Berinde Borcut tripled fixed point theorem in partially ordered (intuitionistic) fuzzy normed spaces

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R E S E A R C H

Open Access

Berinde-Borcut tripled ﬁxed point theorem in

partially ordered (intuitionistic) fuzzy normed

spaces

Poom Kumam

1

_{, Juan Martínez-Moreno}

2*

_{, Antonio-Francisco Roldán-López-de-Hierro}

2

_and

Concepción Roldán-López-de-Hierro

3

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

University of Jaén, Campus Las Lagunillas, s/n, Jaén, 23071, Spain Full list of author information is available at the end of the article

Abstract

In this paper, we prove some tripled ﬁxed point theorems in fuzzy normed spaces. Our results improve and restate the proof lines of the main results given in the paper (Abbaset al.in Fixed Point Theory Appl. 2012:187, 2012).

Keywords: fuzzy metric space; tripled ﬁxed point; intuitionistic space

1 Introduction

Once the notion ofcoupled ﬁxed pointwas given by Gnana Bhaskar and Lakshmikantham in [], the theory of multidimensional ﬁxed points has attracted much attention (see, for instance, [–]), specially in the tripled case (see [–]).

Recently, many authors have shown the existence of tripled fixed points and common tripled fixed points for some contractions in cone metric spaces, partially ordered met-ric spaces, fuzzy metmet-ric spaces, fuzzy normed spaces, intuitionistic fuzzy normed spaces and others. Especially in [], Abbaset al.proved some tripled fixed point theorem for contractive mappings in partially complete intuitionistic fuzzy normed spaces. But the authors found some mistakes in the proof lines of their main result. In this paper we give a corrected version of the main theorem.

At-norm(resp., at-conorm) is a mapping∗: [, ]_→_{[, ] (resp.,}_{: [, ]}_→_{[, ])}

that is associative, commutative, and non-decreasing in both arguments and has  (resp., ) as identity.

Deﬁnition ([, ]) For anya∈[, ], let the sequence{∗na}∞_n_=be deﬁned by∗a=a and∗n_a_{= (}_∗n–_a)_∗_{a. Then a}_t-norm_∗_{is said to be}_{of H}_-type_{if the sequence}_{∗n_a_}∞

n=is

equicontinuous ata= .

Deﬁnition  Afuzzy normed space(brieﬂy, FNS) is a triple (X,μ,∗), whereXis a vector space,∗is a continuoust-norm andμ:X×(,∞)→[, ] is a fuzzy set such that, for all x,y∈Xandt,s> ,

(F) μ(x,t) > ;

(F) μ(x,t) = for allt> if and only ifx= ; (F) μ(αx,t) =μ(x,_|_αt_|)for allα= ;

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(F) μ(x,t)∗μ(y,s)≤μ(x+y,t+s); (F) μ(x,·) : (,∞)→[, ]is continuous; (F) lim_t_→∞μ(x,t) = andlim_t_→_μ(x,t) = .

Using the continuoust-norms andt-conorms, Saadati and Park [] introduced the con-cept of an intuitionistic fuzzy normed space.

Deﬁnition ([, ]) Anintuitionistic fuzzy normed space(brieﬂy, IFNS) is a -tuple (X,μ,ν,∗,) whereX is a vector space,∗is a continuoust-norm,is a continuous t-conorm andμ,ν:X×(,∞)→[, ] are fuzzy sets such that, for allx,y∈Xandt,s> ,

(IF) μ(x,t) +ν(x,t)≤; (IF) μ(x,t) > andν(x,t) < ;

(IF) μ(x,t) = for allt> if and only ifx= if and only ifν(x,t) = for allt> ; (IF) μ(αx,t) =μ(x,_|_αt_|)andν(αx,t) =ν(x,_|_αt_|)for allα= ;

(IF) μ(x,t)∗μ(y,s)≤μ(x+y,t+s)andν(x,t)ν(y,s)≥ν(x+y,t+s); (IF) μ(x,·),ν(x,·) : (,∞)→[, ]are continuous;

(IF) lim_t_→∞μ(x,t) =  =lim_t_→_ν(x,t)andlim_t_→_μ(x,t) =  =lim_t_→∞ν(x,t).

Obviously, if (X,μ,ν,∗,) is a IFNS, then (X,μ,∗) is a FNS. We refer to this space as its support.

Lemma  μ(x,·)is a non-decreasing function on(,∞)andν(x,·)is a non-increasing function on(,∞).

Some properties and examples of IFNS and the concepts ofconvergenceand aCauchy sequencein IFNS are given in [].

Deﬁnition  Let (X,μ,ν,∗,) be an IFNS.

