A note on `Coupled fixed point theorems for alpha-psi contractive-type mappings in partially ordered...

Karapınar and Agarwal Fixed Point Theory and Applications 2013, 2013:216http://www.fixedpointtheoryandapplications.com/content/2013/1/216

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A note on ‘Coupled fixed point theorems forα-ψ-contractive-type mappings in partiallyordered metric spaces’Erdal Karapınar1* and Ravi P Agarwal2,3

*Correspondence:[email protected];[email protected] of Mathematics,Atilim University, Incek, Ankara06836, TurkeyFull list of author information isavailable at the end of the article

AbstractIn this paper, we show that some examples in (Mursaleen et al. in Fixed Point TheoryAppl. 2012:124, 2012) are not correct. Then, we extend, improve and generalize theirresults. Finally, we state some examples to illustrate our obtained results.MSC: 47H10; 54H25

Keywords: coupled fixed point; fixed point; ordered set; metric space; α-ψcontractive mapping; α-admissible; (c)-comparison function; partial order

1 Introduction and preliminariesIn the sequel, let X be a non-empty set. Throughout the text, we use indifferently thenotation Y or X to denote the product space X × X. Let T : X → X and F : X → X betwo mappings. From now on, � will denote a partial order on X, and d will be a metricon X. To determine sufficient conditions in order to ensure the existence of the followingkind of points is the main aim of the present manuscript.

Definition . (See []) An element (x, y) ∈ X is called a coupled fixed point of a mappingF : X → X if F(x, y) = x and F(y,x) = y.

Definition . (See []) Let (X,�) be a partially ordered set, and let T : X → X be a map-ping. Then T is said to be non-decreasing with respect to � if x� y implies Tx� Ty, and itis non-increasing (w.r.t. �) if x� y implies Tx � Ty for every x, y ∈ X.

Definition . (See, e.g., []) Let (X,�) be a partially ordered set. A sequence {xn} ⊂ X issaid to be non-decreasing with respect to � if xn � xn+ for all n.

Definition . (See []) Let (X,�) be a partially ordered set, and let d be a metric on X.We say that (X,�,d) is regular if for every non-decreasing sequence {xn} ⊂ X such thatxn → x ∈ X as n→ ∞, there exists a subsequence {xn(k)} of {xn} such that xn(k) � x for all k.

Definition . (See []) Let (X,�) be a partially ordered set, and let F : X → X be a map-ping. The mapping F is said to have the mixed monotone property if F(x, y) is monotonenon-decreasing in x and monotone non-increasing in y, that is, for any x, y ∈ X,

x,x ∈ X, x � x �⇒ F(x, y)� F(x, y)

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and

y, y ∈ X, y � y �⇒ F(x, y)� F(x, y).

The partial order � on X can be induced on X in the following way

(x, y), (u, v) ∈ X, (x, y)� (u, v) ⇐⇒ x � u and y � v. ()

We say that (x, y) is comparable to (u, v) if either (x, y)� (u, v) or (x, y)� (u, v). Accordingto the definitions above, a sequence {(xn, yn)} ⊂ X is non-decreasing with respect to� if(xn, yn)� (xn+, yn+) for all n.If d is a metric on X, we will consider the metrics d,dmax : Y × Y → [,∞) defined, for

all (x, y), (u, v) ∈ Y , by

d((x, y), (u, v)

)= d(x,u) + d(y, v), dmax

((x, y), (u, v)

)= max

(d(x,u),d(y, v)

).

Notice that (X,�,d) is regular if for every non-decreasing sequence {(xn, yn)} ⊂ X suchthat (xn, yn)

d→ (x, y) ∈ Y as n → ∞, there exists a subsequence {(xn(k), yn(k))} of {(xn, yn)}such that (xn(k), yn(k))� (x, y) for all k.Given a mapping F : X → X, the mapping TF : Y → Y will be defined as follows:

TF (x, y) =(F(x, y),F(y,x)

)for all (x, y) ∈ Y .

The following result can be easily shown.

