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Iterative approximation of fixed points of quasi-contraction mappings in cone metric spaces

Journal of Inequalities and Applications20142014:226

https://doi.org/10.1186/1029-242X-2014-226

Received: 19 February 2014

Accepted: 23 May 2014

Published: 3 June 2014

Abstract

In this paper, we establish a full statement of Ćirić’s fixed point theorem in the setting of cone metric spaces. More exactly, we obtain a priori and a posteriori error estimates for approximating fixed points of quasi-contractions in a cone metric space. Our result complements recent results of Zhang (Comput. Math. Appl. 62:1627-1633, 2011), Ding et al. (J. Comput. Anal. Appl. 15:463-470, 2013) and others.

MSC: Primary 54H25; secondary 47H10, 46A19.

Keywords

Picard iterationcone metric spacesolid vector spacefixed pointsquasi-contractionserror estimates

1 Introduction

In this paper, we study fixed points of quasi-contraction mappings in a cone metric space ( X , d ) over a solid vector space ( Y , ) . Cone metric spaces have a long history (see Collatz [1], Zabrejko [2], Janković et al. [3], Proinov [4] and references therein). A unified theory of cone metric spaces over a solid vector space was developed in a recent paper of Proinov [4]. Recall that an ordered vector space with convergence structure ( Y , ) is called:

  • a solid vector space if it can be endowed with a strict vector ordering ( ) ;

  • a normal vector space if the convergence of Y has the sandwich property.

Every metric space ( X , d ) is a cone metric space over (with usual ordering and usual convergence). On the other hand, every cone metric space over a solid vector space is a metrizable topological space (see Proinov [4] and references therein). It is well known that a lot of fixed point results in cone metric setting can be directly obtained from their metric versions (see Du [5], Amini-Harandi and Fakhar [6], Feng and Mao [7], Kadelburg et al. [8], Asadi et al. [9], Proinov [4], and Ercan [10]).

For instance, for this purpose we can use the following theorem. This theorem shows that every cone metric is equivalent to a metric which preserves the completeness as well as some inequalities.

Theorem 1.1 ([[4], Theorem 9.3])

Let ( X , d ) be a cone metric space over a solid vector space ( Y , ) . Then there exists a metric ρ on X such that the following statements hold true.
  1. (i)

    The topology of ( X , d ) coincides with the topology of ( X , ρ ) .

     
  2. (ii)

    ( X , d ) is complete if and only if ( X , ρ ) is complete.

     
  3. (iii)
    For x , x 1 , , x n X , y , y 1 , , y n X and λ 1 , , λ n R ,
    d ( x , y ) i = 1 n λ i d ( x i , y i ) implies ρ ( x , y ) i = 1 n λ i ρ ( x i , y i ) .
     

In 1922, Banach [11] proved his famous fixed point theorem for contraction mappings. Banach’s contraction principle is one of the most useful theorems in the fixed point theory. It has two versions: a short version and a full version. In a metric space setting its full statement can be seen, for example, in the monograph of Berinde [[12], Theorem 2.1]. Recently, full statements of Banach’s fixed point theorem in a cone metric spaces over a solid vector space were given by Radenović and Kadelburg [[13], Theorem 3.3] and Proinov [[4], Theorem 11.1].

Definition 1.2 ([14])

Let ( X , d ) be a metric space. A mapping T : X X is called a quasi-contraction (with contraction constant λ) if there exists λ [ 0 , 1 ) such that
d ( T x , T y ) λ max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) }
(1)

for all x , y X .

There are a large number of generalizations of Banach’s contraction principle (see, for example, [1418] and references therein). In 1974, Ćirić [14] introduced contraction mappings and proved the following well known generalization of Banach’s fixed point theorem.

Theorem 1.3 Let ( X , d ) be a complete metric space and T : X X be a quasi-contraction with contraction constant λ. Then the following statements hold true:
  1. (i)

    Existence and uniqueness. T has a unique fixed point ξ in X.

     
  2. (ii)

    Convergence of Picard iteration. For every starting point x X the Picard iteration sequence ( T n x ) converges to ξ.

     
  3. (iii)
    A priori error estimate. For every point x X the following a priori error estimate holds:
    d ( T n x , ξ ) λ n 1 λ d ( x , T x ) for all  n 0 .
    (2)
     

Following Zhang [19], in the next definition, we define a useful binary relation between an ordered vector space Y and the set of all subsets of Y. It plays a very important role in this paper as it is used to prove our main result.

Definition 1.4 ([19])

Let ( Y , ) be an ordered vector space, x Y and A Y . We say that x A if there exists at least one vector y A such that x y .

