Skip to main content

Coincidence point theorems for generalized cyclic ( κ h , φ L ) S -weak contractions in partially ordered Menger PM-spaces

Abstract

In this paper, we introduce a new concept of generalized cyclic ( κ h , φ L ) S -weak contraction mappings and establish some coincidence point results for such mappings in complete partially ordered Menger PM-spaces. Our results generalize the main results of Nashine (Nonlinear Anal. 75:6160-6169, 2012) and Gopal et al. (Appl. Math. Comput. 233:955-967, 2014). We also obtain the corresponding coincidence point theorems for generalized cyclic ( κ h , φ L ) S -weak contractions in partially ordered metric spaces.

MSC:47H10, 54H25.

1 Introduction and preliminaries

Fixed point theory is a very useful tool in many fields such as nonlinear operator theory, control theory, game theory, dynamics and economic theory. One of the most fundamental fixed point theorems is the Banach contraction mapping principle. Due to its simplicity and importance, this classical result has been generalized by many authors in different directions (see [15]).

In 2004, Ran and Reurings [6] established a fixed point theorem for Banach’s contraction mappings in partially ordered metric spaces. In 2009, Harjani and Sadarangani [7] proved some fixed point theorems for weakly contractive mappings in complete metric spaces endowed with a partial order. In 2012, Nashine [8] presented some fixed point results for cyclic generalized ψ-weakly contractive mappings in complete metric spaces. Other authors also obtained some important results in this area (see [914]). On the other hand, in 2012, Samet et al. [15] introduced the concept of α-ψ-contractive and α-admissible mappings in metric spaces. In 2013, Berzig and Karapinar [16] proved some fixed point results for (αψ,βφ)-contractive mappings for a generalized altering distance in complete metric spaces.

In 1942, Menger [17] introduced the concept of a probabilistic metric space, and a large number of authors have done considerable work in such field (see, e.g. [1822]). Recently, the extension of fixed point theory to generalized structures as partially ordered probabilistic metric spaces has attracted much attention (see, e.g. [2325]). Gopal et al. [26] established some fixed point results for α-ψ-type contractive mappings and generalized β-type contractive mappings in Menger PM-spaces.

In this paper, we introduce the notion of generalized cyclic ( κ h , φ L ) S -weak contraction mappings to establish some corresponding coincidence point theorems in complete partially ordered Menger PM-spaces. Also, an application is given to show the validity of our results. It is worth pointing out that our results extend and generalize the main results of [8] and [26].

First, we recall some notions, lemmas and examples which will be used in the sequel.

Let R denote the set of reals and R + the nonnegative reals. A mapping F:R R + is called a distribution function if it is nondecreasing and left continuous with inf t R F(t)=0 and sup t R F(t)=1. We will denote by D the set of all distribution functions and D + ={FD:F(t)=0,t0}.

Let H denote the specific distribution function defined by

H(x)= { 0 , x 0 , 1 , x > 0 .

Definition 1.1 ([18])

The mapping Δ:[0,1]×[0,1][0,1] is called a triangular norm (for short, a t-norm) if the following conditions are satisfied:

  • (Δ-1) Δ(a,1)=a, for all a[0,1];

  • (Δ-2) Δ(a,b)=Δ(b,a);

  • (Δ-3) Δ(a,b)Δ(c,d), for ca, db;

  • (Δ-4) Δ(a,Δ(b,c))=Δ(Δ(a,b),c).

Three typical examples of continuous t-norms are Δ 1 (a,b)=max{a+b1,0}, Δ 2 (a,b)=ab and Δ M (a,b)=min{a,b}, for all a,b[0,1].

Definition 1.2 ([18])

A triplet (X,F,Δ) is called a Menger probabilistic metric space (shortly, a Menger PM-space), if X is a nonempty set, Δ is a t-norm and is a mapping from X×X D + satisfying the following conditions (for x,yX, we denote F(x,y) by F x , y ):

  • (MS-1) F x , y (t)=H(t), for all tR, if and only if x=y;

  • (MS-2) F x , y (t)= F y , x (t), for all x,yX and tR;

  • (MS-3) F x , z (s+t)Δ( F x , y (s), F y , z (t)), for all x,y,zX and s,t0.

Definition 1.3 ([19])

(X,F,Δ) is called a non-Archimedean Menger PM-space (shortly, a N.A Menger PM-space), if (X,F,Δ) is a Menger PM-space and Δ satisfies the following condition: for all x,y,zX and t 1 , t 2 0,

F x , z ( max { t 1 , t 2 } ) Δ ( F x , y ( t 1 ) , F y , z ( t 2 ) ) .
(1.1)

Definition 1.4 ([19])

A non-Archimedean Menger PM-space (X,F,Δ) is said to be type of ( D ) g , if there exists a gΩ, such that

g ( Δ ( s , t ) ) g(s)+g(t),

for all s,t[0,1], where Ω={g:g:[0,1][0,+) is continuous,strictly decreasing,g(1)=0}. In fact, we obtain g( F x , z (t))g( F x , y (t))+g( F y , z (t)), for all x,y,zX and t R + .

Example 1.1 Let (X,F,Δ) be a N.A Menger PM-space, and Δ Δ 1 (or Δ(s,t) t s t + s t s for all s,t[0,1]). Then (X,F,Δ) is of ( D ) g -type for gΩ defined by g(t)=1t (or g(t)= 1 t 1 for 0<t1 and g(0)=+).

Remark 1.1 Schweizer and Sklar [18] point out that if (X,F,Δ) is a Menger probabilistic metric space and Δ is continuous, then (X,F,Δ) is a Hausdorff topological space in the (ε,λ)-topology T, i.e., the family of sets { U x (ε,λ):ε>0,λ(0,1]} (xX) is a basis of neighborhoods of a point x for T, where U x (ε,λ)={yX: F x , y (ε)>1λ}.

Definition 1.5 ([1])

The function h:[0,)[0,) is called an altering distance function, if the following properties are satisfied: (a) h is continuous and nondecreasing; (b) h(t)=0 if and only if t=0.

Definition 1.6 ([21])

A function ψ: R + R + is said to be a Ψ-function, if it is nondecreasing and continuous in R + , ψ(t) as t, ψ(t)=0 if and only if t=0.

In the sequel, the class of all Ψ functions will be denoted by Ψ. Any altering distance function h with the additional property lim t + h(t)=+ generalizes a Ψ-function ψ through ψ(0)=0 and ψ(t)=sup{s:h(s)<t} whenever t>0.

