Skip to main content

Advertisement

Research | Open | Published:

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

Abstract

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

2000 Mathematics Subject Classification

47H10

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: XkX 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: Xk → 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: XkX 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: Ek → 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: XkX 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: XkX 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) < kn λn d(f fu, fu). So kn λn d(f fu, fu) - d(f fu, fu) P for all n ≥ 1. Since kn λn → 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) ≤ kn λn 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 = R2, 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: X2X 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 .max { ( 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 xt 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 xt denote a column vector. Then, xt+1 = Axt. 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= xt then xt is a stationary distribution of the Markov Process. Thus, the problem of finding a stationary distribution is equivalent to the fixed point problem Axt = xt.

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 =0d ( 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 < Anx0 > converges to the unique stationary distribution. □

References

  1. 1.

    Presic SB: Sur la convergence des suites, Comptes. Rendus. de 1'Acad des Sci de Paris 1965, 260: 3828–3830.

  2. 2.

    Ciric LjB, Presic SB: On Presic type generalisation of Banach contraction principle. Acta Math Univ Com 2007,LXXVI(2):143–47.

  3. 3.

    Huang LG, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. J Math Anal Appl 2007,332(2):1468–1476. 10.1016/j.jmaa.2005.03.087

  4. 4.

    Rezapour S, Hamlbarani R: Some notes on the paper cone metric spaces and fixed point theorems of contractive mappings. Math Anal Appl 2008,345(2):719–724. 10.1016/j.jmaa.2008.04.049

  5. 5.

    Di Bari C, Vetro P: ϕ -pairs and common fixed points in cone metric spaces. Rendiconti del circolo Matematico di Palermo 2008,57(2):279–285. 10.1007/s12215-008-0020-9

  6. 6.

    Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. J Math Anal Appl 2008,341(1):416–420. 10.1016/j.jmaa.2007.09.070

  7. 7.

    Abbas M, Rhodades BE: Fixed and periodic point results in cone metric spaces. Appl Math Lett 2009,22(A):511–515.

  8. 8.

    Di Bari C, Vetro P: Weakly ϕ -pairs and common fixed points in cone metric spaces. Rendiconti del circolo Matematico di Palermo 2009,58(1):125–132. 10.1007/s12215-009-0012-4

  9. 9.

    Ilic D, Rakocevic V: Common fixed points for maps on cone metric space. Math Anal Appl 2008,341(2):876–882. 10.1016/j.jmaa.2007.10.065

  10. 10.

    Arandjelovic I, Kadelburg Z, Radenovic S: Boyd-Wong type common fixed point results in cone metric spaces. Appl Math Comput 2011, 217: 7167–7171.

  11. 11.

    Raja P, Vaezpour SM: Some extensions of Banch's contraction principle in complete cone metric spaces. Fixed Point Theory Appl 2008, 2008: 11. Article ID 768294

  12. 12.

    Jankovic S, Kadelburg Z, Radenovic S: On cone metric spaces: a survey. Nonlin Anal 2011, 74: 2591–2601. 10.1016/j.na.2010.12.014

  13. 13.

    Simic S: A note on Stone's, Baire's, Ky Fan's and Dugundj's theorem in tvs-cone metric spaces. Appl Math Lett 2011, 24: 999–1002. 10.1016/j.aml.2011.01.014

  14. 14.

    Vetro P: Common fixed points in Cone metric spaces. Rendiconti del circolo Matematico di Palermo, Serie II 2007,56(3):464–468. 10.1007/BF03032097

  15. 15.

    Kadelburg Z, Radenovic S, Rakocevic V: A note on the equivalence of some metric and cone fixed point results. Appl Math Lett 2011, 24: 370–374. 10.1016/j.aml.2010.10.030

Download references

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.

Author information

Correspondence to Reny George.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

RG gave the idea of this work. All authors worked on the proofs and examples. KPR and RR drafted the manuscript. RG read the manuscript and made necessary corrections. All authors read and approved the final manuscript.

Rights and permissions

Reprints and Permissions

About this article

Keywords

  • Coincidence and common fixed points
  • cone metric space; weakly compatible