Lemma . (See, e.g., []) The following properties hold:(a) if (X,d) is complete, then (Y ,d) and (Y ,dmax) are complete;(b) F has the mixed monotone property on (X,�) if, and only if, TF is monotone

non-decreasing with respect to �;(c) (x, y) ∈ X ×X is a coupled fixed point of F if and only if (x, y) is a fixed point of TF .

Let� be the family of functionsψ : [,∞)→ [,∞) satisfying the following conditions:

(�) ψ is non-decreasing;(�)

∑+∞n= ψn(t) <∞ for all t > , where ψn is the nth iterate of ψ .

These functions are known in the literature as Bianchini-Grandolfi gauge functions insome sources and as (c)-comparison functions in others (see, e.g., []). They have a crucialrole in fixed point theory (see, e.g., [–]). It is easily proved that if ψ is a (c)-comparisonfunction, then ψ(t) < t for any t > .Very recently, Samet et al. [] introduced the following concepts.

Definition . Let (X,d) be a metric space, and let T : X → X be a given mapping. Wesay that T is an α-ψ-contractive mapping if there exist two functions α : X ×X → [,∞)and ψ ∈ � such that

α(x, y)d(Tx,Ty)≤ ψ(d(x, y)

)for all x, y ∈ X. ()



Clearly, any contractive mapping (that is, a mapping satisfying the Banach contractionproperty associated to k ∈ (, )) is an α-ψ-contractive mapping with α(x, y) = for allx, y ∈ X and ψ(t) = kt for all t ≥ .

Definition . Let T : X → X, and let α : X ×X → [,∞). We say that T is α-admissibleif for all x, y ∈ X, we have

α(x, y)≥ �⇒ α(Tx,Ty)≥ .

Various examples of such mappings are presented in []. The characterization of thisnotion for the setting ofG-metric spaces was considered in [, ]. The main results in []are the following fixed point theorems.

Theorem . Let (X,d) be a complete metric space, and let T : X → X be an α-ψ-contractive mapping. Suppose that

(i) T is α-admissible;(ii) there exists x ∈ X such that α(x,Tx)≥ ;(iii) T is continuous.Then there exists u ∈ X such that Tu = u.

Theorem . Let (X,d) be a complete metric space, and let T : X → X be an α-ψ-contractive mapping. Suppose that

(i) T is α-admissible;(ii) there exists x ∈ X such that α(x,Tx)≥ ;(iii) if {xn} is a sequence in X such that α(xn,xn+)≥ for all n and xn → x ∈ X as

n→ ∞, then α(xn,x)≥ for all n.Then there exists u ∈ X such that Tu = u.

Theorem . Adding to the hypotheses of Theorem . (resp. Theorem .) the condition‘For all x, y ∈ X, there exists z ∈ X such that α(x, z) ≥ and α(y, z) ≥ ,’ we obtain theuniqueness of the fixed point.

In [], the authors alsomentioned some existing results can be considered as a particularcase of their main results, see e.g. [–]. Later, Karapınar and Samet [] extended andgeneralized the result of Samet et al. [] by stating the following definitions.

Definition . Let (X,d) be a metric space, and let T : X → X be a given mapping. Wesay that T is a generalized α-ψ-contractive mapping of type I if there exist two functionsα : X ×X → [,∞) and ψ ∈ � such that for all x, y ∈ X, we have

α(x, y)d(Tx,Ty)≤ ψ(M(x, y)

), ()

whereM(x, y) = max{d(x, y), d(x,Tx)+d(y,Ty) , d(x,Ty)+d(y,Tx) }.

Definition . Let (X,d) be a metric space, and let T : X → X be a given mapping. Wesay that T is a generalized α-ψ-contractive mapping of type II if there exist two functions



α : X ×X → [,∞) and ψ ∈ � such that for all x, y ∈ X, we have

α(x, y)d(Tx,Ty)≤ ψ(N(x, y)

), ()

where N(x, y) = max{d(x, y), d(x,Tx)+d(y,Ty) }.

Remark . Clearly, since ψ is non-decreasing, every α-ψ-contractive mapping is a gen-eralized α-ψ-contractive mapping of types I and II. Notice also that every generalizedα-ψ-contractive mapping of type II is also a generalized α-ψ-contractive mapping oftype I.

