Abstract
This article is concerned with coupled coincidence points and common fixed points for two mappings in metric spaces and cone metric spaces. We first establish a coupled coincidence point theorem for two mappings and a common fixed point theorem for two w-compatible mappings in metric spaces. Then, by using a scalarization method, we extend our main theorems to cone metric spaces. Our results generalize and complement several earlier results in the literature. Especially, our main results complement a very recent result due to Abbas et al.
Similar content being viewed by others
1 Introduction
Throughout this article, unless otherwise specified, we always suppose that ℕ is the set of positive integers and X is a nonempty set. In addition, for convenience, we denote gx = g(x) for each x ∈ X and each mapping g : X → X.
Recently, Abbas et al. [1] introduced the following concept of w-compatible mappings:
Definition 1.1. The mappings g : X → X and F : X × X → X are called w-compatible if g(F(x, y)) = F(gx, gy) whenever gx = F(x, y) and gy = F(y, x).
Moreover, they established several coupled coincidence point theorems and common fixed point theorems for such mappings. The problem investigated in [1] is interesting. In fact, recently, the existence of coupled fixed points, coupled coincidence points, coupled common fixed points, and common fixed points for nonlinear mappings with two variables has attracted more and more attention. For example, Bhashkar and Lakshmikantham [2] investigated some coupled fixed point theorems in partially ordered sets, and they also discussed an application of their result by investigating the existence and uniqueness of the solution for a periodic boundary value problem; Sabetghadam et al. [3] extended some results in [2] to cone metric spaces; Lakshmikantham and Ćirić [4] proved several coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces; Karapinar [5] extended some results of [4] to cone metric spaces; Zoran and Mitrović [6] considered this topic in normed spaces and established a coupled best approximation theorem; Ding et al. [7] established some coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces under some generalized contraction conditions; etc.
The aim of this article is to make further studies on such problems, and to generalize and complement some known results. Next, let us recall some related definitions:
Definition 1.2. [1]Let g : X → X, F : X × X → X be two mappings.
(I) (x, y) ∈ X × X is called a coupled coincidence point of F and g if gx = F(x, y) and gy = F(y, x).
(II) (x, y) ∈ X × X is called a coupled fixed point of F if x = F(x, y) and y = F(y, x).
(III) x ∈ X is called a common fixed point of F and g if x = gx = F(x, x).
2 Metric spaces
Now, let us present one of our main results.
Theorem 2.1. Let (X, d) be a complete metric space. Assume that g : X → X and F : X × X → X are two mappings satisfying
(H1) there exists a non-decreasing function ϕ : [0,+∞) → [0,+∞) such thatfor each t > 0, and
for all x, y, u, v ∈ X, where
(H2) F(X × X) ⊆ g(X), and g(X) is a closed subset of X.
Then F and g have a coupled coincidence point in X.
Proof. First, let us present some properties about ϕ which will be used in the sequel. We claim that ϕ(t) <t for each t > 0. In fact, if ϕ(t0) ≥ t0 for some t0 > 0, then, since ϕ is non-decreasing, ϕn(t0) ≥ t0 for all n ∈ ℕ, which contradicts the condition .
Moreover, it is easy to see that ϕ(0) = 0, and thus ϕ(t) ≤ t for all t ≥ 0.
Take x0, y0 ∈ X. Since F(X × X) ⊆ g(X), one can construct two sequences {x n }, {y n } in X such that
For any fixed n ∈ ℕ, by (H1), we have
and
where
Since
and
we have
Now, let us prove that for each n ∈ ℕ,
We consider the following three cases:
Case I. If M n = 0 or M n = max{d(gx n , gxn-1), d (gy n , gyn-1)}, then (2.3) obviously holds.
Case II. M n = d(gx n , gxn+1) > 0.
Then, by (2.1),
which is a contradiction.
Case III. M n = d(gy n , gyn+1) > 0.
Similar to Case II, by (2.2), we get a contradiction.
