Open Access

A generalised fixed point theorem of presic type in cone metric spaces and application to markov process

Fixed Point Theory and Applications20112011:85

DOI: 10.1186/1687-1812-2011-85

Received: 23 March 2011

Accepted: 23 November 2011

Published: 23 November 2011

Abstract

A generalised common fixed point theorem of Presic type for two mappings f: XX and T: X k X in a cone metric space is proved. Our result generalises many well-known results.

2000 Mathematics Subject Classification

47H10

Keywords

Coincidence and common fixed points cone metric space; weakly compatible

1. Introduction

Considering the convergence of certain sequences, Presic [1] proved the following:

Theorem 1.1. Let (X, d) be a metric space, k a positive integer, T: X k X be a mapping satisfying the following condition:
d ( T ( x 1 , x 2 , , x k ) , T ( x 2 , x 3 , , x k + 1 ) ) q 1 d ( x 1 , x 2 ) + q 2 d ( x 2 , x 3 ) + + q k d ( x k , x k + 1 )
(1.1)

where x1, x2, ..., xk+ 1are arbitrary elements in X and q1, q2, ..., q k are non-negative constants such that q1 + q2 + · · · + q k < 1. Then, there exists some x X such that x = T(x, x, ..., x). Moreover if x1, x2, ..., x k are arbitrary points in X and for n N x n+k = T(x n , xn+1, ..., xn+k-1), then the sequence < x n > is convergent and lim x n = T(lim x n , lim x n , ..., lim x n ).

Note that for k = 1 the above theorem reduces to the well-known Banach Contraction Principle. Ciric and Presic [2] generalising the above theorem proved the following:

Theorem 1.2. Let (X, d) be a metric space, k a positive integer, T: X k → X be a mapping satisfying the following condition:
d ( T ( x 1 , x 2 , , x k ) , T ( x 2 , x 3 , , x k + 1 ) ) λ . m a x { d ( x 1 , x 2 ) , d ( x 2 , x 3 ) , d ( x k , x k + 1 )
(1.2)

where x1, x2, ..., xk+1are arbitrary elements in X and λ (0,1). Then, there exists some x X such that x = T(x, x, ..., x). Moreover if x1, x2, ..., x k are arbitrary points in X and for n Nx n+k = T(x n , xn+1, ..., xn+k-1), then the sequence < x n > is convergent and lim x n = T(lim x n , lim x n , ..., lim x n ). If in addition T satisfies D(T(u, u, ... u), T(v, v, ... v)) < d(u, v), for all u, v X then x is the unique point satisfying x = T(x, x, ..., x).

Huang and Zang [3] generalising the notion of metric space by replacing the set of real numbers by ordered normed spaces, defined a cone metric space and proved some fixed point theorems of contractive mappings defined on these spaces. Rezapour and Hamlbarani [4], omitting the assumption of normality, obtained generalisations of results of [3]. In [5], Di Bari and Vetro obtained results on points of coincidence and common fixed points in non-normal cone metric spaces. Further results on fixed point theorems in such spaces were obtained by several authors, see [515].

The purpose of the present paper is to extend and generalise the above Theorems 1.1 and 1.2 for two mappings in non-normal cone metric spaces and by removing the requirement of D(T(u, u, ... u), T(v, v, ... v)) < d(u, v), for all u, v X for uniqueness of the fixed point, which in turn will extend and generalise the results of [3, 4].

2. Preliminaries

Let E be a real Banach space and P a subset of E. Then, P is called a cone if
  1. (i)

    P is closed, non-empty, and satisfies P ≠ {0},

     
  2. (ii)

    ax + by P for all x, y P and non-negative real numbers a, b

     
  3. (iii)

    x P and - x P x = 0, i.e. P ∩ (-P) = 0

     

Given a cone P E, we define a partial ordering ≤ with respect to P by xy if and only if y - x P. We shall write x < y if xy and xy, and x y if y - x intP, where intP denote the interior of P. The cone P is called normal if there is a number K > 0 such that for all x, y E, 0 ≤ xy implies || x || ≤ K || y || .

