gmonotone property and commutes with g. Also, suppose that \(T ( X\times X ) \subseteq g ( X ) \). If there exists \(( x_{0},y_{0} ) \in X\times X\) such that \(gx_{0}\preceq T ( x_{0},y_{0} )\) and \(gy_{0}\succeq T ( y_{0},x_{0} )\), then T and g have a coupled coincidence point in X.
Proof
By the given assumption and the proof of [34], Theorem 2.2 (also see Theorem 1.7), we construct the sequences \(\{ x_{n} \} \) and \(\{ y_{n} \} \) in X such that
for which \(\{ gx_{n} \} _{n=0}^{\infty}\) is nondecreasing and \(\{ gy_{n} \} _{0}^{\infty}\) is nonincreasing. Putting \(x=x_{n}\), \(y=y_{n}\), \(u=x_{n+1}\) and \(v=y_{n+1}\) in (2.27), we obtain that
According to the proof of [34], Theorem 2.2, we get that
and
Since ψ is nondecreasing, then by (2.28)(2.30) it is not hard to verify that
Similarly, putting \(x=y_{n+1}\), \(y=x_{n+1}\), \(u=y_{n}\) and \(v=x_{n}\) in (2.27), we acquire that
Further, denote
It follows immediately from (2.31)(2.34) that
Now, we shall prove that
for all \(n\geq1\), where \(\lambda=\frac{1}{s^{ \varepsilon}}\in [0,\frac{1}{s})\).
Indeed, if \(\xi_{n}=\delta_{n}\), then (2.35) means \(s^{ \varepsilon }\delta_{n}\leq\delta_{n}\). This leads to \(\delta_{n}=0\) (because \(s^{ \varepsilon}>1\)) and (2.36) holds trivially. If \(\xi_{n}=\max \{ d ( gx_{n},gx_{n+1} ) ,d ( gy_{n},gy_{n+1} ) \}\), i.e., \(\xi_{n}=\delta_{n1}\), then (2.35) follows (2.36).
Now by (2.36) we get \(\delta_{n}\leq\lambda^{n}\delta_{0}\). Therefore,
Finally, according to [18], Lemma 3.1, the sequences \(\{ gx_{n} \} \) and \(\{ gy_{n} \} \) are bCauchy sequences. The rest of the proof is the same as in [34], Theorem 2.2. □
Remark 2.13
Theorem 2.8 is more superior in several aspects as compared to Theorem 1.7. Indeed, (2.27) dismisses the condition \(\phi(M_{s,T,g}( x,y,u,v))\) of (1.9). This indicates that (2.27) is much broader than (1.9). Further, the constant \(\varepsilon>1\) is much more general in (2.27) because it is not only restricted to \(\varepsilon=3\) in (1.9). In addition, the proof of Theorem 2.8 is simpler than the one of Theorem 1.7 because it ignores Lemma 1.8, but Theorem 1.7 depends on this lemma utterly.
The following examples show the superiority of the obtained results.
Example 2.14
Let \(X=[0,\infty)\) and d on X be given by \(d(x,y)=xy^{2}\) for all \(x,y\in X\). Then d is a bmetric on X, with \(s=2\). Define an ordering ‘⪯’ on X as follows:
Define selfmaps f, g, S and R on X by
Take \(1<\varepsilon<2\). Note that
Make full use of [29], Example 2.6, it is easy to see that all conditions of Theorem 2.1 are satisfied and hence the corresponding conclusions hold. However, if \(1<\varepsilon<2\), then condition (1.2) does not hold but our condition (2.1) holds. As a consequence, this example shows that our theorem is a genuine generalization of Theorem 1.3.