Lemma 1.1 ([19])

Let { x n } be a sequence in (X,F,Δ) such that lim n F x n , x n + 1 (t)=1 for all t>0. If the sequence { x n } is not a Cauchy sequence in X, then there exist ε 0 >0, t 0 >0 and two sequences {k(i)}, {m(i)} of positive integers such that

  1. (1)

    m(i)>k(i), and m(i) as i;

  2. (2)

    F x m ( i ) , x k ( i ) ( t 0 )<1 ε 0 and F x m ( i ) 1 , x k ( i ) ( t 0 )1 ε 0 , for i=1,2, .

2 Main results

In this section, we first introduce the new notions of generalized κ-admissible mappings, weakly comparable mappings and generalized cyclic ( κ h , φ L ) S -weak contraction mappings in Menger PM-spaces.

Definition 2.1 Let X be a nonempty set, S,T:XX be two self-maps and κ:X×X×(0,) R + be a function. S and T are called generalized κ-admissible, if for all x,yX, t>0, and κ(Sx,Sy,t)1, we have κ(Tx,Ty,t)1. κ is called m-transitive on X, if for all t>0, x 0 , x 1 ,, x m , x m + 1 X, κ( x 0 , x 1 ,t)1, κ( x 1 , x 2 ,t)1,,κ( x m , x m + 1 ,t)1, we have κ( x 0 , x m + 1 ,t)1.

Example 2.1 Let X=[0,), Sx=ln(1+ x 2 ) for all xX,

Tx= { x 2 + 1 2 , x [ 0 , 3 ] , x 2 , x ( 3 , ) andκ(x,y,t)= { 1 , x , y [ 0 , 2 ] , 2 , otherwise .

In fact, if x,yX, for all t>0, κ(Sx,Sy,t)=κ(ln(1+ x 2 ),ln(1+ y 2 ),t)1, then x,y[0, e 2 1 ][0,3]. Hence, Tx,Ty[0,2], and so κ(Tx,Ty,t)1. Thus, S and T are generalized κ-admissible. Also, we can verify that κ is m-transitive.

Let (X,) be a partially ordered set, we will write xy whenever x and y are comparable (that is, xy or yx holds).

Definition 2.2 Let (X,) be a partially ordered set, S,T:XX be two self-maps and T(X)S(X). T is called weakly comparable with respect to S, if x,yX such that SxTx=Sy implies Tx and Ty are comparable (that is, TxTy). is called m-transitive on X, if x 0 , x 1 ,, x m + 1 X such that x i x i + 1 for all i{0,1,2,,m} implies x 0 x m + 1 .

Example 2.2 Let X={2,4,6,8,10}, ={2,2,4,4,6,6,8,8,10,10,2,4,2,6,8,4,10,6}, and

S: ( 2 4 6 8 10 6 10 2 8 4 ) ,T: ( 2 4 6 8 10 2 6 4 10 4 ) .

Since S(2)=6T(2)=2=S(6), we have T(2)=2T(6)=4. Since S(4)=10T(4)=6=S(2), we have T(4)T(2)=2. Since S(6)T(6)=4=S(10), we have T(6)T(10). Since S(10)T(10)=S(10), we have T(10)T(10). Note that S(8) and T(8) are not comparable. Hence, T is weakly comparable with respect to S.

Definition 2.3 Let X be a nonempty set, m be a positive integer, A 1 , A 2 ,, A m be subsets of X, Y= i = 1 m A i and S,T:YY be two self-maps. Then Y is said to be a cyclic representation of Y with respect to S and T, if the following two conditions are satisfied:

  1. (i)

    S( A i ), i=1,2,,m, are nonempty closed sets;

  2. (ii)

    T( A 1 )S( A 2 ),T( A 2 )S( A 3 ),,T( A m )S( A 1 ).

Example 2.3 Let X= R + , A 1 =[0,4], A 2 =[0, 7 2 ], A 3 =[0,3], and Y= i = 1 3 A i . Define S,T:YY by Sx= 1 3 + 11 12 x and Tx=1+ 1 2 x, for all xY. Then it is easy to verify that Y= i = 1 3 A i is a cyclic representation of Y with respect to S and T.

Definition 2.4 Let (X,) be a partially ordered set and (X,F,Δ) be a N.A Menger PM-space of type ( D ) g . Let κ:X×X×(0,)[0,) be a function and ϕ:X[0,) be a lower semi-continuous function. Let m be a positive integer, A 1 , A 2 ,, A m be subsets of X, Y= i = 1 m A i , and S,T:YY be two self-maps. T is said to be a generalized cyclic ( κ h , φ L ) S -weak contraction, if Y is a cyclic representation of Y with respect to S and T, A m + 1 = A 1 , and for k{1,2,,m} and for all x,yX, SxS( A k ) and SyS( A k + 1 ) are comparable such that

h [ g ( F T x , T y ( t ) ) + ϕ ( T x ) + ϕ ( T y ) ] κ ( S x , S y , t ) [ h ( M t ( S x , S y ) ) φ ( M t ( S x , S y ) ) ] + L ( N t ( S x , S y ) ) ,
(2.1)

for all t>0 and β(0,1], where h is an altering distance function, φ,L:[0,)[0,) are two continuous functions such that L(0)=0, φ(s)=0 if and only if s=0, φ(s)h(s) for all s R + , N t (Sx,Sy)=min{g( F S x , T x (t)),g( F S x , T y ((2β)t)),g( F S y , T x (βt))}, and

M t ( S x , S y ) = max { g ( F S x , S y ( t ) ) + ϕ ( S x ) + ϕ ( S y ) , g ( F S x , T x ( t ) ) + ϕ ( S x ) + ϕ ( T x ) , g ( F S y , T y ( t ) ) + ϕ ( S y ) + ϕ ( T y ) , 1 2 [ g ( F S x , T y ( ( 2 β ) t ) ) + g ( F S y , T x ( β t ) ) + ϕ ( S x ) + ϕ ( S y ) + ϕ ( T x ) + ϕ ( T y ) ] } .

Now we are ready to state our main results.

