On -contractions in partially ordered fuzzy metric spaces

Abstract

Under some weaker conditions, some coincidence point and common fixed point theorems are established in partially ordered fuzzy metric spaces using weakly compatible mappings. By using the theorems, we obtain some coupled and multidimensional fixed point results, which are generalization and improvement of very recent theorems in the corresponding literature. In order to illustrate our main results, we give three examples.

Wang Fixed Point Theory and Applications (2015) 2015:233 DOI 10.1186/s13663-015-0485-0

*Correspondence: mailto:[email protected]

Web End [email protected] School of Mathematical Sciences, Yancheng Teachers University, Yancheng, Jiangsu 224051,P.R. China

On -contractions in partially ordered fuzzy metric spaces

http://crossmark.crossref.org/dialog/?doi=10.1186/s13663-015-0485-0&domain=pdf

Web End = Shuang Wang*

1 Introduction

In , the notion of coupled xed point was rst introduced by Guo and Lakshmikantham []. Recently, Gnana-Bhaskar and Lakshmikantham [] established some coupled xed point theorems in partially ordered metric space. The fuzzy version of the results of Gnana-Bhaskar and Lakshmikantham [] was studied by Sedghi et al. []. After that, common coupled xed point results in fuzzy metric spaces were established by Hu [] and Hu et al. []. Very recently, Choudhury et al. [] established coupled coincidence point and xed point results for compatible mappings in partially ordered fuzzy metric spaces. Later, Roldn et al. [] obtained multidimensional coincidence point theorems for nonlinear mappings in any number of variables in partially ordered fuzzy metric spaces. Their results generalize, clarify and unify several classical and very recent related results in the literature in the setting of metric spaces.

But many results (see, e.g., []) are obtained under the assumptions: (a) (t) = kt for all t > , where k (, ); or (b) n= n(t) < for all t > . It is obvious that the condition(a) is special. In [], iri [] has pointed out, the condition (b) is very strong and dicult for testing in practice. Then iri introduced the condition (CBW): () = , (t) < t and lim infrt

+ (t) < t for all t > . Later, Jachymski [] presented the condition (c): < (t) < t and limn n(t) = for all t > . In order to weaken the condition (c) further, Fang []

introduced the condition (d): for each t > there exists r t such that limn n(r) = in the context of Menger probabilistic metric spaces and fuzzy metric spaces. In this paper, under the condition (d), we present some coincidence point and common xed

2015 Wang. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 2 of 16

point results for weakly compatible mappings in partially ordered fuzzy metric spaces. By using the theorems, we obtain some coupled and multidimensional xed point results, which are generalization and improvement of very recent theorems in the corresponding literature. In addition, we illustrate our main results with three examples.

2 Preliminaries

In order to x the framework needed to state our main results, we recall the following notions. Let n

N, X be a non-empty set and Xn be the Cartesian product of n copies of X. For brevity, g(x), (y, y, . . . , yn), (ym, ym, . . . , ynm), (zm, zm, . . . , znm), (z, z, . . . , zn), (v, v, . . . , vn)

and (x, x, . . . , xn) will be denoted by gx, Y , Ym, Zm, Z, V , and X, respectively.Throughout this paper, let {A, B} be a partition of the set n = {, , . . . , n}, i.e., A B =

n and A B = . Let , , . . . , n : n n be n mappings from n into itself. We denote A,B = { : n n : (A) A and (B) B}, A,B = { : n n : (A)

B and (B) A},

N = {, , . . . , n, . . .},

I = [, ]. If (X, )

is a partially ordered space, we use the following notation from [], for y, v X and i n

y i v

N = {, . . . , n, . . .},

R+ = [, ), and

y v, if i A, y v, if i B.

Consider on Xn the next natural partial order: for Y, V Xn

Y n V yi i vi ()

for all i. If Y n V or Y n V , then two points Y and V are comparable (denoted by Y V ).

Proposition . ([]) If Y n V , it follows that (y(), y(), . . . , y(n)) n (v(), v(), . . . , v(n)) if AB, (y(), y(), . . . , y(n)) n (v(), v(), . . . , v(n)) if AB.

Denition . ([]) Let (Xn, n) be a partially ordered set, and T and G be self-mappings of Xn. It is said that T is a G-isotone mapping if, for any Y, Y Xn

G(Y) n G(Y) T(Y) n T(Y).

Denition . ([]) Let (X, ) be a partially ordered set and F : X X. We say F has the mixed g-monotone property if F is monotone non-decreasing in its rst argument and is monotone non-increasing in its second argument, that is, for any x, y X,

x, x X, g(x) g(x) implies qF(x, y) F(x, y),

and

y, y X, g(y) g(y) implies F(x, y) F(x, y).