In 2009, Ilić and Rakočević [20] generalized the concept of quasi-contraction to cone metric space as follows: A selfmapping T of a cone metric space ( X , d ) over an ordered vector space ( Y , ) is called a quasi-contraction on X if there exists λ [ 0 , 1 ) such that
d ( T x , T y ) λ { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) }
(3)

for all x , y X . They proved the following result [[20], Theorem 2.1]: Let ( X , d ) be a cone metric space over a normal solid Banach space ( Y , ) ; then every quasi-contraction T of the type (3) has a unique fixed point in X, and for all x X the Picard iterative sequence ( T n x ) converges to this fixed point. Kadelburg et al. [[21], Theorem 2.2] improved this result by omitting the assumption of normality provided that λ [ 0 , 1 / 2 ) . Gajić and Rakočević [[22], Theorem 3] proved this result for any contraction constant λ [ 0 , 1 ) . Rezapour et al. [[23], Theorem 2.1] proved this result in the case when Y is a solid topological vector space and λ [ 0 , 1 ) . Furthermore, Kadelburg et al. [[8], Theorem 3.5(b)] proved that this result is equivalent to the short version of Ćirić’s fixed point theorem.

In 2011, Zhang [19] presented the following new definition for quasi-contractions in cone metric spaces.

Definition 1.5 ([19])

Let ( X , d ) be a cone metric space over an ordered vector space ( Y , ) . A mapping T : X X is called a quasi-contraction (with contraction constant λ) if there exists λ [ 0 , 1 ) such that
d ( T x , T y ) λ co { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) }
(4)

for all x , y X .

By applying Theorem 1.1 to the first two conclusions of Theorem 1.3, we obtain the following fixed point theorem in a cone metric setting.

Theorem 1.6 Let ( X , d ) be a complete cone metric space over a solid vector space ( Y , ) and T : X X be a quasi-contraction. Then the following statements hold true:
  1. (i)

    Existence and uniqueness. T has a unique fixed point ξ in X.

     
  2. (ii)

    Convergence of Picard iteration. For every starting point x X the Picard iteration sequence ( T n x ) converges to ξ.

     

In 2011, Zhang [19] proved Theorem 1.6 in the case when Y is a normal solid Banach space. In 2013, Ding et al. [24] proved this theorem in the case when Y is a solid topological vector space.

In this paper, we establish a full statement of Ćirić’s fixed point theorem in the setting of cone metric spaces. Our result complements Theorem 1.6. Thus it extends and complements the corresponding results of Zhang [19], Ding et al. [24], and others.

For some recent results on the topic, we refer the reader to [2538]. In the papers [37, 38] one can find some applications of cone metric spaces to iterative methods for finding all zeros of polynomial simultaneously.

2 Preliminaries

In this section, we introduce some basic definitions and theorems of cone metric spaces over a solid vector space.

Definition 2.1 ([4])

Let Y be a real vector space and S be the set of all infinite sequences in Y. A binary relation → between S and Y is called a convergence on Y if it satisfies the following axioms:

(C1) If x n x and y n y , then x n + y n x + y .

(C2) If x n x and λ R , then λ x n λ x .

(C3) If λ n λ in and x Y , then λ n x λ x .

A real vector space Y endowed with convergence is said to be a vector space with convergence. If x n x , then ( x n ) is said to be a convergent sequence in Y, and the vector x is said to be a limit of ( x n ) .

Definition 2.2 ([4])

Let ( Y , ) be a vector space with convergence. An ordering on Y is said to be a vector ordering if it is compatible with the algebraic and convergence structures on Y in the sense that the following are true:

(V1) If x y , then x + z y + z .

(V2) If λ 0 and x y , then λ x λ y .

(V3) If x n x , y n y , x n y n for all n, then x y .

A vector space with convergence Y endowed with vector ordering is called an ordered vector space with convergence.

Definition 2.3 ([4])

Let ( Y , , ) be an ordered vector space with convergence. A strict ordering on Y is said to be a strict vector ordering if it is compatible with the vector ordering, the algebraic structure and the convergence structure on Y in the sense that the following are true:

(S1) If x y , then x y .

(S2) If x y and y z , then x z .

(S3) If x y , then x + z y + z .

(S4) If λ > 0 and x y , then λ x λ y .

(S5) If x n x , y n y and x y , then x n y n for all but finitely many n.

It turns out that an ordered vector space can be endowed with at most one strict vector ordering (see Proinov [[4], Theorem 5.1]).

Definition 2.4 (Solid vector space)

An ordered vector space with convergence endowed with a strict vector ordering is said to be a solid vector space.

Let us consider an important example of a solid vector space.

Example 2.5 Let ( Y , τ ) be a topological vector space and K Y be a cone with nonempty interior K . Define the vector ordering on Y and the strict vector ordering on Y, respectively, by means of
x y if and only if y x K , x y if and only if y x K .

Then Y is a solid vector space called a solid topological vector space.

Now let us recall the definition of a cone metric space known also as ‘K-metric spaces’ (see Zabrejko [2], Proinov [4] and references therein).

