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A coupled coincidence point theorem in partially ordered metric spaces with an implicit relation
Fixed Point Theory and Applications volume 2013, Article number: 38 (2013)
Abstract
In this manuscript, we discuss the existence of a coupled coincidence point for mappings F:X\times X\to X and g:X\to X, where F has the mixed gmonotone property, in the context of partially ordered metric spaces with an implicit relation. Our main theorem improves and extends various results in the literature. We also state some examples to illustrate our work.
MSC:47H10, 54H25, 46J10, 46J15.
1 Introduction and preliminaries
It is well known that fixed point theory is one of the crucial and very efficient tools in nonlinear functional analysis. This is because its evergrowing use in this field is very extensive in applications. In particular, the effects of fixed point theory are most apparent in fields like economy, computer sciences and engineering including many branches of mathematics. Historically, in 1886, Poincaré initiated first fixed point results. Then, in 1912, Brouwer published a result in this field, which was equivalent to Poincaré’s theorem, which in the simplest terms states that a continuous function from a disk D to itself has a fixed point. But most of the substantial advances in fixed point theory started after the celebrated fixed point result of Banach, known as Banach’s contraction mapping principle, in 1922. This principle can be stated as follows: any contraction in a complete metric space has a unique fixed point. When compared to Browder’s fixed point theorem, the power of Banach’s principle comes from the fact that it guarantees the uniqueness of a fixed point and gives a method to determine the fixed point. These two strengths of Banach’s contraction mapping principle have attracted attention of many prominent mathematicians who aim to broaden the applications of nonlinear functional analysis via fixed point theory in various quantitative sciences.
In the light of these developments, Guo and Lakshmikantham [1] defined the notion of a coupled fixed point in 1987. Later, GnanaBhaskar and Lakshmikantham [2] improved the idea of a coupled fixed point in the category of partially ordered metric spaces by introducing the notion of a mixed monotone mapping and presented certain applications on the solution of periodic boundary value problems. Interested readers may refer to [3–5] and the references therein to follow the development of fixed point theory on partially ordered metric spaces.
For the sake of completeness, we will review the basic definitions and fundamental results.
Definition 1 (See [2])
Let (X,\u2aaf) be a partially ordered set and F:X\times X\to X. The mapping 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 (See [2])
An element (x,y)\in X\times X is called a coupled fixed point of the mapping F:X\times X\to X if
We state now the main results of GnanaBhaskar and Lakshmikantham in [2].
Theorem 3 (See [2])
Let (X,\u2aaf) be a partially ordered set and suppose there exists a metric d on X such that (X,d) is a complete metric space. Assume that there exists a k\in [0,1) with
for all x\u2ab0u and y\u2aafv. Let either

(a)
F:X\times X\to X be a continuous mapping having the mixed monotone property on X, or

(b)
X have the following property:

(i)
if a nondecreasing sequence \{{x}_{n}\}\to x, then {x}_{n}\u2aafx for all n,

(ii)
if a nonincreasing sequence \{{y}_{n}\}\to y, then y\u2aaf{y}_{n} for all n.