Karapınar and Samet [] proved the following theorems.

Theorem . Let (X,d) be a complete metric space. Suppose that T : X → X is a general-ized α-ψ-contractive mapping of type I (respectively, of type II) and satisfies the followingconditions:

(i) T is α-admissible;(ii) there exists x ∈ X such that α(x,Tx)≥ ;(iii) T is continuous.Then there exists u ∈ X such that Tu = u.

Theorem . Let (X,d) be a complete metric space. Suppose that T : X → X is a general-ized α-ψ-contractivemapping of type I (respectively, of type II), and the following conditionshold:

(i) T is α-admissible;(ii) there exists x ∈ X such that α(x,Tx)≥ ;(iii) if {xn} is a sequence in X such that α(xn,xn+)≥ for all n and xn → x ∈ X as

n→ ∞, then there exists a subsequence {xn(k)} of {xn} such that α(xn(k),x)≥ forall k.

Then there exists u ∈ X such that Tu = u.

For the uniqueness of a fixed point of a generalized α-ψ-contractive mapping, we willconsider the following hypothesis, in which Fix(T) denotes the set of all fixed points of T .(H) For all x, y ∈ Fix(T), there exists z ∈ X such that α(x, z)≥ and α(y, z)≥ .

Theorem . Adding condition (H) to the hypotheses of Theorem . (resp. Theorem .),we obtain that u is the unique fixed point of T .

Inspired by the above mentioned results, Mursaleen et al. [] characterized the idea toprove the existence and uniqueness of a coupled fixed point. Before starting the main the-orem ofMursaleen et al., we recall the basic definition and fundamental results in coupledfixed point theory.The first result in the existence and uniqueness of fixed point of contraction mapping in

partially ordered complete metric spaces was given by Ran and Reurings []. Followingthis initial work, a number of authors have investigated the fixed points of various map-pings and their applications in the theory of differential equations. A notion of coupledfixed point was defined by Guo and Laksmikantham []. After that, Bhaskar and Laksh-mikantham [] proved the existence and uniqueness of a coupled fixed point in the context



of partially orderedmetric spaces by introducing the notion of mixedmonotone property.In that paper, the authors proved the existence and uniqueness of a solution of periodicboundary value problems. After this remarkable paper, the notion of coupled fixed pointhave attracted attention of a number of authors (see, e.g., [–]).Mursaleen et al. [] reconsidered the notion of an α-admissible mapping, introduced

by Samet et al. [], in the following way:

Definition . (See []) Let F : X → X and α : X × X → [,∞) be two mappings.We say that F is α-admissible if for all x, y,u, v ∈ X, we have

α((x, y), (u, v)

) ≥ �⇒ α((F(x, y),

(F(y,x)

)),(F(u, v),

(F(v,u)

))) ≥ .

Remark . Notice that Definition . is exactly the same with Definition . by choos-ing X.

The main results in [] are the following ones.

Theorem . Let (X,�) be a partially ordered set, and suppose that there is a metric d onX such that (X,d) is a complete metric space. Let F : X × X → X be a mapping having themixed monotone property of X. Suppose that there exist ψ ∈ � and α : X × X → [,∞)such that for all x, y,u, v ∈ X, the following holds

α((x, y), (u, v)

)d(F(x, y),F(u, v)

) ≤ ψ

(d(x,u) + d(y, v)

)()

for all x � u and y� v. Suppose also that(i) F is α-admissible;(ii) there exists x, y ∈ X such that

α((x, y),

(F(x, y),F(y,x)

)) ≥ and α((y,x),

(F(y,x),F(x, y)

)) ≥ ;

(iii) F is continuous.If there exist x, y ∈ X such that x � F(x, y) and y � F(y,x), then F has a coupled

fixed point, that is, there exist x, y ∈ X such that

F(x, y) = x and F(y,x) = y.