Definition 2.1. [3] Let X be a non empty set. Suppose that the mapping d: X × XE satisfies:

(d1) 0 ≤ d (x, y) for all x, y X and d (x, y) = 0 if and only if x = y

(d2)d (x, y) = d (y, x) for all x, y X

(d3)d (x, y) ≤ d (x, z) + d (z, y) for all x, y, z X

Then, d is called a conemetric on X and (X, d) is called a conemetricspace.

Definition 2.2. [3] Let (X, d) be a cone metric space. The sequence {x n } in X is said to be:
  1. (a)

    A convergent sequence if for every c E with 0 c, there is n0 N such that for all nn0, d (x n , x) c for some x X. We denote this by limn→∞x n = x.

     
  2. (b)

    A Cauchy sequence if for all c E with 0 c, there is no N such that d (x m , x n ) c, for all m, nn0.

     
  3. (c)

    A cone metric space (X, d) is said to be complete if every Cauchy sequence in X is convergent in X.

     
  4. (d)

    A self-map T on X is said to be continuous if lim n→∞ x n = x implies that lim n→∞ T(x n ) = T(x), for every sequence {x n }in X.

     
Definition 2.3. Let (X, d) be a metric space, k a positive integer, T: X k X and f : XX be mappings.
  1. (a)

    An element x X said to be a coincidence point of f and T if and only if f(x) = T(x, x, ..., x). If x = f(x) = T(x, x, ..., x), then we say that x; is a common fixed point of f and T. If w = f(x) = T(x, x, ..., x), then w is called a point of coincidence of f and T.

     
  2. (b)

    Mappings f and T are said to be commuting if and only if f(T(x, x, ... x)) = T(fx, fx, ... fx) for all x X.

     
  3. (c)

    Mappings f and T are said to be weakly compatible if and only if they commute at their coincidence points.

     

Remark 2.4. For k = 1, the above definitions reduce to the usual definition of commuting and weakly compatible mappings in a metric space.

The set of coincidence points of f and T is denoted by C(f, T).

3. Main results

Consider a function ϕ: E k → E such that
  1. (a)

    ϕ is an increasing function, i.e x1 < y1, x2 < y2, ..., x k < y k implies ϕ(x1, x2, ..., x k ) < ϕ(y1, y2, ..., y k ).

     
  2. (b)

    ϕ(t, t, t, ...) ≤ t, for all t X

     
  3. (c)

    ϕ is continuous in all variables.

     

Now, we present our main results as follows:

Theorem 3.1. Let (X, d) be a cone metric space with solid cone P contained in a real Banach space E. For any positive integer k, let T: X k X and f: XX be mappings satisfying the following conditions:
T ( X k ) f ( X )
(3.1)
d ( T ( x 1 , x 2 , , x k ) , T ( x 2 , x 3 , , x k + 1 ) ) λ ϕ ( d ( f x 1 , f x 2 ) , d ( f x 2 , f x 3 ) , , ( f x k , f x k + 1 ) )
(3.2)
where x1, x2, ..., xk+1are arbitrary elements in X and λ ( 0 , 1 k ) and
f ( X ) i s c o m p l e t e
(3.3)
there exist elements x1, x2, ..., x k in X such that
R = m a x d ( f x 1 , f x 2 ) θ , d ( f x 2 , f x 3 ) θ 2 , , d ( f x k , T ( x 1 , x 2 , , x k ) ) θ k e x i s t i n E
(3.4)

where θ = λ 1 k . Then, f and T have a coincidence point, i.e. C(f, T) ≠ .

