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Fixed point theorems for a generalized almost (ϕ,φ)-contraction with respect to S in ordered metric spaces

Abstract

In this paper, the existence theorems of fixed points and common fixed points for two weakly increasing mappings satisfying a new condition in ordered metric spaces are proved. Our results extend, generalize and unify most of the fundamental metrical fixed point theorems in the literature.

MSC:47H09, 47H10.

1 Introduction and preliminaries

The classical Banach contraction principle is one of the most useful results in nonlinear analysis. In a metric space, the full statement of the Banach contraction principle is given by the following theorem.

Theorem 1.1 Let(X,d)be a complete metric space andT:XX. If T satisfies

d(Tx,Ty)kd(x,y)
(1.1)

for allx,yX, wherek[0,1), then T has a unique fixed point.

Due to its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis and its has many applications in solving nonlinear equations. Then, several authors studied and extended it in many direction; for example, see [119] and the references therein.

Despite these important features, Theorem 1.1 suffers from one drawback: the contractive condition (1.1) forces T to be continuous on X. It was then natural to ask if there exist weaker contractive conditions which do not imply the continuity of T. In 1968, this question was answered in confirmation by Kannan [20], who extended Theorem 1.1 to mappings that need not be continuous on X (but are continuous at their fixed point, see [21]).

On the other hand, Sessa [22] introduced the notion of weakly commuting mappings, which are a generalization of commuting mappings, while Jungck [23] generalized the notion of weak commutativity by introducing compatible mappings and then weakly compatible mappings [24].

In 2004, Berinde [25] defined the notion of a weak contraction mapping which is more general than a contraction mapping. However, in [26] Berinde renamed it as an almost contraction mapping, which is more appropriate. Berinde [25] proved some fixed point theorems for almost contractions in complete metric spaces. Afterward, many authors have studied this problem and obtained significant results (see [2736]). Moreover, in [25] Berinde proved that any strict contraction, the Kannan [26] and Zamfirescu [37] mappings as well as a large class of quasi-contractions are all almost contractions.

Let T and S be two self mappings in a metric space (X,d). The mapping T is said to be a S-contraction if there exists k[0,1) such that d(Tx,Ty)kd(Sx,Sy) for all x,yX.

In 2006, Al-Thagafi and Shahzad [38] proved the following theorem which is a generalization of many known results.

Theorem 1.2 ([[38], Theorem 2.1])

Let E be a subset of a metric space(X,d)and S, T be two selfmaps of E such thatT(E)S(E). Suppose that S and T are weakly compatible, T is an S-contraction andS(E)is complete. Then S and T have a unique common fixed point in E.

Recently Babu et al.[39] defined the class of mappings satisfying condition (B) as follows.

Definition 1.3 Let (X,d) be a metric space. A mapping T:XX is said to satisfy condition (B) if there exist a constant δ(0,1) and some L0 such that

d(Tx,Ty)δd(x,y)+Lmin { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) }
(1.2)

for all x,yX.

They proved a fixed point theorem for such mappings in complete metric spaces. They also discussed quasi-contraction, almost contraction and the class of mappings that satisfy condition (B) in detail.

In recent year, Ćirić et al.[40] defined the following class of mappings satisfying an almost generalized contractive condition.

Definition 1.4 Let (X,d) be a metric space, and let S,T:XX. A mapping T is called an almost generalized contraction if there exist δ[0,1) and L0 such that

d(Tx,Sy)δM(x,y)+Lmin { d ( x , T x ) , d ( y , S y ) , d ( x , S y ) , d ( y , T x ) }
(1.3)

for all x,yX, where M(x,y)=max{d(x,y),d(x,Tx),d(y,Sy), d ( x , S y ) + d ( y , T x ) 2 }.

Definition 1.5 Let (X,) be a partial ordered set. We say that x,yX are comparable if xy or yx holds.