Denition . ([]) Let (X, ) be a partially ordered space. We say that F has the mixed g-monotone property if F is g-monotone non-decreasing in arguments of A and g-monotone non-increasing in arguments of B, i.e., for all x, x, . . . , xn, y, z X and all i

gy gz F(x, . . . , xi, y, xi+, . . . , xn) i F(x, . . . , xi, z, xi+, . . . , xn).

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 3 of 16

Denition . ([]) An element Y Xn is called a coincidence point of the mappings T : Xn Xn and G : Xn Xn if T(Y) = G(Y). Furthermore, if T(Y) = G(Y) = Y , then we say that Y is a common xed point of T and G.

Denition . Let F : Xn X and g : X X be two mappings. A point (x, x, . . . , xn) Xn is:(i) A coupled coincidence point ([]) if n = , F(x, x) = g(x), and F(x, x) = g(x). If g is the identity mapping on X, then (x, x) X is called a coupled xed point of

the mapping F ([]). A coupled common xed point of F and g ([]) if n = , F(x, x) = g(x) = x, and F(x, x) = g(x) = x.(ii) A -coincidence point ([]) of F and g if

F(xi(), xi(), . . . , xi(n)) = gxi

for i n. If g is the identity mapping on X, then (x, x, . . . , xn) Xn is called a

-xed point of the mapping F.

Denition . ([]) A triple (X, , ) is called a partially ordered topological space if is a Hausdor topology on X and is a partial order on X. A partially ordered topological space (X, , ) is said to have the sequential g-monotone property if it veries:

(i) If {xm} is a non-decreasing sequence and {xm} x, then gxm gx for all m.

(ii) If {ym} is a non-increasing sequence and {ym} y, then gym gy for all m.

If g is the identity mapping, then X is said to have the sequential monotone property.

Denition . ([]) A triangular norm (also called a t-norm) is a map :

I that

is associative, commutative, non-decreasing in both arguments and has as identity. A t-norm is continuous if it is continuous in I as mapping. If a, a, . . . , am

I, then

mi=ai = a a am.

For each a [, ], the sequence {ma}m= is dened inductively by a = a and m+a = (ma) a for all m .

Denition . ([]) A t-norm is said to be of H-type if the sequence {ma}m= is equicontinuous at a = , i.e., for all (, ), there exists (, ) such that if a ( , ], then ma > for all m

N.

Denition . ([]) A fuzzy metric space in the sense of Kramosil and Michlek (briey, a FMS) is a triple (X, M, ) where X is a non-empty set, is a t-norm and

M : X X

I is a fuzzy set satisfying the following conditions for all x, y, z X and t, s :

(FM-) M(x, y, ) = ;(FM-) M(x, y, t) = , for all t > if and only if x = y; (FM-) M(x, y, t) = M(y, x, t);(FM-) M(x, y, t) M(y, z, s) M(x, z, t + s);

(FM-) M(x, y, ) :

I is left continuous.

Remark . Note that is continuous in the original denition in [].

I

I

R+

R+

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 4 of 16

Denition . ([]) A triple (X, M, ) is called a fuzzy metric space (in the sense of George and Veeramani) if X is an arbitrary non-empty set, is a continuous t-norm and

M : X X

I is a fuzzy set satisfying, for each x, y, z X and t, s > , conditions (FM-), (FM-), (FM-), (GV-): M(x, y, ) : (, )

I is continuous, and (GV-):

Denition . ([]) Let (X, M, ) be a FMS. A sequence {xn} in X is said to be convergent to x X if limn M(xn, x, t) = for all t > . A sequence {xn} in X is said to an M-Cauchy sequence, if for each (, ) and t > there exists n

N such that M(xn, xm, t) > for all m, n n. A fuzzy metric space is called complete if every M-Cauchy sequence is convergent in X.

Lemma . ([]) If (X, M) is a FMS under some t-norm and x, y X, then M(x, y, ) is a non-decreasing function on (, ).

Denition . ([]) A partially ordered fuzzy metric space (for short, poFMS) is a quadruple (X, M, , ) such that (X, M, ) is a FMS and is a partial order on X.

Denition . ([]) Let p

N and let (X, M, ) be a FMS. A mapping G : Xp X is said to be continuous at a point Y Xp if, for any sequence {Ym}m in Xp converging to Y, the sequence {G(Ym)}m converges to G(Y). If G is continuous at each Y Xp, then G is said continuous on Xp.

