Cho et al. Journal of Inequalities and Applications 2013, 2013:104http://www.journalofinequalitiesandapplications.com/content/2013/1/104

RESEARCH Open Access

Coupled coincidence point theorems in(intuitionistic) fuzzy normed spacesYJ Cho1, J Martínez-Moreno2*, A Roldán3 and C Roldán4

*Correspondence:jmmoreno@ujaen.es2Department of Mathematics,University of Jaén, Jaén, SpainFull list of author information isavailable at the end of the article

AbstractIn this paper, we prove some coupled coincidence point theorems in fuzzy normedspaces. Our results improve and restate the proof lines of the main results given in thepapers (Eshaghi Gordji et al. in Math. Comput. Model. 54:1897-1906, 2011) and(Sintunavarat et al. in Fixed Point Theory Appl. 2011:81, 2011).MSC: Primary 47H10; secondary 54H25; 34B15

Keywords: fuzzy normed space; intuitionistic fuzzy normed space; coupled fixedpoint theorem; t-norm; H-type

1 IntroductionRecently, many authors have shown the existence of coupled fixed points and commonfixed points for some contractions in cone metric spaces, partially ordered metric spaces,fuzzy metric spaces, fuzzy normed spaces, intuitionistic fuzzy normed spaces and others([–]).Especially in [], Sintunavarat et al. proved some coupled fixed point theorems for

contractive mappings in partially complete intuitionistic fuzzy normed spaces, which ex-tended and improved coupled coincidence point theorems in Gordji et al. []. But theauthors found some mistakes in the proof lines of the main result (Theorem .) of []and the same mistakes in [].In Section of this paper, we restate some definitions and the main results in []. In

Section , we give some comments about the incorrect proof lines of themain results givenin [] and [] and explain why the lines of the proofs are wrong. Finally, in Section , weextend and improve some coupled fixed point theorems.

2 PreliminariesA t-norm (resp., a t-conorm) is a mapping ∗ : [, ] → [, ] (resp., � : [, ] → [, ]) thatis associative, commutative and non-decreasing in both arguments and has (resp., ) asidentity.

Definition . [] A fuzzy normed space (in brief, FNS) is a triple (X,μ,∗), where X is avector space, ∗ is a continuous t-norm and μ : X × (,∞)→ [, ] is a fuzzy set such that,for all x, y ∈ X and t, s > ,(F) μ(x, t) > ;(F) μ(x, t) = for all t > if and only if x = ;(F) μ(αx, t) = μ(x, t

|α| ) for all α �= ;

© 2013 Cho et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

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(F) μ(x, t) ∗ μ(y, s) ≤ μ(x + y, t + s);(F) μ(x, ·) : (,∞)→ [, ] is continuous;(F) limt→∞ μ(x, t) = and limt→ μ(x, t) = .

Using the continuous t-norms and t-conorms, Saadati and Park [] introduced the con-cept of an intuitionistic fuzzy normed space.

Definition . An intuitionistic fuzzy normed space (in brief, IFNS) is a -tuple (X,μ,ν,∗,�) where X is a vector space, ∗ is a continuous t-norm, � is a continuous t-conorm andμ,ν : X × (,∞)→ [, ] are fuzzy sets such that, for all x, y ∈ X and t, s > ,(IF) μ(x, t) + ν(x, t)≤ ;(IF) μ(x, t) > and ν(x, t) < ;(IF) μ(x, t) = for all t > if and only if x = if and only if ν(x, t) = for all t > ;(IF) μ(αx, t) = μ(x, t

|α| ) and ν(αx, t) = ν(x, t|α| ) for all α �= ;

(IF) μ(x, t) ∗ μ(y, s)≤ μ(x + y, t + s) and ν(x, t) � ν(y, s) ≥ ν(x + y, t + s);(IF) μ(x, ·),ν(x, ·) : (,∞) → [, ] are continuous;(IF) limt→∞ μ(x, t) = = limt→ ν(x, t) and limt→ μ(x, t) = = limt→∞ ν(x, t).

