- Research
- Open Access
Common fixed points of Ćirić-type contractive mappings in two ordered generalized metric spaces
- M Abbas^{1},
- YJ Cho^{2} and
- T Nazir^{1}Email author
https://doi.org/10.1186/1687-1812-2012-139
© Abbas et al.; licensee Springer 2012
- Received: 16 March 2012
- Accepted: 22 August 2012
- Published: 4 September 2012
Abstract
In this paper, using the setting of two ordered generalized metric spaces, a unique common fixed point of four mappings satisfying a generalized contractive condition is obtained. We also present an example to demonstrate the results presented herein.
MSC:54H25, 47H10, 54E50.
Keywords
- weakly compatible mappings
- compatible mappings
- dominated mappings
- common fixed point
- partially ordered set
- generalized metric space
1 Introduction and preliminaries
The study of a unique common fixed point of given mappings satisfying certain contractive conditions has been at the center of rigorous research activity. Mustafa and Sims [1] generalized the concept of a metric in which a real number is assigned to every triplet of an arbitrary set. Based on the notion of generalized metric spaces, Mustafa et al. [2–5] obtained some fixed point theorems for some mappings satisfying different contractive conditions. The existence of common fixed points in generalized metric spaces was initiated by Abbas and Rhoades [6] (see also [7] and [8]). For further study of common fixed points in generalized metric spaces, we refer to [9–12] and references mentioned therein. Abbas et al. [13] showed the existence of coupled common fixed points in two generalized metric spaces (for more results on couple fixed points, see also [14–21]).
The existence of fixed points in ordered metric spaces has been initiated in 2004 by Ran and Reurings [22] and further studied by Nieto and Lopez [23]. Subsequently, several interesting and valuable results have appeared in this direction [24–30].
The aim of this paper is to study common fixed point of four mappings that satisfy the generalized contractive condition in two ordered generalized metric spaces.
In the sequel, $\mathbb{R}$, ${\mathbb{R}}^{+}$ and $\mathbb{N}$ denote the set of real numbers, the set of nonnegative integers and the set of positive integers respectively. The usual order on $\mathbb{R}$ (respectively, on ${\mathbb{R}}^{+}$) will be indistinctly denoted by ≤ or by ≥.
In [1], Mustafa and Sims introduced the following definitions and results:
- (a)
$G(x,y,z)=0$ if $x=y=z$ for all $x,y,z\in X$;
- (b)
$0<G(x,y,z)$ for all $x,y,z\in X$ with $x\ne y$;
- (c)
$G(x,x,y)\le G(x,y,z)$ for all $x,y,z\in X$ with $y\ne z$;
- (d)
$G(x,y,z)=G(p\{x,y,z\})$, where p is a permutation of $x,y,z\in X$ (symmetry);
- (e)
$G(x,y,z)\le G(x,a,a)+G(a,y,z)$ for all $x,y,z,a\in X$.
Then G is called a G-metric on X and $(X,G)$ is called a G-metric space.
- (1)
a G-Cauchy sequence if, for any $\epsilon >0$, there exists ${n}_{0}\in N$ (the set of natural numbers) such that, for all $n,m,l\ge {n}_{0}$, $G({x}_{n},{x}_{m},{x}_{l})<\epsilon $;
- (2)
G-convergent if, for any $\epsilon >0$, there exist $x\in X$ and ${n}_{0}\in N$ such that, for all $n,m\ge {n}_{0}$, $G(x,{x}_{n},{x}_{m})<\epsilon $;
- (3)
A G-metric space X is said to be G-complete if every G-Cauchy sequence in X is G-convergent in X.
It is known that $\{{x}_{n}\}$ is G-convergent to a point $x\in X$ if and only if $G({x}_{m},{x}_{n},x)\to 0$ as $n,m\to \mathrm{\infty}$.