Theorem 2.1 Let (X,) be a partially ordered set and (X,F,Δ) be a complete N.A Menger PM-space of type ( D ) g . Let m be a positive integer, A 1 , A 2 ,, A m be subsets of X, Y= i = 1 m A i , T:YY be a generalized cyclic ( κ h , φ L ) S -weak contraction satisfying (2.1). Suppose that the following conditions hold:

  1. (i)

    S and T are generalized κ-admissible;

  2. (ii)

    κ and are m-transitive;

  3. (iii)

    T is weakly comparable with respect to S;

  4. (iv)

    there exists x 0 A 1 such that S x 0 T x 0 and κ(S x 0 ,T x 0 ,t)1 for all t>0;

  5. (v)

    if a sequence { y n }Y satisfies y n y n + 1 , κ( y n , y n + 1 ,t)1 for all nN, and t>0 and y n y as n, then y n y and κ( y n ,y,t)1 for n sufficiently large and for all t>0.

Then S and T have a coincidence point in X, that is, there exists xX such that Sx=Tx.

Proof Since T( A 1 )S( A 2 ) and x 0 A 1 , there exists an x 1 A 2 , such that S x 1 =T x 0 . Since T( A 2 )S( A 3 ) and x 1 A 2 , there exists an x 2 A 3 , such that S x 2 =T x 1 . Continuing this process, we can construct two sequences { x n } and { y n } defined by y n + 1 =S x n + 1 =T x n , for all nN, and there exists i n {1,2,,m} such that x n A i n and x n + 1 A i n + 1 .

By condition (iv), we get S x 0 T x 0 =S x 1 and κ(S x 0 ,S x 1 ,t)1 for all t>0. It follows from (i) and (iii) that S x 1 =T x 0 T x 1 =S x 2 and κ(S x 1 ,S x 2 ,t)1 for all t>0. By induction, we obtain

y n =S x n S x n + 1 = y n + 1 andκ( y n , y n + 1 ,t)1,for all t>0.
(2.2)

We will complete the proof by the following three steps.

Step 1. We prove that

lim n g ( F y n , y n + 1 ( t ) ) =0and lim n ϕ( y n )=0,for all t>0.
(2.3)

Without loss of generality, assume that y n + 1 y n , for all nN (otherwise, T x n 0 =S x n 0 + 1 = y n 0 + 1 = y n 0 =S x n 0 for some n 0 N, then x n 0 is the coincidence point of S and T. Hence, the conclusion holds).

Since x n A i n , x n + 1 A i n + 1 , y n S( A i n ), and y n + 1 S( A i n + 1 ) are comparable, for i n {1,2,,m}, by (2.1) and (2.2), we get

h [ g ( F y n , y n + 1 ( t ) ) + ϕ ( y n ) + ϕ ( y n + 1 ) ] κ ( y n 1 , y n , t ) [ h ( M t ( y n 1 , y n ) ) φ ( M t ( y n 1 , y n ) ) ] + L ( N t ( y n 1 , y n ) ) h ( M t ( y n 1 , y n ) ) φ ( M t ( y n 1 , y n ) ) + L ( N t ( y n 1 , y n ) ) ,
(2.4)

for all t>0, where N t ( y n 1 , y n )=min{g( F y n 1 , y n (t)),g( F y n 1 , y n + 1 ((2β)t)),0}=0 and

M t ( y n 1 , y n ) = max { g ( F y n 1 , y n ( t ) ) + ϕ ( y n 1 ) + ϕ ( y n ) , g ( F y n 1 , y n ( t ) ) + ϕ ( y n 1 ) + ϕ ( y n ) , g ( F y n , y n + 1 ( t ) ) + ϕ ( y n ) + ϕ ( y n + 1 ) , 1 2 [ g ( F y n 1 , y n + 1 ( ( 2 β ) t ) ) + ϕ ( y n 1 ) + 2 ϕ ( y n ) + ϕ ( y n + 1 ) ] } max { g ( F y n 1 , y n ( t ) ) + ϕ ( y n 1 ) + ϕ ( y n ) , g ( F y n , y n + 1 ( t ) ) + ϕ ( y n ) + ϕ ( y n + 1 ) , 1 2 [ g ( F y n 1 , y n ( ( 2 β ) t ) ) + g ( F y n , y n + 1 ( ( 2 β ) t ) ) + ϕ ( y n 1 ) + 2 ϕ ( y n ) + ϕ ( y n + 1 ) ] } .

For all β(0,1], we have 2β1. Since F x , y is nondecreasing and g is strictly decreasing, we have g( F y n 1 , y n ((2β)t))g( F y n 1 , y n (t)) for all nN. Hence, M t ( y n 1 , y n )max{g( F y n 1 , y n (t))+ϕ( y n 1 )+ϕ( y n ),g( F y n , y n + 1 (t))+ϕ( y n )+ϕ( y n + 1 )}. On the other hand, it is obvious that M t ( y n 1 , y n )max{g( F y n 1 , y n (t))+ϕ( y n 1 )+ϕ( y n ),g( F y n , y n + 1 (t))+ϕ( y n )+ϕ( y n + 1 )}. Thus, M t ( y n 1 , y n )=max{g( F y n 1 , y n (t))+ϕ( y n 1 )+ϕ( y n ),g( F y n , y n + 1 (t))+ϕ( y n )+ϕ( y n + 1 )}.

Suppose that M t ( y n 1 , y n )=g( F y n , y n + 1 (t))+ϕ( y n )+ϕ( y n + 1 ), by (2.4), we have

h [ M t ( y n 1 , y n ) ] h [ M t ( y n 1 , y n ) ] φ ( M t ( y n 1 , y n ) ) ,for all t>0,

which implies that φ(g( F y n , y n + 1 (t)))+ϕ( y n )+ϕ( y n + 1 )=0. Thus, g( F y n , y n + 1 (t))=0, that is, F y n , y n + 1 (t)=1 for all t>0. Then y n = y n + 1 , which is in contradiction to y n y n + 1 for all nN.

Hence, M t ( y n 1 , y n )=g( F y n 1 , y n (t))+ϕ( y n 1 )+ϕ( y n ). Let Q n (t)=g( F y n 1 , y n (t))+ϕ( y n 1 )+ϕ( y n ). By (2.4), we get

h [ Q n + 1 ( t ) ] h [ Q n ( t ) ] φ ( Q n ( t ) ) h [ Q n ( t ) ] ,for all t>0.
(2.5)

Since h is nondecreasing, it follows from (2.5) that { Q n (t)} is a decreasing sequence and bounded from below, for every t>0. Hence, there exists r t 0, such that lim n Q n (t)= r t .

By using the continuities of h and φ, letting n in (2.5), we get h( r t )h( r t )φ( r t ), which implies that φ( r t )=0. Thus r t =0, that is, lim n g( F y n 1 , y n (t))+ϕ( y n 1 )+ϕ( y n )=0 for all t>0. Hence, (2.3) holds.