Theorem . Let (X,�) be a partially ordered set, and suppose that there is a metric d onX such that (X,d) is a complete metric space. Let F : X × X → X be a mapping having themixed monotone property of X. Suppose that there exist ψ ∈ � and α : X × X → [,∞)such that for all x, y,u, v ∈ X, the following holds

α((x, y), (u, v)

)d(F(x, y),F(u, v)

) ≤ ψ

(d(x,u) + d(y, v)

)()

for all x � u and y� v. Suppose also that(i) F is α-admissible;



(ii) there exists x, y ∈ X such that

α((x, y),

(F(x, y),F(y,x)

)) ≥ and α((y,x),

(F(y,x),F(x, y)

)) ≥ ;

(iii) if {xn} and {yn} are sequences in X such that α((xn, yn), (xn+, yn+))≥ andα((yn,xn), (yn+,xn+))≥ for all n, and limn→∞ xn = x and limn→∞ yn = y, thenα((xn, yn), (x, y))≥ and α((yn,xn), (y,x))≥ for all n.

If there exist x, y ∈ X such that x � F(x, y) and y � F(y,x), then F has a coupledfixed point, that is, there exist x, y ∈ X such that


Theorem . In addition to the hypothesis of Theorem ., suppose that for every(x, y), (s, t) ∈ X ×X, there exists (u, v) ∈ X ×X, such that

α((x, y), (u, v)

) ≥ and α((s, t), (u, v)

) ≥ ,

and also assume that (u, v) is comparable to (x, y) and (s, t). Then F has a unique fixedpoint.

In this paper, we show that coupled fixed point results of Mursaleen et al. [] can beobtained by usual fixed point theorems.Moreover, by giving an example, we conclude thatthe main result of Mursaleen et al. [] is not strong enough to be applied to their ownexamples. The object of this paper is to extend, improve and generalize their results in amore simple set up. Finally, we also note that the remarks and comments of this paper arealso valid for [].

2 Main resultsWe start this section by giving an example to show the weakness of Theorem .. First, wenotice that the function F(x, y) =

xy in Example . in [, ] and the function F(x, y) = ln( + |x|) +

ln( + |y|) in Example . in [, ] do not satisfy the mixed monotoneproperty.Now, we state the following example.

Example . Let X =R and d : X×X → [,∞) be the Euclideanmetric. Consider a map-ping α : X ×X → [,∞) defined by

α((x, y), (u, v)

)=

⎧⎨⎩, if x≥ u, y ≤ v,, otherwise.

Define a mapping F : X ×X → X as

F(x, y) =x – y

for all x, y ∈ X.

It is clear that F is mixed monotone, but we claim that it does not satisfy condition ().Indeed, assume that there exists ψ ∈ � such that

α((x, y), (u, v)

)d(F(x, y),F(u, v)

) ≤ ψ

(d(x,u) + d(y, v)

)()



holds for all x ≥ u and v ≥ y. Let us take x = u, y = v in the previous inequality. Hence,t = |x – u| > and inequality () turns into

t=|x – u|

= d

(F(x, y),F(u, v)

) ≤ ψ

( |x – u|

)=ψ

(t

). ()

Recall that ψ(t) < t for any t > . Hence, inequality () turns into

t

≤ ψ

(t

)<t,

which is a contradiction. Hence, Theorem . is not applicable to the operator F in orderto prove that (, ) is the unique coupled fixed point of F .

We notice that Theorem . is not strong enough to conclude that F has a coupled fixedpoint. Inspired by Example ., we suggest the following statement instead of Theorem ..

Theorem . Let (X,�) be a partially ordered set, and suppose there is a metric d on Xsuch that (X,d) is a complete metric space. Let F : X → X be a mapping having the mixedmonotone property on X. Suppose that there exist ψ ∈ � and α : X × X → [,∞) suchthat for all x, y,u, v ∈ X the following holds

α((x, y), (u, v)

)d(F(x, y),F(u, v)) + d(F(y,x),F(v,u))

≤ ψ

(d(x,u) + d(y, v)

)()

for which x� u and y � v. Suppose also that(i) F is α-admissible;(ii) there exists x, y ∈ X such that

α((x, y),

(F(x, y),F(y,x)

)) ≥ and α((y,x),

(F(y,x),F(x, y)

)) ≥ ;

(iii) F is continuous.If there exist x, y ∈ X such that x � F(x, y) and y � F(y,x), then F has a coupled

fixed point, that is, there exist x, y ∈ X such that


Proof Following the lines of the proof of Theorem ., we conclude the result. To avoidthe repetition, we omit the details. �

Analogously, instead of Theorem ., we state the following.

