International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 376832 | https://doi.org/10.5402/2012/376832

R. D. Daheriya, Rashmi Jain, Manoj Ughade, "Some Fixed Point Theorem for Expansive Type Mapping in Dislocated Metric Space", International Scholarly Research Notices, vol. 2012, Article ID 376832, 5 pages, 2012. https://doi.org/10.5402/2012/376832

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

Academic Editor: G. L. Karakostas
Received25 Aug 2011
Accepted11 Oct 2011
Published15 Feb 2012

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 𝑋 be a nonempty set, and let π‘‘βˆΆπ‘‹Γ—π‘‹β†’[0,∞) be a function, called a distance function. One needs the following conditions:(M1)𝑑(π‘₯,π‘₯)=0, (M2)𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯)=0, then π‘₯=𝑦,(M3)𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯), (M4)𝑑(π‘₯,𝑦)≀𝑑(π‘₯,𝑧)+𝑑(𝑧,𝑦), (M4)’𝑑(π‘₯,𝑦)≀max{𝑑(π‘₯,𝑧),𝑑(𝑧,𝑦)}, for all π‘₯,𝑦,π‘§βˆˆπ‘‹.
If 𝑑 satisfies conditions (M1)–(M4), then it is called a metric on 𝑋. If 𝑑 satisfies conditions (M1), (M2), and (M4), it is called a quasimetric on 𝑋. If it satisfies conditions (M2)–(M4) ((M2) and (M4)), it is called a dislocated metric (or simply 𝑑-metric) (a dislocated quasimetric (or simply π‘‘π‘ž-metric)) on 𝑋, respectively. If a metric 𝑑 satisfies the strong triangle inequality (M4)’, then it is called an ultrametric.

Definition 1.2 (see [3]). A sequence {π‘₯𝑛}π‘›βˆˆβ„• in π‘‘π‘ž-metric space (dislocated quasimetric space) (𝑋,𝑑) is called a Cauchy sequence if, for given πœ–>0,there exists 𝑛0βˆˆπ‘ such that 𝑑(π‘₯π‘š,π‘₯𝑛)<πœ– or 𝑑(π‘₯𝑛,π‘₯π‘š)<πœ–, that is, min{𝑑(π‘₯π‘š,π‘₯𝑛),𝑑(π‘₯𝑛,π‘₯π‘š)}<πœ–for allπ‘š,𝑛β‰₯𝑛0.

Definition 1.3 (see [3]). A sequence {π‘₯𝑛}π‘›βˆˆβ„• in π‘‘π‘ž-metric space [𝑑-metric space] is said to be 𝑑-converge to π‘₯βˆˆπ‘‹ provided that limπ‘›β†’βˆžπ‘‘ξ€·π‘₯𝑛,π‘₯=limπ‘›β†’βˆžπ‘‘ξ€·π‘₯,π‘₯𝑛=0.(1.1) In this case, π‘₯ is called a π‘‘π‘ž-limit [𝑑-limit] of {π‘₯𝑛} and we write π‘₯𝑛→π‘₯.

Definition 1.4 (see [3]). A π‘‘π‘ž-metric space (𝑋,𝑑) is called complete if every Cauchy sequence in it is a dq-convergent.

Example 1.5. Let𝑋=[0,1]. Define π‘‘βˆΆπ‘‹Γ—π‘‹β†’β„+0 by 𝑑(π‘₯,𝑦)=max{π‘₯,𝑦}. Then the pair (𝑋,𝑑) is a dislocated metric space. We define an arbitrary sequence {π‘₯𝑛}in 𝑋 by π‘₯𝑛=2/(2𝑛+1), π‘›βˆˆπ‘βˆͺ{0}. Let πœ€=supπ‘›βˆˆβ„•{2/(2𝑛+1)}, then, for 𝑛,π‘šβˆˆπ‘ and 𝑛>π‘š, we have𝑑(π‘₯𝑛,π‘₯π‘š)=𝑑(2/(2𝑛+1),2/(2π‘š+1))=2/(2π‘š+1)β‰€πœ€. Thus, {π‘₯𝑛} is a Cauchy sequence in 𝑋. Also as π‘›β†’βˆž, then π‘₯𝑛→0βˆˆπ‘‹. Hence, every Cauchy sequence in 𝑋 is convergent with respect to 𝑑. Thus, (𝑋,𝑑) 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 π‘‘π‘ž-metric space.

