A coupled coincidence point theorem in partially ordered metric spaces with an implicit relation
 Selma Gülyaz^{1},
 Erdal Karapınar^{2}Email author and
 İlker Savas Yüce^{3}
https://doi.org/10.1186/16871812201338
© Gülyaz et al.; licensee Springer. 2013
Received: 7 September 2012
Accepted: 6 February 2013
Published: 25 February 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.
Keywords
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])
Definition 2 (See [2])
We state now the main results of GnanaBhaskar and Lakshmikantham in [2].
Theorem 3 (See [2])
 (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)
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])
Definition 5 (See [6])
Definition 6 [7]
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])
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.
$d(gF({x}_{n},{y}_{n}),F(g{x}_{n},g{y}_{n}))=0$, and $d(gF({y}_{n},{x}_{n}),F(g{y}_{n},g{x}_{n}))=0$. Therefore, F and g are Ocompatible.
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.
 (i)
φ is continuous and nondecreasing,
 (ii)
$\phi (t)<t$ for each $t>0$ and $\phi (0)=0$.
 (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:
 (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)
then F and g have a coupled coincidence point in X.
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.
where $h\in [0,1)$.
by (2.9).
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 $gy=F(y,x)$.
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.
Declarations
Authors’ Affiliations
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