ISRN Mathematical AnalysisVolume 2012 (2012), Article ID 376832, 5 pagesdoi:10.5402/2012/376832

Research Article

Some Fixed Point Theorem for Expansive Type Mapping in Dislocated Metric Space

R. D. Daheriya,¹ Rashmi Jain,² and Manoj Ughade²

¹Department of Mathematics, J.H. Govt. College Betul, Betul 460001, India²Department of Mathematics, RKDF Institute of Science & Technology, Bhopal 462026, India

Received 25 August 2011; Accepted 11 October 2011

Academic Editor: G. L. Karakostas

Copyright © 2012 R. D. Daheriya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The purpose of this paper is to present some fixed point theorem in dislocated quasimetric space for expansive type mappings.

1. Introduction and Preliminaries

It is well known that Banach Contraction mappings principle is one of the pivotal results of analysis. Generalizations of this principle have been obtained in several directions. Dass and Gupta [1] generalized Banach’s Contraction principle in metric space. Also Rhoades [2] established a partial ordering for various definitions of contractive mappings. In 2005, Zeyada et al. [3] established a fixed point theorem in dislocated quasimetric spaces. In 2008, Aage and Salunke [4] proved some results on fixed point in dislocated quasimetric spaces. Recently, Isufati [5], proved fixed point theorem for contractive type condition with rational expression in dislocated quasimetric spaces. The following definitions will be needed in the sequel.

Definition 1.1 (see [3]). Let be a nonempty set, and let be a function, called a distance function. One needs the following conditions:(M1), (M2), then ,(M3), (M4), (M4)’, for all If satisfies conditions (M1)–(M4), then it is called a metric on . If satisfies conditions (M1), (M2), and (M4), it is called a quasimetric on . If it satisfies conditions (M2)–(M4) ((M2) and (M4)), it is called a dislocated metric (or simply -metric) (a dislocated quasimetric (or simply -metric)) on , respectively. If a metric satisfies the strong triangle inequality (M4)’, then it is called an ultrametric.

Definition 1.2 (see [3]). A sequence in -metric space (dislocated quasimetric space) is called a Cauchy sequence if, for given there exists such that or , that is, for all.

Definition 1.3 (see [3]). A sequence in -metric space [-metric space] is said to be -converge to provided that In this case, is called a -limit [-limit] of and we write .

Definition 1.4 (see [3]). A -metric space is called complete if every Cauchy sequence in it is a dq-convergent.

Example 1.5. Let. Define by . Then the pair is a dislocated metric space. We define an arbitrary sequence in by , . Let , then, for and , we have. Thus, is a Cauchy sequence in . Also as , then . Hence, every Cauchy sequence in is convergent with respect to . Thus, is a complete dislocated metric space.

2. Main Results

In this paper, we prove some fixed point theorem for continuous mapping satisfying expansion condition in complete -metric space.

Theorem 2.1. Let be a complete dislocated metric space and a continuous mapping satisfying the following condition: for all , , where are real constants and, . Then, has a fixed point in .

Proof. Choose be arbitrary, to define the iterative sequence as follows and for . Then, using (2.1), we obtain The last inequality gives where . Hence, by induction, we obtain Note that, for such that , we have Since , then as, . Hence, as. This forces that is a Cauchy sequence in . But is a complete dislocated metric space; hence, is -converges. Call the -limit. Then, as. By continuity of we have, that is, ; thus, has a fixed point in.

UniquenessLet be another fixed point of in, then and . Now, This implies that This is true only when. Similarly,. Hence, and so . Hence, has a unique fixed point in.

Next we prove Theorem 2.1 for surjective mapping.

Theorem 2.2. Let be a complete dislocated metric space and a surjective mapping satisfying the condition (2.1) for all, , where are real constants and , . Then, has a fixed point in .

Proof. Choose to be arbitrary, and define the iterative sequence as follows: for . Then, using (2.1), we obtain, sequence is a Cauchy sequence in . But is a complete dislocated metric space; hence is -converges. Call the -limit. Then, as. Existence of Fixed PointSince is a Surjective map, so there exists a point in, such that. Consider Taking , we get Similarly, . Hence, and so , that is, is a fixed point of.UniquenessLet be another fixed point of in, then and . Now, This implies that This is true only when. Similarly, . Hence, and so . Hence has a unique fixed point in. The proof is completed.

References

B. K. Dass and S. Gupta, “An extension of Banach contraction principle through rational expression,” Indian Journal of Pure and Applied Mathematics, vol. 6, no. 12, pp. 1455–1458, 1975.

B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257–290, 1977.

F. M. Zeyada, G. H. Hassan, and M. A. Ahmed, “A generalization of a fixed point theorem due to Hitzler and Seda in dislocated quasi-metric spaces,” The Arabian Journal for Science and Engineering A, vol. 31, no. 1, pp. 111–114, 2006.

C. T. Aage and J. N. Salunke, “The results on fixed points in dislocated and dislocated quasi-metric space,” Applied Mathematical Sciences, vol. 2, no. 57–60, pp. 2941–2948, 2008.

A. Isufati, “Fixed point theorems in dislocated quasi-metric space,” Applied Mathematical Sciences, vol. 4, no. 5–8, pp. 217–223, 2010.