Thus, in all cases, (2.3) holds for each n ∈ ℕ. In addition, combining (2.1) and (2.2), we get that for all n ∈ ℕ:
Let ε > 0 be fixed. Since , by (2.5), there exists N ∈ ℕ such that for all n >N,
Throughout the rest of this article, we denote
for each p ∈ ℕ and each n ∈ ℕ.
Let n >N be fixed. Let us show that for all p ∈ ℕ:
By (2.6), we have
By (2.5) and (2.6), we get
Next, let us show that . By (H1), we have
where
If
then by (2.5) and (2.8),
which yields
i.e., . Thus,
If , one can similarly show that a n ≤ ε. Hence, in all cases, a n ≤ ε, so that . Then, by (2.6), we get
In general, in order to prove that , one can first show that , and then by the inequality , the conclusion follows easily.
Now, we have proved that (2.7) holds for all p ∈ ℕ, which means that {gx n } and {gy n } are Cauchy sequences. Then, by the completeness of g(X), there exist x, y ∈ X such that
By (H1) we have
and
where
Now, we claim that gx = F(x, y) and gy = F(y, x). In fact, if this is not true, then
which, together with (2.9), yield that c n = max{d(gx, F(x, y)), d(gy, F(y, x))} when n is sufficiently large. Letting n → ∞ in (2.10) and (2.11), it follows that
and
This is a contradiction. Thus, gx = F(x, y) and gy = F(y, x), i.e., (x, y) is a coupled coincidence point of F and g.
Example 2.2. Let X = [2,+∞), d(x, y) = |x-y|, F(x, y) = x + y, g(x) = x2, and . It is easy to verify that all the assumptions of Theorem 2.1 are satisfied. So F and g have a coupled coincidence point. In fact, we have F(2, 2) = g(2).
If F and g are w-compatible, we have the following result:
Theorem 2.3. Suppose that all of the assumptions of Theorem 2.1 are satisfied, and F and g are w-compatible. Then F and g have a unique common fixed point.
Proof. We give the proof in 3 steps.
Step 1. We claim that if
then gx1 = gx2 = gy1 = gy2. In fact, by (H1), we have
and
where . Then, it follows that
which gives that ω = 0, i.e., gx1 = gx2 and gy1 = gy2.
By a similar argument, in the case of
one can also show that gx1 = gy2 and gy1 = gx2. Then, it follows that
Step 2. By Theorem 2.1, (x, y) is a coupled coincidence point of F and g, i.e., gx = F(x, y) and gy = F(y, x). Then, by Step 1, we have gx = gy. Let u = gx = gy. Since F and g are w-compatible, we have
Again by Step 1, one obtains gu = gx. Thus u = gx = gu = F(u, u), i.e., u is a common fixed point of F and g.
Step 3. Let v = gv = F(v, v). By Step 1, one can deduce that gv = gu. So u = gu = gv = v, which means that u is the unique common fixed point of F and g.
3 Applications to cone metric spaces
In this section, by a scalarization method used in [7], we apply our main results in metric spaces to cone metric spaces, and obtain some new theorems.
In the following, we always suppose that E is a Banach space, P is a convex cone in E with is the partial ordering induced by P, (X, ρ) is a cone metric space with the underlying cone P, e ∈ intP, and ξ e : E → ℝ is defined by
In addition, x ≫ y stands for x - y ∈ intP.
First, let us recall some definitions about cone metric space.
Definition 3.1. [8]Let X be a nonempty set and P be a cone in a Banach space E. Suppose that a mapping d : X × X → E satisfies:
(d1) θ ≼ ρ(x, y) for all x,y ∈ X and ρ(x, y) = θ if and only if x = y, where θ is the zero element of P;
(d2) ρ(x, y) = ρ(y, x) for all x, y ∈ X;
(d3) ρ(x, y) ≼ ρ(x, z) + ρ(z, y) for all x, y, z ∈ X.
Then ρ is called a cone metric on X and (X, ρ) is called a cone metric space.