Theorem 2.1. Let (𝑋,𝑑) be a complete dislocated metric space and 𝑇 a continuous mapping satisfying the following condition: []𝑑(𝑇π‘₯,𝑇𝑦)+𝛼max{𝑑(π‘₯,𝑇𝑦),𝑑(𝑦,𝑇π‘₯)}β‰₯𝛽⋅𝑑(π‘₯,𝑇π‘₯)1+𝑑(𝑦,𝑇𝑦)1+𝑑(π‘₯,𝑦)+𝛾⋅𝑑(π‘₯,𝑦)(2.1) for all π‘₯,π‘¦βˆˆπ‘‹, π‘₯≠𝑦, where 𝛼,𝛽,𝛾β‰₯0 are real constants and𝛽+𝛾>1+2𝛼, 𝛾>1+𝛼. Then, 𝑇 has a fixed point in 𝑋.

Proof. Choose π‘₯0βˆˆπ‘‹ be arbitrary, to define the iterative sequence {π‘₯𝑛}π‘›βˆˆβ„• as follows and 𝑇π‘₯𝑛=π‘₯π‘›βˆ’1 for 𝑛=1,2,3,…. Then, using (2.1), we obtain 𝑑𝑇π‘₯𝑛+1,𝑇π‘₯𝑛+2𝑑π‘₯+𝛼max𝑛+1,𝑇π‘₯𝑛+2ξ€Έξ€·π‘₯,𝑑𝑛+2,𝑇π‘₯𝑛+1𝑑π‘₯ξ€Έξ€Ύβ‰₯𝛽⋅𝑛+1,𝑇π‘₯𝑛+1ξ€·π‘₯ξ€Έξ€Ί1+𝑑𝑛+2,𝑇π‘₯𝑛+2ξ€Έξ€»ξ€·π‘₯1+𝑑𝑛+1,π‘₯𝑛+2ξ€Έξ€·π‘₯+𝛾𝑑𝑛+1,π‘₯𝑛+2ξ€Έξ€·π‘₯βŸΉπ‘‘π‘›,π‘₯𝑛+1𝑑π‘₯+𝛼max𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯,𝑑𝑛+2,π‘₯𝑛𝑑π‘₯ξ€Έξ€Ύβ‰₯𝛽⋅𝑛+1,π‘₯𝑛π‘₯ξ€Έξ€Ί1+𝑑𝑛+2,π‘₯𝑛+1ξ€Έξ€»ξ€·π‘₯1+𝑑𝑛+1,π‘₯𝑛+2ξ€Έξ€·π‘₯+𝛾⋅𝑑𝑛+1,π‘₯𝑛+2ξ€Έξ€·π‘₯βŸΉπ‘‘π‘›,π‘₯𝑛+1𝑑π‘₯+𝛼max𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯,𝑑𝑛+2,π‘₯𝑛𝑑π‘₯ξ€Έξ€Ύβ‰₯𝛽⋅𝑛+1,π‘₯𝑛π‘₯ξ€Έξ€Ί1+𝑑𝑛+1,π‘₯𝑛+2ξ€Έξ€»ξ€·π‘₯1+𝑑𝑛+1,π‘₯𝑛+2ξ€Έξ€·π‘₯+𝛾⋅𝑑𝑛+1,π‘₯𝑛+2ξ€Έξ€·π‘₯βŸΉπ‘‘π‘›,π‘₯𝑛+1ξ€Έξ€·π‘₯+𝛼𝑑𝑛,π‘₯𝑛+2ξ€Έξ€·π‘₯β‰₯𝛽⋅𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯+𝛾⋅𝑑𝑛+1,π‘₯𝑛+2ξ€Έξ€·π‘₯βŸΉπ‘‘π‘›,π‘₯𝑛+1ξ€Έξ€·π‘₯+𝛼𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯+𝛼𝑑𝑛+1,π‘₯𝑛+2ξ€Έξ€·π‘₯β‰₯𝛽⋅𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯+𝛾⋅𝑑𝑛+1,π‘₯𝑛+2ξ€Έξ€·π‘₯⟹(1+π›Όβˆ’π›½)𝑑𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯β‰₯(π›Ύβˆ’π›Ό)𝑑𝑛+1,π‘₯𝑛+2ξ€Έ.(2.2) The last inequality gives 𝑑π‘₯𝑛+1,π‘₯𝑛+2≀1+π›Όβˆ’π›½π‘‘ξ€·π‘₯π›Ύβˆ’π›Όπ‘›,π‘₯𝑛+1ξ€Έξ€·π‘₯=𝛿𝑑𝑛,π‘₯𝑛+1ξ€Έ,(2.3) where 𝛿=(1+π›Όβˆ’π›½)/(π›Ύβˆ’π›Ό)<1. Hence, by induction, we obtain 𝑑π‘₯𝑛+1,π‘₯𝑛+2≀𝛿𝑛+1𝑑π‘₯0,π‘₯1ξ€Έ.(2.4) Note that, for π‘š,π‘›βˆˆβ„• such that π‘š>𝑛, we have 𝑑π‘₯π‘š,π‘₯𝑛π‘₯β‰€π‘‘π‘š,π‘₯π‘šβˆ’1ξ€Έξ€·π‘₯+π‘‘π‘šβˆ’1,π‘₯π‘šβˆ’2ξ€Έξ€·π‘₯+β‹―+𝑑𝑛+1,π‘₯π‘›ξ€Έβ‰€ξ€Ίπ›Ώπ‘šβˆ’1+π›Ώπ‘šβˆ’2+β‹―+𝛿𝑛𝑑π‘₯0,π‘₯1ξ€Έ=𝛿𝑛1+𝛿+β‹―+π›Ώπ‘šβˆ’π‘›βˆ’1𝑑π‘₯0,π‘₯1ξ€Έβ‰€π›Ώπ‘›βˆžξ“π‘Ÿ=0π›Ώπ‘Ÿπ‘‘ξ€·π‘₯0,π‘₯1ξ€Έ=𝛿𝑛𝑑π‘₯1βˆ’π›Ώ0,π‘₯1ξ€Έ.(2.5) Since 0≀𝛿<1, then asπ‘›β†’βˆž, 𝛿𝑛(1βˆ’π›Ώ)βˆ’1β†’0. Hence, 𝑑(π‘₯π‘š,π‘₯𝑛)β†’0 asπ‘š,π‘›β†’βˆž. This forces that {π‘₯𝑛}π‘›βˆˆβ„•is a Cauchy sequence in 𝑋. But 𝑋 is a complete dislocated metric space; hence, {π‘₯𝑛}π‘›βˆˆβ„• is 𝑑-converges. Call the 𝑑-limitπ‘₯βˆ—βˆˆπ‘‹. Then, π‘₯𝑛→π‘₯βˆ— asπ‘›β†’βˆž. By continuity of 𝑇 we have, 𝑇π‘₯βˆ—ξ‚΅=π‘‡π‘‘βˆ’limπ‘›β†’βˆžπ‘₯𝑛=π‘‘βˆ’limπ‘›β†’βˆžπ‘‡π‘₯𝑛=π‘‘βˆ’limπ‘›β†’βˆžπ‘₯π‘›βˆ’1=π‘₯βˆ—(2.6) that is, 𝑇π‘₯βˆ—=π‘₯βˆ—; thus, 𝑇 has a fixed point in𝑋.