Denition . ([]) Let (X, M, ) be a FMS. The mappings F and g where F : X X and g : X X, are said to be compatible if for all t >

lim

n M

and

g F(yn, xn) , F g(yn), g(xn) , t = ,

whenever {xn} and {yn} are sequences in X such that limn F(xn, yn) = limn g(xn) = x and limn F(yn, xn) = limn g(yn) = y for some x, y X.

Denition . ([]) Let (X, M, , ) be a poFMS and let = (, , . . . , n) be an n-tuple of mappings from n into itself. Two mappings F : Xn X and g : X X are said to be -compatible if, for all sequences {xm}m, {xm}m, . . . , {xnm}m X such that {gxm}m, {gxm}m, . . . , {gxnm}m are monotone and

lim

m F

we have

m , xi()m, . . . , xi(n)m , F gxi()m, gxi()m, . . . , gxi(n)m , t =

for all t > and all i.

R+

M(x, y, t) > .

g F(xn, yn) , F g(xn), g(yn) , t =

lim

n M

xi()

m , xi()m, . . . , xi(n)m = lim

m gxim X for all i,

lim

m M

gF xi()

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 5 of 16

Remark . If n = in Denition ., then F, g : X X are compatible w.r.t. (X, M, , ).

Denition . ([]) We will say that the maps f , g : X X are weakly compatible (or the pair (f , g) is w-compatible) if fgx = gfx for all x X such that fx = gx.

Let denote the family of all functions : R+

R+ such that limn n(t) = for all t > , and let w denote the family of all functions : R+

R+ verifying the condition(d), that is, for each t > there exists r t such that limn n(r) = .

It is evident that the condition limn n(t) = for all t > implies the condition

(d). However, the following example shows that the reverse is not true in general. Hence w.

Example . ([]) Let the function : R+

R+ be dened by

(t) =

t+t , if t < , t + , if t , t , if < t < .

()

Notice that w but /

.

Lemma . ([]) Let w, then for each t > there exists r t such that (r) < t.

3 Main results

In this section we establish our main results and use them to obtain some coupled and multidimensional xed point theorems.

Lemma . If (X, M, ) is a FMS with M(x, y, ) :

R+

I is continuous, then M is a contin-

uous mapping on X (, ).

Proof The proof is the same as that for a fuzzy metric space in the sense of George and Veeramani (see Rodrguez-Lpez and Romaguera [], Proposition ).

Lemma . Let (X, M, ) be a FMS such that is a t-norm of H-type. Let {xn} be a sequence in (X, M, ). If there exists a function w satisfying(i) (t) > for all t > ;(ii) M(xn, xm, (t)) M(xn, xm, t) for all n, m

N and t > ;

(iii) limt M(x, x, t) = , then {xm} is a Cauchy sequence.

Proof We proceed with the following steps:

Step . We claim that for any t > ,

M(xn, xn+, t) as n . ()

By (iii), for any (, ), there exists t > such that M(x, x, t) > . Since w, there exists t t such that limn n(t) = . Thus, for each t > , there exists n

N

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 6 of 16

such that n(t) < t for all n n. It is evident that (ii) implies that

M xn, xn+, (t) M(xn, xn, t) for all n

It follows from (i) that n(t) > for all n

that

N and t > . ()

So, by () and the monotonicity of M(x, y, ), we have

M(xn, xn+, t) M xn, xn+, n(t) M(x, x, t) M(x, x, t) >

for all n n. Taking into account that , t > are arbitrary, we conclude that () holds.

Step . We claim that for any t > ,

M(xn, xm, t) mnM xn, xn+, t (r) for all m n + , ()

where r t. Since w, for any t > , there exists r t such that (r) < t by Lemma .. Since M(xn, xn+, t) M(xn, xn+, t (r)) = M(xn, xn+, t (r)), then () holds for m =

n + . Suppose now that M(xn, xm, t) mnM(xn, xn+, t (r)) holds for some xed m n + . By (FM-), (ii) and the monotonicity of , we get

M(xn, xm+, t) = M xn, xm+, t (r) + (r)

M xn, xn+, t (r) M xn+, xm+, (r)

M xn, xn+, t (r) M(xn, xm, r)

M xn, xn+, t (r) M(xn, xm, t)

M xn, xn+, t (r) mnM xn, xn+, t (r)

= m+nM xn, xn+, t (r) .

Thus, we prove that if () holds for some m n + , then it also holds for m + . By induction, we conclude that () holds for all m n + .

Step . We claim that {xn} is a Cauchy sequence. As is a t-norm of H-type, for any

(, ) there exists (, ) such that

if a ( , ], then l a > for all l

N such that M(xn, xn+, t (r)) > for all

N and t > . ()

N and t > . By induction, it follows from ()

M xn, xn+, n(t) M(x, x, t) for all n

N. ()

It follows from () that there exists n

n n. So, by (), we have

mnM xn, xn+, t (r) > ()

for all m > n n. By () and (), we get for each t > and (, ), M(xn, xm, t) > for all m > n n, which implies that {xn} is a Cauchy sequence.

