On Common Fixed Points Results in Cone Rectangular Metric Spaces

1. INTRODUCTION

Recently, Guang and Xian [4] introduced a new metric space known as a cone metric space. Subsequently Azam et al. [1] have given the idea of cone rectangular metric space. Here, we will introduce some new results on cone rectangular metric space. Let E is a real Banch space and P is a subset of E.P. is called a cone if and only if it satisfies the following conditions. (i) P is closed, non-empty and P{0} (ii) (iii) Definition – 1 : Let X be a nonempty set. Suppose that d is a mapping from X × X E, satisfies : (i) (ii) (iii) (iv) Then d is called a cone metric on X and (X, d) is called cone metric space. Definition-2 : Let x be a nomempty set. Suppose the mapping d : x × x  E, satisfies.

(i) and

(ii) (iii) and for all distinct point w,zX{x,y} [rectangular property] Then d is called a cone rectangular metric on X, and (X, d) is called a cone rectangular metric space. Definition -3 : Let {xn} be a sequence in (X, d) and x (X, d). If for every c E, with 0 < < c there is n0 N such that for all n > no, d (xn, x) << c, then {x0} is said to be convergent and converges to x. i.e. Definition-4 : A sequence (xn) is said to be Cauchy in X if for c E with 0 < < c there is n0 N such that for all then {xn} is called Cauchy sequence. Definition – 5: A cone rectangular metric space is said to be complete cone rectangular metric space if evry Canchy sequence is X is convergent.

Theorem (1.1) : Let (X, d) is a complete cone rectangular metric space and P is a normal cone with normal constant K. Let f is self-mapping from X into itself satisfying. Then f has an unique fixed point in X. Proof : Let be an arbitrary point in X. Let us take a sequence {xo} in X such that Now substituting and in inequality we obtain 0112d(x,fx)d(x,x) 120112d(x,x)({d(xx)d(x,x)}

Where Again for we have Thus in general we have for positive integer n Taking the normality of Cone. (1.4) gives Which yields Now we will claim that our inequality satisfies the rectangular property for finding the fixed point in X. Because of this we will calculate the following results. For we have

2d(fy,f)d(y,fy)hd(y,fy)1



Again Thus in general for positive integer n Now from rectangular property we have for Similarly

Thus in general for n > m and from Lemma [3, 5] Now for Applying the normality of cone we obtain., Which implies that Now we can conclude from (1.6) and (1.10) that (xn) is a Cauchy sequence in X. As x is complete cone rectangular metric space then there exists a point x in (X, d) such that Now Letting we get Now applying the normality of cone we have, Hence x is a common fixed point off in x. Now we will prove that x is unique. If possible let there exists another fixed point . Then

Again a contradiction and hence Hence x is an unique common fixed point off in X. This completes the proof of the theorem.

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