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Coupled fixed point theorems for αψcontractive type mappings in partially ordered metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 228 (2012)
Abstract
The object of this paper is to determine some coupled fixed point theorems for nonlinear contractive mappings in the framework of a metric space endowed with partial order. We also prove the uniqueness of a coupled fixed point for such mappings in this setup.
MSC:47H10, 54H25, 34B15.
1 Introduction
Fixed point theory is a very useful tool in solving a variety of problems in control theory, economic theory, nonlinear analysis and global analysis. The Banach contraction principle [1] is the most famous, simplest and one of the most versatile elementary results in fixed point theory. A huge amount of literature is witnessed on applications, generalizations and extensions of this principle carried out by several authors in different directions, e.g., by weakening the hypothesis, using different setups, considering different mappings.
Many authors obtained important fixed point theorems, e.g., Abbas et al. [2], Agarwal et al. [3, 4], Bhaskar and Lakshmikantham [5], Choudhury and Kundu [6], Choudhury and Maity [7], Ćirić et al. [8], Luong and Thuan [9], Nieto and López [10, 11], Ran and Reurings [12] and Samet [13] presented some new results for contractions in partially ordered metric spaces. In [14], Ilić and Rakočević determined some common fixed point theorems by considering the maps on cone metric spaces. Recently, Haghi et al. [15] have shown that some coincidence point and common fixed point generalizations in fixed point theory are not real generalizations. For more detail on fixed point theory and related concepts, we refer to [16–34] and the references therein.
In [5], Bhaskar and Lakshmikantham introduced the notions of mixed monotone property and coupled fixed point for the contractive mapping F:X\times X\to X, where X is a partially ordered metric space, and proved some coupled fixed point theorems for a mixed monotone operator. As an application of the coupled fixed point theorems, they determined the existence and uniqueness of the solution of a periodic boundary value problem. Recently, Lakshmikantham and Ćirić [35] have proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces. Most recently, Samet et al. [36] have defined αψcontractive and αadmissible mapping and proved fixed point theorems for such mappings in complete metric spaces.
The aim of this paper is to determine some coupled fixed point theorems for generalized contractive mappings in the framework of partially ordered metric spaces.
2 Definitions and preliminary results
We start with the definition of a mixed monotone property and a coupled fixed point and state the related results.
Definition 2.1 ([5])
Let (X,\le ) be a partially ordered set and F:X\times X\to X be a mapping. Then a map F is said to have the mixed monotone property if F(x,y) is monotone nondecreasing in x and is monotone nonincreasing in y; that is, for any x,y\in X,
and
Definition 2.2 ([5])
An element (x,y)\in X\times X is said to be a coupled fixed point of the mapping F:X\times X\to X if
Theorem 2.3 ([5])
Let (X,\le ) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X be a continuous mapping having the mixed monotone property on X. Assume that there exists a k\in [0,1) with
for all x\ge u and y\le v. If there exist {x}_{0},{y}_{0}\in X such that
then there exist x,y\in X such that F(x,y)=x and F(y,x)=y.
Theorem 2.4 ([5])
Let (X,\le ) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Assume that X has the following property:

(i)
if a nondecreasing sequence ({x}_{n})\to x, then {x}_{n}\le x for all n;

(ii)
if a nonincreasing sequence ({y}_{n})\to y, then y\le {y}_{n} for all n.
Let F:X\times X\to X be a mapping having the mixed monotone property on X. Assume that there exists a k\in [0,1) with
for all x\ge u and y\le v. If there exist {x}_{0},{y}_{0}\in X such that
then there exist x,y\in X such that F(x,y)=x and F(y,x)=y.
3 Main results
In this section, we establish some coupled fixed point results by considering maps on metric spaces endowed with partial order.
Denote by Ψ the family of nondecreasing functions \psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) such that {\sum}_{n=1}^{\mathrm{\infty}}{\psi}^{n}(t)<\mathrm{\infty} for all t>0, where {\psi}^{n} is the n th iterate of ψ satisfying (i) {\psi}^{1}(\{0\})=\{0\}, (ii) \psi (t)<t for all t>0 and (iii) {lim}_{r\to {t}^{+}}\psi (r)<t for all t>0.