Proposition 1.3 [1]
- (1)
A sequence $\{{x}_{n}\}$ in X is G-convergent to a point $x\in X$;
- (2)
$G({x}_{n},{x}_{m},x)\to 0$ as $n,m\to \mathrm{\infty}$;
- (3)
$G({x}_{n},{x}_{n},x)\to 0$ as $n\to \mathrm{\infty}$;
- (4)
$G({x}_{n},x,x)\to 0$ as $n\to \mathrm{\infty}$.
Definition 1.4 A G-metric on X is said to be symmetric if $G(x,y,y)=G(y,x,x)$ for all $x,y\in X$.
for all $x,y\in X$.
for all $x,y\in X$.
Now, we give an example of a non-symmetric G-metric.
G -metric
(x,y,z) | G(x,y,z) |
---|---|
(1,1,1), (2,2,2) | 0 |
(1,1,2), (1,2,1), (2,1,1) | 0.5 |
(1,2,2), (2,1,2), (2,2,1) | 1.0 |
Note that G satisfies all the axioms of a generalized metric, but $G(x,x,y)\ne G(x,y,y)$ for two distinct points $x,y\in X$.
Definition 1.7 Let f and g be self-mappings on a set X. If $w=fx=gx$ for some $x\in X$, then the point x is called a coincidence point of f and g and w is called a point of coincidence of f and g.
Definition 1.8 [31]
Let f and g be self-mappings on a set X. Then f and g are said to be weakly compatible if they commute at every coincidence point.
Definition 1.9 [8]
Let X be a G-metric space and f, g be self-mappings on X. Then f and g are said to be R-weakly commuting if there exists a positive real number R such that $G(fgx,fgx,gfx)\le RG(fx,fx,gx)$ for all $x\in X$.
The maps f and g are R-weakly commuting on X if and only if they commute at their coincidence points.
- (a)
G is a generalized metric on X;
- (b)
⪯ is a partial order on X.
Definition 1.11 Let $(X,\u2aaf)$ be a partial ordered set. Then two points $x,y\in X$ are said to be comparable if $x\u2aafy$ or $y\u2aafx$.
Definition 1.12 [24]
Let $(X,\u2aaf)$ be a partially ordered set. A self-mapping f on X is said to be dominating if $x\u2aaffx$ for all $x\in X$.
Example 1.13 [24]
Let $X=[0,1]$ be endowed with usual ordering and $f:X\to X$ be a mapping defined by $fx=\sqrt[n]{x}$ for some $n\in \mathbb{N}$. Since $x\le {x}^{\frac{1}{n}}=fx$ for all $x\in X$, f is a dominating mapping.
Definition 1.14 Let $(X,\u2aaf)$ be a partially ordered set. A self-mapping f on X is said to be dominated if $fx\u2aafx$ for all $x\in X$.
Example 1.15 Let $X=[0,1]$ be endowed with usual ordering and $f:X\to X$ be a mapping defined by $fx={x}^{n}$ for some $n\in \mathbb{N}$. Since $fx={x}^{n}\le x$ for all $x\in X$, f is a dominated mapping.
Definition 1.16 A subset $\mathcal{K}$ of a partially ordered set X is said to be well-ordered if every two elements of $\mathcal{K}$ are comparable.
2 Common fixed point theorems
for all $x,y\in X$, where $a,b,c,e\ge 0$ with $a+b+c+2e<1$, then T has a unique fixed point provided that X is T-orbitally complete (for related definitions and results, we refer to [33]).
for all $x,y\in X$, where $0\le q<1$.
In this section, we show the existence of a unique common fixed point of four mappings satisfying Ćirić-type contractive condition in the framework of two ordered generalized metric spaces.
Now, we start with the following result:
- (a)
f, S are compatible, f or S is continuous and g, T are weakly compatible
- (b)
g, T are compatible, g or T is continuous and f, S are weakly compatible,
then f, g, S and T have a common fixed point. Moreover, the set of common fixed points of f, g, S and T is well-ordered if and only if f, g, S and T have one and only one common fixed point.
and so ${y}_{2k+1}={y}_{2k+2}$ since ${k}^{2}<1$.