Example 2.15
Let X be the set of Lebesgue measurable functions on \([0,1]\) such that \(\int_{0}^{1} x(t)\,\mathrm{d}t<1\). Define \(d:X\times X\rightarrow[0,\infty)\) by
Then d is a bmetric on X, with \(s=2\). Also, this space can also be equipped with a partial order given by
The operator \(T: X\rightarrow X\) is defined by
Take \(1<\varepsilon<3\), then
Let \(x_{0}=0\), \(L=0\) and \(g=I_{X} \) (an identity mapping on X). Simple circulations show that all conditions of Theorem 2.7 are satisfied for any \(\psi\in\Psi\) and hence T and g have a coincidence point in X. However, \(1<\varepsilon<3\) never includes \(\varepsilon=3\). That is to say, this example is not applicable for Theorem 1.5. Consequently, our theorem is more convenient in applications.
References
 1.
Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133181 (1922)
 2.
Hussain, N, Khan, AR, Agarwal, RP: Krasnoselśkii and Ky Fan type fixed point theorems in ordered Banach spaces. J. Nonlinear Convex Anal. 11(3), 475489 (2010)
 3.
Agarwal, RP, ElGebeily, MA, ÓRegan, D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 87, 109116 (2008)
 4.
Aliouche, A, Simpson, C: Fixed points and lines in 2metric spaces. Adv. Math. 229, 668690 (2012)
 5.
Imdad, M, Ali, J: A general fixed point theorem in fuzzy metric spaces via an implicit function. J. Appl. Math. Inform. 26(3), 591603 (2008)
 6.
Hadžić, O, Pap, E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, London (2001)
 7.
Kadelburg, Z, Radenović, S, Rakočević, V: A note on the equivalence of some metric and cone metric fixed point results. Appl. Math. Lett. 24(3), 370374 (2011)
 8.
Khan, MS, Swaleh, M, Sessa, S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 30, 19 (1984)
 9.
Popescu, O: Fixed points for \((\psi,\varphi)\)weak contractions. Appl. Math. Lett. 24, 14 (2011)
 10.
Radenović, S, Kadelburg, Z: Quasicontractions on symmetric and cone symmetric spaces. Banach J. Math. Anal. 5(1), 3850 (2011)
 11.
Ðorić, D: Common fixed point for generalized \(( \psi,\phi ) \)weak contractions. Appl. Math. Lett. 22, 18961900 (2009)
 12.
Kadelburg, Z, Radenović, S: On generalized metric space: a survey. TWMS J. Pure Appl. Math. 5(1), 313 (2014)
 13.
Radenović, S, Simić, S, Cakić, N, Golubović, Z: A note on tvscone metric fixed point theory. Math. Comput. Model. 54, 24182422 (2011)
 14.
Kirk, WA, Shahzad, N: Fixed Point Theory in Distance Spaces. Springer, Berlin (2014)
 15.
Li, J, Li, C, Wong, N, Yao, J: Recent development in fixedpoint theory, optimization, and their applications. Abstr. Appl. Anal. 2014, Article ID 293463 (2014)
 16.
Bakhtin, IA: The contraction principle in quasimetric spaces. In: Functional Analysis, vol. 30, pp. 2637 (1989)
 17.
Hussain, N, Ðorić, D, Kadelburg, Z, Radenović, S: Suzukitype fixed point results in metric type spaces. Fixed Point Theory Appl. 2012, Article ID 126 (2012)
 18.
Jovanović, M, Kadelburg, Z, Radenović, S: Common fixed point results in metrictype spaces. Fixed Point Theory Appl. 2010, Article ID 978121 (2010)
 19.
Khamsi, MA, Hussain, N: \(KKM\) mappings in metric type spaces. Nonlinear Anal. 73, 1233129 (2010)
 20.
Bota, MF, Karapınar, E: A note on ‘Some results on multivalued weakly Jungck mappings in bmetric spaces’. Cent. Eur. J. Math. 11(9), 17111712 (2013)
 21.
Aydi, H, Bota, MF, Karapınar, E, Moradi, S: A common fixed point for weak φcontractions on bmetric spaces. Fixed Point Theory 13(2), 337346 (2012)
 22.