Lemma 3.1 If \psi :[0,\mathrm{\infty}]\to [0,\mathrm{\infty}] is nondecreasing and right continuous, then {\psi}^{n}(t)\to 0 as n\to \mathrm{\infty} for all t\ge 0 if and only if \psi (t)<t for all t>0.
Definition 3.2 Let (X,d) be a partially ordered metric space and F:X\times X\to X be a mapping. Then a map F is said to be (\alpha ,\psi )contractive if there exist two functions \alpha :{X}^{2}\times {X}^{2}\to [0,+\mathrm{\infty}) and \psi \in \mathrm{\Psi} such that
for all x,y,u,v\in X with x\ge u and y\le v.
Definition 3.3 Let F:X\times X\to X and \alpha :{X}^{2}\times {X}^{2}\to [0,+\mathrm{\infty}) be two mappings. Then F is said to be (\alpha )admissible if
for all x,y,u,v\in X.
Theorem 3.4 Let (X,\le ) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X be a mapping having the mixed monotone property of X. Suppose that there exist \psi \in \mathrm{\Psi} and \alpha :{X}^{2}\times {X}^{2}\to [0,+\mathrm{\infty}) such that for x,y,u,v\in X, the following holds:
for all x\ge u and y\le v. Suppose also that

(i)
F is (\alpha )admissible,

(ii)
there exist {x}_{0},{y}_{0}\in X such that
\alpha (({x}_{0},{y}_{0}),(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0})))\ge 1\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha (({y}_{0},{x}_{0}),(F({y}_{0},{x}_{0}),F({x}_{0},{y}_{0})))\ge 1, 
(iii)
F is continuous.
If there exist {x}_{0},{y}_{0}\in X such that {x}_{0}\le F({x}_{0},{y}_{0}) and {y}_{0}\ge F({y}_{0},{x}_{0}), then F has a coupled fixed point; that is, there exist x,y\in X such that
Proof Let {x}_{0},{y}_{0}\in X be such that \alpha (({x}_{0},{y}_{0}),(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0})))\ge 1 and \alpha (({y}_{0},{x}_{0}),(F({y}_{0},{x}_{0}),F({x}_{0},{y}_{0})))\ge 1 and {x}_{0}\le F({x}_{0},{y}_{0})={x}_{1} (say) and {y}_{0}\ge F({y}_{0},{x}_{0})={y}_{1} (say). Let {x}_{2},{y}_{2}\in X be such that F({x}_{1},{y}_{1})={x}_{2} and F({y}_{1},{x}_{1})={y}_{2}. Continuing this process, we can construct two sequences ({x}_{n}) and ({y}_{n}) in X as follows:
for all n\ge 0. We will show that
for all n\ge 0. We will use the mathematical induction. Let n=0. Since {x}_{0}\le F({x}_{0},{y}_{0}) and {y}_{0}\ge F({y}_{0},{x}_{0}) and as {x}_{1}=F({x}_{0},{y}_{0}) and {y}_{1}=F({y}_{0},{x}_{0}), we have {x}_{0}\le {x}_{1} and {y}_{0}\ge {y}_{1}. Thus, (3.2) hold for n=0. Now suppose that (3.2) hold for some fixed n, n\ge 0. Then, since {x}_{n}\le {x}_{n+1} and {y}_{n}\ge {y}_{n+1} and by the mixed monotone property of F, we have
and
From above, we conclude that
Thus, by the mathematical induction, we conclude that (3.2) hold for all n\ge 0. If for some n we have ({x}_{n+1},{y}_{n+1})=({x}_{n},{y}_{n}), then F({x}_{n},{y}_{n})={x}_{n} and F({y}_{n},{x}_{n})={y}_{n}; that is, F has a coupled fixed point. Now, we assumed that ({x}_{n+1},{y}_{n+1})\ne ({x}_{n},{y}_{n}) for all n\ge 0. Since F is (\alpha )admissible, we have
Thus, by the mathematical induction, we have
and similarly,
for all n\in \mathbb{N}. Using (3.1) and (3.3), we obtain
Similarly, we have
Adding (3.5) and (3.6), we get
Repeating the above process, we get
for all n\in \mathbb{N}. For \u03f5>0 there exists n(\u03f5)\in \mathbb{N} such that
Let n,m\in \mathbb{N} be such that m>n>n(\u03f5). Then, by using the triangle inequality, we have
This implies that d({x}_{n},{x}_{m})+d({y}_{n},{y}_{m})<\u03f5. Since
and
and hence ({x}_{n}) and ({y}_{n}) are Cauchy sequences in (X,d). Since (X,d) is a complete metric space and hence ({x}_{n}) and ({y}_{n}) are convergent in (X,d). Then there exist x,y\in X such that
Since F is continuous and {x}_{n+1}=F({x}_{n},{y}_{n}) and {y}_{n+1}=F({y}_{n},{x}_{n}), taking limit n\to \mathrm{\infty}, we get
and
that is, F(x,y)=x and F(y,x)=y and hence F has a coupled fixed point. □
In the next theorem, we omit the continuity hypothesis of F.