Definition 1.6 Let (X,) be a partial ordered set. A mapping T:XX is said to be nondecreasing if TxTy, whenever x,yX and xy.

Definition 1.7 Let (X,) be a partial ordered set. Two mappings S,T:XX are said to be strictly increasing if Sx<TSx and Tx<STx for all xX.

In 2004, Ran and Reurings [41] proved the following result.

Theorem 1.8 Let(X,)be a partially ordered set such that every pair{x,y}Xhas a lower and an upper bound. Suppose that d is a complete metric on X. LetT:XXbe a continuous and monotone mapping. Suppose that there existsδ[0,1)such thatd(Tx,Ty)δd(x,y)for all comparablex,yX. If there exists x 0 Xsuch that x 0 T x 0 , then T has a unique fixed pointpX.

Ćirić et al. in [40] established fixed point and common fixed point theorems which are more general than Theorem 1.8 and several comparable results in the existing literature regarding the existence of a fixed point in ordered spaces.

In this paper, we introduce a new class which extends and unifies mappings satisfying the almost generalized contractive condition and establish the result on the existence of fixed points and common fixed points in a complete ordered space. This result substantially generalizes, extends and unifies the main results of Ćirić et al. [[40], Theorems 2.1, 2.2, 2.3, 2.6], Theorem 1.8, and several comparable results in the existing literature regarding the existence of a fixed and a common fixed point in ordered spaces.

2 Fixed point theorems for a generalized almost (ϕ,φ)-contraction

First we introduce the notion of generalized almost (ϕ,φ)-contraction mappings.

Definition 2.1 Let (X,) be a partially ordered set, and let a metric d exist on X. A mapping T:XX is called a generalized almost (ϕ,φ)-contraction if there exist two mappings ϕ:X[0,1) which ϕ(Tx)ϕ(x) and φ:X[0,) such that

d(Tx,Ty)ϕ(x)M(x,y)+φ(x)min { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) }
(2.1)

for all comparable x,yX, where M(x,y)=max{d(x,y),d(x,Tx),d(y,Ty), d ( x , T y ) + d ( y , T x ) 2 }.

Theorem 2.2 Let(X,)be a partially ordered set, and let a complete metric d exist on X. LetT:XXbe a strictly increasing continuous mapping with respect toand a generalized almost(ϕ,φ)-contraction mapping. If there exists x 0 Xsuch that x 0 T x 0 , then T has a unique fixed point in X.

Proof If T x 0 = x 0 , then x 0 is fixed of T and we finish the proof. Now, we may assume that T x 0 x 0 , that is, x 0 <T x 0 . We construct the sequence { x n } in X by

x n + 1 = T n + 1 x 0 =T x n
(2.2)

for all n0. Since T is strictly increasing, we have that T x 0 < T 2 x 0 << T n x 0 < T n + 1 x 0 < . Thus, x 1 < x 2 << x n < x n + 1 < , which implies that the sequence { x n } is strictly increasing. Note that

(2.3)

Since x n and x n + 1 are comparable, we get

d ( x n + 1 , x n + 2 ) = d ( T x n , T x n + 1 ) ϕ ( x n ) M ( x n , x n + 1 ) + φ ( x n ) min { d ( x n , T x n ) , d ( x n + 1 , T x n + 1 ) , d ( x n , T x n + 1 ) , d ( x n + 1 , T x n ) } = ϕ ( x n ) max { d ( x n , x n + 1 ) , d ( x n + 1 , x n + 2 ) } + φ ( x n ) min { d ( x n , x n + 1 ) , d ( x n + 1 , x n + 2 ) , d ( x n , x n + 2 ) , d ( x n + 1 , x n + 1 ) } = ϕ ( x n ) max { d ( x n , x n + 1 ) , d ( x n + 1 , x n + 2 ) } .
(2.4)