Aydi, H, Bota, MF, Karapınar, E, Mitrović, S: A fixed point for setvalued quasicontractions in bmetric spaces. Fixed Point Theory Appl. 2012, Article ID 201 (2012)
 23.
Bota, MF, Karapınar, E, Mlesnite, O: UlamHyers stability results for fixed point problems via αψcontractive mappings in bmetric spaces. Abstr. Appl. Anal. 2013, Article ID 825293 (2013)
 24.
Kutbi, MA, Karapınar, E, Ahmad, J, Azam, A: Some fixed point results for multivalued mappings in bmetric spaces. J. Inequal. Appl. 2014, Article ID 126 (2014)
 25.
AminiHarandi, A: Fixed point theory for quasicontraction maps in bmetric spaces. Fixed Point Theory 15(2), 351358 (2014)
 26.
Czerwik, S: Contraction mappings in bmetric spaces. Acta Math. Inform. Univ. Ostrav. 1, 511 (1993)
 27.
Samreen, M, Kamran, T, Shahzad, N: Some fixed point theorems in bmetric spaces endowed with graph. Abstr. Appl. Anal. 2013, Article ID 967132 (2013)
 28.
Roshan, JR, Parvaneh, V, Kadelburg, Z: Common fixed point theorems for weakly isotone increasing mappings in ordered bmetric spaces. J. Nonlinear Sci. Appl. 7, 229245 (2014)
 29.
Roshan, JR, Parvaneh, V, Altun, I: Some coincidence point results in ordered bmetric spaces and applications in a system of integral equations. Appl. Math. Comput. 226, 725737 (2014)
 30.
Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some application to matrix equations. Proc. Am. Math. Soc. 132, 14351443 (2004)
 31.
Nieto, JJ, López, RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223239 (2005)
 32.
Nieto, JJ, López, RR: Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 23, 22052212 (2007)
 33.
Aghajani, A, Abbas, M, Roshan, JR: Common fixed point of generalized weak contractive mappings in partially ordered bmetric spaces. Math. Slovaca 4, 941960 (2014)
 34.
Aghajani, A, Arab, R: Fixed points of \(( \psi,\phi,\theta ) \)contractive mappings in partially ordered bmetric spaces and applications to quadratic integral equations. Fixed Point Theory Appl. 2013, Article ID 245 (2013)
 35.
Allahyari, R, Arab, R, Haghighi, AS: A generalization on weak contractions in partially ordered bmetric spaces and its applications to quadratic integral equations. J. Inequal. Appl. 2014, Article ID 355 (2014)
 36.
Hussain, N, Parvaneh, V, Roshan, JR, Kadelburg, Z: Fixed points of cyclic \((\psi,\varphi,L,A,B)\)contractive mappings in ordered bmetric spaces with applications. Fixed Point Theory Appl. 2013, Article ID 256 (2013)
 37.
Parvaneh, V, Roshan, JR, Radenović, S: Existence of tripled coincidence points in ordered bmetric spaces and an application to a system of integral equations. Fixed Point Theory Appl. 2013, Article ID 130 (2013)
 38.
Roshan, JR, Parvaneh, V, Shobkolaei, N, Sedghi, S, Shatanawi, W: Common fixed points of almost generalized \((\psi,\varphi )_{s}\)contractive mappings in ordered bmetric spaces. Fixed Point Theory Appl. 2013, Article ID 159 (2013)
Acknowledgements
The authors would like to express their sincere appreciation to the referees and editors for their very helpful suggestions and kind comments. The third author is thankful to the Ministry of Education, Science and Technological Development of Serbia.
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contribute equally and significantly in writing this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Huang, H., Radenović, S. & Vujaković, J. On some recent coincidence and immediate consequences in partially ordered bmetric spaces. Fixed Point Theory Appl 2015, 63 (2015). https://doi.org/10.1186/s1366301503083
Received:
Accepted:
Published:
MSC
 47H10
 54H25
Keywords
 coincidence point
 coupled coincidence point
 partially ordered bmetric space
 weakly compatible
 mixed gmonotone property