(i)
If there exist two elements {x}_{0},{y}_{0}\in X with
then there exist x,y\in X such that
Following this theorem, several coupled coincidence/fixed point theorems and their applications to integral equations, matrix equations and a periodic boundary value problem have been reported (see, e.g., [6–19] and references therein). In particular, Lakshmikantham and Ćirić [6] established coupled coincidence and coupled fixed point theorems for two mappings F:X\times X\to X and g:X\to X, where F has the mixed gmonotone property and the functions F and g commute, as an extension of the fixed point results in [2]. For the sake of completeness, we recall these characterizations.
Definition 4 (See [6])
Let (X,\u2aaf) be a partially ordered set and let F:X\times X\to X and g:X\to X be two mappings. We say F has the mixed gmonotone property if F(x,y) is gnondecreasing in its first argument and is gnonincreasing in its second argument, that is, for any x,y\in X,
and
Definition 5 (See [6])
An element (x,y)\in X\times X is called a coupled coincident point of the mappings F:X\times X\to X and g:X\to X if
Definition 6 [7]
The mappings F and g, where F:X\times X\to X, g:X\to X, are said to be compatible if
and
where \{{x}_{n}\} and \{{y}_{n}\} are sequences in X such that F({x}_{n},{y}_{n})=g{x}_{n}\to x and F({y}_{n},{x}_{n})=g{y}_{n}\to y as n\to \mathrm{\infty} for all x,y\in X are satisfied.
Definition 7 (See [20])
Two selfmappings A and B are said to be weakly compatible if they commute at their coincidence points, i.e., ABu=BAu whenever Au=Bu, u\in X.
Choudhury and Kundu in [7] defined the notion of compatibility and showed the result established in [6] with a different set of conditions. In other words, the authors constructed their result by assuming that F and g are compatible mappings. Later, Luong and Thuan [13] slightly improved the notion of compatible mappings on partially ordered metric spaces, namely Ocompatible mappings. In this paper [13], the authors proved some coupled coincidence point theorems for Ocompatible type mappings in the context of partially ordered generalized metric spaces.
We recall the concept of Ocompatible mappings as follows.
Definition 8 (cf. [13])
Let (X,\u2aaf,d) be a partially ordered metric space. Two mappings F:X\times X\to X and g:X\to X are said to be Ocompatible if
and
where \{{x}_{n}\} and \{{y}_{n}\} are sequences in X such that \{g{x}_{n}\}, \{g{y}_{n}\} are monotone and
and
are satisfied for some x,y\in X.
Let (X,\u2aaf,d) be a partially ordered metric space. If F:X\times X\to X and g:X\to X are compatible, then they are Ocompatible. However, the converse is not true. The following example shows that there exist mappings which are Ocompatible but not compatible.
Example 9 Let X=\{0\}\cup [1/2,2] with the usual metric d(x,y)=xy for all x,y\in X. We consider the following order relation on X:
Let F:X\times X\to X be given by
and g:X\to X be defined by
Then F and g are Ocompatible but not compatible. To discern this, let \{{x}_{n}\} and \{{y}_{n}\} be two sequences in X such that \{g{x}_{n}\} and \{g{y}_{n}\} are monotone and
and
for some x,y\in X. It is easy to see that x=y\in \{0,1\} since F({x}_{n},{y}_{n})=F({y}_{n},{x}_{n})\in \{0,1\}. It is not possible to have x=y=1. Assume otherwise. There exists a positive integer N such that g{x}_{n}=1 and g{y}_{n}=1 for all n>N, since both g{x}_{n} and g{y}_{n} are monotone. Then we have {x}_{n},{y}_{n}\in [1/2,1] for all n>N, which implies that F({x}_{n},{y}_{n})=0 and F({y}_{n},{x}_{n})=0 for all n>N, a contradiction. Thus, x=y=0. In this case, we derive that g{x}_{n}=0 and g{y}_{n}=0 for all n>M for some positive integer M. Then, we get {x}_{n}=0 and {y}_{n}=0 for all n>M. As a result, we obtain
d(gF({x}_{n},{y}_{n}),F(g{x}_{n},g{y}_{n}))=0, and d(gF({y}_{n},{x}_{n}),F(g{y}_{n},g{x}_{n}))=0. Therefore, F and g are Ocompatible.
On the other hand, let
for n=1,2,3,\dots . Observe that
and
as n approaches to ∞, where F({y}_{n},{x}_{n})=F({x}_{n},{y}_{n}) and g{x}_{n}=g{y}_{n}. But we have
which does not approach to 0 as n approaches to ∞. Hence, F and g are not compatible.
Remark 10 In Example 9, if we let a=3/2, we obtain the example presented in [13].
In nonlinear analysis, especially in fixed point theory, implicit relations on metric spaces have been investigated heavily in many articles (see, e.g., [21–24] and references therein). In this paper, by using the following implicit relation, we examine the existence of a coupled coincidence point theorem for mappings F:X\times X\to X and g:X\to X in the context of a partial metric space, where F has the mixed gmonotone property and F, g are Ocompatible.
Let {\mathbb{R}}_{+} denote the set of all nonnegative real numbers. Also, let Φ denote the collection of all functions \phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+} which satisfy