If for some nN, max{d( x n , x n + 1 ),d( x n + 1 , x n + 2 )}=d( x n + 1 , x n + 2 ), then we get

d( x n + 1 , x n + 2 )ϕ( x n )d( x n + 1 , x n + 2 )<d( x n + 1 , x n + 2 ),
(2.5)

which is a contradiction. Thus, we have

max { d ( x n , x n + 1 ) , d ( x n + 1 , x n + 2 ) } =d( x n , x n + 1 )
(2.6)

for all nN. Therefore, from (2.4), we have

d( x n + 1 , x n + 2 )ϕ( x n )d( x n , x n + 1 )

for all nN. Hence,

d ( x n + 1 , x n + 2 ) ϕ ( x n ) d ( x n , x n + 1 ) ϕ ( x n ) ϕ ( x n 1 ) d ( x n 1 , x n ) ϕ ( x n ) ϕ ( x n 1 ) ϕ ( x 1 ) ϕ ( x 0 ) d ( x 0 , x 1 ) = ϕ ( T n x 0 ) ϕ ( T n 1 x 0 ) ϕ ( T x 0 ) ϕ ( x 0 ) d ( x 0 , x 1 ) ϕ ( T n 1 x 0 ) ϕ ( T n 2 x 0 ) ϕ ( x 0 ) ϕ ( x 0 ) d ( x 0 , x 1 ) ϕ ( x 0 ) ϕ ( x 0 ) ϕ ( x 0 ) ϕ ( x 0 ) ( n + 1 ) - term d ( x 0 , x 1 ) = ( ϕ ( x 0 ) ) n + 1 d ( x 0 , x 1 )
(2.7)

for all n1. Now, for positive integers m and n with m>n, we have

d ( x n , x m ) d ( x n , x n + 1 ) + d ( x n + 1 , x n + 2 ) + + d ( x m 1 , x m ) ( ϕ ( x 0 ) ) n d ( x 0 , x 1 ) + ( ϕ ( x 0 ) ) n + 1 d ( x 0 , x 1 ) + + ( ϕ ( x 0 ) ) m 1 d ( x 0 , x 1 ) = ( ( ϕ ( x 0 ) ) n + ( ϕ ( x 0 ) ) n + 1 + + ( ϕ ( x 0 ) ) m 1 ) d ( x 0 , x 1 ) ( ( ϕ ( x 0 ) ) n 1 ϕ ( x 0 ) ) d ( x 0 , x 1 ) .
(2.8)

Since ϕ( x 0 )[0,1), if we take the limit as m,n0, then d( x n , x m )0, which implies that { x n } is a Cauchy sequence. Since X is complete, there exists a zX such that x n z as n. It follows from the continuity of T that T( T n x 0 )= T n + 1 x 0 z implies that Tz=z. Therefore, z is a fixed point of T. □

Example 2.3 Let X=[0,1], the partial order ≤ be defined by ab if and only if ba[0,) and d be the usual metric on X. Let T:XX be defined by Tx= x 2 4 for all xX. It can be easily checked that T is a generalized almost (ϕ,φ)-contraction mapping with ϕ(x)= x + 1 4 and φ(x)=lnx. Moreover, T is strictly increasing on X, and there exists 0X such that 0T0. Therefore, T satisfies the conditions of Theorem 2.2 and thus T has a fixed point 0.

The following corollaries follow immediately from the Theorem 2.2 with ϕ(x)=δ and φ(x)=L.

Corollary 2.4 ([[40], Theorem 2.1])

Let(X,)be a partially ordered set, and let a complete metric d exist on X. LetT:XXbe a strictly increasing continuous mapping with respect to ≤. Suppose that there existδ[0,1)andL0such that

d(Tx,Ty)δM(x,y)+Lmin { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) }
(2.9)

for all comparablex,yX, whereM(x,y)=max{d(x,y),d(x,Tx),d(y,Ty), d ( x , T y ) + d ( y , T x ) 2 }. If there exists x 0 Xsuch that x 0 T x 0 , then T has a fixed point in X.