Theorem . Let (X,�) be a partially ordered set, and suppose that there is a metric don X such that (X,d) is a complete metric space. Let F : X → X be a mapping having themixed monotone property on X. Suppose that there exist ψ ∈ � and α : X ×X → [,∞)such that for all x, y,u, v ∈ X, the following holds

α((x, y), (u, v)

)d(F(x, y),F(u, v)

) ≤ ψ

(d(x,u) + d(y, v)

)()

for which x� u and y � v. Suppose also that



(i) F is α-admissible;(ii) there exists x, y ∈ X such that

α((x, y),

(F(x, y),F(y,x)

)) ≥ and α((y,x),

(F(y,x),F(x, y)

)) ≥ ;

(iii) if {xn} and {yn} are sequences in X such that α((xn, yn), (xn+, yn+))≥ andα((yn,xn), (yn+,xn+))≥ for all n, and limn→∞ xn = x and limn→∞ yn = y, thenα((xn, yn), (x, y))≥ and α((yn,xn), (y,x))≥ for all n.

If there exist x, y ∈ X such that x � F(x, y) and y � F(y,x), then F has a coupledfixed point, that is, there exist x, y ∈ X such that


Proof Following the lines of the proof of Theorem ., we easily conclude the result. Weomit the details. �

Let us reconsider Example ..

Example . Let X =R, and let d : X×X → [,∞) be the Euclideanmetric. Consider themapping α : X ×X → [,∞) defined as

α((x, y), (u, v)

)=

⎧⎨⎩, if x≥ u, y ≤ v or x≤ u, y ≥ v,, otherwise.

Consider the mapping F : X ×X → X defined by

F(x, y) = x – y

for all x, y ∈ X.

Clearly, F has a mixedmonotone property, and we claim that it also satisfies condition ().Indeed, if α((x, y), (u, v)) = , then the result is straightforward. Suppose α((x, y), (u, v)) = .Without loss of generality, assume that x ≥ u and y ≤ v. Then we have that

α((x, y), (u, v)

)d(F(x, y),F(u, v)) + d(F(y,x),F(v,u))

=d(F(x, y),F(u, v)) + d(F(y,x),F(v,u))

=

∣∣(x – u) – (y – v)∣∣ +

∣∣(y – v) – (x – u)

∣∣

≤ [|x – u| + |y – v|] ()

holds for all x≥ u and v≥ y. On the other hand,

d((x, y) + d(u, v))

=[|x – u| + |y – v|]. ()

Hence, it is sufficient to choose ψ(t) = t to provide all conditions of Theorem .. Notice

that the point (, ) is the unique coupled fixed point of F .



Example . LetX = [–, ], and let d : X×X → [,∞) be the Euclideanmetric. Considerthe mapping α : X ×X → [,∞) defined as

α((x, y), (u, v)

)=


Define a mapping F : X ×X → X as

F(x, y) = x – y

for all x, y ∈ X.

Then F is mixed monotone and satisfies all conditions of Theorem .. Indeed, ifα((x, y), (u, v)) = , the result is trivial. Suppose α((x, y), (u, v)) = . Without loss of gen-erality, assume that x≥ y, u≥ v. Then we have that

α((x, y), (u, v)

)d(F(x, y),F(u, v)) + d(F(y,x),F(v,u))

=d(F(x, y),F(u, v)) + d(F(y,x),F(v,u))

=

∣∣(x – u)–

(y – v

)∣∣ +

∣∣(y – v)–

(x – u

)∣∣

≤ [∣∣x – u

∣∣ + ∣∣y – v∣∣] ≤

[|x – u| + |y – v|]

holds for all x≥ u and y≤ v. On the other hand,

d((x, y) + d(u, v))

=[|x – u| + |y – v|].

Hence, it is sufficient to choose ψ(t) = t to provide all conditions of Theorem .. Notice

that the point (, ) is the coupled fixed point of F .

Now, we improve Example . in [] in the following way.