Theorem 3.5 Let (X,\le ) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X be a mapping such that F has the mixed monotone property. Assume that there exist \psi \in \mathrm{\Psi} and a mapping \alpha :{X}^{2}\times {X}^{2}\to [0,+\mathrm{\infty}) such that
for all x,y,u,v\in X with x\ge u and y\le v. Suppose that

(i)
conditions (i) and (ii) of Theorem 3.4 hold,

(ii)
if ({x}_{n}) and ({y}_{n}) are sequences in X such that
\alpha (({x}_{n},{y}_{n}),({x}_{n+1},{y}_{n+1}))\ge 1\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha (({y}_{n},{x}_{n}),({y}_{n+1},{x}_{n+1}))\ge 1
for all n and {lim}_{n\to \mathrm{\infty}}{x}_{n}=x\in X and {lim}_{n\to \mathrm{\infty}}{y}_{n}=y\in X, then
If there exist {x}_{0},{y}_{0}\in X such that {x}_{0}\le F({x}_{0},{y}_{0}) and {y}_{0}\ge F({y}_{0},{x}_{0}), then there exist x,y\in X such that F(x,y)=x and F(y,x)=y; that is, F has a coupled fixed point in X.
Proof Proceeding along the same lines as in the proof of Theorem 3.4, we know that ({x}_{n}) and ({y}_{n}) are Cauchy sequences in the complete metric space (X,d). Then there exist x,y\in X such that
On the other hand, from (3.3) and hypothesis (ii), we obtain
and similarly,
for all n\in \mathbb{N}. Using the triangle inequality, (3.8) and the property of \psi (t)<t for all t>0, we get
Similarly, using (3.9), we obtain
Taking the limit as n\to \mathrm{\infty} in the above two inequalities, we get
Hence, F(x,y)=x and F(y,x)=y. Thus, F has a coupled fixed point. □
In the following theorem, we will prove the uniqueness of the coupled fixed point. If (X,\le ) is a partially ordered set, then we endow the product X\times X with the following partial order relation:
for all (x,y),(u,v)\in X\times X.
Theorem 3.6 In addition to the hypothesis of Theorem 3.4, suppose that for every (x,y), (s,t) in X\times X, there exists (u,v) in X\times X such that
and also assume that (u,v) is comparable to (x,y) and (s,t). Then F has a unique coupled fixed point.