Corollary 2.5 Let(X,)be a partially ordered set, and let a complete metric d exist on X. LetT:XXbe a strictly increasing continuous mapping with respect to ≤. Assume that there exist two mappingsϕ:X[0,1)withϕ(Tx)ϕ(x)andφ:X[0,)such that

d(Tx,Ty)ϕ(x)M(x,y)+φ(x)min { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) }
(2.10)

for all comparablex,yX, whereM(x,y)=max{d(x,y), d ( x , T x ) + d ( y , T y ) 2 , d ( x , T y ) + d ( y , T x ) 2 }. If there exists x 0 Xsuch that x 0 T x 0 , then T has a fixed point in X.

Proof Since (2.10) is a special case of the generalized almost (ϕ,φ)-contraction, the result follows from Theorem 2.2. □

Corollary 2.6 Let(X,)be a partially ordered set, and let a complete metric d exist on X. LetT:XXbe a strictly increasing continuous mapping with respect to ≤. Assume that there exist two mappingsϕ:X[0,1)withϕ(Tx)ϕ(x)andφ:X[0,)such that

d(Tx,Ty)ϕ(x)d(x,y)+φ(x)min { d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) }
(2.11)

for all comparablex,yX. If there exists x 0 Xsuch that x 0 T x 0 , then T has a fixed point in X.

Proof Since (2.11) is a special case of the generalized almost (ϕ,φ)-contraction, the result follows from Theorem 2.2. □

Theorem 2.7 Let(X,)be a partially ordered set, and let a complete metric d exist on X. LetT:XXbe a strictly increasing mapping with respect toand a generalized almost(ϕ,φ)-contraction mapping with a continuous mapping ϕ. If there exists x 0 Xsuch that x 0 T x 0 and for an increasing sequence{ x n }in X converging toxXwe have lim n φ( x n )<and x n xfor allnN, then T has a fixed point in X.

Proof Suppose that Txx for all xX. Following similar arguments to those given in Theorem 2.2, we obtain an increasing sequence { x n } such that x n z as n for some zX. By the given hypothesis, we have x n z for all nN. Therefore,

d ( x n + 1 , T z ) = d ( T x n , T z ) ϕ ( x n ) max { d ( x n , z ) , d ( x n , T x n ) , d ( z , T z ) , d ( x n , T z ) + d ( z , T x n ) 2 } + φ ( x n ) min { d ( x n , T x n ) , d ( z , T z ) , d ( x n , T z ) , d ( z , T x n ) } = ϕ ( x n ) max { d ( x n , z ) , d ( x n , x n + 1 ) , d ( z , T z ) , d ( x n , T z ) + d ( z , x n + 1 ) 2 } + φ ( x n ) min { d ( x n , x n + 1 ) , d ( z , T z ) , d ( x n , T z ) , d ( z , x n + 1 ) } .
(2.12)

Taking the limit as n, we get that d(z,Tz)ϕ(z)d(z,Tz). Since ϕ(z)[0,1), we have d(z,Tz)=0, that is, Tz=z, which is a contradiction. Therefore, there exists z 1 X such that T z 1 = z 1 which implies that z 1 is a fixed point of T. □

3 Common fixed point theorems for a generalized almost (ϕ,φ)-contraction with respect to S

Definition 3.1 Let (X,) be a partially ordered set, and let a metric d exist on X, and S,T:XX. A mapping T is called a generalized almost (ϕ,φ)-contraction with respect to S if there exist two mappings ϕ:X[0,1) with ϕ(Sx)ϕ(x) and ϕ(Tx)ϕ(x) and φ:X[0,) such that

d(Tx,Sy)ϕ(x)M(x,y)+φ(x)min { d ( x , T x ) , d ( y , S y ) , d ( x , S y ) , d ( y , T x ) }
(3.1)

for all comparable x,yX, where M(x,y)=max{d(x,y),d(x,Tx),d(y,Sy), d ( x , S y ) + d ( y , T x ) 2 }.