Obviously, if (X,μ,ν,∗,�) is an IFNS, then (X,μ,∗) is an FNS. We refer to this space asits support.

Lemma . μ(x, ·) is a non-decreasing function on (,∞) and ν(x, ·) is a non-increasingfunction on (,∞).

Some properties and examples of IFNS and the concepts of convergence and a Cauchysequence in IFNS are given in [].

Definition . [] Let (X,μ,ν,∗,�) be an IFNS.() A sequence {xn} ⊂ X is called a Cauchy sequence if, for any ε > and t > , there

exists n ∈N such that μ(xn – xm, t) > – ε and ν(xn – xm, t) < ε for all n,m ≥ n.() A sequence {xn} ⊂ X is said to be convergent to a point x ∈ X denoted by xn → x or

by limn→∞ xn = x if, for any ε > and t > , there exists n ∈ N such thatμ(xn – x, t) > – ε and ν(xn – x, t) < ε for all n≥ n.

() An IFNS in which every Cauchy sequence is convergent is said to be complete.

Most of the following definitions were introduced in [].

Definition . Let F : X ×X → X and g : X → X be two mappings.() F and g are said to be commuting if gF(x, y) = F(gx, gy) for all x, y ∈ X .() A point (x, y) ∈ X ×X is called a coupled coincidence point of the mappings F and g

if F(x, y) = gx and F(y,x) = gy. If g is the identity, (x, y) is called a coupled fixed pointof F .

() If (X,�) is a partially ordered set, then F is said to have themixed g-monotoneproperty if it verifies the following properties:

x,x ∈ X, gx � gx =⇒ F(x, y) � F(x, y), ∀y ∈ X,

y, y ∈ X, gy � gy =⇒ F(x, y) � F(x, y), ∀x ∈ X.

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If g is the identity mapping, then F is said to have themixed monotone property.() If (X,�) is a partially ordered set, then X is said to have the sequential g-monotone

property if it verifies the following properties:(B) If {xn} is a non-decreasing sequence and limn→∞ xn = x, then gxn � gx for all

n ∈N.(B) If {xn} is a non-increasing sequence and limn→∞ yn = y, then gyn � gy for all

n ∈N.If g is the identity mapping, then X is said to have the sequentialmonotone property.

Definition . Let X and Y be two IFNS. A function f : X → Y is said to be continuousat a point x ∈ X if, for any sequence {xn} in X converging to x, the sequence {f (xn)} in Yconverges to f (x). If f is continuous at each x ∈ X, then f is said to be continuous on X.

Definition . [] Let (X,μ,ν,∗,�) be an IFNS. The pair (μ,ν) is said to satisfy the n-property on X × (,∞) if limn→∞[μ(x,knt)]np = and limn→∞[ν(x,knt)]np = wheneverx ∈ X, k > and p > .

The following lemma proved by Haghi et al. [] is useful for our main results.

Lemma . Let X be a nonempty set and g : X → X be a mapping. Then there exists asubset E ⊂ X such that g(E) = g(X) and g : E → X is one-to-one.

In order to state our results, we give the main results given in [] and [].

Lemma . [, Lemma .] Let (X,μ,ν,∗,�) be an IFNS. Let a ∗ b ≥ ab, a � b ≤ ab for alla,b ∈ [, ] and (μ,ν) satisfy the n-property. Suppose that {xn} is a sequence in X such that

μ(xn+ – xn,kt) ≥ μ(xn – xn–, t), ν(xn+ – xn,kt) ≤ ν(xn – xn–, t)

for all t > and n ∈ N, where < k < . Then the sequence {xn} is a Cauchy sequence in X.

