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

Reny George12*, KP Reshma2 and R Rajagopalan1

Author Affiliations

1 Department of Mathematics, College of Science, Al-Kharj University, Al-Kharj, Kingdom of Saudi Arabia

2 Department of Mathematics and Computer Science, St. Thomas College, Ruabandha Bhilai, Durg, Chhattisgarh State, 490006, India

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Fixed Point Theory and Applications 2011, 2011:85  doi:10.1186/1687-1812-2011-85

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2011/1/85

Received:23 March 2011
Accepted:23 November 2011
Published:23 November 2011

© 2011 George et al; licensee Springer.

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.


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

2000 Mathematics Subject Classification


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: Xk X be a mapping satisfying the following condition:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M1">View MathML</a>


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

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:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M2">View MathML</a>


where x1, x2, ..., xk+1 are 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, ..., xk are arbitrary points in X and for n Nxn+k = T(xn, xn+1, ..., xn+k-1), then the sequence < xn > is convergent and lim xn = T(lim xn, lim xn, ..., lim xn). 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 [5-15].

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

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

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

(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 x y if and only if y - x P. We shall write x < y if x y and x y, 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 ≤ x y implies || x || ≤ K || y || .

Definition 2.1. [3]Let X be a non empty set. Suppose that the mapping d: X × X E 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 {xn} in X is said to be:

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

(b) A Cauchy sequence if for all c E with 0 ≪ c, there is no ∈ N such that d (xm, xn) ≪ c, for all m, n n0.

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

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

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

(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.

(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.

(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

(a) ϕ is an increasing function, i.e x1 < y1, x2 < y2, ..., xk < yk implies ϕ(x1, x2, ..., xk) < ϕ(y1, y2, ..., yk).

(b) ϕ(t, t, t, ...) ≤ t, for all t X

(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: Xk X and f: X X be mappings satisfying the following conditions:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M3">View MathML</a>


<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M4">View MathML</a>


where x1, x2, ..., xk+1 are arbitrary elements in X and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M5">View MathML</a>and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M6">View MathML</a>


there exist elements x1, x2, ..., xk in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M7">View MathML</a>


where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M8">View MathML</a>. Then, f and T have a coincidence point, i.e. C(f, T) ≠ ∅.

Proof. By (3.1) and (3.4) we define sequence < yn > in f(X) as yn = fxn for n = 1, 2, ..., k and yn+k = f(xn+k) = T(xn, xn+1, ..., xn+k-1), n = 1, 2, ... Let αn = d(yn, yn+1). By the method of mathematical induction, we will now prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M9">View MathML</a>


for all n. Clearly by the definition of R, (3.5) is true for n = 1, 2, ..., k. Let the k inequalities αn n, αn+1 n+1, ..., αn+k-1 n+k-1 be the induction hypothesis. Then, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M10">View MathML</a>

Thus inductive proof of (3.5) is complete. Now for n, p N, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M11">View MathML</a>

Let 0 ≪ c be given. Choose δ > 0 such that c + Nδ(0) ⊆ P where Nδ(0) = {y E; || y || < δ}. Also choose a natural number N1 such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M12">View MathML</a>, for all n > N1. Then, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M13">View MathML</a> for all n N1. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M14">View MathML</a> for all n N1. Hence, sequence < yn > is a Cauchy sequence in f(X), and since f(X) is complete, there exists v, u X such that limn→∞yn = v = f(u). Choose a natural number N2 such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M15">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M16">View MathML</a> for all n N2.

Then for all n N2

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M17">View MathML</a>

Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M18">View MathML</a> for all m ≥ 1.

So, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M19">View MathML</a> for all m ≥ 1. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M20">View MathML</a> 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: Xk X and f: X X be mappings satisfying (3.1), (3.2), (3.3) and let there exist elements x1, x2, ... xk 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, ...,xk are arbitrary points in X and for n N, yn+k = f(xn+k) = T(xn, xn+1, ... xn+k-1), n = 1, 2, ..., then the sequence < yn > is convergent and lim yn = f(lim yn) = T(lim yn, lim yn, ..., lim yn).

Proof. As proved in Theorem 3.1, there exists v, u X such that limn→∞ yn = 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,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M21">View MathML</a>

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 yn = f(lim yn) = T(lim yn, lim yn, ... lim yn). Now suppose x, y be two fixed points of f and T. Then,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M22">View MathML</a>

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 × X E such that d(x, y) = (|x - y |, |x - y |). Then, d is a cone metric on X. Let T: X2 X and f: X X be defined as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M23">View MathML</a>

T and f satisfies condition (3.2) as follows:

Case 1. x, y, z ∈ [0, 1]

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M24">View MathML</a>

Case 2. x, y ∈ [0, 1] and z ∈ [1,2]

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M25">View MathML</a>

Case 3. x ∈ [0, 1] and y; z [1,2]

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M26">View MathML</a>

Case 4. x, y, z [1,2]

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M27">View MathML</a>

Similarly in all other cases <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M28">View MathML</a>. 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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M29">View MathML</a> denote the n - 1 dimensional unit simplex. Note that any x ∈ Δn-1 may 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 aij 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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M30">View MathML</a> 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 aij ≥ 0 and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M31">View MathML</a> 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 εi = minjaij and define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M32">View MathML</a>.

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

Proof. Let d: Δn-1 × Δn-1 R2 be given by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M33">View MathML</a> for all x, y ∈ Δn-1 and some α ≥ 0.

Clearly d(x, y) ≥ (0, 0) for all x, y ∈ Δn-1 and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M34">View MathML</a> for all i x = y. Also x = y xi = yi for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M35">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M36">View MathML</a>

So Δn-1 is a cone metric space. For x ∈ Δn-1, let y = Ax. Then each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M37">View MathML</a>. Further more, since each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M38">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M39">View MathML</a>, so y ∈ Δn-1. Thus, we see that A: Δn-1 → Δn-1. We will show that A is a contraction. Let Ai denote the ith row of A. Then for any x, y ∈ Δn-1, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2011/1/85/mathml/M40">View MathML</a>

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. □

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.


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.


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