New Conditions of The Existence of Fixed Point in

The main idea is to construct a new algebra and find new necessary and sufficient conditions equivalent to the existence of fixed point. In this work, an algebra is constructed, called ∆ordered Banach algebra, we define convergent in this new space, Topological structure on ∆ − ordered Banach Algebra and prove this as Housdorff space. Also, we define new conditions as ∆ − lipshtiz , , ∆ − contraction conditions in this algebra construct, we prove this condition is the existence and uniqueness results of the fixed point. In this paper , we prove a common fixed point if the self-functions satisfy the new condition which is called φ − contraction .

a new approximate value x2 = T (x1). Thus, a sequence of root values can be generated by applying the formula xn + 1 = T (xn) for n = 0.1,2, ... The fixed point a in the equation above represents the distance of the intersection point of the curves of y = x, y = T (x) for each axis x, y . If x0 is the initial fixed point, then T(x0) is the length of the column from x0 on the x axis until it intersects the curve of the T-function and since the points on the rectangle y = x are equal to the distance from both axes y and x, so the line passing at the point (x0, T (x0)) rectangle the x-axis will intersect the line y=x in the x-axis, represent x1 where x 1=T(x0) In a similar way, we find the remaining points where xn + 1 = T (xn). Here, we ask the following question: How do we choose the function T to ensure that the generated values are converged from the repeated formula xn + 1 = T (xn)?
To answer the question , we can prove the existence and uniqueness of fixed point under some new conditions by constructing a new algebra called ∆ − ℎ .

1-∆-Ordered Banach Algebra
We start this section by a definition of Banach algebra.

"Definition (2.1)[2]
:let is a linear space over field of real numbers . is called Banach algebra if is Banach space with an operation of multiplication is defined as following :for , , ∈ , for all ∈
That is ≺ and this contradiction.

3-Main Results
Then is a unique fixed point in .

5-Conclusion
In this paper, we introduce a new concept which is called ∆ordered Banach algebra. Also, we define ℎ ℎ , − , ∆ − and ∆ − ℎ . In the new work, we prove fixed point theorems satisfying these maps in ∆ordered Banach algebra. Our conditions and results are new in comparison with those of the results of cone metric space .These results can be extended to other spaces.