Theorem . ([, Theorem .], [, Theorem .]) Let (X,�) be a partially ordered setand suppose that a� b≤ ab ≤ a∗ b for all a,b ∈ [, ]. Let (X,μ,ν,∗,�) be a complete IFNSsuch that (μ,ν) has the n-property. Let F : X × X → X and g : X → X be two mappingssuch that F has the mixed g-monotone property and

μ(F(x, y) – F(u, v),kt

) ≥ μ(gx – gu, t) ∗ μ(gy – gv, t),

ν(F(x, y) – F(u, v),kt

) ≤ ν(gx – gu, t) � ν(gy – gv, t),

for which gx � gu and gy � gv, where < k < , F(X × X) ⊆ g(X) and g is continuous.Suppose either(a) F is continuous or(b) X has the sequential g-monotone property.

If there exist x, y ∈ X such that gx � F(x, y) and gy � F(y,x), then there exist x, y ∈ Xsuch that gx = F(x, y) and gy = F(y,x).

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3 Comments and suggestionsIn this section, we show that the conditions of the above Lemma . and Theorem . in[] are inadequate and, furthermore, the proof lines of Theorem . are not correct. Wealso would like to point out that the results in [] can be corrected under the appropriateconditions on the t-norm and the FNS.First of all, in the conditions of Lemma . and Theorem ., we have a t-conorm � such

that a � b ≤ ab for all a,b ∈ [, ]. If we take b = , then a = a � ≤ for all a ∈ [, ]. Itis obviously impossible. Moreover, it is well known and easy to see that if ∗ is a t-normand � is a t-conorm, then a ∗ b ≤ a � b for all a,b ∈ [, ]. In this sense, Lemma . andTheorem . have to be corrected.Secondly, from the property (IF), it follows that a sequence {xn} ⊂ X is a Cauchy se-

quence if, for any ε > and t > , there exists n ∈ N such that μ(xn – xm, t) > – ε for alln,m ≥ n. That is, the sequence {xn} ⊂ X is a Cauchy sequence on the IFNS (X,μ,ν,∗,�)if it also is on the FNS (X,μ,∗). A similar comment is valid for the convergence.Furthermore, the completeness of an IFNS is equivalent to the completeness of its sup-

port FNS and so we can deduce any fixed point theorem for IFNS (when the conditions onμ and ν are splitting) as an immediate consequence of its associated fixed point theoremfor FNS. In particular, it is sufficient to prove Theorem . just for FNS. Therefore, weonly develop Theorem . for FNS.Also, some proof lines of Lemma . are not correct (see p., lines -):

μ(xn – xm, t) ≥[μ

(x – x, ( – k) t

kn

)]m

≥[μ

(x – x, ( – k) t

kn

)]nq

→ ,

where q > such thatm < nq, and

ν(xn – xm, t) ≤[ν

(x – x, ( – k)

tkn

)]m

≤[ν

(x – x, ( – k) t

kn

)]np

→ ,

where p > such that m < np. Hence the sequence {xn} is a Cauchy sequence. This is notcorrect since the same q (or p) would not be valid for all positive integers m > n≥ n. Forinstance, let (X,‖ · ‖) be an ordinary normed space, define μ(x, t) = t

t+‖x‖ for any x ∈ X andt > and a ∗ b = ab for all a,b ∈ [, ]. Then (X,μ, – μ,∗,∗′) is an IFNS. If k = / andm = n, we have

[μ

(x – x, ( – k) t

kn

)]m=

[n–t

n–t + ‖x – x‖]n

→ e–‖x–x‖

t < .

Also, the proof lines of Theorem . that are not correct are the following ones (seep., lines -):

μ(gxn – gxm, t) ∗ μ(gxn – gxm, t)

≥ · · ·

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≥[μ

(gx – gx, ( – k) t

kn

)]m–n∗

[μ

(gy – gy, ( – k) t

kn

)]m–n

≥[μ

(gx – gx, ( – k)

tkn

)]m∗

[μ

(gy – gy, ( – k)

tkn

)]m

≥[μ

(gx – gx, ( – k) t

kn

)]np

∗[μ

(gy – gy, ( – k) t

kn

)]np

→ ,

where q > such thatm < nq. In general, we cannot obtain μ(xn – xm, t) → as n,m → ∞.It is not shown that {xn} is a Cauchy sequence. Moreover, a similar conclusion can beobtained for ν(xn – xm, t)→ . Thus, from the hypothesis of Theorem ., the conclusioncannot be guaranteed.