Similarly, if $m=2k+1$, then one can easily obtain ${y}_{2k+2}={y}_{2k+3}$. Thus $\{{y}_{n}\}$ becomes a constant sequence and ${y}_{2n}$ serves as the common fixed point of f, g, S and T.
Suppose that ${G}_{1}({y}_{2n},{y}_{2n+1},{y}_{2n+1})>0$ for all $n\ge 0$.
where $h=\frac{k}{2-k}$. Obviously, $0\le h<1$.
Therefore, by using the above two inequalities, we have $fz=z$.
Thus (2.9) and (2.10) imply $fz=gv$. Since g and T are weakly compatible, we have $gz=gfz=gTv=Tgv=Tfz=Tz$, and so z is the coincidence point of g and T.
and so $z=gz$. Therefore, $fz=gz=Sz=Tz=z$. The proof is similar when f is continuous. Similarly, if (b) holds, then the result follows.
and hence $z=u$.
The converse follows immediately. This completes the proof. □
Two G -metrices
(x,y,z) | ${\mathit{G}}_{\mathbf{1}}\mathbf{(}\mathit{x},\mathit{y},\mathit{z}\mathbf{)}$ | ${\mathit{G}}_{\mathbf{2}}\mathbf{(}\mathit{x},\mathit{y},\mathit{z}\mathbf{)}$ |
---|---|---|
(0,0,0), (1,1,1), (2,2,2), (3,3,3), | 0 | 0 |
(0,0,2), (0,2,0), (2,0,0), (0,2,2), (2,0,2), (2,2,0), (0,0,1), (0,1,0), (1,0,0), (0,0,3), (0,3,0), (3,0,0), | 4 | 3 |
(0,1,1), (1,0,1), (1,1,0), (0,3,3), (3,0,3), (3,3,0), (1,1,2), (1,2,1), (2,1,1), (1,2,2), (2,1,2), (2,2,1), (1,1,3), (1,3,1), (3,1,1), (1,3,3), (3,1,3), (3,3,1), (2,2,3), (2,3,2), (3,2,2), (2,3,3), (3,2,3), (3,3,2), | 8 | 6 |
(0,1,2), (0,1,3), (0,2,1), (0,2,3), (0,3,1), (0,3,2), (1,0,2), (1,0,3), (1,2,0), (1,2,3), (1,3,0), (1,3,2), (2,0,1), (2,0,3), (2,1,0), (2,1,3), (2,3,0), (2,3,1), (3,0,1), (3,0,2), (3,1,0), (3,1,2), (3,2,0), (3,2,1), | 8 | 6 |
Self maps
x | f(x) | g(x) | S(x) | T(x) |
---|---|---|---|---|
0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 2 | 2 |
2 | 0 | 2 | 2 | 3 |
3 | 0 | 0 | 3 | 3 |
Dominated and dominating maps
x∈X | f is dominated | g is dominated | S is dominating | T is dominating |
---|---|---|---|---|
x = 0 | f(0)=0 | g(0)=0 | 0 = S(0) | 0 = T(0) |
x = 1 | f(1)=0<1 | g(1)=0<1 | 1<2 = S(1) | 1<2 = T(1) |
x = 2 | f(2)=0<2 | g(2)=2 | 2 = S(2) | 2<3 = T(2) |
x = 3 | f(3)=0<3 | g(3)=0<3 | 3 = S(3) | 3 = T(3) |
