FOUR MAPPINGS WITH A COMMON FIXED POINT ON E-b-METRIC SPACES

In this paper, we consider E-b-metric space, which is a generalized vector metric space.The metric is Riesz space valued. Here, we prove some results concerning common fixed point for four mappings on E-b-metric spaces. This generalizes the results of Rahimi, Abbas and Rad [4 ]. Keyword E-b-metric space, Riesz space, weak annihilators, weakly compatible, weakly increasing

Then (X, d, E) is said to be a vector metric space or E-metric space. Vector metric Space generalizes metric space. For x, y, z, w of a vector metric space, the following statements hold: (i) 0 ≤ d(x, y) (ii) d(x, y) = d(y, x) (iii) |d(x, w)  d(y, w)| ≤ d(x, y) (iv) |d(x, w)  d(y, z)| ≤ d(x, y) + d(w, z) A sequence (x n ) in a vector metric space (X, d, E) is said to be vectorial convergent or E-convergent to some x  E, written as d.E n x x  if  (a n ) in E such that a n  0 and d(x n , x) ≤ a n  n.
A sequence (x n ) is said to be ECauchy sequence if there exists a sequence (a n ) in E such that a n  0 and d(x n , x n +p ) ≤ a n holds  n and p.
A vector metric space X is called Ecomplete if each ECauchy sequence in X, vectorial converges to a limit in X.
When E = R(the set of real numbers) the concepts of vectorial convergence and metric convergence, ECauchy sequence and Cauchy sequence are the same. If X = E and d is the absolute valued vector metric on X, then the concepts of vectorial convergence and order convergence are the same. Example 1.9 . A Riesz space E is a vector metric space with d : E  E→ E defined by d(x, y) = | x -y |  x , y X, is called absolute valued metric on E. Definition 1. 10. Let X be a nonempty set and let s  1 be a given real number. A function d : X  X  R + is called b-metric if it satisfies the following conditions : Fixed point theorems on bmetric spaces have been studied by several authors, one can see ( [10], [12]).
I.R. Petre [13] defined vector b-metric or Ebmetric as follows : Definition 1.11. Let X be a nonempty set and s  1. A function d : X  X  E + is called Ebmetric if for any x, y, w X, the following conditions hold: Hence ( X, d, R 2 ) is E-b-metric space with parameter s = 2  1.
Then (X, d, E) is E-b-metric space with parameter s = 2  1.Since the function Thus relaxed triangular inequality holds with s = 2 p-1  1. We give below an example of E-b-metric space which is not a metric space.
Example 1. 15. Let X = {0, 1, 2} , E = R 2 and d : X × X → R 2 be defined as Let (X, d, E) be E-b metric space and P and Q be self maps on X.
Definition 1.16 [15]. Let y = Px = Qx for some x  X then y is said to be a point of coincidence and x a coincidence point of P and Q.
Definition 1.17 [15]. The maps P and Q are called weakly compatible if they commute at every coincidence point.
Lemma 1.18 [15]. The maps P and Q are weakly compitable maps X If P and Q have a unique point of coincidence c = Pc = Qc, then c is the unique common fixed point of Pand Q.
Let (X, ≼) be a partially ordered set and P and Q are self maps on X.
Definition 1.19 [15]. The pair (P, Q) of maps is said to be weakly increasing if Px ≼ QPx and Qx ≼ PQx Definition 1.20 [15]. The pair (P, Q) is said to be partially weakly increasing if Px ≼ QPx Observe that the pair (P, Q) is weakly increasing iff ordered pair (P, Q) and (Q, P) are partially weakly increasing.
Definition 1.21 [15]. The mapping P is called a weak annihilator of Q if PQx ≼ x  x X. Definition 1.22 [15]. The mapping P is called dominating if x Px  x  X.

II. MAIN RESULTS
In this section, we prove some common fixed point theorems for four mappings in Ebmetric space. Proof : Let x 0 be arbitrary point of X. Since A(X)  H(X) and B(X)  G(X), we can choose x 1  X such that Ax 0 = Hx 1 , x 2  X such that Bx 1 = Gx 2 . Continuing this process, construct a sequence {y n }, which is defined as : y 2n-1 = Hx 2n-1 = Ax 2n-2 and y 2n = Gx 2n = Bx 2n-1 , n 0 . . By given assumptions, Thus, for all n we have, x n ≤ x n+1. Firstly, we prove that Therefore  n and p, d(y n , y n+p )≤ s d(y n , y n+1 ) + s 2 d(y n+1 , y n+2 ) +s 3 d(y n+2 , y n+3 )+…….+ s p d(y n+p-1 , y n+p ) ≤ s λ n d(y 0 , y 1 ) + s 2 λ n+1 d(y 0 , y 1 ) +………….+s p λ n+p-1 d(y 0 , y 1 ) It is given that E is Archimedean, so the sequence (y n ) is an ECauchy . It is assumed Thus, there exists a sequence {e n } in E such that e n  0 and d(Gx 2n , u) ≤ e n . Further we can find out a v  X such that Gv = u. Now it remains to prove that Av = u. Although n. There are five possibilities:   We define an ordering on X as x y ⇔ y x x, y ∈ X Let A, B, G, H : X→X be defined by

Clearly, A(X)  H(X) and B (X)  G(X) and the pairs (H, A) and (G, B) partially weakly increasing that
is Hx = ≥ x = AHx, which gives Hx AHx and Gx = x = BGx, which gives Gx BGx.
Also, A and B are dominating maps, that is , Ax = ≤ x and Bx = ≤ x for all x ∈ X implies that x Ax and x Bx for all x ∈ X. Furthormore, A and B are weak annihilators of H and G, respectively, that is, AHx x and BGx x for all x ∈ X.