Research Article | Open Access
K. P. R. Rao, K. R. K. Rao, "A Common Fixed Point Theorem for Two Hybrid Pairs of Mappings in -Metric Spaces", International Journal of Analysis, vol. 2013, Article ID 404838, 4 pages, 2013. https://doi.org/10.1155/2013/404838
A Common Fixed Point Theorem for Two Hybrid Pairs of Mappings in -Metric Spaces
We introduce a condition (W.C.C) and prove a common fixed point theorem for two hybrid pairs of mappings in b-metric spaces.
1. Introduction and Preliminaries
The study of fixed points for set-valued contraction maps using the Hausdorff metric was initiated by Nadler  in 1969, who extended the Banach contraction principle to set-valued mappings. The theory of set-valued maps has many applications in control theory, convex optimization, differential equations, and economics.
Theorem 1. Let be a complete metric space. Suppose satisfy for all , where . Then has a fixed point in .
Aydi et al.  proved a slight version of the above theorem in -metric spaces as follows.
Theorem 2. Let be a complete -metric space. Suppose satisfy for all , where . Then has a fixed point in .
In this paper, we introduce a new condition, namely, “condition (W.C.C)” of mappings which are not continuous and non commuting pairwise. Using this new condition we prove a common fixed point theorem for two pairs of hybrid mappings in -metric spaces.
For the sequel, we need the following concepts.
Czerwik  presented a generalization of the well-known Banach fixed point theorem in -metric spaces by defining the following.
Definition 3. Let be a nonempty set and a given real number. A function is called a -metric if the following axioms are satisfied for all :) if and only if , (), ().
Let be a -metric space. Let be the collection of all non-empty closed and bounded subsets of . For , define where with Then is called the Hausdorff metric induced by the -metric.
We cite the following lemmas from Czerwik . For more details we refer to [7, 8].
Lemma 4. Let be a -metric space. For any and any , we have the following: (i) for any , (ii),(iii) for any ,(iv),(v),(vi),(vii).
Lemma 5. Let be a -metric space. Let . Then for each and for all there exists such that .
Lemma 6. Let be a -metric space. For and , we have .
In this paper, we introduce the following condition on three mappings without continuity and commutativity .
Definition 7. Let be a -metric space. The mappings where and are said to satisfy the condition () if for all .
Remark 8. The mappings satisfying the condition () need not be continuous and commutative in view of the following examples.
Example 9. Let be a -metric space, where and .
Define by We distinguish the following cases.
Case 1. Consider
Case 2. Consider
Case 3. Consider
Case 4. Consider
Thus for all is satisfied.
In this example all the mappings are discontinuous.
Example 10. Let be a -metric on a set . Define , , , and . Clearly the mappings satisfy the condition and the pairs , , and are not commuting.
Now, we prove our main result with the condition for two hybrid pairs of mappings in -metric spaces.
2. Main Result
Theorem 11. Let be a complete -metric space. Let and be mappings satisfying the following:(2.1.1) , where , , , for all , where , (2.1.2),(2.1.3)the mappings or the mappings satisfy the condition .
Then either the pairs , and have coincidence points or the mappings and have a common fixed point in .
Proof. Suppose for some .
Then and . Thus is a coincidence point of and , and is a coincidence point of and .
Assume that for all .
Let and .
Then and .
Let . From (2.1.2), there exist such that .
From (2.1.2), there exist such that and from Lemma 5, we have Again from (2.1.2), there exist such that and from Lemma 5, we have Continuing in this way, we get the sequences and in such that , for all and
For simplification, write .
Now which in turn yields where .
Similarly, we can show that Thus, from (17) and (18), we have for all , so that If , then .
If , then .
Thus Since , from (19), we have Now for with , we have Hence is a Cauchy sequence in . Since is complete, there exists such that which in turn yields and as .
Suppose the mappings satisfy the condition . Then From (23), we have Letting , we get
Now Using (vii) of Lemma 4 and letting and using (21), we get Thus we have so that From Lemma 6, we have . Thus From (23), we get Now from (2.1.1), we have
Hence , since .
From (31), we have . Thus From (30) and (33), it follows that is a common fixed point of , and .
The authors are thankful to the referee for his valuable suggestions.
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Copyright © 2013 K. P. R. Rao and K. R. K. Rao. 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.