- (1)When $x=0$ and $y=2$, then $fx=0$, $gy=2$, $Sx=0$, $Ty=3$ and so$\begin{array}{rcl}{G}_{1}(fx,fx,gy)& =& {G}_{1}(0,0,2)=4\\ <& \frac{3}{4}(6)=\frac{3}{4}{G}_{2}(2,2,3)=\frac{3}{4}{G}_{2}(gy,gy,Ty)\\ \le & kmax\{{G}_{2}(Sx,Sx,Ty),{G}_{2}(fx,fx,Sx),{G}_{2}(gy,gy,Ty),\\ [{G}_{2}(fx,fx,Ty)+{G}_{2}(gy,gy,Sx)]/2\}\end{array}$
- (2)When $x=1$ and $y=2$, then $fx=0$, $gy=2$, $Sx=2$, $Ty=3$ and so$\begin{array}{rcl}{G}_{1}(fx,fx,gy)& =& {G}_{1}(0,0,2)=4\\ <& \frac{3}{4}(6)=\frac{3}{4}{G}_{2}(2,2,3)=\frac{3}{4}{G}_{2}(gy,gy,Ty)\\ \le & kmax\{{G}_{2}(Sx,Sx,Ty),{G}_{2}(fx,fx,Sx),{G}_{2}(gy,gy,Ty),\\ [{G}_{2}(fx,fx,Ty)+{G}_{2}(gy,gy,Sx)]/2\}\end{array}$
- (3)When $x=2$ and $y=2$, then $fx=0$, $gy=2$, $Sx=2$, $Ty=3$ and so$\begin{array}{rcl}{G}_{1}(fx,fx,gy)& =& {G}_{1}(0,0,2)=4\\ <& \frac{3}{4}(6)=\frac{3}{4}{G}_{2}(2,2,3)=\frac{3}{4}{G}_{2}(Sx,Sx,Ty)\\ \le & kmax\{{G}_{2}(Sx,Sx,Ty),{G}_{2}(fx,fx,Sx),{G}_{2}(gy,gy,Ty),\\ [{G}_{2}(fx,fx,Ty)+{G}_{2}(gy,gy,Sx)]/2\}\end{array}$
- (4)Finally, when $x=3$ and $y=2$, then $fx=0$, $gy=2$, $Sx=3$, $Ty=3$ and so$\begin{array}{rcl}{G}_{1}(fx,fx,gy)& =& {G}_{1}(0,0,2)=4\\ <& \frac{3}{4}(6)=\frac{3}{4}{G}_{2}(2,2,3)=\frac{3}{4}{G}_{2}(gy,gy,Ty)\\ \le & kmax\{{G}_{2}(Sx,Sx,Ty),{G}_{2}(fx,fx,Sx),{G}_{2}(gy,gy,Ty),\\ [{G}_{2}(fx,fx,Ty)+{G}_{2}(gy,gy,Sx)]/2\}\end{array}$
Thus, for all cases, the contractions (2.1) and (2.2) are satisfied. Hence all of the conditions of Theorem 2.1 are satisfied. Moreover, 0 is the unique common fixed point of f, g, S and g.
for any $k\in [0,1)$. So corresponding results in ordinary metric spaces cannot be applied in this case.
Theorem 2.1 can be viewed as an extension of Theorem 2.1 of [8] to the case of two ordered G-metric spaces.
Since the class of weakly compatible mappings includes R-weakly commuting mappings, Theorem 2.1 generalizes the comparable results in [8].
- (a)
f, S are compatible, f or S is continuous and g, T are weakly compatible
- (b)
g, T are compatible, g or T is continuous and f, S are weakly compatible,
then f, g, S and T have a common fixed point in X. Moreover, the set of common fixed points of f, g, S and T is well-ordered if and only if f, g, S and T have one and only one common fixed point in X.