Example . Let X = [,∞), and let d : X ×X → [,∞) be a Euclidean metric. Considerthe mapping α : X ×X → [,∞) defined by

α((x, y), (u, v)

)=


Define the mapping F : X ×X → X as

F(x, y) =

⎧⎨⎩

[ln( + x) – ln( + y)], if x ≥ y,, otherwise.

Then F is mixed monotone and satisfies all conditions of Theorem .. Indeed, ifα((x, y), (u, v)) = , the result follows trivially. Suppose α((x, y), (u, v)) = . Without loss of



generality, assume that x ≥ u and y ≤ v. Then we have

α((x, y), (u, v)

)d(F(x, y),F(u, v)) + d(F(y,x),F(v,u))

=d(F(x, y),F(u, v)) + d(F(y,x),F(v,u))

=∣∣(ln( + x) – ln( + u)

)–

(ln( + y) – ln( + v)

)∣∣

+∣∣(ln( + y) – ln( + v)

)–

(ln( + x) – ln( + u)

)∣∣

≤

[∣∣∣∣ln(( + x) + u

)∣∣∣∣]+

[∣∣∣∣ln(( + v) + y

)∣∣∣∣]

+

[∣∣∣∣ln(( + y) + v

)∣∣∣∣]+

[∣∣∣∣ln(( + u) + x

)∣∣∣∣]

≤

[ln

( + |x – u|) + ln

( + |y – v|)],

≤

ln

( +

[|x – u| + |y – v|]

)()

holds for all x≥ u and v≥ y. On the other hand,

d((x, y) + d(u, v))

=[|x – u| + |y – v|]. ()

Hence, it is sufficient to choose ψ(t) = ln( + t), t > to provide all conditions of Theo-

rem .. Notice that the point (, ) is the coupled fixed point of F .

For the uniqueness of the coupled fixed point, we state the following theorem.

Theorem . In addition to the hypothesis of Theorem ., suppose that for every(x, y), (s, t) ∈ X ×X, there exists (u, v) ∈ X ×X such that

α((x, y), (u, v)

) ≥ and α((s, t), (u, v)

) ≥ ,


To avoid the repetition, we omit the proof, since the result is easily obtained by followingthe lines of the proof of Theorem ..

3 From coupled fixed point theorem to usual fixed point theoremAccording to the definitions above, we reconsider Definition . in the following way.

Definition . Let F : X → X, and let α∗ : X × X → [,∞). The operator F is α∗-admissible if for all (x, y), (u, v) ∈ X, we have

α∗((x, y), (u, v)) = α∗((y,x), (v,u)) and ()

α∗((x, y), (u, v)) ≥ �⇒ α∗((F(x, y), (F(y,x))), (F(u, v), (F(v,u)))) ≥ . ()



Lemma . Let F : X → X, and let α∗ : X × X → [,∞). If F is α∗-admissible, thenthe operator TF : X → X, defined by TF (x, y) = (F(x, y),F(y,x)) for all (x, y) ∈ X, is α-admissible in the sense of Definition ., that is,

α(z,w)≥ �⇒ α(TF (z),TF (w)

) ≥

for all z = (x, y),w = (u, v) ∈ Y = X, where α∗ = α : Y × Y → [,∞).

We omit the proof, since it is straightforward.

Remark . Regarding the definition of d : X ×X → [,∞), that is,

d((x, y), (u, v)

)= d(x,u) + d(y, v) = d(y, v) + d(x,u) = d

((y,x), (v,u)

),

one can conclude the assumption () is very natural but not necessary. For example, con-sider a partially ordered set (X,�), then, we set (X ×X,�) as in (). Now, one can define

α((x, y)(u, v)

)=

⎧⎪⎪⎨⎪⎪⎩, if (x, y)� (u, v),, if (x, y)� (u, v),, otherwise,

which is clearly not equal to α∗((x, y)(u, v)).