Definition 2.6 (Cone metric space)

Let X be a nonempty set, and let ( Y , ) be an ordered vector space with convergence. A vector-valued function d : X × X Y is said to be a cone metric on Y if the following conditions hold:
  1. (i)

    d ( x , y ) 0 for all x , y X and d ( x , y ) = 0 if and only if x = y ;

     
  2. (ii)

    d ( x , y ) = d ( y , x ) for all x , y X ;

     
  3. (iii)

    d ( x , y ) d ( x , z ) + d ( z , y ) for all x , y , z X .

     

The pair ( X , d ) is called a cone metric space over Y.

Let ( X , d ) be a cone metric space over a solid vector space ( Y , , ) , x 0 X and r Y with r 0 . Then the set U ( x 0 , r ) = { x X : d ( x , x 0 ) r } is called an open ball with center x 0 and radius r.

Every cone metric space X over a solid vector space Y is a Hausdorff topological space with topology generated by the basis of all open balls. Then a sequence ( x n ) of points in X converges to x X if and only if for every vector c Y with c 0 , d ( x n , x ) c for all but finitely many n.

Recall also that a sequence ( x n ) in X is called a Cauchy sequence if for every c Y with c 0 there is N N such that d ( x n , x m ) c for all n , m > N . A cone metric space X is called complete if each Cauchy sequence in X is convergent.

In order to prove our main result we need the following two theorems.

Theorem 2.7 ([4])

Let ( X , d ) be a complete cone metric space over a solid vector space ( Y , ) . Suppose ( x n ) is a sequence in X satisfying
d ( x n , x m ) b n for all  n , m 0  with  m n ,
where ( b n ) is a sequence in Y which converges to 0. Then ( x n ) converges to a point ξ X with error estimate
d ( x n , ξ ) b n for all  n 0 .

Theorem 2.8 ([4])

Let ( X , d ) be a cone metric space over a solid vector space ( Y , ) and T : X X . Suppose that for some x X , the Picard iteration ( T n x ) converges to a point ξ X . Suppose also that there exist nonnegative numbers α and β such that
d ( ξ , T ξ ) α d ( x , ξ ) + β d ( T x , ξ ) for each  x X .
(5)

Then ξ is a fixed point of T.

3 Auxiliary results

Let A be a subset of a real vector space Y. Recall that the convex hull of A, denoted coA, is the smallest convex set including A. Suppose x , x 1 , , x n Y . It is well known that x co { x 1 , , x n } if and only if there exist nonnegative numbers α 1 , , α n such that i = 1 n α i = 1 and x = i = 1 n α i x i .

Lemma 3.1 Let ( Y , ) be an ordered vector space. Suppose that x, y, x 1 , , x n , y 1 , , y m are vectors in Y and λ is a real number. Then:

(P1) x co { x 1 , , x n } x co { x 1 , , x n , y } ;

(P2) x co { x 1 , , x n } and x i y i for all i x co { y 1 , , y n } ;

(P3) x co { x 1 , , x n , y } and y co { y 1 , , y m } x co { x 1 , , x n , y 1 , , y m } ;

(P4) x co { 0 , x 1 , , x n } x co { x 1 , , x n } if x i 0 for some i;

(P5) x co { λ x , x 1 , , x n } x co { x 1 , , x n } if λ < 1 and x i 0 for some i;

(P6) x co { x 1 , , x n , y } x co { x 1 , , x n } if y = x i for some i.

Proof We only prove the necessity of (P5) since the proofs of the other properties are similar. The inequality x co { λ x , x 1 , , x n , } implies that there exist nonnegative numbers α , α 1 , , α n such that α + i = 1 n α i = 1 and x α λ x + i = 1 n α i x i . From this inequality and α λ < 1 , we deduce
x i = 1 n β i x i ,
(6)
where β i = α i / ( 1 α λ ) . We have i = 1 n β i = ( 1 α ) / ( 1 α γ ) < 1 . By the assumptions, we have x i 0 for some i. Without loss of generality we may assume that x 1 0 . Define the nonnegative numbers γ 1 , , γ n by γ 1 = 1 j = 2 n β j and γ i = β i for i 2 . From (6) and β 1 γ 1 , we obtain
x i = 1 n γ i x i .

This implies x co { x 1 , , x n } since i = 1 n γ i = 1 . □

Remark 3.2 Note that Lemma 3.1 remains true if we omit the expression ‘co’ from its formulation.

The following lemma was given by Zhang [[19], Lemma 6] in a slightly different form. We give a simple proof of this lemma.

Lemma 3.3 Let ( X , d ) be a cone metric space over an ordered vector space ( Y , ) , T : X X be a quasi-contraction with contraction constant λ [ 0 , 1 ) , and let x X . Then for every m N , we have
d ( T i x , T m x ) λ i co { d ( x , T x ) , , d ( x , T m x ) } for  i = 1 , , m .
(7)
Proof We prove the statement by induction on m. It is obviously true for m = 1 . Assume that n N and assume that (7) is satisfied for any natural number m n . We have to prove that
d ( T i x , T n + 1 x ) λ i co { d ( x , T x ) , , d ( x , T n + 1 x ) } for  i = 1 , , n + 1 .
(8)

We divide the proof of (8) into three steps.