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 7 of 16

Now, we state and prove some xed point results for weakly compatible mappings in partially ordered fuzzy metric spaces.

Theorem . Let (X, M, , ) be a complete poFMS such that is a t-norm of H-type. Let T : X X and G : X X be two mappings such that T is a G-isotone mapping and

T(X) G(X). Assume that there exists w such that, for all t > and y, v X with G(y) G(v),

M T(y), T(v), (t) M G(y), G(v), t . ()

Also suppose that either(C) T and G are continuous and compatible and M(x, y, ) :

I is continuous or (C) (X, M, ) has the sequential monotone property and G(X) is closed.

If there exists y X such that G(y) T(y) and limt M(G(y), T(y), t) = . Then T and G have a coincidence point.

Proof Let y X such that G(y) T(y) and limt M(G(y), T(y), t) = . Since T(X) G(X), there exists y X such that G(y) = T(y). Recursively, we see that, for every m

N,

there exists ym+ X such that G(ym+) = T(ym). Set z = G(y) and zm+ = G(ym+) = T(ym) for every m

N.

Since G(y) T(y), we suppose that G(y) T(y), i.e., z z (the case G(y) T(y) is treated similarly). Assume that zm zm for some m

N, that is, G(ym) G(ym).

Since T is a G-isotone mapping, we get zm = T(ym) T(ym) = zm+. This actually means that the sequence {zm} is non-decreasing. Using () and monotonicity of {zm}, we get

M zn, zm, (t) = M T(yn), T(ym), (t)

M G(yn), G(ym), t = M(zn, zm, t)

for all m, n

N and t > . Obviously, the inequality () implies that (t) > for all t > . Indeed, if there exists some t > such that (t) = . It follows from () that

= M T(y), T(y), (t) M G(y), G(y), t = ,

which is a contradiction. Since limt M(G(y), T(y), t) = , we have limt M(z, z, t) = . So, by Lemma ., {zm} is a Cauchy sequence.

Now suppose that the condition (C) holds. Since (X, M, , ) is complete, there exists z X such that limm zm = z, that is,

lim

m T(ym) =

G G(ym+) , T G(ym) , t = lim

m M

R+

m G(ym) = z. ()

Since T and G are compatible, we have

lim

m M

lim

G T(ym) , T G(ym) , t = ()

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 8 of 16

for all t > . As G is continuous, we have

lim

m G

G(ym) = G(z). ()

Using Lemma ., we nd that M is a continuous mapping on X (, ). By the continuity of M and ()-(), we have = limm M(G(G(ym+)), T(G(ym)), t) = M(G(z), T(z), t)

for all t > , which implies that G(z) = T(z) and z is a coincidence point of T and G.

Now suppose that the condition (C) holds. Since (X, M, , ) is complete and G(X) is closed, there exists z X such that limm T(ym) = limm G(ym) = G(z). Since (X, M, )

has the sequential monotone property, we have G(ym) G(z) for all m

N. Since

w, for each t > there exists r t such that (r) < t by Lemma .. So, by () and the monotonicity of M(x, y, ), we have

M T(ym), T(z), t M T(ym), T(z), (r) M G(ym), G(z), r M G(ym), G(z), t

for all t > and m

N. Letting m in the above inequality, we get T(ym) T(z) as m . By the uniqueness of the limit, we conclude that G(z) = T(z) and z is a coincidence point of T and G.

Theorem . In addition to the hypotheses of Theorem ., let G be weakly compatible with T if assumption (C) holds. Suppose that for all coincidence points y, v X of mappings

T and G, there exists u X such that

(C) G(u) is comparable to G(y) and G(v);(C) limt M(G(u), G(y), t) = limt M(G(u), G(v), t) = .

Then T and G have a unique common xed point.

Proof Put u = u and dene a sequence {G(um)} by G(um+) = T(um) for m

N. We may as

sume that G(y) G(u) (the case G(y) G(u) is treated similarly). Since T is a G-isotone mapping, we have G(y) = T(y) T(u) = G(u). By induction we obtain G(y) G(um) for all m

N. Owing to limt M(G(u), G(y), t) = , for any (, ), there exists t > such that M(G(u), G(y), t) > . Since w, there exists t t such that m(t) as m . Thus, for each t > , there exists m

N such that m(t) < t for all m m. So, by () and the monotonicity of M(x, y, ), we get for all m m and t > ,

M G(um), G(y), t M G(um), G(y), m(t)

= M T(um), T(y), m(t)

M G(um), G(y), m(t)

M G(u), G(y), t

M G(u), G(y), t > .

