Abstract
A generalised common fixed point theorem of Presic type for two mappings f: X → X and T: X^{k }→ X in a cone metric space is proved. Our result generalises many wellknown results.
2000 Mathematics Subject Classification
47H10
Keywords:
Coincidence and common fixed points; cone metric space; weakly compatible1. 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:
where x_{1}, x_{2}, ..., x_{k+1 }are arbitrary elements in X and q_{1}, q_{2}, ..., q_{k }are nonnegative constants such that q_{1 }+ q_{2 }+ · · · + q_{k }< 1. Then, there exists some x ∈ X such that x = T(x, x, ..., x). Moreover if x_{1}, x_{2}, ..., x_{k }are arbitrary points in X and for n ∈ N x_{n+k }= T(x_{n}, x_{n+1}, ..., x_{n+k1}), 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 wellknown 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:
where x_{1}, x_{2}, ..., x_{k+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 x_{1}, x_{2}, ..., x_{k }are arbitrary points in X and for n ∈ Nx_{n+k }= T(x_{n}, x_{n+1}, ..., x_{n+k1}), 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 nonnormal 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 nonnormal 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, nonempty, and satisfies P ≠ {0},
(ii) ax + by ∈ P for all x, y ∈ P and nonnegative 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:
(d_{1}) 0 ≤ d (x, y) for all x, y ∈ X and d (x, y) = 0 if and only if x = y
(d_{2})d (x, y) = d (y, x) for all x, y ∈ X
(d_{3})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:
(a) A convergent sequence if for every c ∈ E with 0 ≪ c, there is n_{0 }∈ N such that for all n ≥ n_{0}, d (x_{n}, x) ≪ c for some x ∈ X. We denote this by lim_{n→∞ }x_{n }= x.
(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, n ≥ n_{0}.
(c) A cone metric space (X, d) is said to be complete if every Cauchy sequence in X is convergent in X.
(d) A selfmap 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 : 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: E^{k }→ E such that
(a) ϕ is an increasing function, i.e x_{1 }< y_{1}, x_{2 }< y_{2}, ..., x_{k }< y_{k }implies ϕ(x_{1}, x_{2}, ..., x_{k}) < ϕ(y_{1}, y_{2}, ..., y_{k}).
(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: X^{k }→ X and f: X → X be mappings satisfying the following conditions:
where x_{1}, x_{2}, ..., x_{k+1 }are arbitrary elements in X and and
there exist elements x_{1}, x_{2}, ..., x_{k }in X such that
where . 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}, x_{n+1}, ..., x_{n+k1}), n = 1, 2, ... Let α_{n }= d(y_{n}, y_{n+1}). By the method of mathematical induction, we will now prove that
for all n. Clearly by the definition of R, (3.5) is true for n = 1, 2, ..., k. Let the k inequalities α_{n }≤ Rθ^{n}, α_{n+1 }≤ Rθ^{n+1}, ..., α_{n+k1 }≤ Rθ^{n+k1 }be the induction hypothesis. Then, we have
Thus inductive proof of (3.5) is complete. Now for n, p ∈ N, we have
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 N_{1 }such that , for all n > N_{1}. Then, for all n ≥ N_{1}. Thus, for all n ≥ N_{1}. Hence, sequence < y_{n }> is a Cauchy sequence in f(X), and since f(X) is complete, there exists v, u ∈ X such that lim_{n→∞}y_{n }= v = f(u). Choose a natural number N_{2 }such that and for all n ≥ N_{2}.
Then for all n ≥ N_{2}
So, for all m ≥ 1. Since 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: X → X be mappings satisfying (3.1), (3.2), (3.3) and let there exist elements x_{1}, x_{2}, ... 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 x_{1}, x_{2}, ...,x_{k }are arbitrary points in X and for n ∈ N, y_{n+k }= f(x_{n+k}) = T(x_{n}, x_{n+1}, ... x_{n+k1}), 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 lim_{n→∞ }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,
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,
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 × X → E such that d(x, y) = (x  y , x  y ). Then, d is a cone metric on X. Let T: X^{2 }→ X and f: X → X be defined as follows:
T and f satisfies condition (3.2) as follows:
Case 1. x, y, z ∈ [0, 1]
Case 2. x, y ∈ [0, 1] and z ∈ [1,2]
Case 3. x ∈ [0, 1] and y; z ∈ [1,2]
Similarly in all other cases . Thus, f and T satisfy condition (3.2) with ϕ(x_{1}, x_{2}) = max{x_{1}, x_{2}}. 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 denote the n  1 dimensional unit simplex. Note that any x ∈ Δ_{n1 }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 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 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 for each j. If for any period t, x^{t+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 .
Theorem 4.1. Under the assumption a_{i,j }> 0, a unique stationary distribution exist for the Markov process.
Proof. Let d: Δ_{n1 }× Δ_{n1 }→ R^{2 }be given by for all x, y ∈ Δ_{n1 }and some α ≥ 0.
Clearly d(x, y) ≥ (0, 0) for all x, y ∈ Δ_{n1 }and for all i ⇒ x = y. Also x = y ⇒ x_{i }= y_{i }for all
So Δ_{n1 }is a cone metric space. For x ∈ Δ_{n1}, let y = Ax. Then each . Further more, since each , we have , so y ∈ Δ_{n1}. Thus, we see that A: Δ_{n1 }→ Δ_{n1}. We will show that A is a contraction. Let A_{i }denote the ith row of A. Then for any x, y ∈ Δ_{n1}, we have
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 x^{0 }∈ Δ_{n1}, the sequence < A^{n}x^{0 }> 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.
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.
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