Step 2. We prove that { y n } is a Cauchy sequence. To prove this fact, we first prove the following claim.

Claim: for every t>0 and ε>0, there exists n 0 N, such that p,q n 0 with pq1modm then F y p , y q (t)>1ε, that is, g( F y p , y q (t))<g(1ε).

In fact, suppose this is not true, then there exist t 0 >0 and ε 0 >0, such that for any nN, we can find p(n)>q(n)n with p(n)q(n)1modm satisfying F y p ( n ) , y q ( n ) ( t 0 )1 ε 0 , that is, g( F y p ( n ) , y q ( n ) ( t 0 ))g(1 ε 0 ).

Now, take n>2m. Then corresponding to q(n)n, we can choose p(n) in such a way that it is the smallest integer with p(n)>q(n) satisfying p(n)q(n)1modm and g( F y p ( n ) , y q ( n ) ( t 0 ))g(1 ε 0 ). Therefore, g( F y p ( n ) m , y q ( n ) ( t 0 ))<g(1 ε 0 ). Using the non-Archimedean Menger triangular inequality and Definition 1.4, we have

g ( 1 ε 0 ) g ( F y q ( n ) , y p ( n ) ( t 0 ) ) g ( Δ ( F y q ( n ) , y q ( n ) + 1 ( t 0 ) , F y q ( n ) + 1 , y p ( n ) ( t 0 ) ) ) g ( F y q ( n ) , y q ( n ) + 1 ( t 0 ) ) + g ( F y q ( n ) + 1 , y p ( n ) ( t 0 ) ) g ( F y q ( n ) , y q ( n ) + 1 ( t 0 ) ) + g ( F y q ( n ) + 1 , y p ( n ) + 1 ( t 0 ) ) + g ( F y p ( n ) + 1 , y p ( n ) ( t 0 ) ) 2 g ( F y q ( n ) , y q ( n ) + 1 ( t 0 ) ) + g ( F y q ( n ) , y p ( n ) + 1 ( t 0 ) ) + g ( F y p ( n ) + 1 , y p ( n ) ( t 0 ) ) 2 g ( F y q ( n ) , y q ( n ) + 1 ( t 0 ) ) + g ( F y q ( n ) , y p ( n ) ( t 0 ) ) + 2 g ( F y p ( n ) + 1 , y p ( n ) ( t 0 ) ) 2 g ( F y q ( n ) , y q ( n ) + 1 ( t 0 ) ) + g ( F y q ( n ) , y p ( n ) m ( t 0 ) ) + g ( F y p ( n ) m , y p ( n ) ( t 0 ) ) + 2 g ( F x p ( n ) + 1 , x p ( n ) ( t 0 ) ) 2 g ( F y q ( n ) , y q ( n ) + 1 ( t 0 ) ) + g ( 1 ε 0 ) + i = 1 m g ( F y p ( n ) i , y p ( n ) i + 1 ( t 0 ) ) + 2 g ( F y p ( n ) + 1 y p ( n ) ( t 0 ) ) .
(2.6)

Since lim n g( F y n , y n + 1 (t))=0 for all t>0, letting n in (2.6), we obtain

g ( 1 ε 0 ) = lim n g ( F y q ( n ) , y p ( n ) ( t 0 ) ) = lim n g ( F y q ( n ) , y p ( n ) + 1 ( t 0 ) ) = lim n g ( F y q ( n ) + 1 , x p ( n ) ( t 0 ) ) = lim n g ( F y q ( n ) + 1 , y p ( n ) + 1 ( t 0 ) ) .
(2.7)

By p(n)q(n)1modm, we know that y q ( n ) and y p ( n ) lie in different adjacently labeled sets S( A i ) and S( A i + 1 ), for 1im. Since and κ are m-transitive, we obtain y p ( n ) y q ( n ) and κ( y p ( n ) , y q ( n ) ,t)1 for all t>0. Using the fact that T is a generalized cyclic ( κ h , φ L ) S -weak contraction and letting β=1, we have

h [ g ( F y q ( n ) + 1 , y p ( n ) + 1 ( t 0 ) ) + ϕ ( y q ( n ) + 1 ) + ϕ ( y p ( n ) + 1 ) ] κ ( y q ( n ) , y p ( n ) , t ) [ h ( M t 0 ( y q ( n ) , y p ( n ) ) ) φ ( M t 0 ( y q ( n ) , y p ( n ) ) ) ] + L ( N t 0 ( y q ( n ) , y p ( n ) ) ) h ( M t 0 ( y q ( n ) , y p ( n ) ) ) φ ( M t 0 ( y q ( n ) , y p ( n ) ) ) + L ( N t 0 ( y q ( n ) , y p ( n ) ) ) ,
(2.8)

where N t 0 ( y q ( n ) , y p ( n ) )=min{g( F y q ( n ) , y q ( n ) + 1 ( t 0 )),g( F y q ( n ) , y p ( n ) + 1 ( t 0 )),g( F y p ( n ) , y q ( n ) + 1 ( t 0 ))} and

M t 0 ( y q ( n ) , y p ( n ) ) = max { g ( F y q ( n ) , y p ( n ) ( t 0 ) ) + ϕ ( y q ( n ) ) + ϕ ( y p ( n ) ) , g ( F y q ( n ) , y q ( n ) + 1 ( t 0 ) ) + ϕ ( y q ( n ) ) + ϕ ( y q ( n ) + 1 ) , g ( F y p ( n ) , y p ( n ) + 1 ( t 0 ) ) + ϕ ( y p ( n ) ) + ϕ ( y p ( n ) + 1 ) , 1 2 [ g ( F y q ( n ) , y p ( n ) + 1 ( t 0 ) ) + g ( F y p ( n ) , y q ( n ) + 1 ( t 0 ) ) + ϕ ( y q ( n ) ) + ϕ ( y p ( n ) ) + ϕ ( y q ( n ) + 1 ) + ϕ ( y p ( n ) + 1 ) ] } .

By (2.3) and (2.7), we have lim n M t 0 ( x q ( n ) , x p ( n ) )=max{g(1 ε 0 ),0,0, 1 2 [g(1 ε 0 )+g(1 ε 0 )]}=g(1 ε 0 ) and lim n N t 0 ( y q ( n ) , y p ( n ) )=min{0,g(1 ε 0 ),g(1 ε 0 )}=0. According to the continuities of h and φ, letting n in (2.8), we get

h [ g ( 1 ε 0 ) ] h [ g ( 1 ε 0 ) ] φ ( g ( 1 ε 0 ) ) .