(i)
φ is continuous and nondecreasing,

(ii)
\phi (t)<t for each t>0 and \phi (0)=0.
We reserve ℍ for the class of all continuous functions H:{\mathbb{R}}_{+}^{6}\to \mathbb{R} satisfying

(H1)
H({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6}) is nonincreasing in {t}_{3} and {t}_{6},

(H2)
if H(z,u,u+v,v,w,u+v)\le 0, then z+u\le h(v+w), where h\in [0,1).
Example 11 The following functions lie in ℍ:

{H}_{1}({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6})={t}_{1}+{t}_{2}\alpha {t}_{3}\beta {t}_{4}\gamma {t}_{5}\theta {t}_{6}, where α, β, γ, θ are nonnegative real numbers satisfying \alpha +\beta +\gamma +\theta <1.

{H}_{2}({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6})={t}_{1}k\{{t}_{3}+{t}_{4}\}, where k\in (0,\frac{1}{2}).

{H}_{3}({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6})={t}_{2}k\{{t}_{3}+{t}_{4}\}, where k\in (0,\frac{1}{2}).

{H}_{4}({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6})={t}_{1}+{t}_{2}h\{{t}_{3}+{t}_{4}\}, where h\in (0,1).

{H}_{5}({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6})={t}_{1}+{t}_{2}\alpha ({t}_{4}+{t}_{5}), where \alpha \in [0,1).

{H}_{6}({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6})={t}_{1}+{t}_{2}\alpha {t}_{5}\beta ({t}_{4}+{t}_{5})\gamma ({t}_{1}+{t}_{6}), where α, β, γ are nonnegative real numbers satisfying \alpha +\beta +2\gamma <1.

{H}_{7}({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6})={t}_{1}+{t}_{2}\alpha max\{\frac{{t}_{3}}{2},\frac{{t}_{4}+{t}_{5}}{2},{t}_{4}+{t}_{5}{t}_{2}\}, where \alpha \in [0,1).
In this paper, we prove a coupled coincidence point theorem for mappings satisfying such implicit relations.
2 Main result
We start by stating our primary theorem:
Theorem 12 Let (X,d,\u2aaf) be a partially ordered complete metric space. Suppose that F:X\times X\to X and g:X\to X are two mappings such that F has the mixed gmonotone property. Assume that there exists H\in \mathbb{H} such that
for all x,y,u,v\in X with gx\u2ab0gu and gy\u2aafgv. Suppose also that F(X\times X)\subseteq g(X) and g is continuous on X and Ocompatible with F. Additionally, suppose that either

(a)
F is continuous, or

(b)
X has the properties

(i)
if a nondecreasing sequence {x}_{n}\to x, then g{x}_{n}\u2aafgx for all n, and

(ii)
if a nonincreasing sequence {y}_{n}\to y, then gy\u2aafg{y}_{n} for all n.