() A sequence{xn} ⊂X is called aCauchy sequenceif, for any>  andt> , there

existsn∈Nsuch thatμ(xn–xm,t) >  –andν(xn–xm,t) <for alln,m≥n.

() A sequence{xn} ⊂Xis said to beconvergent toa pointx∈X, denoted byxn→xor

bylimn→∞xn=x, if, for any>  andt> , there existsn∈Nsuch thatμ(xn–x,t) >  –

andν(xn–x,t) <for alln≥n.

() An IFNS in which every Cauchy sequence is convergent is said to becomplete.

Deﬁnition ([]) LetF:X→Xandg:X→Xbe two mappings.

• We say thatFandgarecommutingifgF(x,y,z) =F(gx,gy,gz)for allx,y,z∈X. • A point(x,y,z)∈X_{is called a}_{tripled coincidence point of the mappings}_F_and_g_if

F(x,y,z) =gx,F(y,x,y) =gyandF(z,y,x) =gz. Ifgis the identity,(x,y,z)is called a tripled ﬁxed point ofF.

• If(X,)is a partially ordered set, thenFis said to have themixedg-monotone propertyif it veriﬁes the following properties:

x,x∈X, gxgx ⇒ F(x,y,z)F(x,y,z), ∀y∈X,

y,y∈X, gygy ⇒ F(x,y,z)F(x,y,z), ∀x∈X,

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Ifgis the identity mapping, thenFis said to have themixed monotone property. • If(X,)is a partially ordered set, thenXis said to have thesequentialg-monotone

propertyif it veriﬁes the following properties:

(B) If{xn}is a non-decreasing sequence andlimn→∞xn=x, thengxngxfor all

n∈N.

(B) If{xn}is a non-increasing sequence andlimn→∞yn=y, thengyngyfor all

n∈N.

Ifgis the identity mapping, thenXis said to have the sequentialmonotone property.

Deﬁnition  LetXandY be two IFNS. A functionf :X→Y is said to becontinuous at a point x∈Xif, for any sequence{xn}inXconverging tox, the sequence{f(xn)}inY

converges tof(x). Iff is continuous at eachx∈X, thenf is said to becontinuous on X.

The following lemma proved by Haghiet al.[] is useful for our main results:

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

Deﬁnition  Let (X,μ,ν,∗,) be an IFNS. The pair (μ,ν) is said to satisfy then-property onX×(,∞) iflim_n_→∞[μ(x,kn_t)]np_{=  and}_lim

n→∞[ν(x,knt)]n

p

=  wheneverx∈X,k>  andp> .

In order to state our results, we recall the main result given in [].

Theorem (Abbaset al., Theorem .) Let(X,)be a partially ordered set and suppose that aa=a,ab≤a∗b for all a,b∈[, ].Let(X,μ,ν,∗,)be a complete IFNS such that (μ,ν)has the n-property.Let F:X×X×X→X and g:X→X be two mappings such that F has the mixed g-monotone property and

μF(x,y,z) –F(u,v,w),kt≥μ(gx–gu,t)∗μ(gy–gv,t)∗μ(gz–gw,t),

νF(x,y,z) –F(u,v,w),kt≤ν(gx–gu,t)ν(gy–gv,t)ν(gz–gw,t),

for which gxgu and gygv and gzgw,where <k< .Suppose either (a) Fis continuous or

(b) Xhas the sequentialg-monotone property.

If there exist x,y,z ∈X such that gx F(x,y,z), gy F(y,x,y) and gz

F(z,y,x),then F and g have a tripled coincidence point.

2 Comments and revised tripled ﬁxed point theorem

Firstly, we show that the conditions of Theorem  are inadequate and, further, the proof lines of Theorem  are not correct. We also would like to point out that the results in [] can be corrected under the appropriate conditions on thet-norm and the FNS. The proof lines of Theorem  are not correct (see pp. and ):

μ(xn–xm,t)≥

μ

x–x, ( –k)

t kn

m

≥

μ

x–x, ( –k)

t kn

np

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where p>  such that m<np_{. Hence the sequence} _{_x

n} is a Cauchy sequence. This is

not correct since the samepwould not be valid for all positive integersm>n≥n. For

example, let (X, · ) be an ordinary normed space, deﬁneμ(x,t) = _t₊t_xfor anyx∈Xand t>  anda∗b=abfor alla,b∈[, ]. Then (X,μ,  –μ,∗,∗) is an IFNS. Ifk= / and m= n_{, we have}

μ

x–x, ( –k)

t kn

m

=

n–_t

n–_t₊_x –x

n

→e–x–tx _{< .}

Now, by replacing in Theorem  the hypothesis thatμsatisfies then-property with the one that thet-norm is ofH-type, we state and prove a tripled fixed point theorem as a modification.