Thus, φ(g(1 ε 0 ))=0, that is, g(1 ε 0 )=0. Then ε 0 =0, which is in contradiction to ε 0 >0.

Therefore, our claim is proved. Next, we will prove that { y n } is a Cauchy sequence.

By the continuity of g and g(1)=0, we have lim a 0 + g(1aε)=0, for any given ε>0. Since g is strictly decreasing, there exists a>0, such that g(1aε) g ( 1 ε ) 2 .

For any given t>0 and ε>0, there exists a>0, such that g(1aε) g ( 1 ε ) 2 . By the claim, we can find n 0 N, such that if p,q n 0 with pq1modm, then

F y p , y q (t)>1aεandg ( F y p , y q ( t ) ) <g(1aε) g ( 1 ε ) 2 .
(2.9)

Since lim n g( F y n , y n + 1 (t))=0, we can also find n 1 N, such that

g ( F y n , y n + 1 ( t ) ) g ( 1 ε ) 2 m ,
(2.10)

for any n> n 1 .

Suppose that r,smax{ n 0 , n 1 } and s>r. Then there exists k{1,2,,m} such that srkmodm. Therefore, sr+j1modm, for j=mk+1, j{0,1,,m1}. Thus we have

g ( F y r , y s ( t ) ) g ( F y r , y s + j ( t ) ) +g ( F y s + j , y s + j 1 ( t ) ) ++g ( F y s + 1 , y s ( t ) ) .
(2.11)

By (2.9), (2.10), and (2.11), we get

g ( F y r , y s ( t ) ) < g ( 1 ε ) 2 +j g ( 1 ε ) 2 m g ( 1 ε ) 2 + g ( 1 ε ) 2 =g(1ε).
(2.12)

Since g is strictly decreasing, by (2.12), we have F y r , y s (t)>1ε. This proves that { y n } is a Cauchy sequence.

Step 3. We show that S and T have a coincidence point in X.

Since { y n }X is a Cauchy sequence and (X,F,Δ) is a complete Menger PM-space, there exists y X, such that y n y . Since S(Y)=S( i = 0 m A i )= i = 0 m S( A i ) is closed and { y n }S(Y), we know that y S(Y). Hence, there exists zY, such that y =Sz. As Y= i = 1 m A i is a cyclic representation of Y with respect to S and T, the sequence { y n } has infinite terms in each S( A i ) for i{1,2,,m}.

First, suppose that y S( A i ), then TzS( A i + 1 ), and we can choose a subsequence { y n k } of { y n } with y n k S( A i 1 ) (the existence of this subsequence is guaranteed by the above discussion). Since y n y n + 1 and y n y =Sz, by (v), we obtain the result that y n k S( A i 1 ) and y S( A i ) are comparable, κ( y n k , y ,t)1 for all t>0 and k sufficiently large. Letting β=1, by (2.1), we have

h [ g ( F y n k + 1 , T z ( t ) ) + ϕ ( y n k + 1 ) + ϕ ( T z ) ] κ ( y n k , y , t ) [ h ( M t ( y n k , y ) ) φ ( M t ( y n k , y ) ) ] + L ( N t ( y n k , y ) ) h [ M t ( y n k , y ) ] φ ( M t ( y n k , y ) ) + L ( N t ( y n k , y ) ) ,
(2.13)

where N t ( y n k , y )=min{g( F y n k , y n k + 1 (t)),g( F y n k , T z (t)),g( F y n k + 1 , y (t))} and

M t ( y n k , y ) = max { g ( F y n k , y ( t ) ) + ϕ ( y n k ) + ϕ ( y ) , g ( F y n k , y n k + 1 ( t ) ) + ϕ ( y n k ) + ϕ ( y n k + 1 ) , g ( F y , T z ( t ) ) + ϕ ( y ) + ϕ ( T z ) , 1 2 [ g ( F y n k , T z ( t ) ) + g ( F y n k + 1 , y ( t ) ) + ϕ ( y n k ) + ϕ ( y n k + 1 ) + ϕ ( y ) + ϕ ( T z ) ] } .

Since ϕ is lower semi-continuous and y n y , by (2.3), we have

ϕ ( y ) lim inf n ϕ( y n )=0.

Let G 0 be the set of all discontinuous points of F x , T x (). Since g, h, φ, and L are continuous, we find that G 0 is also the set of all discontinuous points of g( F x , T x ()), h[g( F x , T x ())+ϕ(Tz)], φ(g( F x , T x ())+ϕ(Tz)), and L(g( F x , T x ())). Moreover, we know that G 0 is a countable set. Let G= R + G 0 . If tG{0} (t is a continuous point of F x , T x ()), by (2.3), we have lim k N t ( y n k , y )=min{0,g( F y , T z (t)),0}=0 and

lim k M t ( y n k , y ) = max { 0 , 0 , g ( F y , T z ( t ) ) + ϕ ( T z ) , 1 2 [ g ( F y , T z ( t ) ) + ϕ ( T z ) ] } = g ( F y , T z ( t ) ) + ϕ ( T z ) .

Letting n in (2.13), we get

h [ g ( F y , T z ( t ) ) + ϕ ( T z ) ] h [ g ( F y , T z ( t ) ) + ϕ ( T z ) ] φ ( g ( F y , T z ( t ) ) + ϕ ( T z ) ) ,

which implies that φ(g( F y , T z (t))+ϕ(Tz))=0. Hence, g( F y , T z (t))=0 and ϕ(Tz)=0. Then

F y , T z (t)=1,for all tG{0}.
(2.14)

If t G 0 with t>0, by the density of real numbers, there exist t 1 , t 2 G, such that 0< t 1 <t< t 2 . Since the distribution is nondecreasing, we have

1=H( t 1 )= F y , T z ( t 1 ) F y , T z (t) F y , T z ( t 2 )=1.
(2.15)

Hence, from (2.14) and (2.15), we have F y , T z (t)=1 for any t>0. Thus, Sz= y =Tz, that is, z is the coincidence point of S and T. □

Corollary 2.1 Let (X,) be a partially ordered set and (X,F,Δ) be a complete N.A Menger PM-space of type ( D ) g , κ:X×X×(0,)[0,) be a function, S,T:XX be two self-maps and T(X)S(X). Suppose that for x,yX, Sx and Sy are comparable, we have

g ( F T x , T y ( t ) ) κ(Sx,Sy,t)Φ ( M t ( S x , S y ) ) +L ( N t ( S x , S y ) ) ,
(2.16)

for all t>0, where Φ:[0,)[0,) is a continuous function, Φ(t)<t for t>0 and Φ(0)=0, L is the same as the one in Theorem  2.1,

M t (Sx,Sy)=max { g ( F S x , S y ( t ) ) , g ( F S x , T x ( t ) ) , g ( F S y , T y ( t ) ) , 1 2 [ g ( F S x , T y ( t ) ) + g ( F S y , T x ( t ) ) ] }

and

N t (Sx,Sy)=min { g ( F S x , T x ( t ) ) , g ( F S x , T y ( t ) ) , g ( F S y , T x ( t ) ) } .