Definition 3.2. Let (X, ρ) be a cone metric space. Let {x n } be a sequence in X and x ∈ X. If ∀c ≫ θ, there exists N ∈ ℕ such that for all n >N, ρ(x n , x) ≪ c, then we say that {x n } converges to x, and we denote it byor x n → x, n → ∞. If ∀c ≫ θ, there exists N ∈ ℕ such that for all n, m >N, ρ(x n , x m ) ≪ c, then {x n } is called a Cauchy sequence in X. In addition, (X, ρ) is called complete cone metric space if every Cauchy sequence is convergent.
Recall that it has been of great interest for many authors to study fixed point theorems in cone metric spaces, and there is a large literature on this topic. We refer the reader to [1, 3, 5, 7, 9–28] and the references therein for some recent developments on this topic.
Next, let us recall some notations and basic results about the scalarization function ξ e .
Lemma 3.3. [[7], Lemma 1.1] The following statements are true:
(i) ξ e (·) is positively homogeneous and continuous on E;
(ii) y, z ∈ E with y ≼ z implies ξ e (y) ≤ ξ e (z);
(ii) ξ e (y + z) ≤ ξ e (y) + ξ e (z) for all y, z ∈ E.
Combining Theorems 2.1 and 2.2 of [7] and, we have the following results:
Theorem 3.4. Let (X, ρ) be a cone metric space with underlying cone P. Then, ξ e ○ ρ is a metric on X. Moreover, if (X, ρ) is complete, then (X, ξ e ○ ρ) is a complete metric space.
By using Theorems 2.1 and 2.3, one can deduce many results on cone metric spaces. For example, we have the following theorem:
Theorem 3.5. Let (X, ρ) be a cone metric space with underlying cone P. Assume that g:X → X and F:X × X → X are two mappings satisfying that F(X × X) ⊆ g(X), g(X) is a complete cone metric space, and there exists a constant λ ∈ (0,1) such that for each x, y, u, v ∈ X, there is awith
where
and co denotes the convex hull. Then F and g have a coupled coincidence point in X. Moreover, if F and g are w-compatible, then F and g have a unique common fixed point.
Proof. Let d = ξ e ○ ρ. By Theorem 3.4, d is a metric on X and (g(X), d) is a complete metric space. Then, by Lemma 3.3, we have
where is defined in Theorem 2.1. Now, letting
it is easy to see that all of the assumptions of Theorem 2.1 are satisfied. Thus F and g have a coupled coincidence point in X. In addition, if F and g are w-compatible, by Theorem 2.3, F and g have a unique common fixed point.
Remark 3.6. Theorem 3.5 is a complement of [[1], Theorem 2.4]. Moreover, Theorem 3.5 extends some existing results. For example, one can deduce [[3], Theorem 2.2] from Theorem 3.5. In addition, note that Theorems 3.4 and 3.5 are true and in the context of tvs-cone metric spaces (for details see [23, 28]).
Remark 3.7. It is needed to note that one can also get Theorem 3.5 by using the method of Minkowski functional, which is introduced in [22].
References
Abbas M, Ali Khan M, Radenović S: Common coupled fixed point theorems in cone metric spaces for w-compatible mappings. Appl Math Comput 2010, 217: 195–202. 10.1016/j.amc.2010.05.042
Gnana Bhaskar T, 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
Sabetghadam F, Masiha HP, Sanatpour AH: Some coupled fixed point theorems in cone metric space. Fixed Point Theory Appl 2009, 2009: 8. (Article ID 125426)
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
Karapinar E: Couple fixed point theorems for nonlinear contractions in cone metric spaces. Comput Math Appl 2010, 59: 3656–3668. 10.1016/j.camwa.2010.03.062
Mitrović ZD: A coupled best approximations theorem in normed spaces. Nonlinear Anal 2010, 72: 4049–4052. 10.1016/j.na.2010.01.035
Ding HS, Li L, Radojević S: Coupled coincidence point theorems for generalized nonlinear contraction in partially ordered metric spaces. Fixed Point Theory Appl 2012, 2012: 96.
Du WS: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal 2010, 72: 2259–2261. 10.1016/j.na.2009.10.026
Huang LG, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. J Math Anal Appl 2007, 332: 1468–1476. 10.1016/j.jmaa.2005.03.087
Abbas M, Jungck G: Common fixed point results of noncommuting mappings without continuity in cone metric spaces. J Math Anal Appl 2008, 341: 418–420.