Step 1. We claim that for every natural number i n the following inequality holds:
d ( T i x , T n + 1 x ) co { λ i d ( x , T x ) , , λ i d ( x , T n x ) , λ d ( T n x , T n + 1 x ) , λ d ( T i 1 x , T n + 1 x ) } .
By the definition of the quasi-contraction mapping, we obtain
d ( T i x , T n + 1 x ) = d ( T ( T i 1 x ) , T ( T n x ) ) λ co { d ( T i 1 x , T n x ) , d ( T i 1 x , T i x ) , d ( T n x , T n + 1 x ) , d ( T i 1 x , T n + 1 x ) , d ( T i x , T n x ) } .
From the induction hypothesis and properties (P1) and (P2), we get the following three inequalities:
d ( T i 1 x , T n x ) λ i 1 co { d ( x , T x ) , , d ( x , T n x ) } , d ( T i 1 x , T i x ) λ i 1 co { d ( x , T x ) , , d ( x , T n x ) } , d ( T i x , T n x ) λ i 1 co { d ( x , T x ) , , d ( x , T n x ) } .

From the last four inequalities and properties (P3) and (P6), we obtain the desired inequality.

Step 2. We claim that for every natural number i n the following inequality holds:
d ( T i x , T n + 1 x ) co { λ i d ( x , T x ) , , λ i d ( x , T n + 1 x ) , λ d ( T n x , T n + 1 x ) } .
We prove this by finite induction on i. Setting i = 1 in the claim of Step 1, we immediately arrive at the following inequality:
d ( T x , T n + 1 x ) co { λ d ( x , T x ) , , λ d ( x , T n x ) , λ d ( T n x , T n + 1 x ) , λ d ( x , T n + 1 x ) } ,
which proves the claim of Step 2 for i = 1 . Assume that for some i n , the claim of Step 2 holds. Now we shall show that
d ( T i + 1 x , T n + 1 x ) co { λ i + 1 d ( x , T x ) , , λ i + 1 d ( x , T n + 1 x ) , λ d ( T n x , T n + 1 x ) } .
(9)
It follows from Step 1 that
d ( T i + 1 x , T n + 1 x ) co { λ i + 1 d ( x , T x ) , , λ i + 1 d ( x , T n + 1 x ) , λ d ( T n x , T n + 1 x ) , λ d ( T i x , T n + 1 x ) } .
(10)
By the finite induction hypothesis and property (P2), we have
d ( T i x , T n + 1 x ) co { λ i d ( x , T x ) , , λ i d ( x , T n + 1 x ) , d ( T n x , T n + 1 x ) } .
(11)

From (10), (11), and properties (P3) and (P6), we obtain (9).

Step 3. Now we shall prove (8). From the claim of Step 2 with i = n , we get
d ( T n x , T n + 1 x ) co { λ n d ( x , T x ) , , λ n d ( x , T n + 1 x ) , λ d ( T n x , T n + 1 x ) } .
According to the property (P5), this inequality is equivalent to
d ( T n x , T n + 1 x ) λ n co { d ( x , T x ) , , d ( x , T n + 1 x ) } ,
which by (P2) implies
d ( T n x , T n + 1 x ) λ i 1 co { d ( x , T x ) , , d ( x , T n + 1 x ) } .
(12)

Finally, by the claim of Step 2 and the inequality (12), taking into account the properties (P3) and (P6), we obtain (8). This completes the proof of the lemma. □

In the following lemma, we show that if T is a quasi-contraction of a cone metric space X, then for every starting point x X , the Picard iteration sequence ( T n x ) is bounded in the space X.

Lemma 3.4 Let ( X , d ) be a cone metric space over an ordered vector space ( Y , ) , T : X X be a quasi-contraction with contraction constant λ [ 0 , 1 ) , and let x X . Then for every m N , we have
d ( x , T m x ) 1 1 λ d ( x , T x ) .
(13)
Proof We prove the statement by induction on m. If m = 1 , then inequality (13) holds since 0 λ < 1 . Assume that n N and assume that (7) is satisfied for any natural number m n . Then we have to prove that
d ( x , T n + 1 x ) 1 1 λ d ( x , T x ) .
(14)
From the triangle inequality, we obtain
d ( x , T n + 1 x ) d ( x , T x ) + d ( T x , T n + 1 x ) .
(15)
By Lemma 3.3, we get
d ( T x , T n + 1 x ) λ co { d ( x , T x ) , , d ( x , T n + 1 x ) } .
(16)
By the induction hypothesis, we have that (13) holds for all m n . Then it follows from (16), (P2), and (P6) that
d ( T x , T n + 1 x ) co { λ 1 λ d ( x , T x ) , λ d ( x , T n + 1 x ) } .
This inequality implies that there exists α [ 0 , 1 ] such that
d ( T x , T n + 1 x ) α λ 1 λ d ( x , T x ) + ( 1 α ) λ d ( x , T n + 1 x ) .
(17)
Combining (15) and (17), we get
d ( x , T n + 1 x ) d ( x , T x ) + α λ 1 λ d ( x , T x ) + ( 1 α ) λ d ( x , T n + 1 x ) ,
which is equivalent to the following inequality:
( 1 λ + α λ ) d ( x , T n + 1 x ) 1 λ + α λ 1 λ d ( x , T x ) .