Also, assume that the conditions (i)-(v) of Theorem  2.1 are satisfied, where m=1.

Then S and T have a coincidence point in X, that is, there exists xX, such that Sx=Tx.

Proof Letting h(x)=x, ϕ(x)0, β1, and φ(t)=tΦ(t) in Theorem 2.1, the conclusion follows immediately. □

Definition 2.5 Let X be a nonempty set, S,T:XX be two self-maps and α:X×X×(0,)(0,) be a function. S and T are called generalized α-admissible, if for all x,yX, t>0, α(Sx,Sy,t)1 implies α(Tx,Ty,t)1. α is called 1-transitive on X, if for all t>0, x 0 , x 1 , x 2 X, α( x 0 , x 1 ,t)1, α( x 1 , x 2 ,t)1 implies α( x 0 , x 2 ,t)1.

Theorem 2.2 Let (X,) be a partially ordered set and (X,F,Δ) be a complete N.A Menger PM-space, Δ be a continuous t-norm and Δ= Δ M , α:X×X×(0,)(0,) be a function, S,T:XX be two self-maps and T(X)S(X). Suppose that for x,yX, Sx and Sy are comparable, we have

α(Sx,Sy,t) [ ( 1 F T x , T y ( ψ ( c t ) ) 1 ) L ( N ψ ( t ) ( S x , S y ) ) ] Φ ( M ψ ( t ) ( S x , S y ) ) ,
(2.17)

for all t>0, such that min{ F S x , S y (ψ(t)), F S x , T x (ψ(t)), F S y , T y (ψ(t)), F S x , T y (ψ(t)), F S y , T x (ψ(t))}>0, where c(0,1), ψΨ, L and Φ are the same as the ones in Corollary  2.1,

M ψ ( t ) ( S x , S y ) = max { 1 F S x , S y ( ψ ( t ) ) 1 , 1 F S x , T x ( ψ ( t ) ) 1 , 1 F S y , T y ( ψ ( t ) ) 1 , 1 2 [ 1 F S x , T y ( ψ ( t ) ) + 1 F S y , T x ( ψ ( t ) ) 2 ] }

and

N ψ ( t ) (Sx,Sy)=min { 1 F S x , T x ( ψ ( t ) ) 1 , 1 F S x , T y ( ψ ( t ) ) 1 , 1 F S y , T x ( ψ ( t ) ) 1 } .

Also assume that the following conditions hold:

  1. (i)

    S and T are generalized α-admissible;

  2. (ii)

    α and are 1-transitive;

  3. (iii)

    T is weakly comparable with respect to S;

  4. (iv)

    there exists x 0 A 1 such that S x 0 T x 0 and α(S x 0 ,T x 0 ,t)1 for all t>0;

  5. (v)

    if a sequence { y n }Y satisfies y n y n + 1 , α( y n , y n + 1 ,t)1 for all nN and t>0, and y n y as n, then y n y and α( y n ,y,t)1 for n sufficiently large and for all t>0.

Then S and T have a coincidence point in X.

Proof Let g:[0,1][0,+) be a function defined by g(t)= 1 t 1 for t>0 and g(0)=+. Then gΩ. Since Δ= Δ M , we have g(Δ(s,t))g(s)+g(t) for all s,t[0,1]. Hence, (X,F,Δ) is a N.A Menger PM-space of ( D ) g . Let κ:X×X×(0,)[0,) be a function and κ(x,y,t)= 1 α ( x , y , t ) for all x,yX and t>0.

Since ψ and F x , y are both nondecreasing, we have F T x , T y (ψ(ct)) F T x , T y (ψ(t)). Hence, 1 F T x , T y ( ψ ( c t ) ) 1 1 F T x , T y ( ψ ( t ) ) 1. By the definition of g, we have

g ( F T x , T y ( ψ ( t ) ) ) g ( F T x , T y ( ψ ( c t ) ) ) .
(2.18)

For x,yX, Sx and Sy are comparable, by (2.17) and (2.18), we get

g ( F T x , T y ( ψ ( t ) ) ) κ ( S x , S y , ψ ( t ) ) Φ ( M ψ ( t ) ( S x , S y ) ) +L ( N ψ ( t ) ( S x , S y ) ) .
(2.19)

Since ψ is continuous and ψ(t)+ as t+, it follows from (2.19) that (2.16) holds. Also, S and T also satisfy (i)-(v) of Theorem 2.1.

Thus, all the conditions of Corollary 2.1 are satisfied. Therefore, S and T have a coincidence point in X. □

3 Coincidence point results in partially ordered metric spaces

In this section, we will apply the results obtained in Section 2 to establish the corresponding coincidence point theorems for generalized cyclic ( κ h , φ L ) S -weak contractions in partially ordered metric spaces. We first introduce a new notion in metric spaces that we will use in Theorem 3.1.

Definition 3.1 Let S and T be two self-maps of a metric space (X,d), κ:X×X R + be a function. S and T are called generalized κ-admissible, if for all x,yX, κ(Sx,Sy)1 implies κ(Tx,Ty)1. κ is called m-transitive on X, if x 0 , x 1 ,, x m , x m + 1 X, κ( x 0 , x 1 )1,κ( x 1 , x 2 )1,,κ( x m , x m + 1 )1 implies κ( x 0 , x m + 1 )1.