Since , t > are arbitrary, we deduce that M(G(um), G(y), t) as m . This shows that limm G(um) = G(y). Similarly, we nd that limm G(um) = G(v). The uniqueness of the limit proves that G(y) = G(v).

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 9 of 16

Denote w = T(y) = G(y). Since T and G are weakly compatible mappings, we have T(w) = TG(y) = GT(y) = G(w). So, w is also a coincidence point of T and G. Therefore, G(w) = G(y) = w and w is a common xed point of T and G. In order to prove the uniqueness, assume that w is another common xed point of T and G. Then we have w = G(w) = G(w) = w. This completes the proof.

Example . Let (X, ) be the partially ordered set with X = [, ] and the natural ordering of the real numbers as the partial ordering . Dene M : X X

R+

I by

M(x, y, t) =

, if t = ,

e

|xy|

t , if t > .

Then M(x, y, ) :

I is continuous. Let x y = min{x, y} for all x, y X. Then (X, M, ) is a complete FMS with M(x, y, t) as t , for all x, y X. Consider T, G : X X dened by T(x) = x + and G(x) = x.

It is easy to verify the following statements.(i) T(X) G(X) and T is a G-isotone mapping.

(ii) The condition (C) holds.(iii) There exists y = such that G(y) = = T(y).

Let y, v X such that G(y) G(v), that is, y v. Next, we show that the inequality () is satised with (t) = t, for all t > . If () does not hold, then there exists t > such that

M

R+

T(y), T(v),

t

< M G(y), G(v), t ,

that is,

e|y/v/|/(t/) < e|yv|/t,

that is,

y v > |y v|.

Since y, v [, ],

|y v| <

y v =

|y v|(y + v)

|y v|,

which is impossible. Hence () holds.

By Theorems . and ., T and G have a unique common xed point, which is z = . In this example, computing according to z = G(y) and zm+ = G(ym+) = T(ym) for every m

N, we obtain {z = , z = , z = , z = , . . .}. Thus the sequence {zn} is a nontrivial sequence.

Example . Let X = {, ., ., ., ., } and dene M : X X

R+

I as follows:

M(x, y, t) =

, if |x y| < t,

t

|xy|+t , if |x y| t.

()

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 10 of 16

As Gregori et al. have pointed out in [], any FMS(X, M) is equivalent to Menger space in the sense that M(x, y, t) = Fx,y(t) for all x, y X and t . Thus, (X, M) is a FMS under = min (see Example in []).

Let M be the discrete topology on X. Note that the metric space (X, d) is complete, where d is dened by d(x, y) = |x y|. Next, we claim that if {xn} is a Cauchy sequence in (X, M, ), then {xn} is also a Cauchy sequence in (X, d). Indeed, if {xn} is not a Cauchy sequence in (X, d), then there exist > and two sequences {ni} and {mi} such that mi >

ni i and d(xmi, xni) for all i

N. Taking (, ], it follows from () that

M(xmi, xni, ) =

which contradicts that {xn} is a Cauchy sequence in (X, M, ). So, {xn} is also a Cauchy sequence in (X, d). Since (X, d) is complete, there exists x X such that xn x as n , and so for any t > there exists n

N such that |xn x| < t for all n n. Thus, for any t > and (, ), by (), we have M(xn, x, t) = > for all n n, which implies that (X, M, ) is complete.

Endow X with the following partial order:

x, y X, x y x = y or (x, y) (, .), (, .) .

Let : R+

R+ be dened by (). Consider T, G : X X dened by

T(x) =

It is not dicult to prove the following statements.(i) T(X) G(X).(ii) The condition (C) holds (since M is the discrete topology on X).(iii) There exists y = such that G(y) T(y) and limt M(G(y), T(y), t) = .(iv) All conditions of Theorem . hold. In fact, y = and v = . are all coincidence points of T and G. Since TG() = GT() and TG(.) = GT(.), by Denition ., G is weakly compatible with T. In addition, there exists u = . such that G(y) G(u) and G(v) G(u). It follows from () that (C) holds.

(v) T is a G-isotone mapping. Indeed, let y, v X such that G(y) G(v).(a) If G(y) = G(v) then y = v or y, v {, .} or y, v {., ., }. Thus, T(y) = T(v).

(b) If (G(y), G(v)) = (, .), then y {, .} and v {., ., }. Thus, T(y) = T(v).

(c) If (G(y), G(v)) = (, .), then y {, .} and v = .. Thus, (T(y), T(v)) = (, .).