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

μF(x,y,z) –F(u,v,w),kt≥μ(x–u,t)∗μ(y–v,t)∗μ(z–w,t), ()

for which xu,yv and zw.Suppose that either: (a) Fis continuous or

(b) Xhas the sequential monotone property.

If there exist x,y,z∈X such that xF(x,y,z),yF(y,x,y)and zF(z,y,

x),then F has a tripled ﬁxed point.Furthermore,if xand yare comparable,then x=y,

that is,x=F(x,x).

Proof As in [] starting withx,y,z∈Xsuch thatxF(x,y,z),yF(y,x,y)

andzF(z,y,x), one can deﬁne inductively three sequences{xn},{yn},{zn} ⊂Xsuch

thatxn+=F(xn,yn,zn),yn+=F(yn,xn,yn) andzn+=F(zn,yn,xn).

Deﬁne

δn(t) =μ(xn–xn+,t)∗μ(yn–yn+,t)∗μ(zn–zn+,t).

Continuing as in [], we have

μ(xn–xn+,kt)≥δn–(t), μ(yn–yn+,kt)≥δn–(t) and

μ(zn–zn+,kt)≥δn–(t)∗δn–(t).

()

Sincea∗a≥afor alla∈[, ], it follows that

δn(kt) =μ(xn–xn+,kt)∗μ(yn–yn+,kt)∗μ(zn–zn+,kt)≥δn–(t).

This implies that

≥δn(t)≥δn–

t k

≥δn–

t k

≥ · · · ≥δ

t kn

.

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Now, we claim that, for anyp≥ andn≥,

μ(xn–xn+p,t)≥ ∗pδn–(t–kt),

μ(yn–yn+p,t)≥ ∗pδn–(t–kt), and

μ(zn–zn+p,t)≥ ∗pδn–(t–kt).

()

In fact, it is obvious forp=  by (),a∗a≥aand Lemma  sincet/k≥t–ktandδn–is

non-decreasing. Assume that () holds for somep≥. 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 (), () anda∗a≥a, we have

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

≥ ∗p_δ

n–(t–kt). ()

Hence, by the monotonicity of thet-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)∗

∗p_δ

n–(t–kt)

=∗p+_δ

n–(t–kt).

Similarly, we have

μ(yn+–yn+p+,kt)≥ ∗p+δn–(t–kt) and

μ(zn+–zn+p+,kt)≥ ∗p+δn–(t–kt).

()

Therefore, by induction, () holds for allp≥. Suppose thatt>  and∈(, ] are given. By hypothesis, since∗is at-norm ofH-type, there exists  <η<  such that∗p_{(a) >  –}_for

alla∈( –η, ] andp≥. Sincelim_n_→∞δn(t) = , there existsnsuch thatδn(t–kt) >  –η

for alln≥n. Hence, from (), we get

μ(xn–xn+p,t) >  –, μ(yn–yn+p,t) >  –, μ(zn–zn+p,t) >  –, ∀n≥n.

Therefore,{xn},{yn}and{zn}are Cauchy sequences. We can continue as in [] to complete

the proof.

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X and g:X→X be two mappings such that F has the mixed g-monotone property and

μF(x,y,z) –F(u,v,w),kt≥μ(gx–gu,t)∗μ(gy–gv,t)∗μ(gz–gw,t),

νF(x,y,z) –F(u,v,w),kt≤ν(gx–gu,t)ν(gy–gv,t)ν(gz–gw,t),

for which gxgu and gygv and gzgw,where <k< .Suppose either (a) Fis continuous or

(b) Xhas the sequentialg-monotone property.

If there exist x,y,z ∈X such that gx F(x,y,z), gy F(y,x,y) and gz

F(z,y,x),then F and g have a tripled coincidence point.

Proof As in Theorem . in [].

Of course, all the results are valid ifXis intuitionistic.

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.

Author details

1_{Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod,}

Thrung Khru, Bangkok, Thailand. 2_{Department of Mathematics, University of Jaén, Campus Las Lagunillas, s/n, Jaén,}

23071, Spain.3_{Department of Statistics and Operations Research, University of Granada, Granada, Spain.}

Acknowledgements

The ﬁrst author was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Oﬃce of the Higher Education Commission (Under NRU-CSEC Project No. NRU56000508). The last three authors have been partially supported by Junta de Andalucía, by projects FQM-268, FQM-178 and FQM-235 of the Andalusian CICYE.

Received: 1 October 2013 Accepted: 8 January 2014 Published:30 Jan 2014

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