Remark 3.2 If we take ϕ(x)=δ where δ[0,1) and φ(x)=L where L[0,), then the generalized almost (ϕ,φ)-contraction with respect to S reduces to an almost generalized contraction of Ćirić et al. in [40].

The following theorem deals with the existence of a common fixed point of two weakly increasing mappings which is more general and covers more than the result of Ćirić et al. in [40].

Theorem 3.3 Let(X,)be a partially ordered set, and let a complete metric d exist on X. LetS,T:XXbe two strictly weakly increasing mappings with respect toand T be a generalized almost(ϕ,φ)-contraction with respect to S. If either S or T is continuous, then there exists a common fixed point of S and T in X.

Proof Let x 0 be an arbitrary point in X. We construct the sequence { x n } in X such that

x 2 n + 1 =T x 2 n and x 2 n + 2 =S x 2 n + 1
(3.2)

for all n0. Since S and T are strictly weakly increasing, we get

x 2 n + 1 =T x 2 n <ST x 2 n =S x 2 n + 1 = x 2 n + 2
(3.3)

and

x 2 n + 2 =S x 2 n + 1 <TS x 2 n + 1 =T x 2 n + 2 = x 2 n + 3
(3.4)

for all n0. Therefore, x 1 < x 2 << x n < x n + 1 < , that is, the sequence { x n } is strictly increasing. Note that

(3.5)

for all n0. Since x 2 n and x 2 n + 1 are comparable, we get

(3.6)

If for some nN, max{d( x 2 n , x 2 n + 1 ),d( x 2 n + 1 , x 2 n + 2 )}=d( x 2 n + 1 , x 2 n + 2 ), then we get

d( x 2 n + 1 , x 2 n + 2 )ϕ( x 2 n )d( x 2 n + 1 , x 2 n + 2 )<d( x 2 n + 1 , x 2 n + 2 ),
(3.7)

which is a contradiction. Thus,

max { d ( x 2 n , x 2 n + 1 ) , d ( x 2 n + 1 , x 2 n + 2 ) } =d( x 2 n , x 2 n + 1 )
(3.8)

for all nN. Therefore, from (3.6), we have

d( x 2 n + 1 , x 2 n + 2 )ϕ( x 2 n )d( x 2 n , x 2 n + 1 )

for all nN. Similarly, it can be proved that

d( x 2 n + 3 , x 2 n + 2 )ϕ( x 2 n + 2 )d( x 2 n + 2 , x 2 n + 1 )

for all nN. It follows from the proof of Theorem 2.2 that

d( x n + 1 , x n + 2 ) ( ϕ ( x 0 ) ) n + 1 d( x 0 , x 1 )
(3.9)

for all n1. Similarly to the proof of Theorem 2.2, we get { x n } is a Cauchy sequence. It follows from the completeness of X that there exists a zX such that x n z as n. Suppose that T is continuous, then T( T n x 0 )= T n + 1 x 0 z. This implies that Tz=z. Therefore, z is a fixed point of T. Suppose that Szz. Since zz. Therefore,

d ( z , S z ) = d ( T z , S z ) ϕ ( z ) max { d ( z , z ) , d ( z , T z ) , d ( z , S z ) , d ( z , S z ) + d ( z , T z ) 2 } + φ ( z ) min { d ( z , T z ) , d ( z , S z ) , d ( z , S z ) , d ( z , T z ) } ϕ ( z ) d ( z , S z ) ,
(3.10)

which is a contradiction. Therefore, Sz=z and then z is a common fixed point of S and T. Similarly, it can be proved that S and T have a common fixed point if S is continuous. □

The following corollaries are a generalization and extension of Corollaries 2.4 and 2.5 in Ćirić et al. in [40].