Proof. By (3.1) and (3.4) we define sequence < y n > in f(X) as y n = fx n for n = 1, 2, ..., k and y n+k = f(x n+k ) = T(x n , xn+1, ..., xn+k-1), n = 1, 2, ... Let α n = d(y n , yn+1). By the method of mathematical induction, we will now prove that
α n R . θ n
(3.5)
for all n. Clearly by the definition of R, (3.5) is true for n = 1, 2, ..., k. Let the k inequalities α n n , αn+1n+1, ..., αn+k-1n+k-1be the induction hypothesis. Then, we have
α n + k = d ( y n + k , y n + k + 1 ) = d ( T ( x n , x n + 1 , , x n + k - 1 ) , T ( x n + 1 , x n + 2 , , x n + k ) ) λ ϕ ( d ( f x n , f x n + 1 ) , d ( f x n + 1 , f x n + 2 ) , , d ( f x n + k - 1 , f x n + k ) ) = λ ϕ ( α n , α n + 1 , , α n + k - 1 ) λ ϕ ( R θ n , R θ n + 1 , , R . θ n + k - 1 ) λ ϕ ( R θ n , R θ n , , R θ n ) λ R θ n = R . θ n + k .
Thus inductive proof of (3.5) is complete. Now for n, p N, we have
d ( y n , y n + p ) d ( y n , y n + 1 ) + d ( y n + 1 , y n + 2 ) + + d ( y n + p - 1 , y n + p ) , R θ n + R θ n + 1 + + R θ n + p - 1 R θ n ( 1 + θ + θ 2 + ) = R θ n 1 - θ

Let 0 c be given. Choose δ > 0 such that c + N δ (0) P where N δ (0) = {y E; || y || < δ}. Also choose a natural number N1 such that R θ n 1 - θ N δ ( 0 ) , for all n > N1. Then, R θ n 1 - θ c for all nN1. Thus, d ( y n , y n + p ) R θ n 1 - θ c for all nN1. Hence, sequence < y n > is a Cauchy sequence in f(X), and since f(X) is complete, there exists v, u X such that limn→∞y n = v = f(u). Choose a natural number N2 such that d ( y n , y n + 1 ) c λ ( k + 1 ) and d ( x , y n + 1 ) c k + 1 for all nN 2 .

Then for all nN2
d ( f u , T ( u , u , u ) ) d ( f u , y n + k ) + d ( y n + k , T ( u , u , u ) ) = d ( f u , y n + k ) + d ( T ( x n , x n + 1 , x n + k - 1 ) , T ( u , u , u ) ) d ( f u , y n + k ) + d ( T ( u , u , u ) , T ( u , u , x n ) ) + d ( T ( u , u , x n ) , T ( u , u , x n , x n + 1 ) ) + d ( T ( u , x n , x n + k - 2 ) , T ( x n , x n + 1 x n + k - 1 ) d ( f u , y n + k ) + λ ϕ { d ( f u , f u ) , d ( f u , f u ) , , d ( f u , f x n ) } + λ ϕ { d ( f u , f u ) , d ( f u , f u ) , , d ( f u , f x n ) , d ( f x n , f x n + 1 ) } + + λ ϕ { d ( f u , f x n ) , d ( f x n , f x n + 1 ) , d ( f x n + k - 2 , f x n + k - 1 ) } . = d ( f u , y n + k ) + λ ϕ ( 0 , 0 , , d ( f u , f x n ) ) + λ ϕ ( 0 , 0 , , d ( f u , f x n ) , d ( f x n , f x n + 1 ) ) + + λ ϕ ( d ( f u , f x n ) , d ( f x n , f x n + 1 ) , d ( f x n + k - 2 , f x n + k - 1 ) ) . c k + 1 + λ ϕ ( c λ ( k + 1 ) , c λ ( k + 1 ) , , c λ ( k + 1 ) ) + λ ϕ ( c λ ( k + 1 ) , c λ ( k + 1 ) , , c λ ( k + 1 ) ) + + λ ϕ ( c λ ( k + 1 ) , c λ ( k + 1 ) , , c λ ( k + 1 ) ) c k + 1 + λ c λ ( k + 1 ) + λ c λ ( k + 1 ) = c .

Thus, d ( f u , T ( u , u , u ) ) c m for all m ≥ 1.