Theorem 3.1 Let (X,d,) be an ordered complete metric space and κ:X×X×(0,)[0,) be a function. Let m be a positive integer, A 1 , A 2 ,, A m be subsets of X, Y= i = 1 m A i , S and T:YY be two self-maps, Y be a cyclic representation of Y with respect to S and T. Suppose that A m + 1 = A 1 , and for k{1,2,,m} and for all x,yX, SxS( A k ) and SyS( A k + 1 ) are comparable, we have

h ( d ( T x , T y ) ) κ(Sx,Sy) [ h ( M ( S x , S y ) ) φ ( M ( S x , S y ) ) ] +L ( N ( S x , S y ) ) ,
(3.1)

for all t>0, where h is a continuous and nondecreasing linear function, h(s)=0 if and only if s=0, φ,L:[0,)[0,) are two continuous functions such that L(0)=0, φ(t)=0 if and only if t=0, φ ( s ) t φ( s t ) and L ( s ) t L( s t ) for all t>0, φ(t)h(t) for all t R + ,

N(Sx,Sy)=min { d ( S x , T x ) , d ( S x , T y ) , d ( S y , T x ) }

and

M(Sx,Sy)=max { d ( S x , S y ) , d ( S x , T x ) , d ( S y , T y ) , 1 2 [ d ( S x , T y ) + d ( S y , T x ) ] } .

Also, assume that the following conditions hold:

  1. (i)

    S and T are generalized κ-admissible;

  2. (ii)

    κ and are m-transitive;

  3. (iii)

    T is weakly comparable with respect to S;

  4. (iv)

    there exists x 0 A 1 such that S x 0 T x 0 and κ(S x 0 ,T x 0 )1;

  5. (v)

    if a sequence { y n }Y satisfies y n y n + 1 , κ( y n , y n + 1 )1 for all nN, and y n y as n, then y n y and κ( y n ,y)1 for n sufficiently large.

Then S and T have a coincidence point in X, that is, there exists xX such that Sx=Tx.

Proof Let (X,F,Δ) be the induced PM-space, where is defined by F(x,y)(t)= F x , y (t)= t t + d ( x , y ) , for all t>0 and x,yX.

In fact, for 0< t 1 t 2 , F x , z (max{ t 1 , t 2 })= F x , z ( t 2 )= t 2 t 2 + d ( x , z ) and F x , y ( t 1 ) F y , z ( t 2 ) F x , y ( t 1 ) + F y , z ( t 2 ) F x , y ( t 1 ) F y , z ( t 2 ) = t 1 t 2 t 1 t 2 + t 1 d ( y , z ) + t 2 d ( x , y ) , by d(x,z)d(x,y)+d(y,z), we have

t 1 t 2 + t 1 d ( y , z ) + t 2 d ( x , y ) t 1 t 2 =1+ d ( x , y ) t 1 + d ( y , z ) t 2 1+ d ( x , z ) t 2 = t 2 + d ( x , z ) t 2 ,

which implies that (1.1) holds and Δ(s,t) t s t + s t s . Hence, by Example 1.1, we know that (X,F,Δ) is a N.A Menger PM-space of ( D ) g -type for gΩ defined by g(t)= 1 t 1 for 0<t1 and g(0)=+. It is not difficult to prove that a sequence { x n } in (X,d) converges to a point x X if and if only { x n } in (X,F,Δ) τ-converges to x . Since (X,d) is a complete metric space, (X,F,Δ) is a τ-complete N.A Menger PM-space of type ( D ) g .

For x,yX, SxS( A i ) and SyS( A i + 1 ) are comparable, by (3.1) and the properties of h, φ, L, for t>0, we have

h ( d ( T x , T y ) t ) = h ( d ( T x , T y ) ) t κ ( S x , S y ) [ h ( M ( S x , S y ) ) t φ ( M ( S x , S y ) ) t ] + L ( N ( S x , S y ) ) t κ ( S x , S y ) [ h ( M ( S x , S y ) t ) φ ( M ( S x , S y ) t ) ] + L ( N ( S x , S y ) t ) .
(3.2)

Since F x , y (t)= t t + d ( x , y ) for t>0, we have g( F x , y (t))= d ( x , y ) t for t>0. It follows from (3.2) that (2.1) holds. In fact, S and T also satisfy (i)-(v) of Theorem 2.1.

Thus, all the conditions of Theorem 2.1 are satisfied when ϕ(x)0. Therefore, the conclusion holds. □

4 An illustration

In this section, we give an example to demonstrate Theorem 2.1.

Example 4.1 Let X= R + , Δ(s,t)= t s t + s t s for all s,t[0,1], g(t)= 1 t 1 for all 0<t1 and g(0)=+. Define F:X×X D + by

F(x,y)(t)= F x , y (t)= { t t + | x y | , t > 0 , 0 , t 0 ,

for all x,yX. Then (X,F,Δ) is a complete N.A Menger PM-space of ( D ) g -type. Suppose that A 1 =[0,4], A 2 =[0,2], A 3 =[0, 2 ], and Y= i = 1 3 A i =[0,4]. Define S,T:YY and κ:X×X×(0,)(0,) by

Sx=x,Tx= { x 2 , x [ 0 , 1 ) , x , x [ 1 , 4 ] , κ(x,y,t)= { 1 , x , y [ 0 , 1 )  or  x , y [ 1 , 4 ] , 4 , otherwise .

Then S and T are generalized κ-admissible and Y is a cyclic representation of Y with respect to S and T, and T is weakly comparable with respect to S, and κ and are 3-transitive.

Let ϕ:X[0,), ϕ(x)0 for all xX, β1, L(t)=0, φ(t)= t 2 , h(t)=t, for all t[0,). Now, we verify inequality (2.1) in Theorem 2.1. By the definitions of F, g, ϕ, h, φ, and L, we only need to prove that

| T x T y | t 1 2 κ(x,y,t)max { | x y | t , | T x x | t , | T y y | t , 1 2 [ | T x y | t + | T y x | t ] } ,

for all t>0, that is,

|TxTy| 1 2 κ(x,y,t)max { | x y | , | T x x | , | T y y | , 1 2 [ | T x y | + | T y x | ] } ,
(4.1)

where κ(x,y,t)=1 if x,y[0,1) or x,y[1,4], and κ(x,y,t)=4 if otherwise.

We consider the following cases:

Case 1. If x,y[0,1), then 1 2 κ(x,y,t)= 1 2 . By the definition of T, we have

|TxTy|= 1 2 |xy| 1 2 κ(x,y,t)|xy|,

which implies that (4.1) holds.

Case 2. If x[0,1) and y[1,4], then 1 2 κ(x,y,t)=2 and x< y y. By the definition of T, we have

| T x T y | = y 1 2 x y x + y 1 2 x = | y 1 2 x | + | y x | = 1 2 κ ( x , y , t ) ( 1 2 [ | T x y | + | T y x | ] ) ,

which implies that (4.1) holds. Similarly, if x[1,4] and y[0,1), we also have (4.1) holds.