Corollary 3.4 Let(X,)be a partially ordered set, and let a complete metric d exist on X. LetS,T:XXbe two strictly weakly increasing mappings with respect to ≤. Assume that there exist two mappingsϕ:X[0,1)withϕ(Sx)ϕ(x)andϕ(Tx)ϕ(x)andφ:X[0,)such that

d(Tx,Sy)ϕ(x)M(x,y)+φ(x)min { d ( x , T x ) , d ( y , S y ) , d ( x , S y ) , d ( y , T x ) }
(3.11)

for all comparablex,yX, whereM(x,y)=max{d(x,y), d ( x , T x ) + d ( y , S y ) 2 , d ( x , S y ) + d ( y , T x ) 2 }. If either S or T is continuous, then there exists a common fixed point of S and T in X.

Proof Since (3.11) is a special case of the generalized almost (ϕ,φ)-contraction with respect to S, the result follows from Theorem 3.3. □

Corollary 3.5 Let(X,)be a partially ordered set, and let a complete metric d exist on X. LetS,T:XXbe two strictly increasing mappings with respect to ≤. Assume that there exist two mappingsϕ:X[0,1)withϕ(Sx)ϕ(x)andϕ(Tx)ϕ(x)andφ:X[0,)such that

d(Tx,Sy)ϕ(x)d(x,y)+φ(x)min { d ( x , T x ) , d ( y , S y ) , d ( x , S y ) , d ( y , T x ) }
(3.12)

for all comparablex,yX. If either S or T is continuous, then there exists a common fixed point of S and T in X.

Proof Since (3.12) is a special case of the generalized almost (ϕ,φ)-contraction with respect to S, the result follows from Theorem 3.3. □

Now, we have the following result of the continuity on the set of common fixed points. Let F(S,T) denote the set of all common fixed points of S and T.

Theorem 3.6 Let(X,)be a partially ordered set, and let a complete metric d exist on X. LetS,T:XXand T be a generalized almost(ϕ,φ)-contraction mapping with respect to S. IfF(S,T)and for any sequence{ x n }in X with x n xasnfor somexX, we have x n xfor allnN, then S and T are continuous atzF(S,T).

Proof Fix zF(S,T). Let { x n } be any sequence in X converging to z and then x n z for all nN. From the notion of a generalized almost (ϕ,φ)-contraction with respect to S, we get

d ( T z , S x n ) ϕ ( z ) max { d ( z , x n ) , d ( z , T z ) , d ( x n , S x n ) , d ( z , S x n ) + d ( x n , T z ) 2 } + φ ( z ) min { d ( z , T z ) , d ( x n , S x n ) , d ( z , S x n ) , d ( x n , T z ) } = ϕ ( z ) max { d ( z , x n ) , 0 , d ( x n , S x n ) , d ( z , S x n ) + d ( x n , z ) 2 } + φ ( z ) min { 0 , d ( x n , S x n ) , d ( z , S x n ) , d ( x n , T z ) } = ϕ ( z ) max { d ( z , x n ) , d ( x n , S x n ) , d ( z , S x n ) + d ( x n , z ) 2 }
(3.13)

for all n1. Letting n, we have S x n z=Sz. Therefore, S is continuous at zF(S,T). Similarly, it can be shown that T is continuous at zF(S,T). □

Remark 3.7 In fixed point theory, after the remarkable paper of Huang and Zhang [42], cone metric spaces have been considered by several authors. The results and theorems in this paper can be also generalized to cone metric spaces.

References

  1. Chis A: Fixed point theorems for multivalued generalized contractions on complete gauge spaces. Carpath. J. Math. 2006, 22(1–2):33–38.

    MathSciNet  Google Scholar 

  2. Ćirić LB: On contraction type mappings. Math. Balk. 1971, 1: 52–57.

    Google Scholar 

  3. Guran L: Fixed points for multivalued operators with respect to a w -distance on metric spaces. Carpath. J. Math. 2007, 23(1–2):89–92.