Therefore, T(y) T(v).

Next, we shall prove that () holds. Let y, v X such that G(y) G(v). It follows from (a) and (b) that = |T(y) T(v)| < (t) for all t > . Thus, M(T(y), T(v), (t)) =

M(G(y), G(v), t) for all t > , i.e., () holds. By (c), we have |T(y) T(v)| = . and

+ |xmi xni|

for all i

N,

, if x {, ., ., ., }, ., if x = .;

G(x) =

, if x {, .}, ., if x = .,

., if x {., ., }.

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 11 of 16

|G(y) G(v)| = .. If (t) > ., by (), it is evident that () holds. Suppose that (t) .. From (), it is easy to see that (t)

t+t for all t . So, we have t . Therefore, |G(y) G(v)| > t. By (), we have

M T(y), T(v), (t) =

, by (), we have (t) = < = |T(y) T(v)|. By (), we have

M T(y), T(v), (t) =

So, () does not hold.

If G is the identity mapping on X in Theorems . and ., then the following corollary is obtained immediately.

Corollary . Let (X, M, , ) be a complete poFMS such that is a t-norm of H-type. Let T : X X be a non-decreasing mapping. Assume that there exists w such that, for all t > and y, v X with y v,

M T(y), T(v), (t) M(y, v, t).

Also suppose that either T and M(x, y, ) :

I are continuous or (X, M, ) has the sequential monotone property. If there exists y X, such that y T(y) and

limt M(y, T(y), t) = , then T has a xed point.

Furthermore, suppose that for all xed points y, v X of T, there exists u X such that u is comparable to y and v and limt M(u, y, t) = limt M(u, v, t) = . Then T has a unique xed point.

Remark . Corollary . can be considered as a partially ordered version of Theorem . in [].

Example . Let (X, M, , ), T, be the same as in Example .. Using a similar argument to Example ., we deduce that the conditions of Corollary . are satised. So, T has a unique xed point, which is z = . However, Theorem . in [] cannot be applied to this example because the condition M(T(y), T(v), (t)) M(y, v, t) for all y, v X and t > does not hold. In fact, if (y, v) = (., .), then |T(y) T(v)| = . Taking t = , by (), we have (t) = . Thus, by (), we have

M T(y), T(v), (t) =

(t)

|T(y) T(v)| + (t)

= M G(y), G(v), t ,

i.e., () holds.

By Theorems . and ., T and G have a unique common xed point, which is z = .

However, the totally ordered version of Theorems . and . cannot be applied to this example (since () does not hold). In fact, if (G(y), G(v)) = (., .), then y = . and v

{., ., }. Taking t =

(t)

+ (t)

t +t +

t +t

= t

t + +t

tt +

(t)

|T(y) T(v)| + (t)

=

< = M

G(y), G(v), t .

R+

= M(y, v, t).

Next, we give some basic concepts and results that we will need to obtain some coupled and multidimensional xed point results.

=

<

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 12 of 16

Denition . Let F : Xn X and g : X X be two mappings. A point (x, x, . . . , xn) Xn is a common xed point of F and g if F(xi(), xi(), . . . , xi(n)) = gxi = xi for i n.

Denition . Given n , the mappings F : Xn X and g : X X are weakly compatible (or the pair (F, g) is w-compatible) if

F(xi(), xi(), . . . , xi(n)) = gxi, i n

gF(xi(), xi(), . . . , xi(n)) = F(gxi(), gxi(), . . . , gxi(n)), i n.

Lemma . Let (X, M, ) be a FMS such that is a continuous t-norm. Let Mn : Xn Xn

I be given by

Mn(A, B, t) = ni=M(ai, bi, t) ()

for all A = (a, a, . . . , an), B = (b, b, . . . , bn) Xn, and all t . Then the following properties hold:(i) (Xn, Mn, ) is also a FMS.

(ii) Let {Am = (am, am, . . . , anm)} be a sequence on Xn and let A = (a, a, . . . , an) Xn.

Then {Am} A if, and only if, {aim} ai for all i {, , . . . , n}.

(iii) If (X, M, ) is complete, then (Xn, Mn, ) is complete.

Proof The proofs of (i) and (ii) in Lemma . are the same as Lemma in []. Next, we shall prove that (iii) holds. Suppose that {Ym} is a Cauchy sequence in (Xn, Mn, ). Thus, for any (, ) and t > , there exists n

N such that Mn(Yn, Ym, t) > for all n, m n.

yin, yim, t ni=M yin, yim, t > for all n, m n.