4 Themodification in FNSIn this section, by replacing the hypothesis that μ satisfies the n-property with the onethat the t-norm is of H-type, we state and prove a coupled fixed point theorem as a mod-ification.

Definition . [] For any a ∈ [, ], let the sequence {∗na}∞n= be defined by ∗a = aand ∗na = (∗n–a) ∗ a. Then a t-norm ∗ is said to be of H-type if the sequence {∗na}∞n= isequicontinuos at a = .

Theorem . Let (X,�) be a partially ordered set and (X,μ,∗) be a complete FNS suchthat ∗ is of H-type and a ∗ b ≥ ab for all a,b ∈ [, ]. Let k ∈ (, ) be a number and F :X ×X → X be a mapping such that F has the mixed monotone property and

μ(F(x, y) – F(u, v),kt

) ≥ [μ(x – u, t)

]/ ∗ [μ(y – v, t)

]/, (.)

for which x� u and y � v. Suppose that either(a) F is continuous or(b) X has the sequential monotone property.

If there exist x, y ∈ X such that x � F(x, y) and y � F(y,x), then F has a coupledfixed point. Furthermore, if x and y are comparable, then x = y, that is, x = F(x,x).

Proof Let x, y ∈ X be such that x � F(x, y) and y � F(y,x). Since F(X×X) ⊆ X, wecan choose x, y ∈ X such that x = F(x, y) and y = F(y,x). Again, from F(X ×X) ⊆ X,we can choose x, y ∈ X such that x = F(x, y) and y = F(y,x). Continuing this process,we can construct two sequences {xn} and {yn} in X such that, for each n ≥ ,

xn+ = F(xn, yn), yn+ = F(yn,xn). (.)

The proof is divided into two steps.Step . Prove that {xn} and {yn} are Cauchy sequences. Firstly, we show by induction that,

for each n≥ ,

xn � xn+, yn � yn+. (.)

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For n = , (.) holds trivially. Suppose that, for some fixed n ≥ , (.) holds. Sincexn � xn+ and xn � xn+ and F has the mixed monotone property, it follows from (.) that

xn+ = F(xn, yn) � F(xn+, yn), yn+ = F(yn,xn) � F(yn+,xn). (.)

Similarly, we have

xn+ = F(xn+, yn+) � F(xn+, yn), yn+ = F(yn+,xn+) � F(yn+,xn). (.)

Thus, combining (.) and (.), (.) holds.Let δn(t) = [μ(xn – xn+, t)]/ ∗ [μ(yn – yn+, t)]/ for all n≥ . Then it follows from (.),

(.) and (F) that

μ(xn – xn+,kt) = μ(F(xn–, yn–) – F(xn, yn),kt

)≥ [

μ(xn– – xn, t)]/ ∗ [

μ(yn– – yn, t)]/

= δn–(t) (.)

and

μ(yn – yn+,kt) = μ(F(yn,xn) – F(yn–,xn–),kt

)≥ [

μ(yn – yn–, t)]/ ∗ [

μ(xn – xn–, t)]/

= δn–(t). (.)

Then it follows from the t-norm and a ∗ b ≥ ab that δn(kt) ≥ δn–(t) for all n ≥ . Thisimplies that

≥ δn(t) ≥ δn–

(tk

)≥ δn–

(tk

)≥ · · · ≥ δ

(tkn

). (.)

Since limn→∞ δ( tkn ) = for all t > , we have limn→∞ δn(t) = for all t > .