Multiplying both sides of this inequality by 1 / ( 1 λ + α λ ) , we obtain (14). This completes the proof of the lemma. □

Lemma 3.5 Let ( X , d ) be a cone metric space over an ordered vector space ( Y , ) , and let T : X X be a quasi-contraction with contraction constant λ [ 0 , 1 ) . Then for all x , y X , we have
d ( x , T x ) α d ( x , y ) + β d ( x , T y ) ,
(18)

where α = λ / ( 1 λ ) and β = ( 1 + λ ) / ( 1 λ ) .

Proof Let x , y X be fixed. First we shall prove that
d ( T x , T y ) λ ( d ( x , y ) + d ( x , T x ) + d ( x , T y ) ) .
(19)
It follows from Definition 1.5 that there exist five nonnegative numbers α , β , γ , μ , ν such that α + β + γ + μ + ν = 1 and
d ( T x , T y ) λ ( α d ( x , y ) + β d ( x , T x ) + γ d ( y , T y ) + μ d ( x , T y ) + ν d ( y , T x ) ) .
From this and the inequalities
d ( y , T y ) d ( x , y ) + d ( x , T y ) and d ( y , T x ) d ( x , y ) + d ( x , T x ) ,
we obtain
d ( T x , T y ) λ ( ( α + γ + ν ) d ( x , y ) + ( β + ν ) d ( x , T x ) + ( γ + μ ) d ( x , T y ) ) .
This inequality yields (19) since α + γ + ν 1 , β + ν 1 and γ + μ 1 . Now we are ready to prove (18). From the triangle inequality, we get
d ( x , T x ) d ( x , T y ) + d ( T x , T y ) .
From this and (19), we obtain
d ( x , T x ) d ( x , T y ) + λ ( d ( x , y ) + d ( x , T x ) + d ( x , T y ) ) ,
which can be presented in the following equivalent form:
( 1 λ ) d ( x , T x ) λ d ( x , y ) + ( 1 + λ ) d ( x , T y ) .

Multiplying both sides of this inequality by 1 / ( 1 λ ) , we get (18). □

Lemma 3.6 Let ( X , d ) be a cone metric space over an ordered vector space ( Y , ) . Then every quasi-contraction T : X X has at most one fixed point in X.

Proof Suppose that x and y are two fixed points of T. It follows from the inequality (4) and properties (P4) and (P6) that d ( x , y ) λ d ( x , y ) which implies d ( x , y ) 0 . On the other hand, d ( x , y ) 0 . Hence, d ( x , y ) = 0 , which yields x = y . □

4 Main result

Now we are ready to state the main result of this paper. Let ( X , d ) be a complete cone metric space over an ordered vector space Y. Recall that for a point x 0 X and a vector r Y with r 0 , the set U ¯ ( x 0 , r ) = { x X : d ( x , x 0 ) r } is called a closed ball with center x 0 and radius r.

Theorem 4.1 Let ( X , d ) be a complete cone metric space over a solid vector space ( Y , ) , and let T : X X be a quasi-contraction with contraction constant λ [ 0 , 1 ) . Then the following statements hold true:
  1. (i)
    Existence, uniqueness and localization. T has a unique fixed point ξ which belongs to the closed ball U ¯ ( x , r ) with radius
    r = 1 1 λ d ( x , T x ) ,
     
where x is any point in X.
  1. (ii)

    Convergence of Picard iteration. Starting from any point x X the Picard sequence ( T n x ) remains in the closed ball U ¯ ( x , r ) and converges to ξ.

     
  2. (iii)
    A priori error estimate. For every point x X the following a priori estimate holds:
    d ( T n x , ξ ) λ n 1 λ d ( x , T x ) for all  n 0 .
    (20)
     
  3. (iv)
    A posteriori error estimates. For every point x X the following a posteriori estimate holds:
    d ( T n x , ξ ) 1 1 λ d ( T n x , T n + 1 x ) for all  n 0 ,
    (21)
    d ( T n x , ξ ) λ 1 λ d ( T n x , T n 1 x ) for all  n 1 .
    (22)
     