Uniqueness
Let π‘¦βˆ—be another fixed point of 𝑇 in𝑋, then π‘‡π‘¦βˆ—=π‘¦βˆ—and 𝑇π‘₯βˆ—=π‘₯βˆ—. Now, 𝑑𝑇π‘₯βˆ—,π‘‡π‘¦βˆ—ξ€Έξ€½π‘‘ξ€·π‘₯+𝛼maxβˆ—,π‘‡π‘¦βˆ—ξ€Έξ€·π‘¦,π‘‘βˆ—,𝑇π‘₯βˆ—π‘‘ξ€·π‘₯ξ€Έξ€Ύβ‰₯π›½β‹…βˆ—,𝑇π‘₯βˆ—ξ€·π‘¦ξ€Έξ€Ί1+π‘‘βˆ—,π‘‡π‘¦βˆ—ξ€Έξ€»1+𝑑(π‘₯βˆ—,π‘¦βˆ—)ξ€·π‘₯+π›Ύβ‹…π‘‘βˆ—,π‘¦βˆ—ξ€Έ.(2.7) This implies that 𝑑π‘₯βˆ—,π‘¦βˆ—ξ€Έξ€·π‘₯+π›Όπ‘‘βˆ—,π‘¦βˆ—ξ€Έξ€·π‘₯β‰₯π›Ύβ‹…π‘‘βˆ—,π‘¦βˆ—ξ€Έξ€·π‘₯thatisπ‘‘βˆ—,π‘¦βˆ—ξ€Έξ€·π‘₯β‰₯(π›Ύβˆ’π›Ό)π‘‘βˆ—,π‘¦βˆ—ξ€Έξ€·π‘₯βŸΉπ‘‘βˆ—,π‘¦βˆ—ξ€Έβ‰€1𝑑π‘₯π›Ύβˆ’π›Όβˆ—,π‘¦βˆ—ξ€Έ.(2.8) This is true only when𝑑(π‘₯βˆ—,π‘¦βˆ—)=0. Similarly,𝑑(π‘¦βˆ—,π‘₯βˆ—)=0. Hence, 𝑑(π‘₯βˆ—,π‘¦βˆ—)=𝑑(π‘¦βˆ—,π‘₯βˆ—)=0and so π‘₯βˆ—=π‘¦βˆ—. Hence, 𝑇 has a unique fixed point in𝑋.

Next we prove Theorem 2.1 for surjective mapping.

Theorem 2.2. Let (𝑋,𝑑) be a complete dislocated metric space and 𝑇 a surjective mapping satisfying the condition (2.1) for allπ‘¦βˆˆπ‘‹, ≠𝑦, where 𝛼,𝛽,𝛾β‰₯0 are real constants and 𝛽+𝛾>1+2𝛼, 𝛾>1+𝛼. Then, 𝑇 has a fixed point in 𝑋.