Now, we claim that, for any p ≥ ,

μ(xn – xn+p, t)≥ ∗pδn–(t – kt), μ(yn – yn+p, t) ≥ ∗pδn–(t – kt), ∀n≥ . (.)

In fact, it is obvious for p = by (.), (.) and Lemma . since t/k ≥ t – kt and δn– isnon-decreasing. Assume that (.) holds for some p ≥ . By (.), we have

μ(xn – xn+, t) ≥ μ(xn – xn+,kt)≥ δn–(t)

and so

μ(xn – xn+, t – kt) ≥ δn–(t – kt).

Thus, from (.), (.) and a ∗ b ≥ ab, we have

μ(xn+ – xn+p+,kt)≥[μ(xn – xn+p, t)

]/ ∗ [μ(yn – yn+p, t)

]/ ≥ ∗pδn–(t – kt).

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Hence, by the monotonicity of the t-norm ∗, we have

μ(xn – xn+p+, t) = μ(xn – xn+p+, t – kt + kt)

≥ μ(xn – xn+, t – kt) ∗ μ(xn+ – xn+p+,kt)

≥ δn–(t – kt) ∗ (∗pδn–(t – kt))= ∗p+δn–(t – kt).

Similarly, we have

μ(yn+ – yn+p+,kt) ≥ ∗p+δn–(t – kt).

Therefore, by induction, (.) holds for all p≥ . Suppose that t > and ε ∈ (, ] are given.By hypothesis, since ∗ is a t-normofH-type, there exists < η < such that ∗p(a) > –ε forall a ∈ ( –η, ] and p ≥ . Since limn→∞ δn(t) = , there exists n such that δn(t – kt) > –η

for all n ≥ n. Hence, from (.), we get

μ(xn – xn+p, t) > – ε, μ(yn – yn+p, t) > – ε, ∀n≥ n.

Therefore, {xn} and {yn} are Cauchy sequences.Step .We prove that F has a coupled fixed point. SinceX is complete, there exist x, y ∈ X

such that limn→∞ xn = x and limn→∞ yn = y. Suppose that the assumption (a) holds. By thecontinuity of F , we get

x = limn→∞xn+ = lim

n→∞F(xn, yn) = F(

limn→∞xn, lim

n→∞ yn)= F(x, y).

Similarly, we can show that F(y,x) = y.Suppose now that (b) holds. Since {xn} is a non-decreasing sequence with xn → x and

{yn} is a non-increasing sequence with xy → y, from (B) and (B), we have xn � x andyn � y for all n≥ . Then, by (.), we obtain

μ(xn+ – F(x, y),kt

) ≥ μ(F(xn, yn) – F(x, y),kt

) ≥ [μ(xn – x, t)

]/ ∗ [μ(yn – y, t)

]/.Letting n → ∞, we have limn→∞ xn = F(x, y). Hence F(x, y) = x.Similarly, we can show that F(y,x) = y.Suppose that x � y. By induction and the mixed monotone property of F , it follows

that xn = F(xn–, yn–) � F(yn–,xn–) = yn. From (.), it follows that

μ(xn+ – yn+,kt) = μ(F(xn, yn) – F(yn,xn,kt)

≥ [μ(xn – yn, t)

]/ ∗ [μ(yn – xn, t)

]/= μ(xn – yn, t).

By the iterative procedure, we have

μ(xn – yn,kt)≥ μ(xn– – yn–, t)≥ μ

(xn– – yn–,

tk

)≥ · · · ≥ μ

(x – y,

tkn–

).

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Taking n→ ∞, since limn→∞ μ(x – y, tkn– ) = for all t > , we conclude that μ(x– y, t) ≥

for all t > , i.e., x = y. This completes the proof. �

Next, we prove the existence of a coupled coincidence point theorem, where we do notrequire that F and g are commuting.