Proof Let x be an arbitrary point in X. By Lemma 3.3, for all m , n N with m n , we have
d ( T n x , T m x ) λ n co { d ( x , T x ) , , d ( x , T m x ) } .
From this, Lemma 3.4, and properties (P2) and (P6), we deduce
d ( T n x , T m x ) b n , where  b n = λ n 1 λ d ( x , T x ) .
(23)
Note that ( b n ) is a sequence in Y which converges to 0 since λ n 0 in . Now applying Theorem 2.7 to the Picard sequence ( T n x ) , we conclude that there exists a point ξ X such that ( T n x ) converges to ξ and d ( T n x , ξ ) b n for every n 0 . The last inequality coincides with the estimate (20). Setting n = 0 in (20), we get
d ( x , ξ ) 1 1 λ d ( x , T x )
(24)
which means that ξ U ¯ ( x , r ) . The inequality (24) holds for every point x X . Applying (24) to the point T n x , we obtain (21). Setting n = 1 in (20), we get
d ( T x , ξ ) λ 1 λ d ( x , T x ) .
(25)

Applying (25) to the point T n 1 x , we get (22). Setting n = 0 in (23), we obtain d ( x , T m x ) b 0 for every m 0 . Hence, the sequence ( T n x ) lies in the ball U ¯ ( x , r ) since r = b 0 .

It follows from Lemma 3.5 that ξ satisfies condition (5). Hence, by Theorem 2.8, we conclude that ξ is a fixed point of T. The uniqueness of the fixed point follows from Lemma 3.6. □

Theorem 4.1 extends and complements the recent results of Ding et al. [[24], Theorem 3.1] and Zhang [[19], Theorem 3] as well as previous results due to Ilić and Rakočević [[20], Theorem 2.1], Kadelburg et al. [[21], Theorem 2.2], Rezapour et al. [[23], Theorem 2.1] and Kadelburg et al. [[8], Theorem 3.5(b)] who have studied quasi-contraction mappings of the type (3).

Theorem 4.1 also extends and complements the results of Abbas and Rhoades [[39], Corollary 2.3], Olaleru [[40], Theorem 2.1], Azam et al. [[41], Theorem 2.2], Song et al. [[42], Corollary 2.1]. These authors have studied the class of mappings satisfying a contractive condition of the type
d ( T x , T y ) α d ( x , y ) + β d ( x , T x ) + γ d ( y , T y ) + μ d ( x , T y ) + ν d ( y , T x )
(26)

for all x , y X , where α, β, γ, μ and ν are five nonnegative constants such that α + β + γ + μ + ν < 1 . In this case Theorem 4.1 holds with λ = α + β + γ + μ + ν since condition (26) implies condition (4) with this λ. Let us note that in this special case Theorem 4.1 holds even with λ = ( α + δ ) / ( 1 δ ) , where δ = ( β + γ + μ + ν ) / 2 .

5 Examples

Zhang [[19], Example 1] gives an example showing that the set of all quasi-contractions of the type (3) is a proper subset of the set of all quasi-contractions defined by (4). In order to prove this, he considers a selfmapping of a cone metric space X over a normal solid vector space Y. Ding et al. [[24], Example 4.1] provide a similar example, but for the case of a non-normal solid vector space Y.

The aim of this section is to unify these two examples. Let denote the set of all quasi-contractions of the type (3), and let denote the set of all quasi-contractions defined by (4), that is,
B = { T : X X d ( T x , T y ) λ { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } } , C = { T : X X d ( T x , T y ) λ co { d ( x , y ) , d ( x , T x ) , d ( y , T y ) , d ( x , T y ) , d ( y , T x ) } } .

Now we shall construct a family of examples which show that is a proper subset of . In particular, this family contains both the example of Zhang [19] and the example of Ding et al. [24].

Definition 5.1 Let ( Y , ) be an ordered vector space, and let a , b , c 0 be three vectors in Y. We say that the triple ( a , b , c ) satisfies property (C) if the following two statements hold:

  • a λ co { b , c } for some λ [ 0 , 1 ) , b a + c and c a + b .

  • a k { b , c } is wrong for every k [ 0 , 1 ) .

Proposition 5.2 Let Y = R be endowed with the usual ordering ≤. Then there are no triples ( a , b , c ) in Y satisfying property (C).

Proof Assume that there is a triple ( a , b , c ) in Y with property (C). Then a λ max { b , c } for some λ [ 0 , 1 ) . On the other hand, a k max { b , c } is wrong for every k [ 0 , 1 ) . This is a contradiction which proves the proposition. □

Proposition 5.3 Let Y = R n ( n 2 ) be endowed with coordinate-wise ordering . Then in Y there exist infinitely many triples ( a , b , c ) satisfying property (C).

Proof Choose three real numbers α, β and γ such that
β < γ < 3 β and max { β , γ β } α < β + γ 2 .

Then the vectors a = ( α , , α ) , b = ( β , , β , γ ) and c = ( γ , , γ , β ) satisfy property (C) with λ = 2 α β + γ . □

Proposition 5.4 Let Y = C n [ 0 , 1 ] ( n 2 ) be endowed with point-wise ordering . Then in Y there exist infinitely many triples ( a , b , c ) satisfying property (C).

Proof Choose three real numbers α, β and γ as in the proof of Proposition 5.3, then choose a real number δ such that
γ α δ < γ and δ α + γ β 2 .