Proof. Choose π‘₯0βˆˆπ‘‹ to be arbitrary, and define the iterative sequence {π‘₯𝑛}π‘›βˆˆβ„• as follows: 𝑇π‘₯𝑛=π‘₯π‘›βˆ’1 for 𝑛=1,2,3,…. Then, using (2.1), we obtain, sequence {π‘₯𝑛}π‘›βˆˆβ„• is a Cauchy sequence in 𝑋. But 𝑋 is a complete dislocated metric space; hence {π‘₯𝑛}π‘›βˆˆβ„• is 𝑑-converges. Call the 𝑑-limitπ‘₯βˆ—βˆˆπ‘‹. Then, π‘₯𝑛→π‘₯βˆ— asπ‘›β†’βˆž. Existence of Fixed Point
Since 𝑇 is a Surjective map, so there exists a point 𝑦 in𝑋, such thatπ‘₯=𝑇𝑦. Consider 𝑑π‘₯𝑛,π‘₯=𝑑𝑇π‘₯𝑛+1𝑑π‘₯,𝑇𝑦β‰₯βˆ’π›Όmax𝑛+1ξ€Έξ€·,𝑇𝑦,𝑑𝑦,𝑇π‘₯𝑛+1𝑑π‘₯ξ€Έξ€Ύ+𝛽⋅𝑛+1,𝑇π‘₯𝑛+1ξ€Έ[]1+𝑑(𝑦,𝑇𝑦)ξ€·π‘₯1+𝑑𝑛+1ξ€Έξ€·π‘₯,𝑦+𝛾⋅𝑑𝑛+1ξ€Έ.,𝑦(2.9) Taking π‘›β†’βˆž, we get []𝑑(π‘₯,π‘₯)β‰₯βˆ’π›Όmax{𝑑(π‘₯,π‘₯),𝑑(𝑦,π‘₯)}+𝛽⋅𝑑(π‘₯,π‘₯)1+𝑑(𝑦,π‘₯)1+𝑑(π‘₯,𝑦)+𝛾⋅𝑑(π‘₯,𝑦)⟹0β‰₯βˆ’π›Όπ‘‘(π‘₯,𝑦)+𝛾𝑑(𝑦,π‘₯)⟹(π›Ύβˆ’π›Ό)𝑑(π‘₯,𝑦)≀0βŸΉπ‘‘(π‘₯,𝑦)=0as𝛾>𝛼.(2.10) Similarly, 𝑑(𝑦,π‘₯)=0. Hence, 𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯)=0β‡’π‘₯=𝑦 and so 𝑇π‘₯=π‘₯, that is, π‘₯ is a fixed point of𝑇.
Uniqueness
Let π‘¦βˆ—be another fixed point of 𝑇 in𝑋, then π‘‡π‘¦βˆ—=π‘¦βˆ—and 𝑇π‘₯βˆ—=π‘₯βˆ—. Now, 𝑑𝑇π‘₯βˆ—,π‘‡π‘¦βˆ—ξ€Έξ€½π‘‘ξ€·π‘₯+𝛼maxβˆ—,π‘‡π‘¦βˆ—ξ€Έξ€·π‘¦,π‘‘βˆ—,𝑇π‘₯βˆ—π‘‘ξ€·π‘₯ξ€Έξ€Ύβ‰₯π›½β‹…βˆ—,𝑇π‘₯βˆ—ξ€·π‘¦ξ€Έξ€Ί1+π‘‘βˆ—,π‘‡π‘¦βˆ—ξ€Έξ€»1+𝑑(π‘₯βˆ—,π‘¦βˆ—)ξ€·π‘₯+π›Ύβ‹…π‘‘βˆ—,π‘¦βˆ—ξ€Έ.(2.11) This implies that 𝑑π‘₯βˆ—,π‘¦βˆ—ξ€Έξ€·π‘₯+π›Όπ‘‘βˆ—,π‘¦βˆ—ξ€Έξ€·π‘₯β‰₯π›Ύβ‹…π‘‘βˆ—,π‘¦βˆ—ξ€Έξ€·π‘₯thatis,π‘‘βˆ—,π‘¦βˆ—ξ€Έξ€·π‘₯β‰₯(π›Ύβˆ’π›Ό)π‘‘βˆ—,π‘¦βˆ—ξ€Έξ€·π‘₯βŸΉπ‘‘βˆ—,π‘¦βˆ—ξ€Έβ‰€1𝑑π‘₯π›Ύβˆ’π›Όβˆ—,π‘¦βˆ—ξ€Έ.(2.12) This is true only when𝑑(π‘₯βˆ—,π‘¦βˆ—)=0. Similarly, 𝑑(π‘¦βˆ—,π‘₯βˆ—)=0. Hence, 𝑑(π‘₯βˆ—,π‘¦βˆ—)=𝑑(π‘¦βˆ—,π‘₯βˆ—)=0and so π‘₯βˆ—=π‘¦βˆ—. Hence 𝑇 has a unique fixed point in𝑋. The proof is completed.

References

  1. B. K. Dass and S. Gupta, β€œAn extension of Banach contraction principle through rational expression,” Indian Journal of Pure and Applied Mathematics, vol. 6, no. 12, pp. 1455–1458, 1975. View at: Google Scholar | Zentralblatt MATH
  2. B. E. Rhoades, β€œA comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257–290, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. F. M. Zeyada, G. H. Hassan, and M. A. Ahmed, β€œA 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. View at: Google Scholar
  4. C. T. Aage and J. N. Salunke, β€œThe results on fixed points in dislocated and dislocated quasi-metric space,” Applied Mathematical Sciences, vol. 2, no. 57–60, pp. 2941–2948, 2008. View at: Google Scholar | Zentralblatt MATH
  5. A. Isufati, β€œFixed point theorems in dislocated quasi-metric space,” Applied Mathematical Sciences, vol. 4, no. 5–8, pp. 217–223, 2010. View at: Google Scholar

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.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views974
Downloads582
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.