Theorem . Let (X,�) be a partially ordered set and let (X,μ,∗) be a complete FNSsuch that ∗ is of H-type and a ∗ b ≥ ab for all a,b ∈ [, ]. Let k ∈ (, ) be a number andF : X × X → X and g : X → X be two mappings such that F has the mixed g-monotoneproperty and

μ(F(x, y) – F(u, v),kt

) ≥ [μ(gx – gu, t)

]/ ∗ [μ(gy – gv, t)

]/, (.)

for which gx� gu and gy � gv. Suppose that F(X ×X)⊆ g(X), g is continuous and either(a) F is continuous or(b) X has the sequential g-monotone property.

If there exist x, y ∈ X such that gx � F(x, y) and gy � F(y,x), then there exist x, y ∈ Xsuch that gx = F(x, y) and gy = F(y,x), that is, F and g have a coupled coincidence point.

Proof Using Lemma ., there exists E ⊂ X such that g(E) = g(X) and g : E → X is one-to-one.We define a mappingA : g(E)× g(E)→ X byA(gx, gy) = F(x, y). Since g is one-to-oneon g(E),A is well defined. Thus it follows from (.) that

μ(A(gx, gy) –A(gu, gv),kt

) ≥ [μ(gx – gu, t)

]/ ∗ [μ(gy – gv, t)

]/, (.)

for which gx� gu and gy� gv. Since F has the mixed g-monotone property, we have

gx, gx ∈ g(X), gx � gx

=⇒ A(gx, gy) = F(x, y) � F(x, y) =A(gx, gy), ∀gy ∈ g(X),

gy, gy ∈ g(X), gy � gy

=⇒ A(gx, gy) = F(x, y) � F(x, y) =A(gx, gy), ∀gx ∈ g(X),

which implies that A has the mixed monotone property.Suppose that the assumption (a) or (b) holds. Using Theorem . with themappingA, it

follows thatA has a coupled fixed point (u, v) ∈ g(X)×g(X), i.e., u =A(u, v) and v =A(v,u).Since (u, v) ∈ g(X) × g(X), there exists (u, v) ∈ X × X such that gu = u and gv = v. Thusgu = u =A(u, v) =A(gu, gv). Similarly, gv =A(gv, gu). This completes the proof. �

Now, we show the existence and uniqueness of coupled coincidence points. Note that if(S,�) is a partially ordered set, then we endow the product S×S with the following partialorder:

(x, y) � (u, v) ⇐⇒ x � u, y� v, ∀x, y,u, v ∈ S.

We say that (x, y) and (u, v) are comparable if (x, y) � (u, v) or (x, y)� (u, v).

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Theorem . In addition to the hypotheses of Theorem ., suppose that, for any pair ofcoupled coincidence points (x, y), (x∗, y∗) ∈ X × X, there exists a point (u, v) ∈ X × X suchthat (gu, gv) is comparable to (gx, gy) and (gx∗, gy∗). Then F and g have a unique coupledcoincidence point, that is, there exists a unique (x, y) ∈ X ×X such that x = gx = F(x, y) andy = gy = F(y,x).

Proof From Theorem ., the set of coupled coincidences is nonempty. Now, we showthat if (x, y) and (x*, y*) are coupled coincidence points, that is, gx = F(x, y), gy = F(y,x) andgx* = F(x*, y*), gy* = F(y*,x*), then

gx = gx*, gy = gy*. (.)

Put u = u and v = v and choose u, v ∈ X such that gu = F(u, v) and gv = F(v,u).Then, as in the proof of Theorem ., we can inductively define the sequences {un} and{vn} such that

gun+ = F(un, vn), gvn+ = F(vn,un). (.)