Altun I, Durmaz G: Some fixed point results in cone metric spaces. Rendiconti del Circolo Mathematico di Palermo 2009, 58: 319–325. 10.1007/s12215-009-0026-y
Altun I, Damjanović B, Djorić D: Fixed point and common fixed point theorems on ordered cone metric spaces. Appl Math Lett 2010, 23: 310–316. 10.1016/j.aml.2009.09.016
Ding HS, Li L: Coupled fixed point theorems in partially ordered cone metric spaces. Filomat 2011, 25: 137–149.
Ding HS, Li L, Long W: Coupled common fixed point theorems for weakly increasing mappings with two variables. J Comput Anal Appl to appear
Dorić D, Kadelburg Z, Radenović S: Coupled fixed point for mappings without mixed monotone property. Appl Math Lett 2012. doi: 10.1016/j.aml.2012.02.022
Golubović Z, Kadelburg Z, Radenović S: Coupled coincidence points of mappings in ordered partial metric spaces. Abstr Appl Anal 2012, 2012: 18. (Article ID 192581)
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
Ilić D, Rakočević V: Quasi-contraction on a cone metric space. Appl Math Lett 2009, 22: 728–731. 10.1016/j.aml.2008.08.011
Janković S, Kadelburg Z, Radenović S: Rhoades BE: Assad-Kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces. Fixed Point Theory Appl 2009, 2009: 16. (Article ID 761086)
Jungck G, Radenović S, Radojević S, Rakočević V: Common fixed point theorems for weakly compatible pairs on cone metric spaces. Fixed Point Theory Appl 2009, 2009: 13. (Article ID 643840)
Kadelburg Z, Radenović S, Rakočvić V: Remarks on quasi-contraction on a cone metric space. Appl Math Lett 2009, 22: 1674–1679. 10.1016/j.aml.2009.06.003
Kadelburg Z, Pavlović M, Radenović S: Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces. Comput Math Appl 2010, 59: 3148–3159. 10.1016/j.camwa.2010.02.039
Kadelburg Z, Radenović S, Rakočević V: A note on the equivalence of some metric and cone metric fixed point results. Appl Math Lett 2011, 24: 370–374. 10.1016/j.aml.2010.10.030
Kadelburg Z, Radenović S: Coupled fixed point results under tvs-cone metric and w -cone-distance. Adv Fixed Point Theory 2012, in press.
Nashine HK, Kadelburg Z, Radenović S: Coupled common fixed point theorems for w*-compatible mappings in ordered cone metric spaces. Appl Math Comput 2012, 218: 5422–5432. 10.1016/j.amc.2011.11.029
Radenović S, Rhoades BE: Fixed point theorem for two non-self mappings in cone metric spaces. Comput Math Appl 2009, 57: 1701–1707. 10.1016/j.camwa.2009.03.058
Rezapour Sh, Hamlbarani R: Some note on the paper "Cone metric spaces and fixed point theorems of contractive mappings". J Math Anal Appl 2008, 345: 719–724. 10.1016/j.jmaa.2008.04.049
Rezapour Sh, Haghi RH, Shahzad N: Some notes on fixed points of quasi-contraction maps. Appl Math Lett 2010, 23: 498–502. 10.1016/j.aml.2010.01.003
Zhang X: Fixed point theorem of generalized quasicontractive mapping in cone metric spaces. Comput Math Appl 2011. doi:10.1016/j.camwa.2011.03.107
Acknowledgements
The authors thank the referees for their valuable comments that helped to improve the text. Wei Long acknowledges support from the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province of China (20114BAB211002), and the Foundation of Jiangxi Provincial Education Department (GJJ12205). Third author is thankful to the Ministry of Science and Technological Development of Serbia.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed 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 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Long, W., Rhoades, B.E. & Rajović, M. Coupled coincidence points for two mappings in metric spaces and cone metric spaces. Fixed Point Theory Appl 2012, 66 (2012). https://doi.org/10.1186/1687-1812-2012-66
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2012-66