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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 483059, 5 pages
Common Fixed Point Theorems for Nonlinear Contractive Mappings in Dislocated Metric Spaces
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
Received 7 January 2013; Accepted 1 February 2013
Academic Editor: Ru Dong Chen
Copyright © 2013 Yijie Ren 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.
In 1986, Matthews generalized Banach contraction mapping theorem in dislocated metric space that is a wider space than metric space. In this paper, we established common fixed point theorems for a class of contractive mappings. Our results extend the corresponding ones of other authors in dislocated metric spaces.
Fixed point theory is an important branch of nonlinear analysis and can be used to many discipline branches, such as control theory, convex optimization, differential inclusion, and economics. Banach proved a celebrated fixed point theorem for contraction mappings in complete metric space which is one of the pivotal results of analysis. Dass and Gupta  generalized Banach contraction mapping in metric space. Also Rhoades  established a partial ordering for various definitions of contractive mapping. The concept of metric spaces, as an environmental space in fixed point theory, has been generalized in several directions. Some of such generalizations are dislocated spaces, quasimetric spaces, dislocated quasimetric spaces, and generalized quasimetric spaces. The concept of dislocated spaces is treated differently by different authors. Matthews  generalized Banach contraction mapping theorem in dislocated metric space. Hitzler  presented variants of Banach contraction principle for various modified forms of a metric space including dislocated metric space and applied them to semantic analysis of logic programs. In this context, Hitzler and Seda  raised some related questions on the topological aspects of dislocated metrics. In 2005, Zeyada et al.  generalized a fixed point theorem in dislocated quasimetric spaces. In 2008, Aage and Salunke  proved some results on fixed points in dislocated quasimetric space. Recently, Isufati  proved fixed point theorem for contractive type condition with rational expression in dislocated quasimetric space. In this paper, we study the mapping refereed by Xia Dafeng and obtained fixed point theorems in dislocated metric space. For fixed point theorems, see [9, 10]. The following definition is introduced by Xia et al. .
Definition 1 (see ). Let . Let , , satisfying the following:(1)if , then there exists , such that ;(2)if , then there exists , such that .
Theorem 2 (see ). Let be complete metric spaces, let be continuous mappings, and for all , such that or then have a unique common fixed point.
Definition 3 (see ). Let be a nonempty set and let be a function called a distance function. If for all ,(1)nonnegativity: ;(2)faithful: ;(3)the triangle inequality: ,So, here is a quasimetric on , and is called a quasimetric space.
Definition 4 (see ). Let be a nonempty set and let be a function called a distance function. If for all ,(1)nonnegativity: ;(2)indistancy implies equality: implies ;(3)the triangle inequality: ,so, here is called a dislocated quasimetric or -metric on , and is called a dislocated quasimetric space.
Definition 5 (see ). Let be a nonempty set and let be a function called a distance function. If for , (1)nonnegativity: ;(2)indistancy implies equality: implies ;(3)symmetry: ;(4)the triangle inequality: ,so, here is called a dislocated metric or -metric on and the pair is called a dislocated metric space.
Definition 6 (see ). A sequence in -metric space (dislocated quasimetric space) is called a Cauchy sequence, if for a given , there exists such that or ; that is, for all .
Definition 7 (see ). A sequence in -metric spaces is said to be -converged to provided that In this case, is called a -limit of and we write .
Definition 8 (see ). A -metric space is called -complete if every -Cauchy sequence in converges with respect to in .
Lemma 9. Every converging sequence in a -metric space is a Cauchy sequence.
Proof. Let be a sequence which converges to some , and let be arbitrarily given. Then there exists with for all . For , then we obtain that . Hence is a Cauchy sequence.
Lemma 10. Limits in dislocated metric spaces are unique.
Lemma 11. Limits in dislocated quasimetric spaces are unique.
Proof. Let and be limits of the sequence . Then and as . By the triangle inequality, it follows that . As , we have . Similarly, . Hence ; that is, . Also as . That is, , and . By the property (2) of Definition 4, we conclude that .
Example 12. Let . Define by . Then the pair is a dislocated metric space. We define an arbitrary sequence in ; if , there exists an positive integer that satisfies . Then, for any , 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.
3. Main Results
In this section, now we establish that common fixed points for mapping satisfying contractive condition are proved in the frame of dislocated metric spaces.
Definition 13 (see ). There exist that satisfy the condition , if one lets be nondecreasing and non-negative, then , for a given .
Lemma 14 (see ). If satisfy the condition , then , for a given .
Lemma 15 (see ). Let , and satisfy the condition ; for all , if or or , then .
Theorem 16. Let be a complete dislocated metric space, and let be self-mapping, if(1)either or is continuous;(2)there exists satisfying the condition , for all , such that then have unique common fixed points.
Proof. Let be continuous, arbitrary in , , the sequence of , and
By the given condition, we have
By Lemma 15, we have
By Lemma 14, we have
Hence, by induction, for all , we obtain
If , we have
Note that by the condition we know, for such that , we have
Hence, as . This forces that is a Cauchy sequence in . But is a completely dislocated metric space; hence, is -converges. Call the -limit . Then, as . By the continuity of , as , . So as , , is the fixed point of .
By the given condition, we have
Hence, , so is a common fixed point of .
Uniqueness. Let be another common fixed point of . Then by the given condition, we have Since , is the unique fixed point of ; similarly, we prove that is also the unique fixed point of . Thus the fixed point of is unique, and we prove the theorem.
Theorem 17. Let be a complete dislocated metric space; and let be continuous mapping, if(1)there exists satisfying the condition , for all , if , such that (2)there exists such that have a condensation point, then have a unique common fixed point.
Proof. Let be the sequence of , and for all , ,
Suppose that is the condensation point of ; there exists the subsequence of such that . Since is continuous, .
Also consider Thus
Hence, we know that is decreasing. Let and . Since is the subsequence of , we have
Hence, we conclude that , a contradiction. So , is the fixed point of . Similarly, is the fixed point of .
Uniqueness. Let be another common fixed point of . Then by the given condition, we have
Since , is the unique fixed point of . Similarly, we prove that is also the unique fixed point of . Thus the fixed point of is unique.
This research was supported by NSFC Grants No: 11071279.
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