Since (F(x, y),F(y,x)) = (gx, gy) and (F(u, v),F(v,u)) = (gu, gv) are comparable, we can sup-pose that gx � gu and gy � gv. It is easy to show, by induction and g-monotonicity, thatgx� gun and gy� gvn for all n ≥ . From (.), we obtain

μ(gx – gun,kt) = μ(F(x, y) – F(un–, vn–),kt

)≥ [

μ(gx – gun–, t)]/ ∗ [

μ(gy – gvn–, t)]/ (.)

and

μ(gvn – gy,kt) = μ(F(vn–,un–) – F(y,x),kt

)≥ [

μ(gvn– – gy, t)]/ ∗ [

μ(gun– – gx, t)]/. (.)

Now, let βn(t) = [μ(gx – gun, t)]/ ∗ [μ(gy – gvn, t)]/. By (.) and (.), we have

βn(t)≥ βn–

(tk

)≥ · · · ≥ β

(tkn

)

and

μ(gx – gun,kt)≥ β

(tkn

), μ(gy – gvn,kt)≥ β

(tkn

).

Since limn→∞ β( tkn ) = , we conclude that limn→∞ gun = gx and limn→∞ gvn = gy.

Similarly, limn→∞ gun = gx* and limn→∞ gvn = gy*. Hence gx = gx* and gy = gy* and so(.) is proved.Since gx = F(x, y) and gy = F(y,x), if z = gx and w = gy, by the commutativity of F and g ,

we have

gz = ggx = gF(x, y) = F(gx, gy) = F(z,w)

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and

gw = ggy = gF(y,x) = F(gy, gx) = F(w, z),

i.e., (z,w) is a coupled coincidence point. In particular, from (.), we have z = gx = gz andw = gy = gw. Therefore, (z,w) is a coupled common fixed point of F and g .To prove the uniqueness of the coupled common fixed point of F and g , assume that

(p,q) is another coupled common fixed point. Then, by (.), we have p = gp = gz = z andq = gq = gw = w. This completes the proof. �

Finally, we present an intuitionistic version of Theorem . with the dual conditions ont-conorms. The proof is just reduced to apply Theorem . to the support FNS.

Corollary . Let (X,�) be a partially ordered set and suppose that a ∗ b ≥ ab and ( –a)� ( – b) ≤ –ab for all a,b ∈ [, ]. Let (X,μ,ν,∗,�) be a complete IFNS such that ∗ and� are of H-type. Let F : X × X → X and g : X → X be two mappings such that F has themixed g-monotone property,

μ(F(x, y) – F(u, v),kt

) ≥ [μ(gx – gu, t)

]/ ∗ [μ(gy – gv, t)

]/

and

ν(F(x, y) – F(u, v),kt

)≤ (

–[ – ν(gx – gu, t)

]/) � ( –

[ – ν(gy – gv, t)

]/), (.)

for which g(x) � g(u) and g(y) � g(v),where < k < , F(X×X) ⊆ g(X) and g is continuous.Suppose that either(a) F is continuous or(b) X has the sequential g-monotone property.

If there exist x, y ∈ X such that g(x) � F(x, y) and g(y) � F(y,x), then there existx, y ∈ X such that g(x) = F(x, y) and g(y) = F(y,x), that is, F and g have a coupled coincidencepoint.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors read and approved the final manuscript.

Author details1Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, 660-701, Korea. 2Departmentof Mathematics, University of Jaén, Jaén, Spain. 3Department of Statistics and Operations Research, University of Jaén,Jaén, Spain. 4Department of Statistics and Operations Research, University of Granada, Granada, Spain.

AcknowledgementsThis work was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170) and by the Junta deAndalucía and projects FQM-268, FQM-178 of the Andalusian CICYE, Spain.

Received: 20 September 2012 Accepted: 24 February 2013 Published: 18 March 2013

Cho et al. Journal of Inequalities and Applications 2013, 2013:104 Page 11 of 11http://www.journalofinequalitiesandapplications.com/content/2013/1/104

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doi:10.1186/1029-242X-2013-104Cite this article as: Cho et al.: Coupled coincidence point theorems in (intuitionistic) fuzzy normed spaces. Journal ofInequalities and Applications 2013 2013:104.