A Unique Common Fixed Point Theorem for Four Maps under Contractive Conditions in Cone Metric Spaces

In this paper, we prove existence of coincidence points and a common fixed point theorem for four maps under contractive conditions in cone metric spaces for non continuous mappings and relaxation of completeness in the space. These results extend and improve several well known comparable results in the existing literature. AMS Subject Classification: 47H10, 54H25.


Introduction and preliminaries
In 2007 Huang and Zhang [3] have generalized the concept of a metric space, replacing the set of real numbers by an ordered Banach space and obtained some fixed point theorems for mapping satisfying different contractive conditions. Subsequently, Abbas and Jungck [1] and Abbas and Rhoades [2] have studied common fixed point theorems in cone metric spaces (see also [3,4] and the references mentioned therein). Recently, Abbas and Jungck [1] have obtained coincidence points and common fixed point theorems for two mappings in cone metric spaces .The purpose of this paper is to extend and improves the fixed point theorem of [6].
Throughout this paper, E is a real Banach space, N = {1,2,3,…} the set of all natural numbers. For the mappings f,g :X→X, let C(f,g) denotes set of coincidence points of f,g, that is C(f,g):= {z∈X : fz = gz }.
We recall some definitions of cone metric spaces and some of their properties [3]. Definition 1.1. Let E be a real Banach Space and P a subset of E .The set P is called a cone if and only if: (a) P is closed, nonempty and P }; (c) x∈P and -x ∈P implies x = 0.

Definition 1.2.
Let P be a cone in a Banach Space E , define partial ordering ' ≤ 'on E with respect to P by x ≤ y if and only if y-x P ∈ .We shall write x<y to indicate x y ≤ but x y ≠ while x<<y will stand for y-x∈Int P , where Int P denotes the interior of the set P. This Cone P is called an order cone. Definition 1.3. Let E be a Banach Space and P ⊂ E be an order cone .The order cone P is called normal if there exists L>0 such that for all x,y∈E, 0 y x ≤ ≤ implies ║x║ ≤ L ║y║. The least positive number L satisfying the above inequality is called the normal constant of P. Definition 1.4. Let X be a nonempty set of E .Suppose that the map d: X× X → E satisfies : for all x,y X ∈ ; for all x,y,z X ∈ . Then d is called a cone metric on X and (X,d) is called a cone metric space .
It is obvious that the cone metric spaces generalize metric spaces.
where α ≥ 0 is a constant .Then (X,d) is a cone metric space. Definition 1.5. Let (X,d) be a cone metric space .We say that {x n } is

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(a). a Cauchy sequence if for every c in E with 0 << c , there is N such that for all n , m > N, d(x n, x m ) << c ; (b). a convergent sequence if for any 0 << c, there is N such that for all n > N, d(x n, x) << c, for some fixed x∈ X.
A Cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X. Then the pair (f, g) is said to be (IT)-Commuting at z X ∈ if f(g(z)) = g(f(z)) with f(z) = g(z ).

(2). Common fixed point theorem
In this section, we obtain existence of coincidence points and a common fixed point Theorem for four maps on a cone metric space.
Which implies that ║d(y n , y m )║ → 0 as n , m → ∞ . Hence {y n } is a Cauchy sequence. Let us suppose that S(X) is complete subspace of X. Completeness on S(X) implies existence of z∈S(X) such that ∞ → n lim y 2n = Sx 2n = z.
That is, for any 0 << c, for sufficiently large n, we have d (y n, y ) << c.
Since z∈T(X) ⊆ I(X), then there exists a point u∈X such that z = Iu.
≤ k (0) + 0 = 0.That is Su =z. Therefore z = Su = Iu, that is u is a coincidence point of S and I.
Therefore z = Tv = Jv , that is v is a coincidence point of T and J. In view of (2.6) and (2.7) it follows S,T,I and J have a common fixed point namely z. Uniqueness, let z 1 be another common fixed point of S,T,I and J . Then ║d(z, z 1 )║ = ║d(Sz,Tz 1 )║ ≤ k║d(Iz , Jz 1 )║ ≤ k║d(z, z 1 )║ <║d(z, z 1 )║, (Since k<1) which is a contradiction. Implies z = z 1.