International Journal of Applied Mathematical Research Assignment

Assignment Task

Introduction

Huang and Zhang, replaced the real numbers by an ordering Banach space, and defined a cone metric spaces(X, d) of contractive mappings and also discussed some properties of convergence of sequences; many authors have established and extend different types of contractive mappings in cone metric spaces see for instance., and also generalized the results by .The author proved fixed point results in cone metric spaces. The purpose of this paper is to obtain the generalization of results in [1] and 2.1, 2.2 of, by using non-normality of cone.

1.1. Let E be a real Banach space and P be a subset of E. P is called a cone if and only if:

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

(ii) a + b ∊ P for all , ∊ P and non – negative real number a, b;

(iii) ∊ P and - ∊ P => = 0 <=> P ∩ (-P) = {0}. Given a cone P ⊂ E,

we define a partial ordering ≤ on E with respect to P by x ≤ y if and only if – ∊ P.

We shall write ≪ y if – x ∊ intP, where int P denotes the interior of P.

The cone P is called normal if there is a number K > 0 such that , ∊ E, 0 ≤ x ≤ y implies || x || ≤ K || y ||.

The least positive number satisfying the above is called the normal constant P. The cone p is called regular if every increasing sequence which is bounded from above is convergent . That is , if { xn } is sequence such that x1 ≤ 2 ≤ ... n ≤ …≤ for some , then there is such that ⟶ 0 ( ⟶∞). Equivalently the cone p is regular if and only if every decreasing sequence which is bounded from below is convergent.

1.2. Let X be a non – empty set. Suppose the mapping satisfies

(i) d (x ,y ) = d ( (x ,y) for all , ∊ X;

(ii) d ((x ,y) ≤ d (x , z) + d (z, x) for all , ∊ X.

Then d is called a cone metric, on X and pair (X, d) is called a cone metric space. It is obvious that cone metric spaces generalize metric space.

1.3. Let (X, d) be a cone metric space, ∊ X and { n}n ≥ 1 a sequence in X. Then,

(i) { n} n ≥ 1 converges to x whenever for every c ∊ E with o ≪ c, there is a natural number N such that d ( n, ) ≪c for all n ≥ N. We denote this by limn→∞ n = or n , (n ∞).

(ii) { n}n ≥ 1 is said to be a Cauchy sequence if for every c ∊ E with o ≪ c, there is a natural number N such that d ( n, m) ≪ c for all n, m ≥ N.

(iii) (X, d) is called a complete cone metric space if every Cauchy sequence in X is convergent

1.4. Cone P is called minihedral cone if sup {x, y} exists for all x, y ∊ E and strongly minihedral if every subset of E which is bounded from above has a supremum. Lemma

1.5. Every strongly minihedral normal cone is regular.

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