Then the functions a ( t ) = α , b ( t ) = β + δ t and c ( t ) = γ δ t satisfy property (C) with λ = 2 α β + γ . □

Example 5.5 Let ( Y , ) be an arbitrary solid vector space, and let ( a , b , c ) be any triple in Y with property (C). Furthermore, let X = { x , y , z } , and let d : X × X Y be defined by
d ( x , y ) = a , d ( x , z ) = b , d ( y , z ) = c ,
d ( u , v ) = d ( u , v ) and d ( u , u ) = 0 for u , v X . Then ( X , d ) is a complete cone metric space over Y since the triple ( a , b , c ) satisfies property (C). Consider now the mapping T : X X defined by
T x = x , T y = x , T z = y .

Using Lemma 3.1, it is easy to prove that T C if and only if a λ co { b , c } for some λ [ 0 , 1 ) . Analogously, taking into account Remark 3.2, one can easily prove that T B if and only if a k { b , c } for some k [ 0 , 1 ) . Now taking into account that the triple ( a , b , c ) satisfies property (C), we conclude that T C and T B . Hence, is a proper subset of .

Declarations

Acknowledgements

The research is supported by Project NI13 FMI-002 of Plovdiv University.

Authors’ Affiliations

(1)
Faculty of Mathematics and Informatics, University of Plovdiv, Plovdiv, Bulgaria