Theorem . Let (X,d) be a complete metric space, and let F : X ×X → X be a mapping.Suppose that there exist ψ ∈ � and α∗ : X ×X → [,∞) such that the following holds

α∗((x, y), (u, v))d(F(x, y),F(u, v)) + d(F(y,x),F(v,u))

≤ ψ

(d(x,u) + d(y, v)

)()

for all x, y,u, v ∈ X. Suppose also that(i) F is α∗-admissible;(ii) there exists (x, y) ∈ X such that

α∗((x, y), (F(x, y),F(y,x))) ≥ ;

(iii) F is continuous.Then, there exist x, y ∈ X such that


Remark . Theorem . coincides with Theorem . if we replace α with α∗ in the state-ment of Theorem ..

Theorem . Theorem . follows from Theorem ..

Proof From (), for all (x, y), (u, v) ∈ X ×X, we have

α∗((x, y), (u, v))d(F(x, y),F(u, v)) + d(F(v,u),F(y,x))

≤ ψ

(d(x,u) + d(y, v)

),



that is,

α∗((x, y), (u, v))δ(TF (x, y),TF (u, v)) ≤ ϕ

(δ((x,u), (y, v)

))

for all (x, y), (u, v) ∈ Y , where δ : Y × Y → [,∞) is the metric on Y given by

δ((x, y), (u, v)

)=d((x, y), (u, v))

for all (x, y), (u, v) ∈ Y ,

and TF (x, y) = (F(x, y),F(y,x)) for all (x, y) ∈ X. Thus, we proved that the mapping TF sat-isfies condition () and hence all conditions of Theorem . are satisfied. Then TF has afixed point, which implies that F has a coupled fixed point. �

Theorem . Let (X,d) be a complete metric space, and let F : X ×X → X be a mapping.Suppose that there exist ψ ∈ � and α∗ : X ×X → [,∞) such that the following holds

α∗((x, y), (u, v))d(F(x, y),F(u, v)) + d(F(y,x),F(v,u))

≤ ψ

(d(x,u) + d(y, v)

)()

for all x, y,u, v ∈ X. Suppose also that(i) F is α-admissible;(ii) there exists (x, y) ∈ X such that

α((x, y),

(F(x, y),F(y,x)

)) ≥ ;

(iii) if {xn} and {yn} are sequences in X such that α∗((xn, yn), (xn+, yn+))≥ for all n andlimn→∞ xn = x and limn→∞ yn = y, then α∗((xn, yn), (x, y))≥ .

Then, there exist x, y ∈ X such that


Remark . Theorem . coincides with Theorem . if we replace α with α∗ in the state-ment of Theorem ..

Theorem . Theorem . follows from Theorem ..

Proof Following the lines of the proof of Theorem ., we observe that T satisfies condi-tion () and hence all conditions of Theorem . are satisfied. Then TF has a fixed point,which implies that F has a coupled fixed point. �

Theorem . In addition to the hypothesis of Theorem ., suppose that for every(x, y), (s, t) ∈ X ×X, there exists (u, v) ∈ X ×X such that

α((x, y), (u, v)

) ≥ and α((s, t), (u, v)

) ≥ ,




Proof Following the lines of the proof of Theorem ., we observe that T satisfies condi-tion () and hence all conditions of Theorem . are satisfied. Then TF has a unique fixedpoint, which implies that F has a unique coupled fixed point. �

Fixed point theorems onmetric spaces endowed with a partial orderIn the last decades, one of the most attractive research topics in fixed point theory wasto prove the existence of fixed point on metric spaces endowed with partial orders. Thefirst result in this direction was reported by Turinici [] in . Following this inter-esting paper, Ran and Reurings in [] characterized the Banach contraction principle inpartially ordered sets with some applications to matrix equations. Later, the results in [,] were further extended and improved by many authors (see, for example, [–] andthe references cited therein). In this section, wewill deduce thatmore general formof The-orem . (respectively, Theorem .) can be obtained from our Theorem . (respectively,Theorem .).We obtain the following result whose analog can be found in [].