So, c m - d ( f u , T ( u , u , u ) ) P for all m ≥ 1. Since c m 0 as m → ∞ and P is closed, -d(fu, T(u, u, ... u)) P, but P ∩(-P) = /0/. Therefore, d(fu, T(u, u, ... u)) = 0. Thus, fu = T(u, u, u, ..., u), i.e. C(f, T) ≠ . □

Theorem 3.2. Let (X, d) be a cone metric space with solid cone P contained in a real Banach space E. For any positive integer k, let T: X k X and f: XX be mappings satisfying (3.1), (3.2), (3.3) and let there exist elements x1, x2, ... x k in X satisfying (3.4). If f and T are weakly compatible, then f and T have a unique common fixed point. Moreover if x1, x2, ...,x k are arbitrary points in X and for n N, y n+k = f(x n+k ) = T(x n , xn+1, ... xn+k-1), n = 1, 2, ..., then the sequence < y n > is convergent and lim y n = f(lim y n ) = T(lim y n , lim y n , ..., lim y n ).

Proof. As proved in Theorem 3.1, there exists v, u X such that limn→∞y n = v = f(u) = T(u, u, u ... u). Also since f and T are weakly compatible f(T(u, u, ... u) = T(fu, fu, fu ... fu). By (3.2) we have,
d ( f f u , f u ) = d ( f T ( u , u , u ) , T ( u , u , u ) ) = d ( T ( f u , f u , f u , f u ) , T ( u , u , u ) ) d ( T ( f u , f u , f u , f u ) , T ( f u , f u , f u , u ) ) + d ( T ( f u , f u , f u , u ) , T ( f u , f u , , u , u ) ) + + d ( T ( f u , u , u , u ) , T ( u , u , u ) ) λ ϕ ( d ( f f u , f f u ) , d ( f f u , f f u ) , d ( f f u , f u ) ) + λ ϕ ( d ( f f u , f f u ) , d ( f f u , f u ) , d ( f u , f u ) ) + λ ϕ ( d ( f f u , f u ) , d ( f u , f u ) , d ( f u , f u ) ) = λ ϕ ( 0 , 0 , 0 , d ( f f u , f u ) ) + λ ϕ ( 0 , 0 0 , d ( f f u , f u ) , 0 ) + . λ ϕ ( d ( f f u , f u ) , 0 , 0 0 ) = k λ d ( f f u , f u ) .
Repeating this process n times we get, d(f fu, fu) < k n λ n d(f fu, fu). So k n λ n d(f fu, fu) - d(f fu, fu) P for all n ≥ 1. Since k n λ n → 0 as n → ∞ and P is closed, --d(f fu, fu) P, but P ∩ (-P) = {0}. Therefore, d(f fu, fu) = 0 and so f fu = fu. Hence, we have, fu = f fu = f(T(u, u, ... u)) = T(fu, fu, fu ... fu), i.e. fu is a common fixed point of f and T, and lim y n = f(lim y n ) = T(lim y n , lim y n , ... lim y n ). Now suppose x, y be two fixed points of f and T. Then,
d ( x , y ) = d ( T ( x , x , x x ) , T ( y , y , y y ) ) d ( T ( x , x , x ) , T ( x , x , x , y ) ) + d ( T ( x , x , x , y ) , T ( x , x , x x , y , y ) ) + + d ( T ( x , y , y , y ) , T ( y , y , y ) ) λ ϕ { d ( f x , f x ) , d ( f x , f x ) , , d ( f x , f y ) } + λ ϕ { d ( f x , f x ) , d ( f x , f x ) , d ( f x , f y ) , d ( f y , f y ) } + + λ ϕ { d ( f x , f y ) , d ( f y , f y ) , d ( f y , f y ) } . = λ ϕ ( 0 , 0 , , d ( f x , f y ) ) + λ ϕ ( 0 , 0 , d ( f x , f y ) , 0 ) + + λ ϕ ( d ( f x , f y ) , 0 , 0 , 0 , 0 ) ) . = k λ d ( f x , f y ) = k λ d ( x , y ) .

Repeating this process n times we get as above, d(x, y) ≤ k n λ n d(x, y) and so as n → ∞d(x, y) = 0, which implies x = y. Hence, the common fixed point is unique. □

Remark 3.3. Theorem 3.2 is a proper extension and generalisation of Theorems 1.1 and 1.2.

Remark 3.4. If we take k = 1 in Theorem, 3.2, we get the extended and generalised versions of the result of [3] and [4].

