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On some recent coincidence and immediate consequences in partially ordered b-metric spaces
- Huaping Huang^{1},
- Stojan Radenović^{2}Email author and
- Jelena Vujaković^{3}
https://doi.org/10.1186/s13663-015-0308-3
© Huang et al.; licensee Springer. 2015
Received: 13 January 2015
Accepted: 14 April 2015
Published: 1 May 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 b-metric spaces. Two examples are also provided to support the superiority of the obtained results.
Keywords
- coincidence point
- coupled coincidence point
- partially ordered b-metric space
- weakly compatible
- mixed g-monotone property
MSC
- 47H10
- 54H25
1 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, 2-metric spaces, fuzzy metric spaces, probabilistic metric spaces, G-metric spaces, cone metric spaces, cone Banach spaces, b-metric spaces or metric type spaces, etc. (see [2–27]). One of the most influential spaces is b-metric 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 b-metric spaces have appeared (see [17–27]).
The following definitions and results will be needed in what follows.
Definition 1.1
([28])
- (b1)
\(d ( x,y ) =0\) if and only if \(x=y\);
- (b2)
\(d ( x,y ) =d (y,x )\);
- (b3)
\(d ( x,z ) \leq s [ d ( x,y ) +d (y,z ) ]\).
Otherwise, for more concepts such as b-convergence, b-completeness, b-Cauchy sequence and b-closed set in b-metric spaces, we refer the reader to [20–29] and the references mentioned therein.
Definition 1.2
- (1)
elements \(x,y\in X\) are called comparable if \(x\preceq y\) or \(y\preceq x\) holds;
- (2)
f is called monotone g-nondecreasing 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 g-weakly 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 b-metric 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 b-metric spaces (see [3, 28, 29, 33–38]).
Throughout this paper, we introduce the denotations Ψ, ϒ, Φ, Θ as follows.
- (a1)
ψ is continuous,
- (a2)
ψ is nondecreasing,
- (a3)
\(\psi ( 0 ) =0<\psi ( t ) \) for every \(t>0\).
- (b1)
φ is right continuous,
- (b2)
φ is nondecreasing,
- (b3)
\(\varphi(t)< t\) for every \(t>0\).
- (c1)
ϕ is lower semi-continuous,
- (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.
Theorem 1.3
([29])
Theorem 1.4
([29])
Similarly, in [35] authors introduced and proved the following results.
Theorem 1.5
([35])
Suppose that \((X,\preceq,d)\) is a partially ordered complete b-metric 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 g-nondecreasing 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 b-metric 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 g-nondecreasing 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 b-closed. 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 well-known notions and proved the following new results.
- (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 g-monotone property if F is nondecreasing g-monotone in its first argument and is nonincreasing g-monotone in its second argument, that is, for any \(x,y\in X\),and$$ x_{1},x_{2}\in X,\quad gx_{1}\preceq gx_{2}\quad\Rightarrow\quad F ( x_{1},y ) \preceq F ( x_{2},y ) $$In particular, if g is an identity mapping, then F is said to have the mixed monotone property.$$ y_{1},y_{2}\in X,\quad gy_{1}\preceq gy_{2}\quad\Rightarrow\quad F ( x,y_{1} ) \succeq F ( x,y_{2} ) . $$
Theorem 1.7
([34])
Suppose that \((X,\preceq,d)\) is a partially ordered complete b-metric 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 g-monotone 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])
2 Main results
In this section, we improve and generalize coincidence and coupled coincidence point results of Theorems 1.3-1.7 in several directions without utilizing Lemma 1.8 in the proofs.
Theorem 2.1
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.
Now, combining (2.4) and (2.6), we get that (2.2), where \(\lambda =\frac{1}{s^{ \varepsilon}}\in[0,\frac{1}{s})\).
Step II. We show that f, g, R and S have a coincidence point.
Theorem 2.2
Proof
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
- (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
Theorem 2.7
Suppose that \((X,\preceq,d)\) is a partially ordered complete b-metric 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 g-nondecreasing 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
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 b-metric 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 g-nondecreasing 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 b-closed. 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
Corollary 2.9
- (a)
\(( X,d,\preceq ) \) is regular, or
- (b)
T is continuous.
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
Theorem 2.12
Suppose that \((X,\preceq,d)\) is a partially ordered complete b-metric 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 g-monotone 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
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., \(\xi_{n}=\delta_{n-1}\), then (2.35) follows (2.36).
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
Example 2.15
Declarations
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.
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.
Authors’ Affiliations
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