Case 3. If x,y[1,4], then 1 2 κ(x,y,t)= 1 2 and x + y 2. By the definition of T, we have

|TxTy|=| x y | x + y 2 | x y |= 1 2 |xy|,

which implies that (4.1) holds.

Thus, all the conditions of Theorem 2.1 are satisfied. Therefore, S and T have a coincidence point in X, indeed, x=0 and x=1 are coincidence points of S and T.

References

  1. Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 1984, 30(1):1–9. 10.1017/S0004972700001659

    Article  MathSciNet  MATH  Google Scholar 

  2. Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 2003, 4: 79–89.

    MathSciNet  MATH  Google Scholar 

  3. Zhu CX, Yin JD: Calculations of a random fixed point index of a random semi-closed 1-set-contractive operator. Math. Comput. Model. 2010, 51: 1135–1139. 10.1016/j.mcm.2009.12.023

    Article  MathSciNet  MATH  Google Scholar 

  4. Zhu CX: Research on some problems for nonlinear operators. Nonlinear Anal. 2009, 71: 4568–4571. 10.1016/j.na.2009.03.014

    Article  MathSciNet  MATH  Google Scholar 

  5. Vetro C, Vetro F: Metric or partial metric spaces endowed with a finite number of graphs: a tool to obtain fixed point results. Topol. Appl. 2014, 164: 125–137.

    Article  MathSciNet  MATH  Google Scholar 

  6. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4

    Article  MathSciNet  MATH  Google Scholar 

  7. Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 2009, 71: 3403–3410. 10.1016/j.na.2009.01.240

    Article  MathSciNet  MATH  Google Scholar 

  8. Nashine HK: Cyclic generalized ψ -weakly contractive mappings and fixed point results with applications to integral equations. Nonlinear Anal. 2012, 75: 6160–6169. 10.1016/j.na.2012.06.021

    Article  MathSciNet  MATH  Google Scholar 

  9. Roshan JR, Parvaneh V, Altun I: Some coincidence point results in ordered b -metric spaces and applications in a system of integral equations. Appl. Math. Comput. 2014, 226: 725–737.

    Article  MathSciNet  Google Scholar 

  10. Shatanawi W, Postolache M: Common fixed point results for mappings under nonlinear contraction of cyclic form in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 60

    Google Scholar 

  11. Amini-Harandi A: Coupled and tripled fixed point theory in partially ordered metric spaces with applications to initial value problem. Math. Comput. Model. 2013, 57: 2343–2348. 10.1016/j.mcm.2011.12.006

    Article  MATH  Google Scholar 

  12. Esmaily J, Vaezpour SM, Rhoades BE: Coincidence and common fixed point theorems for a sequence of mappings in ordered metric spaces. Appl. Math. Comput. 2013, 219: 5684–5692. 10.1016/j.amc.2012.11.015

    Article  MathSciNet  MATH  Google Scholar 

  13. Esmaily J, Vaezpour SM, Rhoades BE: Coincidence point theorem for generalized weakly contractions in ordered metric spaces. Appl. Math. Comput. 2012, 219: 1536–1548. 10.1016/j.amc.2012.07.054

    Article  MathSciNet  MATH  Google Scholar 

  14. Agarwwal RP, Sintunavarat W, Kumam P: Coupled coincidence point and common coupled fixed point theorems lacking the mixed monotone property. Fixed Point Theory Appl. 2013., 2013: Article ID 22

    Google Scholar 

  15. Samet B, Vetro C, Vetro P: Fixed point theorems for α - ψ -contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014

    Article  MathSciNet  MATH  Google Scholar 

  16. Berzig M, Karapinar E:Fixed point results for (αψ,βφ)-contractive mappings for a generalized altering distance. Fixed Point Theory Appl. 2013., 2013: Article ID 205

    Google Scholar 

  17. Menger K: Statistical metrics. Proc. Natl. Acad. Sci. USA 1942, 28: 535–537. 10.1073/pnas.28.12.535

    Article  MathSciNet  MATH  Google Scholar 

  18. Schweizer B, Sklar A: Probabilistic Metric Spaces. North-Holland, Amsterdam; 1983.

    MATH  Google Scholar 

  19. Chang SS, Cho YJ, Kang SM: Probabilistic Metric Spaces and Nonlinear Operator Theory. Sichuan University Press, Chengdu; 1994.

    Google Scholar 

  20. Wu ZQ, Zhu CX, Li J: Common fixed point theorems for two hybrid pairs of mappings satisfying the common property (E.A) in Menger PM-spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 25

    Google Scholar 

  21. Choudhury BS, Das K: A new contraction principle in Menger spaces. Acta Math. Sin. 2008, 24: 1379–1386. 10.1007/s10114-007-6509-x

    Article  MathSciNet  MATH  Google Scholar 

  22. Ume JS: Common fixed point theorems for nonlinear contractions in a Menger space. Fixed Point Theory Appl. 2013., 2013: Article ID 166

    Google Scholar 

  23. Hu XQ, Ma XY: Coupled coincidence point theorems under contractive conditions in partially ordered probabilistic metric spaces. Nonlinear Anal. 2011, 74: 6451–6458. 10.1016/j.na.2011.06.028

    Article  MathSciNet  MATH  Google Scholar 

  24. Ćirić LB, Agarwal RP, Samet B: Mixed monotone generalized contractions in partially ordered probabilistic metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 56

    Google Scholar 

  25. Dorić D: Nonlinear coupled coincidence and coupled fixed point theorems for not necessary commutative contractive mapping in partially ordered probabilistic metric spaces. Appl. Math. Comput. 2013, 219: 5926–5935. 10.1016/j.amc.2012.12.047

    Article  MathSciNet  MATH  Google Scholar 

  26. Gopal D, Abbas M, Vetro C: Some new fixed point theorems in Menger PM-spaces with application to Volterra type integral equation. Appl. Math. Comput. 2014, 232: 955–967.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the editor and the referees for their valuable comments and suggestions. The research was supported by the National Natural Science Foundation of China (11361042, 11071108, 11326099, 11461045) and the Provincial Natural Science Foundation of Jiangxi, China (20132BAB201001, 2010GZS0147, 20142BAB211016).

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Chuanxi Zhu or Wenqing Xu.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as 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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, C., Xu, W. & Wu, Z. Coincidence point theorems for generalized cyclic ( κ h , φ L ) S -weak contractions in partially ordered Menger PM-spaces. Fixed Point Theory Appl 2014, 214 (2014). https://doi.org/10.1186/1687-1812-2014-214

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1812-2014-214

Keywords