Proof From Theorem 3.4, the set of coupled fixed points is nonempty. Suppose (x,y) and (s,t) are coupled fixed points of the mappings F:X\times X\to X; that is, x=F(x,y), y=F(y,x) and s=F(s,t), t=F(t,s). By assumption, there exists (u,v) in X\times X such that (u,v) is comparable to (x,y) and (s,t). Put u={u}_{0} and v={v}_{0} and choose {u}_{1},{v}_{1}\in X such that {u}_{1}=F({u}_{1},{v}_{1}) and {v}_{1}=F({v}_{1},{u}_{1}). Thus, we can define two sequences ({u}_{n}) and {v}_{n} as
Since (u,v) is comparable to (x,y), it is easy to show that x\le {u}_{1} and y\ge {v}_{1}. Thus, x\le {u}_{n} and y\ge {v}_{n} for all n\ge 1. Since for every (x,y),(s,t)\in X\times X, there exists (u,v)\in X\times X such that
Since F is (\alpha )admissible, so from (3.10), we have
Since u={u}_{0} and v={v}_{0}, we get
Thus,
Therefore, by the mathematical induction, we obtain
for all n∈ and similarly, \alpha ((y,x),({v}_{n},{u}_{n}))\ge 1. From (3.10) and (3.11), we get
Similarly, we have
Adding (3.12) and (3.13), we get
Thus,
for each n\ge 1. Letting n\to \mathrm{\infty} in (3.14) and using Lemma 3.1, we get
This implies
Similarly, one can show that
From (3.15) and (3.16), we conclude that x=s and y=t. Hence, F has a unique coupled fixed point. □
Example 3.7 (Linear case)
Let X=[0,1] and d:X\times X\to \mathbb{R} be a standard metric. Define a mapping F:X\times X\to X by F(x,y)=\frac{1}{4}xy for all x,y\in X. Consider a mapping \alpha :{X}^{2}\times {X}^{2}\to [0,+\mathrm{\infty}) be such that
Since xyuv\le xu+yv holds for all x,y,u,v\in X. Therefore, we have
It follows that
Thus (3.1) holds for \psi (t)=t/2 for all t>0, and we also see that all the hypotheses of Theorem 3.4 are fulfilled. Then there exists a coupled fixed point of F. In this case, (0,0) is a coupled fixed point of F.
Example 3.8 (Nonlinear case)
Let X=\mathbb{R} and d:X\times X\to \mathbb{R} be a standard metric. Define a mapping F:X\times X\to X by F(x,y)=\frac{1}{4}ln(1+x)+\frac{1}{4}ln(1+y) for all x,y\in X. Consider a mapping \alpha :{X}^{2}\times {X}^{2}\to [0,+\mathrm{\infty}) be such that
Then we get
Thus,
Therefore (3.1) holds for \psi (t)=\frac{1}{2}ln(1+t) for all t>0, and also the hypothesis of Theorem 3.4 is fulfilled. Then there exists a coupled fixed point of F. In this case, (0,0) is a coupled fixed point of F.
4 Concluding remark
The author of [33] recently established some coupled fixed point theorems in partially ordered metric spaces shortly by using some usual corresponding fixed point theorems on the metric space M=X\times X. Note that if the righthand side of the αψcontractive type condition (3.1) is replaced by \frac{1}{2}(d(x,u)+d(y,v)), then a very short proof similar to what followed in [33] can be provided for a coupled fixed point theorem of Theorem 3.4 type by making just use of the results in [36]. However, since the righthand side of (3.1) is not of the form \frac{1}{2}(d(x,u)+d(y,v)), specially for nonlinear functions ψ, then it is not possible to apply the method [33]. In this connection, notice that Example 3.7 works for both when the righthand side is either \frac{1}{2}(d(x,u)+d(y,v)) or as in (3.1), but Example 3.8 works only for (3.1). Hence, our results are more interesting and different from the existing results of [33] and [36].
References
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.
Abbas M, Nazir T, Radenović S: Fixed points of four maps in partially ordered metric spaces. Appl. Math. Lett. 2011, 24: 1520–1526. 10.1016/j.aml.2011.03.038
Agarwal RP, Meehan M, O’Regan D: Fixed Point Theory and Applications. Cambridge University Press, Cambridge; 2001.