References

  1. Collatz L: Functional Analysis and Numerical Mathematics. Academic Press, New York; 1966.MATHGoogle Scholar
  2. Zabrejko PP: K -Metric and K -normed linear spaces: survey. Collect. Math. 1997, 48: 825–859.MathSciNetMATHGoogle Scholar
  3. Janković S, Kadelburg Z, Radenović S: On cone metric spaces: a survey. Nonlinear Anal. 2011, 74: 2591–2601. 10.1016/j.na.2010.12.014MathSciNetView ArticleMATHGoogle Scholar
  4. Proinov PD: A unified theory of cone metric spaces and its applications to the fixed point theory. Fixed Point Theory Appl. 2013., 2013: Article ID 103Google Scholar
  5. Du W-S: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 2010, 72: 2259–2261. 10.1016/j.na.2009.10.026MathSciNetView ArticleMATHGoogle Scholar
  6. Amini-Harandi A, Fakhar M: Fixed point theory in cone metric spaces obtained via the scalarization method. Comput. Math. Appl. 2010, 59: 3529–3534. 10.1016/j.camwa.2010.03.046MathSciNetView ArticleMATHGoogle Scholar
  7. Feng Y, Mao W: The equivalence of cone metric spaces and metric spaces. Fixed Point Theory 2010, 11: 259–263.MathSciNetMATHGoogle Scholar
  8. Kadelburg Z, Radenović S, Rakočević V: A note on the equivalence of some metric and cone metric fixed point results. Appl. Math. Lett. 2011, 24: 370–374. 10.1016/j.aml.2010.10.030MathSciNetView ArticleMATHGoogle Scholar
  9. Asadi M, Rhoades BE, Soleimani H: Some notes on the paper ‘The equivalence of cone metric spaces and metric spaces’. Fixed Point Theory Appl. 2012., 2012: Article ID 87Google Scholar
  10. Ercan Z: On the end of the cone metric spaces. Topol. Appl. 2014, 166: 10–14.MathSciNetView ArticleMATHGoogle Scholar
  11. Banach S: Sur les opérations dans les ensembles abstraits et leurs applications aux équations integrals. Fundam. Math. 2009, 3: 133–181.MATHGoogle Scholar
  12. Berinde V Lecture Notes in Mathematics 1912. In Iterative Approximation of Fixed Points. Springer, Berlin; 2007.Google Scholar
  13. Radenović S, Kadelburg Z: Quasi-contractions on symmetric and cone symmetric spaces. Banach J. Math. Anal. 2011, 5: 38–50. 10.15352/bjma/1313362978MathSciNetView ArticleMATHGoogle Scholar
  14. Ćirić LB: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 1974, 45: 267–273.MATHGoogle Scholar
  15. Proinov PD: A generalization of the Banach contraction principle with high order of convergence of successive approximations. Nonlinear Anal. 2007, 67: 2361–2369. 10.1016/j.na.2006.09.008MathSciNetView ArticleMATHGoogle Scholar
  16. Proinov PD: New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems. J. Complex. 2010, 26: 3–42. 10.1016/j.jco.2009.05.001MathSciNetView ArticleMATHGoogle Scholar
  17. Zhang X: Fixed point theorem of generalized operator quasi-contractive mapping in cone metric spaces. Afr. Math. 2014, 25: 135–146. 10.1007/s13370-012-0105-7View ArticleMATHGoogle Scholar
  18. Ali MU, Kamran T, Karapinar E: A new approach to ( α , ψ ) -contractive nonself multivalued mappings. J. Inequal. Appl. 2014., 2014: Article ID 71Google Scholar
  19. Zhang X: Fixed point theorem of generalized quasi-contractive mapping in cone metric space. Comput. Math. Appl. 2011, 62: 1627–1633. 10.1016/j.camwa.2011.03.107MathSciNetView ArticleMATHGoogle Scholar
  20. Ilić D, Rakočević V: Quasi-contraction on a cone metric space. Appl. Math. Lett. 2009, 22: 728–731. 10.1016/j.aml.2008.08.011MathSciNetView ArticleMATHGoogle Scholar
  21. Kadelburg Z, Radenović S, Rakočević V: Remarks on ‘Quasi-contraction on a cone metric space’. Appl. Math. Lett. 2009, 22: 1674–1679. 10.1016/j.aml.2009.06.003MathSciNetView ArticleMATHGoogle Scholar
  22. Gajić L, Rakočević V: Quasi-contractions on a nonnormal cone metric spaces. Funct. Anal. Appl. 2012, 46: 62–65. 10.1007/s10688-012-0008-2View ArticleMATHGoogle Scholar
  23. Rezapour S, Haghi RH, Shahzad N: Some notes on fixed points of quasi-contraction maps. Appl. Math. Lett. 2010, 23: 498–502. 10.1016/j.aml.2010.01.003MathSciNetView ArticleMATHGoogle Scholar
  24. Ding H-S, Jovanović M, Kadelburg Z, Radenović S: Common fixed point results for generalized quasicontractions in tvs-cone metric spaces. J. Comput. Anal. Appl. 2013, 15: 463–470.MathSciNetMATHGoogle Scholar
  25. Karapinar E: Fixed point theorems in cone Banach spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 609281Google Scholar
  26. Karapinar E: Some nonunique fixed point theorems of Ćirić type on cone metric spaces. Abstr. Appl. Anal. 2010., 2010: Article ID 123094Google Scholar
  27. Karapinar E: Couple fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl. 2010, 59: 3656–3668. 10.1016/j.camwa.2010.03.062MathSciNetView ArticleMATHGoogle Scholar
  28. Khamsi MA: Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 315398Google Scholar
  29. Dukić D, Paunović L, Radenović S: Convergence of iterates with errors of uniformly quasi-Lipschitzian mappings in cone metric spaces. Kragujev. J. Math. 2011, 35: 399–410.MathSciNetMATHGoogle Scholar
  30. Karapinar E, Kumam P, Sintunavarat W: Coupled fixed point theorems in cone metric spaces with a c -distance and applications. Fixed Point Theory Appl. 2012., 2012: Article ID 194Google Scholar
  31. Radenović S: A pair of non-self mappings in cone metric spaces. Kragujev. J. Math. 2012, 36: 189–198.MathSciNetMATHGoogle Scholar
  32. Aydi H, Karapinar E, Mustafa Z: Coupled coincidence point results on generalized distance in ordered cone metric spaces. Positivity 2013, 17: 979–993. 10.1007/s11117-012-0216-2MathSciNetView ArticleMATHGoogle Scholar
  33. Du W-S, Karapinar E: A note on cone b -metric and its related results: generalizations or equivalence? Fixed Point Theory Appl. 2013., 2013: Article ID 210Google Scholar
  34. Kadelburg Z, Radenović S: A note on various types of cones and fixed point results in cone metric spaces. Asian J. Math. Appl. 2013., 2013: Article ID 0104Google Scholar
  35. Popović B, Radenović S, Shukla S: Fixed point results to tvs-cone b -metric spaces. Gulf J. Math. 2013, 1: 51–64.Google Scholar
  36. Xu S, Radenović S: Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebra without assumption of normality. Fixed Point Theory Appl. 2014., 2014: Article ID 101Google Scholar
  37. Proinov PD, Cholakov SI: Semilocal convergence of Chebyshev-like root-finding method for simultaneous approximation of polynomial zeros. Appl. Math. Comput. 2014, 236: 669–682.MathSciNetView ArticleMATHGoogle Scholar
  38. Proinov, PD, Cholakov, SI: Convergence of Chebyshev-like method for simultaneous computation of multiple polynomial zeros. C. R. Acad. Bulg. Sci. 67 (2014, in press)Google Scholar
  39. Abbas M, Rhoades BE: Fixed and periodic point results in cone metric spaces. Appl. Math. Lett. 2009, 22: 511–515. 10.1016/j.aml.2008.07.001MathSciNetView ArticleMATHGoogle Scholar
  40. Olaleru JO: Some generalizations of fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 657914Google Scholar
  41. Azam A, Beg I, Arshad M: Fixed point in topological vector space-valued cone metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 604084Google Scholar
  42. Song G, Sun X, Zhao Y, Wang G: New common fixed point theorems for maps on cone metric spaces. Appl. Math. Lett. 2010, 23: 1033–1037. 10.1016/j.aml.2010.04.032MathSciNetView ArticleMATHGoogle Scholar

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© Proinov and Nikolova; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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