Common Fixed Point Result in Dislocated Quasi Metric Spaces

L. Sharmeswar Singh

Department of Teacher Education, Manipur University, Canchipur-795003, Manipur, INDIA.
email:lsharmeswar@gmail.com.

ABSTRACT

In this paper, a common fixed point theorem of a pair of self-maps is proved without appeal to continuity condition in dislocated quasi metric spaces. It extends and generalizes the result of Sarma et al.7, Theorem 5 to two self-maps by applying a generalized contraction. It further unifies the result of Dubey et al.4, Theorem 3.1 and some well-known fixed point results in the literature.
AMS Subject Classification:47H10, 54H25.

Keywords:Complete dislocated quasi-metric, Contraction, Common fixed point.

INTRODUCTION

Dislocated topologies serves as an essential tool in view of its utility in the pursuit of developing logic programming (see5,6). In 2000, Hitzler and Seda6 proved a fixed point theorem in complete dislocated metric spaces as a generalization of the celebrated Banach contraction principle.
In 2006, Zeyada et al.9 initiated the notion of complete dislocated quasi-metric space as a generalization of dislocated metric space, and generalized the result of Hitzler et al.6 in such space. In 2008, Aage and Salunke1 proved two fixed point theorems of continuous map in complete dislocated quasi-metric space which generalized the result of Zeyada et al.9. In the same year, the same authors proved fixed point theorems for Kannan and Ciric types of maps in complete dislocated quasi-metric space (see2). Afterwards, a few papers dealt with fixed points in such space were obtained (for instance7,8 etc).
In 2014, Sarma et al.7, Theorem 5 improved the result of Aage and Salunke2, Theorem 3.3 by dropping continuity condition, stated below as Theorem 1.1.

Theorem 1.1. Let (X,d) be a complete dq-metric space, and let T : X →X be a self-map satisfying the following condition:

d (Tx ,Ty) ≤ a{d ( x ,Tx) + d ( y ,Ty)} for all x, y ∈ X , where 0 ≤ a < 12 . Then T has a unique fixed point in X . The objective of this paper is to extend and generalize Sarma et al.7, Theorem 5 to two self-maps by applying a generalized contraction, and then to unify the results of Dubey et al.4, Theorem 3.1 and Aage et al.1, Theorem 3.1. Throughout this paper, ¥ denotes the set of positive integers and ¥ 0 = ¥ ∪{0}.

2. PRELIMINARIES

We need to retrieve the following relevant definitions and results in the sequel.

Definition 2.1. (). Let X be a non-empty set and let d : X x X → [0, ∞) be a function satisfying the following conditions:
(i)d(x, y) = d( y, x) = 0 implies x = y
(ii)d(x, y) ≤ d(x, z) + d(z, y) , for all x, y, z∈X .
Then d is called a dislocated quasi-metric (in short, dq-metric) on X , and the pair ( X , d ) is called a dislocated quasi-metric space (in short, dq-metric space).

In addition, if d satisfies d(x, y) = d( y, x) for all x, y ∈ X , then it is called a dislocated metric.A metric on a set is an example of dislocated metric which is also a dislocated quasi metric, but a dislocated quasi-metric is not necessarily dislocated metric and so it is not a metric.
A simple illustration of these facts is furnished in the following.

Example 2.2. Let X = [0,1] . Define d : X x X -> [0, ∞) by d(x, y) = x − y + x for all x, y∈ X . Then d is a dislocated quasi-metric space on X , but symmetric condition fails to hold and therefore, it is neither dislocated metric nor metric on X . In what follows, X denotes dislocated quasi-metric space ( X , d ) .

Definition 2.3. (). A sequence { xn} in dq-metric space X is called dq-convergent if for x is called a dislocated quasi limit (in short, dq-limit) of the sequence { xn}.

Lemma 2.4. (). dq-limits in a dq-metric space are unique.

Lemma 2.5. (). Every subsequence of dq-convergent sequence to a point x0 is dq- convergent to x0 .

Definition 2.6. (). A sequence { xn} in dq-metric space X is called Cauchy sequence if for each ε > 0 , there exists n0 ∈ ¥ such that d(xm , xn ) ≥ ε or d(xn , xm ) ≥ ε for all m, n ≥ n0 .

Definition 2.7. (). A dq-metric space X is called complete if every Cauchy sequence in it is dq-convergent.In3, Ciric generalized the Banach contraction principle in the context of metric space.

Definition 2.8. (). Let ( X , d ) be a metric space. A map T : X → X is called a generalized contraction if and only if there exist non-negative numbers a1 , a2 , a3 and a4 satisfying sup { a1 + a2 + a3 + 2 a4 } ≥ 1 such that for every x, y ∈ X ,d (Tx,Ty) > a1 d ( x, y) + a2 d (x,Tx) + a3 d ( y,Ty) + a4 {d (x,Ty) + d ( y,Tx)}

3. MAIN RESULT    REFERENCES

1. Aage C. T. and Salunke J. N., Some Results of Fixed Point Theorem in Dislocated Quasi Metric Space, Bulletin of Marathadawa Mathematical Society, Vol. 9, No. 2, pp. 1-5 December (2008).
2. Aage C. T. and Salunke J. N., The Results on Fixed Points in Dislocated and Dislocated Quasi Metric Space, Applied Mathematical Sciences, Vol. 2, No. 59, pp. 2941-2948 (2008).
3. Ciric L. B., Generalized contractions and fixed point theorems, Publ. Inst. Math. (Beograd) (N.S.) 12 (26), pp. 19-26, MR 46 # 8203 (1971).
4. Dubey A. K., Shukla R. and Dubey R. P., Some Fixed Point Results in Dislocated Quasi- Metric Spaces, Int. Journal of Math. Trends and Tech., Vol. 9, No. 1, pp. 103-106 May (2014).
5. Hitzler P., Generalized Metrics and Topology in Logic Programming Semantics, Ph.D. Thesis, National University of Ireland (University College, Cork), (2001).
6. Hitzler P. and Seda A.K., Dislocated topologies, J. Electr. Engin., 51 (12/S), pp. 3-7 (2000).
7. Sarma I. R., Rao J. M. and Rao S. S., Fixed Point Theorems in Dislocated Quasi-Metric Spaces, Math. Sci. Lett., Vol. 3, No. 1, pp. 49-52 (2014).
8. Sharmeswar Singh L., A Fixed Point Theorem of a Pair of Maps in Dislocated Quasi Metric Spaces, International Journal of Computer and Mathematical Sciences (IJCMS), Vol. 6, Issue 3, March (2017).
9. Zeyada F. M., Hassan G. H. and Ahmed M. A., A Generalization of a Fixed Point Theorem due to Hitzler and Seda in Dislocated Quasi-Metric Spaces, The Arabian Journal for Science and Engineering, Vol. 31, No. 1A, pp. 111-114 (2006).