    MathSciNet  Google Scholar 

  4. Rhoades BE: A biased discussion of fixed point theory. Carpath. J. Math. 2007, 23(1–2):11–26.

    MathSciNet  Google Scholar 

  5. Rakotch E: A note on contractive mappings. Proc. Am. Math. Soc. 1962, 13: 459–465. 10.1090/S0002-9939-1962-0148046-1

    MathSciNet  Article  Google Scholar 

  6. Reich S: Kannan’s fixed point theorem. Boll. Unione Mat. Ital. 1971, 4: 1–11.

    Google Scholar 

  7. Reich S: Some remarks concerning contraction mappings. Can. Math. Bull. 1971, 14: 121–124. 10.4153/CMB-1971-024-9

    Article  Google Scholar 

  8. Reich S: Fixed point of contractive functions. Boll. Unione Mat. Ital. 1972, 5: 26–42.

    Google Scholar 

  9. Reich S, Zaslavski AJ: Approximating fixed points of contractive set-valued mappings. Commun. Math. Anal. 2010, 8: 70–78.

    MathSciNet  Google Scholar 

  10. Reich S, Zaslavski AJ: The set of noncontractive mappings is σ -porous in the space of all nonexpansive mappings. C. R. Acad. Sci., Sér. I Math. 2001, 333: 539–544.

    MathSciNet  Google Scholar 

  11. Reich S, Zaslavski AJ: A note on Rakotch contraction. Fixed Point Theory 2008, 9: 267–273.

    MathSciNet  Google Scholar 

  12. Rus IA: Generalized contractions. Semin. Fixed Point Theory 1983, 3: 1–130.

    MathSciNet  Google Scholar 

  13. Sintunavarat W, Kumam P:Weak condition for generalized multi-valued (f,α,β)-weak contraction mappings. Appl. Math. Lett. 2011, 24: 460–465. 10.1016/j.aml.2010.10.042

    MathSciNet  Article  Google Scholar 

  14. Sintunavarat W, Kumam P: Gregus type fixed points for a tangential multi-valued mappings satisfying contractive conditions of integral type. J. Inequal. Appl. 2011., 2011: Article ID 3

    Google Scholar 

  15. Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c -distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040

    MathSciNet  Article  Google Scholar 

  16. Sintunavarat W, Kumam P: Common fixed point theorems for hybrid generalized multi-valued contraction mappings. Appl. Math. Lett. 2012, 25: 52–57. 10.1016/j.aml.2011.05.047

    MathSciNet  Article  Google Scholar 

  17. Sintunavarat W, Kumam P:Common fixed point theorems for generalized JH-operator classes and invariant approximations. J. Inequal. Appl. 2011., 2011: Article ID 67

    Google Scholar 

  18. Sintunavarat W, Kumam P: Fixed point theorems for a generalized intuitionistic fuzzy contraction in intuitionistic fuzzy metric spaces. Thai J. Math. 2012, 10(1):123–135.

    MathSciNet  Google Scholar 

  19. Sintunavarat W, Kumam P: Generalized common fixed point theorems in complex valued metric spaces and applications. J. Inequal. Appl. 2012., 2012: Article ID 84

    Google Scholar 

  20. Kannan R: Some results on fixed points. Bull. Calcutta Math. Soc. 1968, 10: 71–76.

    Google Scholar 

  21. Rhoades BE: Contractive definitions and continuity. Contemp. Math. 1988, 72: 233–245.

    MathSciNet  Article  Google Scholar 

  22. Sessa S: On a weak commutativity condition of mappings in fixed point consideration. Publ. Inst. Math. 1982, 32: 149–153.

    MathSciNet  Google Scholar 

  23. Jungck G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 1986, 9(4):771–779. 10.1155/S0161171286000935

    MathSciNet  Article  Google Scholar 

  24. Jungck G: Common fixed points for noncontinuous nonself maps on nonmetric spaces. Far East J. Math. Sci. 1996, 4: 199–215.