Agarwal RP, ElGebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151
Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
Choudhury BS, Kundu A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal. 2010, 73: 2524–2531. 10.1016/j.na.2010.06.025
Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011, 54: 73–79. 10.1016/j.mcm.2011.01.036
Ćirić L, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 131294
Luong NV, Thuan NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal. 2011, 74: 983–992. 10.1016/j.na.2010.09.055
Nieto JJ, RodríguezLópez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s1108300590185
Nieto JJ, RodríguezLópez R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23(12):2205–2212. 10.1007/s1011400507690
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002993903072204
Samet B: Coupled fixed point theorems for a generalized MeirKeeler contraction in partially ordered metric spaces. Nonlinear Anal. 2010, 72: 4508–4517. 10.1016/j.na.2010.02.026
Ilić D, Rakočević V: Common fixed points for maps on cone metric space. J. Math. Anal. Appl. 2008, 341: 876–882. 10.1016/j.jmaa.2007.10.065
Haghi RH, Rezapour S, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 2011, 74: 1799–1803. 10.1016/j.na.2010.10.052
Haghi RH, Rezapour S: Fixed points of multi functions on regular cone metric spaces. Expo. Math. 2010, 28: 71–77. 10.1016/j.exmath.2009.04.001
Derafshpour M, Rezapour S, Shahzad N: Best proximity points of cyclic φ contractions on reflexive Banach space. Topol. Methods Nonlinear Anal. 2011, 37(1):193–202.
Aleomraninejad SMA, Rezapour S, Shahzad N: Some fixed point results on a metric space with a graph. Topol. Appl. 2012, 159: 659–663. 10.1016/j.topol.2011.10.013
Ghorbanian V, Rezapour S, Shahzad N: Some ordered fixed point results and the property (P). Comput. Math. Appl. 2012, 63: 1361–1368. 10.1016/j.camwa.2011.12.071
Mohiuddine SA, Alotaibi A: On coupled fixed point theorems for nonlinear contractions in partially ordered G metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 897198
Mohiuddine SA, Alotaibi A: Some results on tripled fixed point for nonlinear contractions in partially ordered G metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 179
Nashine HK, Kadelburg Z, Radenović S:Coupled common fixed point theorems for{w}^{\ast}compatible mappings in ordered cone metric spaces. Appl. Math. Comput. 2012, 218: 5422–5432. 10.1016/j.amc.2011.11.029
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.2011.06.040
Aydi H, Samet B, Vetro C: Coupled fixed point results in cone metric spaces for W compatible mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 27
Jleli M, Cojbasic Rajic V, Samet B, Vetro C: Fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations. J. Fixed Point Theory Appl. 2012. doi:10.1007/s11784–012–0081–4
Nashine HK, Samet B, Vetro C: Coupled coincidence points for compatible mappings satisfying mixed monotone property. J. Nonlinear Sci. Appl. 2012, 5(2):104–114.
Samet B, Vetro C: Coupled fixed point, F invariant set and fixed point of N order. Ann. Funct. Anal. 2010, 1(2):46–56.
Samet B, Vetro C: Coupled fixed point theorems for multivalued nonlinear contraction mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 4260–4268. 10.1016/j.na.2011.04.007
Sintunavarat W, Kumam P, Cho YJ: Coupled fixed point theorems for nonlinear contractions without mixed monotone property. Fixed Point Theory Appl. 2012., 2012: Article ID 170
Karapinar E, Samet B: Generalized α  ψ contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012., 2012: Article ID 793486
Abdeljawad T, Karapinar E, Aydi H: A new MeirKeeler type coupled fixed point on ordered partial metric spaces. Math. Probl. Eng. 2012., 2012: Article ID 327273
Abdeljawad T: Coupled fixed point theorems for partially contractive type mappings. Fixed Point Theory Appl. 2012., 2012: Article ID 148
AminiHarandi, A: Coupled and tripled fixed point theory in partially ordered metric spaces with applications to initial value problem. Math. Comput. Model. doi:10.1016/j.mcm.2011.12.006 (in press)
Shatanawi W, Samet B, Abbas M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model. 2012, 55: 680–687. 10.1016/j.mcm.2011.08.042
Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020
Samet B, Vetro C, Vetro P: Fixed point theorems for α  ψ contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014
Acknowledgements
The work of the second author was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. He acknowledges with thanks DSR technical and financial support.
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An erratum to this article is available at http://dx.doi.org/10.1186/168718122013127.
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Mursaleen, M., Mohiuddine, S.A. & Agarwal, R.P. Coupled fixed point theorems for αψcontractive type mappings in partially ordered metric spaces. Fixed Point Theory Appl 2012, 228 (2012). https://doi.org/10.1186/168718122012228
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DOI: https://doi.org/10.1186/168718122012228
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