(i)
If there exist two elements {x}_{0},{y}_{0}\in X with
then F and g have a coupled coincidence point in X.
Proof Let {x}_{0},{y}_{0}\in X be such that g{x}_{0}\u2aafF({x}_{0},{y}_{0}) and g{y}_{0}\u2ab0F({y}_{0},{x}_{0}). We will construct the iterative sequences \{{x}_{n}\} and \{{y}_{n}\} in X as follows:
The sequences \{{x}_{n}\} and \{{y}_{n}\} are well defined since F(X\times X)\subseteq g(X). Using the mathematical induction and the fact that F has the mixed gmonotone property, we obtain
for all n\ge 0. If there is a number {n}_{0}\in {\mathbb{N}}^{\ast}=\{0,1,2,3,\dots \} such that g{x}_{{n}_{0}}=g{x}_{{n}_{0}+1} and g{y}_{{n}_{0}}=g{y}_{{n}_{0}+1}, then g{x}_{{n}_{0}}=g{x}_{{n}_{0}+1}=F({x}_{{n}_{0}},{y}_{{n}_{0}}) and g{y}_{{n}_{0}}=g{y}_{{n}_{0}+1}=F({y}_{{n}_{0}},{x}_{{n}_{0}}). In this case, the theorem follows since ({x}_{{n}_{0}},{y}_{{n}_{0}}) is a coupled coincidence point of F and g.
Assume that g{x}_{n}\ne g{x}_{n+1} or g{y}_{n}\ne g{y}_{n+1} for all n\in {\mathbb{N}}^{\ast}. Since g{x}_{n+1}\u2ab0g{x}_{n} and g{y}_{n+1}\u2aafg{y}_{n}, we have
by (2.1). Using the definitions of \{{x}_{n}\} and \{{y}_{n}\} in 2.2, we obtain
By combining the property (H1) of H with the triangle inequality, inequalities in (2.4) turn into
Then, the property (H2) of H implies that
where h\in [0,1).
Set {d}_{n}:=d(g{x}_{n+1},g{x}_{n})+d(g{y}_{n+1},g{y}_{n}). Thus, the inequality (2.5) can be written as
We will show that the sequences \{g{x}_{n}\} and \{g{y}_{n}\} are Cauchy. Without loss of generality, we may assume that m>n. Then, by the triangle inequality, we have
Letting n,m\to \mathrm{\infty} in (2.6), we have
Consequently, {lim}_{n,m\to \mathrm{\infty}}d(g{x}_{m},g{x}_{n})=0 and {lim}_{n,m\to \mathrm{\infty}}d(g{y}_{m},g{y}_{n})=0. Hence the sequences \{g{x}_{n}\} and \{g{y}_{n}\} are Cauchy. Since X is complete, there exist x,y\in X such that
and
Thus, we obtain
as n\to \mathrm{\infty}. Since F and g are Ocompatible, we have
and
by (2.9).
Now, assume that (a) holds, i.e., F is continuous. By taking the limit in the following inequality:
as n\to \mathrm{\infty}, we obtain
by (2.7), (2.10) and the continuity of F and g. Similarly, we can show that d(gy,F(y,x))=0. As a result, F and g have a coupled coincidence point in X, i.e., gx=F(x,y) and gy=F(y,x).
Assume that (b) holds. Since \{g{x}_{n}\} is a nondecreasing sequence and g{x}_{n}\to x, we have gg{x}_{n}\u2aafgx for all n\in {\mathbb{N}}^{\ast} by (i). Similarly, since \{g{y}_{n}\} is a nonincreasing sequence and g{y}_{n}\to y, we also have gg{y}_{n}\u2ab0gy for all n\in {\mathbb{N}}^{\ast}. Since g is continuous, using (2.7), (2.10) and (2.11), we derive that
and
By the hypothesis of the theorem, we know that
for every n\in {\mathbb{N}}^{\ast}. In the inequality above, we let n\to \mathrm{\infty} and use (2.12) and (2.13) to obtain
which implies that d(gy,F(y,x))+d(gx,F(x,y))\le h(0+0)=0 by (H2) for h\in [0,1). Finally, we find that gx=F(x,y) and gy=F(y,x) completing the proof. □
Example 13 Let (X,d,\u2aaf), F and g be defined as in Example 9. We see that

X is complete and X has the properties

(i)
if a nondecreasing sequence {x}_{n}\to x, then g{x}_{n}\u2aafgx for all n\in {\mathbb{N}}^{\ast},

(ii)
if a nonincreasing sequence {y}_{n}\to y, then gy\u2aafg{y}_{n} for all n\in {\mathbb{N}}^{\ast}.