    MathSciNet  Google Scholar 

  25. Berinde V: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 2004, 9(1):43–53.

    MathSciNet  Google Scholar 

  26. Berinde V: General constructive fixed point theorems for Ćirić-type almost contractions in metric spaces. Carpath. J. Math. 2008, 24(2):10–19.

    MathSciNet  Google Scholar 

  27. Beg I, Abbas M: Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition. Fixed Point Theory Appl. 2006., 2006: Article ID 74503

    Google Scholar 

  28. Berinde V: Some remarks on a fixed point theorem for Ćirić-type almost contractions. Carpath. J. Math. 2009, 25(2):157–162.

    MathSciNet  Google Scholar 

  29. Berinde V: Common fixed points of noncommuting almost contractions in cone metric spaces. Math. Commun. 2010, 15(1):229–241.

    MathSciNet  Google Scholar 

  30. Berinde V: Approximating common fixed points of noncommuting almost contractions in metric spaces. Fixed Point Theory 2010, 11(2):179–188.

    MathSciNet  Google Scholar 

  31. Berinde V: Approximating common fixed points of noncommuting discontinuous weakly contractive mappings in metric spaces. Carpath. J. Math. 2009, 25(1):13–22.

    MathSciNet  Google Scholar 

  32. Ćirić LB: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 1974, 45: 267–273.

    Google Scholar 

  33. Ćirić L, Hussain N, Cakić N: Common fixed points for Ćirić type f -weak contraction with applications. Publ. Math. (Debr.) 2010, 76(1–2):31–49.

    Google Scholar 

  34. Ćirić L, Rakočević V, Radenović S, Rajović M, Lazović R: Common fixed point theorems for non-self mappings in metric spaces of hyperbolic type. J. Comput. Appl. Math. 2010, 233: 2966–2974. 10.1016/j.cam.2009.11.042

    MathSciNet  Article  Google Scholar 

  35. Pacurar M: Remark regarding two classes of almost contractions with unique fixed point. Creat. Math. Inform. 2010, 19(2):178–183.

    MathSciNet  Google Scholar 

  36. Pacurar M: Iterative Methods for Fixed Point Approximation. Risoprint, Cluj-Napoca; 2009.

    Google Scholar 

  37. Zamfirescu T: Fixed point theorems in metric spaces. Arch. Math. 1972, 23: 292–298. 10.1007/BF01304884

    MathSciNet  Article  Google Scholar 

  38. Al-Thagafi MA, Shahzad N: Noncommuting selfmaps and invariant approximations. Nonlinear Anal. 2006, 64: 2777–2786.

    MathSciNet  Article  Google Scholar 

  39. Babu GVR, Sandhya ML, Kameswari MVR: A note on a fixed point theorem of Berinde on weak contractions. Carpath. J. Math. 2008, 24(1):8–12.

    MathSciNet  Google Scholar 

  40. Ćirić L, Abbas M, Saada R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 2011, 217: 5784–5789. 10.1016/j.amc.2010.12.060

    MathSciNet  Article  Google Scholar 

  41. 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

    MathSciNet  Article  Google Scholar 

  42. Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 2007, 332: 1468–1476. 10.1016/j.jmaa.2005.03.087

    MathSciNet  Article  Google Scholar 

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Acknowledgements

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission. The first author would like to thank the Research Professional Development Project Under the Science Achievement Scholarship of Thailand (SAST) for financial support during the preparation of this manuscript. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042138). This project was partially completed during the first and third authors’ last visit to Kyungnam University.

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Correspondence to Poom Kumam.

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Sintunavarat, W., Kim, J.K. & Kumam, P. Fixed point theorems for a generalized almost (ϕ,φ)-contraction with respect to S in ordered metric spaces. J Inequal Appl 2012, 263 (2012). https://doi.org/10.1186/1029-242X-2012-263

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Keywords

  • coincidence points
  • common fixed points
  • generalized multi-valued f-weak contraction
  • generalized multi-valued (f,α,β)-weak contraction
  • T-weakly commuting