Thus, for n, m n, we have M(yin, yim, t) > for each i n. Therefore, {yim} is a Cauchy sequence in (X, M, ) for each i n. Since (X, M, ) is complete, then {yim} converges to a point yi of X for each i n. Thus, {Ym} converges to a point Y of Xn. That is, (Xn, Mn, )

is complete. This completes the proof.

The following multidimensional xed point theorem is an immediate consequence of Theorems . and ..

Theorem . Let (X, M, , ) be a complete poFMS with a continuous t-norm of H-type. Let = (, , . . . , n) be an n-tuple of mappings from n into itself verifying i A,B if i A and i A,B if i B. Suppose that F : Xn X and g : X X are two mappings such that F has the mixed g-monotone property on X and F(Xn) g(X). Assume that there exists w such that

M F(x, x, . . . , xn), F(y, y, . . . , yn), (t) ni=M(gxi, gyi, t) ()

for all t > and all x, x, . . . , xn, y, y, . . . , yn X with gxi i gyi for i n, where : [, ] [, ] is a mapping such that n (a) a for each a [, ]. Suppose that

ni=M(gxj(i), gyj(i), t) ni=M(gxi, gyi, t) for j n, ()

R+

By (), we have

min

in M

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 13 of 16

for all x, x, . . . , xn, y, y, . . . , yn X with gxi i gyi for i n. Suppose that either

(C) F and g are continuous and -compatible and M(x, y, ) :

I is continuous, or (C) (X, M, ) has the sequential monotone property and g(X) is closed.

If there exist x, x, . . . , xn X satisfying

gxi i F xi(), xi(), . . . , xi(n) and lim

t M

gxi, F xi(), xi(), . . . , xi(n) , t =

for i n, then F and g have, at least, one -coincidence point.

Furthermore, assume that for all pairs of -coincidence points (x, x, . . . , xn), (y, y, . . . , yn) Xn of F and g there exists (u, u, . . . , un) Xn such that

(C) (gu, gu, . . . , gun) is comparable to (gx, gx, . . . , gxn) and (gy, gy, . . . , gyn); (C) limt M(gui, gxi, t) = limt M(gui, gyi, t) = for i n.

Also, assume that F is weakly compatible with g if assumption (C) holds. Then F and g have a unique common xed point.

Proof Since (X, M, , ) is a complete poFMS such that is a continuous t-norm of H-type, so is (Xn, Mn, , n) by Lemma .. Let T : Xn Xn and G : Xn Xn be two mappings dened by

T(Y) = F(y(), y(), . . . , y(n)), . . . , F(yi(), yi(), . . . , yi(n)), . . . ,

F(yn(), yn(), . . . , yn(n)) ()

and

G(Y) = (gy, gy, . . . , gyn) ()

for Y Xn. It follows from F(Xn) g(X) that T(Xn) G(Xn). By (), () and the continuity of , there exists X such that G(X) n T(X) and limt Mn(G(X), T(X), t) = .

Suppose that {Ym}m Xn such that {G(Ym)}m is monotone and the following limit exists: limm T(Ym) = limm G(Ym) Xn. From () and (), we see that, for sequences {ym}m, {ym}m, . . . , {ynm}m X such that {gym}m, {gym}m, . . . , {gynm}m are monotone and the following limit exists: for all i n, limm F(yi()m, yi()m, . . . , yi(n)m) =

limm gyim X. Since F and g are -compatible, we have

lim

m Mn

gF yi()

m , gyi()m, . . . , gyi(n)m , t = .

Therefore, T and G are compatible.

Now, we show that T is a G-isotone mapping. Suppose that G(Y) n G(V) for Y, V Xn. By () and (), we have gyj gvj when j A and gyj gvj when j B. For each i A, we have i A,B. So, for xed i A, we have gyi(j) gvi(j) when j A and gyi(j) gvi(j)

when j B. Thus, by the mixed g-monotone property of F, for xed i A, we have

F(yi(), . . . , yi(j), yi(j), yi(j+), . . . , yi(n))

F(yi(), . . . , yi(j), vi(j), yi(j+), . . . , yi(n)) ()

R+

()

GT(Ym), TG(Ym), t = ni= lim

m M

m , yi()m, . . . , yi(n)m ,

F gyi()

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 14 of 16

when j A. Similarly, if j B, then inequality () holds for xed i A. So, for xed i A, inequality () holds for all j. From this, we have

F(yi(), yi(), . . . , yi(n)) F(vi(), yi(), yi(), . . . , yi(n))

F(vi(), vi(), yi(), . . . , yi(n))

. . .

