Some common fixed point results in complete cone metric space with W- distance

Introduction

Fixed point theory is a basic tool in application of various branches of mathematics from linear algebra, elementary calculus to topology and analysis. This theory is very closed to other disciplines like Game theory, Optimization theory, Physics, Economics etc. Fixed point theory has been developed in various spaces in last century. In modern day this theory is very favourite for researchers. The concept of fixed point theory was firstly propounded by Poincare [9] in 1886. After that in 1912, Brower [2] proved fixed point theorem for solution of equation f(x) = x. In 1922, Polish mathematician Stefan Banach [1] who was one of the founders of functional analysis gave a principle which was one of the fundamental principles in the field of functional analysis. In that principle S. Banach stated as follow-"Every contraction mapping of a complete metric space X into itself has a unique fixed point." (Bonsall 1962). In 2007, concept of a metric space was generalized by Huang and Zhang [5] by replacing the set of real numbers by an ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. On the other hand during last decade of 20 th century Osama kada [6] and Naoki Shioji [11] introduced the concept of metric space with w-distance, gave some examples, properties of w-distance and improved some known results also. By using this concept proved a complete metric space which was generalized the fixed point theorems of Subrahmanyan [12] , Kannan [7] and Ciric [3] . In consecutive years various mathematician established some fixed point theorems. In this sequel in 2011, Dhanorkar and Salunke [4] and in 2012 Manish Sharma and Rajesh Shrivastva [10] obtained some fixed point results for cone metric space with w-distance. In 2013, S. k. Tiwari, R. P. Dubey [15] proved some fixed point theorems for generalized contractive mapping in cone metric space and with A. K. Dubey [16] they gave common fixed point results in cone metric spaces. Recently in (July 2017), S. K. Tiwari, and Kaushik Das [14] extended some common fixed point results for contractive mappings in cone metric spaces. In this research paper we will define w-distance, cone metric space, normality of cone and proof some new common fixed point theorems for contractive mapping on cone metric space with w-distance. This result improved and modified some well known results in literatures also.

Preliminaries

We recall some standard notations and definitions

where α≥0 is a constant. Then (X, d) is cone metric space.

Definition 1.2.5. Let X be cone metric space with d. Suppose the mapping: ,( ,) .

x sequence is called a p-convergent to a point  whenever for every ,0 , E   there is a positive integer N such that, for all , ( , ) . (3) (X, d) is a complete cone metric space with w-distance if every Cauchy sequence is p-convergent.

Lemma 1.2.7:

There is not normal cone with normal constant M<1.

Main Result

In this section we will establish following three theorems which improved various existed fixed point results.

1-ℎ 𝐾𝑝(𝑥 0 , 𝑥 1 ) (since P is normal).

Therefore {𝑥 2𝑘 } is a Cauchy sequence in X. Since (𝑋, 𝑑) is a complete cone metric space. Therefore, there exist 𝑥 * ∈ 𝑋 such that 𝑥 2𝑘 → 𝑥 * as 𝑘 → ∞. Now, 𝑝(𝑇 1 𝑥 * , 𝑥 * ) ≤ 𝑝(𝑇 1 𝑥 2𝑘+1 , 𝑇 1 𝑥 * ) + 𝑝((𝑇 1 𝑥 2𝑘+1 , 𝑥 * ) 0  as 𝑘 → ∞.

𝑆𝑖𝑛𝑐𝑒 𝑥 2𝑘 → 𝑥 * 𝑎𝑛𝑑 𝑇 1 𝑥 2𝑘+1 → 𝑇 1 𝑥 * 𝑎𝑠 𝑘 → ∞. Therefore 𝑝(𝑇 1 𝑥 * , 𝑥 * ) = 0 . This implies that, 𝑇 1 𝑥 * =𝑥 * . Hence 𝑥 * is a fixed point of 𝑇 1 .

Similarly, it can be established that 𝑇 2 𝑥 * = 𝑥 * . Therefore, 𝑇 1 𝑥 * = 𝑥 * =, 𝑇 1 𝑥 * . Thus 𝑥 * is the common fixed point of pair of maps 𝑇 1 and 𝑇 2 .This completes the proof of this given theorem. Theorem 1.3.2: Let (𝑋, 𝑑) be a complete cone metric space with w-distance p and let P be a normal cone with normal constant K. Suppose 𝑇 1 , 𝑇 2 ∶ 𝑋 → X be any two continuous and surjective mappings satisfying the contractive condition

for all x, y ∈ X, where 𝛼 ≥ 0, 𝛽 < 1 𝑎re constant, with 3𝛼 + 2 𝛽 < 1. Then 𝑇 1 and 𝑇 2 have a common fixed point in X.

Proof: Let xo be an arbitrary point in 𝑋. Since 𝑇 1 and 𝑇 2 be onto mappings. Then, there exist xo ∈X and x1 ∈X such that,

In this way, we define the sequences {𝑥 2𝑘 } and {𝑥 2𝑘+1 } by Similarly, it can be establish that 𝑇 2 𝑥 * = 𝑥 * . Therefore, 𝑇 1 𝑥 * = 𝑥 * =, 𝑇 1 𝑥 * . Thus 𝑥 * is the common fixed point of pair of maps 𝑇 1 and 𝑇 2 . This completes the proof.

Theorem 1.3.3: Let (𝑋, 𝑑) be a complete cone metric space with w-distance p and consider P be a normal cone with normal constant K. Suppose 𝑇 1 , 𝑇 2 ∶ 𝑋 → be any two continuous and onto mappings satisfying the contractive condition. Therefore {𝑥 2𝑘 } is a Cauchy sequence in X. Since (𝑋, 𝑑) is a complete cone metric space. Therefore, there exist 𝑥 * ∈ 𝑋 such that 𝑥 2𝑘 → 𝑥 * as 𝑘 → ∞. Now, 𝑝(𝑇 1 𝑥 * , 𝑥 * ) ≤ 𝑝(𝑇 1 𝑥 2𝑘+1 , 𝑇 1 𝑥 * ) + 𝑝((𝑇 1 𝑥 2𝑘+1 , 𝑥 * ) 0  𝑎𝑠 𝑘 → ∞.

𝑆𝑖𝑛𝑐𝑒 𝑥 2𝑘 → 𝑥 * 𝑎𝑛𝑑 𝑇 1 𝑥 2𝑘+1 → 𝑇 1 𝑥 * 𝑎𝑠 𝑛 → ∞. Therefore 𝑝(𝑇 1 𝑥 * , 𝑥 * ) = 0 . This implies that, 𝑇 1 𝑥 * =𝑥 * . Hence 𝑥 * is a fixed point of 𝑇 1 Similarly, it can be establish that 𝑇 2 𝑥 * = 𝑥 * . Therefore, 𝑇 1 𝑥 * = 𝑥 * =, 𝑇 1 𝑥 * . Thus 𝑥 * is the common fixed point of pair of maps 𝑇 1 and 𝑇 2 . This completes the proof.

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