(i)

F(X\times X)=\{0,1\}\subset \{0\}\cup [1/2,1]=g(X).

g is continuous and g and F are Ocompatible.

There exist {x}_{0}=0 and {y}_{0}=1 such that g{x}_{0}\u2aafF({x}_{0},{y}_{0}) and g{y}_{0}\u2ab0F({y}_{0},{x}_{0}).

F has the mixed gmonotone property, which can be proved as follows: Let y,{x}_{1},{x}_{2}\in X such that g{x}_{1}\u2aafg{x}_{2}. There are two cases to consider:

(1)
If g{x}_{1}=g{x}_{2}, then {x}_{1}=0 and {x}_{2}=0 or {x}_{1},{x}_{2}\in [1/2,1] or {x}_{1},{x}_{2}\in (1,a] or {x}_{1},{x}_{2}\in (a,2]. Thus, F({x}_{1},y)=0=F({x}_{2},y) if y\in \{0\}\cup [1/2,1] and {x}_{1}=0 and {x}_{2}=0 or {x}_{1},{x}_{2}\in [1/2,1]. Otherwise, F({x}_{1},y)=1=F({x}_{2},y).

(2)
If g{x}_{1}\prec g{x}_{2}, then g{x}_{1}=0 and g{x}_{2}=1, i.e., {x}_{1}=0 and {x}_{2}\in [1/2,1]. Thus, F({x}_{1},y)=0=F({x}_{2},y) if y\in \{0\}\cup [1/2,1] and F({x}_{1},y)=1=F({x}_{2},y) if y\in (1,2].
Therefore, F is gnondecreasing in its first argument. Similarly, it can be shown that F is gnonincreasing in its second argument.

(1)

There exists H({t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5},{t}_{6})={t}_{1}max\{{t}_{3},{t}_{4}\}/2\in \mathbb{H} such that (2.1) holds. To discern this, for any x,y,u,v\in X with gx\u2ab0gu and gy\u2aafgv, we need to show that d(F(y,x),F(v,u))=0. Indeed,

(1)
if gx\succ gu and gy\prec gv, then y=u=0 and x,v\in [1/2,1]. Thus
d(F(y,x),F(v,u))=d(F((0,x),F(v,0))=d(0,0)=0; 
(2)
if gx=gu and gy\prec gv, then y=0 and v\in [1/2,1]. Either x=u=0, or x,u\in [1/2,1]. In any case,
d(F(0,x),F(v,u))=d(0,0)=0.Otherwise, we get
d(F(0,x),F(v,u))=d(0,0)=0.Similarly, if gx\succ gu and gy=gv, then d(F(y,x),F(v,u))=0;

(3)
if gx=gu and gy=gv, then both x, u are in one of the sets \{0\}, [1/2,1], (1,a] or (a,2] and both y, v are also in one of the sets \{0\}, [1/2,1], (1,a] or (a,2]. Thus
d(F(y,x),F(v,u))=d(0,0)=0.If x=u=0 or x,u\in [1/2,1] and y=v=0 or y,v\in [1/2,1], otherwise,
d(F(y,x),F(v,u))=d(1,1)=0.

(1)
Therefore, all of the conditions of Theorem 12 are satisfied. Therefore, we conclude that F and g have a coupled coincidence point.
Note that we cannot apply the result of Choudhury and Kundu [7], the result of Choudhury, Metiya and Kundu [25] as well as the result of Lakshmikantham and Ciric [6] to this example.
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Gülyaz, S., Karapınar, E. & Yüce, İ.S. A coupled coincidence point theorem in partially ordered metric spaces with an implicit relation. Fixed Point Theory Appl 2013, 38 (2013). https://doi.org/10.1186/16871812201338
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DOI: https://doi.org/10.1186/16871812201338