F(vi(), vi(), . . . , vi(n)) ()

for i A. Similarly, we have

F(yi(), yi(), . . . , yi(n)) F(vi(), vi(), . . . , vi(n)) ()

for i B. Thus, by (), (), and (), we deduce that T is a G-isotone mapping.

The conditions (C) and (C) imply that (C) and (C) hold w.r.t. (Xn, Mn, , n). It is easy to deduce that T and G are weakly compatible if assumption (C) holds w.r.t.

(Xn, Mn, , n). If F and g are continuous, then T and G are continuous.

Given G(Y) n G(V), by Proposition ., (gyi(), gyi(), . . . , gyi(n)) and (gvi(), gvi(), . . . , gvi(n)) are comparable by n. Therefore, () and () can be applied to these points, and it follows that for all t > ,

Mn T(Y), T(V), (t) = ni=M F(y

i(), yi(), . . . , yi(n)), F(vi(), vi(), . . . , vi(n)), (t)

ni= nj=M(gyi(j), gvi(j), t)

ni= nj=M(gyj, gvj, t)

= ni= Mn G(Y), G(V), t

= n Mn G(Y), G(V), t

Mn G(Y), G(V), t . ()

Next we shall prove that the condition (C) of Theorem . holds w.r.t. (Xn, Mn, , n). Since g(X) is closed, so is G(X). Suppose that {Zm} is a non-decreasing sequence in Xn such that Zm Z (m ). Using Lemma ., we have zim zi (m ) for all i n. Since

Zm n Zm+ for all m

N, then (zim)mN is a non-decreasing sequence when i A and (zim)mN is a non-increasing sequence when i B. If i A, as (X, M, ) has the sequential monotone property, then we have zim zi for all m

N. Similarly, if i B, then zim zi

N. The other case is treated similarly.

Therefore, all conditions of Theorems . and . hold w.r.t. (Xn, Mn, , n). Theorem . implies that T and G have a coincidence point, which is a -coincidence point of F and g. Moreover, it follows from Theorem . that T and G have a unique common xed point, which is a unique common xed point of F and g.

Remark . Theorem . improves Theorems and in []:(i) We use w, and w is a class of more general functions than (t) = kt, k (, ).(ii) The continuity of g and -compatible of F and g are omitted if assumption (C) holds. The continuity of is not necessary.

N. That is, Zm n Z for every m

for all m

Wang Fixed Point Theory and Applications (2015) 2015:233 Page 15 of 16

(iii) The condition limt M(x, y, t) = for all x, y X is weakened by conditions () and (C).

(iv) Our result is valid for fuzzy metric spaces in the sense of Kramosil and Michlek, so it is also valid for fuzzy metric spaces in the sense of George and Veeramani. The completeness is a weaker kind of completeness in Theorem . (see []).

Taking n = , A = {}, and B = {} in Theorem ., we deduce the following coupled xed point theorem improving Theorem . in [].

Corollary . Let (X, M, , ) be a complete poFMS such that is a continuous t-norm of H-type. Let F : X X and g : X X be two mappings such that F has the mixed g-monotone property on X and F(X) g(X). Assume that there exists w such that

M F(x, x), F(y, y), (t) M g(x), g(y), t M g(x), g(y), t

for all t > and x, x, y, y X with g(x) g(y) and g(x) g(y), where : [, ] [, ] is a mapping such that (a) (a) a for each a [, ]. Suppose that either F and g are continuous and compatible and M(x, y, ) :

I is continuous or condition (C) holds. If there are x, x X such that g(x) F(x, x), g(x) F(x, x) and

limt M(g(x), F(x, x), t) = limt M(g(x), F(x, x), t) = , then F and g have a coupled coincidence point in X.

Furthermore, assume that for all pairs of coupled coincidence points (x, x), (y, y) X of F and g there exists (u, u) X such that (gu, gu) is comparable to (gx, gx) and (gy, gy), limt M(gui, gxi, t) = limt M(gui, gyi, t) = for i . Also, assume that F are weakly compatible with g if assumption (C) holds. Then F and g have a unique common coupled xed point.

Remark . Corollary . is better than Theorem . in [] in two senses.(i) In Corollary ., we use w, and w is a class of more general functions than in Theorem . of [].

(ii) Corollary . is valid for partially ordered fuzzy metric spaces in the sense of Kramosil and Michlek, so it is also valid for fuzzy metric spaces in the sense of George and Veeramani.

Competing interests

The author declares that she has no competing interests.

Authors contributions

SW completed the paper herself. The author read and approved the nal manuscript.

Acknowledgements

The author thanks the editor and the referees for their useful comments and suggestions. Supported by the Natural Science Foundation of Jiangsu Province under grant (13KJB110028).

Received: 18 August 2015 Accepted: 9 October 2015

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