Example 3.5. Let E = R 2 , P = {(x, y) E\x, y ≥ 0}, X = [0, 2] and d: X × XE such that d(x, y) = (|x - y |, |x - y |). Then, d is a cone metric on X. Let T: X 2 X and f: XX be defined as follows:
T ( x , y ) = ( x 2 + y 2 ) 4 + 1 2 i f ( x , y ) [ 0 , 1 ] × [ 0 , 1 ] T ( x , y ) = ( x + y ) 4 + 1 2 i f ( x , y ) [ 1 , 2 ] × [ 1 , 2 ] T ( x , y ) = ( x 2 + y ) 4 + 1 2 i f ( x , y ) [ 0 , 1 ] × [ 1 , 2 ] T ( x , y ) = ( x + y 2 ) 4 + 1 2 i f ( x , y ) [ 1 , 2 ] × [ 0 , 1 ] f ( x ) = x 2 i f x [ 0 , 1 ] f ( x ) = x i f x [ 1 , 2 ]

T and f satisfies condition (3.2) as follows:

Case 1. x, y, z [0, 1]
d ( T ( x , y ) , T ( y , z ) ) = ( T ( x , y ) - T ( y , z ) , T ( x , y ) - T ( y , z ) ) = ( x 2 - z 2 4 , x 2 - z 2 4 ) ( x 2 - z 2 4 + y 2 - z 2 4 , x 2 - y 2 4 + y 2 - z 2 4 ) 1 2 . m a x { d ( f x , f y ) , d ( f y , f z ) }
Case 2. x, y [0, 1] and z [1, 2]
d ( T ( x , y ) , T ( y , z ) ) = ( x 2 + y 2 4 - y 2 + z 4 , x 2 + y 2 4 - y 2 + z 4 ) ( x 2 - y 2 4 + y 2 - z 4 , x 2 - y 2 4 + y 2 - z 4 ) 1 2 . m a x { ( f x , f y ) , d ( f y , f z ) }
Case 3. x [0, 1] and y; z [1, 2]
d ( T ( x , y ) , T ( y , z ) ) = ( x 2 + y 4 - y + z 4 , x 2 + y 4 - y + z 4 ) = ( x 2 - z 4 , x 2 - z 4 ) ( x 2 - y 4 1 + y - z 4 , x 2 - y 4 + y - z 4 ) 1 . 2 . m a x { d ( f x , f y ) , d ( f y , f z ) }
Case 4. x, y, z [1, 2]
d ( T ( x , y ) , T ( y , z ) ) = ( x + y 4 - y + z 4 , x + y 4 - y + z 4 ) ( x - y 4 + y - z 4 , x - y 4 + y - z 4 ) 1 2 . max { ( f x , f y ) , d ( f y , f z ) } .

Similarly in all other cases d ( T ( x , y ) , T ( y , z ) ) 1 2 . m a x { ( f x , f y ) , d ( f y , f z ) } . Thus, f and T satisfy condition (3.2) with ϕ(x1, x2) = max{x1, x2}. We see that C(f, T) = 1, f and T commute at 1. Finally, 1 is the unique common fixed point of f and T.

4. An application to markov process

Let Δ n - 1 = { x R + n : Σ i = 1 n x i = 1 } denote the n - 1 dimensional unit simplex. Note that any x Δn-1may be regarded as a probability over the n possible states. A random process in which one of the n states is realised in each period t = 1, 2, ... with the probability conditioned on the current realised state is called Markov Process. Let a ij denote the conditional probability that state i is reached in succeeding period starting in state j. Then, given the prior probability vector x t in period t, the posterior probability in period t + 1 is given by x i t + 1 = a i j x j t for each i = 1, 2, .... To express this in matrix notation, we let x t denote a column vector. Then, x t+1 = Ax t . Observe that the properties of conditional probability require each a ij ≥ 0 and i = 1 n a i j = 1 for each j. If for any period t, xt+ 1= x t then x t is a stationary distribution of the Markov Process. Thus, the problem of finding a stationary distribution is equivalent to the fixed point problem Ax t = x t .

For each i, let ε i = min j a ij and define ε = i = 1 n ε i .

Theorem 4.1. Under the assumption a i,j > 0, a unique stationary distribution exist for the Markov process.