- (1)If $x,y\in [0,\frac{1}{2})$, then we have$\begin{array}{rcl}{G}_{1}(fx,fx,gy)& =& \frac{1}{12}|x-3y|\le \frac{1}{12}(x+3y)\\ \le & \frac{3}{10}\left(\frac{17}{12}x\right)+\frac{3}{10}\left(\frac{9}{4}y\right)\\ =& {a}_{2}{G}_{2}(fx,fx,Sx)+{a}_{3}{G}_{2}(gy,gy,Ty)\\ \le & {a}_{1}{G}_{2}(Sx,Sx,Ty)+{a}_{2}{G}_{2}(fx,fx,Sx)+{a}_{3}{G}_{2}(gy,gy,Ty)\\ +{a}_{4}[{G}_{2}(fx,fx,Ty)+{G}_{2}(gy,gy,Sx)].\end{array}$
- (2)If $x\in [0,\frac{1}{2})$ and $y\in [\frac{1}{2},1]$, then we have$\begin{array}{rcl}{G}_{1}(fx,fx,gy)& =& \frac{1}{12}|x-2y|\le \frac{1}{12}(x+2y)\\ \le & \frac{3}{10}\left(\frac{17}{12}x\right)+\frac{3}{10}\left(\frac{14}{6}y\right)\\ =& {a}_{2}{G}_{2}(fx,fx,Sx)+{a}_{3}{G}_{2}(gy,gy,Ty)\\ \le & {a}_{1}{G}_{2}(Sx,Sx,Ty)+{a}_{2}{G}_{2}(fx,fx,Sx)+{a}_{3}{G}_{2}(gy,gy,Ty)\\ +{a}_{4}[{G}_{2}(fx,fx,Ty)+{G}_{2}(gy,gy,Sx)].\end{array}$
- (3)If $y\in [0,\frac{1}{2})$ and $x\in [\frac{1}{2},1]$, then we have$\begin{array}{rcl}{G}_{1}(fx,fx,gy)& =& \frac{1}{12}|x-3y|\le \frac{1}{12}(x+3y)\\ \le & \frac{3}{10}\left(\frac{17}{12}x\right)+\frac{3}{10}\left(\frac{9}{4}y\right)\\ =& {a}_{2}{G}_{2}(fx,fx,Sx)+{a}_{3}{G}_{2}(gy,gy,Ty)\\ \le & {a}_{1}{G}_{2}(Sx,Sx,Ty)+{a}_{2}{G}_{2}(fx,fx,Sx)+{a}_{3}{G}_{2}(gy,gy,Ty)\\ +{a}_{4}[{G}_{2}(fx,fx,Ty)+{G}_{2}(gy,gy,Sx)].\end{array}$
- (4)If $x,y\in [\frac{1}{2},1]$, then we obtain$\begin{array}{rcl}{G}_{1}(fx,fx,gy)& =& \frac{1}{12}|x-2y|\le \frac{1}{12}(x+2y)\\ \le & \frac{3}{10}\left(\frac{17}{12}x\right)+\frac{3}{10}\left(\frac{14}{6}y\right)\\ =& {a}_{2}{G}_{2}(fx,fx,Sx)+{a}_{3}{G}_{2}(gy,gy,Ty)\\ \le & {a}_{1}{G}_{2}(Sx,Sx,Ty)+{a}_{2}{G}_{2}(fx,fx,Sx)+{a}_{3}{G}_{2}(gy,gy,Ty)\\ +{a}_{4}[{G}_{2}(fx,fx,Ty)+{G}_{2}(gy,gy,Sx)].\end{array}$
Thus (2.13) is satisfied with ${a}_{1}={a}_{4}=\frac{1}{10}$ and ${a}_{2}={a}_{3}=\frac{3}{10}$, where ${a}_{1}+{a}_{2}+{a}_{3}+2{a}_{4}<1$. Similarly, (2.14) is satisfied. Thus all the conditions of Corollary 2.3 are satisfied. Moreover, 0 is the unique common fixed point of f and g.