Proposition . Let (X,�) be a partially ordered set, and let d be a metric on X such that(X,d) is complete. Let F : X ×X → X have a mixed monotone property. Suppose that thereexists a function ψ ∈ � such that

d(F(x, y),F(u, v)) + d(F(y,x),F(v,u))

≤ ψ

(d(x,u) + d(y, v)

)

for all (x, y), (u, v) ∈ Y with (x, y)� (u, v). Suppose also that the following conditions hold:(i) there exists (x, y) ∈ Y such that (x, y)� (F(x, y),F(y,x));(ii) F is continuous or (X,�,d) is regular.Then F has a coupled fixed point.Moreover, if for all (x, y), (u, v) ∈ Y there exists (z,w) ∈ Y

such that (x, y)� (z,w) and (u, v)� (z,w), we have uniqueness of the fixed point.

Proof Define the mapping α∗ : Y × Y → [,∞) by

α∗((x, y)(u, v)) =⎧⎨⎩ if (x, y)� (u, v) or (x, y)� (u, v),, otherwise.

Clearly, TF is an α-ψ-contractive mapping, that is,

α∗((x, y), (u, v))d(TF (x, y),TF (u, v)) ≤ ψ

(d

((x, y), (u, v)

))

for all (x, y), (u, v) ∈ Y . From condition (i), we have α∗((x, y), (TF (x, y),TF (y,x))) ≥ .Due to Lemma ., TF is non-decreasing mapping with respect to �. Moreover, for all(x, y), (u, v) ∈ Y , from the monotone property of TF , we have

α∗((x, y), (u, v)) ≥ �⇒ (x, y)� (u, v) or

(x, y)� (u, v) �⇒ TF (x, y)� TF (u, v) or

TF (x, y)� TF (u, v) �⇒ α∗(TF (x, y),TF (u, v)) ≥ .



Thus F is α∗-admissible. Hence, due to Lemma . TF is α-admissible. Now, if F is contin-uous, the existence of a fixed point follows from Theorem .. Suppose now that (X,�,d)is regular. Let {xn} be a sequence in X such that α∗((xn, yn), (xn+, yn+)) ≥ for all n and(xn, yn)→ (x, y) ∈ X as n→ ∞. From the regularity hypothesis, there exists a subsequence{xn(k)} of {xn} such that xn(k) � x and yn(k) � y for all k. This implies from the definitionof α∗ that α∗((xn(k), yn(k)), (x, y)) ≥ for all k. In this case, the existence of a fixed pointfollows from Theorem .. To show the uniqueness, let (x, y), (u, v) ∈ X. By hypothesis,there exists (z,w) ∈ X such that (x, y) � (z,w) and (u, v) � (z,w), which implies fromthe definition of α∗ that α∗((x, y), (z,w))≥ and α∗((u, v), (z,w))≥ . Thus, we deduce theuniqueness of the fixed point by Theorem .. �

Now, we state the result of [] as an easy consequence of Proposition ..

Corollary . Let (X,�) be a partially ordered set, and d be a metric on X such that (X,d)is complete. Let F : X ×X → X have a mixed monotone property. Suppose that there existsa function k ∈ [, ) such that

[d(F(x, y),F(u, v)

)+ d

(F(y,x),F(v,u)

)] ≤ k[d(x,u) + d(y, v)

]

for all (x, y), (u, v) ∈ Y with (x, y) � (u, v), where � is defined as in (). Suppose also thatthe following conditions hold:

(i) there exists (x, y) ∈ Y such that (x, y)� (F(x, y),F(y,x));(ii) F is continuous or (X,�,d) is regular.Then F has a coupled fixed point.Moreover, if for all (x, y), (u, v) ∈ Y there exists (z,w) ∈ Y

such that (x, y)� (z,w) and (u, v)� (z,w), we have uniqueness of the fixed point.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors read and approved the final manuscript.

Author details1Department of Mathematics, Atilim University, Incek, Ankara 06836, Turkey. 2Department of Mathematics, Texas A&MUniversity-Kingsville, Kingsville, Texas 78363, USA. 3Department of Mathematics, Faculty of Science, King AbdulazizUniversity, Jeddah, 21589, Saudi Arabia.

Received: 6 June 2013 Accepted: 30 July 2013 Published: 14 August 2013

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doi:10.1186/1687-1812-2013-216Cite this article as: Karapınar and Agarwal: A note on ‘Coupled fixed point theorems for α-ψ -contractive-typemappings in partially ordered metric spaces’. Fixed Point Theory and Applications 2013 2013:216.


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