Proof. Let d: Δn-1× Δn-1R2 be given by d ( x , y ) = ( i = 1 n x i - y i , α i = 1 n x i - y i ) for all x, y Δn-1and some α ≥ 0.

Clearly d(x, y) ≥ (0, 0) for all x, y Δn-1and d ( x , y ) = ( 0 , 0 ) ( i = 1 n x i - y i , α i = 1 n x i - y i ) = ( 0 , 0 ) x i - y i = 0 for all i x = y. Also x = y x i = y i for all i x i - y i = 0 i = 1 n x i - y i = 0 d ( x , y ) = ( 0 , 0 )
d ( x , y ) = ( i = 1 n x i - y i , α i = 1 n x i - y i ) = ( i = 1 n y i - x i , α i = 1 n y i - x i ) = d ( y , x ) d ( x , y ) = ( i = 1 n x i - y i , α i = 1 n x i - y i ) = ( i = 1 n ( x i - z i ) + ( z i - y i ) , α i = 1 n ( x i - z i ) + ( z i - y i ) ) ( i = 1 n ( x i - z i ) + ( z i - y i ) , α i = 1 n ( x i - z i ) + ( z i - y i ) ) = ( i = 1 n ( x i - z i ) , α i = 1 n ( x i - z i ) ) + ( i = 1 n ( z i - y i ) , α i = 1 n ( z i - y i ) ) = d ( x , z ) + d ( z , x ) .
So Δn-1is a cone metric space. For x Δn-1, let y = Ax. Then each y i = j = 1 n a i j x j 0 . Further more, since each i = 1 n a i j = 1 , we have i = 1 n y i = i = 1 n j = 1 n a i j x j = j = 1 n x j i = 1 n a i j = j = 1 n x j = 1 , so y Δn-1. Thus, we see that A: Δn-1→ Δn-1. We will show that A is a contraction. Let A i denote the i th row of A. Then for any x, y Δn-1, we have
d ( A x , A y ) = ( i = 1 n | ( A x ) i ( A y ) i | , α i = 1 n | ( A x ) i ( A y ) i | ) = ( i = 1 n | j = 1 n a i j x j a i j y j | , α i = 1 n | j = 1 n a i j x j a i j y j | ) = ( i = 1 n | j = 1 n ( a i j ε i ) ( x j y j ) + ε i ( x j y j ) | , α i = 1 n | j = 1 n ( a i j ε i ) ( x j y j ) + ε i ( x j y j ) | ) ( i = 1 n ( | j = 1 n ( a i j ε i ) ( x j y j ) | + ε i | j = 1 n ( x j y j ) | ) , α ( i = 1 n ( | j = 1 n ( a i j ε i ) ( x j y j ) | + ε i | j = 1 n ( x j y j ) | ) ( i = 1 n j = 1 n ( a i j ε i ) | x j y j | , α i = 1 n j = 1 n ( a i j ε i ) | x j y j | ) ( S i n c e j = 1 n ( x j y j ) = 0 ) = ( j = 1 n | x j y j | i = 1 n ( a i j ε i ) , α j = 1 n | x j y j | i = 1 n ( a i j ε i ) = ( j = 1 n | x j y j | ( 1 ε ) , α j = 1 n | x j y j | ( 1 ε ) ) ( S i n c e i = 1 n a i j = 1 and i = 1 n ε i = ε ) = d ( x , y ) ( 1 ε )

which establishes that A is a contraction mapping. Thus, Theorem 3.2 with k = 1 and f as identity mapping ensures a unique stationary distribution for the Markov Process. Moreover for any x0 Δn-1, the sequence < A n x0 > converges to the unique stationary distribution. □

Declarations

Acknowledgements

The authors would like to thank the learned referees for their valuable comments which helped in bringing this paper to its present form. The first and third authors are supported by Ministry of Education, Kingdom of Saudi Arabia.

Authors’ Affiliations

(1)
Department of Mathematics, College of Science, Al-Kharj University
(2)
Department of Mathematics and Computer Science, St. Thomas College

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© George et al; licensee Springer. 2011

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/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.