3 Application
- (i)For each $s,t\in \mathrm{\Omega}$,${\int}_{\mathrm{\Omega}}{q}_{1}(t,s,u(s))\phantom{\rule{0.2em}{0ex}}ds\le u(s)$
- (ii)There exists $r:\mathrm{\Omega}\to \mathrm{\Omega}$ such that${\int}_{\mathrm{\Omega}}|{q}_{1}(t,s,u(t))-{q}_{2}(t,s,v(t))|\phantom{\rule{0.2em}{0ex}}dt\le r(t)|u(t)-v(t)|$
for each $s,t\in \mathrm{\Omega}$ with ${sup}_{t\in \mathrm{\Omega}}r(t)\le k$ where $k\in [0,1)$.
Then the integral equations (3.1) have a common solution in ${L}^{2}(\mathrm{\Omega})$.
is satisfied. Now we can apply Theorem 2.1 by taking S and T as identity maps to obtain the common solutions of integral equations (3.1) in ${L}^{2}(\mathrm{\Omega})$. □
- (5)
A G-metric naturally induces a metric ${d}_{G}$ given by ${d}_{G}(x,y)=G(x,y,y)+G(x,x,y)$. If the G-metric is not symmetric, then the inequalities (2.1), (2.2), (2.13) and (2.14) do not reduce to any metric inequality with the metric ${d}_{G}$. Hence our results do not reduce to fixed point problems in the corresponding metric space $(X,\u2aaf,{d}_{G})$.
Declarations
Acknowledgements
The authors thank the referees for their appreciation and suggestions regarding this work.
Authors’ Affiliations
References
- Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7: 289–297.MathSciNetGoogle Scholar
- Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorem for mapping on complete G -metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 189870Google Scholar
- Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175Google Scholar
- Mustafa Z, Shatanawi W, Bataineh M: Existence of Fixed point Results in G -metric spaces. Int. J. Math. Math. Sci. 2009., 2009: Article ID 283028Google Scholar
- Mustafa Z, Awawdeh F, Shatanawi W: Fixed point theorem for expansive mappings in G -metric spaces. Internat. J. Contemp. Math. Sci. 2010, 5: 2463–2472.MathSciNetGoogle Scholar
- Abbas M, Rhoades BE: Common fixed point results for non-commuting mappings without continuity in generalized metric spaces. Appl. Math. Comput. 2009, 215: 262–269. 10.1016/j.amc.2009.04.085MathSciNetView ArticleGoogle Scholar
- Abbas M, Nazir T, Radenović S: Some periodic point results in generalized metric spaces. Appl. Math. Comput. 2010, 217: 4094–4099. 10.1016/j.amc.2010.10.026MathSciNetView ArticleGoogle Scholar
- Abbas M, Khan SH, Nazir T: Common fixed points of R -weakly commuting maps in generalized metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 41Google Scholar
- Abbas M, Cho YJ, Nazir T: Common fixed point theorems for four mappings in TVS-valued cone metric spaces. J. Math. Inequal. 2011, 5: 287–299.MathSciNetView ArticleGoogle Scholar
- Chugh R, Kadian T, Rani A, Rhoades BE: Property p in G -metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 401684Google Scholar
- Saadati R, Vaezpour SM, Vetro P, Rhoades BE: Fixed point theorems in generalized partially ordered G -metric spaces. Math. Comput. Model. 2010, 52: 797–801. 10.1016/j.mcm.2010.05.009MathSciNetView ArticleGoogle Scholar
- Shatanawi W: Fixed point theory for contractive mappings satisfying Φ-maps in G -metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 181650Google Scholar
- Abbas M, Khan AR, Nazir T: Coupled common fixed point results in two generalized metric spaces. Appl. Math. Comput. 2011, 217: 6328–6336. 10.1016/j.amc.2011.01.006MathSciNetView ArticleGoogle Scholar
- Aydi H, Damjanović B, Samet B, Shatanawi W: Coupled fixed point theorems for nonlinear contractions in partially ordered G -metric spaces. Math. Comput. Model. 2011, 54: 2443–2450. 10.1016/j.mcm.2011.05.059View ArticleGoogle Scholar
- Chang SS, Cho YJ, Huang NJ: Coupled fixed point theorems with applications. J. Korean Math. Soc. 1996, 33: 575–585.MathSciNetGoogle Scholar
- Cho YJ, He G, Huang NJ: The existence results of coupled quasi-solutions for a class of operator equations. Bull. Korean Math. Soc. 2010, 47: 455–465.MathSciNetView ArticleGoogle Scholar
- Cho YJ, Rhoades BE, Saadati R, Samet B, Shatanawi W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type. Fixed Point Theory Appl. 2012., 2012: Article ID 8. doi:10.1186/1687–1812–2012–8Google Scholar
- Cho YJ, Shah MH, Hussain N: Coupled fixed points of weakly F -contractive mappings in topological spaces. Appl. Math. Lett. 2011, 24: 1185–1190. 10.1016/j.aml.2011.02.004MathSciNetView ArticleGoogle Scholar
- Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011, 54: 73–79. 10.1016/j.mcm.2011.01.036MathSciNetView ArticleGoogle Scholar
- Gordji ME, Cho YJ, Baghani H: Coupled fixed point theorems for contractions in intuitionistic fuzzy normed spaces. Math. Comput. Model. 2011, 54: 1897–1906. 10.1016/j.mcm.2011.04.014View ArticleGoogle Scholar
- Sintunavarat W, Cho YJ, Kumam P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 81Google Scholar
- Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some application to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4MathSciNetView ArticleGoogle Scholar
- Nieto JJ, Lopez RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5MathSciNetView ArticleGoogle Scholar
- Abbas M, Nazir T, Radenović S: Common fixed points of four maps in partially ordered metric spaces. Appl. Math. Lett. 2011, 24: 1520–1526. 10.1016/j.aml.2011.03.038MathSciNetView ArticleGoogle Scholar
- Cho YJ, Saadati R, Wang S: Common fixed point theorems on generalized distance in order cone metric spaces. Comput. Math. Appl. 2011, 61: 1254–1260. 10.1016/j.camwa.2011.01.004MathSciNetView ArticleGoogle Scholar
- Guo D, Cho YJ, Zhu J: Partial Ordering Methods in Nonlinear Problems. Nova Science Publishers, New York; 2004.Google Scholar
- Huang NJ, Fang YP, Cho YJ: Fixed point and coupled fixed point theorems for multi-valued increasing operators in ordered metric spaces. 3. In Fixed Point Theory and Applications. Edited by: Cho YJ, Kim JK, Kang SM. Nova Science Publishers, New York; 2002:91–98.Google Scholar
- Nieto JJ, Pouso RL, Rodríguez-López R: Fixed point theorems in ordered abstract sets. Proc. Am. Math. Soc. 2007, 135: 2505–2517. 10.1090/S0002-9939-07-08729-1View ArticleGoogle Scholar
- Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23: 2205–2212. 10.1007/s10114-005-0769-0MathSciNetView ArticleGoogle Scholar
- Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c -distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040MathSciNetView ArticleGoogle Scholar
- Jungck G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 1986, 9: 771–779. 10.1155/S0161171286000935MathSciNetView ArticleGoogle Scholar
- Kannan R: Some results on fixed points. Bull. Calcutta Math. Soc. 1968, 60: 71–76.MathSciNetGoogle Scholar
- Ćirić Lj: Generalized contractions and fixed-point theorems. Publ. Inst. Math. 1971, 12(26):19–26.Google Scholar
- Ćirić Lj: Fixed points for generalized multi-valued contractions. Mat. Vesnik 1972, 9(24):265–272.MathSciNetGoogle Scholar
- Abbas M, Nazir T, Radenović S: Common fixed point of generalized weakly contractive maps in partially ordered G -metric spaces. Appl. Math. Comput. 2012, 218: 9383–9395. 10.1016/j.amc.2012.03.022MathSciNetView ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.